The Science of Everything Podcast - Episode 114: Special Relativity Part 1

Episode Date: February 8, 2021

The first of a two part series on special relativity, I provide a brief overview of the historical development of relativity theory, including a discussion of the role of the Michelson-Morley experiment, and considerations from classical electromagnetism. I then outline Einstein's two postulates, discuss their meaning, and provide some explanation for how to interpret a constant speed of light. I also provide an introduction to Lorentz transformations and the notion of spacetime. Recommended pre-listening is Episode 13: Newtonian Mechanics. 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 114. Special Relativity. I'm your host, James Fodor. So, it's been a long time coming. I've been wanting to do this episode for probably about 10 years, and I've spent a long time trying to understand special relativity, both formally in my physics education, and then doing my own reading and research and playing around with things.

And I'll be working on this episode for quite a long time now. And finally, I've decided that, Yeah, it's come to the stage where I'm happy to put together the material and, well, this is the resulting podcast. So, special relativity is a notoriously difficult topic to understand. The mathematics of it are fairly simple, but conceptually it's just very hard. So this episode is an attempted introduction to special relativity that doesn't assume really a lot of math background. And really what I'm focusing on is the conceptual understanding of the principles of relativity and why they happen, why it works that way. and resolving certain paradoxes and misunderstandings and so forth.

So recommended pre-listing for this episode is episode 13, Newtonian mechanics, and that's basically just to sort of give a bit of a background as to what comes before special relativity or what it's sort of building on. In general, it might be helpful to have a bit of background in some other physics concepts such as electromagnetism for listening to this episode, just to sort of give you some, a bit of a physics basis to work on,

although not strictly necessary. And this is an episode that more than others will repay repeated listening, I think. So in this episode, we're going to talk about the historical background of relativity, a little bit about Einstein and how he came up with the idea and how it relates to electromagnetism and so forth. And then I'm going to discuss the basic principles of special relativity, including the idea of inertial reference frames, Einstein's key postulates, the constancy of the speed of light in particular.

We'll talk about Lorenz transformations and the notion, of space time and why the speed of light is constant. Then we'll look at some of the consequences or implications of special relativity, including the relativity of simultaneity, time dilation, length contraction, mass energy equivalence, and look at some of the experimental tests of special relativity. Finally, I'll discuss some of the paradoxes that come out, or at least apparent paradoxes, that come out from special relativity, including the twin paradox and the latter paradox. So there's a lot to get through, but first, before talking about the historical background, I want to emphasize that here I'm only talking about special relativity. I'm not going to talk

about general relativity, because although the two are very closely related, general relativity builds upon special relativity. In general relativity, you have to talk about gravity and curved space time, and that's an additional layer of complexity that I don't want to get into here. So here, I'm only going to talk about special relativity. So basically, for those of you who have some familiarity with what that means, I'm going to just assume that space is flat and ignore the effects of gravity. That being said, let's get started and talk about the historical background to special relativity. The historical background to the idea of relativity goes back at least to Galileo. But I'm not going to cover all of that. I'm just going to pick on a few key points that are relevant and seem important.

So, as you probably know, special relativity was proposed by Albert Einstein in 1905, in a paper called On the Electrodynamics of Moving Bodies. And that name already gives you a hint about what he was thinking about, because Electrodynamics is basically a concept from electromagnetism, and he really developed the idea of relativity theory while thinking about some of the ideas of electromagnetism and their implications. But we'll get to that in a moment. First, a word on what the special refers to in special relativity. Well, I've already indicated that special relativity means that the theory only applies in the case where space time is flat. That is, there's no curvature. And essentially, that occurs when you have minimal gravitation. effects. This wasn't exactly known at the time the full, obviously, general theory of relativity

came out later, and as I said, I'll talk about that in a later episode. But just in case there's any confusion about what special means, it essentially means it's a special case of a more general theory. Now, the defining feature of special relativity is that time and space cannot be considered separately, as had previously been thought by Newton and Maxwell and others coming before Einstein. Rather, space and time are interwoven or mixed up together in a single continuum, which is called space time, which is written as a single word. This means that events that occur at the same time for one observer can occur at different times for another observer.

That also applies for space as well. Events that occur at the same space for one observer occur at a different space or at a different location for another observer. And we'll talk about this more subsequently, but this is really the idea of relativity, that time and distance are relative to your reference frame, and so vary from one observer to another. So that's what the name refers to. Special means basically not gravitational, flat space time,

and relative means that distances and times are relative or depend upon the observer or a reference frame you're measuring it from. Now, a bit more background about what Einstein was thinking about. In the 19th century, the theory of electromagnetism was developed, and in this theory, light travels, as a propagation of electromagnetic waves. And because all other waves that we know about require a medium, such as water waves,

sound waves in air, and so forth, 19th century physicists also hypothesized the existence of a substance called the ether, which was a material that light propagated through as it moved, or at least the electromagnetic radiation propagated through to give rise to light as it propagated along. And the idea of the ether was that it was so fine that it didn't interact with ordinary matter,

but provided just enough matter or substance to carry light waves. So basically, according to this theory, electromagnetic waves were waving the ether. That was what was sort of waving to give rise to the propagations. Now, with this idea came the question of whether the ether was moving relative to the earth or whether the earth maybe dragged the surrounding ether along with it or something like that, as a ship drags along some of the surrounding water. And to investigate this question, while a number of experiments were conducted, but one of the most famous ones is called the Michelson-Morley experiment, which was

conducted in 1887. And the idea here was to determine the speed at which the Earth was moving relative to the ether, so this hypothetical medium that light was supposed to travel through. Now, they used a device called an interferometer, and although the precise setup was rather complicated, the basic idea is that you have a shining light through a half-tempered mirror, so that means that basically half the light can pass through, and then half of it's reflected off the surface. And this allows you to split up a beam of light into two components. One passes through and sort of travels along in the same direction. And then the other is sort of deflected to the side. So basically you can separate out a single light wave into two perpendicular

beams, one sort of moving vertically and one moving horizontally. And then basically what they did is that they rotated this around, or this was sort of rotated around as the earth moved. and the interference pattern between the two beams of light was compared to see if it changed essentially when the earth was as the earth was moving around the sun, or also as the earth was rotating. So again, the details are complicated there, but the long and the short of it is that they expected to see a difference

in the interference pattern between these beams of light as the earth moved. Basically because if the earth is moving in one direction, then the speed of light as measured with respect to the earth will differ depending on whether you, you're measuring it along the direction of the Earth's motion or perpendicular to the direction of the Earth's motion. Now, the surprising result here was that the interference pattern was found never to change at all, meaning that the speed of light with respect to the Earth was the same

in all directions. It didn't matter whether you were looking at in the direction of the Earth was moving, in the opposite direction, or sort of side-to-side perpendicular. It was always the same. Now, this didn't really make sense with respect to the idea of the ether, because the ether was supposed to be something that just existed throughout space, and the Earth moved respect to it, and so the speed of light should depend on the Earth's motion with respect to the ether, and therefore that should be measurable as a change in the speed of light. But they didn't find anything. And this was an idea that it seems that Einstein took up on, and it was one of the ideas, not the only one, but one of the ideas that prompted him to think about relativity. Now,

I should say there is some dispute about this, because Einstein said that he didn't know about this experimental result, and it didn't play any role in his thinking, but it seems that it might have. at any rate, it's certainly important for thinking about the general development of the ideas around this time, sort of turn of the 20th century, even if we can dispute the exact extent to which it influenced Einstein specifically. Now, the other big thing that influenced the thinking around this time, apart from the failure to detect the ether, was electromagnetism, as I mentioned before. Electromagnetism is a very interesting theory. It was the most successful physical theory known at that time, that is late-90th, early 20th century.

But there was one very interesting implication of this theory, which related to the relationship between magnetic and electric fields. Now, this is something that we've discussed in a previous episode on this, so go back to listen to some of those if you're interested. But the long and the short of it is that there seems to be some interesting asymmetries resulting from Maxwell's equations that don't appear to be inherent. That is, they seem to depend on our description of the phenomena rather than the phenomena itself. So let me explain this in a little bit more detail. This is sometimes described as the paradox, or maybe not paradox, but at least the situation of a moving magnet and a conductor. So if I have a magnet that's stationary, and then I move a coil of wire with respect to the magnet, then from the perspective of the magnet, the wire is moving, right?

The magnetic field is constant, but the wire is moving. According to Maxwell's equations, the electrons that are in that wire, will, because of their motion relative to the magnetic field, experience a force that pushes them along, causing a current in that wire. All well and good. But what happens if I look at the same situation, but instead of from the perspective of the magnet,

being stationary and the wire moving, I now imagine I'm sitting on top of the wire. And so from my perspective, I'm not moving. The magnet is moving towards me, meaning that the magnetic field is changing. In this situation, Maxwell's equations predict that an electric field will be induced around the magnet and thereby the electric field will cause a current to flow in the wire.

In this situation, the result is the same. Current will flow in the wire. It will be in the same direction and the same magnitude, assuming you the motion is the same in both cases. So the phenomena is the same, but the description of it, according to Maxwell's equations, varies. In one case, it's a force exerted by the magnetic field on the charge. And in the other case, the changing magnetic field gives rise to an electric field, which then exerts a force on the charges inside the conductor. So what's the deal with this? How is it that we have sort of two different descriptions of the same phenomena in the same situation? That seems a bit strange, and it seems strange to Einstein and others at the time as well. Basically, there was a desire

to try to understand why this seemed to be the case. It seems strange that you should have two different descriptions that seem to be describing exactly the same thing. Maybe there's something fundamental that's going on here, which might be able to unify these descriptions and give a more fundamental description. And so this was one of the key things that got Einstein and others, but particularly Einstein, thinking about ideas of relativity. So that's some of the historical background behind relativity. We talked mainly about two ideas. First, the failure in the Michelson-Morley experiment to discover the ether, the hypothesized medium that the electromagnetic radiation would travel through, and the second was apparent strange coincidences or symmetries in Maxwell's

electromagnetic theory between magnetic and electric fields. So thinking about these sorts of ideas and some others as well that we won't discuss here, Einstein came up with two postulates. And this is the way that he sort of originally articulated special relativity, and it's still the way it's widely taught today, although it's not the only way you can describe it. But for our purposes here, I'll start with the two postulate formulation. It's sometimes called. So here were his postulates. These are just things that he asserted as postulates, and then sort of we'll see what follows from them. So the first is that the laws of physics are invariant, meaning the same, in all inertial frames of reference.

Okay, well, what's an inertial frame of reference? Well, to understand that, we need to understand what a reference frame is, and this is just a coordinate system. So think of your Cartesian plane, your X and your Y axis. That's a coordinate system. You can extend that into three dimensions, if you like, X, Y, and Z. And you can apply that to sort of any situation you're describing, and this is widely done. in physics, right? And indeed, other disciplines as well, even economics uses coordinate systems to describe how, you know, variables change over time and so forth. So a given observer

will have a given reference frame associated with them, which describes the location and times of different events. Now, there are many different reference frames, really any object or entity in the universe has its own reference frame that moves along with it. But there are special types of reference frames that are called inertial reference frames. And the reason they're call this is because in inertial reference frames, Newton's Law of Inertia holds, which means that an object continues to travel in the same direction and at the same velocity, unless and until a force acts upon it. So an inertial reference frame is one that is not accelerating or changing direction. It's just continuing along the same direction at the same speed. So it can be moving,

it's just that it can't be accelerating or changing direction. Also, an inertial reference frame is not subjected to any external gravitational forces, although again, that's a little bit beyond special relativity. So for the most part, I'll just talk about an inertial reference frame as one that's not accelerating. So that's what Einstein was talking about when he meant an inertial reference frame, something that can be moving, but it's not accelerating. So according to Einstein, the laws of physics are invariant in all inertial reference frames. This is really not too controversial. It just means that the laws of physics don't change if you perform experiments, say, out in a field compared to if you're on a train performing the same experiments. It doesn't

matter how fast you're moving, all the results that you calculate and predictions that you make should be the same. And that kind of makes sense, right? It's the second postulate that's really different and surprising. And Einstein's second postulate is that the speed of light in a vacuum is the same in every inertial reference frame. Or in other words, it's the same for all observers, regardless of the motion between the light source and the observer. Now, on the face of it, this is a completely ridiculous idea. That just isn't how it works Einstein, surely, you might be thinking. After all, if I sit on the back of a truck, let's say, well, actually, let's have me sit on the front of a truck, makes it a little bit easier. If the truck is going at 60 kilometers an hour and I throw a basketball forwards at, say, five kilometers an hour relative to the truck,

then that basketball will now be traveling, at least initially, before it gets slowed down by air resistance and so on. It will initially travel at 65 kilometers an hour relative to the ground, because I have to add the velocities of the truck to the velocity that I've thrown the ball at with respect to the truck. I mean, that's just common sense, right? Another example of this is if I have two cars driving towards each other in a headlong collision, if each car is going at 80 kilometres an hour, but in opposite directions, then the relative velocity of the two cars to each other will be 160 kilometers an hour, because again, you have to add the velocities. So how is it that Einstein can be saying that if I were to imagine being on a spaceship

that's traveling at 90% of the speed of light, and then I'm sitting on the front of the spaceship and I turn on a torch, then light comes out of the torch, Surely the speed of light of those light rays coming out of the torch will be at 1.9 times the speed of light relative to someone who's just stationary watching on the side, because there's the 90% of the speed of light of the spaceship plus the 100% of the speed of light from the light itself. So surely it's 1.9 times the speed of light will be the speed at which they will see the light travelling. Well, Einstein says no. Einstein says, again in a vacuum, the speed of light's always the same. So what Einstein says is that that light ray will travel, yeah, sure, at the speed of light away from the observer on the spaceship. But then the observer who's just standing along the side or hovering, I guess, along the side watching, that light will also be traveling at the speed of light.

Not at 1.9 times the speed of light. And this applies to any other configuration you come up with as well, regardless of whether they're traveling away or towards each other or parallel or whatever. The speed of light is always to any observer in an inertial reference frame, the same. same. Now, there's a few points to this. Some people get confused about the notion of the speed of light, and for this reason, in physics is often just a note of a lower case C, rather than called the speed of light, although it's called that as well. The speed of light is about 300 million metres per second, or approximately 300,000 kilometres per second. And it is not just the speed at which electromagnetic radiation travels, again, in a vacuum, it's also the

speed at which all mass-less particles and perturbations in fields travel in a vacuum. And so that includes, for example, electromagnetic radiation as well as gravitational waves. So it's not just the speed at which light or electromagnetic radiation is travelling, it's the speed that's sort of anything that doesn't have mass will travel in a vacuum. Now, once you start talking about through water or air or other media, then the speed can change. But again, we're always talking about in a vacuum here. Any particle or object that has non-zero rest mass can increase in velocity and approach C, you can get arbitrarily close to sea, but you can't actually reach it. And we'll explain why that is in a moment. So again, when we talk about the speed of light, it's not the

case that light has a speed and that happens to be the speed limit for the whole universe. Rather, it is that there is a speed limit that comes out of the equations of physics that all mass-less fields and objects will travel at, and light just happens to be one of those things. Gravitational waves and some others travel at that speed as well. But light's just the most familiar thing that travels in that speed. So at this point, it's just important to understand when we talk about the speed of light, it's the speed that light happens to travel at, not so much that it's sort of defined by being light's speed, if you see what I mean. Baring that in mind, though, still, how can we understand that the speed of light in a vacuum

is the same for all observers? It just doesn't make any sense. Well, that is what gives rise to the strange results and apparent paradoxes of special relativity. They follow as logical consequences of these two initial postulates, especially the constancy of the speed of light. Before I start to explain the consequences of this postulate and sort of the strange phenomena that it gives rise to, I want to explain a different way of understanding relativity, which is more widely used these days. And this is the notion of a Lorentz transformation. So let's go back to the idea of a reference frame or a coordinate system, essentially the same thing, that we introduced just earlier. So basically this is a set of numbers that describes the distance between any two events. in special relativity we talk about events as just a thing that happens at a specific point in space at a specific time.

So, I mean, a balloon popping or a torch being turned on or, you know, two cars crashing into each other, whatever. Those are all events. Now, you know, technically they're extended objects, so, you know, you could break that down into which particle exactly we're talking about. But because we're generally talking about, you know, the size and space of the universe and long periods of time, you know, whether it's a fraction of a centimeter here or there or whatever, it doesn't matter that much. I'll just talk about all these things as if they're just point events that occur at a precise point at a precise time. Now, the thing about different reference frames is that different observers will have different numbers that they'll attach or different coordinates that they will ascribe to the same events. This is familiar, and the classic way that we transform coordinate systems between observers is called Galalayan transformations.

A Galilean transformation works as follows. If you are traveling, or if you're moving, suppose I'm in an aircraft, for example, and I'm, looking at the ground. I'm moving with respect to the ground. The location at which events appear to occur to me, traveling in the aircraft, depends not just on where they are with respect to the ground, but also on the time that I measure them and how fast I'm traveling. So as I travel, basically, it looks like things on the ground are moving away from me, because I'm moving. From my perspective, it looks like the ground's moving away from me as I travel past. Of course, from the ground, it's the opposite. It looks like the plane is moving, and, and you're going.

stationery if you're looking up from your point of view your stationery and the plane is moving. And that's just the difference in reference frames. The basic point though is that you transform coordinates in the Galilean transformation simply by multiplying the speed by the time taken and then subtracting that from the distance. So essentially if I'm traveling at 100 kilometres an hour for one hour then an object that appeared in one location will now appear 100 kilometers behind me because I've moved 100 kilometers in that time. From my point of view I'm still in the same place, but that thing's now moved 100 kilometres away. So again, a simple example of this, suppose that I'm driving a car and I'm travelling at 10 metres per second,

and suppose that I pass some sort of light on the road that goes off just as I pass it. So that's an event, that light switching on. From my reference frame, that happens, let's say at 0.0. It's where I am. And let's just call it time zero for sake of argument. Then I travel 10 metres in one second. Let's suppose that the light blinks on and off very quickly, and then one second later it blinks on again. But now I've moved. I'm 10 meters ahead of where that light turned on. From my point of view, it looks like that the lights move 10 meters behind me, because it's one second. I'm moving at 10 meters per second, so that's 10 meters. And so the new coordinate of that event in my reference frame is now minus 10 instead of zero, because essentially it looks like it's moved

10 meters behind me from my reference frame. And that is a simple example of the Galilean transformation. It's how we transform coordinates in everyday ordinary context. And this is how Galileo, remember, talked about relativity. That's why it's named that. That's a very simple transformation. It's a simple linear transformation. And that is what we generally assume whenever we imagine or think about how time and place is affected by traveling at different speeds. In special relativity, the Galilean transformation goes out the window.

It can't be correct because we've assumed that the speed of light has to be constant. So in order for that to hold, what we need is a new transformation, a new transformation, A new way of transforming coordinates in one reference frame to coordinates in a different reference frame. And this transformation is called the Lorentz transformation after the guy came out with it, essentially. Now, obviously, we're not going to go through the math here, but you can derive the Lorentz transformations from Einstein's postulates. They're not really different things. They're just two ways of describing the same thing. Basically, you can derive the Lorentz transformation from Einstein's postulates, or you can start with the Lorentz transformation and then get to Einstein's postulates.

It's all the same thing. So it doesn't sort of matter which you start with. I guess these days, I think physicists would typically see the Lorentz transformations as being sort of more fundamental. Basically, physicists assume that all of the laws of nature must be Lorentz invariant, which means they stay the same under Lorentz transformations, and the constancy of the speed of light just follows out of that, essentially. And so that's considered to be fundamental. So it's not so much that nature decided that the speed of light would be the same.

It's rather that nature is such that the Lorentz transformation always holds. And as to why that is, we'll talk about a little bit later. But I'm just emphasizing that these are not different things, that just two sides of the same coin, two different ways of describing the same thing. But what does the Lorentz transformation look like you might be wondering? If the Galilean is just sort of a simple, you times the speed by the time, and then you subtract that from the distance, well, it's more complicated than the narence transformation, because basically you have to adjust according to your relative velocity between the two reference frames. Now, there's a formula for doing that, which is a little bit complicated. It's 1 over the square root of 1 minus v squared over C squared,

where V is your velocity and C is the speed of light. But, you know, that's a little bit complicated. So I'm not going to talk about that each time. I'll just talk about the Lorentz transformation. The basic idea here, though, is that in the Lorentz transformation, it's not just space that changes, it's also time. In the Galilean transformation, you never change the time at which events occur. It's always the same.

The only thing you change is the spatial coordinates. As I move, it looks like things are moving with respect. to me, and so I assign them a new coordinate value in my reference frame compared to a different reference frame. It's the same thing in the Lorentzian transformations. It's just that I now need to transform space and time. What this means is that not only will different observers disagree about where an event occurred, but they'll also disagree about when it occurred. And this is called the relativity of simultaneity. That is, observers in different inertial reference frames will disagree about which events are simultaneous to one another.

Now, this is a pretty wacky concept, because, again, we don't experience this in everyday life. An event that occurs simultaneous with another event for someone in China will also occur simultaneous with that event in Chile or in Czech Republic or wherever else, right? But that's not true in general. That's only true because the relative velocity of observers in different parts of Earth is pretty much zero, relative to the speed of light, at least. Remember, this is all relative to the speed of light. The speed of light is very, very fast. It takes light about one second to get to the moon, or is that the moon and back? I think that's to the moon and back. Well, whichever way it is, that's pretty quick. It took the Apollo missions, I think, three days to get to the moon. So that gives you an idea about how fast light is. It's many, many times faster than the speed of sound. So anything that's traveling at sort of everyday speeds, which even includes extremely rapid speeds like spacecraft and aircraft and aircraft and so on, even that is peanuts compared to the speed of light. So for all intents and purposes on Earth,

everything is so slow that you can just ignore the Lorentz transformations. And one important property of the Lorentz transformations is that when your relative speed is very low, as I said everything on Earth essentially is, then they just basically transform back into the ordinary Galilean transformations. And essentially you just get Newton's laws back out of it. Newton's laws are a special case of Einstein's special relativity when speeds are fairly low. So special relativity doesn't exactly replace classical Newtonian physics, it's more that it extends it into a new domain, into the domain of

very high speeds, where Newtonian mechanics goes wrong. But anyway, that's a description of sort of how these transformations work, and it's sort of how we operate in special relativity. What we do is we take an event, or generally two events, and then we ask, what are the coordinates of these events in one reference frame, and then we apply the Lawrence transformation and work out what the coordinates of those same events are in a different reference frame, and the reference frames are just defined by their relative velocity. Remember that there's no absolute velocity in special relativity. It doesn't make sense to say how fast is something going in special relativity. And this is one thing that I think is often not taught very well, especially at

high school level, when special relativity is discussed, because a lot of the examples will talk about the velocity of something as if it's absolute, as if there's an absolute velocity that the spacecraft is traveling at or whatever. But this is false. There's no absolute velocity in special relativity, and this follows from Einstein's first postulate that the laws of are the same in all inertial reference frames. You can't distinguish between something that's stationary and something that's moving by doing any experiment, Einstein says, because the laws of physics are all the same, so the results are always going to be the same. As long as you're not accelerating, the results are always the same. So there is no way to tell whether you're moving in an absolute

sense. You can only define motion relative between one reference frame and another reference frame. Therefore, the difference between reference frames is really just purely defined in terms of their relative velocity. One way to think about that is that if you imagine that you have two spaceships traveling towards each other at say 100 kilometers an hour, then in terms of special relativity, it doesn't really matter whether they're both traveling at 100 kilometers an hour towards each other or whether one is stationary and the other is traveling at 200 kilometers an hour, because it's really the same thing. All that matters is their relative velocity, not their absolute velocity, because according to special relativity, there is no absolute velocity.

So always bear that in mind. When we talk about velocity, we just mean velocity compared from one reference frame to another, not some sort of absolute velocity compared to some sort of absolute reference frame. Now, to further expand on this idea, I need to introduce the idea of space time. I've kind of already alluded to this. Early on, I mentioned that in relativity, space and time cannot be treated separately. You have to consider them as part of a whole. They're sort of fused together in this sort of entity that we call space time, or technically it's a manifold, but that's more for general relativity. Space time is four-dimensional. There's the regular three spatial dimensions, and there's a time dimension. Time dimensions are converted into spatial dimensions, it's so-called

spatialized, by multiplying it, multiplying all time, by C, the speed of light. And that's not arbitrary. There's a very good reason to do that, and basically because that's what allows the space-time interval to be Lorentz invariant. So if you used any other speed, it wouldn't make sense. So, again, it might seem arbitrary, well, how do we know that converting time to space, the correct way to do that is by multiplying by speed of light? Why not the speed of anything else? Well, again, it's not because it's the speed of light, it's because it is the upper limit on speed that is possible in the universe. And it's the only speed that will give you Lorentz invariance. So, basically, we can compare all distances between events to each other, whether it's time differences or space differences,

distances, distance, and space, they can all be compared to each other just by multiplying time intervals by C. So this is why light year is used as a unit of distance in astronomy, because basically it's just converting the time that light travels in a year, converting that into a distance. So it's a very natural unit of measurement. Now, when we do that, we can construct something that's called a space time interval, which I just mentioned before. The space time interval, technically the square of the interval, is the time distance, multiplied by C, squared, minus the space distance squared, and the space distances has three components,

you know, X, Y, and Z. Why do it this way? Well, again, it turns out that the space time interval, as defined in this way, is invariant, is Lorentz invariant. That means that every inertial observer agrees on the space-time interval between two events. This is very important to understand,

because what we've said, according to the principles of relativity, observers in different inertial reference frames will not agree on the spatial distance between two events. Or the time distance between two events. So they were not agree on whether events are simultaneous, for example. Simultaneous events have a time distance of zero. Spatial events that have a space distance of zero are at the same location. So different observers won't agree on that. But there's one thing that they always will agree on. And they always agree on the

space time interval. So the space time interval is invariant. It's the same for all inertial observers. And it's sort of analogous to the sort of distance measures that we use in everyday life. It's just that it turns out under special relativity, you have to factor in time distances differences in there as well, not just spatial distances. So in the world we are used to, all observers who do their measurements properly will agree on the size of different objects, if you measure them with tape measure, whereas in relativity, all observers will agree on the distances between events, but only if you measure them according to the spacetime interval, not if you just measure space or time separately.

The space time interval is what's invariant, not space or time separately. They must be considered together. Now, this leads to a further question, why is it that the speed of light is constant, or perhaps another way to look at that, is why is the spacetime interval invariant? So what's going on here? Why does it work this way? The way relativity is often taught is as if this all follows from Einstein's postulates, which I mean it does, and that's how Einstein derive these results. But it's not like the laws of nature are the way they are because Einstein said that they were. Sometimes in some books that I've read, it describes length of contraction and time dilation and other phenomena.

that we'll get to in a moment, as if it's sort of like, well, time has to be dilated in order to preserve the constancy of the speed of light. But that's not really correct. The speed of light's not, it's not like nature's just trying to ensure that the speed of light's constant and it dilates time or whatever to preserve that. I think that's a misdescription. What's happening fundamentally? And I think this is what is often missing in descriptions of special relativity. So at one level, you can just say, well, just follow what the equation say. Just take the Lorentz transformation and apply that and do the maths and it all works. It makes predictions, which will get to in a moment, which are validated, and that's all there is to it.

Don't ask any questions about what's sort of deeper than that, because there isn't anything deep in that. It's just a predictive theory. And that's an attitude that some people take, a lot of physicists to take that line. Personally, I think that there probably is more to it than that. It's predictive because it's getting at something that's sort of true about reality, and that's a whole question in philosophy of science that I don't want to get into here. But if you talk to different physicists about this, they will say slightly different things, depending on, hey, frame the question. So I'm just sort of mentioning that fact. but for those who might want something a little bit more,

that is why does it work this way? What's going on here? Why is the speed of light constant? Well, a good way to understand this is to think about it this way. The space time interval between any two events, again, events are just things that happen in the universe. The space time interval between any two events is always invariant. As I've just said, it's always the same. It's a constant for any initial reference frames. However, there are different ways of experiencing that spacetime interval. If you are moving very slowly relative to those two events that occur, then you will experience a long spatial interval and also a long temporal interval.

So think about traveling very slowly between two planets. Just say that planets are stationary with respect to each other just to make it easier. That's not essential, but it just helps the analogy. If I'm traveling very slowly, it takes me a long time to get there, and it's also a long distance. light years apart. So that's long spatial, long temporal distance. By contrast, if I move very rapidly with respect to those events, so instead of moving very slowly with respect to the planet, I move at nearly the speed of light with respect to those planets, then what I'll experience is a short

spatial and a short temporal interval. Obviously, I'll experience a shorter temporal interval because I'm traveling faster and I get there quicker, but also it turns out that the space between those planets is contracted according to length contraction, which we'll get to in a moment. The point here is that there's no correct or objective way to experience the interval, the space time interval between events. It purely depends on your relative velocity to those events. However, the magnitude of the space time interval itself is always fixed. So it's just a question as to how that's experienced, or if you prefer, because it doesn't require an observer, that's just one way to put it, it's how that space time interval is sort of parceled up. You can have the

space time interval with large time intervals and large space intervals or small time intervals and small space intervals. That author might say odd because you might think, well, wouldn't it be the other way around? Wouldn't it be like a seesaw if you have big space, wouldn't you have small time interval and then vice versa? No, and the reason for that is because of the minus sign in the space time interval equation. And that's very important. It's basically it's the time interval squared minus the space interval squared. And that minus sign is important. And it's, you know, it's a space interval squared. And It's why you have this strange phenomena whereby both of them can be small. So basically, if you subtract a small number from another small number, then you can get a small number and result.

Or you can have both of them big. So if you subtract a big number from a big number, you can still get a small number as a result. So it just depends on the difference between them, not the overall magnitudes. So this has a very interesting consequence, which I think is very important to understand in special relativity. And that is that in any reference frame that measures a long spatial interval between two events, is a long distance between them, that inertial reference frame will also measure a long temporal interval between those events. Whereas an inertial reference frame that measures a short spatial interval between events will also measure a short temporal interval between those events.

And that's required in order for the space-time interval to be invariant. So basically there's different ways of experiencing spacetime differences. The space-time interval is fixed and invariant, and that's just the way it is. That's the way space-time works, apparently. not obvious, but that's the way it works, and you can experience that interval differently depending on your relative velocity to those events. Now there's another way to think about this,

which doesn't contradict with anything I've said, but it's just a different way to think about it. And this is described in an excellent book called Relativity Visualized by Lewis Carroll Epstein. This is a classic, so if you're interested in more in relativity, I'd recommend it. It's not very technical, but it does explain it quite well. Here's what he says.

He says, and he describes this as a myth, So it's not literally science itself. It's a way of understanding or thinking about the science. Now, the way he says it is that everything is always moving at the speed of light in four-dimensional space time. So obviously everything's not always moving at the speed of light in space, because we know that the speed of light's the upper limit, and most things can't get to the speed of light. Only massless things can move at the speed of light.

But what he's saying is that in four dimensions, everything is moving at the speed of light. So basically the idea is that light moves at the speed of light entirely in the spatial direction. And for it, if you imagine sitting on a photon, is it travel through space, which is what Einstein imagined, sitting on a ray of light and thinking about what it would see, time is stationary for light. That is, light exists sort of at all parts of its temporal existence, that sort of, quote unquote, the same time, because it doesn't see any temporal intervals. It's all spatial for light. However, slow objects, you know, like planets and people and so forth, are almost stationary in the spatial dimension, because we're barely moving in space. But in time,

we're moving very rapidly. In fact, you could say we're moving at the speed of light in the time direction. And those are the extremes, not moving in space, but moving maximally in time, or not moving in time, like a photon, but moving maximally in space. But you can also have intermediate levels where you're traveling, you know, fairly fast, but not fully the speed of light. So you travel at some amount of speed in space and some amount in time. It's important to understand that when we say you might think the speed of light in time doesn't make a whole lot of sense, because how can you have a speed in time? Well, the way to understand this is in terms of thinking about how many units of one elaps for the unit of the other.

So that is, we normally define speed as distance over time, right? The amount of units of distance you travel per unit of time. But you can kind of, again, in a special relativity way, and sort of in a vague way as well, just sort of to understand it, you can kind of flip it around and ask how fast is time passing in terms of time relative to number of units of space that you've traveled? So photons don't move in time. That is, the amount of time that elapses for them relative to the distance that they move in that time is effectively zero. So just like an object that's not moving, that's not changing location in time is sort of stationary in space. A photon that's moving at the speed of light, but never really changes location, is stationary with respect to time. And so that's the sense in which we can talk about speeds.

It's sort of about the balance between how you traverse the four-dimensional. space time interval in terms of whether that's more in the space or more in the time directions or dimensions, maybe a better way to say it. Remember, of course, that there are three spatial dimensions. I'm just simplifying here, imagining if there's only one spatial dimension. So again, that's not exactly part of the physics. That's a interpretation or a myth, as he describes it, that Epstein offers in his relativity visualized book. And I think that is a good way to think about this, as if photons are sort of stationary in time, just as a sort of stationary object is stationary in space. And all objects sort of between, traveling at velocities between those

extremes of zero and sea have sort of intermediate velocity. So everything is sort of traveling at the speed of light, but they're just doing it in sort of different dimensions. Are you traveling more in the space dimension or more in the time dimension? And you can, you can change that sort of trade-off, but you can't ever slow down. As he says, you're always traveling at the speed of light. It just depends on sort of how you're directed. And this is another way of, I think, putting what I said before about the space-time interval between events is always fixed. You can just change how you experience it. You can experience it as short spatial and short temporal intervals or as long-spatial and long-temporal intervals. And whether you experience it in one

or the other depends on your relative velocity to the events in question that you're moving between. So hopefully those interpretive moves help for understanding what's actually happening here. fundamentally this all arises out of the way space and time are at the sort of most basic level. And they seem very strange by our standards because we don't experience space and time at the most fundamental level. We experience it in the ordinary everyday, from our point of view, you know, primates living at the bottom of a 10 kilometre thick atmosphere on a planet orbiting the sun. You know, at a very specific part of the universe with a very specific set of phenomena that we're familiar with. So that's why these things seem very strange to us. but they're not contradictory, and in fact that these results are, theories are very well confirmed

by a huge range of experiments. Special relativity is one of the best verified theories in all of physics. Now, at this point, I just want to go back to something that we discussed earlier, which is about how Maxwell's equations led Einstein to think about some of the initial ideas of special relativity, particularly the fact that there seems to be this strange way in which electric and magnetic fields can kind of vary depending on your point of view or the reference frame in question, you know, with the example of the Y that's moved towards the magnet, whether you consider a electrical or magnetic filter to be formed, depends upon which frame of reference you consider. And the way that this is currently understood is through what's called a covariant

formulation of classical electromagnetism. So it's really the same as Maxwell's equations and, you know, classical electromagnetism of the 19th century. It's just essentially written or formulated slightly differently. And the basic idea is that there is this construct, which is called the electromagnetic tensile, and without going to the details of the maths, it's basically like a four-by-four matrix, which describes the components both of electric and magnetic fields, and essentially, depending on your reference frame, or depending on the Lorentz transformation from one reference frame to another, one person will observe something exhibiting electric fields, another will observe it in terms of magnetic fields, and then some reference frames will be a combination of the two.

So this is essentially a generalization of Maxwell's equations that allows them to be, what I said, manifestly covariant, which means Maxwell's equations were always Lorentz invariant. That is, if you apply the Lorentz transformation formula to the equations and you work through the math, you get the same results. So in terms of the predictions about the fourth. that apply to the particles and the changes in velocities and accelerations and so on, every frame of reference agrees. The only area where they weren't invariant was in terms of whether it's described in terms of electric or magnetic forces. And this covariant formulation or manifestly covariant formulation of Maxwell's equations fills in that loophole.

Basically, it ensures that the description is always the same or consistent from one reference to another. And that's done through the electromagnetic tensile. and it actually reduces the number of Maxwell's equations from four to two as well, because now you can combine some of them together. Because basically there's no fundamental difference at a relativistic level between electric fields and magnetic fields. That is dependent on which reference frame you are. Everyone will agree on the accelerations and the forces

and the positions of the particles and then so forth, but different observers will disagree as to whether a given phenomenon is due to electric or magnetic forces. So that part of it is relative, but the electromagnetic, tensor and the forces and accelerations, that is invariant or Lorentz invariant, so the same for reference frames. So that is how special relativity is able to sort of resolve this paradox that led Einstein to start thinking about these questions in the first place. So originally I'd intended to do special relativity as a single episode, but it seems that

there's still a lot more to talk about. So what I'm going to do is divide this up into two episodes. So next episode we'll talk about the key phenomena that special relativity gives rise to including length, contraction, time dilation, and mass energy equivalence, as well as looking at some of the key paradoxes in relativity, including the Twins Paradox. So look forward to that next time, but for the moment I'll leave it here for this episode. So thanks very much for listening. If you want to get in touch, you can email me at Fods12.g.com. That's FODS1.2 at gmail.com.

Also, if you'd like to support the show, you can give me a donation via PayPal, or you can become a patron-supporter. Google Science of Everything podcast, Patreon, you'll be able to find that, or you should be able to see a link in the show notes. I really appreciate everyone who's made a donation in one form or other. It does help me to spend more time on the show, so that is immensely appreciated. That's all for now, though, so thanks for listening, and I'll talk to you next time.