How Does Gravity Warp the Flow of Time? (Script) (Patreon)

Over the course of your life your feet will age approximately 1 second more than your head due to gravitational time dilation  -  and that’s assuming that your life is long and that you’re quite tall. But that tiny difference in flow of time may be what keeps you stuck to this planet at all.

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Albert Einstein enjoyed imagining people falling off buildings. He said it himself - he described his happiest thought as the following: “For an observer falling freely from the roof of a house, the gravitational field does not exist.” We now know this as the equivalence principle - it states that there’s no experiment that you can do to distinguish a frame of reference in freefall within a gravitational field from a frame of reference floating off in space in the absence of gravity. Provided of course you’re in a lab with no windows, and there’s no air resistance, and you haven’t hit the ground yet. But otherwise, as far as the universe is concerned, the sense of floating you feel in both circumstances is exactly the same. Likewise, the sense of weight you feel stationary on the surface of the Earth is identical to the sense of weight you would feel accelerating at 1-g distant from any gravitational field - at least as far as the laws of physics are concerned.

Einstein had his happy thought in 1907, a couple of years after he started his scientific revolution with the special theory of relativity. It took him another 8 years and a lot of help to grow this simple idea into his full theory of gravity - the general theory of relativity.

General relativity, or “GR” explains the force of gravity as being due to curvature in space and time. Mass and energy change the lengths of rulers and the speeds of clocks - and somehow those changes lead to objects being attracted to each other. John Archibald Wheeler put this notion the most pithily: Spacetime tells matter how to move; matter tells spacetime how to curve. A common way to depict this is with the classic balls-on-rubber-sheet analogy. Balls are constrained to move only on the sheet, and will move in straight lines if the sheet is flat - but if the sheet is curved then there are no straight lines. But the rubber sheet picture is at best a crude analogy. For one thing, it implies that curvature in the fabric of space is the cause of gravitation - but that’s only half - less than half the picture. Matter tells space AND time how to curve, and it’s the curvature of time that’s mostly responsible for telling matter how to move.

There’s a deep connection between gravity and time - gravitational fields seem to slow the pace of time in what we call gravitational time dilation. And today we’ll explore the origin of this effect. And ultimately, we’ll use what we learn to understand how curvature in time - this gradient of time dilation - can be thought of as the true source of the force of gravity. It would actually be really helpful if you’ve already seen our recent video on paradoxes in special relativity. You could watch it now if you haven’t. I’d wait, but you know where the pause button work.

We’re going to start out by me totally convincing you that time must run slow in a gravitational field - an effect we call gravitational time dilation. But to do that I need to give you a quick refresher on regular old time dilation, which tells us moving clocks must appear to tick slowly. This is from special relativity - which itself seems a near miraculous insight. Einstein did have help and built on prior and contemporary wisdom. But it’s fair to say that relativity was discovered in his own imagination - in his brilliant thought- or gedankenexperiments.

Einstein’s thought laboratory - his gedankenlab - was filled with many incredible imaginary devices, but one of his favorites was the photon clock. This is a simple pair of perfectly reflective, massless mirrors between which bounces a single photon of light. A counter ticks over every time the photon does a full cycle. The photon clock represents the simplest possible clock, and anything we conclude for it also applies to any other clock. And, in fact, to any matter - anything that experience time, which in practice means anything with mass. We’ve talked about why this is the case previously.

The amount of time taken for one tick of the photon clock is the distance the photon travels divided by its speed - so twice separation of the mirrors divided by the speed of light. But let’s say the gerdankenlab is moving at a constant velocity past a stationary physicist. They see the photon clock ticking, but the photon travels a longer path. How long does it take to execute that one tick? Here we have to invoke the great founding axiom of special relativity - that the speed of light is always measured to be the same for all observers, no matter their personal speed. From the stationary perspective, the photon seems to travel further but it has to keep the same speed - so it appears to take longer to complete a single up-down tick. Add an identical but stationary photon clock. It seems to tick more than once for a single tick of the moving clock. And this apparent slowing of time applies to everything in the moving lab.

But the whole situation is symmetric. For an observer in the moving lab, it appears that you and your clock are ticking slow. That’s because there’s no preferred notion of “stillness” in relativity. They see the world as moving, and themselves stationary.

Time dilation due to motion is inevitable if we accept the axiom of the constancy of the speed of light. To get to gravitational time dilation all we need to do is add in the equivalence principle as our second axiom. It tells us that whatever we conclude about the passage of time in an accelerating frame must also be true in a gravitational field.

To get an accelerating frame we could strap rockets to our gerdankenlab - and don’t worry, we will. But first, let’s try this - build our lab into a giant, ring-shaped space station. If we set it rotating at the right speed then centripetal acceleration leads to some nice artificial gravity. Let’s also suit up a physicist and have them float in space at one spot as the space station turns. They’re in a non-accelerating, or inertial frame of reference. We have a photon clock in the lab and an identical one with the physicist. One tick of either clock is very short, which means that over that interval the lab moves only a tiny arc of the full circle. So we can approximate its motion as a straight line. Over that brief interval we know perfectly well what the time difference is between the two frames of reference. Both observers see the other’s time has slowed.

But after a full revolution, both observers ask each other how many ticks their clock ticked. And it turns out that the stationary clock did tick less - time slowed for the rotating case. This seems paradoxical, but the solution is the same is as for the twin paradox from our previous episode. The summary is this: two observers moving in straight lines to each other do perceive the other as time-dilated - slowed. But as soon as one of those observers changes direction, the symmetry is broken.

In the twin paradox, the twin traveling to a nearby star and back has aged less even though both could see the other’s clock ticking slowly. We can see that when we use a spacetime diagram to show how the traveler tracks the passage of time back on Earth. Her perception of what is “simultaneous” to current moment flips at the turnaround point, so that she misses a bunch of the ticks of her brothers clock.

Here’s the spacetime diagram for our rotating lab. Now 2 dimensions of space instead of one. The spacetime path or worldline of the lab is a helix, and the lab’s perception of “now” is this shifting plane. It’s easier to see if we just take a slice out of this - one dimension of space again. Now the worldline is.a sine wave. The lines of constant time for the moving clock tilt back and forth, and as that line tilts it fast-fowards over the clock ticks of the stationary clock.

The source of acceleration doesn’t matter. You get the same result if you do strap rockets to the gendankenlab. The photon in the accelerating clock has to chase the upper mirror some, increasing the distance it needs to travel. On the way down the lower mirror catches up to it, reducing the down-tick distance. But overall, the distance for a single up-down tick is larger in a linearly accelerating frame compared to an inertial frame.

On the spacetime diagram the spaceship’s worldline looks like this, and its evolving line of “now” looks like this. You can think of the time dilation due to acceleration as resulting from this line of simultaneity slowly sweeping forward on the inertial clock’s worldline, so that it appears to tick faster.

OK, so what does all this have to do with gravity? The equivalence principle demands that there’s no experiment that can distinguish between acceleration and gravity. Ergo someone standing in a gravitational field must experience the same sense of weight AND the same time dilation that you would get from being spun in a circle at the right radius and speed, or accelerated with linear acceleration equal to the gravitational acceleration. If both of our axioms are true - the constancy of the speed of light and the equivalence of acceleration and gravity, then time must run slow in gravitational fields.

It kind of blows me away that you can calculate the difference of the flow of time between an inertial and accelerating frame using pure special relativity with its kinematic time dilation plus shifting reference frames, OR you can use general relativity to calculate the gravitational time dilation for the equivalent gravitational acceleration. You get the same answers. You do have to be careful to choose the right relative distances between observers. In the case of the twin paradox, gravitational time dilation gives the right relative time flows if you consider the traveling twin to be in a gravitational well with constant acceleration equal to her spaceship’s acceleration. But how deep in the well? As deep as the distance back to Earth - which is why the time dilation is so huge, even if the acceleration is mild. Another note of caution: be aware that circular orbital motion IN a gravitational field is very different from our rotating space station- then both gravitational time dilation and kinematic time dilation play separate roles.

So is it some sort of cosmic coincidence that you get the same number with shifting reference frames as with artificial gravity? No, it’s telling us that the source of the time dilation is fundamentally the same.

OK, this is all fine and good. We’ve reasoned our way to seeing that gravitational time dilation must be a thing if our axioms are right. But that doesn’t feel entirely satisfying - it doesn’t seem explanatory. What really is it about the gravitational field that’s causing time to tick slow?

Perhaps the photon - or whatever light-speed quantum components make up matter - actually have to travel further - between mirrors or between the forces binding matter. So that photon clocks and matter evolve more slowly in gravitational fields.  Or is it that if you’re inside a gravitational field, your sense of “now” is continually sweeping forward compared regions further outside the gravitational field? Sure, there are these and even more ways to think about this - and no one of them is closer to reality - they are, in a sense, just our way to map the math to our intuition.

But ultimately, asking “why does gravity slow time” is a bit backwards. A better question may be “why does slowed time cause gravity”. The curvature of space by matter isn’t nearly enough to give gravity at the strength we feel it. You’re held in your chair right now by curvature in time. In short, you’re held down because your butt is ticking faster than your head. And I’ll show you exactly why that’s true real soon, when we explore the tangled connections between time and gravity in a curved spacetime.