Can Time Really Slow Down? (Patreon)

[This is a transcript with references.]

You may live in a different time-zone than I, but one hour for me is one hour for you. An hour might feel longer if it’s a job interview, but even the most awkward conversation doesn’t actually slow down time. And yet, Einstein said, time can slow down – it doesn’t always pass at the same rate. Just what did Einstein say about the passage of time? What’s the resolution of the twin paradox, what’s up with Newton’s bucket, and what does it all mean? That’s what we’ll talk about today.

When I was a teenager, I was super interested in Einstein’s theories of space and time. And I read a lot of popular science books about them. I didn’t understand a thing.I then went on to do a PhD in physics.  And today it’s my turn trying to explain it to you.

The most important part of Einstein’s theories is that they combine space and time to one common entity, space-time. This idea didn’t come from Einstein but from Minkowski, but Einstein was the one to understand what it means. Which is why today you can buy a bobble-head Einstein but not a bobble-head Minkowski. Sorry Minkowski.

Einstein was the one to understand that if you combine space with time, then time becomes a coordinate, like space. That space is a coordinate means we can put distance markers on it, and we can choose them as we want. Once we have chosen some agreed-upon markers, we can use them to communicate where we are. You can give me you GPS coordinates, and I’ll know where to find you. But those coordinates don’t tell me what the length of the path is that will take me there. Because there could be many different paths. The coordinates alone don’t tell me.

If time becomes a coordinate, the same happens to time. If you give me your coordinates in space and in time, I will know where and when to find you. But the coordinates don’t tell me the length of the path that brings me to you, and they also don’t tell me how long it’ll take me to get there.

Wait, you may say, if it’s 5am now and I tell you we’re meeting at 5pm then that’ll take 12 hours right? No, wrong. Those 12 hours are your coordinate time. They are just convenient markers. They are convenient for you because they agree with the time that actually passes for you. But how much time passes for me while I get to our meeting depends on how I get there. It’s just that, the difference between the passage of your time and my time as I come to meet you is normally so small we don’t notice. It’d only become noticeable if I was moving close to the speed of light, and my doctor advised me against it.

If you’ve been keeping track, three dimensions of space plus one dimension of time makes 4 dimensions. Unfortunately YouTube doesn’t support 4-D graphics, at least not yet, so we’ll have to settle on just one dimension of space and one dimension of time. Just imagine that it describes something that can move only left or right. Like American politics maybe.  

Here’s what this space-time looks like. Not particularly impressive, I know. Actually, it looks exactly like it’s just two dimensions of space. Like a map or something.

That’s what it looks like, but looks can be deceiving, and in this case they are. For one thing, in space-time, if you don’t move, then you still make a line, namely one that just goes straight up. If you move at some constant velocity, then you can describe that by a straight line tilted at some angle. By convention, a 45 degree angle marks the speed of light. Why 45 degrees? Why not? Maybe it’s physicists’ idea of having fun.

But more importantly, space-time differs from space in one crucial way, which is how you calculate distances in it. If you have two dimensions of space, one called x and the other y, with two points in them, then calculating the length of a straight line between them is straight forward.

You take the distances between the coordinate labels of x, call that delta x, and the same for y, call that delta y, and then the distance between the points is the square root of the sum of the squares, as we learned in kindergarten. This is called the “Euclidean Distance”.

In space-time it doesn’t work like this. In space-time we deal with combinations of coordinates in space and time, which we call “events”.  If you have two events, each with a position in x and a time in t, and you want to calculate the space-time distance between them, then again you take the differences between the coordinate labels.

In this case, that’s delta x and delta t. And then you take the square root of the square of delta t MINUS the square of delta x divided by c, where c is the speed of light. This minus makes all the difference. Basically, the rest of this video is to explain what the consequences of this minus are. This is also called the Lorentzian Distance, after Hendrik Lorentz, that’s Lorentz with t z.

Why is there a minus there? Because it works. This is how theories in physics come about. You have an idea, you check the predictions, you stick with your idea or toss it out. This one was a keeper.

To see what this means, let’s look at points that have the same distance from a reference point that we’ll take to be the origin of the coordinate system. Let’s say for example the distance is equal to one, in whatever units you have chosen. If these were coordinates in a two-dimensional space, then all points at the same distance from the origin would be on a circle. Different distances correspond to circles with different radii.

But if you do this in space-time, then all points at the same space-time distance from the origin are hyperbole. You can’t move on those lines because that’d require you to move faster than light. But you could move on one of those. That would require a constant acceleration.  

The key to making sense of space-time is now this. The time that passes for an observer moving on any curve in this space-time is the length of that curve, calculated using this peculiar notion of Lorentzian distance that we just discussed. It’s called the “proper time”. It’s the proper way of calculating time according to Einstein.

If you move between two events at constant velocity, then the time which passes on the way is, as we previously saw, the square root of delta t square minus delta x over c square. If the observer doesn’t move on a straight line, you break up the line into small straight pieces, sum over them, and take a limit. So you integrate over the curve.

Don’t worry if you don’t know how to do that integral, there’s no exam at the end of the video. You just need to know that this peculiar distance in space-time measures how much time passes as an observer moves from one event to another. The time that passes between two events is not the time on the coordinate axis. That’s just a label, it’s convention, it has no physical meaning. So the consequence of making time into a coordinate is that the physically relevant time becomes a personal thing. It depends on how you move around.

In the special case when an observer sits still according to these space coordinates, the time that passes for them is the same as the coordinate time. Because then you have the proper time is the square root of the square of delta t, which is just delta t. But in general, the two are not the same.

Confusing the coordinate time with the proper time is the reason for most misunderstandings of Einstein’s theories that I have come across. Let’s draw two observers into this diagram, one who just sits at rest, let’s call her Alice. And one who moves with constant velocity to the right, that’s Bob. You can then ask, for example, how much time has passed for Bob until he reaches the same coordinate time as Alice, that would be when he’s up here. Remember that the time that passed is the length of this line. And the length is constant on a hyperbole. So it’s the same on all those hyperbole.

Then you come to the odd conclusion that when Bob reaches the same coordinate time less proper time has passed for him than for Alice. Yes, the line that looks longer is actually shorter according to this space-time distance! This is what people often call “time dilation”. I would call that pseudo-time dilation because it’s a meaningless comparison. Why would you care about the coordinate time? It’s just a label, it has no physical meaning for Bob.

If you confuse the two times you run into a problem because Einstein also taught us that absolute velocities have no physical meaning. It is only physically meaningful to talk about differences in velocities.

If you go for a walk, you can say you are moving relative to earth at a speed of 5 kilometers per hour. Or you could say the Earth, including the ground and air and everything, is moving relative to you at 5 kilometers per hour. Physically that means the same thing. Another way to say this is that there’s no such thing as absolute rest. No, not even in the cemetery.

And because absolute velocities are not physically meaningful, both Alice and Bob could claim to be at rest. At rest with themselves. According to Bob, he is in rest and Alice moves to the left. Now he could ask how much time passed for Alice until she reaches the same coordinate time as I. And he would come to the conclusion that less time has passed for Alice. So now what? Has less time passed for Bob or less time passed for Alice? Well, that’s a meaningless question because you’re just comparing coordinate times for different events.

This confusion of coordinate time with proper time is the origin of the “twin paradox” which goes as follows. Alice and Bob are identical twins. Genderfluid, I guess. Bob goes on a trip to Andromeda, takes a selfie as he passes by the supermassive black hole, and hurries back home to post it on twitter. When he comes back, who is older, Bob or Alice?

If you think that time dilation comes from the relative velocity between them, then each must be older than the other which makes no sense.

The resolution of the twin paradox is usually to point out that actually the situation is not symmetric. Because for Bob to make a round trip he cannot move at constant velocity. He needs to accelerate to turn around. And just as a reminder, acceleration is a change of velocity, not a change of speed. Velocity has a direction, so changing direction at constant speed is also an acceleration. And acceleration is not relative, acceleration is absolute.

What does it mean that acceleration is absolute? It means you can measure whether you are accelerated or not. One way to do this is with a spring. A spring will stretch when acceleration is acting on it, not when it’s moving at constant velocity. Bob could take a spring along to figure out if his motion is physically different from Alice’s.

If you want to know how much time passes for Bob on his trip, you have to integrate this weird distance measure over the curve he is moving on. And the time that passes on Bob’s trip will always be shorter than the time that passes for Alice at home, doesn’t matter just which way he travels This is the real time dilation. It comes from the acceleration.

Why is the time on Bob’s trip always shorter than Alice’s? I briefly mentioned this previously in my video about the principle of least action. In space-time, if you want to know what path an object takes from one event to another then that’s always the path with the largest proper time. Now we know that Alice is at rest and no forces act on her and therefore her path between the two events is that with the largest proper time. Consequently, any other path would have to have a force acting on the object, and it would have a shorter proper time. And there’s nothing paradoxical about it because they are physically different.

I sometimes read that the resolution for the twin paradox is that one must include gravity, but that’s wrong. It’s got nothing to do with gravity. I believe this wrong explanation comes from another confusion, which is to think that Einstein’s Special Relativity can only deal with constant velocities, and if you want to deal with acceleration, you need General Relativity. This is just wrong. One can totally describe acceleration in special relativity. The difference between the two is that in Special Relativity space-time is flat, whereas in General Relativity it is, in general, curved. If it’s curved, that describes gravity. You don’t need either General Relativity or gravity to solve the twin paradox.

I also sometimes read that Bob must start at rest with Alice, so he has to speed up first. And that’s correct indeed if you want the story to work. But it’s not relevant for sorting out the physics. It’s sufficient if Bob and Alice meet at the same initial and final place, they don’t also need to have the same velocity.

By the way, that acceleration is absolute is also why Newton’s bucket paradox isn’t a paradox. What’s up with Newton’s bucket?

Newton pointed out the following. Take a bucket and fill it half with water. The water will sit there with an even surface. So far, so unsurprising. Now spin the bucket. The water will be pushed against the sides and form a dip in the middle. It’ll also start spinning with the bucket. Newton now said, when you filled the water into the bucket, the water was at rest with the bucket. When they both spin, they are also at rest with each other. And yet the water has a different shape. If the relative velocity is zero in both cases, then there must be something else explaining this shape. Mach later argued that the reason is that the bucket with the water moves relative to the rest of the universe.

It's arguably true that the spinning bucket moves relative to rest of the universe, but this isn’t the reason why the water behaves differently in both cases. The reason it behaves differently is that a change of direction is also an acceleration. The water is accelerated because it goes in a circle. And acceleration is absolute. The one case is accelerated, the other not. There’s nothing paradoxical about both cases being different.

Okay, now that we have figured out the twin paradox and Newton’s bucket, what’s with time that slows down near black holes? To understand that, you have to know one more thing: gravity is not a force. Gravity is caused by the curvature of space-time. What does it mean that gravity is not a force? It means that if you fall in a gravitational field, you are not accelerated. Because there’s no force acting on you.

Einstein allegedly had this insight when a man fell off a roof and reported feeling weightless. Whether that anecdote is true or not, he captured it in his thought experiment with the elevator. If you’re in an elevator that falls with you in a gravitational field, you have no idea if there’s gravity at all. And likewise, if the elevator is pulled up and accelerates upwards, then you can’t distinguish that from being in a gravitational field. If you are being pulled up, there is a force that acts on you, but it’s not gravity. Because gravity is not a force! The force that acts on you is that from the floor pushing you up.

Same thing with earth’s gravity. At the moment, there is no gravitational force acting on you. Because gravity is not a force. There’s a force acting on you from the ground, which is the reason why you are not freely falling into the center of earth. Since there’s a force acting on you, you are accelerated. Acceleration is absolute. And acceleration causes time-dilation. So being near a gravitating body causes time to slow down. Again you can actually measure this acceleration with something as simple as a spring. Bring in a spring from outer space, place it at rest with the surface of earth, and it will stretch. Why? Because it’s accelerated.

But wait, I hear you say, you’re at rest, so you’re at a constant velocity. And acceleration is a change of velocity. So you can’t possibly be accelerated.

No, no. You’re not at rest. There’s no such thing as rest, not even on the cemetery. You are at rest relative to the surface of Earth. And you are accelerating at the same rate as the surface of earth. The relative velocity is zero. Flat earthers actually have this part right. What they get wrong is that they think this acceleration is the same everywhere.

But actually, the acceleration on a mountain top is somewhat smaller than that on sea level. That’s because the curvature of space is a little smaller up there. And that’s because, well, the Earth is round.

And since the acceleration is somewhat higher at sea level, time passes somewhat slower there than on a mountain top. This effect has been measured to great precision in a series of experiments starting as early as the 1950s. It’s one of the major predictions of Einstein’s theory of General Relativity. It proves that time actually does slow down near gravitating bodies. And, yeah, it also proves that the earth isn’t flat.

This effect becomes particularly strong if you hover near the horizon of a black hole. But again, it’s not gravity that causes time to slow down, because gravity is not a force. It’s the acceleration you need to not fall into the black hole that slows time down.

Okay, I know, lots to digest. The main takeaways are this: (1) Acceleration is absolute. (2) The reason time slows down is acceleration. (3) It’s a real effect and has been measured. (4) Special Relativity does describe acceleration, but only in flat space-time. General Relativity can describe curved space-times and with that, gravity. (5) Gravity is not a force which is why being at rest with the surface of a gravitating body requires an acceleration and that too slows down time.

Was that any better than those incomprehensible books I read 30 years ago? Let me know in the comments.