The Science of Everything Podcast - Episode 115: Special Relativity Part 2

Episode Date: February 28, 2021

In the second and final part of this series, I discuss the major consequences of special relativity, including the relativity of simultaneity, time dilation, length contraction, and mass-energy equivalence. I then provide an overview of some of the experimental tests of special relativity, and conclude with an analysis of some alleged paradoxes in relativity, including the twins paradox, and the ladder paradox. Recommended pre-listening is Episode 114: Special Relativity Part 1. If you enjoyed the podcast please consider supporting the show by making a PayPal donation or becoming a Patreon supporter. https://www.patreon.com/jamesfodor https://www.paypal.me/ScienceofEverything

You're listening to The Science of Everything podcast episode 115, Special Relativity Part 2. I'm your host, James Fodor. So this episode is going to pick up directly from where the previous episode left off in talking about special relativity. So remember last time I introduced the basic concepts of relativity and gave a bit of historical background as to how Einstein sort of came up with the key ideas, particularly the constancy of the speed of light, and some of the relevance of results such as the Michaelson-Morley experiment, as well as looking at some of the phenomena of the classical electromagnetism that were part of his intellectual journey towards devising the theory of special relativity. What we're going to do in this episode is discuss some of the key phenomena that result from, essentially from the postulates of special relativity, particularly the postulate of the consternity of the constantity of the speed of light. So we're going to look at important phenomena such as the relativity of simultaneity, length contraction, time dilation, and the interconversion of energy and matter. We'll also look at some of the paradoxes or alleged paradoxes of relativity, including the twin paradox and the barn ladder paradox.

So recommended pre-listening is, unsurprisingly, episode 114, Special Relativity Part 1. Pretty much required to listen to this because I'm just going, picking up straight. from where I left off there, because as I said, it was originally intended to be a single episode. So that being said, let's jump straight in and start talking about some of the key phenomena of special relativity. So I've already mentioned some of these, and so I'll move past them fairly quickly, but some of the others will spend a bit more time on. So one is the relativity of simultaneity, and I've already talked about this before. Whenever you say an event happens at a given time, so if I say, you know, I work up at 7 a.m., what I'm in there is my waking up and the clock reading

7 a.m. occur simultaneously. So those are two events. They happen simultaneously. So that's what we mean when we say an event happens at a time. It just means two events occur simultaneously. So in a sense, that's all you have, because there's no cosmic clock, right? There's just events, and you can say, are they simultaneous or what's the time interval between them? Now, if all observers agreed on the simultaneous of two events, then you could talk about a universal time. And that's what Newton thought, and people thought before Einstein. But according to Einstein, there is no universal time.

There's no universal simultaneous of events. So there's no mechanism for determining who is really moving. and so there's no way to define an absolute time. Different frames of reference will keep time differently, and they'll measure different temporal intervals between two events. This leads to a phenomena called time dilation, which is one of the important implications of special relativity. I guess the sort of three big ones are time dilation, length contraction,

and mass energy equivalence. So I've already mentioned some of these, but let's talk about them in a bit more detail. So time dilation refers to the fact that time intervals change, depending on relative velocity between inertial reference frames. Time dilation is calculated using Lorentz transformations, as shouldn't be surprising. So basically what you do is you start with a time difference in one reference frame,

time interval between two events, say it could be the car passing this tree and a car passing another tree. Whatever the events are, you measure the time interval between them from the frame of reference of someone standing on the street next to the tree. And then you throw them into this formula, which is a Lorenz transformation formula, and you calculate the time that will be measured between those same events, the car passing the first tree and then the car passing the second tree,

in the frame of reference of someone inside the car. Well, that will give you a new equation, which will give you a new coordinate for the time interval between those events. And the fact that these are different, that is different observers measure different time intervals depending on their relative velocity, is called time dilation. This is sometimes summarized by saying that moving clocks run slowly. this is really not a very good way of remembering what happens there.

So, I mean, it can be helpful as a starting point. So as a starting point, you can think of it like this. If a spaceship goes to the moon and travels at, say, well, let's say Mars, it's further away. If a spaceship travels to Mars at some sizable fraction at the speed of light, you start to see relativistic effects, I don't know, 10% or more of the speed of light, something like that, you know, significant relativistic effects. So if we travel to Mars at half the speed of light and then turn around and come back, what you'll find is that the person on the spaceship who was traveling to Mars,

experienced less time than we did back on Earth. So their clocks are moving relative to us, and so they move slowly. One way to put that is that moving clocks tick at a slower rate. They experience fewer ticks compared to the same clock running on Earth. That's not moving. But again, there's no such thing as absolute velocity

in special relativity. So we shouldn't really say that moving clocks run slowly because there's no such thing as a moving clock. There's only a relatively moving clock. So a better way to say this, and a better way to understand what's happening here is to introduce the notion of proper time. So the proper time between two events is the time distance or the time difference between two events as measured by a clock that's present at both events.

The proper time is always the shortest time interval that any observer will measure between those two events. So different observers in different inertial reference frames will measure different time intervals, as we've said. That's time dilation. but proper time, which is measured by someone who is present at both events, is always the shortest one. You can't get any shorter, no one will measure a shorter time interval than the proper time. So if we think about our car travelling between the first and the second tree, let's think about this. The two events are the car passing tree one, and then the car passing tree two. So what frame of reference is present at both of those events?

Clearly it's not the person who's standing next to one of the trees, because they're only present at one of those events. the other event happens over there at the other tree. The only reference frame that's present at both events is the reference frame that moves along with the car, because then it's present at the first event and present at the second event as well, exactly when the car reaches the second tree. So this is an important point. Whenever you have an object that's sort of moving relative to some other objects, the proper time will be measured in the reference frame of the moving object.

And this will be the shortest time interval. Any other observer will measure a longer time interval. So from this point of view, thinking about now that spaceship traveling to Mars and then traveling back, although everyone on Earth will agree that they've experienced, say, 12 months, whereas the person who's traveled to Mars has experienced eight months or how long it is, depending on the velocity. And so I guess from that point of view, everyone else on Earth is sort of going to be, quote, quote, correct. But really, if you think about the proper time, it's actually the time on Earth that's longer.

The proper time is measured by the person in the spaceship, because the two events in question, which is, let's say, leaving Earth and then arriving at Mars, and then I guess there's a third event as to arriving back on Earth. Technically, you do need to accelerate to change directions, and we'll get to that in a little bit. But just to simplify things for the moment, the only person who's present at all of those events is the guy in the spaceship. And so his time measurements will be the proper time, and therefore they will be the shortest time interval. Everyone else's time measurements will be longer than that. So it's actually Earth's time that's been dilated, that is lengthened, relative to the proper time measured by the spaceship. Now, it's important to understand that when we talk about proper time, and there's proper length as well that's coming up,

that doesn't mean it's like the true objective time or length. It just means it's just a name for the time as measured by one particular reference frame. It's just a particularly important one because it's always the shortest one. And it's measured by the observer or the reference frame that moves along with the events, and therefore it is that the events are at the same location for that observer. So it's not like that's the true time interval. The laws of physics aren't any different in that reference frame to any of the others, so you can't distinguish that reference frame from any of the others.

It's not special. It's just a particularly important one from a laws of physics point of view to understand that it's always the shortest time interval. So the way I like to think about time dilation is not that moving clocks run slowly. It's rather that moving clocks measure longer time intervals relative to clocks, that are stationary with respect to the events in question. Obviously, that's a lot more of a mouthful, but it's much more correct. So moving clocks measure longer time intervals between events compared to clocks that are stationary with respect to those events.

Again, think about the car traveling between the trees example. The clock that is stationary with respect to those events is the clock in the car. Clocks that are moving with respect to that car, such as the stopwatch of the guy standing on the street, next to one of the trees, That clock is moving with respect to the first clock, and therefore it will measure a longer time interval, and that is time dilation. Obviously, the faster you're going relative to the speed of light, the longer the dilation gets. Technically, dilations approach infinity as you approach the speed of light, because you can never get to the speed of light. Again, we mentioned that before. You can't accelerate up to the speed of light. You can get arbitrary close, but you can ever actually get there.

The only things that can actually travel at the speed of light are massless particles or objects or fields or so. force, but not anything that has mass. Okay, now let's talk about the second big consequence of special relativity, which is length contraction. Time dilation on length contraction are basically pairs. They're two sides of the same coin. One is how time changes, the other is how space changes. And as I said, because of space time being wound up together, you're going to have to have both. So along with time dilation goes length contraction. Now, the sort of popular way of describing length contraction is that moving objects shrink, or they contract. hence the name, but only in the direction in which they're moving.

So this is important. If I have a spaceship that's moving very quickly, it's approaching the speed of light, let's say, it will contract, it will get shorter, and I'll explain what that means in a moment, but only in the direction that it's traveling. It doesn't contract, you know, lengthwise, perpendicular to the direction of travel, or sort of up and down, only sort of alongwise with respect to the axis that it's traveling parallel towards. So that's why we often only talk about one spatial dimension in special relativity,

even though there's three. It's just because whatever direction you're traveling in, you can treat that as a single dimension, and then the others will be the two directions that are perpendicular to that direction of travel. And you don't have any length contraction or, in fact, any relativistic effects in those directions. It's only in the direction you're traveling. So that's why it's often very useful to think about as if there's one spatial dimension and one time dimension. But anyway, coming back to length contraction. Length contraction is the phenomena whereby a moving object's length is measured to be shorter than its proper length. So it contracts. Now, what is this proper length? Well, the proper length is the analog to proper time. The proper length is the distance between two events, or let's say the length of an object. The proper length is the length as measured in the object's own reference frame. And here's an important point, because remember, the proper time is the shortest time interval between events, between those two events. But the proper length is the longest spatial interval between two events, or to put it another way, it's the longest that any observer will measure that object to be.

So, let's think back about my spacecraft again. If the spacecraft is traveling in a significant fraction of the speed of light, me sitting on Earth, or even just sort of hovering outside another spacecraft and watching, I will see that moving spacecraft contracted. It will look skinnier and skinnier as it approaches the speed of light. However, for the person on the spacecraft, it won't look like to them anything's happened. It will just look totally normal. In fact, from their point of view, you will be shorter because you are moving and not them. Again, this is all the relativity point of view. There's no way to detect absolute motion. And so from each point of view, the other one is shrinking or has been contracted. But within their reference frame,

they are always as long as they'll ever be. That is their proper length. So again, that's the length as measured from the object's own rest frame. Time dilation and length contraction always happen together. So it's not one or the other. They're sort of two sides of the same coin. They're two aspects of the transformation of space time intervals. And I'll talk a little bit more about how they connect or integrate to each other in a little bit. But first, we need to cover the final major implication of special relativity, which is mass energy equivalence. So as an object speed approaches the speed of light from an external observer's point of view, remember because from an observer, say, sitting on the rocket or whatever it is, they're not moving.

You're always at rest with respect to yourself, right? So it's always from an external observer's point of view. But anyway, so as an object approaches speed of light from some observers' point of view, the relativistic mass of that object keeps increasing. In fact, it increases asymptotically. So it doesn't increase linearly, which is what is expected by Newtonian physics. In special relativity, it doesn't just go linearly. It goes asymptotically.

Your energy guts higher and higher and higher and higher, until it goes to hypothetically infinity when you reach the speed of light. Of course, you can never reach the speed of light. In fact, this is why, because it would take infinite amount of energy to accelerate any massive object to the speed of light. It doesn't matter how light you are. As long as you have any mass at all, it would take infinite energy to accelerate you up to the speed of light. Hypothetically, it's actually possible that there are particles that always travel faster than the speed of light. And these hypothetical particles are called tachions.

You may have heard before because they're often used in science fiction. Most physicists don't think they exist. They've never been detected, and they're not really consistent with current laws of physics. physics, the only reason I mention them is just because they're not explicitly ruled out by special relativity. So hypothetically, you could have a particle that just always travels faster than the speed of light, but what you couldn't have is a massive object or particle that is accelerated up to the speed of light, because that would require infinite energy. Now, this mass energy equivalence, the fact that your mass, that your energy increases

asymptotically as you get close and closer to the speed of light, implies a mass energy equivalence. There's a bunch of formula for showing exactly how that works, which I get into. But the basic idea is as you accelerate, your energy increases, and therefore your inertia increases, which is sort of the same as increasing your mass. So this is a way sort of loosely of seeing that. If an object is hard to move or hard to accelerate, then that's basically what it means to have a high mass. And so using this and some other derivations, Einstein showed that there's an equivalence between mass and energy. They're actually just two variants of the same thing, or two sides of the same coin. And this is where the famous equation, E equals M.C. Squibre

comes from, because that is the energy content of a given amount of mass. Basically, you take the mass and multiply it by the speed of light squared, and you get the amount of energy that is sort of congealed in that mass. I've heard mass described as congealed energy, which is sort of a useful way to think about it. So this is very interesting, because the equivalence principle implies that when energy is lost, or I mean, it can't be lost, but it's transformed in chemical reactions or nuclear reactions or any type of energy transformation, then the system will lose the corresponding amount of mass, So whenever potential energy is lost in any system, say a battery, that it will actually lose mass.

Although because it takes a lot of energy to make a very small amount of mass, it means that it's probably not going to be detectable. But this is how nuclear energy works. You're basically extracting energy from changes in mass of certain isotopes. Anyway, there's a lot more to say about that, but I'm going to leave that there for the moment, because I just wanted to mention that in brief as a third key consequence of relativity. So we've talked about time dilation, length contraction, and mass energy equivalence. Now, I'm just going to briefly go through a few experimental tests or verifications of special relativity. There have been really hundreds of these, and you can look on Wikipedia or other sources for tons more information about this.

One classic test of relativistic time dilation is called the I of Stillwall experiment, which tested the contribution of relativistic time dilation to Doppler Shift. Doppler shift occurs when there are a relativistic time dilation is called the Ives stillwall experiment, which tested the contribution of relativeistic time dilation to Doppler shift. Doppler shift occurs when there is relative motion between the emitter of some sort of series of waves and the source. And this occurs due to relativity as well, so relativistic Doppler shift. I haven't discussed it in much detail, but it occurs as well. And using rays of particular charged ions, they were able to measure the change basically in the Doppler shift as the angle was changed. And they found, in fact, that time dilation led to the exact amount of Doppler shift.

that was predicted by special relativity. Another experiment was conducted by William Batozzi in the 60s, where he used an electron accelerator to initiate five runs of electrons, so just spitting electrons basically at a screen, but with different kinetic energies. And these electrons traveled about 8 meters until they hit a disc, and then the energy that was absorbed by some of the electrons hitting the disc, they weren't all absorbed, but for those that were,

you could measure the energy that was absorbed by the disc using a calerolera, as a direct measurement of their kinetic energy, because the kinetic energy is going to be converted into, basically, heat energy when they're absorbed and the energy dissipates. And this is a direct way to measure the effect of mass energy equivalence and the dramatic increase in kinetic energy as predicted by relativity. And in fact, the results agreed with expectation to within about 10% of Marta's vera. So that's direct confirmation of relativistic effects on kinetic energy. Another famous experiment was conducted by Hathelay and Kee, heating in 71, where they took four atomic clocks aboard commercial airliners and flew around the

world twice, first eastward and then westward, and they compared the clocks against those that had remained on the ground. What they found is that the clocks disagreed, and because atomic clocks are very precise, of course, this isn't just because they lost time, this was because of a number of effects. One, of course, is that the fact is that the clocks were moving relative to those on Earth. In fact, in interpreting this experiment, you can think of the center of the Earth as an inertial reference frame. I mean, the Earth is rotating around the sun, but that effect is sort of the same for regardless of where you're around on the Earth, so it can be neglected here. So relative to the center of the Earth, the clocks on the surface of the Earth are moving

with the rotation of the Earth. So if you travel eastward, which is in the direction of Earth's rotation, that's going to be an increase in velocity. Whereas if you travel westward against the rotation of the earth, you're going to have a lower velocity, again, relative to the stationary frame in the center of the earth. And because, as we've said, clocks that are in relatively, that have a relatively high velocity move slower relative to those with a lower velocity. What you'd expect is that the eastward clock that's moving faster than those that are stationary on the Earth's surface will experience a time loss, whereas the westward clock, which is moving slower relative to those on the Earth's surface will gain time. That is, the clock will

tick more slowly. Now there's also an effect due to gravitation, basically because as you move away from a massive object, the gravitational effect is smaller, and therefore the clocks will also gain time because of that. Basically, the clocks slow down as they move into a gravitational field. So that's part of general relativity, which I haven't talked about here, but just know that you have to subtract off that effect as well. And once you consider both a gravitational and the kinematic, called special relativity effects, the change in the time that was expected was very close to the measured changes in time to within a few percent of error. And since then, similar, more precise versions of that experiment have been done. So you can directly measure this

time dilation effect, as well as effects due to general relativity, by moving atomic clocks around, which I think is pretty cool. So those are just some of the experimental verifications of relativity. Again, there are literally hundreds more. Now, before finishing up, I want to talk about some of the paradoxes or apparent paradoxes, depending on how you wanted to find a paradox precisely, that emerge from special relativity and can be barriers to understanding what's happening here. So first of all, let's talk about travelling to a distant planet. This is not so much a paradox, it's just an illustration of what happens when you combine time dilation and length contraction. So imagine I've got a spaceship travelling from Earth to a planet that's 10 light

years away, and imagine that I'm travelling at 0.8c, so 80% of the speed of light. Now, from the perspective of Earth, which will call the stationary reference frame, again, it's not station in an absolute sense, it's just convenient to talk about it in that way. So from Earth's point of view, this journey will take 12.5 years. That's just 10 divided by 0.8. From the perspective of the rocket, the journey will only take 7.5 years, and the measured distance will be 6 light years. Now, those come out of the Lorentz transformations. If you do the math, that's what you come out with. So what we see is that the rocket, the spaceship that's traveling to the distant planet, not only takes less time in its own reference frame compared to how much time it takes.

in Earth's Justin Frame, but it also doesn't even have to travel as far. It only travels six light years compared to 10 from the Earth's point of view. So, let's think about how all this works. In understanding this, it's useful to think about proper length and proper time. So remember, the proper time is measured in a reference frame that is present at both events. And the two events here are basically the spaceship leaving Earth and the spaceship arriving at the distant planet. Now, clearly Earth is not present at both of those events.

It's present at the first one, but not the second one. It's 10 light years away from the second one. Indeed, the reference frame that is present at both events is clearly the one, the reference frame of the spaceship. It's the only one that would be present at the leaving and at the arriving at just the right time. And so proper time between the two events is measured in the reference frame of the spaceship. So therefore, the proper time between the leaving and the arriving is going to be that measured by the spaceship, which is 7.5 years. Not the 12.5 years that Earth measures. Earth measures a longer time interval because of time time.

violation. Earth is moving with respect to the rocket, and therefore it measures a longer time with respect to the proper time, which is measured by the rocket. Now, let's think about proper distance. The proper distance is measured in a reference frame that is at rest with respect to the events in question, and here the events in question are arriving at the distant planet and leaving Earth. And let's just suppose that the Earth and the distant planet are at rest with respect to each other, just for simplicity. Now, clearly the spaceship is not at rest with respect to those events because it's moving. That's the whole point of it, right? But Earth is at rest with respect to those events. And therefore, the proper distance is measured in the reference frame of Earth.

And that distance will be 10 light years. That's the distance between the planets. So when we talk about distances at an astronomical scale, you know, we're talking about with respect to Earth's reference frame. So therefore, this distance, the 10 light years, as we recall, is the proper distance and therefore is also the longest distance that any reference frame can measure. Any other reference frame will measure shorter distances, including the reference frame of the spaceship, which is moving between them. And that is length contraction. So the length contraction occurs with respect to the proper distance that is measured in the Earth's reference frame, and leads to a shorter distance as measured by the spaceship. That's how you get six light years distance compared

to the 10 light years. By contrast, time dilation is measured with respect to the proper time, which is always the shortest time that can be measured, and that occurs in the moving reference frame that's present at both events, which is the spaceship. So the proper time is 7.5 years, the dilated time as measured in the Earth's reference frame, is 12.5 years instead of the 7.5 years from the rocket's point of view.

So the key point here is not so much the numbers. The point is that proper length and proper time are measured in different reference frames, and you need to consider time dilation and length contraction with respect to those correct reference frames. If you muddle it and try to dilate with respect to the same reference frames,

so suppose you try to delay, time dilate the 12.5 years and space contract the 10 light years, then you'll get nonsensical answers that will contradict each other. No, the correct way to do this is to ensure that you get the right proper distance and the correct proper time

and dilate and contract those appropriately. And so that's a key important point to remember when you're looking at these things, that proper time and proper length will generally be defined in different reference frames, at least when you're talking about this sort of setup where you've got sort of a spaceship traveling to a distant planet or something like that. So that's the first illustration there. Now, let's use that and move to a more complicated example, which is often described as the twin paradox. So here's the setup of the twin paradox. I think Einstein talked about this, so it's very old. The basic idea is you have two twins.

One of them says goodbye to the other gets in a spaceship and leaves, and they travel to a distant planet at some substantial fraction of the speed of light. When they reach their destination, they turn around and then come back again at the same speed, but just in reverse. And then at some point they'll return to Earth, and then they will meet their twin. Now, the issue here is that the travelling twin can say that the stationary twin is moving, just as the stationary twin can say that the traveling twin is moving. Think about it this way. from the perspective of the twin on Earth, the twin and the spaceship leaves, travels away, and then comes back. So obviously, from the stationary twins' point of view, the travelling twin is moving. And therefore, the twin on Earth would expect the travelling twin to experience time dilation and therefore measure less time.

On the other hand, the twin in the spaceship sees the exact opposite. They see, you know, they close the hatch and they see their twin recede away as they travel from Earth and Earth recede away. and then at some point they turn around and then Earth comes back and then they see their twin again. So from the perspective of the travelling twin, Earth moves away and then comes back. So there's a symmetry here. Both twins see themselves as stationary, because again, you're always stationary with respect to your own reference frame. That's the way reference frames work. And the other reference frame as moving.

But when the twins come back to each other and meet each other again, they can't both be older than the other one. One of them has to be older and the other has to be younger. I mean, all they could be at the same age, but they can't both have experienced a longer time interval compared to the other. So what's going on here? How do you resolve this paradox? How do you break the symmetry, which seems to say that each twin should be older than the other?

Because both should have experienced time dilation, and therefore both be simultaneously younger and older than the other twin. Again, there's a few different ways that you can explain the resolution of this paradox, and you'll see slightly different answers. It's often said that the resolution of this paradox is that the traveling twin, the twin in the spaceship, is not in a inertial reference frame. because they have to accelerate, because eventually they have to turn around. We've said that they travel away and then come back. Well, that requires turning around. But that's a little bit premature,

because you can set up this same paradox by just imagining that the traveling twin doesn't accelerate at all. They're just magically, not physically possible, but they magically just flip their velocity and somehow turn around in an incident of time and then travel back. I mean, one way you could do this is just to have a third, well, I guess a triplet, who's traveling at the same velocity, but towards the earth, such that the one twin meets the third twin, just as the third twin is sort of traveling towards Earth. So basically, it's like the twin flipped around, but actually they never changed their velocity.

It was another twin that's been traveling. That makes it more complicated to imagine. But the point is that you can set up the situation so you don't actually have to have acceleration. Acceleration is not critical to the thought experiment, although in practice, if you did take a spaceship out and then turn around, there would be acceleration. So whenever you have acceleration, you're in a non-inertial reference frame.

And when you're in a non-inertial reference frame, you can detect effects due to acceleration. What Einstein said is that all inertial reference frames are equivalent. That is, the laws of physics are the same. But non-inertial reference frames are not equivalent. If you're accelerating, you can feel that acceleration, basically as a force acting on you. This is like if you're in an aircraft or even in a car that's going up a hill or down a steep hill. You can feel that on your stomach, right? That's the acceleration that you're experiencing.

the forces due to acceleration. And those are objective real forces that affect the measurements that you make and affect your predictions in conducting experiments. And so those are objectively measurable. And they provide a symmetry breaker between the Earth twin and the Traveling Twin. Because the Earth Twin will never, I mean, they're in Earth's gravitational well, but put that to the side for the moment because that's general relativity. But they don't experience any acceleration forces due to changing direction,

whereas the traveling twin does. And so that's a difference here. That's a symmetry break. But that's really not all that's going on, again, because you can formulate this paradox in a way that doesn't really involve acceleration. It's more just that the velocity flips sign rather than you actually have to experience acceleration. Again, that's unphysical, but you can imagine situations where that would apply or, you know, with a third twin or whatever situation. So how do we resolve the paradox? Well, fundamentally, it's not about the acceleration per se.

It's about how simultaneous works. remember that one of the key results or consequences of special relativity is that different observers will not agree on which events are simultaneous with each other. So basically what happens is once the space twin leaves and the other twin is on Earth, each will measure the others' events as if they are dilated. Remember, clocks that are moving with respect to you travel more slowly than clocks in your own reference frame. So that is, the Earth twin will see the space twin as if their clock is travelling slowly, whereas the Space Twin will see the Earth Twins' clock as if it's travelling slowly. And at the moment there's no paradox there, because the twins haven't come back. But the way that you can reconcile this is that they don't agree on which events are simultaneous.

Think about it this way. Suppose that it takes 10 years for the journey, just as an example. And in this case, we'll imagine that the Space Twins traveling at 0.6C. At the five-year point on Earth, the Earth twin will measure, will measure five years having elapsed, as I've said, but they will see the space twin at that same time as over than having had four years elapse. So they will say from their point of view, oh, the space twin's clock is running slowly, as you would expect. But let's think about it from the point of view of the space twin. So the space twin sees himself as having travel for four years.

He does these numbers and works it out. and says, well, at this time, so simultaneous with this event, as me marking my halfway milestone, simultaneous with this, the Earth twin is only marking his 3.2 year milestone. But the Earth twins, when the Earth twin is marking his five-year milestone, he thinks that the Space Twin is marking his four-year milestone. So the Earth twin says, now the Space Twin, my Space Twin, is measuring time slowly, because only four years to have a lapse for him, whereas the Space Twin says, well, my Earth twin is experiencing time slowly because only 3.2 years have elapsed for him.

And this is because they don't agree on which events are simultaneous. You can sort of continue this process here because the 3.2 years, obviously, the guy on Earth, the Earth twin doesn't agree that that 3.2 years is simultaneous with the 4 years for the halfway point for the Space Twin. No, they'll say that that 3.2 years for them is simultaneous with an event which for the Space Twin is something like 2 and a half years or so. The Earth Twin is going to say, no, no, you've got it wrong space twin. When I'm experiencing 3.2 years, you're not experiencing your halfway point. That's simultaneous with only about two and a half years for you.

But then the Space Twin says, no, no, no, you've got it wrong Earth Twin. When I'm experiencing my two and a half years, you're experiencing your 1.7 years, or whatever it is exactly. And so it goes. They keep disagreeing about which events are simultaneous. So if you pick an interval between two events for one reference frame, and then convert that to simultaneous events as measured in the other reference frame, each twin will always think that their interval between those events is longer than the other guys, and therefore that the other guy's time, the other guy is measuring time slowly.

So again, the example that we gave is five for the Earth guy, four for the Space Guy, but 3.2 for the Earth guy, as determined by the Space Guy, when projecting back to an event that's simultaneous with his four. So the Earth twin says, four and five are simultaneous, So when I reach five, which is halfway, twin also reaches four. But the space twin says, no, those aren't simultaneous. My four are simultaneous with your 3.2. So you've got it all wrong.

In fact, you're the one whose clock is running slowly. But then Earth twin says, no, no, no, my 3.2 is simultaneous with your 2.5. Your clock is running slowly. But then the space twin says, no, my 2.5 is simultaneous with your 1.7. So your clock is running slowly. You see how it goes. Whenever you get a pairing like this, they don't agree on what's simultaneous. and so when you compare events that one observer thinks are simultaneous,

they always think that their time interval is longer than the other guy's time interval. And so this is how it works. This is how the apparent discrepancy is reconciled. That's only half of the journey. Let's think about what happens when they come back. Well, from the point of view of the travelling twin, when their velocity flips, so when they go from traveling away to now traveling back, there is what is called a simultaneous gap.

So remember I said that when the travelling twin is at four years out on their own time, they think that the Earth twin is 3.2 years in, Earth time. But as soon as they start traveling back, you can show it from the diagram, they also think that their events, so just after four years, is now simultaneous with about 6.8 years elapsed from the Earth's time. So they jump straight from 3.2 to 6.8, not in their own time, but from the Earth's point of view. the reason that happens is because they somehow just flip velocity.

They flip from traveling away to traveling towards. In practice, of course, you can't do that. You'd have to gradually accelerate. So you wouldn't just flip directly from one to another. You'd quickly, what they basically do is very rapidly observe lots of events happening on Earth that they were simultaneous with. So it would be like, you know, three and a half years or so of Earth time just rapidly pass for them. They would just see all these events very quickly as they kind of rotated around.

round and came backwards. But in the extreme case where they just somehow flip and immediately come backwards, which again is non-physical, because that require an infinite acceleration to, an infinite force to accelerate you that quickly. But again, just for the purpose of an example, they actually just sort of skip all of that history. Again, not from their own point of view. It's not like they don't experience anything. It's just that there are no events in the travellers time that are simultaneous with the events that Earth experience between 3.2 and 6.8 years, just because of this flip point. You can see this on a diagram. I'll post these up on the, on the Facebook. There's a gap in the simultaneous. So what this means is that from the perspective

of the space twin, their final four years corresponds to only 3.2 years of Earth time. And then when the Space Twin arrives back, both the Space Twin and the Earth twin agree that 10 years elapsed from the Earth's point of view, but only eight years elapsed from the Space Twin's point of view. And therefore, the Space Twin will be younger. the Earth twin will be older because of time dilation. The way to reconcile the fact that there should be a symmetry here is that in fact, all the time that the Space Twin can account for is only 6.4 years on Earth. It seems to them that 3.2 years have sort of disappeared.

I mean, they still elaps. It's just that they can't account for them because there are no events in their time that were simultaneous with those events. Again, this is just because of the magical flipping of velocities. In practice, what would have happened if they accelerated very quickly is that they would have just quickly fast-forwarded, if you like, through all of these events. But that involves moving through a sequence of different inertial frames, which makes it quite complicated, but you can do these calculations. It just makes it hard to understand. So, again, this is difficult to comprehend, but here's the takeaway.

Both observers, when they arrive back, agree that the Earth observer has experienced 10 years and the Space Twin has experienced 8 years, and therefore the Space Twin will be younger than the Earth Twin. How is that reconciled with the symmetry of the situation? Well, there isn't a perfect symmetry because one of the twins flips their velocity, whereas the other twin doesn't. And that flipping the velocity gives rise to a simultaneous gap. Basically, the space twin experiences four years going out and then four years coming back, and drawing simultaneous lines, connecting events that they regard as simultaneous in their reference frame, they can account for 3.2 years on Earth, and then the four years going back is 6.8 years on Earth, but that only adds up to about 6.4 years.

there's 3.6 years of simultaneous where there are no events that are simultaneous in the Space Twins point of view for events on Earth with events that happen to them because they just sort of skipped over all of those events. So the entire time the Space Twin is traveling, they see or they regard the clock of the Earth Twin as ticking more slowly because for each of the intervals

it's 3.2 compared to 4. That's a slower clock, right? So out interval is 3.2 to 4. back into Villeen it's 3.2 to 4. So in both cases the space twin sees the Earth's clock is running slower. But the Earth twin sees the Space Twins' clock is running slower because for them it's 5 to 4 on the outward journey and then 5 to 4 on the inward journey.

So both twins see the other twin throughout the entire journey see their clock is running slower. The difference is that because of the acceleration here instantaneous, just as hypothetical, a bunch of time is skipped over giving rise to a simultaneity gap which means that overall the total amount of time elapsed on Earth is 10 years instead of the 6.4,

which is actually accounted for by the travelling twin. So the symmetry is broken by the change in the velocity, not so much the acceleration itself, but by the flip in the sign of the velocity of the travelling twin. That gives rise to a simultaneous 80 gap, which breaks the symmetry, even while for the actual time taken during the journey, each twin sees the other as having a slower clock. The simultaneous gap sort of adds in the missing time that you need to reconcile these. Again, if this actually happened in reality, you wouldn't see an instantaneous flip in the sign because that's impossible.

You would see a gradual acceleration, and that complicates the story because you have to move through different reference frames, each corresponding to a different relative velocity of the two observers. But conceptually, this analysis here is correct. So, again, a simultaneously gap is not physical, but it's a way of thinking about what's happening here. Now, there's one more paradox that I want to discuss before finishing out here, and this is called the latter paradox. The ladder paradox is a thought experiment involving a ladder, which has held parallel to the ground, moving horizontally at a relativistic speed. And therefore, it undergoes Lorentz's length contraction, so it sort of gets shorter, quote unquote. Now, what we imagine is that the ladder passes through an open front door of a garage or a barn. I'll call it a barn.

Now, imagine the barn is shorter. It's sort of narrower than the rest length of the ladder. So if it would just place the ladder in the barn, it wouldn't fit. but if the ladder is travelling at a significant fraction of the speed of light, it will contract, whereas the barn being stationary won't contract, and therefore the ladder will actually be able to fit inside the barn. You'll be able to shut the front and the rear doors, fitting the ladder inside, whereas you couldn't do that if the ladder was just stationary

because it would be too wide to fit inside the barn. So the apparent paradox here is that, from the perspective of someone standing from the side, it looks like the ladder should be able to fit inside the barn. And indeed, you can imagine shutting both the doors, the front door and the rear door, door to trap the ladder inside, and so proving that it fits inside. But from the perspective of someone riding along with the ladder, they experienced the ladder as having its proper length. Remember, that's the longest possible length, and it's measured along the reference frame that's stationary with respect to that object. The proper length of that ladder never changes, and is, by assumption,

longer than the width of the barn. And actually, from their point of view, it's the barn that contracts, not the ladder. So the question is, which is it? Because if the ladder contracts, then it can fit inside the barn. But if the barn contracts, then the ladder won't fit inside the barn. So, either the doors fit around it or they won't, right? So, you know, which is the case? It doesn't seem you can have it both ways. And the answer is you can have it both ways. You can have it both ways because, and again, this is pretty much always the resolution of these sort of paradoxes, the two reference frames do not agree on the simultaneousity of the events. In particular, whilst an observer standing on the side might say that the back of the barn door and the front

barn door were shut at the same time, thereby trapping the ladder inside and proving that it is in fact shorter than the barn. From their point of view, that works because those events are simultaneous and the ladder is contracted due to length contraction. However, from the perspective of the ladder, someone riding along the ladder, the ladder is too wide to fit inside the barn, and from their point of view, the barn actually contracts. So obviously it's not going to be possible to shut both doors around the ladder. So what happens? Well, this will be reconciled because they won't agree that both barn doors were shut at the same time. From their point of view, it would look like that the rear door was opened before the front door was shut. So basically,

the ladder started going out the other side before it all entered one end. So there was never a time when the ladder was wholly contained inside the barn. And so there's no contradiction here. The events are reconciled by the fact that the two observers, either watching from the side or riding along with the ladder, do not agree on whether the barn doors opened and closed simultaneously or not. So one will see them simultaneous and the ladder are shorter than the barn, the other will see them as not simultaneous, and the barn shorter than the latter. No contradiction here, just different ways of perceiving what's happened, depending on your reference frame. So basically all of these paradoxes of relativity are resolved by considering the fact that events are not

simultaneous, are not universally simultaneous. Whether to events are simultaneous is relative to your reference frame. And that is a hard thing to understand because we're not used to that in ordinary, you know, discourse and interaction. If something simultaneous, then it's always simultaneous. for everyone, but not true with special relativity. So, that concludes what I wanted to say about special relativity. Let me just go through a brief summary before we finish out the episode. So the key idea of special relativity is that the speed of light is invariant and constant in all inertial reference frames, meaning reference frames that are not accelerating.

From this key postulate comes a number of consequences, including time dilation, length contraction, and mass energy equivalence. Time dilation means that the proper time between two events, which is, measured by a clock that's present at both events, is always the shortest time interval, and any other observer measures a longer time interval between those events. Length contraction says that the proper length of an object, which is the length of that object, as measured in its own reference frame, is always the longest spatial interval, or length, of that object, and any other reference frame that's moving with respect to that object

will measure a shorter interval, thereby giving rise to length contraction. These two phenomena of time dilation plus length contraction give rise to, give rise to, to many strange and counterintuitive results of special relativity, such as what we talked about in the twin paradox, whereby one twin stays on Earth, the other twin travels to a distant planet at a significant fraction at the speed of light, and then returns. It seems that by symmetry, each twin should be younger than the other, which is impossible, but in fact, this is reconciled because the travelling twin, who flips their velocity to travel backwards halfway through the journey, experiences a simultaneously gap. So there is a bunch of events on Earth

they just skip over and never experience. Or, to put it more clearly, there are no events in the journey of the space twin that are simultaneous with a bunch of events that occur on Earth. And thereby each twin sees the entirety of the journey of the other twin as being shorter than they themselves measure that interval. And yet, because of the sum, it's an 80 gap, the travelling twin ends up returning to Earth being younger than the stationary twin.

And that ultimately drives from the fact that there's an asymmetry caused by the change in the velocity. When there's no change in velocity like that, the symmetry is simply not broken, and there's simply a disagreement as to how to understand or perceive the situation, as illustrated by the ladder paradox, whereby a ladder may fit inside a barn, if, from the point of view of an observer who sees that ladder contracting as it travels rapidly, whereas an observer traveling along the ladder won't see the ladder contracting. Instead, the barn will contract, but that will be reconciled by the fact that they won't agree that the barn doors were both shut at the same time, thereby the ladder.

never actually fitted inside the barn. So all of these paradoxes that arise, or apparent paradoxes that arise as a result of time dilation and length contraction, are resolved by considering that Samuelsenady is relative to the reference frame that you are in. And fundamentally, all of this occurs as a result of the underlying structure of space time, which is the, I guess, fabric of the universe, if you want to use that flowery language, whereby space and time are sort of linked up together, and the intervals between events, which is called a spacetime interval are invariant, and therefore the same for all inertial reference frames,

but can be experienced differently, such that in one reference frame you can experience a long time interval between events and also a long spatial interval between events, while a different reference frame can experience a short spatial interval and a short temporal interval between those same events. Same spacetime interval, which is invariant, but different ways of experiencing that. The special theory of relativity is one of the best experimentally verified theories in the history of science, and I mentioned briefly some of the experiments that have been used to test things like length contraction and time dilation and mass energy equivalence and so forth. And so what we're left with is a very puzzling, but also very sort of beautiful and fascinating

theory, which is the theory of special relativity. So, thanks for listening. Hopefully you found that comprehensible and interesting. I tried to make that as clear as possible. as I said, it may be worth listening to this a couple of times to sort of get your head around it. I will post up some diagrams to help with, particularly the Twin Paradox case, on the Facebook, for you to have a look at there, so you can understand what's going on there. If you enjoyed the podcast, feel free to send me an email. My address is Fods12.gm.com. That's FODDS1.2 at gmail.com.

And you can give feedback or suggestions for future episodes, or just let me know that you've been listening. I always like to get feedback from listeners. also I always appreciate reviews of the podcast on iTunes or your preferred aggregator of choice. Those are a good way to spread information about the show. And finally, if you'd like to support the show financially to help me get out more podcasts, you can become a Patreon supporter by just Googling Science of Everything podcast, Patreon, and you should be able to find that. Or you can make a one-off donation through PayPal,

and the links to that should also be found in the show information. Thanks again for listening, and I'll talk to you next time. Thank you.