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. Albert Einstein really 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 - actually 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 would 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, rather than general relativity, but even special relativity
seems a near miraculous insight. Einstein also had help and built on prior
and contemporary wisdom to develop it. 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 that we conclude for it also applies to any other clock. And, in fact, to any matter - anything that can 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 appears for everything in the moving lab. But the whole situation is symmetric. For an observer in the moving lab, it appears that the stationary clock is 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 more - time slowed for the rotating case. This seems paradoxical, but the solution is
the same as it is 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 the inertial frame. 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. And 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 a constant acceleration equal to her spaceship’s acceleration. But how deep in the well? Well, as deep as the distance back to Earth - which is why the time dilation in this case 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? Well 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 do have to travel further - between mirrors or between the forces binding matter. So that photon clocks and matter do 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 to regions further outside the gravitational field? Sure, both of there are valid and there are 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 the 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.
Ok, silly thought experiment time. If gravity = slower time (thus slower time = gravity), and you somehow managed to reverse time, would you end up with gravity as a repulsive force?