Daniel and Kelly’s Extraordinary Universe - What is a space-time interval?

Episode Date: July 28, 2026

Daniel and Kelly break space, twist time, and then put it back together to explain the most important concept in relativity you've never heard of.See omnystudio.com/listener for privacy information.

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thinking that. Listen to Podlandia, AEO rewatch on the IHeartRadio app, Apple Podcasts, or wherever you get your podcasts. Special Relativity brings space and time together and in the darkness binds them into space time. That sounds pretty awesome and it was fun to say, like something that could only happen in Mordor or during the Big Bang. But what does it actually mean? What is space time? How do you measure distance? in four-dimensional space-time, and why would you? Physicists are always saying that space and time are like two sides of the same coin, making more sense together. What are they talking about?

In what way do they make more sense together? Today, we're going to blow up all of those analogies and give you a real conceptual understanding of how relativity breaks space and twists time, but then clicks them back together into a harmonious hole, We'll talk about what that really means and the crucial, but rarely explained concept at the heart of it all, the spacetime interval. Welcome to Daniel and Kelly's spacey, timey universe. I study parasites and space, and I have fun in all spaces and at all times. Hi, I'm Daniel. I'm a particle physicist who likes to think about aliens, and I was having fun this morning until Kelly showed me the crusher.

Oh, I wasn't sure you were going to mention the Crusher on the show. It's the only thing I can think about now. Thank you very much. Kelly, tell everybody, what does the Crusher crush? The tube that goes to the testicle and connects it to the rest of the body, I am looking into methods for what will say weathering our male goats. And I'm trying to figure out what method I would feel comfortable doing or if I need to call the vets. And wow, the crusher is a big piece of equipment that once I have it in my hands, I am not sure that I am up for actually using.

But you were up for brandishing it on our video call this morning with glee. Oh, I mean, this thing is huge. Everybody needs to see it. Oh, stop it. Put it away. Put it away. The crusher. All right.

But today we are not crushing goats or any of their bits. We are crushing your confusion. We are explaining the way the universe works. You always pull us back somehow. Okay. So today we're talking about space time. Yes. Which got me thinking about all of space and all of time.

And so, Daniel, if you could snap your fingers and for five minutes, you would be in a protective bubble and you could go anywhere in space. But you couldn't change the time. Like you had to just pop up right there. Now, what would you want to see? Kelly, you need this episode because if you had heard this episode already, you would know that that question is incoherent because there is no universal now. That's right. Right?

There's no sense of time across the whole universe where you can say this now is the same as that now. It's observer dependent as we're going to talk about. And you know what? I should have known that from our what is time and what is space episode series. But okay, so is there some way to constrain it so I can say like you can't go back to the beginning of the universe? I was trying to like. Why do you want to stop me from going back to the beginning of the universe?

That's what I thought you'd say. totally right. That is what I was going to say. But you've said that on the show before. So I'm trying to figure out what you think is the most interesting thing happening now. And my guess is that you were going to say black holes. I'm going to go to the factory that makes the crusher and I'm going to burn it down. Okay. No, but if I could go somewhere in the modern universe, then for sure I would like to see not only what's inside a black hole, but the immediate vicinity of a black hole would be super fascinating.

to see frame dragging and to see all that time dilation and all the extreme events around a black hole. I think my second choice would be the heart of a neutron star because when every other defense against gravity has already collapsed and matter becomes insanely dense so that protons and electrons eat each other and become neutrons, then you have this fascinating environment where quantum mechanics and gravity are at about the same strength. and we don't know what's going on at the heart of neutron stars because we can't calculate that stuff because we don't have a theory of quantum gravity. We'll never go inside a black hole. So that's sort of just like fun to say but hopeless. But the inside of a neutron star is out there and real and on the outside of event horizon. So in principle, we could one day see it. And that could be the key to unlocking quantum gravity.

Whoa. Okay. So how could I have worded the question to constrain your ability to see the Big Bang? What would have been the right way to word that? Daniel, what would you like to see in the universe except for the Big Bang? Oh, okay. I don't know. Wouldn't you have to go backwards in time to see the Big Bang?

It depends a little bit of what you mean by C. I mean, the information from the Big Bang is out there. In some respects, people often say seeing the C&B is like seeing the Big Bang, though it actually comes from 380,000 years later. There is information from the Big Bang out there in gravitational waves. And so, like, seeing it means detecting those and reconstructing what happened. We can do that today without, like, being there during the Big Bang. In that sense, everything you see happened in the past already.

But I feel like you understand that that's not what I mean. Like, yes, you can see evidence from it. Yes. But, like, I don't, you know, so if I go to a civil war site and I find a, like, bullet case, that's, like, evidence that it happened. Yeah. But it's not the same as being there and watching both sides run across a field and start fighting each other, right? No, but it's sort of like going really, really far away instantaneously in capturing the light that left that Civil War battlefield contemporaneously and using it to watch in real time the battle, which is something you could literally do, right?

All right, all right, all right. Well, I'm going to have to be more careful next time I ask you a question. But we've all learned something from Kelly's bad question. which is why Kelly's here. And these are really fun topics to think about because our intuition is broken about how all of this stuff works. We tend to think that time flows the same way for everyone. We tend to think that distances are real and they're universal. When it turns out the universe operates in a way that's fundamentally in conflict with our intuition,

but amazingly does make sense. So we have this task ahead of us to incorporate this new intuition about the universe. universe into our minds. But often this means just like read a bunch of equations and use them to predict what happens. But I think it's possible to develop a new kind of intuition, a conceptual understanding for how the universe really does work. And that's what we're going to try to do today by introducing you to one of the most important concepts in special relativity that almost nobody talks about and that we've never covered on this show. What? What? Yes. Well, this is a very exciting day here at DK.EU. And let's see if our audience,

heard about this thing that pretty much no one talks about. And so we asked our audience, what is a space-time interval? Well, I would expect that that would be the reduction of Einstein's space-time into a quantum-type interval, but I'm really not too sure. It's the time between two objects and space, maybe? I don't know. Could a space-time interval be the point? where one max plank length ends and one begins? I'm not sure. A space-time interval is the required time frame between semi-convincing scientific mumbo jumbo that you hear on sci-fi shows and movies. But seriously, maybe it's some form of the smallest unit of space time, length, and distance, kind of almost a

singularity but not quite. I've never actually heard the term space-time interval before, but guessing from context, I would imagine a space time interval is one frame of the universe. So, you know, you move forward one unit of time and look at the way everything else changed. That's one interval of space time. All right, Daniel, I asked a question that belied my ignorance. And so I'm not going to even attempt to say if I think the audience was correct or not with this question. How close were they? Pretty much zero points all around. A few points for people thinking this is a kind of distance because that's the closest. It has nothing to do with the plank length or frames of the universe or minimum distances or

anything like that. It's really an analog to what we think of as distance, but now in four-dimensional space time with an uncomfortable twist in it that we're going to get to. Good news, everybody. We all need this episode. So here we go. And we are going to start from the beginning because clearly Kelly needs to go back to first principles. So what I want you to explain to me, Daniel, so space, time sounds like space and time and distance. And there's like, so what is distance? Let's start there. Yeah. Let's build our intuition with stuff that makes sense to us. And let's start easy. Let's start in one dimension, right? So imagine, for example, you have. the pencil and you have a ruler and you want to know how long is my pencil what do you do well you put it along the ruler you're operating in one dimension right so the pencil and the ruler are lined up and you say well the back of the pencil you know is at two centimeters on the ruler and the front of the pencil is at eight centimeters on the ruler and then you subtract he's like eight minus two

six centimeters long i don't know why you wouldn't just put it at the beginning of the ruler and then you don't have to do any subtraction exactly you could put it at the beginning of the ruler so that you get zero and six. But what happens? You get the same answer, right? If you put the pencil at the beginning or if you slide it along, it doesn't matter. The answer you get does not depend

on where on the ruler your pencil is, right? It's invariant. Okay. That's cool. That all makes sense. It makes so much sense. It's kind of blindingly obvious, which is where we want to begin, right?

With like firmly grounded in our intuition. This is very comforting, unlike the crusher. Okay. So why do you keep bringing that up, Kelly. You just like, really want to throw me off. I wasn't going to mention it on this episode, but you brought it up. Because you did the whole crushing thing on video before the episode, and now there's nothing else in my mind. I'm trying to get space time in my brain, but that's all I can see. I'm so sorry. I'm sorry. It's fine. All right. So now let's go to two dimensions. How do things change?

Well, imagine you have like a piece of graph paper. The little squares on the graph paper are one centimeter by one centimeter. and you drop your pencil onto the graph paper and ask you, how long is your pencil? So how do you figure out how far it is from one side of the pencil to the other? Pythagorean theorem! Yes, exactly. So you say, well, how many squares does a cross in one direction and how many squares in the other direction? And now you have a right triangle where the pencil is the hypotenuse, right? It's the long part of the right triangle. And we can calculate one side, because it's just counting squares. We can calculate the length of the length.

the other side, just counting squares. And then the diagonal, which is the length of the pencil, that's the Pythagorean theorem, right? So A squared plus B squared equals C squared, boom, we have an answer. And just like in the one-dimensional case, it doesn't matter where the pencil was on the graph paper. If you dropped it in a different direction, you might have gotten more squares on one side and fewer on the other, but the answer wouldn't change. Or even if by chance you happen to drop the pencil so it lined up perfectly in one direction, right, like along the x-axis conceptually. Then it's basically back to the one-dimensional problem or the other axis. The point is, in this picture, distance is real.

It's invariant. It does not depend on where you drop the pencil or like the size of these squares. If you made them two centimeters or five centimeters or like a furlong or whatever. All those are just arbitrary choices humans have imposed. The pencil distance is the real thing. Right? Yes, this is very comforting. Can we stay here for the rest of the episode?

No, let's push our boundaries. Let's keep going. All right. And so now it's easy to imagine how we might extend this to three dimensions or really any number of dimensions. Like if you have your pencil and it's not on the paper, but it's like floating in space. We're on the space station now, right? And the pencil is floating in space.

And you imagine like not graph paper, but like laser lines through space. And you want to measure how long is it from one edge of my pencil to the other? Well, you play the same game, but now you have three directions of counting squares. And you can still do Pythagorean theorem. It's just A squared plus B squared plus C squared equals D squared, right? It's the same game fundamentally that the geometry has not changed because the nature of these dimensions are all the same. And so we can just keep doing this. And if you wanted to work in 11-dimensional space or 26-dimensional space, the concept of distance wouldn't change mathematically,

or intuitively because the distance is the real thing, right? Everything else is just bookkeeping. Okay, so I'm trying to imagine a pencil floating in space, and now A squared plus B squared plus C squared equals the length of the pencil squared. Yeah. My gut feels like the more dimensions you add, the bigger the pencil's going to get because you've got all these extra squared terms that are going to keep getting added. But that's not true because why?

Yeah, that's not true because. the pencil is a certain length. And if you add squares in one of the dimensions, the other ones will shrink. Just like if the pencil falls to the floor and it's now on a two-dimensional surface, you only have two contributions. Or the third one is just zero, right? And those are longer than if it's floating up above the graph and it gets a third dimension, the first two dimensions are going to shrink a little bit. So they all act in harmony to make sure the distance never changes. You like flip a pencil from one astronaut to another, so it's like rotating through space, you know, your intuition tells you the length of the pencil is not changing. The difference between its front

and its back along some axis might change, right? It might go up and down and be negative or whatever, but the length of the pencil doesn't change because all three dimensions are working in harmony to contribute various pieces to the length, and it always adds up to the same number, because that's the physical thing. Everything else is just bookkeeping. It wouldn't make it. It wouldn't make sense if the length of the pencil depended on like where the graph paper was, right? Or which rule are you used? It's a real thing in the universe. All right. But I bet that if I were to say, Daniel, you can travel a billion, a quadrillion miles away in any direction in space. What is it that you want to see within that area? Everybody should know what I'm talking about. Are you going

to tell me that something about that doesn't work like the pencil? I see. You're asking me to be the physics wet blanket here. I mean, we have to take turns there, right? Well, if you said I could go a quadrillion miles with respect to Earth, then you've given me a frame of reference in which that is well defined. Okay. So, yeah, I could totally do that. All right, that's what I should have asked. Anyway, all right, moving on.

And so we have three dimensions of space. We're comfortable with the idea of distance in space. Cool. Let's put that aside for now and think about time. Time is one dimensional, right? We don't have like two clocks going on. We don't think so. It's actually a super fun theory we can explore.

And we did an episode a couple of years ago about two dimensions of time. But for now, let's just think about one dimension of time, right? And so, for example, you can click your stopwatch when this episode starts and when the episode ends. And you can ask how long is this episode of the pod? And you can look at the difference and that's the length, right? Totally obvious, basic subtraction. Perfect. Yes, for our poor audio guy, it probably feels like an hour is infinite, but it is still just an hour.

Exactly. And in the same way, it doesn't actually matter when you hit the stopwatch to start as long as you noted when the episode started. If you'd accidentally hit your stopwatch like an hour before, as long as you wrote down, oh, the episode started at one hour in, you could still measure the length of the episode by looking at the difference, right? Because when you started it is arbitrary. what you're interested in is the physical thing, which is the difference between the start and the end of the episode. So intervals and time are also physical in that sense, right?

They're real. It's not just bookkeeping. What does physical mean in this case? Like, physical, I think of as something I can touch. But you mean physics all. I mean it's a part of the universe. It's not part of our measurement system. And it's invariant in that way. And here so far, all these definitions are in a classical world.

right, before relativity. This is Newton's idea about how time and distance worked, right? That anybody could measure the difference between two things that happened in the universe and the time difference or the space difference, and they would always get the same answer. That's the way Newton thought the universe worked, and that's the way I operate and you operate. And this whole segment of the pod has been like, let's confirm our intuition about how distances work. And it's all very comforting until we learn that it's all wrong. Oh, no, you pulled the soft blanket of Newtonian physics off. Yeah, exactly.

And now it's cold and unwelcoming. Let's take a break before we have to sit with our discomfort for too long. And when we get back, we'll hear what Einstein had to add. Hey, I'm Ruby Carr, the host of the podcast, Encore. Check out our brand new episodes featuring music for. From the show that everyone is reheating as we speak, heated rivalry. Join me as I go behind the songs that brought Shane and Elia together. I'll tell you the stories of Fice, My Moon, My Man, Wolf forades, I'll Believe in Anything,

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We've had seasons that have shaken everything. thing. The question is, what remains when the shaking stops? Hi, I'm Gary Valenzano, and on my podcast, Unshaken with Gary Vee, I sit down with amazing personalities and we talk about the seasons that nearly broke them, the lows that carried them to their highs. Listen to Unshaken with Gary V on the IHeart Radio app, Apple Podcasts, or wherever you get your podcasts. This is Michael Rappaport, and my podcast, the I Am Rapaport Stereo podcast, is unlike anyone you've ever heard. We're a variety show, and if you're looking for strong opinions, funny opinions about sports, entertainment, politics, pop culture, and whatever else catches my attention, then subscribe now. This kid, Jafar Jackson, is as good as Rami Malik as Freddie Mercury, and it's as good as Timothy Shammalay as, as

Bob Dylan, and I say that with love and respect for both of those actors. And I don't know how many Oscar nominations they give out, I don't know if it's five, six for best actor. 150% this kid Jafar Jackson should absolutely positively get nominated for his portrayal as Michael Jackson. Listen to I Am Rap Report on the I Heart Radio app, Apple Podcast, or wherever you get your podcast. And we're back. We had a very comforting discussion about distance and time where everything felt like it made sense. And now we are at the point of the episode where we are going to expand our brains and learn something new. So, all right, Daniel, make us uncomfortable.

All right. So not crusher level uncomfortable. Well, in 1905, Einstein pulled out the metaphorical crusher. And he said to Newton, he said, you know, put your theories in this, buddy, because here I come. It took me a second. Sorry for the delay. I was drinking water. It almost totally destroyed my computer. All right, go on. And these intuitive notions of how the universe works, they do work for us at our basic level. But Einstein discovered that when speeds get high, relative speeds get high, then they don't work. The universe is broken. It actually works in a fundamentally different way.

And it's important to understand where this comes from. It all comes from the same source, which is the fact that the speed of light is constant for all observers. That is at the heart of everything. If you accept that the speed of light is the same number, no matter who measures it, no matter how fast they are moving with respect to like the flashlight that gave off that beam, then you'll see that everything breaks. And we're going to walk you through that and how it crushes your intuition. All right. Crush away. So imagine, here's the setup.

Kelly gives two flashlights to her goats and puts those goats in a crate and puts that crate on a self-driving tractor to the vet to, you know, have their problems taken care of with great anesthetic, of course, and very humanely. Good, good. All right. And so they have two flashlights and one is pointed towards the front of the crate and one is pointed towards the back of the crate. Okay. And they turn the flashlights on at the same moment. So they click both buttons. Okay.

And then we're interested in like, well, how long does it take for that beam of light to hit the front of the crate versus the back of the crate? So what would you say, Kelly? Ghost are in the center of the crate. They turn on both flashlights at the same time. Do those beams reach the front and the back of the crate at the same time? Yes or no? I feel like they should, but it's moving forward, right? And so does that mess up the answer?

That's very good intuition. But we're asking right now from the goat's point of view, right? So yes, the goat is in the crate and it's on the back of the self-driving tractor and it's driving itself towards the vet. But the goat doesn't know how fast the tractor is moving relative to you. You're outside the tractor. It's inside the crate and the crate is not moving with respect to the goat. Okay, so it reaches the front and the back at the same time. Yes, you're exactly right.

The goat says that the light reaches the front and the back at the same time. And that makes perfect sense because it's going at light speed forward and light speed backwards. and it hits the front and the back, which are the same distance away, in the same time. Distance is rate times time. Very simple, right? Nothing complicated. That's right.

All right. But Kelly is also watching. She sent the goats on this self-driving tractor, and she doesn't want them to get hurt except maybe get crushed when they get there. But she's paying attention to what happened. And for her, unlike the goats who see the walls of the crate at rest, for Kelly, the crate is moving forward. And she says, hold on a sense.

second, I also see light traveling at the speed of light forwards and at the speed of light backwards, but I see the front of the crate is moving forward from the point where the light was emanated. So it's moving away from the source at the same time as the light is chasing it, so it takes longer. In the back of the crate, you see moving towards where the light was emitted. So that distance is effectively shorter. And so you say, no, no, no, the light hits the back of the crate first and the front of the crate second. And then I look at it. at the goats and I say, you guys are bad at this. And meanwhile, Zach, who's in his Ferrari, is zipping by at 100 miles an hour to do who knows what. And he says, I see the opposite, right?

Because from his point of view, the crate is going backwards. And so he says, no, no, the back of the crate is running away from the light and the front of the crate is moving towards the light. So Zach sees the front of the crate get hit before the back of the crate. Ever since that lucrative Dorito sponsorship, he's been able to, Ford fast cars. Exactly. And so even though he's got one hand in the Doritos bag and one hand on the wheel, he's able to make these measurements.

And so what's going on here? The issue comes down to the fact that everybody, the goat, Zach, and Kelly, all see light moving at the same speed no matter what. If you replicated this experiment with sound instead of with light, then everybody would agree on what happened. Why is that? Because the speed of sound is not the same for all of them. observers. Like, if the ghosts are bleeding, if they're going, bha, inside the crate, then they're going to

see their sound hit the front and the back at the same time. Kelly, who sees the crate moving, is going to see the sound going forward faster than it's going backwards, because the sound is moving with the same speed with respect to the air in the crate. And so, like, if Zach has a speed relative to that air or Kelly has a speed relative to that air, then they see the sound moving at different speeds, right? So the front and the back move at different speeds, which accounts for everything and closes the loop. So because light moves at the same speed with respect to all observers, everybody sees light moving forwards or backwards at the same speed, even if they have velocity relative to the goats, then that's why people see things happen in different orders, right?

That's why they can't agree on whether the light hits the front or the back of the crate at the same time. So the takeaway here is the fact that the speed of light is constant for everybody breaks this notion of simultaneity, that if two things happen at the same time for Kelly, they also happen at the same time for the goats, and they happen at the same time for Zach or for anybody else paying attention. That's no longer true because of this one crucial fact that the speed of light is the same for all observers. I don't like it, but I accept it. It's confusing. It's not confusing. You explained it clearly. It remains sort of counterintuitive. It is. But I followed that explanation.

I'm with you. It is. And it's somehow reassuring if you know where it comes from. Like it comes from the speed of light fact. It's not just some like arbitrary thing about the universe. In fact, in order for the universe to make sense to accommodate this speed of light being universal, time has to not be universal for everybody. It doesn't make sense for time to be universal.

And this is why we have things like time dilation. right? When we say moving clocks run slow, it's really another statement of the same thing. If I see somebody in a spaceship and they have a clock and they're moving at half the speed of light relative to me, I will see their clock ticking slowly and they will see my clock ticking slowly. And all of the surprising counterintuitive facts about relativity come from this one fact that the speed of light is constant for all observers, which means simultaneity is not universal, which means clocks run slow or fast depending on your speed. speed relative to them. And so what that means is what we said earlier about how everybody will agree about how long the podcast is. That is not true. The distance in time between two events is not universal. And it's not just because somebody's listening to the pod at 1.5x or whatever, or at half X because we talk too fast or whatever. It's because for them, clocks run faster or

slower depending on their velocity relative to them. Okay. So it can be nearly infinite for Matt when he's editing our episodes. Depending on his velocity with respect to the equipment. Yeah, exactly. So we've gone over all of this in the past, which I'm, and it is always useful to go over this again, especially when your co-host is as smart as a rock. But when do we get to how this is different? So remember at the end of the last segment, we were saying distance and time,

classical physics are physical. And you were like, what is physical mean? I was saying it's a part of the universe. It doesn't depend on. where your graph paper is or who's doing the measurement. It's a real thing everybody should agree on. Now we're saying that's not true, right? Time is not universal.

It's not physical. It's observer dependent. It does depend on where you put the graph paper. It does depend on how fast the graph paper is going. So now we're sort of unmoored. We're like, oh, no, what's real out there in the universe? Daniel was gaslighting me. So what we're going to get to is that time is broken and space is broken, but we can put

them back together. Oh, good. And there is a concept of distance in space time, which is called the space time interval, which is physical, which is real, which everybody can agree about. So don't worry, we're going to get there. Yes. We've just broken time, and we're going to use that to break space, and then we're going to put them back together with super glue.

Awesome. All right. Now let's talk about space then. Yeah. So remember we were saying earlier, like the length of the pencil, that's physical. It doesn't matter if it's rotating. It doesn't matter if your graph is moving. It doesn't matter what rule you use.

it is a number and everybody should agree on it, right? Well, it turns out that's not true, and it's for the same fundamental reason that the speed of light is universal, which breaks time, which is then going to break distance. Because think about how we measure distance. How do we measure distance is you measure where the back end of the pencil is and you measure where the front end of the pencil is and then you subtract. Very basic, right? How could that go wrong? How could that depend on clocks in any way? Well, implicit in that definition is I'm measuring the back and the front end of the pencil at the same time because if the pencil is moving and I measure the back into the pencil now and the front end of the pencil later, then I'm going to get two numbers that

are much further apart and the pencil is going to look too long, right? And so I have to measure the front and the back at the same time. And you can't cheat by saying, I'm in the tractor, the tractor's moving forward, but I'm going to take a picture of the pencil so that I can measure from the front and the back at the same time. Now you've got a different frame of reference because it's a picture and that's cheating. Is that right? Okay, yeah, great question.

And I love the idea of using a camera. The camera actually makes this more complicated instead of simpler because now you have the additional issue of how long did it take the information to arrive from the back or the front of the pencil to the camera. And if your camera's not like centered along the pencil,

then you're going to get like out of date information in one direction. And usually when we talk about special Relativity, we try to remove all of those effects, the effect of light traveling from the event to the observer, because that's a completely separate, different kind of effect. And so all the things we're talking about today assume that basically you have an infinite number of observers at every location who can always measure things locally. So there's no time delay due to the speed of light.

So the camera is really tempting, but it actually adds a bunch of confusion. And we can dig into special relativity, including those time delays, but it's sort of like a different rabbit hole. All right, different episode. So let's instead say Kelly has two goats and she can measure the front and back of the pencil with her two goat assistants. And they record, you know, where on the ruler it was. And then she subtracts it and she gets a number, right? Thanks, shortbread and numino.

That's right. But Zach is watching from his Ferrari and he says, look, one goat was off. Your goats didn't measure the front and the back at the same time. Because remember, Zach doesn't think two things happening at the same time for Kelly or happening at the same time for him. him. Typical. So see how the speed of light being constant breaks the universality of simultaneous, which breaks the universality of distance, right? People disagree about when you measure the front and when you measure the back, and that changes your measured distance of these things. To give just one more simple example, go back to the pencil. If you measured the back of the pencil

to be at two centimeters and the front to be at eight centimeters, so you said it was six centimeters long, 8 minus 2. But what if the pencil was moving and you had measured the front before you measured the back, not at the same time? Then the reason the pencil is at 8 centimeters when you measured it is because it moved. So because different people disagree about what at the same time means, because speed breaks simultaneity, then people disagree about when to measure the front and the back. And so they disagree about length, because length depends on simultaneity. Hamilton 80. That's why breaking time also breaks distance. All right. So we can't depend on time. We can't depend on distance. Yeah. I'm with you. So all of these things are now broken, right?

Einstein revealed that time is not universal. Space is not universal. So the comfortable footing we were on a few minutes ago seems to have evaporated behind us, right? And so who comes to the rescue, but math profs. Daniel. Oh. Math props. All right, none of us come to DKEU to be comfortable. We come here to be challenged. So tell me what the math prof had to say. So this is after Einstein had introduced special relativity, which showed us that distance is not universal, that time is not universal, right? And so then Herman Minkowski, who's famous for introducing this concept of Minkowski space, and he was a math professor. He actually taught Einstein. And Einstein famously skipped some of his lectures. And Minkowski thought Einstein was a quote, lazy dog. Anyway. So Minkowski gives this talk a few years later, and he starts off with this famous line saying, quote, henceforth, space by itself and time by itself are doomed to fade away into

mere shadows, and only a kind of union of the two will preserve an independent reality. So it's a famous line because what he's saying is space is not real, time is not real, themselves. But you know what we can do is we can link them together and build something new, which is physical, which is real, which we can count on. I think 99% of us can work without space time, but I'm glad the physicists have space time. It's sort of like if you're studying elephants, you know, and elephants are lined up so you can either see the head or the tail, right? And you're like measure the number of heads and you measure the number of elephants' tails. And you have no idea that heads and tails are connected. And you just, you're just, you're

like noting, okay, today I see 90 heads and 10 tails. You come back tomorrow and you see 60 tails and 40 heads and you're like, what is going on? The number of elephant heads is changing. The number of elephant tails is changing. What is going on with these elephant heads and tails? And then somebody comes along and tells you heads and tails, neither of these are fundamental. They're both parts of the same thing. And what's really constant here is the number of elephants, the sum of the heads and the tails of the elephants. They just turn around and so you can measure are different numbers of heads or tails that you can see, but that doesn't change the number of elephants. That's what's physical, what's real. And that's what Minkowski did. He said, let me bring

space and time together. The reason space looks broken and the reason time looks broken is that you're just looking at one part of a larger object that's rotating. So as you see space changing, time is changing in a way that's compensating for space changing. You're just not paying attention to it. It's sort of like your question earlier about the pencil and three-dimensional space, right? You're like, if the pencil floats up from the graph paper and now it has three dimensions instead of two, how do I know that those two dimensions are compensating correctly to make sure the distance is the same? A two-dimensional physicist who only could see like two of those dimensions as the pencil floats up, he would say, no, the pencil is getting shorter. You'd be like,

that's just because you forgot to account for the third dimension, right? The pencil is the same length. That's what's happening here. Space may be getting weird and short, but time. is out there compensating for it. So if we put space and time together in some way, then we can build a new sense of distance, which really is universal, which really is invariant, which really is physical. Well, one, I love when you re-contextualize my dumb questions to make them seem insightful, much appreciated. And two, let's take some time to think about space, and we're going to take a break. And when we get back, Daniel is going to really dig in to what a space time interval is. Hey, I'm Ruby Carr, the host of the podcast, Encore.

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So space and time makes sense for all of us again. All right. So we are building space time. We have three dimensions. We want to add time to it. Earlier I said it's no big deal to add more. mentions you can just add them to our Pythagorean theorem, right? You have X squared plus Y squared.

No problem. We can add Z squared. It's all cozy. It's all happy. It's easy. So probably you're thinking, great. Now we just have to add time. We have X squared plus Y squared plus Z squared plus T squared, right?

Yes. Minkowski says, no. That's wrong. We do have to account for time, but time enters with a negative. What? So the concept of a space time interval is distance. in space minus the distance in time. It's X squared plus Y squared plus Z squared minus T squared.

What? This weird negative sign. Yes, exactly. All right. So I'm thinking about the lights in the box. I'm thinking about my goats in the box. Does subtracting time depend on like what distance you're moving in? Because like I'm imagining, you know, when we measured the pencil and the distance was shorter

because you had been moving in a certain direction. Does that have anything to do with why time is negative? or is my intuition, yet again, totally off here? No, your intuition is right on. It all comes from the same origin of how space and time connect to each other, which is fundamentally because of how the speed of light is invariant for all observers. But there's a little bit of a twist here. What we're building, this space time interval,

is not exactly like a four-dimensional analog of distance, right? It's not like how far apart these two points. are. So let's make that explicit. Let's say, what are we talking about here? What is it, space time interval? Imagine you have some event. So a goat is born. That happens at a location and a time. Yay, we all celebrate. Then you have another event. A goat is crushed, right? That happens at some location and at some time, right? And we can talk about the distance between those two events, right? Maybe the goat was born on Kelly's farm and the goat was crushed at the vet. There's a certain distance there, and we can talk about the time interval between those two, right? The goat was

born, and three weeks later, it was crushed or whatever. Now we can also talk about the space time interval between those two events, those space time events and now four-dimensional coordinates in space time. We can talk about the space time interval, and that has the distance component and the time component, but in a negative, right? So the time adds in a negative. It's not just like a straight up distance. And you might think, well, why would you do that? Why would you add the negative? That doesn't give me some concept of distance. Now it's like space and time playing off each other, right? Well, the reason is that that's the physical thing. That's what the universe wants. That's the thing that's invariant. If I calculate the distance in space and I calculate the distance in time from my point of view,

I'm going to get a number for the interval. If Kelly does it, she's going to get the same number. If Zach and his Ferrari does it, he's going to get the same number. We're all going to disagree on distances. We're all going to disagree on times. But when we put them together into the space time interval, we're all going to get the same number. The same way that the pencil has a length in space, it doesn't matter, right, who's measuring it or how it's rotating or whatever. The space time interval is the physical thing in space time.

It's what the universe respects. So us little humans, we want senses of distance. And we'd love to add time to space with a positive number. So we have like a four-dimensional Pythagorean theorem. But no, the universe says this is the thing that's important. Is it reasonable to think about it as like distance standardized by time or not quite? I feel like the answer is not quite. Right.

So we've answered one question, which is like, what is the space time interval? It's this weird mathematical construction where you have space and time playing against each other. And we've said it's important because the universe respects it. Now the question is like, well, what is this thing? Right? Like, what do I do with this thing? How do I think about it? So it's not a distance, again, because time is playing in the interval.

The way to think about it is causal separation, all right? Because there are three different kinds of space time intervals. You can have a space time interval that's zero, which is what we call light-like. And that's when the space and the time balance each other. So like the distance minus the time, because, you know, you have a positive and negative. they can be zero. Two events have a space time interval of zero if light can leave one and arrive at the other. So, for example, Kelly turns on a flashlight and then the flashlight arrives at the end of a barn, those two events have a zero space time interval because they're exactly causally connected, right?

The interval between them is exactly how long it would take light to go from one to the other. So events have zero space time intervals if they are perfectly causally. closely linked, just enough time and not too much. A message departing and arriving are two spacetime events with zero spacetime interval. They have a distance in space and a distance in time, but those balance to give no interval. Events that have a zero spacetime interval from you are events that you can exactly reach with your beam of light. Close by things in the near future have a zero space time interval, far away things in the far future. It's basically your forward light cone. Okay. You can also have positive space time intervals, right? So for example,

Andromeda right now is too far away for us to affect it causally. Kelly turns on her death ray from her farm. She cannot zap in the aliens in Andromeda right now because the death ray is still traveling there. So the distance between Kelly's farm and Andromeda has a positive. space time interval. It's outside the realm of anything she can influence because her death ray, which moves at the speed of light, can't reach it until the future. Even moving at the speed of light is not fast enough. It's just too far away. But so when I'm standing in a barn, I turn on my flashlight, and the light hits the opposite wall. That happens in the future too, but it's just a shorter future. It happens in the future, but you're right, it's a shorter future. And so you can imagine,

we talk sometimes about a light cone, right? Think about the future that. that you can influence, okay? You can influence exactly where you are right now, or you can influence things that are one light second away in one light second, or things that are one light year away in a year, right? That's your future light cone. That light cone has space-time interval of zero.

Because the speed of light is the same for all observers. Everybody agrees on what Kelly's light cone is. You turn on the flashlight, and it does hit the side of the barn, And that event where the light reaches the other side of the barn is on your future light cone. It's in your causal reach. It has zero space time interval. And an event much further away at that same moment is outside your light cone. You can't affect it.

Light doesn't have time to reach it. It has a positive space time interval. And everyone agrees on that. Unlike distance and time, the space time interval is the invariant thing. Can Kelly zap this alien and when is a number everybody agrees on no matter where they are in the universe or how fast they're going. Everything outside that cone, Kelly cannot zap. It's beyond her causal reach.

Everything within that cone has a negative space time interval. So that's what the space time interval is about. It's about what can you influence? Where is your causal reach? Space time intervals that are negative are time-like, things you can reach. Space time intervals that are zero. zero are where light will reach and space time intervals that are positive are outside of your reach. You cannot influence them because light cannot get from you to them in time. So it's sort of

like a sense of distance, but it's more like a sense of causal distance. It's like, is this something I can influence in the future? I think I'm still a little caught up on like, so if I turn on a laser, it's going to get to an endromeda eventually. Yes. And so why can't we just wait more time. You can wait more time, absolutely. But you have to wait more time to get further. And so you can't zap Andromeda now, you have to zap it in the future. Just like you can't zap the other side of your barn.

Now you have to wait for light to get there. So there's no limit on how far you can zap, but there is a limit on how far you can zap in a given time in a second or in 10 seconds or whatever. If the universe wasn't expanding, then the whole universe is eventually in your light cone, your future light cone, right? You could zap anybody. It just takes time for those death rays to get there. But everywhere along the bleeding edge of your death rays has the same space time interval.

That's zero because the distance and the time portions balance perfectly. Because we're adding up distance and we're subtracting time. And the space time conversion we use to subtract time is the speed of light that enters into that equation. And so the space time interval between two events basically tells you, could light have gotten here between these two events or not? If it's zero, then light has just enough time to get there. If it's positive, then it's too far for light to get to. You can't influence it. If it's negative, then light can definitely influence it.

So is Andromeda moving away from me faster than my laser is going? Andromeda is actually moving towards us, right, because of gravity. So it's going to collide with us in a few years. But that's what I didn't. A few billion years. I say a few years. We got time. Don't worry about it.

And please don't build a laser and point it at the aliens. We hope the aliens are friendly. I'm just using that as an example. Yeah, yeah. And this isn't about Indromeda moving. It's not that it's escaping your laser. It's just that it's too far away for you to hit it now or in a year. Your laser won't get there for millions of years.

And so the events your laser can visit are only in the future. those are the ones with space time interval zero. Okay, so I get one spacetime value for shining a laser at my barn and a different space time interval value by shining a laser at Andromeda. And I'm still not understanding why they have different signs. I can't remember if one was zero, one was positive. I've lost track. But like, why are those different?

Okay, well, let's stick with your barn for now and then we'll get to Andromeda. Okay. So if you turn your flashlight on now and you shine it at the other wall, right, then you can ask, what is the space time interval between me turning on my light and the other side of my barn right now before the light has a chance to get there? And that is a positive space time interval because the light has not had a chance to get to the other side of the barn. Okay. Right. But if you ask, what is the space time interval between me turning on my flashlight and the other side of my barn in,

one millisecond, right, or however long it takes light to get there, that has a zero space time interval. So there's two different spacetime intervals there. There's the one between your flashlight now and the other side of the barn now, or between your flashlight now and the other side of your barn when the light gets there in the future. Okay. So one is zero because light has had a chance to get there, but if you ask what is the space time interval between my flashlight and the other side of the barn now, before light has had a chance to get there, that's a positive spacetime interval that's outside of your light cone. So the same location can be inside or on or outside the light cone depending on the time. Okay.

Because the light cone expands with time. Same is true for Andromeda, right? If you fire your death ray right now and ask, is Indromeda within my light cone? Well, it isn't right now, but in five million years or whatever, it will be. Okay. So that has a spacetime interval that's zero. But firing your death ray right now and Andromeda right now, Now that has a positive space time interval.

So basically just tells you which parts of the universe can use them, which parts are accessible to you because of this limited speed of light. Got it. So that's something interesting about the universe, right? Like we should listen when the universe tells us, hey, this is the physical thing. This is what's invariant. This is what's important. Like when we notice that momentum is conserved, we say, okay, that means something useful

about the universe. And it does. It tells us something fascinating about space. Check out our episode about no-thor's theorem to understand the connections between symmetries and conservation laws. But this is a really good way to get deep insights about the nature of the universe is to say, what does the universe respect? What does it conserve? What does it keep constant? In this case, we're not talking about conservation.

We're talking about invariance. What does the universe say everybody has to agree on? No matter how fast your Ferrari or your spaceship is, everybody's going to agree on Kelly's light cone. Can she reach Andromeda with her death? not now, but yes, in the future. That's something everybody agrees on no matter how fast they're going. And so that's interesting. That's fascinating. And it's connected to the idea of distance, right? But it's clear that like spatial distance itself is not something the universe insists everybody agree on. It's all twisted and broken by high speeds and relativity. So this is something the universe

respects. And that means that it's closer to the truth. It's like at the heart. of the universe. Fundamentally, it tells us something about the geometry of the universe. Remember when we're adding up the pieces of the Pythagorean theorem for space, and as we add more dimensions, we just add them the same way because all the dimensions are the same, X and Y and Z, they don't matter. Well, time is a dimension, and it's a part of space time, which is four-dimensional, but it's different. The geometry of time and space are different with respect to each other. That's what the universe is saying here. Okay. So as you said, this is like fundamental and important. But when we get the explanation about how distance is invariant and time is invariant, it's not usually followed by this explanation for, well, how do you make it all variant or whatever? Why don't we usually get that other step? Is it just because it's hard to explain? Or what do you think?

Yeah, I think this concept of a space time interval is a little bit abstract because there's no real intuitive an analog for it. Like, what is this thing we're talking about? Like, yeah. Yes, the universe respects it, but it's not something that lines up with our notions very nicely. And so I think that's why it's not typically like introduced in Popsai. Whereas, you know, like the universe respects energy or doesn't respect energy or momentum. These are things we can get handle on. So it's easier to talk about. But the space time interval really is foundational to the whole concept of the geometry of the universe, which is what relativity is all about. That's really what Minkowski was saying to Einstein.

Yeah, you broke space and you broke time, but really you showed us that these are two sides of a more complicated object. You don't stick them together trivially. You stick them together backwards. And the universe has this weird geometry. And that's how it organizes itself. That's the internal mechanism of the mathematics of the universe. And this turned out to be really crucial for Einstein's next step in general relativity, because in special relativity, these interval formulas are pretty simple. You just take the differences in each spatial dimension. You take the difference in the time dimension. You add them up with a negative and you do the Pythagorean theorem. You're good. In general relativity, the locations don't add up all the same way. Some locations add up more and some locations add up less. That's what we mean by the metric we talk about sometimes in general relativity. How is space curved in this location? How much space time interval do you accrue by passing through this space versus passing through that space? And so this concept of having a geometry, to the universe and then having that geometry be non-trivial. That's how you go from special relativity to general relativity. But you've got to get your brain to like the geometry of the universe first

before you can even get to general relativity. Well, this conversation didn't crush my spirits. It lifted them up. And now I have, I guess you can't really make a positive space time because then that would be too far to travel. And so I'm still going to be thinking about this tonight, but I totally followed the explanation. And that's awesome. I learned something new. today. And one more speed bump to avoid is you often hear people saying that photons do not experience time. And here, what we're saying is light can go from A to B with no space time interval, right? It's light like. It's a zero space time interval along the light cone. That's not the same as saying photons do not experience time, right? Saying photons have a zero space time interval is very

different from saying photons do not experience time. There's a whole separate pop-soling. rabbit hole, but photons do take time to move across the universe. They just don't take space time. I'm realizing I want to ask if space time intervals have units. Does it make sense to think of it as a measurement with a unit? Yes, it does. And the units we use are distance, right? Like length, like meters, for example. And time enters with a conversion factor, which is the speed of light. So it's not really time. It's the speed of light times time. C, T, T. which is why photons have a space-time interval of zero because they move at the speed of light.

So that conversion factor ensures that the space and the time parts balance perfectly. Way to go, mass. Harmonious mathematics. Love it. All right, well, thank you, everybody, for going on this journey through space and time and through space time. Thank you, Kelly, for all the great questions and for not using the crusher on today's episode.

Thank you, Daniel, for the great explanation. I will keep the crusher at home with me. You guys hear the sound of that thing? Oh, my gosh. Oh, gosh. All right. And that's me signing off with a shiver. Yep, that's me signing off as well.

I don't think I'm going to be able to use this thing on anything actually living without. Yeah, I'm not doing it. All right, everybody, until next time. Have a good space and have a good time. Thanks, everybody for listening. Please go and do us a favor and rate the show on whatever podcast app you're using. It really helps people find us. Daniel and Kelly's extraordinary universe is edited by the amazing Matt Kesselman.

He really is a wizard. You can also find us online on Blue Sky, Instagram, and X, D&K Universe. Come engage with us. You can email us at Questions at Danielandkelly.org. We really do want to hear from you. And you can find our website, www. Daniel and Kelly.org, where you'll also find an invitation to join our Discord, where everybody comes and talks about the information.

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