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

0:07
Speaker 1: Special relativity brings space and time together and in the
0:12
Speaker 1: darkness binds them into space time. That sounds pretty awesome,
0:17
Speaker 1: and it was fun to say, like something that could
0:19
Speaker 1: only happen in Mordoor or during the Big Bang. But
0:23
Speaker 1: what does it actually mean? What is space time? How
0:27
Speaker 1: do you measure distances in four dimensional space time? And
0:32
Speaker 1: why would you? Physicists are always saying that space and
0:36
Speaker 1: time are like two sides of the same coin making
0:39
Speaker 1: more sense together. What are they talking about? In what
0:43
Speaker 1: way do they make more sense together? Today We're gonna
0:46
Speaker 1: blow up all of those analogies and give you a
0:48
Speaker 1: real conceptual understanding of how relativity breaks space and twists time,
0:55
Speaker 1: but then clicks them back together into a harmonious hole.
0:59
Speaker 1: We'll talk about what that really means and the crucial
1:02
Speaker 1: but rarely explain concept at the heart of it all
1:05
Speaker 1: the space time interval. Welcome to Daniel and Kelly's spacey
1:11
Speaker 1: timey Universe.
1:26
Speaker 2: Hello.
1:26
Speaker 3: I'm Kelly Wiener Smith. I study parasites and space, and
1:29
Speaker 3: I have fun in all spaces and at all times. Hi.
1:39
Speaker 1: I'm Daniel. I'm a particle physicist who likes to think
1:41
Speaker 1: about aliens, and I was having fun this morning until
1:44
Speaker 1: Kelly showed me the crusher.
1:46
Speaker 3: Oh was it sure you were going to mention the
1:49
Speaker 3: crusher on the show.
1:50
Speaker 1: It's the only thing I can think about now, Thank
1:52
Speaker 1: you very much. Kelly. Tell everybody what does the crusher crush.
1:59
Speaker 3: The tube that goes to the testicle and connects it
2:02
Speaker 3: to the rest of the body. I am looking into
2:04
Speaker 3: methods for what we'll say, weathering are male goats, and
2:10
Speaker 3: I'm trying to figure out what method I would feel
2:11
Speaker 3: comfortable doing, or if I need to call the vets
2:14
Speaker 3: and Wow, the crusher is a big piece of equipment
2:16
Speaker 3: that once I have it in my hands, I am
2:18
Speaker 3: not sure that I am up for actually using.
2:22
Speaker 1: But you were up for brandishing it on our video
2:25
Speaker 1: call this morning.
2:28
Speaker 3: Everybody needs to stop it.
2:31
Speaker 1: Away the crusher, all right, But today we are not
2:34
Speaker 1: crushing goats or any of their bits. We are crushing
2:37
Speaker 1: your confusion. We are explaining the way the universe works.
2:43
Speaker 3: You always pull us back somehow, Okay, So today we're
2:47
Speaker 3: talking about space time, yes, which got me thinking about
2:50
Speaker 3: all of space and all of time. And so, Daniel,
2:53
Speaker 3: if you could snap your fingers and for five minutes,
2:57
Speaker 3: you would be in a protective bubble, and you could
2:59
Speaker 3: go anywhere in space, but you couldn't change the time,
3:01
Speaker 3: like you had to just pop up right there. Now,
3:05
Speaker 3: what would you want to see?
3:07
Speaker 1: Kelly? You need this episode because if you had heard
3:10
Speaker 1: this episode already, you would know that that question is
3:12
Speaker 1: incoherent because there is no universal now. That's right, right,
3:16
Speaker 1: there's no sense of time across the whole universe where
3:19
Speaker 1: you can say this now is the same as that now.
3:22
Speaker 1: It's observer dependent. We're gonna talk about and.
3:25
Speaker 3: You know what, I should have known that from our
3:26
Speaker 3: What Is Time and What Is Space? Episode series? But okay,
3:31
Speaker 3: so is there some way to constrain it so I
3:33
Speaker 3: can say, like, you can't go back to the beginning
3:35
Speaker 3: of the universe. I was trying to, like, why do.
3:37
Speaker 1: You want to stop me from going back to the
3:38
Speaker 1: beginning of the.
3:39
Speaker 3: Universe, because that's what I thought you'd say.
3:41
Speaker 1: You were totally right, that is what I was gonna.
3:42
Speaker 3: Say, But you said that on the show before. So
3:46
Speaker 3: trying to figure out what you think is the most
3:47
Speaker 3: interesting thing happening now. And my guess is that you
3:50
Speaker 3: were gonna say black holes.
3:52
Speaker 1: I'm gonna go to the factory that makes the crusher
3:54
Speaker 1: and I'm gonna burn it down. Okay, No, but if
3:59
Speaker 1: I could go somewhere in the modern universe, then for
4:03
Speaker 1: sure I would like to see not only what's inside
4:06
Speaker 1: a black hole, but the immediate vicinity of a black
4:09
Speaker 1: hole would be super fascinating to see frame dragging and
4:13
Speaker 1: to see all that time dilation and all the extreme
4:16
Speaker 1: events around a black hole. I think my second choice
4:20
Speaker 1: would be the heart of a neutron star, because when
4:24
Speaker 1: every other defense against gravity has already collapsed and matter
4:28
Speaker 1: becomes insanely dense so the protons and electrons eat each
4:32
Speaker 1: other and become neutrons, then you have this fascinating environment
4:36
Speaker 1: where quantum mechanics and gravity are at about the same strength.
4:40
Speaker 1: And we don't know what's going on at the heart
4:42
Speaker 1: of neutron stars because we can't calculate that stuff because
4:44
Speaker 1: we don't have a theory of quantum gravity. We'll never
4:47
Speaker 1: go inside a black hole, so that's sort of just
4:49
Speaker 1: like fun to say, but hopeless. But the inside of
4:52
Speaker 1: a neutron star is out there and real and on
4:55
Speaker 1: the outside of event horizon, so in principle we could
4:58
Speaker 1: one day see it, and that be the key to
5:00
Speaker 1: unlocking quantum gravity.
5:02
Speaker 3: WHOA Okay, So, how could I have worded the question
5:05
Speaker 3: to constrain your ability to see the Big Bang? What
5:09
Speaker 3: would have been the right way to word that?
5:11
Speaker 1: Of Daniel? What would you like to see in the
5:12
Speaker 1: universe except for the Big Bang?
5:18
Speaker 3: Well, wouldn't you have to go backwards in time to
5:20
Speaker 3: see the Big Bang?
5:21
Speaker 1: It depends a little bit of what you mean by sea.
5:23
Speaker 1: I mean the information from the Big Bang is out
5:26
Speaker 1: there in some respects. People often say seeing the CMB
5:30
Speaker 1: is like seeing the Big Bang, though it actually comes
5:32
Speaker 1: from three hundred and eighty thousand years later. There is
5:35
Speaker 1: information from the Big Bang out there and gravitational waves,
5:38
Speaker 1: and so like seeing it means detecting those and reconstructing
5:42
Speaker 1: what happened. We can do that today without like being
5:45
Speaker 1: there during the Big Bang. In that sense, everything you
5:48
Speaker 1: see happened in the past already.
5:51
Speaker 3: So but I feel like you understand that that's not
5:54
Speaker 3: what I mean like that, like, like, yes, you can
5:58
Speaker 3: see evidence from it, yes, but like I don't you know,
6:02
Speaker 3: So if I go to a civil war site and
6:03
Speaker 3: I find a like bullet case, that's like evidence that
6:07
Speaker 3: it happened. Yeah, but it's not the same as being
6:10
Speaker 3: there and watching both sides run across a field and
6:13
Speaker 3: start fighting each other.
6:15
Speaker 1: Right, No, but it's sort of like going really really
6:18
Speaker 1: far away instantaneously and capturing the light that left that
6:21
Speaker 1: Civil War battlefield contemporaneously and using it to watch in
6:26
Speaker 1: real time the battle, which is something you could literally.
6:29
Speaker 3: Do, right, All right, all right, all right, Well I'm
6:32
Speaker 3: gonna have you be more careful next time I ask you
6:33
Speaker 3: a question. But we've all learned something from Kelly's bad question,
6:38
Speaker 3: which is why Kelly's here.
6:40
Speaker 1: So, and these are really fun topics to think about
6:42
Speaker 1: because our intuition is broken about how all of this
6:45
Speaker 1: stuff works. We tend to think that time flows the
6:48
Speaker 1: same way for everyone. We tend to think that distances
6:51
Speaker 1: are real and they're universal. When it turns out the
6:55
Speaker 1: universe operates in a way that's fundamentally in conflict with
6:58
Speaker 1: our intuition but amazingly does make sense. So we have
7:03
Speaker 1: this task ahead of us to incorporate this new intuition
7:07
Speaker 1: about the universe into our minds. But often this means
7:10
Speaker 1: just like, read a bunch of equations and use them
7:11
Speaker 1: to predict what happens. But I think it's possible to
7:14
Speaker 1: develop a new kind of intuition, a conceptual understanding for
7:18
Speaker 1: how the universe really does work. And that's what we're
7:21
Speaker 1: going to try to do today by introducing you to
7:23
Speaker 1: one of the most important concepts in special relativity that
7:26
Speaker 1: almost nobody talks about and that we've never covered on
7:29
Speaker 1: this show.
7:29
Speaker 3: What. Yes, Well, this is a very exciting day here
7:33
Speaker 3: at dk EU, and let's see if our audience heard
7:38
Speaker 3: about this thing that pretty much no one talks about.
7:40
Speaker 3: And so we asked our audience, what is a space
7:44
Speaker 3: time interval?
7:46
Speaker 2: Well, I would expect that that would be the reduction
7:51
Speaker 2: of Einstein's space time into a quantum type interval, but
7:57
Speaker 2: I'm really not too sure. It's the between two objects
8:02
Speaker 2: and space.
8:03
Speaker 4: Maybe I don't know.
8:05
Speaker 1: Could a space time interval be the point where one
8:09
Speaker 1: max plank length ends and one begins.
8:13
Speaker 5: I'm not sure a space time interval is the required
8:16
Speaker 5: time frame between semi convincing scientific mumbo jumbo that you
8:21
Speaker 5: hear on sci fi shows and movies. But seriously, maybe
8:25
Speaker 5: it's some form of the smallest unit of space time
8:30
Speaker 5: length and distance, kind of almost a singularity, but not quite.
8:35
Speaker 4: I've never actually heard the term space time interval before,
8:38
Speaker 4: but guessing from context, I would imagine a space time
8:42
Speaker 4: interval is one like frame of the universe. So you know,
8:49
Speaker 4: you move forward one unit of time and look at
8:53
Speaker 4: the way everything else changed. That's one interval of space type.
8:56
Speaker 3: All right, Daniel, I asked a question that belied my ignorance,
9:00
Speaker 3: and so I'm not going to even attempt to say
9:03
Speaker 3: if I think the audience was correct or not with
9:07
Speaker 3: this question. How close were they?
9:09
Speaker 1: Pretty much? Zero points all around? A few points for
9:13
Speaker 1: people thinking this is a kind of distance, because that's
9:16
Speaker 1: the closest. It has nothing to do with the plank
9:19
Speaker 1: length or frames of the universe or minimum distances or
9:24
Speaker 1: anything like that. It's really an analog to what we
9:27
Speaker 1: think of as distance, but now in four dimensional space
9:31
Speaker 1: time with an uncomfortable twist in it. That we're going
9:34
Speaker 1: to get.
9:34
Speaker 3: To good news everybody. We all need this episode. So
9:39
Speaker 3: here we go, and we are going to start from
9:42
Speaker 3: the beginning, because clearly Kelly needs to go back to
9:44
Speaker 3: first principles. So what I want you to explain to me, Daniel?
9:48
Speaker 3: So space time sounds like space and time and distance,
9:51
Speaker 3: and there's like, so what is distance? Let's start there.
9:56
Speaker 1: Yeah, let's build our intuition with stuff that makes sense
9:58
Speaker 1: to us. And let's start easy. Let's start in one dimension, right.
10:02
Speaker 1: So imagine, for example, you have a pencil and you
10:06
Speaker 1: have a ruler, and you want to know how long
10:09
Speaker 1: is my pencil? What do you do? Well, you put
10:11
Speaker 1: it along the ruler. You're operating in one dimension, right,
10:14
Speaker 1: So the pencil and the ruler are lined up, and
10:16
Speaker 1: you say, well, the back of the pencil, you know,
10:19
Speaker 1: is that two centimeters on the ruler, and the front
10:21
Speaker 1: of the pencil is that eight centimeters on the ruler.
10:23
Speaker 1: And then you subtract it's like eight minus two six
10:26
Speaker 1: centimeters long.
10:27
Speaker 3: I don't know why you wouldn't just put it at
10:28
Speaker 3: the beginning of the ruler and then you don't have
10:30
Speaker 3: to do any subtraction.
10:32
Speaker 1: Exactly. You could put it at the beginning of the
10:33
Speaker 1: ruler so that you get zero and six. But what
10:37
Speaker 1: happens You get the same answer, right, if you put
10:39
Speaker 1: the pencil at the beginning or if you slide it along,
10:41
Speaker 1: It doesn't matter. The answer you get does not depend
10:44
Speaker 1: on where on the ruler of your pencil is. Right,
10:47
Speaker 1: It's invariant. Okay, that's cool. That all makes sense. It's
10:50
Speaker 1: makes so much sense. It's kind of blindingly obvious, which
10:53
Speaker 1: is where we want to begin, right with, like firmly
10:55
Speaker 1: grounded in our intuition.
10:56
Speaker 3: This is very comforting, unlike the crusher.
11:01
Speaker 1: Okay, so why do you keep bringing that up? Kelly?
11:04
Speaker 1: You just like I don't want to throw me off.
11:06
Speaker 3: I wasn't going to mention it on this episode, but
11:09
Speaker 3: you brought it up because you did the whole crushing
11:12
Speaker 3: thing on the video before the episode, And now there's
11:14
Speaker 3: nothing else in my mind.
11:15
Speaker 1: I'm trying to get space time in my brain, but
11:18
Speaker 1: that's all I can see you.
11:19
Speaker 3: I'm so sorry. I'm sorry.
11:22
Speaker 1: It's fine, all right. So now let's go to two dimensions.
11:26
Speaker 1: How do things change? Well, imagine you have like a
11:28
Speaker 1: piece of graph paper and the little squares on the
11:30
Speaker 1: graph paper are one centimeter by one centimeter, and you
11:34
Speaker 1: drop your pencil onto the graph paper and ask you
11:36
Speaker 1: how long is your pencil? So how do you figure
11:39
Speaker 1: out how far it is from one side of the
11:41
Speaker 1: pencil to the other.
11:43
Speaker 3: But the Agrean theorem.
11:45
Speaker 1: Yes exactly, So you say, well, how many squares does
11:49
Speaker 1: it cross in one direction and how many squares in
11:51
Speaker 1: the other? Direction, and now you have a right triangle
11:54
Speaker 1: where the pencil is the hyphpot news right, it's the
11:56
Speaker 1: long part of the right triangle, and we can calculate
11:59
Speaker 1: one side because it's just counting squares. We can calculate
12:02
Speaker 1: the length may the other side just counting squares, and
12:05
Speaker 1: then the diagonal, which is the length of the pencil.
12:07
Speaker 1: That's the Pythagorean theorem. Right, So a squared plus b
12:10
Speaker 1: squared equal c square. Boom, we have an answer. And
12:14
Speaker 1: just like in the one dimensional case, it doesn't matter
12:17
Speaker 1: where the pencil was on the graph paper. If you
12:20
Speaker 1: dropped it in a different direction, you might have gotten
12:23
Speaker 1: more squares on one side and few or on the other,
12:26
Speaker 1: but the answer wouldn't change. Or even if by chance
12:29
Speaker 1: you happen to drop the pencil so it liigned up
12:31
Speaker 1: perfectly in one direction right like along the X axis
12:35
Speaker 1: conceptually right, then it's basically back to the one dimensional
12:38
Speaker 1: problem or the other axis. The point is in this picture,
12:43
Speaker 1: distance is real. It's invariant. It does not depend on
12:47
Speaker 1: where you drop the pencil, or like the size of
12:49
Speaker 1: these squares, if you made them two centimeters or five centimeters,
12:52
Speaker 1: or like a furlong or whatever. All those are just
12:55
Speaker 1: arbitrary choices humans have imposed. The pencil distance is the real.
13:00
Speaker 3: Thing, right, Yes, this is very comforting. Can we stay
13:04
Speaker 3: here for the rest of the episode. No, let's push
13:07
Speaker 3: our boundaries.
13:08
Speaker 1: Let's keep going, all right, And so now it's easy
13:11
Speaker 1: to imagine how we might extend this to three dimensions,
13:14
Speaker 1: or really any number of dimensions. Like if you have
13:17
Speaker 1: your pencil and it's not on the paper, but it's
13:20
Speaker 1: like floating in space. We're on the space station now, right,
13:23
Speaker 1: and the pencils floating in space, and you imagine like
13:26
Speaker 1: not graph paper, but like laser lines through space, and
13:29
Speaker 1: you want to measure how long is it from one
13:31
Speaker 1: edge of my pencil to the other. Well, you play
13:33
Speaker 1: the same game, but now you have three directions of
13:36
Speaker 1: counting squares and you can still do the pythagoranth theorem.
13:39
Speaker 1: It's just A squared plus B squared plus C squared
13:42
Speaker 1: equals D squared. Right, it's the same game fundamentally that
13:45
Speaker 1: the geometry has not changed because the nature of these
13:48
Speaker 1: dimensions are all the same, And so we can just
13:50
Speaker 1: keep doing this. And if you wanted to work in
13:53
Speaker 1: eleven dimensional space or twenty six dimensional space, the concept
13:56
Speaker 1: of distance wouldn't change mathematically or in intuitively, because the
14:01
Speaker 1: distance is the real thing, right, Everything else is just bookkeeping.
14:05
Speaker 3: Okay, So I'm trying to imagine a pencil floating in space,
14:09
Speaker 3: and now a squared plus B squared plus b squared
14:12
Speaker 3: equals the length of the pencil squared. Yeah, my gut
14:16
Speaker 3: feels like the more dimensions you add, the bigger the
14:19
Speaker 3: pencil's going to get because you've got all these extra
14:21
Speaker 3: squared terms that are gonna keep getting added. But that's
14:25
Speaker 3: not true because why.
14:28
Speaker 1: Yeah, that's not true because the pencil is a certain length,
14:31
Speaker 1: and if you add squares in one of the dimensions,
14:35
Speaker 1: the other ones will shrink. Okay, Just like if the
14:37
Speaker 1: pencil falls to the floor and it's now on a
14:39
Speaker 1: two dimensional surface, you only have two contributions, or the
14:42
Speaker 1: third one is just zero, right, and those are longer
14:46
Speaker 1: than if it's floating up above the graph and it
14:49
Speaker 1: gets a third dimension, the first two dimensions are going
14:51
Speaker 1: to shrink a little bit. So they all act in
14:54
Speaker 1: harmony to make sure the distance never changes. You like
14:57
Speaker 1: flip a pencil from one astronaut to another se so
15:00
Speaker 1: it's like rotating through space. You know, your intuition tells
15:03
Speaker 1: you the length of the pencil is not changing. The
15:06
Speaker 1: difference between its front and its back along some axis
15:10
Speaker 1: might change, right, it might go up and down and
15:12
Speaker 1: be negative or whatever, But the length of the pencil
15:14
Speaker 1: doesn't change because all three dimensions are working in harmony
15:18
Speaker 1: to contribute various pieces to the length, and it always
15:22
Speaker 1: adds up to the same number. Because that's the physical thing.
15:25
Speaker 1: Everything else is just bookkeeping. It wouldn't make sense if
15:28
Speaker 1: the length of the pencil depended on like where the
15:31
Speaker 1: graph paper was right, or which ruler you used. It's
15:34
Speaker 1: a real thing in the universe.
15:36
Speaker 3: All right. But I bet that if I were to say, Daniel,
15:40
Speaker 3: you can travel a billion, a trillion, a quadrillion miles
15:44
Speaker 3: away in any direction in space. What is it that
15:46
Speaker 3: you want to see within that area? Everybody should know
15:50
Speaker 3: what I'm talking about. Are you going to tell me
15:52
Speaker 3: that something about that doesn't work? Like the pencil?
15:57
Speaker 1: I see, you're asking me to be the physics sweat
15:59
Speaker 1: blanket here.
16:00
Speaker 3: I mean we have to take turns there, right.
16:04
Speaker 1: Well, if you said I could go at quadrillion miles
16:08
Speaker 1: with respect to Earth, then you've given me a frame
16:11
Speaker 1: of reference. In which that is well defined. Okay, so yeah,
16:14
Speaker 1: I could totally do that, all.
16:15
Speaker 3: Right, that's what I should have asked anyway.
16:16
Speaker 1: All right, moving on, and so we have three dimensions
16:20
Speaker 1: of space. We're comfortable with the idea of distance in space. Cool,
16:24
Speaker 1: let's put that aside for now and think about time.
16:27
Speaker 1: Time is one dimensional, right, We don't have like two
16:31
Speaker 1: clocks going on, we don't think, so it's actually a
16:33
Speaker 1: super fun theory we can explore. And we did an
16:36
Speaker 1: episode a couple of years ago about two dimensions of time.
16:39
Speaker 1: But for now, let's just think about one dimension of time, right,
16:42
Speaker 1: And so, for example, you can click your stop watch
16:45
Speaker 1: when this episode starts and when the episode ends, and
16:47
Speaker 1: you can ask how long is this episode of the
16:50
Speaker 1: pod and you can look at the difference, and that's
16:52
Speaker 1: the length, right, totally obvious, basic subtraction.
16:56
Speaker 3: Perfect. Yes, for our poor audio guy, it probably feels
16:59
Speaker 3: like an hour is infinite, but it is still just
17:02
Speaker 3: an hour exactly.
17:03
Speaker 1: And in the same way, it doesn't actually matter when
17:06
Speaker 1: you hit the stopwatch to start, as long as you
17:09
Speaker 1: noted when the episode started. If you'd accidentally hit your
17:11
Speaker 1: stopwatch like an hour before, as long as you wrote down. Oh,
17:14
Speaker 1: the episode started at one hour in you could still
17:17
Speaker 1: measure the length of the episode by looking at the difference, right,
17:22
Speaker 1: because when you started it is arbitrary. What you're interested
17:25
Speaker 1: in is the physical thing, which is the difference between
17:28
Speaker 1: the start and the end of the episode. So intervals
17:31
Speaker 1: and time are also physical in that sense, right, They're real,
17:34
Speaker 1: It's not just bookkeeping.
17:35
Speaker 3: What does physical mean in this case? Like physical I
17:38
Speaker 3: think of as something I can touch. Oh, I see,
17:41
Speaker 3: you mean physics all.
17:44
Speaker 1: I mean it's a part of the universe. It's not
17:46
Speaker 1: part of our measurement system, and it's invariant in that way.
17:50
Speaker 1: And here so far, all these definitions are in a
17:53
Speaker 1: classical world, right before relativity. This is Newton's idea about
17:58
Speaker 1: how time and distance were worked, Right, that anybody could
18:02
Speaker 1: measure the difference between two things that happened in the
18:04
Speaker 1: universe and the time difference or the space difference, and
18:08
Speaker 1: they would always get the same answer. That's the way
18:10
Speaker 1: Newton thoughts the universe worked, and that's the way I
18:12
Speaker 1: operate and you operate. And this whole segment of the
18:15
Speaker 1: pod has been like, let's confirm our intuition about how
18:18
Speaker 1: distances work and that's all very comforting until we learn
18:22
Speaker 1: that it's all wrong.
18:24
Speaker 3: Oh no, you pulled the soft blanket of Newtonian physics off. Yeah,
18:29
Speaker 3: and now it's cold and unwelcoming. Let's take a break
18:34
Speaker 3: before we have to sit with our discomfort for too long.
18:37
Speaker 3: And when we get back, we'll hear what Einstein had
18:39
Speaker 3: to add. And we're back. We had a very comforting
19:02
Speaker 3: discussion about distance and time where everything felt like it
19:05
Speaker 3: made sense. And now we're at the point of the
19:07
Speaker 3: episode where we are going to expand our brains and
19:10
Speaker 3: learn something new. So all right, Daniel, make us uncomfortable,
19:13
Speaker 3: all right, So not crusher level uncomfortable.
19:17
Speaker 1: Well, in nineteen oh five, Einstein pulled out the metaphorical
19:20
Speaker 1: crusher and he said to Newton, he said, you know,
19:23
Speaker 1: put your theories in this, buddy, because here I come.
19:28
Speaker 3: It took me a second. Sorry for the delay. I
19:31
Speaker 3: was drinking water. It almost totally destroyed my computer.
19:34
Speaker 1: All right, go on, and you know these intuitive notions
19:36
Speaker 1: of how the universe works, they do work for us
19:39
Speaker 1: at our basic level. But Einstein discovered that when speeds
19:43
Speaker 1: get high relative speeds get high, then they don't work.
19:47
Speaker 1: The universe is broken. It actually works in a fundamentally
19:50
Speaker 1: different way. And it's important to understand where this comes from.
19:53
Speaker 1: It all comes from the same source, which is the
19:56
Speaker 1: fact that the speed of light is constant for all observed.
20:00
Speaker 1: That is at the heart of everything. If you accept
20:04
Speaker 1: that the speed of light is the same number no
20:06
Speaker 1: matter who measures it, no matter how fast they are
20:09
Speaker 1: moving with respect to like the flashlight that gave off
20:12
Speaker 1: that beam, then you'll see that everything breaks. And we're
20:15
Speaker 1: going to walk you through that and how it crushes
20:18
Speaker 1: your intuition, all.
20:19
Speaker 3: Right, crush away.
20:22
Speaker 1: So imagine here's the setup. Kelly gives two flashlights to
20:25
Speaker 1: her goats and puts those goats in a crate and
20:28
Speaker 1: puts that crate on a self driving tractor to the
20:31
Speaker 1: vet to you know, have their problems taken care of
20:34
Speaker 1: with great anesthetic, of course, and very humanely good, all right.
20:39
Speaker 1: And so they have two flashlights and one is pointed
20:42
Speaker 1: towards the front of the crate and one is pointed
20:44
Speaker 1: towards the back of the crate. Okay, and they turn
20:47
Speaker 1: the flashlights on at the same moment, so they click
20:49
Speaker 1: both buttons. And then we're interested in, like, well, how
20:53
Speaker 1: long does it take for that beam of light to
20:55
Speaker 1: hit the front of the crate versus the back of
20:57
Speaker 1: the crate. So what would you say, Kelly ghost during
21:00
Speaker 1: the center of the crate, they turn on both flashlights
21:02
Speaker 1: at the same time. Do those beams reach the front
21:05
Speaker 1: and the back of the crate at the same time? Yes?
21:07
Speaker 1: Or no?
21:08
Speaker 3: I feel like they should. But it's moving forward, right,
21:13
Speaker 3: and so does that mess up the answer?
21:15
Speaker 1: That's very good intuition. But we're asking right now from
21:17
Speaker 1: the goat's point of view, right, So, yes, the goat
21:20
Speaker 1: is in the crate, and it's on the back of
21:23
Speaker 1: the self driving tractor and it's driving itself towards the vet.
21:26
Speaker 1: But the goat doesn't know how fast the tractor is
21:29
Speaker 1: moving relative to you. You're outside the tractor. It's inside
21:32
Speaker 1: the crate, and the crate is not moving with respect
21:34
Speaker 1: to the goat.
21:35
Speaker 3: Okay, so it reaches the front and the back at
21:37
Speaker 3: the same time.
21:38
Speaker 1: Yes, you're exactly right. The goat says that the light
21:41
Speaker 1: reaches the front and the back at the same time,
21:43
Speaker 1: and that makes perfect sense because it's going at light
21:45
Speaker 1: speed forward and light speed backwards, and it hits the
21:48
Speaker 1: front and the back, which are the same distance. Away
21:50
Speaker 1: in the same time. Distance is great, time's time, very simple, right,
21:54
Speaker 1: nothing complicated, that's right, all right. But Kelly is also watching.
21:58
Speaker 1: She sent the goats this self driving tractor, and she
22:01
Speaker 1: doesn't want them to get hurt, except maybe get crushed
22:04
Speaker 1: when they get there. But she's paying attention to what happened.
22:07
Speaker 1: And for her, unlike the goats who see the walls
22:10
Speaker 1: of the crate at rest, for Kelly, the crate is
22:13
Speaker 1: moving forward, and she says, hold on a second. I
22:17
Speaker 1: also see light traveling at the speed of light forwards
22:21
Speaker 1: and at the speed of light backwards. But I see
22:23
Speaker 1: the front of the crate is moving forward from the
22:25
Speaker 1: point where the light was emanated, so it's moving away
22:29
Speaker 1: from the source at the same time as the light
22:31
Speaker 1: is chasing it, so it takes longer. And the back
22:34
Speaker 1: of the crate you see, moving towards where the light
22:37
Speaker 1: was emitted, so that distance is effectively shorter. And so
22:40
Speaker 1: you say, no, no, no. The light hits the back
22:43
Speaker 1: of the crate first and the front of the crate second.
22:46
Speaker 3: And then I look at the goats and I say,
22:47
Speaker 3: you guys are bad at this.
22:50
Speaker 1: And meanwhile, Zach, who's in his ferrari is zipping by
22:53
Speaker 1: it one hundred miles an hour to do who knows what.
22:55
Speaker 1: And he says, I see the opposite right, because from
22:59
Speaker 1: his point of view, the crate is going backwards, and
23:02
Speaker 1: so he says, no, no, the back of the crate
23:04
Speaker 1: is running away from the light and the front of
23:06
Speaker 1: the crate is moving towards the light. So Zach sees
23:10
Speaker 1: the front of the crate get hit before the back
23:12
Speaker 1: of the crate.
23:13
Speaker 3: Ever since that lucrative Dorito sponsorship, he's been able to
23:16
Speaker 3: afford fast cars.
23:20
Speaker 1: Exactly, And so even though he's got one hand in
23:23
Speaker 1: the Dorito's bag and one hand on the wheel, he's
23:25
Speaker 1: able to make these measurements. And so what's going on here?
23:29
Speaker 1: The issue comes down to the fact that everybody, the goat, Zach,
23:32
Speaker 1: and Kelly all see light moving at the same speed,
23:35
Speaker 1: no matter what. If you replicated this experiment with sound
23:38
Speaker 1: instead of with light, then everybody would agree on what happened.
23:43
Speaker 1: Why is that because the speed of sound is not
23:45
Speaker 1: the same for all observers, Like if the goats are bleeding,
23:49
Speaker 1: if they're going back inside the crate, then they're going
23:53
Speaker 1: to see their sound hit the front in the back
23:55
Speaker 1: of the same time, Kelly, who sees the crate moving,
23:59
Speaker 1: is going to see the sound going forward faster than
24:02
Speaker 1: it's going backwards, because the sound is moving with the
24:05
Speaker 1: same speed with respect to the air in the crate,
24:08
Speaker 1: and so like, if Zack has a speed relative to
24:10
Speaker 1: that air or Kelly has a speed relative to that air,
24:13
Speaker 1: then they see the sound moving at different speeds. Right,
24:16
Speaker 1: So the front and the back move at different speeds,
24:18
Speaker 1: which accounts for everything and closes the loop. So, because
24:22
Speaker 1: light moves at the same speed with respect to all observers,
24:25
Speaker 1: everybody sees light moving forwards or backwards at the same speed,
24:29
Speaker 1: even if they have velocity relative to the goats. Then
24:33
Speaker 1: that's why people see things happen in different orders, right,
24:36
Speaker 1: That's why they can't agree on whether the light hits
24:38
Speaker 1: the front or the back of the crate at the
24:40
Speaker 1: same time. So the takeaway here is the fact that
24:44
Speaker 1: the speed of light is constant for everybody. Breaks this
24:47
Speaker 1: notion of simultaneity that if two things happen at the
24:50
Speaker 1: same time for Kelly, they also happen at the same
24:52
Speaker 1: time for the goats, and they happen at the same
24:54
Speaker 1: time for Zack or for anybody else. Paying attention. That's
24:57
Speaker 1: no longer true because of this one crucial fact that
25:02
Speaker 1: the speed of light is the same for all observers.
25:05
Speaker 3: I don't like it, but I accept it. It's confusing,
25:09
Speaker 3: it's not confusing. You explained it clearly. It remains sort
25:12
Speaker 3: of counterintuitive. It is, but I followed that explanation. I'm
25:16
Speaker 3: with you.
25:17
Speaker 1: It is, and it's somehow reassuring if you know where
25:19
Speaker 1: it comes from, like it comes from this speed of
25:22
Speaker 1: light fact. It's not just some like arbitrary thing about
25:25
Speaker 1: the universe. In fact, in order for the universe to
25:29
Speaker 1: make sense, to accommodate this speed of light being universal,
25:33
Speaker 1: time has to not be universal for everybody. It doesn't
25:35
Speaker 1: make sense for time to be universal. And this is
25:39
Speaker 1: why we have things like time dilation. Right when we
25:42
Speaker 1: say moving clocks run slow, it's really another statement of
25:46
Speaker 1: the same thing. If I see somebody in a spaceship
25:49
Speaker 1: and they have a clock and they're moving at half
25:51
Speaker 1: the speed of light relative to me, I will see
25:53
Speaker 1: their clock ticking slowly, and they will see my clock
25:55
Speaker 1: ticking slowly. And all of the surprising counterintuitive facts about
25:59
Speaker 1: relativetivity come from this one fact that the speed of
26:03
Speaker 1: light is constant for all observers, which means simultaneidi is
26:06
Speaker 1: not universal, which means clocks run slow or fast depending
26:11
Speaker 1: on your speed relative to them. And so what that
26:13
Speaker 1: means is what we said earlier about how everybody will
26:16
Speaker 1: agree about how long the podcast is. That is not true.
26:19
Speaker 1: What the distance in time between two events is not universal.
26:24
Speaker 1: And it's not just because somebody is listening to the
26:26
Speaker 1: pod at one point five X or whatever, or at
26:29
Speaker 1: half X because we talked too fast or whatever. It's
26:31
Speaker 1: because for them, clocks run faster or slower depending on
26:35
Speaker 1: their velocity relative to them.
26:37
Speaker 3: Okay, so it can be nearly infinite for Matt when
26:40
Speaker 3: he's editing our.
26:41
Speaker 1: Episodes, depending on his velocity with respective of the equipment.
26:46
Speaker 2: Yeah, exactly.
26:47
Speaker 3: So we've gone over all of this in the past,
26:50
Speaker 3: which i'm and i'm it is always useful to go
26:53
Speaker 3: over this again, especially when your co host is as
26:56
Speaker 3: smart as a rock. But when do we get to
26:59
Speaker 3: how this right is different?
27:00
Speaker 1: So remember at the end of last segment we were
27:02
Speaker 1: saying distance in time and classical physics are physical, and
27:05
Speaker 1: we're like, what does physical mean? And I'm saying it's
27:07
Speaker 1: it's a part of the universe. It doesn't depend on
27:10
Speaker 1: where your graph paper is or who's doing the measurement.
27:13
Speaker 1: It's a real thing everybody should agree on. Now we're
27:16
Speaker 1: saying that's not true. Right. Time is not universal, it's
27:19
Speaker 1: not physical, it's observer dependent. It does depend on where
27:22
Speaker 1: you put the graph paper. It does depend on how
27:24
Speaker 1: fast the graph paper is going. So now we're sort
27:26
Speaker 1: of unmoored. We're like, oh no, what's real out there
27:29
Speaker 1: in the universe.
27:30
Speaker 3: Daniel was gaslating this.
27:32
Speaker 1: So what we're going to get to is that time
27:35
Speaker 1: is broken and space is broken, but we can put
27:37
Speaker 1: them back together. And there is a concept of distance
27:41
Speaker 1: in space time which is called the space time interval,
27:44
Speaker 1: which is physical, which is real, which everybody can agree about.
27:48
Speaker 1: So don't worry. We're going to get there. Yes, we've
27:51
Speaker 1: just broken time and we're going to use that to
27:53
Speaker 1: break space and then we're going to put them back
27:55
Speaker 1: together with super glue.
27:57
Speaker 3: Awesome, All right, now let's talk about space then.
28:00
Speaker 1: Yeah. So remember we were saying earlier, like the length
28:03
Speaker 1: of the pencil, that's physical. It doesn't matter if it's rotating,
28:05
Speaker 1: it doesn't matter if your graph is moving, it doesn't
28:07
Speaker 1: matter what rule you use. It is a number and
28:10
Speaker 1: everybody should agree on it.
28:11
Speaker 2: Right.
28:12
Speaker 1: Well, it turns out that's not true, and it's for
28:15
Speaker 1: the same fundamental reason that the speed of light is universal,
28:18
Speaker 1: which breaks time, which is then going to break distance.
28:21
Speaker 1: Because think about how we measure distance. How do we
28:25
Speaker 1: measure distance? Is you measure where the back end of
28:27
Speaker 1: the pencil is, and you measure where the front end
28:29
Speaker 1: of the pencil is, and then you subtract. Very basic, right,
28:32
Speaker 1: How could that go wrong? How could that depend on
28:35
Speaker 1: clocks in any way? Well, implicit in that definition is
28:39
Speaker 1: I'm measuring the back and the front end of the
28:41
Speaker 1: pencil at the same time. Because if the pencil is
28:44
Speaker 1: moving and I measure the back end of the pencil
28:46
Speaker 1: now in the front end of the pencil later, then
28:49
Speaker 1: I'm going to get two numbers that are much further apart,
28:51
Speaker 1: and the pencil is going to look too long, right,
28:53
Speaker 1: And so I have to measure the front in the
28:55
Speaker 1: back at the same time.
28:57
Speaker 3: And you can't cheap by saying I'm in the track
29:00
Speaker 3: the tractors moving forward. But I'm going to take a
29:02
Speaker 3: picture of the pencil so that i can measure from
29:04
Speaker 3: the front in the back at the same time. Now
29:06
Speaker 3: you've got a different frame of reference because it's a picture,
29:09
Speaker 3: and that's cheating, is that right?
29:10
Speaker 1: Okay, yeah, great question. And I love the idea of
29:12
Speaker 1: using a camera. The camera actually makes this more complicated
29:15
Speaker 1: instead of simpler, because now you have the additional issue
29:19
Speaker 1: of how long did it take the information to arrive
29:23
Speaker 1: from the back or the front of the pencil to
29:26
Speaker 1: the camera, And if your camera is not like centered
29:28
Speaker 1: along the pencil, then you're going to get like out
29:30
Speaker 1: of date information in one direction. And usually when we
29:33
Speaker 1: talk about special relativity, we try to remove all of
29:35
Speaker 1: those effects, the effect of light traveling from the event
29:39
Speaker 1: to the observer, because that's a completely separate, different kind
29:43
Speaker 1: of effect, and so all the things we're talking about
29:45
Speaker 1: today assume that basically you have an infinite number of
29:49
Speaker 1: observers at every location who can always measure things locally,
29:52
Speaker 1: so there's no time delay due to the speed of light.
29:55
Speaker 1: So the camera is really tempting, but it actually adds
29:58
Speaker 1: a bunch of confusion, and we can dig into special relativity,
30:00
Speaker 1: including those time delays, but it's sort of like a.
30:02
Speaker 3: Different rabbit hole, all right, different episode.
30:04
Speaker 1: So let's instead say Kelly has two goats, and she
30:08
Speaker 1: can measure the front and back of the pencil with
30:10
Speaker 1: her two goat assistants, and they record, you know where
30:14
Speaker 1: on the ruler it was, and then she subtracts it
30:16
Speaker 1: and she gets a number.
30:17
Speaker 3: Right, thanks short Bread and Newman.
30:19
Speaker 1: No, that's right. But Zack is watching from his ferrari
30:22
Speaker 1: and he says, look, one goat was off. Your goats
30:25
Speaker 1: didn't measure the front in the back of the same time.
30:28
Speaker 1: Because remember Zach doesn't think two things happening at the
30:31
Speaker 1: same time for Kelly are happening at the same time
30:33
Speaker 1: for him.
30:34
Speaker 3: Typical.
30:37
Speaker 1: So see how the speed of light being constant breaks
30:40
Speaker 1: the universality of simultaneti, which breaks the universality of distance. Right,
30:45
Speaker 1: people disagree about when you measure the front and when
30:48
Speaker 1: you measure the back, and that changes you measured distance
30:52
Speaker 1: of these things. So you give just one more simple example.
30:55
Speaker 1: Go back to the pencil. If you measured the back
30:58
Speaker 1: of the pencil to be a two centimeter and the
31:00
Speaker 1: front to be eight centimeters, so you said it was
31:02
Speaker 1: six centimeters long eight minus two. But what if the
31:05
Speaker 1: pencil was moving and you had measured the front before
31:09
Speaker 1: you measured the back, not at the same time. Then
31:12
Speaker 1: the reason the pencil is that eight centimeters when you
31:14
Speaker 1: measured it is because it moved. So because different people
31:19
Speaker 1: disagree about what at the same time means. Because speed
31:23
Speaker 1: breaks simultaneity, then people disagree about when to measure the
31:28
Speaker 1: front and the back, and so they disagree about length
31:31
Speaker 1: because length depends on simultaneity. That's why breaking time also
31:37
Speaker 1: breaks distance.
31:39
Speaker 3: All right, So we can't depend on time, we can't
31:41
Speaker 3: depend on distance.
31:43
Speaker 1: Yeah, so all of these things are now broken. Right.
31:46
Speaker 1: Einstein revealed that time is not universal, space is not universal.
31:50
Speaker 1: So the comfortable footing we were on a few minutes
31:52
Speaker 1: ago seems to have evaporated behind us, right, and so
31:56
Speaker 1: who comes to the rescue but mass.
31:59
Speaker 3: Price math proffs, all right, none of us come to
32:06
Speaker 3: DKAU to be comfortable. We come here to be challenged.
32:09
Speaker 3: So tell me what the math proff had to say.
32:12
Speaker 1: So this is after Einstein had introduced special relativity, which
32:15
Speaker 1: showed us that distance is not universal, that time is
32:18
Speaker 1: not universal. Right, and so then Hermann Minkowski, who's famous
32:23
Speaker 1: for introducing this concept of Minkowski space, and he was
32:26
Speaker 1: a math professor. He actually taught Einstein and Einstein famously
32:30
Speaker 1: skipped some of his lectures, and Minkowski thought Einstein was
32:33
Speaker 1: a quote lazy dog. Anyway, So Minkowski gives this talk
32:40
Speaker 1: a few years later, and he starts off with his
32:43
Speaker 1: famous line saying, quote, henceforth, space by itself and time
32:48
Speaker 1: by itself are doomed to fade away into mere shadows,
32:52
Speaker 1: and only a kind of union of the two will
32:55
Speaker 1: preserve an independent reality. So it's a famous line because
32:59
Speaker 1: what he I'm saying is space is not real. Time
33:02
Speaker 1: is not real by themselves. But you know what we
33:05
Speaker 1: can do is we can link them together and build
33:08
Speaker 1: something new which is physical, which is real, which we
33:12
Speaker 1: can count on.
33:14
Speaker 3: I think ninety nine percent of us can work without
33:17
Speaker 3: space time. But I'm glad the physicists have space suns.
33:21
Speaker 1: It's sort of like if you're studying elephants, you know,
33:24
Speaker 1: and elephants are lined up, so you can either see
33:25
Speaker 1: the head or the tail right, and you like measure
33:28
Speaker 1: the number of heads, and you measure the number of
33:30
Speaker 1: elephants tails, and you have no idea that heads and
33:32
Speaker 1: tails are connected, and you're just like noting, Okay, today,
33:35
Speaker 1: I see ninety heads and ten tails. You come back
33:37
Speaker 1: tomorrow and you see sixty tails and forty heads, and
33:40
Speaker 1: you're like, what is going on the number of elephant
33:42
Speaker 1: heads is changing, the number of elephant tails is changing.
33:45
Speaker 1: What is going on with these elephant heads and tails?
33:47
Speaker 1: And then somebody comes along and tells you heads and tails,
33:51
Speaker 1: neither of these are fundamental. They're at both parts of
33:53
Speaker 1: the same thing. And what's really constant here is the
33:56
Speaker 1: number of elephants, the sum of the heads and the tails.
33:59
Speaker 1: At the elephant. They just turn around, and so you
34:01
Speaker 1: can measure different numbers of heads or tails that you
34:04
Speaker 1: can see, but that doesn't change the number of elephants.
34:08
Speaker 1: That's what's physical, what's real, And that's what Minkowski did.
34:11
Speaker 1: He said, let me bring space and time together. The
34:15
Speaker 1: reason space looks broken and the reason time looks broken
34:18
Speaker 1: is that you're just looking at one part of a
34:20
Speaker 1: larger object that's rotating. So as you see space changing,
34:24
Speaker 1: time is changing in a way that's compensating for space changing.
34:28
Speaker 1: You're just not paying attention to it. It's sort of
34:30
Speaker 1: like your question earlier about the pencil in three dimensional space, right,
34:34
Speaker 1: You're like, if the pencil floats up from the graph
34:36
Speaker 1: paper and now it has three dimensions instead of two.
34:39
Speaker 1: How do I know that those two dimensions are compensating
34:42
Speaker 1: correctly to make sure the distance is the same. A
34:46
Speaker 1: two dimensional physicist who only could see like two of
34:49
Speaker 1: those dimensions. As the pencil floats up, he would say, no,
34:51
Speaker 1: the pencil is getting shorter. You'd be like, that's just
34:54
Speaker 1: because you forgot to account for the third dimension. Right,
34:56
Speaker 1: the pencil is the same length. That's what's happening here.
35:00
Speaker 1: Spa may be getting weird and short, but time is
35:02
Speaker 1: out there compensating for it. So if we put space
35:04
Speaker 1: and time together in some way, then we can build
35:08
Speaker 1: a new sense of distance, which really is universal, which
35:12
Speaker 1: really is invariant, which really is physical.
35:15
Speaker 3: Well, One, I love when you recontextualize my dumb questions
35:19
Speaker 3: to make them seem insightful, much appreciated. And two, let's
35:23
Speaker 3: take some time to think about space. And we're going
35:25
Speaker 3: to take a break, and when we get back, Daniel
35:27
Speaker 3: is going to really dig in to what a space
35:30
Speaker 3: time interval is. All Right, we are having fun at
35:53
Speaker 3: this space in this time, and Daniel is going to
35:56
Speaker 3: help us understand the complicated concept that is space time interval.
36:00
Speaker 3: So space and time makes sense for all of us again.
36:03
Speaker 1: All right, so we are building space time. We have
36:06
Speaker 1: three dimensions. We want to add time to it. Earlier
36:09
Speaker 1: I said, it's no big deal to add more dimensions.
36:12
Speaker 1: You can just add them to our Pythagorea and theorem. Right,
36:15
Speaker 1: you have X squarre plus y squared. No problem, we
36:17
Speaker 1: can add z squared. It's all cozy, it's all happy,
36:20
Speaker 1: it's easy. So probably you're thinking, great, now we just
36:23
Speaker 1: have to add time. We have X squared plus y
36:26
Speaker 1: squared plus D squared plus t squared. Right, Yes, Minkowski says, no,
36:32
Speaker 1: that's wrong. We do have to account for time, but
36:35
Speaker 1: time enters with a negative. So the concept of a
36:39
Speaker 1: space time interval is distance in space minus the distance
36:44
Speaker 1: in time. It's x squared plus y squared plus z
36:47
Speaker 1: squared minus t squared. What this weird negative sign? Yes, exactly,
36:53
Speaker 1: all right, So.
36:53
Speaker 3: I'm thinking about the lights in the box. I'm thinking
36:56
Speaker 3: about my goats in the box. Do subtracting time depend
36:59
Speaker 3: on like what distance you're moving in because, like I'm imagining,
37:04
Speaker 3: you know, when we measured the pencil and the distance
37:06
Speaker 3: was shorter because you had been moving in a certain direction.
37:09
Speaker 3: Does that have anything to do with why time is
37:11
Speaker 3: negative or is my intuition yet again totally off here.
37:16
Speaker 1: Now your intuition is right on. It all comes from
37:19
Speaker 1: the same origin of how space and time connect to
37:22
Speaker 1: each other, which is fundamentally because of how the speed
37:26
Speaker 1: of light is invariant for all observers. But there's a
37:30
Speaker 1: little bit of a twist here. What we're building, this
37:32
Speaker 1: space time interval is not exactly like a four dimensional
37:36
Speaker 1: analog of distance, right, It's not like how far apart
37:40
Speaker 1: these two points are. So let's make that explicitly. Say
37:43
Speaker 1: what are we talking about here? What is it space
37:45
Speaker 1: time interval? Imagine you have some event, So a goat
37:48
Speaker 1: is born, that happens at a location and a time.
37:51
Speaker 3: Yay, we all celebrate.
37:52
Speaker 1: And you have another event, a goat is crushed. Right,
37:56
Speaker 1: That happens at some location and at some time. Right,
38:00
Speaker 1: And we can talk about the distance between those two events. Right,
38:03
Speaker 1: Maybe the goat was born in Kelly's farm and the
38:05
Speaker 1: goat was crushed at the vet. There's a certain distance there,
38:08
Speaker 1: and we can talk about the time interval between those two. Right,
38:12
Speaker 1: the goat was born and three weeks later it was
38:14
Speaker 1: crushed or whatever. Now we can also talk about the
38:17
Speaker 1: space time interval between those two events, those space time events,
38:21
Speaker 1: and now four dimensional coordinates in space time, we can
38:24
Speaker 1: talk about the space time interval, and that has the
38:28
Speaker 1: distance component and the time component, but in a negative, right,
38:32
Speaker 1: So the time adds in a negative. It's not just
38:34
Speaker 1: like a straight up distance. And you might think, well,
38:37
Speaker 1: why would you do that? Why would you add the negative?
38:40
Speaker 1: That doesn't give me some concept of distance? Now, it's
38:43
Speaker 1: like space and time playing off each other, right, Well,
38:46
Speaker 1: the reason is that that's the physical thing. That's what
38:48
Speaker 1: the universe wants. That's the thing that's invariant. If I
38:52
Speaker 1: calculate the distance in space and I calculate the distance
38:55
Speaker 1: in time, from my point of view, I'm going to
38:57
Speaker 1: get a number for the interval. Does it, She's going
39:00
Speaker 1: to get the same number. If Zach and his Ferrari
39:03
Speaker 1: does it, he's going to get the same number. We're
39:05
Speaker 1: all going to disagree on distances, we're all going to
39:08
Speaker 1: disagree on times, but when we put them together into
39:10
Speaker 1: the space time interval, we're all going to get the
39:12
Speaker 1: same number. The same way that, like the pencil has
39:15
Speaker 1: a length in space, it doesn't matter right who's measuring
39:19
Speaker 1: it or how it's rotating or whatever. The space time
39:22
Speaker 1: interval is the physical thing in space time. It's what
39:25
Speaker 1: the universe respects. So us little humans we want senses
39:29
Speaker 1: of distance, and we'd love to add time to space
39:32
Speaker 1: with a positive number, so we have like a four
39:34
Speaker 1: dimensional pythagorea in theorem. But no, the universe says this
39:38
Speaker 1: is the thing that's important.
39:40
Speaker 3: Is it reasonable to think about it as like distance
39:43
Speaker 3: standardized by time or not quite? I feel like the
39:46
Speaker 3: answer is not quite right.
39:48
Speaker 1: So we've answered one question, which is like what is
39:50
Speaker 1: the space time interval. It's this weird mathematical construction where
39:54
Speaker 1: you have space and time playing against each other. And
39:56
Speaker 1: we've said it's important because the universe respects it. Now
40:00
Speaker 1: the question is like, well what is this thing? Right?
40:03
Speaker 1: Like what do I do with this thing? How do
40:05
Speaker 1: I think about it? So it's not a distance again
40:08
Speaker 1: because time is playing in the interval. The way to
40:11
Speaker 1: think about it is causal separation, right, because there are
40:15
Speaker 1: three different kinds of space time intervals. You can have
40:19
Speaker 1: a space time interval that's zero, which is what we
40:22
Speaker 1: call light light, and that's when the space and the
40:24
Speaker 1: time balance each other. So like the distance minus the
40:28
Speaker 1: time because you have a positive and negative they can
40:30
Speaker 1: be zero. Two events have a space time interval of
40:34
Speaker 1: zero if light can leave one and arrive at the other.
40:37
Speaker 1: So for example, Kelly turns on a flashlight and then
40:41
Speaker 1: the flashlight arrives at the end of a barn. Those
40:44
Speaker 1: two events have a zero spacetime interval because they're exactly
40:49
Speaker 1: causally connected. Right the interval between them is exactly how
40:53
Speaker 1: long it would take light to go from one to
40:55
Speaker 1: the other. So events have zero space time intervals if
40:58
Speaker 1: they are perfectly causally linked, just enough time and not
41:02
Speaker 1: too much a message. Departing and arriving are two space
41:07
Speaker 1: time events with zero space time interval. They have a
41:11
Speaker 1: distance in space and a distance in time, but those
41:15
Speaker 1: balance to give no interval. Events that have a zero
41:18
Speaker 1: space time interval from you are events that you can
41:22
Speaker 1: exactly reach with your beam of light. Close by things
41:26
Speaker 1: in the near future have a zero space time interval.
41:28
Speaker 1: Far away things in the far future. It's basically your
41:32
Speaker 1: forward light cone. Okay, you can also have positive space
41:36
Speaker 1: time intervals right So, for example, Andromeda right now is
41:41
Speaker 1: too far away for us to affect it causally, Kelly
41:44
Speaker 1: turns on her death ray from her farm. She cannot
41:47
Speaker 1: zap in the aliens in Andromeda right now because the
41:50
Speaker 1: death ray is still traveling there. So the distance between
41:53
Speaker 1: Kelly's farm and Andromeda has a positive space time interval.
41:58
Speaker 1: It's outside the realm of anything she can influence because
42:03
Speaker 1: her death ray, which with the speed of light, can't
42:05
Speaker 1: reach it until the future. Even moving at the speed
42:08
Speaker 1: of light is not fast enough. It's just too far away.
42:11
Speaker 3: But so when I'm standing in a barn, I turn
42:13
Speaker 3: on my flashlight and the light hits the opposite wall.
42:17
Speaker 3: That happens in the future too, but it's just a
42:19
Speaker 3: shorter future.
42:20
Speaker 1: It happens in the future. But you're right, it's a
42:21
Speaker 1: shorter future. And so you can imagine. We talk sometimes
42:24
Speaker 1: about a light cone. Right, think about the future that
42:26
Speaker 1: you can influence. Okay, you can influence exactly where you
42:30
Speaker 1: are right now, or you can influence things that are
42:34
Speaker 1: one light second away in one light second, or things
42:37
Speaker 1: that are one light year away in a year. Right,
42:41
Speaker 1: that's your future light cone. That light cone has space
42:45
Speaker 1: time interval of zero because the speed of light is
42:47
Speaker 1: the same for all observers. Everybody agrees on what Kelly's
42:50
Speaker 1: light cone is you turn on the flashlight and it
42:53
Speaker 1: does hit the side of the barn, and that event
42:56
Speaker 1: where the light reaches the other side of the barn
42:59
Speaker 1: is on your future l like cone. It's in your
43:02
Speaker 1: causal reach. It has zero space time interval, and an
43:07
Speaker 1: event much further away at that same moment is outside
43:11
Speaker 1: your light cone. You can't affect it. Light doesn't have
43:14
Speaker 1: time to reach it. It has a positive space time interval,
43:18
Speaker 1: and everyone agrees on that. Unlike distance and time, the
43:22
Speaker 1: space time interval is the invariant thing. Can Kelly zapp
43:27
Speaker 1: this alien and when is a number everybody agrees on,
43:30
Speaker 1: no matter where they are in the universe or how
43:32
Speaker 1: fast they're going. Everything outside that cone, Kelly cannot zap.
43:37
Speaker 1: It's beyond her causal reach. Everything within that cone has
43:41
Speaker 1: a negative space time interval. So that's what the space
43:45
Speaker 1: time interval is about. It's about what can you influence
43:48
Speaker 1: where is your causal reach. Space time intervals that are
43:51
Speaker 1: negative are time like things you can reach. Space time
43:55
Speaker 1: intervals that are zero are where light will reach, and
43:57
Speaker 1: space time intervals that are positive are outside of your reak.
44:01
Speaker 1: You cannot influence them because life cannot get from you
44:04
Speaker 1: to them in time. So it's sort of like a
44:08
Speaker 1: sense of distance, but it's more like a sense of
44:10
Speaker 1: causal distance. It's like, is this something I can influence
44:14
Speaker 1: in the future.
44:16
Speaker 3: I think I'm still a little caught up on Like,
44:18
Speaker 3: so if I turn on a laser, it's gonna get
44:21
Speaker 3: to Andrameda eventually, Yes, And so why can't we just
44:25
Speaker 3: wait more time?
44:26
Speaker 1: You can wait more time, absolutely, but you have to
44:29
Speaker 1: wait more time to get further. And so you can't
44:33
Speaker 1: zap Andrameda now. You have to zap it in the future,
44:37
Speaker 1: just like you can't zap the other side of your
44:39
Speaker 1: barn now. You have to wait for life to get there.
44:42
Speaker 1: So there's no limit on how far you can zap,
44:45
Speaker 1: but there is a limit on how far you can
44:48
Speaker 1: zap in a given time, in a second or in
44:50
Speaker 1: ten seconds or whatever. If the universe wasn't expanding, then
44:54
Speaker 1: the whole universe is eventually in your light cone, your
44:58
Speaker 1: future light cone, right, is that anybody? It just takes
45:01
Speaker 1: time for those death rays to get there. But everywhere
45:05
Speaker 1: along the bleeding edge of your death rays has the
45:08
Speaker 1: same space time interval. That's zero because the distance and
45:12
Speaker 1: the time portions balance perfectly. Because we're outing up distance
45:16
Speaker 1: and we're subtracting time, and the space time conversion we
45:19
Speaker 1: used to subtract time is the speed of light that
45:21
Speaker 1: enters into that equation. And so the space time interval
45:25
Speaker 1: between two events basically tells you, kid, light have gotten
45:28
Speaker 1: here between these two events or not. If it's zero,
45:31
Speaker 1: then light has just enough time to get there. If
45:33
Speaker 1: it's positive, then it's too far for light to get to.
45:36
Speaker 1: You can't influence it. If it's negative, then light can
45:39
Speaker 1: definitely influence it.
45:40
Speaker 3: So is Andromeda moving away from me faster than my
45:44
Speaker 3: laser is going.
45:45
Speaker 1: And Drama is actually moving towards us right because of gravity,
45:49
Speaker 1: So it's going to collide with us in a few years.
45:51
Speaker 1: But that's sort of adding a few billion years. I
45:54
Speaker 1: say a few years. We got time, don't worry about it.
45:58
Speaker 1: And please don't build a laser in points at the aliens.
46:00
Speaker 1: We hope the aliens are friendly. I'm just using that
46:02
Speaker 1: as an example. Yeah, yeah, And this isn't about Andromeda moving.
46:06
Speaker 1: It's not that it's escaping your laser. It's just that
46:09
Speaker 1: it's too far away for you to hit it now
46:12
Speaker 1: or in a year, your laser won't get there. For
46:15
Speaker 1: millions of years, and so the events your laser can
46:19
Speaker 1: visit are only in the future. Those are the ones
46:22
Speaker 1: with space time interval zero.
46:24
Speaker 3: Okay, So I get one space time value for shining
46:28
Speaker 3: a laser at my barn and a different space time
46:32
Speaker 3: interval value by shining a laser at Andromeda. And I'm
46:36
Speaker 3: still not understanding why they have different signs. I can't
46:39
Speaker 3: remember if one was zero one was probably I've lost track,
46:42
Speaker 3: but like, why are those different?
46:44
Speaker 1: Okay, Well, let's stick with your barn for now, and
46:46
Speaker 1: then we'll get to Andromeda.
46:47
Speaker 2: Okay.
46:48
Speaker 1: So if you turn your flashlight on now and you
46:52
Speaker 1: shine it at the other wall right, then you can
46:55
Speaker 1: ask what is the space time interval between me turning
46:59
Speaker 1: on my life and the other side of my barn
47:02
Speaker 1: right now before the light has a chance to get there?
47:05
Speaker 1: And that is a positive space time interval because the
47:08
Speaker 1: light has not had a chance to get to the
47:09
Speaker 1: other side of the barn. Right. But if you ask,
47:12
Speaker 1: what is this space time interval between me turning on
47:14
Speaker 1: my flashlight and the other side of my barn in
47:18
Speaker 1: one millisecond right or wherever long it takes light to
47:21
Speaker 1: get there, that has a zero space time interval. So
47:24
Speaker 1: there's two different space time intervals there. There's the one
47:27
Speaker 1: between your flashlight now and the other side of the
47:30
Speaker 1: barn now, or between your flashlight now and the other
47:33
Speaker 1: side of your barn when the light gets there in
47:35
Speaker 1: the future. Okay, so one is zero because light has
47:39
Speaker 1: had a chance to get there. But if you ask,
47:42
Speaker 1: what is the space time interval between my flashlight and
47:45
Speaker 1: the other side of the barn now before light has
47:47
Speaker 1: had a chance to get there, that's a positive space
47:49
Speaker 1: time interval. That's outside of your light cone. So the
47:52
Speaker 1: same location can be inside or on or outside the
47:55
Speaker 1: light cone, depending on the time, because the light cone
47:58
Speaker 1: expands with Time's true for Andromeda, right if you fire
48:02
Speaker 1: your deathray right now and ask is Andromeda within my
48:05
Speaker 1: light cone, well it isn't right now, but in five
48:08
Speaker 1: million years or whatever, it will be, So that has
48:11
Speaker 1: a space time interval that's zero. But firing your deathray
48:15
Speaker 1: right now and Andromeda right now, that has a positive
48:18
Speaker 1: space time interval. So basically just tells you which parts
48:21
Speaker 1: of the universe can use ZAM, which parts are accessible
48:24
Speaker 1: to you because of this limited speed of light.
48:27
Speaker 3: Got it.
48:28
Speaker 1: So that's something interesting about the universe, right, Like we
48:32
Speaker 1: should listen when the universe tells us, hey, this is
48:35
Speaker 1: the physical thing, this is what's invariant, this is what's important.
48:38
Speaker 1: Like when we notice that momentum is conserved, we say, okay,
48:41
Speaker 1: that means something useful about the universe, and it does.
48:44
Speaker 1: It tells us something fascinating about space. Check out our
48:47
Speaker 1: episode about Noether's theorem to understand the connections between symmetries
48:51
Speaker 1: and conservation laws. But this is a really good way
48:53
Speaker 1: to get deep insights about the nature of the universe
48:56
Speaker 1: is to say, what does the universe respect, what does
48:59
Speaker 1: it conserve, what does it keep constant? In this case,
49:02
Speaker 1: we're not talking about conservation, we're talking about invariance. What
49:05
Speaker 1: does the universe say everybody has to agree on no
49:08
Speaker 1: matter how fast your ferrari or your spaceship is, everybody's
49:12
Speaker 1: going to agree on Kelly's light cone, can she reach
49:15
Speaker 1: Andromeda worth her death ray? Not now, but yes, in
49:19
Speaker 1: the future. That's something everybody agrees on, no matter how
49:22
Speaker 1: fast they're going. And so that's interesting, that's fascinating, and
49:26
Speaker 1: it's connected to the idea of distance, right. But it's
49:29
Speaker 1: clear that like spatial distance itself is not something the
49:33
Speaker 1: universe insists everybody agree on. It's all twisted and broken
49:38
Speaker 1: by high speeds and relativity. So this is something the
49:42
Speaker 1: universe respects, and that means that it's closer to the truth.
49:46
Speaker 1: It's like at the heart of the universe. Fundamentally, it
49:49
Speaker 1: tells us something about the geometry of the universe. Remember
49:53
Speaker 1: when we were adding up the pieces of the Pythagorean
49:55
Speaker 1: theorem for space, and as we add more dimensions, we
49:57
Speaker 1: just add them the same way. Because all the dimensions
50:00
Speaker 1: are the same x and y and z, it don't matter. Well,
50:03
Speaker 1: time is a dimension and it's a part of space time,
50:07
Speaker 1: which is four dimensional, but it's different. The geometry of
50:10
Speaker 1: time and space are different with respect to each other.
50:14
Speaker 1: That's what the universe is saying here.
50:16
Speaker 3: Okay, so, as you said, this is like fundamental and important.
50:20
Speaker 3: But when we get the explanation about how distance is
50:23
Speaker 3: invariant and time is invariant, it's not usually followed by
50:27
Speaker 3: this explanation for well, how do you make it all
50:30
Speaker 3: variant or whatever? Why don't we usually get that other step?
50:34
Speaker 3: Is it just because it's hard to explain or what
50:36
Speaker 3: do you think?
50:37
Speaker 1: Yeah, I think this concept of a space time interval
50:39
Speaker 1: is a little bit abstract because there's no real intuitive
50:42
Speaker 1: and analog for it, Like, what is this thing we're
50:44
Speaker 1: talking about, Like, yes, the universe respects it, but it's
50:48
Speaker 1: not something that lines up with our notions very nicely.
50:51
Speaker 1: And so I think that's why it's not typically like
50:54
Speaker 1: introduced in POPSI, whereas you know, like the universe respects
50:58
Speaker 1: energy or doesn't respect energy or momentum. These are things
51:00
Speaker 1: we can get a handle on, so it's easier to
51:03
Speaker 1: talk about. But the space time interval really is foundational
51:07
Speaker 1: to the whole concept of the geometry of the universe,
51:09
Speaker 1: which is what relativity is all about. That's really what
51:12
Speaker 1: Menkowski was saying to Einstein. Yeah, you broke space and
51:16
Speaker 1: you broke time, but really you showed us that these
51:19
Speaker 1: are two sides of a more complicated object. You don't
51:22
Speaker 1: stick them together trivially. You stick them together backwards, and
51:26
Speaker 1: the universe has this weird geometry and that's how it
51:29
Speaker 1: organizes itself. That's the internal mechanism of the mathematics of
51:33
Speaker 1: the universe. And this turned out to be really crucial
51:36
Speaker 1: for Einstein's next step in general relativity, because in special relativity,
51:41
Speaker 1: these interval formulas are pretty simple. You just take the
51:44
Speaker 1: differences in each spatial dimension, you take the difference in
51:46
Speaker 1: the time dimension, you add them up with a negative,
51:49
Speaker 1: and you do the pythegonorary and theorem. You're good. In
51:52
Speaker 1: general relativity, the locations don't add up all the same way.
51:55
Speaker 1: Some locations add up more and some locations add up less.
51:58
Speaker 1: That's what we mean by the metrical talk about sometimes
52:00
Speaker 1: in general relativity, how is space curved in this location?
52:04
Speaker 1: How much space time interval do you crew by passing
52:07
Speaker 1: through this space versus passing through that space? And so
52:11
Speaker 1: this concept of having a geometry to the universe and
52:13
Speaker 1: then having that geometry be non trivial, that's how you
52:17
Speaker 1: go from special relativity to general relativity. But you got
52:20
Speaker 1: to get your brain like the geometry of the universe
52:23
Speaker 1: first before you can even get to general relativity.
52:26
Speaker 3: Well, this conversation didn't crush my spirits. It lifted them up.
52:30
Speaker 3: And now I have I guess you can't really make
52:32
Speaker 3: a positive space time because then that would be too
52:35
Speaker 3: far to travel. And so I'm still going to be
52:38
Speaker 3: thinking about this tonight. But I totally followed the explanation,
52:41
Speaker 3: and that's awesome. I learned something new today.
52:44
Speaker 1: And one more speed bump to avoid is you often
52:46
Speaker 1: hear people saying that photons do not experience time. And
52:49
Speaker 1: here what we're saying is light can go from A
52:52
Speaker 1: to B with no space time interval. Right, it's lightlight,
52:57
Speaker 1: it's a zero space time interval along the light. That's
53:01
Speaker 1: not the same as saying photons do not experience time.
53:04
Speaker 2: Right.
53:05
Speaker 1: Saying photons have a zero space time interval is very
53:08
Speaker 1: different from saying photons do not experience time. There's a
53:11
Speaker 1: whole separate POPSI rabbit hole. But photons do take time
53:15
Speaker 1: to move across the universe. They just don't take space time.
53:19
Speaker 3: I'm realizing. I want to ask, if space time intervals
53:24
Speaker 3: have units, does it make sense to think of it
53:26
Speaker 3: as a measurement with a unit.
53:29
Speaker 1: Yes, it does. In the units we use are distance
53:32
Speaker 1: right like length like meters for example, and time enters
53:36
Speaker 1: with a conversion factor, which is the speed of light.
53:39
Speaker 1: So it's not really time, it's the speed of light.
53:41
Speaker 1: Time's time ct which is why photons have a space
53:45
Speaker 1: time interval of zero, because they move at the speed
53:48
Speaker 1: of light, so that conversion factor ensures that the space
53:51
Speaker 1: and the time parts balance perfectly. Way to go, mass
53:54
Speaker 1: harmonious mathematics. Love it all right? Well, thank you everybody
53:59
Speaker 1: for going on this journey through space and time and
54:02
Speaker 1: through space time. Thank you Kelly for all the great
54:04
Speaker 1: questions and for not using the crusher on today's episode.
54:08
Speaker 3: Thank you Daniel for the great explanation. I will keep
54:10
Speaker 3: the crusher at home with me.
54:13
Speaker 1: You guys, hear the sound of that thing.
54:15
Speaker 2: Oh my gosh, oh gosh.
54:18
Speaker 1: All right, and that's me signing off with a shiver.
54:22
Speaker 3: Yep, that's me signing off as well. I don't think
54:24
Speaker 3: I'm gonna be able to use this thing on anything
54:26
Speaker 3: actually living without Yeah, I'm not doing it. All right, everybody,
54:30
Speaker 3: Until next time, have a good space and have a
54:33
Speaker 3: good time.
54:41
Speaker 1: Thanks everybody for listening. Please go and do us a
54:43
Speaker 1: favor and rate the show on whatever podcast app you're using.
54:46
Speaker 1: It really helps people find us.
54:48
Speaker 3: Daniel and Kelly's Extraordinary Universe is edited by the amazing
54:52
Speaker 3: Matt Kesselman.
54:53
Speaker 1: He really is a wizard. You can also find us
54:57
Speaker 1: online on Blue Sky, Instagram, and d and K Universe.
55:02
Speaker 1: Come engage with us.
55:03
Speaker 3: You can email us at questions at Danielankelly dot org.
55:07
Speaker 3: We really do want to hear from you, and you.
55:09
Speaker 1: Can find our website www dot danieland Kelly dot org,
55:13
Speaker 1: where you'll also find an invitation to join our discord
55:16
Speaker 1: where everybody comes and talks about the amazing universe.
55:20
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55:24
Speaker 3: an iHeart podcast. Thanks for joining us.