Let's Find Out - The Basics of Relativity: Light Speed, Time Dilation, and the Expanding Universe | ASMR

Episode Date: August 9, 2023

Tonight we explore the strange reality of Einstein's General Theory of Relativity and what it means for 3 dimensional beings born into a 4 dimensional Universe. It predicts our birth from a singularity in an expanding Universe, the contraction of space at the speed of light, and both black holes and the infinite time dilation at their horizon. It's bizarre that space truly compresses and time actually slows down deep inside gravitational wells and near the speed of light. We literally observe supernovae billions of light years away exploding at half their normal speed. So let's find out about the history and basics of relativity. ▸ Want to leave a tip or connect?: https://linktr.ee/letsfindoutasmr Video credits: Carrol and Ostlie (textbook I used): https://amzn.to/3QtiABm TNG Simulations https://www.tng-project.org/media/ NASA/STSci/ESA Davis and Lineweaver (2003) https://arxiv.org/abs/astro-ph/0310808 Nolan, C. (2014). Interstellar. Paramount Pictures. The Eagle Simulation, Durham University https://youtu.be/Fim6dWJhxz4 Doppler Shifts, Prof. Richard Pogge, The Ohio State University https://youtu.be/xCTUgpwuP0I @joshborup 2 Hour Jupiter time lapse with Io transit, https://youtu.be/EBBhvY8ZRwE @eigenchris Relativity 110d: Cosmology - FLRW Geodesics, Cosmological Redshift, Horizons, Comoving Coordinates, https://youtu.be/OxkY4Wqdpp8 @eigenchris Relativity 103a: Galilean Relativity - Spacetime Diagrams, https://youtu.be/powCBsDOa8U @howfarawayisit How Fast Is It - 03 - Special Relativity (4K), https://youtu.be/IyxX8LAvkdQ Dr. David Bradstreet (Eastern University), Ole Roemer and the Speed of Light, https://youtu.be/XNJw0B1o8cQ Mythbusters, Soccer ball shot from a truck https://youtu.be/ZH7GpYJoptU Time Stamps: 0:00 Relativity and the expanding Universe (intro) 5:54 Galileo and Roemer discover light isn't infinitely fast (1600's) 19:55 The mystery of the constant speed of electromagnetic field waves 32:17 Galileo discovered relativity 50:26 Einstein discovers Special Relativity 1:11:39 Time dilation and space contraction 1:33:08 Galaxy redshifts and expansion 2:16:24 General Relativity (gravity now explained by relativity) 2:57:41 Light cones (future and past causality) 3:12:21 Relativity in Cosmology: discovery of expanding spacetime #educational #letsfindout #ASMR #relaxing #space #science

Two little fact toys about the universe that are often tossed around in these popular shows and articles are that one, the age of the universe is about 14 billion years old. And then two, the speed of light is the speed limit of the universe, and nothing can exceed it. So if we think about that for a minute, put those two facts together, that should mean that from our central vantage point, light should have only had time to travel a distance of 14 billion light years. In other words, the observable universe from our perspective should only be about 28 billion light years in diameter, right? No, it's 93 billion light years across. In fact, at the edge of this 93 billion light year across universe, light emitted at the beginning of time that's taken the age of the universe to reach us is just now piercing through our particle horizon and has traveled from matter that is now 45 billion light years from us.

and at the time of emission at the beginning of the universe was traveling 58 times the speed of light. So how is this possible? How is light able to overcome and overtake matter? It's being emitted from traveling 60 times its speed in the opposite direction of us. Well, apparently, even within the cosmology community, itself, there's been some confusion about this exact topic to the point where there's literally been technical papers written about the implications of superluminal or faster in light expansion of matter in the universe and the causal contact between them. And me, as the resident

supremely unqualified non-scientist, am going to break it. it down for you so let's find out what the weird effects of relativity taking place on cosmic scales can reveal about this little conundrum take a quick detour into this book here introduction of modern astrophysics and as intimidating as it is there's a ton of really well a ton of insight in this book and it's a really good background to at least you know dip our toe into to have a better understanding of how time space mass and movement through space velocity and acceleration are all relative and unlike newton's and galileo's paradigm there is nothing absolute no reference frame no point

in space or time that we can point to as an objective state of reference around which all other things move and time moves forward. The relativity behind special and general relativity are so non-intuitive that I really am still having trouble wrapping my head around the reality of this paradigm. One affected, for instance, is time dilation between atomic clocks. Identical atomic clocks. One was put on a plane and one was set on the ground. So you had one moving with respect to the other. That plane orbited, traveled around the earth at a constant altitude for maybe a day, 24 hours, something like around that. And there was a difference between the times that those clocks measured. That is an effect of time dilation due to special relativity. And then not only that, but there was another aspect of time

dilation due to a general relativistic time dilation due to the effect of the plane being at a higher position in the gravitational well of Earth being at five miles 35,000 feet above sea level where the other atomic clock was. So these concepts, it's so baffling that They're true, yet so counterintuitive. And despite the fact that the Earth is orbiting the sun, that is orbiting the galaxy, that is orbiting a central gravitational point among a collection of galaxies in our local group, like Indromeda and Triangulum, which itself is moving gravitationally with respect to other clusters of galaxies in the larger supercluster that we exist within.

There's no discernible effects of time dilation and space dilation due to relativity that we can detect, well, with our eyes, without very precise instruments. Yet it's a reality. And we know it because we've been able to detect it. So let's begin with the speed of light. Let's travel 400 years back in time and start with Galileo.

Brought up Galileo because he actually sent the scientific revolution regarding the investigation of light on its way. He had a fascinating thought experiment that he wrote down in which he talked about. he was the first that really demarcated the fact that we can't know that light is infinite simply by the fact that we are unable to measure its speed because it moves too fast for our instruments and his instruments in the 1600s his experiment his thought experiment was that essentially if we're trapped on an island unfortunately attack by a boat, a ship with a cannon aimed right for our head. What happens when you, as this unfortunate little fella,

experiences and observes the cannon being shot towards him, is that there's an explosion, we see the ball travel some distance, and then, only then when it's about here, do we hear the sound the compressed atmosphere the oscillations created by the energy that compresses the atmosphere that hits our ears and we interpret as sound but by the time the ball is here that's when we hear what's going on that launched the ball all the while the what we're observing the light the information traveling fastest information about this you know unfortunate situation for us that travels fastest is the light with no delay we can track the position of this right up until we can't but the gap between the sound you know if we were to put a little bell on this thing we would observe the Doppler effect that we now know in which the gap between the sound we're observing would the sound that

we're observing and the sound that it's actually emitting if we're we could travel along with it and hear the actual emission of the sound the pitch that we're hearing would appear to us to get higher and higher and higher until it's right outside our eardrum and that represents the difference between the sound, how fast it travels, which presumably if this is subsonic, not faster than sound, sound would get to us before the ball would. So there would always be a gap between when the sound arrives and then when the ball lags behind it. Similarly, Galileo proposed that although light seems to travel instantaneously, no matter how far away we're seeing it from, that doesn't necessarily prove

that the speed of light often denoted by the variable C it doesn't prove that it's infinite it doesn't prove that it's not it just proves it's a good way of reasoning that that is not proof

that it's infinitely fast and sure enough less than about 50 years later when that ended up happening was a Danish astronomer by the name of Old Romer he looked at the very moons that Galileo actually

discovered first through his telescope so the interior moon I think it was Io that they were looking at Galileo discovered there's four large moons on Jupiter these are the

orbits not the rings and by 50 or so years half a century after Galileo had discovered them their orbits their orbital periods were pretty well known and it was known that I.O we're going to pretend that's the one is it has about a 42 hour orbital period now if simple schematic of our solar system we pretend that this is the orbit of earth and this is the much further and definitely not to scale orbit of Jupiter far beyond Earth. And what was also known based on Kepler's laws of planetary motion, we had an idea of roughly how fast we traveled in our orbit around the Earth. And so we knew that if we observed something at a particular point in our orbit

that has the orientation towards Jupiter of traveling towards it and a key component of this little experiment here was the fact that it was well known that planets closer to the sun and it was known that Earth is closer than Jupiter is even if the exact distances weren't known orbit much faster

so as the as Jupiter slowly orbits out here Earth is completing multiple years it's year it's a lot long longer than Earth. Well, this tells us that Jupiter won't have moved very far, given any change in Earth's orbit over a short enough period of time. And so it was known that, it was known that if we blow up Jupiter here a little bit, as Io travels around orbits Jupiter, for a huge fraction of its time, I.O. is going to be immersed, or I think it's occulted by Jupiter. It's going to go behind Jupiter. And then some, maybe, you know, a third of its orbit is going to be behind Jupiter.

And then sometime later, 12 hours or so maybe, it will emerge from behind Jupiter. And what Romer's genius observation in deduction was, was that if he recognizes that Earth at a particular time of year, let's say this is winter, is oriented so that Earth's motion around the Sun, most of the motion, the velocity, is directly towards Jupiter, so that over any period of a few days, Earth is approaching Jupiter at the speed of its orbit. We now know it's about 35,000 kilometers per hour.

And then six months later, if this is summer, we know that it's receding from Jupiter. And maybe I should, the differences between here or two is to make the lines I'm about to draw a little more visible. But we knew that relative, to, you know, if this is winter and this is spring and fall, motion here, from here to here, isn't going to have any change in distance between Earth and Jupiter,

and likewise in the fall. But motion from during the winter months, from here to here, your orientation in the orbit is such that all your motion is radially pointed towards Jupiter versus the perpendicular change in, your orbit, which produces no distance change. So there is a distance change when you travel from... Oh, that's no good, is it?

Let's back up. So there is a distance change from this position to a day later or so, to this position. And that distance change was known, even back then. So there's a change in distance and a change in time. Now, what Romer discovered, what he thought to observe, was the time, the predicted time that Io would be occulted by Jupiter and the known time that it should have emerged from behind Jupiter. And as he measured this from this position and then some time

later from the new position that Earth is in its orbit, he recognized that maybe if the gap had closed significantly enough, that would mean that light would take less time to travel by the time it had emerged from behind Jupiter. And therefore, there should be a noticeable decrease in the time that IOS spends behind Jupiter. And there's a lot of other nuances and subtleties to the experiment that I'm not getting at here. But the gist of it is that he measured and observed the time it took at the winter that season. And then six months later, when instead of approaching Jupiter, Earth was receding from Jupiter. So the measurement here will have taken light shorter to travel and then the measurement here.

And so again there's another change in distance. And therefore the corresponding lengthening of the time it took between the IOS immersion behind Jupiter occultation and its emergence from behind Jupiter. and the fact that over here when light when earth is traveling away increasing its distance it's going to lengthen the time that the apparent time of i.O. behind jupiter because light is going to take longer to transfer the information of i.O.'s position around jupiter to earth. Anyways from that, Romer was able to deduce that the speed of light

was something like 2.2 times 10 to the 8th meters per second which is roughly 220,000 meters per second

which is only 25 percent difference a 25% change from the known value of 300,000 meters per second. So as time goes on, this is in maybe, you know, 1610. This is in 1675, I think.

By the year roughly the mid-1850, mid-1800s, there had been breakthroughs. There had been the measurements of the parallaxes, in the distance to Venus known I think in 1750 and then 1850 we knew the parallax we'd been able to take this man named Bessel had taken the parallax to one of the nearest stars

and found out that it's 650,000 times further than our distance to the sun it was 650,000 times further than that accurately. So he found out that all the stars that told him that if that star had

one of the largest parallaxes in the sky, that must be the closest and therefore we had a rough idea of just how far some of the stars could possibly be. And the universe by 1850 had opened up to be somewhere between 100,000

light years across because we had now had an idea of roughly how fast light should travel and somewhere between 100,000 light years across and at about the same time we had men in the mid and late 1800s really unveiling the true nature of light we had Hertz we had Maxwell coming up with his famous equations studying the electromagnetic radiation that he revealed and he found out was actually the same substance of the same nature as light itself and light was just a particular sub-band

of frequencies that he was measuring in the lab ranging from radio waves and microwaves through infrared light ultraviolet light and eventually x-reveillance rays and gamma rays were later discovered. But Maxwell had, through his study of electromagnetic radiation in his lab, had arrived

through his equations at the fact that light travels roughly 300,000 miles per second, meters per second. Maxwell had geniusly been able to consolidate everything that was known up into that point roughly around this time about

electricity and magnetism into one framework. He created his famous four Maxwell's equations, some of the most famous equations in physics, that took these experimental constants that had to do with electricity and magnetism, that described those two phenomena. And he deduced that,

given that there's electric fields and magnetic fields, and that they're actually, two sides of the same coin. He also deduced through his equations that there are electromagnetic disturbances or waves that propagate through these fields and fascinatingly using just his experimental constants, these

one, I think it's epsilon and mu, this is the electric constant and this is the magnetic constant associated with the electric and magnetic fields he was able to derive a theoretical speed of light that was equivalent to one over the square root

of those two constants and this worked out to be almost exactly what we know today as the speed of light so by the late 1800s roughly let's say in 1900 light had been probed to the degree that it was found to be not only did we find at speed but we found out that nothing no other radiation or particles or anything traveled faster than light we found that in certain media like glass and water it would slow down but it would never travel faster than this theoretical theoretical speed here or be measured at that it was becoming to be starting to be this speed limit of the universe that's where it the known laws of classical physics before

Einstein that's what is known as Newtonian physics is all physics and what we've known prior to the quantum-based particle physics of and relativity of the 20th century both of which Einstein was key in integral in pioneering and discovering. We found out that light was an ever-looming constant that dominated physics. And in the meantime, so that was leading up to Einstein's thought experiments with toying around with the possibility of what exactly physics would result, what would physical experiments yield if you could be traveling with a velocity of the speed of light

fast as any known phenomena in the universe? And meanwhile, in the 1800s, throughout the 1800s, beginning as early as maybe the 1810s, we'd been looking at the, sun's light trying to reveal any information that might exist within it and as

Newton did he took he took a prism the sun's light actually gets separated into different colors and then eventually we discovered that

if this is blue and this is red there's actually waves radiation invisible beyond the red infrared and beyond the blue

called ultraviolet or UV light and we found out this was in the early 1800s 1700s even we found out that the ultraviolet was more

energetic than the infrared so it moved light and eventually it was revealed that light was electromagnetic radiation was characterized by a spectrum across which waves wavelengths increased and the frequency decreased and with a wavelength increase and frequency decrease we had an energy decrease so the energy light waves got or electromagnetic waves got more and more energetic the closer we got to the blue end of the

the spectrum. And we found out also that in splitting up the visible light, as instruments got more and more sophisticated, they became full-blown, you know, very highly technical instruments, they were able to actually split the sunlight into very discreet lines. And we discovered through experiment guys like Fresnel he was the first to break these down or he was one of the first to take a lens and break get a high resolution enough separation of the sunlight into a spectrum that you could start to see these little dark bands in it and then within 50 or so years we had kirk off and bunsen of burner fame and kirkoff current law, they were able to distinguish by the mid to late 1800s thousands of discrete

patterns of dark absorption lines, they called them, in the sun's, in the sun's light. And also happening in the 1800s, we had Doppler discovering that sound waves like we talked about up here get compressed. So as your sound is traveling away from a moving object, the sound waves don't quite propagate out radially from the object, and they get compressed in the direction of motion of that object. And so he discovered that maybe you could also apply that to light. And by the late 1800s, early 1900s,

they were looking at the stars, the nearby stars, in noticing that these patterns of light of lines absorption lines were actually being shifted one way or another these two phenomena the speed limit of light the observed shifting of spectral lines from starlight and later on galaxies this was before we even knew other galaxies existed or that we even lived in a galaxy We just thought until Einstein and Hubble in the early 1900s, we lived in just a field of stars. No other island universes, no other fields of stars, just, you know, separate and distinct from our own. These two phenomena would come to play a key role in the creation of relativity, and the application of relativity and specifically general relativity to the cosmos and the evolution of matter and energy across space and time. So I should really make this 19...

Oh no. As I was saying, I should really make this 1905 because not only had the seeds for Einstein's relativity, which would come to dominate our view of the universe, universe again had been planted but they were really to continue the analogy they were starting to sprout and blossom and in fact Einstein would later remark that he thought that by 1905 special relativity itself was ripe for discovery and one of the four major well, one of the four papers that all four of which were

essentially groundbreaking that Einstein published in one single year, 1905, was the paper called On the Electrodynamics of Moving Bodies in which or out of which this equation came. The relation between mass of matter and energy contained within it in proportion to the speed of light.

It's time to open up our chapter on special relativity here. And I browse this a lot, so it's pretty beat up. Maybe time runs a little faster in this chapter. But they open up this chapter talking about Maxwell's equations and just how... Just how influenced by the Greeks the concept of ether, the medium through which Maxwell thought, light,

or his electromagnetic radiation waves traveled. And he even said that, so the Greeks believed there was four earthly elements, earth air, fire, and water, and one heavenly element, the fifth perfect element, called the ether. And Maxwell echoed, they said, their ancient belief, the Greeks,

by saying there can be no doubt that the interplanetary and interstellar spaces are not empty, but are occupied by a material substance or body, which is certainly the largest and probably the most uniform body of which we have any knowledge. And one of the major impulses for

trying to find this ether, which later on was never discovered and was attempted to be discovered by what's called the Michelson Morley experiment, was to find a universal frame of reference. So the impulse to seek out a universal reference frame from which all motion and all physical measured physical phenomena, including light, could be measured. because really had its roots in another genius concept by Galileo and Galileo's thought experiment showed pretty logically and intuitively that we can't know whether or not we're traveling with any constant velocity whether we're stationary or traveling at any velocity, no matter how fast it goes,

because all you would have to do is, as the equations here show, is translate between two coordinate systems, one stationary, you know, from our little perspective here again, it's all relative, one moving to the right, or, you know, you move to the right, and one stays the previously stationary one appears to move to the left, doesn't matter what matters is that you can easily translate the position velocity and acceleration between them and come to find out taking the time derivative of the position and velocity the acceleration for any oh shoot and I wasn't even in frame I poked a hole right through my paper the acceleration is going to be constant between the two reference frames regardless

if the frames are moving with constant velocity. If we have two coordinate systems, we have one, two X, Y coordinate systems, which the X, Y, prime coordinate system, is actually moving with respect to the other. It was Galileo who used this to come up with a theory, of relativity 300 years

before Einstein. If you have two systems that are moving or two systems one of which a ball is moving let's say

two meters per second to the right and this coordinate system starts moving itself coordinate system itself moves two meters

to the right two meters per second Well, from this coordinate system's perspective, you would simply add the velocity of the ball to the velocity of the coordinate system. And you would get the apparent velocity of that ball over there as the observed velocity would simply be the velocity of the coordinate system. So the V-prime plus the velocity of the ball. velocity of the ball. You'd get four meters per second in this instance. Well, and then if the ball was moving the other way towards you,

you would simply subtract the... Well, if the ball was moving this way and the coordinate system in a reference frame, so like those experiments you see were a ball was shot out of the back, of a moving truck against its, the moving truck's direction, the opposite direction, the ball actually looks stationary. So if the truck was moving this way, the, a ball was shot out with an exactly equal but opposite velocity,

or speed, equal velocity, opposite direction. This way, the ball itself would just maintain a, uh, static position. So Galileo discovered if you have coordinate system or two different coordinate systems in which one is moving away with respect to the other,

there are no experiments you could do to prove that it was this one that was moving away and not, in fact, this one, moving away. So we could have this one moving away like that or okay here we go or we could switch reference frames from from this one if we're in this reference frame we see this one move away or we could switch reference frames into this one and see this one move away at the exact same velocity

Although Newton in the 16 and 1700s had essentially cemented in the foundations of science, that there was absolute time, time ticked at the same rate, regardless of how fast you were moving or how your position in the universe and space was equally static and uniform. But Galileo kind of planted the seats of doubt with regards to that he said you could do any experiment

you want whether it's on land or he theorized being in the ship the lower decks of a smoothly sailing ship with no chop and no air resistance or anything like that

no other movement other than the straightforward constant velocity and you wouldn't be able to tell in either lab you're whether or not you were at rest or had a constant velocity and all we knew was you could be at rest with respect to the land on earth or you could be at rest with respect to the sun but then it turns out you know we find more and more

relative motion the more we probe the universe and we find that we're orbiting the sun but as we orbit the sun we're also orbiting a sun that is itself orbiting a galaxy which itself is moving with respect to a central you know gravitational center of mass

among a collection of local galaxies that is itself part of a larger cluster and super cluster and you know filaments of the universe as you go up and up and up and scale and Galileo he proposed that you could transform

between coordinate systems Galileo was essentially the first person to come up with or formalize some equations that were that Einstein's relativity was based on but as the opening of this chapter on

special relativity in our book here says so Maxwell Well, when he discovered that his electromagnetic waves disperse or radiate through electric and magnetic fields at the speed of light, which he initially didn't correlate with light itself, he didn't initially make that connection, but he later did. And he said that there must be a luminiferous ether, an ether through which no other objects except light experience resistance. And so they scientists, physicists, tried to perform, or they did perform an experiment to try to prove or find some sort of universal reference frame. Because the Galilean

transformations, his version of relativity, when you extrapolate it up to speeds of light, started to get strange. The extremes as you approach the speed of light, What that would mean is that if you have one reference frame and from this reference frame are watching someone else travel at the speed of light, their velocity equals C, and they emit light, just a flash of light, well, that would mean that, according to these equations here, from the, if we say the prime, reference frame we count this as the you know call the frames s in s prime and the s prime reference frame is moving at the speed of light along with its flashlight that's stationary within it the flashlight and the reference frame s prime are moving along moving at the speed of light from the perspective of s so if you know from both coordinate systems, let's say they started right on top of each other.

So any initial X, Y, Z positions, and the time is all equal. And then when you start the clock, then the S-prime reference frame shoots off instantaneously at the constant speed of light to the right. Well, any position, if you're going to measure any position of the flashlight with respect to this prime S-prime reference frame from within it, It's simply going to be whatever the position is, whatever the initial position was, subtracting the velocity or the added new position that would occur if you accounted for the velocity of the reference frame itself.

So, in other words, this flash that is at rest with respect to this reference frame, so there shouldn't be any change in position. So if we multiply the velocity times time, and velocity is meters per second, time is seconds. You can put that over one. Those two units cancel out, and you're left with just meters. So that gives you distance, in other words. That's what velocity times time is going to equal distance. Or, in other words, that's what these coordinates represent.

coordinates in space and so there's in other words there's no change in the X position with respect to the moving reference frame however with respect to the S not the S prime but

the S reference frame that you know we're considering stationary in the flashlight and S prime are moving away from that flashlight is going to be moving so its X coordinate is going to change at a rate defined by the velocity of the speed of light.

So its equation is now, we're going to have to solve for the S reference frame. We'll solve for X instead of X prime. The X coordinate of the flashlight, in other words, as seen from S, stationary S, is just going to be flipped. Instead of the X coordinate being the initial coordinate, subtracting the distance, traveled by the frame, it's going to be the initial x-cordinate plus the distance underwent, undergone by the reference frames velocity, multiply by whatever time you consider.

And what that means is that if you're moving at the speed of light and a flashlight goes off, releasing photons at the speed of light, that means that these photons are going to have a velocity of the speed of light plus the reference frame. So a new velocity, speed of light will say, call the reference frame velocity VR plus the speed of light. In other words, some weird stuff happens according to, from the perspective of S, because that would mean that the photons or way light waves traveling with in the direction of the S prime reference frame would be traveling at two times the speed of light.

But the photons traveling back towards our stationary reference frame S would be the speed of light emitted from a source traveling away from it at the speed of light. and if we use our Galilean transformation equations, the speeds cancel each other out. We subtract the speed of light from the source velocity, which is the speed of light. And that would mean that there is a reference frame from which the speed of light can appear

to be at a complete standstill. And this is, early up until the 1900s, there hadn't been any experiment to ever prove that the speed of light is anything other than C. It might slow down in a medium, but in a vacuum, speed of light had never been observed to change speed. And that was the question that Einstein wanted to know. He said if you were traveling with this reference frame at the speed of light, and you had a mirror

and you looked at yourself in the mirror what would you see would you be able to see your reflection would light be able to traverse to the mirror and then reflect back to you would it be affected by the velocity at which you were

going if that velocity was the speed of light itself and he wanted to know how the laws of beyond just the you know kind of boring question of just stopping there. Einstein, as all geniuses do,

really went down the rabbit hole of considering speculating about what the laws of physics as known up until that point would do in a reference frame moving or laboratory moving at the speed of light. And how would you even be able to tell? Well, this Michelson-Morley experiment here was trying to discover a ether, which would be like a field,

like an electric kind of considered like a magnetic electric field in the sense that it just permeates the entire universe. And this ether would be able to, if discovered, act as a universal reference frame relative that all other objects could be measured relative to if we had the precise enough instruments

to be able to detect it and so that's what they did they set out to detect it in the late 1800s think over a period of five years culminating in 1887 they essentially took the orbit of Earth again and I got my number wrong earlier

it's instead of 35,000 kilometers per or miles per hour it's 30,000 kilometers per second. So they took earth and essentially as earth is the velocity of 30 kilometers per second. What they thought was that if there is a universal ether and so maybe it has some sort of velocity that would appear to us but regardless what they did was

shine light laboratory in the direction of earth's motion and then they shine it in the direction perpendicular to earth's motion so they said their experiment I think if I understand it right

essentially was meant to predict in one direction or the other. When things go perpendicular to motion of objects, that usually completely insulates them from the effects of that motion. By setting an experiment in which two lasers were oriented perpendicular to each other, they were hoping to discover that the motion of Earth would allow a significant enough effect on the light beam traveling along with the earth

versus the light beam orthogonal or perpendicular to the earth's motion that one or the other beams would slow down or have its velocity, its measured velocity affected by this hypothesized background ether and they figured they could

detect our motion and put a number on our motion with respect to the ether, saying that we're flowing along with it or at 30 degrees against it at 500,000 miles an hour, whatever they found. They would at least have positive results if there were a difference in the speed of light.

But they found that the experiment was consistent with the velocity of earth through the ether of zero. But as the book goes on to say, they're, according to the Galilean transformations, there should have been a noticeable detection if Earth is traveling at a significant speed and you shoot a laser, shoot, you know, a beam of light, along with it, you should detect some noticeable different value for the speed of light

relative to shooting a laser perpendicular to it. And they said that, you know, those transformations should hold true. and what they found that although they hold true for values much less than the speed of light or ratios of the velocity of a measured object to light being much less than one they found that they hold true to that but as you approach the speed of light a sharp disagreement appeared in the experiments involving relativistic or what we now call relativistic velocities close to the speed of light and a crisis in the Newtonian paradigm was developing so this is where

relativity comes into play and it's to show us that Einstein he revolutionized not only the conception of time the in absolute time but absolute space and the relation between the two he fused them together into something we now literally call space time. No hyphen. I think it took a few years before they removed the hyphen. So right on the cusp of the inability to discover any evidence for an ether,

a medium through which light could travel, Einstein popped onto the scene. And he, after much reflection, Einstein finally rejected the notion of an all-per-rejected the notion of an all-pervading ether. And his paper here, the one in which he proposed special relativity and derived the E-equals mc-squared equation

as the name of the paper was the electro on the electrodynamics of moving bodies. So this was not just simply moving bodies, but this was an attempt to, and this is what set them apart from, those that had partially formulated all the ideas that he used and employed in this paper before him, but he was the first to actually synthesize them into a cohesive, unified whole paradigm of physics. And he was including Maxwell's equations on electricity and magnetism and trying to synthesize them,

trying to derive equations for motions through the universe that would, hold true regardless of what you were talking about whether it was electricity and magnetism propagation of light propagation of matter as it radiates light and that's where the principle of relativity came into play he says that the phenomena of electrodynamics as well as the mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that the same laws of electrodynamics and optics, the physics of light,

will be valid for all frames of reference, for which the equations of mechanics hold good. In other words, Newton's inertial reference frames were constant velocity reference frames. And his two principles here were the one that Galileo employed, which was that the laws of physics are the same in all reference frames moving out a constant velocity,

whether zero or some constant number, they didn't accelerate or decelerate. But the second one, the most crucial one, again that he didn't come up with himself, but he was the first to synthesize and put all together. He added that, the important postulate only apparently irreconcilable with the former that light is always propagated in empty space meaning vacuum with no atmosphere or any physical matter to obstruct it with a definite speed see which is in this is the crucial part here independent of the state of motion of the emitting body what that means is that light is always going to propagate out from a flashlight,

regardless of whether that is considered as still or is moving close to the speed of light away from us. We are always going to detect its light, both moving with the frame and back towards us away from its direction of movement, as being the speed of light. What that comes down to is that if light is going to be constant,

no matter what the speed of its reference frame, of the source of it the length of let's just say we'll stick with the flashlight that flashlight if it's pointed at us it's moving away at the speed of light we are going to observe its length to contract and this and this second part might be even crazier the clock relative to the flashlight if you were sitting on that flashlight looking down at the clock you would and you're riding along with the clock you would observe it to tick at normal speeds but to us we would be observing right here we would be observing that clock on the flashlight to be ticking slower and slower at slower and slower rates the closer to the speed of light it got they go on to derive

equations here and they start with Galileo's transformations but then they say at this point these equations are just consistent with the Galilee and transformations however only one of Einstein's postulates has been employed up to this point that the laws of physics remain the same

regardless of how fast you're going as long as you're going to constant velocity now the argument introduces the second postulate that everyone measures the exact same value for the speed of light

and they go on to set up again two coordinate systems. But you got to remember the crucial part I think is that these are spacetime coordinate systems and where did he write it? Four-dimensional space time Einstein's teacher himself wrote that Minkowski. Henceforth space by itself

and time by itself are doomed to fade away into mere shadows only a kind of union between the two will preserve any kind of independent reality and the authors of this physicist go of this book here and go on to say that the drama of the physical world unfolds now on the stage of a four-dimensional space time instead of objects moving across space measured with the movement measured with respect to a absolute universal ticking clock that never changes rates of ticks, you have to consider space and time as a four-dimensional coordinate system where events are identified not by just their XYZ and time separately, but by all

four coordinates. This to the, what they call the downfall of universal simultaneous. No more could we consider objects to be considered objects separated by space some distance apart to have any sort of simultaneity even if you're 10 feet apart and you have two light bulbs and you rig them the flash at the exact same time from our perspective when you're moving at the speed of light if your direction is anything other than perfectly symmetric with respect to those two lights, right down the middle of them, you will measure those lights as going off at different times. They're going to be real subtle differences, but they will be going off at different times. And when you scale that up to the universe, you start getting some really, really strange things happening.

when you try to talk about things happening at a certain time or even a certain distance away. Really quick going back to this. The point was, is that once they introduced the fact that light, the speed right here, velocity C represents the speed of light, that number never changes, regardless of which frame you're measuring from.

so unlike the situation where we were talking about Galileo measuring an object's velocity if you measure it with you know being shot out of a cannon while it's moving with this reference frame it's going to be twice as fast as if you're going along you know like that you're just going to measure the normal cannon speed if you're going along with the reference frame. You won't have it added. You won't have the velocity added to it. But in the case of light,

you don't add the velocity of the emitting object. And space and time itself has to contract by a factor called the Lorentz factor, which is an equation or an expression

that relates the velocity you of the emitting object to the velocity of light. And it says here that relativistic equations relating the coordinates and space and time of an object differ from regular Newtonium mechanics by only about 1% when this Lorentz factor is 1.01 at a ratio of going about 1-7th, the speed of light.

but it starts to increase. In this graph here is the exact expression of that increase. As you approach the speed of light, essentially the ratio of your velocity to light approaches one. You don't even get a factor of two change until you're all the way at about 90% of the speed of light. It's between the 90th and as you approach the speed of light, 90% all the way to 100%.

although you never reach 100%, that you see a dramatic divergence of relativistic equations and what you would predict for Newtonian mechanics. And one of the, two of the features of that

are that time gets dilated as you increase this, cause this number to go towards one as you increase your velocity and then this whole denominator gets smaller and smaller. And if you have one over a small denominator, or well, anytime your denominator gets smaller than your numerator, the entire number, or in this case the Lorentz factor, grows.

And that shifts dramatically, how space and time starts to look to an observer, both moving with, at that speed and the person on stationary with respect to it. So space and time start to look really, really strange. And because you have no, again, no absolute reference frame, we don't have any way to say that we aren't the ones moving away at the speed of light. All we know is that relative to the matter around us in our galaxy,

our stars in our galaxy we are not moving at very any significant fraction of the speed of light so that leads us into the equation here talking about time dilation and it's saying how the easiest way I found to quickly summarize it is that the shortest time interval between flashing lights if you have a light that just flashes at a constant rate and once per sec one hurts and you have that light what will the shortest time interval between those flashes is always going to be observed from the person moving along with that flashlight that blinking light and so as that light regardless of what direction it's moving away from you in as it moves away from you at an ever-increasing speed it's dilation its perceived gap between the the blinks or its period, its frequency, will appear to start to elongate and stretch.

In other words, it's time of the rate at which time ticks at the location of that blinking object will start to appear to an observer not moving along with it to get slower and slower and slower. And that is called time dilation. And just as the shortest, just a measured time interval between blinking lights is measured when you go along with it. The shortest measured distance of an object, length of an object.

If you have a meter stick, you will measure exactly one meter as long as you're traveling along with that meter. But if that meter stick starts to travel away from you, as its velocity, away from you or relative to you starts to increase so will its length contract shorter and shorter and shorter so in other words as you're watching an object move away from you at an ever-increasing speed as that speed approaches the speed of light you're going to notice not only the object start to get appear shorter and shorter but it's going to appear to slow down Any clocks on that ship will appear to us to start ticking slowly more and more slowly. Now these two effects here, this is a crazy example that they have.

They essentially say that muon, when it's coming from, it's moving nearer the speed of light, something like 99%. Was it here? Muons collide with the nuclei of atoms in Earth's upper atmosphere. Or cosmic rays collide and produce. muons, rather. So they say that muons are unstable, and so they decay in a really, really short period of time, 2.2 millionths of a second.

And so what they said is that this weird thing happens. At the top of Mount Washington, muons were brought into existence through the cosmic ray collision, but the distance from the top of Mount Washington, a detector at the top and a detector at the bottom, is so far that even moving at the speed of light or that light speed or the 99% that they measured it moving at it wouldn't have time to reach the bottom detector you know they show here it only getting about you know two-thirds or three-quarters of the way down the mountain before it just annihilates so if its lifespan from me is only 2.2.2 millionths of a second and you know maybe takes three millions of a second to get all the way to the bottom they noticed that they were measuring particles still existing at the bottom and what they attributed that to

was the relativistic length contraction and time dilation of the entire earth and the entire mountain with respect to the reference frame of the muon and what the that means is that from our perspective the muon is traveling close to the speed of light so just like we talked about that would mean that its clock moving along with it its rate of time would tick slower and its length would appear to be even would appear to be shortened if we can measure its length. But it's the time element that matters from this reference frame. Because

if its clock is ticking slower, that means it's going to survive longer before annihilating. And they said they measured the muons lifetime to actually be essentially a hundred or no, ten times longer than measured from

in the lab, I guess. So they knew its baseline should only be two microseconds or millions of a second. but instead due to relativistic effects plug in the equation you multiply by the Lorentz factor you get a lifespan of 22.5 millionths of a second

that's enough time for it to last all the way to the bottom of the mountain and if that wasn't crazy enough the second reference frame now is the muons reference frame in which if we're moving along at 99.52% of the speed of light, then our lifespan should still only be 2.2 microseconds. So that's not enough to reach the bottom of the mountain now.

What's going on? I'm sure if you guys are looking at this, watching alone, you can see that from the muons near speed of light or relativistic reference frame, the entire height of the mountain contracts again by this Lorentz factor here and it goes all the way down from 1907 meters to 186 meters right there it says this result shows that the effect due to time dilation as measured in one frame may instead be attributed to length contraction as measured in another and it even emphasizes don't think of these as being due to some sort of optical illusion.

Taking, you know, if you have a really long object, looking like it's contracting, I thought it was pretty clever to address that concern. You might think it's just taking light different amounts of time to reach an observer from different parts of the object. But he said the language used here was careful to include measurements of an event's basetime coordinates using meter sticks and clocks located at that event, so there is no time delay in the difference in distance it takes.

They said, of course, no actual laboratory has infinite, you know, clocks and meter sticks to instantaneously measure, take all these measurements. And so time delays caused by finite light travel times have to be taken into consideration. And this is important in determining the relativistic, Doppler shift formula. And so this formula here takes into account time dilation and length contraction and it takes into account the recession velocity and what that would do to the period or frequency

of a light wave being emitted from an object moving at that velocity. And it's really this is what I've been trying to get to. the Doppler effect is like we talked about a little bit before having an object's sound waves be compressed in the direction of its motion and lengthened in the opposite direction of its motion but there is no medium through which the compression and lengthening can occur in light because that only was only found for sound and ways waves moving through the air and waves, waves moving through water. And they said that the Doppler shift is qualitatively different from its counterpart for sound waves. So although the equation actually turns out to be almost identical at low velocities, we need to take into account relativity when we're moving at velocities close to the speed of light. So here is an overview of how they derived the explanation for how light red shifts.

When you have objects relatively cosmologically speaking close to each other within about 10 to 100 million light years, which is close by cosmologically. And that means you can still have objects moving through space at speeds. close you know appreciably an appreciable fraction of the speed of light and you'll notice a red shift that shifts their wavelength that's lambda means wavelength there and the proportion at which their wavelength is shifted relative to the known wavelength of light that is emitted and we can tell by the fingerprints of atoms and you know hydrogen dominates the makeup of stars and if let's just say they have specific lines there's hundreds of lines per atom so it's uh

got um getting ahead of myself there but you have a spectrum and you know that these wavelengths are like you know let's just say 450 500 and what happens is that you measure in a lab hydrogens these are a hypothetical wavelengths the specific wavelengths of absorption lines in the light that is being split by a prism in a lab

you isolate hydrogen as its own element so there's no other elements being or atoms types of atoms polluting or whatever you call it the mixture so it's a pure mixture of hydrogen so you know that all the lines specifically correspond to only hydrogen.

Now you look at starlight, and you split that into a spectrum. James Webb has multiple spectrographs. And what you find, and there's ways that we won't get into talking about here, but there's ways other than parallax of measuring distances to nearby, even nearby stars, but even nearby galaxies, within a couple million light years of us. So we know to a fairly high degree of precision how far away some nearby objects are.

And we can also see that these fingerprints, so we know it's the exact pattern of lines repeating. We see this fingerprints, and again, that's 450, that's 500, And we now we see these three have shifted over from here to here. And we see the whole pattern itself shift over. That is called a red shift. The shift over, this being the blue end of the spectrum, this being the red end, shifts over towards the red end.

And that relationship here, we measure the amount shifted, the actual numbers, relative to what we know it to be at rest in the lab here on Earth. And due to Einstein's equations here, we're able to measure, we're able to derive the actual velocity. I have it pointed to down here. From this equation here, this radial motion red shift equation, they input, so Z is essentially the percent of the shift of wavelengths of light. So it's the percentage, how much it's shifted along the electromagnetic spectrum. They get a number output called denoted by the variable Z. For instance, here they have a quasar.

They know that one specific line that they always look for, some hydrogen line, a very prominent one, so it's a good standard, is the 121.6 nanometer hydrogen line. mission line on a distant quasar they noticed that it's shifted all the way red shifted so its length is getting the wavelength is getting longer to 885.2 nanometers so therefore the difference divided by the known 121 number of the wavelength um 121 nanometer wavelength the The difference that ratio is 6.28, a shift of 6.28. They plug that in for here, and they solve for the velocity.

That's the recession velocity. And here, multiply that by the speed of light, you get a velocity of point, or 96.3% of the speed of light. and also using that same number, that same redshift equation number from up here, from the Doppler formula, you can also simply equate that number to the ratio of rates of time at which objects occur at that object versus how fast you know it should be happening if you were at that object in its own rest frame, moving along with that object simply take that ratio subtract 1 and you get Z and if you know say a supernovae you've been able to measure a distance to that and we know it should take 20 days to peak and dissipate and we know you can measure

its red shift by seeing the difference in hydrogen emission lines then you can solve to get the to to find out how much that object, that object's time has been dilated from our vantage point. And the chapter goes on to derive the famous E equals MC squared equation right here. The rest energy of a particle is when the particle has zero kinetic energy and all its energy is entirely in its mass. and it's from the redshift that Hubble was able to figure out that the universe originally at the beginning at the time Einstein thought came up with his both his special relativity and ten years later in nineteen fifteen his general relativistic equations Einstein and everybody else most people thought that the universe was only about three hundred thousand light years across at most and then Hubble comes along and discovers that the universe is not only

much larger than we thought by measuring the distances to these things called these stars called sepheid variable stars in indromeda which most people thought was inside our galaxy or you know just inside of the field of stars maybe extending out to 200,000 light years away Hubble took careful measurements and determined that the sefied variable stars relative to other ones he'd measured more nearby, whose parallax he'd taken so he knew a definite distance to. We're so far away that they couldn't have even been in the realm of the nearby stars that had so far been measured. And he noticed that they were situated in the nebulae, the one called Indromeda, specifically. but others he measured as well and he found out in 1927 or 25-ish that that nebulae was not only 100 outside the 100 or even 300,000 light-year span that the field of stars were thought to extend to

up until that moment but it was millions of light years away and this blew the perception, the paradigm of how large the universe was up until to the order of tens of millions of light years across which was really hard to believe and then only four or so years later in 1929 he comes up by measuring the red shift in these distances using sefied variables to

different galaxies nearby he comes up up with an equation now called the Hubble or Hubble-Lamatra equation because someone else had discovered it a little bit before him but uh oh here it is he so we have this right here he found a trendline among the distance versus velocity graphs of all the galaxies he measured and what it turned out to be was almost a perfect relation between the distance of galaxies and how fast they were going. And so as you go further and further out, they start to look like they're receding, the red shifts using the redshift equation, which can be at a certain point thought of as just being Z at that low non-relativistic velocity.

is the velocity of the galaxy relative to the speed of light. And then it gets, there's a relativistic version that we looked at too. But he noticed that it's peculiar that so many distant galaxies appeared to be receding

further and faster and faster. And then here's a... So this was from 1929 here. And this little red spot right here, if you guys can see. This little red spot is this entire graph. So parsec is about 3.25 light years. 10 to the 6, that's, is that 2 million?

Yes, 2 million. So that would be about 6 to 7 million light years distant. And this is million parsecs. This is billions of light years. This is about going up to about 2.5 billion light years away. Increased it by... an order of magnitude about, I don't know, six orders of magnitude or so. Or maybe three.

This entire graph is in that tiny red square. And what's crazy is that that trend line has continued all the way out. And what's even crazier is that our graph, graph de jour, extends out 20, 40, 60, 5. Redshift is one of the most important. concepts to understanding cosmology and our graph tonight. I'd say redshift and relativity are two of the key concepts that inform our what we understand about the universe at the deepest levels. And redshift is particularly important because relativity helps us interpret redshift. but Redshift in itself, other than gravitational waves and neutrinos and some other relativistic particles coming in from the cosmos,

light is the only direct data we have from the universe outside of any place in the solar system that we're ever going to send a spacecraft. And Redshift, as we touched upon earlier, is an aspect of the, emission and absorption lines, those lines represent energies of photons, who we can kind of place on the electromagnetic spectrum, and photon energies are directly related to wavelengths or frequencies along the electromagnetic spectrum. And all atoms, hydrogen, calcium, oxygen, carbon has its own specific configurations of electrons around its nucleus. As the electrons get excited, get injected with outside energy from photons, the electrons jump up, they absorb energy of photons coming in, and that will create a dark spot

like this, dark lines on a otherwise light background, or when electrons collapse down towards the nucleus, as they tend to do after a certain period of time, electrons have a tendency to want to be closer to the nucleus, and as they collapse from a higher potential energy state to a lower one, a more relaxed, more at-rest state, They release a photon consisting of energy exactly proportional to the difference in energy states between the higher energy electron, higher potential electron, further away from the nucleus, and the lower energy electron, the new lower energy state or position of the electron in its orbit, the orbital. I'm holding this candle here because it's a good instance of the emission of photons.

It's emitting right now, right to the camera lens and the phone is projecting a simulation of this light onto your eye. I'm holding this camera because the... It's just interesting, or the candle, that the wax is the initial spark of this lighter. the flame of the spark injects heat that candle wax into a liquid form and then eventually a vapor form and when it turns into vapor the the wax burns it to burn and I think the the wick is just a medium for the wax to get transported up into a like a column but the heat vaporizes the wax and the actual atoms or molecules in the wax interacts volatilely with the oxygen in our atmosphere.

And the exothermic reaction of the fuel in this candle here with the oxygen snaps together and they release energy which adds further energy into the system that continues the vaporization of the wax. And it's just, that's pretty cool. It doesn't specifically have anything to do with our video here, but the point is that we had a Okay, so I'm back from that little mishap. I was trying to angle the Candle so it didn't burn my camera lens up, but the heat of this reaction, I guess

Most of the photons are apparently just simply the energy released from the Oxygen wax interaction reaction letting out more energy and and exothermic reaction. They're injecting energy into the local atmospheric atoms of, we have oxygen, nitrogen, and every time the electrons get excited, after a brief period of time on human scales,

the electron jumps back down from its excited state and releases photons when it does that. So the heat, as long as the wax fuel lasts, this reaction and the flame is a self-perpetuating phenomenon and it's actually plasma I didn't realize it's not you know it's a fourth state of matter like we think like lasers are and the interior of the sun is so that's I guess a good segue because they

are related to red shifts because any atom emitting a photon emits photons at specific wavelengths that are proportional to that are directly related to the very unique possible configurations of atomic of electron orbitals around their nucleus. So, in other words, what that tells us is that every atom is going to have their own unique set of possibly thousands of emission lines and absorption lines because they also, they only absorb photons that perfectly match the, the energy of their emission lines. So, these, this really unique property, unique pattern that characterizes each individual type of atom, allows us to identify the atomic makeup of stars.

And the atoms involved in the, in this reaction here, given off by this flame, if we had a spectral prism we could break this flame right here into its specific absorption lens that would tell us exactly what atoms are involved

in the creation and emission of these photons that we're looking at right there and light is the only it's the only hard data that we have from the universe light is the only information we can

say that we definitively have in its raw form without any interpretation required. And red shifts are extremely important because as far back as 18, the 1840s, Doppler had, he had given us a definition in an easy algebraic equation, that the redshift of light, just like the Doppler shifting of sound waves and even water waves is going to tell us direct information about the speed at which that light is traveling

and relative to the known constant speed of light in a laboratory setting when you're at rest with respect to that light when Doppler what's his first name I forget

discovered this relationship they actually, I didn't realize this, but they immediately applied it to light. I thought they were only focused on sound waves mostly. But yeah, even as early as, you know, the 1850s, I think, they were focused on light. And as, you know, lots of discoveries regarding the character of light between its emission lines from the early 1800s that were able to be, they were able to be observed through filtering the sun's light through prisms. And two Maxwell and Faraday's experiments with electricity and magnetism, and Maxwell eventually understanding that light was an electromagnetic phenomenon.

It was a wave propagating through electric and magnetic fields. they I think Doppler himself actually thought to apply the Doppler formula which you could shift to say that the velocity of an object

is simply you multiply both sides by sea simply the speed of light times the observed red shift any thought to say okay well if we look at the spectral lines of the sun or distant stars

we might be able to see that if they're shifted from what they're known to be observed in a stationary laboratory with respect to the observer, then we could tell their velocity, whether they're coming towards us or away from us. If they're shifted towards the blue end of the spectrum, higher energies, they're going to come towards us, the waves are compressed. As it's moving, the waves are being emitted, the wavefronts of the individual wavelengths, the crests of the waves, like we talked about are emitted closer and closer together.

And then conversely, they're longer and longer with each successive wavelength, being on the order of billions of times per second for visible light, and then radio devices that were able to record very long wavelengths from stars and radio emissions being only thousands or millions, millions of times a second.

Discovery of this relationship, this allowed astronomers to start having an idea of how fast stars were moving in relation to us. And eventually they were able to look at the use Kepler's Laws of Motion and discover that just like

the interiors of, well, the solar system, this planets that orbit Let's say the frequency of their orbit or the inverse of their period with speeds proportional to how close they are to the star. So closer planets orbit much faster, as we talked about with Earth and Jupiter, then further planets. Well, they started looking at the velocities of stars and they were able to figure out that stars closer to the interior of the Milky and, you know,

way the orbits the the the velocities told to us by the red shifts the measured red shifts of the light were greater in the interior there were maybe you know 150 so here for maybe maybe like 150 kilometers per second and versus much further significantly lower more like 100 meters per second. And apparently there's, if stars are much more than, much more than these numbers, they actually would reach escape velocity

and they wouldn't be, they would be ejected out of the galaxy. So as early as 1912, I think, this guy named Vesto Sliffer. He was the guy who Hubble relied upon, who was like a predecessor to Hubble

and anticipated his Hubble's discoveries of the receding galaxies and even well using Doppler shifts to start to look at these nebulae in the sky because so what happened was that as we look at our galaxy

well we are in the galaxy so we only see it edge on like that And as we look up, down, or outward, because we're kind of at the edge, we see these nebulae are as telescopes towards the beginning of the 1900s, the early 20th century. We got more and more advanced and larger and higher resolutions could see further and more faint objects. we're noticing that nebulae were the what would become galaxies were visible all over the night sky except in the field the line of sight looking towards the center of our Milky Way so as we looked at the center of our Milky Way we noticed that there weren't any nebulae there weren't any definitely weren't any spot Nebula because it was the big debate all the way up until the early 20s and the late 20s when Hubble It was only resolved when Hubble firmly was able to tell that the distances to other you know what galaxies were actually way beyond any of the observed distances to the stars to any actual specific stars and

But there was also nebula like the Orion nebula that were in in fact are gase clouds, diffuse clouds, nebulous clouds in our actual galaxy that are nearby and relative to the distances we now know to galaxies. And because we had some diffuse nebulae that were looking like blobs and clouds, and then we had others that were much more definite, you know spiral shapes it wasn't certain that these spiral shapes weren't inside the galaxies and you know roughly equivalently distant to these more irregularly shaped nebulae but one of the things was was that we could detect the more irregularly shaped nebulae towards the center of the galaxy.

But any time you looked at the center of the Milky Way, these more spiral shapes disappeared. That was one of the bits of evidence that told us that maybe they were outside our local group of stars. and in 1912 Vesto Sliffer started looking at these galaxies and over about the next 15 years what he noticed he measured about very accurately about 20 to 30 galaxies in their spectral lines and observed the redshift and how they'd been shifted and he noticed that other than Andromeda and maybe one or two other really close galaxies

the red shifts of these spectral lines from our vantage point in the Milky Way showed that they were rapidly instead of this they were showing that they were rapidly receding away from us the red shifts were extremely were extreme in telling us that the velocities were very extreme I don't think he quite put together that, well, he didn't know. It still took until Hubble.

Hubble was the... Hubble spearheaded the discovery of the actual distances, the measurement of not parallaxes, but the luminosity period curves of Cepheid variables and distant galaxies that enabled us to determine based on how faint they were in relating that to the period. and how bright we knew that nearby sepheids were with the same period.

We were able to detect with the law of the luminosity. Actually, I think it was Newton's law, where the luminosity is proportional to the inverse square of the distance. So when Sliffer took 20-30 galaxies, and he noticed that, you know, Andromeda was... Did appear actually quite blue-shifted. It was one of the only galaxies who if this is the you know lab line So this is the line measured in a lab so we know that the line should look like that it was way over here

That's without you know andromed was way blue shifted telling us that it was actually approaching us at a speed significantly, even faster than the higher speeds at the center of the Milky Way. Indromeda was actually approaching us, approaching us at a velocity of 300 kilometers per second. Whereas these other galaxies were moving away at about the same speed, but because they were redshift, we knew that they were, you know, about. But while Sliffer pioneered the redshift analysis of spectral lines of spiral nebula, which indicated that they probably were pretty far outside our galaxy, it was Hubble who pioneered the application or the addition of distance

to give us a definite size, a definite understanding of just how large the universe was based on how far away these galaxies were. So sliffer, you know, getting velocity, you can't, that doesn't tell us distance. It didn't tell us exactly how far away these galaxies were. And it didn't give us any relationship between how far away they were and how fast from us they were traveling. It was, it was Hubble on these spiral galaxies. It was Hubble who put these two together.

I mean, and it was, although he apparently greatly relied upon Sliffer's actual data, he was the one who took the time to really carefully measure the distance to endromeda, for instance. So, endometa is much larger, but it's far away. Measureed that right here on the galaxy, or galaxy. He measured that. Andromeda, like we said, he's moving away at about 300 kilometers per second. But he also measured a sepheed variable star in it that said, Andromeda was about 2 million light years away.

Look here says that sephiates can, they have a range of accurate measurement. They can be measured up all the way up to about 29 megaparsecs, million parsecs. in a parsec is 3.25 years. So roughly almost 100 million light years away, we still get accurate sepheed measurements, and they're so bright that they're not as bright as supernovae, but they're brighter than novi. So if Hubble, what he did was take the measurements of further,

increasingly further and further galaxies. As he was able to make my light bend a little bit here to get to us, as he was able to determine their distances. Let's do a log scale here. We'll say that this

increases by order of magnitude with each step here. That's 2 million to Andromeda. It's 20 million to this one. And 200 million, which is actually the furthest. I think it was about 180, 140 light years away. 140 million light years away was the furthest hubbalax it was able to measure, but it's roughly on that same order of magnitude. And he noticed that the red shift of Andromeda, yes, it was showing 300 kilometers per second

towards us, so it was a blue shift. But there were other galaxies, a little bit further than Andromeda, much closer than the 20 million light years. But as he started taking measurements, we see our graph here shows that at distances a million, two million light years away. You had an increase in velocity. Andromeda just happens to be traveling towards us, but the velocity was getting further away. So the velocity here might be 600 kilometers per second away. And the velocity here might be...

The Hubble's big breakthrough was not only discovering that the distance to the spiral nebulae were way further than any distances measured to stars, so they had to have been outside, and therefore it blew up the universe. It blew our paradigm apart, saying that now there's not only... are a field of stars which extends to an enormous maybe two three hundred thousand light years across but now there's other fields of stars isolated in their own you know gravitational local areas in the universe that themselves might be that far that might be that large and millions of light years away so he starts focusing on these trying to find the

distances to them. He gets 2 million, 20 million, 200 million, almost 200 million light years away, and he starts looking at the red shifts. He starts, and he specifically picked in the galaxies that Sliffer had already made red shift measurements for, being fairly intelligent, they combined data, and he discovered that as he was able to find the distances to these galaxies, the further the further the galaxies were the larger the redshift became and that's Hubble's legacy was this relationship of redshift his relationship of redshift to redshift being equated to distant to velocity to the distance through a constant right here which now we know that we've taken measurements of distances billions of light years away.

We know that this actually isn't a constant. It's not a universal constant. And that actually changes with time, which is going to be important in a little bit. But for the local, I think any less than maybe 100 or 500 million light years on that order, this value doesn't change. And in fact, he found it to be a little larger than it ended up being, I think he said, I think it was 500 kilometers per second per megaparsec. It's now known to be more like 70 kilometers per second per megaparsec. And what, so he came up with this relationship, v.

the recessional velocity of any galaxy is a Hubble constant multiplied by its distance which means that the further or the larger the distance becomes the faster the galaxy is receding from us and if

H.O. is about 70 kilometers per second per megaparcy and that means that for every megaparsec away you get the recessional velocity of any object increases by another 70 kilometers per second and so that means that if we do have a distance you know if we have our distances here and the again the closer these local ones you know relatively local in our local group

within 2 to 5 million light years or maybe even 10 million light years away from us they are gravitationally bound we are part of the same local cluster and so what's called the peculiar velocities created from the gravitational attraction of us to other galaxies in our local group is going to affect this grander trend of the universe but anything else the local group does increasingly strictly conform to the local the the the the the the Hubble Ible of the relation called now Hubble's law or the Hubble Lamatra law and we find that if we extend out to 20 million light years for instance take our little calculator here. So, million light years, you know,

roughly that's seven mega parsecs, we can say. Roughly, it's three light years per parsec. So we can say 70 kilometers per second.

In our little example here, 20 million light years is roughly 7 mega parsecs or million parsecs. So if we were to take that, the velocity of our 20 million light year the recessional velocity of our 20 million light year distant galaxy is going to be 70 kilometers per second the parsec 70 times 7 is going to be 490 a parsecs cancel out you're left with the velocity

490 and if you just add a zero to it so this velocity no Hubble this was the about the extent of how far Hubble got this range is a little more like I said so the speed of light is 300,000 kilometers per second at distances within tens of millions of light years you aren't at any appreciable fraction of the speed of light. But when you start getting out beyond 100 to 300 million light years away, the velocities, the recessional velocities that Hubble was finding on his graph here, start approaching a significant portion of the speed of light. And that's going to affect, well, that's where special relativity is going to come in.

Hubble here, you noticed on the graph too, because I was skeptical at first, like, That's a pretty loose trend line, but the closer we are, closer the galaxies are to us here, we notice that the more divergent their velocities start to become from that trend line, but they do start to go here to it other than maybe that little guy right there. But again, we said that this only goes out to 2 million parsecs, or about 6 million light years. So that's well within the local gravitational cluster that we exist within. And it's still amazing that, you know, despite these deviations, just from the peculiar velocities that are created in our local gravitational well,

the attractor created by the center of mass of all our local gravities here, we still notice a trend line and proof of Hubble's law the proof in the pudding here is like I said the trend line only gets more and more accurate this whole graph 6 million light year wide graph fits in that tiny red square right there in measurements of supernovae going out to 2.1 billion light years, 700 megaparsecs, conform almost exactly to that trend line. And what this trend line allowed us to do was beyond the distances, 100 million light years, that Cepheid variables and other distance measurement methods started losing their power. this trend line

allowed us to simply measure the red shift of an object and figure out and directly infer its distance. But the weird part here we notice is that if relativity is correct we shouldn't see the velocities increase more

than 300,000 kilometers per second. It's 10 to the fifth power. So these distances are still within that range here since 1996 there's been measurements made conforming to the same trend line roughly that doesn't seem to allow the in which the velocities don't deviate the way special relativity predicts that they should and so in other words it allows galaxies to be apparently receding faster than the speed of light,

faster than 300,000 kilometers per second. 3 times 10 to the 4th would be 30,000. This trend line continues on out until as far as we can see in the universe. Far beyond 2 billion light years. This galaxy today is about 34 billion. and light years away. But red shifts are incredibly important because of this relationship

that allows us to see the distances and velocities to objects simply by looking at their light. And this opened up the universe from being 300,000 light years across to a couple hundred million light years across and then through the later discoveries with cosmic microwave back.

background and even what this diagram infers here, distant, distant supernovae, we started to understand that the universe is on the order of billions and tens of billions of light years across, and then even beyond that, beyond what we're ever going to be able to see, the universe is probably at least 20 billion or 20 times wide as that, if not infinitely wide in the world. space. But this opened up this expansion. One of the elements here, we, one of the key aspects was that coordinates. We actually see Einstein. This opened up the door for the relativistic interpretation of the cosmos. It deviated from special relativity, but 10 years, around the same time Sliffer was making his original

observations of red shifts in 1914 there Einstein in 1915 had published his general theory of relativity that went beyond simple constant velocity frames and started to

include or did include in a unified framework that's considered the most elegant scientific paradigm he included acceleration and it was a equation, an equivalency of acceleration to gravity, a theory of gravity that superseded Newton's theory of gravity through multiple experiments and observations within

the following years, one of which was Hubble's discovery of the expanding universe that confirmed Einstein's general theory of relativity. And here's Einstein. observing visiting Hubble actually at Mount Wilson Observatory here and a funny little anecdote I want to add here was that on the tour apparently Einstein had brought along his wife Elsa Einstein and they were being told how the telescope was being used by Hubble and others to explore the structure of the entire universe to which Elsa apparently replied well my husband does that on the back of an old envelope talk about a cheerleader thought that was pretty endearing the maybe one thing I left out was that the expansion the recession velocities of these galaxies appeared the same regardless of what orientation where you looked over here over there didn't matter it wasn't a flow like a river so it was

what they call isotropic and homogenous. It was or homogeneous. But instead all observations indicated that, and still do indicate in a weird elegance to our universe, the structure of our universe, that all galaxies are expanding away from us and away from each other too. So as they expand away from us,

they're also, if each of my fingertips represents a galaxy, they're also expanding away from each other. And this was not only a confirmation of Einstein's general theory of relativity, but it allowed the simplifying features of homogeneity and isotropy, actually allowed this simplifying universe, allowed Einstein and others like DeSitter and Friedman and Lamatra to discover that the implications of his field equations of general relativity

are that the universe should either be collapsing through the gravitational pull of all the matter on itself or more likely expanding through the curvature of space created by the matter and energy that fills it and this is really counterintuitive

this is where intuition for me and I would assume most non-physicists really breaks down because you as far as we know

gravitationalism gravity is an attractive force if anything everything should collapse on itself but in Einstein's new paradigm in which space and time again are not separate concepts but are unified what he added to that was a paradigm of gravity a theory of gravity that said space and time would in fact be warped by matter it would be curved by matter an energy present in it. And therefore, it would affect light, how it travels through it, and therefore how we perceive events, given that we perceive events as the relaying of information by light. And it was through the paradigm of Einstein's general relativity and what that implied for

physics, that Hubble and all the other astronomers and cosmologists of the 20s and 1920s and 30s, found was the best most elegant way to interpret redshifts and everything that redshift implied about the expansion of the universe and galaxies not only from us but from each other and it's out of the concept of space time that the diagram space time diagram tonight the big it to is uh comes and is based on so 1905 he had a zanus mirabulous that miracle year he published starting the fields of quantum theory and general relativity um between the years of 1907 and 1915 Einstein had begun the searched for finding a framework in which relatively relativity can explain gravity as well because it wasn't able to explain gravity. It was only able to explain constant, again, constant velocities

and the weird effects of time dilation and how space time is distorted by extreme velocities approaching the speed of light. But that didn't account for acceleration as far as I understand. I know there's a lot of nuances that I am not aware of and I hope I don't screw anything up here, but what I understand is that when he was trying to discover, trying to formulate his general theory of relativity,

he was trying to, I think there's one anecdote here that said he said, he said I had the happiest thought of my life in 1907. I was sitting in a chair in the patent office in Byrne, when all of a sudden I, a thought occurred to me. If a person falls freely, he will not feel his own weight. I was startled. This simple thought made a deep impression on me.

It impelled me towards a theory of gravitation. So up until Einstein came on the scene, flat Euclidean space was dominating, a dominated science, and Newton had... established that the separation between any two points in three-dimensional space was invariant. It never changed, and that's the distance between them. The length of objects, if the number of atoms in them wasn't physically removed or altered anyway, that also didn't change.

That was invariant. And likewise, another invariance was the time between events. As time clicks forward, the rate at which time clicks. forward that would measure the events never changes it was always invariant and then Einstein changed that with special relativity that showed that not only space but also time they were both able to vary in dynamic and be dynamic and the values measured depending on your frame of reference and your speed could change but what didn't change what remained invariant was the new concept of the unification

between space and time in space time the separation between between two events being marked by its three-dimensional coordinate and its time four dimensions total so you have two events with two separate sets of four of those coordinates so you can measure the using the Pythagorean theorem the distance between those two events in physical 3d space but also add a fourth coordinate like we talked about with the earlier we would touch upon special relativity and have a distance in time between those coordinates and like we said before if you have if you have a a distance a distance

between two events between its different coordinates its three-dimensional coordinates it could plus and minus I'm a little unsure about but it's roughly the difference between its two coordinates in the XYZ three-dimensional frame so you know if this is starts I'll mess that one up if one coordinate is at zero both that's x Y and Z. If one coordinate is at zero, it's at the origin, so its x, y, and z coordinates are all zero. And another one is maybe one, two, three, and one, two. So its position is three, two, tenets stays zero on the z axis. That means its distance between them would be used using the Pythagorean theorem,

that's squared, I think you gotta square all these. But yeah, so if the origin represented by the first particle, that's particle one and that's particle two, is just for simplicity's sake, we say that particle one's at the origin, so all three of its three-dimensional coordinates, marking its location in three-dimensional space are zero, that makes the math really easy for us.

for us and we can just say it's 3 minus 0, 2 minus 0 and 0 so it has no displacement in the z axis so that's just going to be 3 squared is 9 plus 2 squared 4 and that's going to equal 13 and so the distance between them would be the square root of 13 so that's the maybe I should have D because S is going to represent the space time distance and that's the concept this is a fundamentally based on the Pythagorean theorem and then they add this is way way oversimplifying it but they add a a fourth coordinate now take

the time the difference in time between two events so we're in normal situation you're used to measuring the distance between two events without considering the time. You just say, or two objects really in space, usually you just either don't consider the time, or it seems like it's implied that the events or the objects are being measured at the exact same time. But now you measure the time between them, between two events. you say well how can you have a distance between how can you get units of distance with

you know time being in units of of time seconds whatever unit you choose well that's where you multiply it by the speed of light and this is fundamentally how they get and then like we said another page here speed of light

is a velocity but it's fundamentally multiplying a velocity by that time is going to translate it into units of distance. So if C is the distance light travels

per unit time 3 million meters per second and we multiply it by time you can worry about the conversions later but if we have time in seconds and you multiply it by seconds the seconds cancel out in your left with meters so you got to square that if we're

going to add that into the squared Pythagorean theorem there and then you're going to have a time dimension there which is C times the change in time and this space time this then becomes instead of a distance space becomes a distance in time. In this overly simplified example here, this represents a you know a line. These are just very linear coordinates, very simple straight line, but general relativity, the effects of it are that matter, anything with a mass, has not only a gravitational mass, but one of the genius insights of Einstein was that Newton's famous equation here,

the force between two objects is proportional to their masses, mass 1 and mass 2 divided, and is inversely proportional to the radius or the distance between them. And then another of Newton's laws was that the acceleration of any given mass multiplied by that mass is going to equal that same force. But Einstein realized that this mass has nothing to do with this mass in this equation.

So these both equal units of force, but it's kind of misleading to set them all equal to each other. It's not the exact same force because you can have an inertial mass which could be completely away from a gravitational

well, it could be floating far away from any other matter in space, so it's not experiencing any gravitational force on it, and yet it will show resistance to acceleration simply through proportional to how massive it is, and that tells you how much force you have to impart on it. And Einstein's insight in general relativity, one of them, one of his core insights, was that the mass and this law of motion, equation was distinct. That's why I put a subscript M sub i there from this mass.

So one of the Newton's main Einstein's main insight was that the gravitation of earth's, the force imparted to an object by Earth's gravitational field is actually not a force at all. Newton had thought, or at least use the fiction, the sort of, you know, kind of placeholder narrative about how particles interact gravitationally with each other as a concept of force. And he said that, so it was basically Newton who established that it was a force, some type of gravitational force. But Einstein was the one after years and helped from our own. other physicists like Minkowski, he derived the fact in general relativity that it's actually more of a geometric concept. Newton's theories assumed that motion takes place against the backdrop

of a rigid Euclidean universe, a reference frame that extends throughout all space and time. It is static, time and space, things happen through it and throughout it. But the axes of time and space do not change. And to Newton, gravity was mediated by a mysterious force, acting instantaneously across the distance, and whose actions are independent of the intervening space. In contrast, though, Einstein denied that there's any background Euclidean reference frame that extends throughout space. He only worked in small, very, very small local reference frames across which interactions happened. And essentially, the way I understand is that the universe in relativity involves the systematic stitching together of an infinite number of local reference frames into a more general picture of space.

time so not only did Einstein erase the Euclidean universe of absolutes but he also said that there is no gravitational force this force is completely fictitious the acceleration experienced by two objects two massive objects two objects two objects with the mass whether it's the earth and a, you know, golf ball or two golf balls or whatever, the ratio of masses between them are, is simply a curvature or a law of geometry. It's simply a geometric feature of spacetime itself. And that curvature, created by space time, creates the natural course or the natural trajectory, along which objects naturally fall called a geodesic and again this is in not in just space but this is in space time

so it's a four dimensional coordinate system and a ton of really strange things and very exciting things came about from this Newton what it turns out is that our entire solar system, our entire existence, happens on non-relativistic scales of time and space. We don't do anything at the speed of light, and even the planets, as fast as they move around, the Earth moving at 30 kilometers a second. even that is very small, very slow compared to the speed of light. And Newton, his laws that predict the orbits and perfectly characterize the positions and the periods of orbits and the distances from the sun of all the planets only work because they are only valid for very non-relativistic reference frames.

And that is what our, that's characteristic of our solar system. However, reference frames in which you are, in which objects are moving close to the speed of light, and reference frames in which objects are very, very massive. of, in other words, distorting, creating that a distortion in the space time around them, Newton's laws start to break down

and one of the quote, quote unquote, blemishes, the sole blemish, in fact, of Newtonian gravitation, as the book says here, was the inexplicably large rate of shift in the orientation of Mercury's orbit.

And that was the procession of Mercury's orbit. it's a very again all these measurements in physics and in it inexplicably large rate of shift isn't large in the sense that most people would think about it as think of it as but it was only 43 seconds a shift of 43 seconds per century so it's a very very small shift probably so all the planets have a the orbit themselves the shape is not a perfect circle, but it's usually typically in our solar system, it's so near a circle that we, even if you blew it up to scale, or even if you had it downsized, you would hardly be able to tell the deviation from a circle, but if we exaggerated it to have more of an egg-shaped ellipse, that ellipse itself appears to be turning around. So instead of just staying in a stable orientation with respect to, you know, distant stars, that ellipse, itself slowly rotates and that ellipse again defines the orbit of mercury and

Newton's equations of motions Kepler's planetary laws of planetary motion did not explain that they were unable to account for it to the point where they a lot of astronomers considered and there was a frantic I think two-decade long search for the planet called Vulcan which Star Trek planet Volcan is named after Thinking that a lot of astronomers had thought that a mysterious Unseen planet lie existed between Mercury and the Sun for curbing Mercury's orbit Because they had actually found that's actually how Neptune

Was found or was it Uranus and Neptune in Neptune's existence was only a discovered after a few decades of being observed was seen to actually have a perturbation in its orbit that couldn't really be explained except by the possible existence of a further planet that was gravitationally tugging at it and so it was wobbling its orbit and eventually um Neptune was famously by Laverier Laverier in the late 1800s found at the tip of a pen

I think is the phrase often used because it was found purely through gravitational equations working out very complex calculations and again I think if I'm remembering this right the very night

that Laverier finished his calculations, predicting the location of this mysterious planet that was pulling on Uranus. It was handed to an astronomer whose name I can't remember right now, who found it at exactly the predicted coordinates. So it was found purely through mathematical calculations, which is stunning when you think about it. So Mercury was not being perturbed by another planet, though. What happened with Mercury was that the geometry of space-time itself was warped just enough from the non-relativistic Newtonian equations that this procession was, as far as I understand, it was far enough in the sun's gravitational field, that its orbit was being,

perturbed by effects that were only decipherable through understanding general relativity and that the sun wasn't imparting of gravitational force but in fact it was just distorting space and time itself and mercury was way more susceptible to that being much closer to it so the equations of the equations of general relativity

define cosmology on the grandest scale and just to scare you guys off. Well, it scared me. Look at that term right to it. This chapter, at the end of this book on astrophysics cosmology, it's about 80 or 90 pages long. It's the longest chapter in the most math dense. Let's see here.

I wrote here just probably doesn't do it justice, just flipping through it. But they say, the behavior of the universe, the scale factor we're going to get into really shortly here the scale at which the universe expanded can be investigated here after a brief flurry of mathematics to produce the needed tools and then about a hundred pages later of math equations and a hundred and eighty one equations later we finally arrive at our goal they say the goal is the proper distance

This is the distance that allows us to discover and realize, or at least according to the most up-to-date models, that the universe is 93 billion light years across, our observable universe and much, much larger. The distance between galaxies at the edge at opposite sides of our observable or field of view in the universe is 93 billion light years across. cross but it took 181 equations on top of the background of basic physics calculus differential equations and then the general relativistic differential geometry tensor calculus basic classical physics that this sits on top of so I am very unqualified to even begin to try to get into the mathematics

because I just don't understand it most of it I would like to if I ever have time which I don't see that happening but so a lot of weird effects come out of general relativity one is that light bends its path around massive objects

and that's due to the curvature creating a condition in space that causes the shortest path for light in that region of space to actually appear curved to outside observers and apparently to local observers if you were to zoom in on any specific local section of this that local region of space time would be flat and have a straight line of light. But on the grander scheme of things, light gets bent. And here is an example where if you zoomed way out

and you see exaggerate the arcing of the bent light beam, you could create theoretically draw a straight line that would, in principle, from an outdated, point of view of physics should be a shorter path for the light to take. But this picture here doesn't take into account the third dimensional. In our example here, the 2D represents 3D space and then the third dimension of the bowling ball sinking into the rubber sheet represents the fourth dimension of space time. That straight line, if we oriented it to a little bit of the rubber sheet, it represents the fourth dimension of space time,

oriented it to an oblique angle so that you could see in curved space what appeared from directly overhead to be straight axi would mean that from a to c the light would have to travel down further into the sun's gravitational well and time ticks more slowly according to general relativity the deeper in that object's gravitational well that you are. That's why an interstellar, that surface of that planet was, for every hour on that surface of that planet, about seven years would go by for the spaceship that he came from that was orbiting much further away from the black hole,

and therefore much higher in the gravitational well, and therefore in a position that, whose time is ticking much more fast. So it appears when you're down there, like time is ticking normally. But if you do emerge back into a gravitational potential or a curvature in space, that is much less you will have experienced much more time

than any observer who had stayed at this radius away from the center of the gravitational. object, the sun, for instance, here. And what that means is that not only does the path through the curved space physically have longer spatial distance between points A and B, the actual path is longer, but the time while you had been in, if you were in that light beam,

would have run slower, and therefore more time would have elapsed. And so nature wants to take the most efficient route light travels straight because that's the most efficient the quickest shortest distance quickest route between two points but in curved space that shortest route between two points begins to appear bent his theory of relativity was confirmed in 1919 and this is an incredible feat not only did it perfectly predict the exact, not only did the math of general relativity, perfectly predict the aberration of Mercury's perihelian orbit,

but it also predicted the aberration or the distortion, the bending of light from stars around our sun. When there was an eclipse, you could see that there were, the apparent positions of stars appeared out here. But then later on in the year, those stars being far enough away where their positions, their distance on the sky, the angle between them to not ever change over any appreciable amount of time or any human scale time, what was observed was that. During an eclipse, looking at stars just to the, and this is exaggerated, looking at stars just to the right and just to the left of the sun, close enough that their light would have to come very close to the sun, and they wouldn't be normally observable to us from Earth,

because we'd have to look right into the sun. The sun's light is blocking it out. During an eclipse, we could see those stars measure the angle between them, And it was observed that six months later, when we look back at the positions of those stars in the sky without the sun in between us and them, their positions were much closer together. Again, this is way distorted. The positions in reality were you couldn't tell without instruments, but the fact that they were distorted by exactly the amount predicted by general relativity was another confirmation in the early 1900s of the,

accuracy, that gravity is not a force between objects, but instead is a geometric phenomenon in a four-dimensional space time. It goes on to say that the mass in the F-equals M.A., if any of you guys have taken basic physics, is a different mass from the mass in Newton's universal law of universal gravitational gravitation. Makes a distinction between

inertial and gravitational mass. And that's allowed Einstein to set up a situation

sort of analogous to Galileo's initial relative concept of relativity of velocity

that special relativity used in which there would be a relativity of acceleration

you wouldn't be able to, if you had this one representing the inertial mass, this reference frame, and then this one over here, representing the gravitational mass, or the forces. Einstein made the analogy that you wouldn't know if you were in an accelerating spaceship, one that was increasing its velocity every, or let's say gravity, which is 10 meters per second square. So at every second increases its velocity at 10 meters per second squared, you wouldn't know if you dropped an apple, if you were standing on Earth under its, now what we know as a curvature caused by Earth, You wouldn't be able to distinguish that from the acceleration of the actual ship you were in, imparting its force on the apple that's being dropped. Some other effects here is that from the reference frame of someone standing on Earth,

a photon traveling straight up, exerting energy moving out of the curved well of space time. a photon will actually undergo gravitational redshift. So not only are there red shifts that are caused by the velocity approaching the speed of light of an object moving away from you, but also gravitational redshift of an object moving directly away or any component of the motion away from a, directly away from a gravitational center of a massive object will cause it to undergo gravitational redshift, meaning its energy, it loses energy, which lengthens its wavelengths.

Typically, the energy of a photon or its wavelength is equivalent to its energy, the more energy it gets, the bluer, the shorter the wavelengths become, higher frequency, or if it's losing energy moving out of a gravitational well, it will increase its wavelength or decrease its frequency. And in the same way, this is exactly proportional. It's a consequence of time running slower

at a rate near a massive object. this gravitational redshift is a consequence of time moving more slowly for that photon as its position deeper in the gravitational well

and so a photon moving out of the more curved space into less curved space means that it's clock while it's in curved space it experiences more time So all these, those are just all effects, and there is a lot more math behind it. But the, here, the united concepts of space and time, expressed in space time,

specifying each event, allowed Einstein to deduce what this book, a lot of people call his crowning achievement. The field equations, calculating the geometry of space. space time produced by a given distribution of mass and energy. So you have mass and energy on one side, and you have the distortions, the curvatures in space and time that those create on the other side of the equation. And in all these equations, it's always, E equals MC squared,

and this one here, they always look so succinct and therefore simple to adults like myself. but what you've got to realize is that in most of these variables other than, you know, pi and C, I guess G is a constant too, but the T and G here, and it says, T represents the stress energy tensor, which evaluates the effect of a given distribution of mass and energy. Remember special relativity told us that mass and energy are the same thing. So a massive object at rest has an equivalent amount of energy, kinetic energy implied in it. That stress energy tensor evaluates the effect of a given distribution of mass and energy

on the curvature of space time, as described mathematically by the Einstein tensor, G, for gravity on the left. So mass and energy, whether it's radiation of light, or any other forms of energy, whether it's motions of objects, or just static stationary mass at rest, given, you know, with respect to any reference frame, that all contributes to the curvature of space and therefore what we call gravity.

And those, these equations here are all hidden in these variables. So when you see these really tight-looking, neat equations, A lot of times these variables represent lines, equations with like multiple variables inside them. Schwartz shield metric, short shield metric. So this whole equation is, you know, essentially inside this side of the equation. I think it's that side. Not 100%. But in the simplest scenario where spacetime is flat, there's no gravity, no curvature,

light is traveling in straight lines and we have these neat, you know, nice and tidy little equations where everything is linear and straight. But they get very, very complicated when you start having curved space time. And these are going to represent how distances between time and space change in proportion to, to a large mass. So the last little part of this book here is the world lines. And now this is someone if we're eliminating the third dimension of space because we can only have three dimensions projected onto a two dimensional page in images. So we eliminate the third dimension of space and just pretend everything is a flat plane. So you can only move, you know, this way along the x-axis.

or the y axis, time moves up. This is someone, a man at rest, and he's just staying perfectly at rest, he's not moving this way or that way, and as time goes, his position stays the same. Let's do that, doing that. This is a woman running in the Y direction, so not moving this direction, she's just moving this direction,

but she's doing it over a period of time, So as time increases in our T-axis, she's not stationary, but she's actually... And then here is a satellite orbiting Earth. Again, in two dimensions. As it orbits Earth, it spirals up towards the positive T-axis. And this, in the very simplest projection here, allows... It sets the stage for a concept of a light cone. So here it says that if we think of photons.

If you flash a light coming out of our beam here, you think of photons, of a photons are gonna radiate. And again, if we simplify this into just two dimensions, so if we have, you know, let's do, Let's turn this into just a light bulb from overhead. We're looking at this from overhead. And it flashes out in every direction. Then after a certain time, it's the distance light will have traveled.

That distance D is going to equal, you know, if we say a one second, if this sphere, concentric sphere, represents one second, that distance is going to be the speed of light times one second. That's going to be three times 10 to the 8 meters per second times one second. So this distance is going to be 300 million or 300 million meters. As we go out, this represents

two seconds, three seconds. Sorry, my circle game is off because I'm falling off the book here. Yeah, the distance to that. So D1 is 300 million meters.

D2 is going to be twice that if it's you know, two seconds. three seconds. That's going to be 600 million meters. And then it will be at D3. D3 is going to be 1.2 billion.

If we now add the time dimension that would go up like this, instead of looking directly light bulb there, we're going to put our light bulb flat. well we're gonna project it onto a third dimension here like this so I'm just gonna redraw this one right here but if we have a light bulb the origin axis is go to the y axis X this way I guess and then T right there what you have is that as T goes up we say this is one second two seconds and So the circle is going to...

Light is going to expand out in two dimensions, just like here. So we see the cone shape starts emerging. I guess it's probably smartest for me to just draw the outline of the cone and just go to that. Wouldn't it? Isn't it? Isn't it? In it.

So at each successive second in time, this is one second, two, second. two seconds three seconds four seconds you see that the cone is going further in space and this is the furthest possible point each of these at any given time this line this horizon here this edge of this light cone is the furthest possible it represents the what's called the horizon of our possible

all are possible if we're going into the future this is a future causality and then going into the past is the past causality future causality is you know light is the fastest

we don't move at the speed of light we can shine a light and that light beam will expand at the speed of light, such as in this instance right here. That light beam coming from our position in space and time, time and space and time, is the fastest possible way we could ever interact in a causal way, allow, cause something to happen at a distance away from us in the universe.

And therefore, this line represents the horizon, of future causality. At one second, we established here, 300 million meters is the furthest possible distance we can have any effect on anything. And then four seconds,

going up to maybe 2.4 billion meters away is, two seconds away is the moon, actually. So we can't ever have an effect. fact on anything faster than this horizon here and that actually represents what in the graph here they call elsewhere so anything outside that so if we you know projected a um an imaginary horizon with a i guess we'd say less steep on the time axis but um projecting outward if we follow this that means that in one second it will have gone further out here all the way to almost the three second mark you know

whatever three times the speed of light that's impossible we can't ever have any effect on anything faster than the speed of light so in other words what that means is that philosophically here and physically we are limited to within our future light cone and this exact thing extends out into infinity here but it does with each passing moment itself project a new light cone outside of here this is the elsewhere we don't have any effect on anything out here and it says down here it may come as a surprise to realize that vast regions out here these regions out here outside our past light cone. Vast regions of space time are hidden from us.

They're just completely inaccessible to us. And we can never have an effect on them. And likewise, if we project this into the past and we project our T-axis downwards, meaning into the past as well, one second into the past all the way down, into the beginning of the universe. Nothing outside the light cone here

can ever have an effect on us. Or in this example, this little flashlight being lit, doesn't matter if you have a wire, you know, attached to us, a battery that is theoretically a billion light years away. The signal, the current running through the, that wire would not be able to travel faster than the speed of light. I think it would be less than that too. So it would not be able to have any causal impact beyond, you know, at most, a billion

or at earliest a billion light years into the past. So if it's a billion light years away, let's change scales radically here. We'll pretend that down here is, you know, negative one billion, giga for a billion, like gigabyte. years away, that means out here will be one billion light years away. So nothing beyond that, 0.1 billion light years away at this point in time, we'll have ever been able to have a causal impact on our light going off and, you know, if we expand that onto the human race. So it's really interesting to consider these light cones that define, that define, and our causal contact and our causal relation

to the rest of the entire universe. And we'll never be able to escape, as far as we know. You can't travel faster than speed of light and, you know, maybe in 100 or 1,000 or a million years, AI will have helped us be able to juggle complex ideas and put them together so rapidly in such a complex way that we'll unlock insights into further, you know, features of space time and the laws and particle physics and maybe fuse gravity with the other three fundamental forces,

but as of now, we do not, we do not understand any possible way that would allow us to get into the elsewhere right there. Up until now we've had a description of space time and world lines that is going to help us understand our graph, but one important aspect of these graphs here is that you'll notice, although the vertical axis is time, zero to infinity essentially from the birth of our universe up through our present point and beyond, and the horizontal axis is space. to find out that's more of a dynamic concept as the space time we've been talking about kind of implies. But the other side of the horizontal axis here or vertical axis is the scale factor.

And this is something that we haven't really discussed yet. This is another way of talking about the expansion of the universe. It's incredibly hard to overstate just how paradigm shifting, how revolutionary, really, the breakthroughs in physics and therefore cosmology of the early 20th century really was. From the Plank and his predecessors leading up to the 1905 revolutionary set of papers, Einstein published, fusing space in time, washing away or at least

greatly muddying the waters about the nature of light, whether it's a wave or a particle. Given that at certain scales, it has wave-like properties, and then Einstein's, one of his papers, or multiple papers, said that the photon or particle-like description of light

as a propagating massless unit of, energy with its own momentum is a, was actually a better descriptor of a lot of previously undecipherable phenomena in the universe. Specifically the photoelectric effect. These papers revolutionized the way we understood how matter and energy interact through space and time. Einstein fused not only space and time, but he also fused and equated matter and energy. So now, what did this imply for the universe? It's incredible to think about all of that was fundamentally because of Einstein.

In 1905, he established and laid the groundwork for what would lead to among many things our understanding our breakthrough understanding of just how much not only how much larger the universe is but how much more dynamic and really how foreign the characteristics

of its space and time and its matter and energy and the way those two or those four concepts depending on how you slice it interact and distort and manipulate each other and what those consequences for the universe were. And so while Hubble discovered the expansion of the universe in his famous law, the velocity, the recessional velocity of any galaxy is proportional to its distance by a constant H.

It turns out, and this is the amazing part of Einstein's genius and the breakings, through of just the the potency the power of his general relativistic equations this constant was predicted and explained 15 years before Hubble ever made this relation in his famous equation now known as Hubble's law while Hubble was while he was able to make the observation and form this empirical equation that fit the data, the red shifts, and how if we know that all red shifts can be interpreted as the difference between the observed or wavelength. So lambda is the wavelength of light, and as they're redshifted, it's the ratio, I guess, of the difference between the two wavelengths to the original, the initial wavelengths at the source of emission. So all red shifts correspond to, that is the definition of a redshift, that number Z comes out of this relation right here. So as Hubble was able to use Doppler's relativistic law, sorry, non-relativistic law,

his measurements didn't indicate that the velocity was anywhere near the speed of light, although it was much higher than the stars' velocities we were measuring in our local galaxy. what we now call our galaxy they were still much less than the speed of light so there weren't any relativistic effects that were expected to be going on there there wasn't any time dilation any significant

space contraction and so he was able to use Doppler's law which was that the red shift should be exactly proportional to the velocity the velocity of the emitting object to the speed of light. So that's what the red shift equals in terms of Doppler. Doppler's equation.

That's the Doppler shift right there. While Hubble was able to derive using his earlier famous discoveries of the distances to galaxies outside or outside, her own galaxy breaking the debated paradigm up until the early 20s, whether or not there were any other fields and groups of stars, more distant than just a few thousand, 100,000 light years. He was able to use the fact that he could determine distances to up to a few, you know, tens of millions of light years away

using supernovae and sepheid variables in particular. He was able to determine the distance. to these galaxies and when he was able to measure the red shifts of the light coming from those same galaxies he applied this equation here measuring its redshift from this relation between the observed light and the known wavelength that should be at the source of emission millions of light years away to the velocity C the constant speed of light in a vacuum he was simply able to

rearrange that and get the galaxy's recessional velocity and say that it's equal to the redshift times the speed of light and knowing that he was able to also if he knew that and the distance he was able to say that some constant of proportionality

here would be able to equate essentially the recessional velocity divided by the distance there And that was Hubble's constant. While he was able to come up with Hubble's constant, and although we got the wrong values, it was still a breakthrough.

Just the discovery that the universe was expanding, and it was expanding in a very easy, a very simple, linear manner. While that was a breakthrough, what was even more of a breakthrough? It was that Einstein's field equations were able to not only predict this relation right here, but they were able to explain it.

Once Einstein in 1915 came up with his general relativity, what this did was G sub mu and new, the Greek letters, is equal to 8 pi times Newton's gravitational constant. This is a variable here. We're going to see exactly how wild this variable actually is, how deceptively simple. These two terms right here really are. We touched upon it before, but it's worth noting that, again, related to the left side here in this variable here describes the curvature of space time. So this is the geometry.

This is the part of the equation that can be described by the metric we previously talked about. This metric, this curvature, relates to the distance between a light component, the time component that's bounded by the speed of light to the spatial component, which would be Z of the distance between an event. So space-time events, this equation here is affected by the curvatures described in this geometry, which is informed itself by what's called the stress energy tensor over here. This is the stress energy tensor and is itself defined by, by this tells you what the matter and energy and radiation in the universe, how it affects the geometry or the curvature of the universe that it's in.

And in turn, this curvature is going to affect how this matter and energy travels through space. And so these two are the two key elements to understand how the universe works on the largest, grandest scales. And now this equation, importantly, only works at scales over 100 megaparsecs or about 300 billion light years. While Einstein's, the other equations we looked at, the Schwarzschild metric, defines local curvatures from stars and planets

and even galaxies on space, distances well beyond even our local group not two you know one or two million or even ten million but a hundred million and more light years across 100 to 300 million light years is where the universe becomes homogeneous and isotropic enough meaning that it is its density is roughly uniform

and it has no preferred direction so it resembles a a very, just a very homogenous, a very uniform distribution of matter and energy on scales that large. And that is able to define that is able to start

this equation, Einstein's field equation, is able to start perfectly describing the expansion of space. And it's out of this equation that he came up with in a very formal, a very general form in 1950,

that in 1922, about seven years later, this Russian meteorologist turned cosmologist named Alexander Friedman, he was able to teach himself general relativity, interestingly enough, and he actually was the first to derive a specific solution that could actually be directly applied to the universe, an observational way. And he predicted that this relation here

arranged for a universe in which space was in a more simple configuration, homogeneous and isotropic, fairly uniform, especially on the vastest scales. He was able to determine based purely

off the mathematics that The universe should be expanding. And although we think about gravity, someone like myself at least, thinks about gravity on the largest scales as causing the universe to collapse more than anything. It doesn't make sense. It's not intuitive that there is a repulsive force

that acts counter to gravity. What I learned was that this equation here, the reason I called it deceptively seen, simple I'm going to show you guys up here is because it expands into this. This equation right here, someone at profoundphysics.com, this guy, Bill Hervonen, he broke it down because I had to really understand what, because this book here says just how complex it is. she mentions the deceptively simple

field equation is actually a 4x4 tensor that describes the curvature of space time at every location T, X, Y, and Z it's a symmetric tensor so 4 by 4 should give you 16 equations but it's symmetric so 4 are superfluous

so it actually breaks into 10 it expands into 10 separate second order non-linear differential equations, which is daunting in and of itself, let alone having ten of them. And the right side is also a four-by-four symmetric tensor. So this, what this expands into, and just to give you guys an idea of how complicated this

really is. This is what the fully expanded metric becomes right here. And it doesn't stop there. He says even this doesn't really describe how complicated these actually are. Using the summation, the so-called Einstein summation convention. And if you want to

explicitly write out these summations, this is what it would look like. He shows you right here. so instead of instead of this right here it would look more like this which helps us understand that this and so these are 10 equations here equal to you know the right side in which remember this side would also be equally expanded into 10 equations so there would be 20 total combinations of equations in one single or 20 expressions really per equation to really fully expand out all the terms that would allow you to explain the

curvature of all the elements the three spatial elements and the one-time element combined total as a four-dimensional space-time manifold every at every point in space at every where and every when in space time. you would have to take these 20 terms expand them into this this term right here is expanded into this this equals all of these summations you can see the mu and new subscripts there so it's like it looks like so each of the ten terms on the left side of the equation would expand into roughly 60 equations which maybe might be able to reduce into ten equations but nonetheless

it's still one simple equation expands into close to a hundred expressions but that gives you an idea that notation of just how many variables and interactions between and relations between variables is actually going on in Einstein's description so this deceptively simple equation was used by Friedman to derive what's called one of the most famous equations in cosmology

the Friedman equation they can be written written in many different forms but one of the standard forms of the Friedman equation is this right here it's actually a bubble parameter this epsilon here

is the mass the energy density of the universe. This k is the curvature. You can see the speed of light pops up many times in this equation. And the k and the r

both are elements of the curvature of the universe k, representing whether it's positive, like a sphere, or negative, like that saddle. You always see. R is how intense, how tight the curvature is.

And this H is in fact the expansion, the exact same expansion predicted or derived empirically as Hubble's expansion here, Hubble's constant. And what was found out was that if you take the derivation of a distance function, if you have some function that's, you know, say it could be anything, it could be a quadratic function, x squared plus y x plus c and that describes the position of a particle at any moment if you derive that using calculus take the derivative of it you get the velocity at any moment in time of that same particle that's described by that equation and so this humble constant here from hubbell's velocity and distance relation is actually another way of writing that is h o

Instead of the velocity over distance, it could be the derivative of the velocity. It's just one notation way of writing a derivative over its distance. And in an expanding universe, the distance to objects is a representative of how large the universe is expanding. or in other words the scale of the universe and what we knew by observing galactic expansion all galaxies expanding away from us and in a way that also tells us that they're expanding away from each other so we're equally expanding away from other galaxies as they are from us outside our gravitationally bounded local group and local uh super cluster at least. So on scales of, you know, larger than 300, 1 to 300 billion light years, all space is expanding and the galaxies are accordingly expanding away from each other. This relation here also means that at a certain point in the past, the scale of the universe was smaller, the amount of space between these large distances, these large volumes of space,

was also much less. And this is to say that the distance, the derivative of distance, meaning velocity, and the distance, can also be written in terms of a scale of the universe. So if a galaxy is 100 million light years away and receding due to the expansion of the universe, well, that also means that that distance is exactly proportional to the overall scale of the universe, how the expansion of that spherical volume of space and cosmologists use the variable the letter A

to represent scale what's called the scale factor and so this relation right here the Hubble constant equals the rate of change of the expansion of the universe of the scale factor to a new device by the scale factor at any given time and these are functions of time this is known as the Hubble parameter and it's from this Hubble parameter that we get this other vertical label on our on our space-time diagram of the entire universe the scale factor as you go back in time

is the scale of the universe relative to the origin, the birth of the universe. It's scaled up, and usually they simply define it kind of arbitrarily as one, in other words, 100%.

Today's scale of the universe is 100%, and any time in the past is just a fraction or a percent of the current size of the universe. And what that does is help us this Friedman equation here it was from this equation

and with this Hubble constant or this Hubble parameter here being the scale factor of the universe Friedman was able to predict that space was in fact expanding now he did that in 1922

but he did it purely off the mathematics of Einstein's field equations and he modeled a universe in his head that was especially simple and a guy named William De Sitter actually a couple years before Friedman had modeled a universe that was incredibly simple with no matter at all and it I think he found that it was expanding as well but Friedman was the first to come up with this equation that incorporates more realistic more testable parameters

such as the radiation, the density in the space of the distribution of matter in energy and matter is energy, remember in the where to write it,

in Einstein's E equals MC squared this also, keep in mind is deceptively simple, it's elegant because it displays a lot of information in a very compact form, but it too can be expanded very, very much. But this relates the fact that all matter contains energy and can be conveyed, can be described solely in terms of its energy.

So this term here can be essentially thought of, instead of just the matter and energy in the universe, you can just think of the energy, distribution of energy in the entire universe, whether it's matter, radiation, whatever, cosmic rays. And Friedman is still to this day, his equation is so important in cosmology because it is the first equation to use Einstein's to actually come up with a very specific solution to Einstein's more general form here. that allows us to put real values in that we can measure for the scale factor of the universe, the mass or energy distribution or density of the universe. And these equations are all generally a function of time

because matter, the universe is now thought to have come from a big bang, which was mostly all radiation before matter even formed. it was all radiation, and even after protons and neutrons were synthesized out of the primordial elements, they were a very trivial effect on the overall dynamics of the universe. For the first few, I think 100,000 years, it was radiation, the momentum of massless particles, photons, that really shaped the expansion and evolution of the universe. and then once the universe cooled down matter came to dominate and they think about five billion years ago the universe went into and transitioned into a third phase the first being radiation dominated

second being matter dominated third now is dark energy dominated after the expansion became reached a critical tipping point they thought that enough space had been created between distant regions of the universe, that a vacuum energy, now known as dark energy, is still a very, very mysterious, very poorly understood phenomenon, dominated the universe, causing the already expanding universe from determined by these equations to accelerate. But in 1922, just seven years,

after Einstein's equations of relativity, his general, very formal equations came out. And remember, this was already, they were already having an impact. In 1919, they were used to accurately predict how the sun's, the sun had distorted starlight, like I said. our sun and this is us looking at stars and there is an eclipse so the moon

perfectly blocked the sunlight and the starlight as it threw the gravitational well of the sun was bent by the sun towards us so the light otherwise

would have just carried on that in that trajectory but the sun's gravitational the mass of the sun distorted space time itself causing the light to curve towards us looking at starlight coming like this

to us it would look like it's on this spot in the sky but in reality after the sun passes and after the sun goes over here and have a direct line of sight without the sun being in the way we see that the stars are actually these are their full positions the observed positions

right there. So Einstein's field equations formalizing general relativity and gravity perfectly told us how much this difference should be. It also told us the

orbit of mercury how it processes around the sun perfectly accounted and predicted the deviation from Newton's equations. And so you got to remember this is I think the coolest part about this is that after Einstein came out with this

he didn't have any any data initially and he was really eager to apply it to all physical processes and it's fascinating to know that it's really cool that up until today even in 2023 Einstein's predictions have never been disproven general relativity holds true for every known experiment for which it's ever been tested. So Friedman predicts the expansion of space. It's funny because even Einstein, I think, can find it here. Yeah, it says that one of the effects without going into the details of his field equations was that since the ten differential equations are second order,

this means that space time can have non-zero curvature, meaning it can actually be something other than flat, or Euclidean. Even in spacetime neighborhoods where the stress energy tensor, in other words, the matter, the distribution of matter and energy, is zero. So what that means is that even in the absence of matter and energy, which typically Einstein thought of as being the sole cause of any curvature or distortion, in space time, distortion, variation from flat, even in the absence of matter and energy,

Einstein's own equations predicted that the universe would be curved and have some sort of possible curvature to it. And the book even goes on to say that in one way or another, that is the goal of all of cosmology. it's maybe the other one is to discover these features of the universe much of modern cosmology is devoted in one way or another defining the values of the scale the curvature constant and R is the again how how extreme the curvature is now all this it boils down again to red shifts and relativity everything

we know about our horizons and about how light matter and space time dynamically evolve in the universe is due to these equations Hubble derived his his constant in this nice simple little relation velocity is equal to the constant times however far away it is things further away are receding increasingly faster. And from that we can figure out that same relation can give us a relation between the universe's scale factor and its derivative.

But even Hubble himself famously said to DeSitter who was working before Freeman came up with these equations. He was working with Einstein. And Hubble said, in a letter to William DeSitter, Hubble wrote Mr. Hummison and I, Hubble's partner, Hummison,

are both deeply sensible of your gracious appreciation of the papers on velocity and distances of nebula, which he came up with his Hubble's law. We use the term apparent, though, apparent velocities, to emphasize the empirical features of the correlation. The interpretation we feel should be left to you and the very few others who are competent to discuss the matter with a third. authority, meaning Einstein and DeSitter and the very few others who actually understood general relativity in all its glory.

And that's the amazing part about all this, is that the Hubble made the observation to find that there is a constant between the expansion of the universe, how far away things are, and how fast they're receding, but it was Einstein's equations 15 years before. for Hubble even discovered his observational relation that implied and had predicted and explained just that prediction Friedman he made predictions about his expansion here but he didn't have any observation and as tragically I think he died in the war he died September 25 so only three years after he came out with his equations from typhoid fever the bacteria on his way back from his honeymoon in cremia we ate and see his theory his prediction validated but it was

ended up being true he came up with a value for the Hubble constant that was not accurate but the idea was still sound in 1927 only five years after that and about 12 years after Einstein came out with general relativity we had the Matra who independently derived this relation between energy density and the curvature of the universe that predicts an expanding space and he also published the relation of actual observed redshifts and he correctly interpreted them as being due to this cosmological expansion so he not only derived a more accurate prediction than Friedman but he also beat Hubble to

the punch which is why today it's actually called the Hubble the Matra law so what does all this tell us about the actual universe and how light travels through it Well, just like slope of a line is the rate of change of, you know, the rise over run. It's the rate of change on one axis and relative as a function of the rate of change of values on the other axis. We say that the world lines, S squared, coming from Pythagorean theorem, where a squared plus b squared equals c squared you can break this the path of light up in its simplest form into the horizontal over the vertical change well that is s squared equals a square plus b squared

in our world line spacetime diagram here world lines of light or past light cone future light cone the trajectory of light through space is exactly where the speed of light multiplying by the time between any two space time coordinates the difference between that and the actual coordinates themselves and here we only have one dimension of space so I'll eliminate the other two this fundamental relation here where the The path instead of being a path in a path of a line on a two-dimensional graph being one dimension squared plus the other dimension squared, the lengths give you the summation, the vector sum of that path. So vector sum are the metric that allows us to describe and measure events. in space and time, in space time, four-dimensional space time,

is called the Robertson Walker metric. In the 1930s, Robertson Walker, they devised a way of incorporating the radius, the curvature, and the scale factor of the universe into an equation out of which most of the data that we're going to be showing on our graph tonight comes. So this Robertson Walker metric is actually right here. So this spacetime interval, it's not distance, but it's a distance in space time.

So this, the S just represents the distance. The D is the derivative to show that the intervals are only valid over small regions of space time because the scale factor in the rest of the equation is always constantly changing of space time. This can be written the other way around. It's the difference between them. That's why this is negative, and the space component is positive, so it's space subtracting the time component. They incorporated the scale factor.

Everything is squared because it's following the basic Pythagorean theorem. with a instead of a linear relationship where you have x, Y, Z coordinates in a typical very rectilinear grid they have more spherical coordinates because the universe, the way they model it is as an expanding homogeneous isotropic sphere

so they use spherical coordinates where it's the angle across, any angle across instead of an X axis, you have theta and then you have you have so the angle cross and then the angle up is another angle fee and then the radius out to that whatever point on the edge of the sphere you're trying to get to and those are the

three coordinates of spherical um of a spherical coordinate system so you have the radius out plus this which represents the the angles that give you the direction along that sphere and these are defined by the parameters in the universe this omega here this s sub capa this term here is defined by the radius the positive or negative or null curvature of the universe and if it is curved how much it's curved again and this Robertson Walker metric here combination of this freedman's equations give a give us not only the relation between the scale of the universe and the redshift but from that Hubble parameter there equal to the scale this

relation of the scale of the universe we can find the emission the distance to the emitted emitting galaxy, whether it doesn't matter how far away it is, billions of years away, we find the time and that distance changes over time.

We find the time since the Big Bang from which that light was emitted and we know the time since the Big Bang that we're now currently in our region of space time observing that light

and we know the distance not only between us and that galaxy when the light was emitted but we know the distance between us and that galaxy theoretically at least now we know if that galaxy carry along with the expansion how far away it should be now

and that's all in these these equations purely mathematically derived from Einstein's field equations and only confirmed empirically and is continued to continue to be confirmed to further and further distances in space and in the past. So as a brief run-through of how to kit, our graphs depend on red shifts

and interpreting those red shifts to give us information about the scale of the universe. Einstein or Hubble, he used Doppler's non-religious. Doppler shift, which was just a very simple relation saying that whatever percent of the speed of light the velocity is is going to give you the red shift. Well, there's also a Doppler, a relativistic Doppler shift that goes to infinity as as the speed of an object approaches the speed of light. That is conveyed by this equation here where it's the proportion of the velocity, object's velocity, of the speed of light, it's one plus that divided by one minus that ratio. Square root of those, that ratio, minus one.

And as velocity approaches the speed of light, and this approaches unity, this value is going to get closer to two. It's going to approach two, and this value is going to approach zero. And whenever you have a denominator getting increasingly smaller, that makes the entire value increasingly, large, infinitely large. And so this is the relativistic

Doppler shift because it doesn't allow for any velocities beyond the speed of light. And it says that the red shift will tend to infinity as the velocity tends

to the speed of light. But this equation here, in the expansion of the universe allows for and this is beyond this is where these the complexity of Einstein's field equations i felt the need to mention them because this is beyond my actual intuition to grasp any of this and beyond the mathematical mathematical capabilities anyway this is where space allows for matter to move beyond the speed of light because it's no longer moving through space with a peculiar velocity peculiar coming from uh in one of these footnotes here the root word of

meaning private property here the latin word for peculiar even private property which implies which is uh because the motion of that particle belongs to that particle alone and not to the global expansion of the universe so it's peculiar because it's private it's motion discrete from the general motion of the expansion of the universe, which is, by the way, called the Hubble Flow, as objects move outward with the expansion of space, it's called the Hubble Flow. Objects are far enough away about this far

to have its recessional velocity relative to us be almost entirely, or essentially entirely, due to the expansion of the universe, It's the point at which, it's the distance at which we no longer significantly detect any peculiar velocity of any, of any galaxy moving with respect to its more local counterparts. And in fact, it's just the whole cluster, just moving away with the expansion of the universe. So this relativistic Doppler shift wasn't even applied by Hubble because he was made. measuring distance is so nearby and it wasn't until we got better instruments and telescopes to measure

that we could measure further away and detect that the red shifts were shifting far beyond speed of light in fact it wasn't until later in the 20th century and we were able to do that which is about at a speed it's about a distance of 14 billion light years away. That's the distance at which this equation predicts the velocity, the recession velocity of a galaxy, to be greater, equal to or greater than the speed of light. So we have a new type of redshift that can go beyond, not just tend rapidly to infinity as,

the velocity approaches light. But if this velocity is allowed to go past the speed of light relative from our vantage point, it's apparent velocity. Again, that's why Hubble, even back then before he was measuring speeds close to the speed of light,

he knew that eventually technology might reach a point at which we could measure far enough away and the red shifts might start to indicate that the recession velocities were approaching or even exceeding the feet of light. And so he

already was ahead of the game saying, I'm just going to call these apparent velocities. And it fairly went to his grave. Never truly believing that the velocities were true velocities, he possibly just thought they were some

artifact that we of some characteristic of the universe that we weren't certain about that was causing the light to redshift. And he wasn't really certain that galaxies were all expanding away from each other. But the cosmologists that Hubble left this interpretation up to

themselves, Friedman, Lamatra, Robertson Walker, they determined that it was in fact the expansion of space itself. This metric here, the scale factor of the entire universe itself, was expanding. And that expansion would cause, the light traveling through it to itself expand in very, very small amounts, but over billions of light years and billions of years, those amounts became noticeable and detectable as redshifts. And this equation becomes, instead of a relation between the velocity and the speed of light, it now is purely a relation between the scale of the universe now, A sub-zero meaning now,

and the A sub E to the scale of the universe when this measured light was emitted. Then there's a little minus one added for mathematical reasons. This is derived from this differential equation here. And this tells us that again we set the current scale of the universe to one. So this just transforms to one. And therefore, if we know the redshift, we can immediately find out the scale of the universe, the size of the universe,

when that light was emitted. And that tells us the time it was emitted. That tells us the distance of the galaxy. It tells us the distance of the galaxy now, how far that light has had to travel across the ever-expanding space since it was emitted and reaching us now, which is different from the distance that that galaxy was at the time of the mission

or at it at the current age of the universe. So it's this equation here that I've been blabbering about trying to work my way to from Hubble's empirical observations of a Hubble constant that informed us that implied an expansion of the universe which was completely independently

found from empirical means from the mathematical formal deduction of its existence from Einstein's genius insight into the nature of space time and how it is intimately linked to the energy in the distribution of that energy in the form of again radiation and matter throughout the entire universe and now my friend friends, it's time to draw our space-time diagram. So we're going to start drawing this graph, and we're going to be tying everything that we do into this variable known as the scale factor.

It itself is derived from the Hubble constant measured at different times in the universe. So locally it's a constant. Turns out that it's actually better described as a parameter is at different ages of the universe, it had different values. So the better we can, that's why the web telescope, WMAP, Planck, all these different instruments and telescopes, one of their main goals is to measure all the properties of the universe that these equations allow us to derive from the Hubble constant, the

a Hubble parameter at different values in time. Currently, it's somewhere between 68 and 72 kilometers per second per megaparsec. Once we know that, we know that from, that we can relate that, cross-reference that with observed red shifts, and be able to interpret the different scales of the universe and all the characteristics of the universe, such as time and distances and different types of distances, in velocities of the objects that we're observing whose light we're observing and it allows us to interpret the evolution and dynamics of the universe which in turn allows to interpret and check our models of the universe for accuracy the big bang model currently known as the freedman lamatra robertson walker big bang model of the universe and it's the constant attempt to measure the values of the Hubble constant and

related to all these other factors that defines much of cosmology today so this is a very important relation here between the Redshift and the scale of the universe at different epics and this is called cosmological redshift which itself is distinct from Doppler's redshift, the relativistic Doppler redshift, and even another aspect of general relativity called gravitational redshift. This cosmological redshift is unique in that it is from the expansion of the entire universe as a whole, in the creation of space, essentially, the expansion of space itself, which don't ask me what that even means. me I've tried to understand which I don't think you can do unless you put in the work to

understand the math which I unfortunately don't have time or probably the probably the skill to be able to pull off well thank you guys for watching and I really want to thank my patrons and everybody who supports me financially and all the moral support I get from you guys in the comments is hugely uplifting to me and it means a lot so I want to thank all you you guys everybody who shows love and sends encouragement my way really does mean a lot hope you guys enjoyed this one i'll see you next time