What is a space-time interval?

24:11Do better or do less so I don't have to do so much. So join Yamanika each episode as she answers one question. Who's the problem? I'm Yamanika and I'm out. Listen to You're the Problem with Yamanika on the iHeartRadio app, Apple Podcasts, or wherever you get your podcasts.

24:32I'm Nick Turturro. You probably know me from NYPD Blue, The Longest Yard, or Spike Lee's Black Klansman. But I'm also from Queens, which means I know pizza. And on my new podcast, Delivering Happiness with Nick Turturro, I deliver pizza from Prince Street Pizza to a new guest. Are you delivering pizza now? Then we sit down for a real conversation. I've shared a slice with everyone, from Seth Rollins. What are you doing with my belt? To Alex Rodriguez. Just to let you people know, we're talking to one of the goats here. To Bill Burr. I don't think I've ever met somebody so exactly out of their mind as I am.

25:06And now we even have more great guests coming up, including the great John Turturro. It's called happiness. Delivering happiness. Mariano Rivera. Oh my God. And many, many more. Open your free iHeartRadio app, search Delivering Happiness with Nick Turturro, and listen now.

25:33And we're back. We had a very comforting discussion about distance and time where everything felt like it made sense. And now we're 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. Oh, no. And he said to Newton, he said, you know, put your theories in this, buddy, because here I come. Oh. It took me a second. Sorry for the delay. I was drinking water. It almost totally destroyed my computer. All right, go on.

26:08And 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.

26:48And 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.

27:23Okay. 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? Ghosts 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. 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.

27:59But 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.

28:28All 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 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.

29:05So it takes longer. And 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 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.

29:41So Zach sees the front of the crate get hit before the back of the crate. Ever since that lucrative Doritos sponsorship, he's been able to afford 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 observers.

30:20Like if the goats are bleeding, if they're going 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 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. 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?

31:09That'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.

31:44It 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 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.

32:29And 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 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.

33:10Okay, 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, 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 last segment, we were saying distance and time in classical physics are physical. And we're like, what does physical mean? And 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.

33:46It'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.

34:21So 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 superglue. 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 ruler 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.

34:55Because 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 end of 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?

35:27And 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.

35:59And if your camera is not centered along the pencil, then you're going to get 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 could 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.

36:32And 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 are happening at the same time for him.

37:07Typical. So see how the speed of light being constant breaks the universality of simultaneity, 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 eight centimeters, so you said it was six centimeters long, eight minus two. But what if the pencil was moving and you had measured the front before you measured the back, not at the same time?

37:45Then the reason the pencil is at eight 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. That's why breaking time also breaks distance. All right. So we can't depend on time. We can't depend on distance. Yeah. 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?

38:29And so who comes to the rescue but math profs. Daniel! Oh.

38:37Math profs. 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.

39:31So it's a famous line because what he's saying is space is not real, time is not real by 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 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.

40:06and you're just 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 different numbers of heads or tails that you can see, but that doesn't change the number of elephants.

40:41That'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 in three-dimensional space, right? You were 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?

41:19A 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 recontextualize my dumb questions to make them seem insightful.

41:53Much 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.

42:30She became the first female solo rower to go from California to Hawaii. My first thought is, like, what's up with the snacks? Like, what are we eating? The highlights, the rivalries, the breakout stars, the moments that take over your entire timeline. And the conversations that start during the game and somehow keep going all week. Every week, we're breaking down the biggest stories across women's sports. Naomi Osaka showing out. She beat Sabalenka. Shout out to you, Naomi. You get the palm, Naomi. You get the palm for that. Because we're not just interested in what happened, we're interested in why everyone's talking about it.

43:04Because everyone watches women's sports. Listen to Everyone Watches Women's Sports on the iHeartRadio app, Apple Podcasts, or wherever you get your podcasts.

43:14Hey, Portlandia fans. Carrie Brownstein and Fred Armisen here. You know us, or rather, you know them. Tony and Candice, Nina and Lance, Spike, and yes, the chicken. We've played a lot of iconic characters over the years, but today we're showing up as ourselves to tell you about Podlandia, AO Rewatch, our brand new podcast. Each week, we'll revisit an episode of Portlandia from the very beginning, breaking down the sketches, exploring the backstories of our most iconic characters, revisiting the Portland locations you know and love, and opening up about our creative process. How did any of this get made?

43:47Why do we think that was a good idea? We're ready to talk about it. And we'll also be joined by the people who helped bring it all to life. guest stars, collaborators, and friends, including director Jonathan Kreisel, the mayor himself, Kyle McLaughlin, legendary musician Amy Mann, and many more. Kyle is going for it here. You fully improvised, not just words, but a song, a melody. Well, I thought he was going to write, I thought you were all going to write a song. I remember you thinking that. Listen to Podlandia A-O Rewatch on the iHeartRadio app, Apple Podcasts, or wherever you get your podcasts.

44:22I'm Jake Brennan, and on my podcast, Disgraceland, I tell the stories behind music's biggest names, the moments that shaped them, haunted them, and changed music history forever. Like how the story of the Foo Fighters and Dave Grohl isn't just about music. Imagine that. You're in the biggest band on the planet, as Dave Grohl was in 1994 in Nirvana. And the phone rings. and you learn that your singer, your friend, the reluctant voice of a generation, Kurt Cobain, is dead. This is a story of fame, pressure, friendship, and the weight of fulfilling your destiny. Learn more about the moment everything changed.

45:08Disgraceland is part of the Exactly Right Network. Listen to new episodes every Tuesday, bonus episodes Thursday, and rewinds on Sunday on the iHeartRadio app, Apple Podcasts, or wherever you get your podcasts.

45:48with Yamanika. Do you know I just found out who Sydney Sweeney was? New episodes weekly every Wednesday as part of my new network, the Dear Chelsea Network. If he got a bunch of women, then I should have a bunch of men. Do better or do less so I don't have to do so much. So join Yamanika each episode as she answers one question. Who's the problem? I'm Yamanika and I'm out. Listen to You're the Problem with Yamanika on the iHeartRadio app, Apple Podcasts, or wherever you get your podcasts.

46:33All right, we are having fun at this space in this time, and Daniel is going to help us understand the complicated concept that is space-time interval. So space and time make 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 dimensions. 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?

47:11Yes. 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.

47:51Does 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. 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 this space-time interval?

48:28Imagine 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.

49:03So 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.

49:41If 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 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 a four-dimensional Pythagorean theorem, but no, the universe says this is the thing that's important.

50:21Is 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?

50:56Because 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 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.

51:37So events have zero space-time intervals if they are perfectly causally linked. Just enough time and not too much. A message departing and arriving are two space-time events with zero space-time interval. They have a distance in space and a distance in time, but those balance to give no interval. Events that have a zero space-time 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?

52:19So 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 any 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 and 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.

53:01It 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 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. Okay. 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.

53:37And 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.

54:18It'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 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?

54:57I think I'm still a little caught up on like, so if I turn on a laser, it's going to get to Andromeda 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. 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?

55:41You 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.

56:15If 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 sort of adding - A few billion years. Did I say a few years? A few billion 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 Andromeda moving.

56:48It'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 space-time 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.

57:29Okay. So if you turn your flashlight on now and you shine it at the other wall, 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. 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, or however long it takes light to get there, that has a zero space-time interval. So there's two different space-time intervals there.

58:08There'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 space-time 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.

58:41Same is true for Andromeda, right? If you fire your death ray right now and ask, is Andromeda within my light cone? Well, it isn't right now, but in 5 million years or whatever, it will be. So that has a space-time interval that's zero. But firing your death ray right now and Andromeda right now, that has a positive space-time interval. So basically just tells you which parts of the universe can you zap, which parts are accessible to you because of this limited speed of light. Got it. So that's something interesting about the universe, right? We should listen when the universe tells us, hey, this is the physical thing.

59:17This is what's invariant. This is what's important. 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 Noether'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, right?

59:50No 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 ray? 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 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.

1:00:27It's at the heart of the universe. Fundamentally, it tells us something about the geometry of the universe. Remember when we were 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?

1:01:14Why 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 and analog for it. Like, what is this thing we're talking about? Like, 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 introduced in Popsi. Whereas the universe respects energy or doesn't respect energy or momentum. These are things we can get a 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.

1:01:53That'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.

1:02:33In 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 got to get your brain to like the geometry of the universe first before you can even get to general relativity.

1:03:08Well, 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?

1:03:46Saying photons have a zero space-time interval is very different from saying photons do not experience time. There's a whole separate pop-si 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, ct, which is why photons have a space-time interval of zero because they move at the speed of light.

1:04:30So that conversion factor ensures that the space and the time parts balance perfectly. Way to go, math. 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.

1:05:09Yeah, I'm not doing it. All right, everybody, until next time. Have a good space and have a good time.

1:05:43universe. 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.danielandkelly.org where you'll also find an invitation to join our Discord where everybody comes and talks about the amazing universe. And we also have the most amazing moderators. This is an iHeart Podcast. Thanks for joining us. On Raiders of the Lost podcast, we explore cinema like no one else, including huge interviews with stars like Ryan Gosling on Project Hail Mary. It was like the Jaws shark. Didn't always work, came with its own problems.

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