Happy 2023 everyone :)

Albert Einstein stated his happiest thought to be "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 - meaning there's no experiment you can come up with to distinguish a frame of reference in free fall within a gravitational field to one in the vacuum of space with no gravity. Similarly, the sense of weight you feel stationary on Earth's surface would feel exactly the same as you accelerating at 1g distant from any gravitational field (where g is the gravitational acceleration of the Earth). Einstein's general relativity describes this thing we call 'gravity' as curvature in the fabric of space and time itself; the two are conjoined as 'spacetime' - spacetime tells matter how to move and matter tells spacetime how to curve. So, gravity and time seem to be inseparably linked. Gravitational fields actually slow the ticking of time in what's known as gravitational time dilation. 

To begin, we must understand that kinematic time dilation results in moving clocks ticking more slowly. Within special relativity, there exists axioms from which this idea of kinematic time dilation derives from -- 1) the laws of physics are the same in all inertial (non-accelerating) frames and 2) the speed of light is the same for all observers, is constant, and is equal to c (299,792,458 metres per second). To arrive at gravitational time dilation, we add the equivalence principle as our 3rd axiom of relativity. First, let's imagine we have a laboratory inside a ring-shaped space station. If we set it rotating at the right speed, centripetal acceleration leads to an effect of 'artificial gravity'. There is a physicist suited up outside the space station floating (in an inertial frame of reference) as the space station rotates. We also have a photon clock in both the laboratory and in the hands of the floating astronaut. One tick of either clock is very very short, meaning that over a tick the rotating lab moves a tiny arc of the entire circle. Over this small interval, both observers will see the opposing clock as ticking more slowly (via the laws of special relativity). After a full revolution, the photon clock in the rotating lab will have less total ticks than the one with the stationary astronaut (since it has to travel further overall due to the motion of the space station). Time has slowed for the physicists in the rotating lab. This is exactly analogous to our special relativity result - an observer which is accelerating travels more slowly through time. We would get the same outcome if we strapped accelerating rockets to our space lab; the photon in the photon clock would have to travel an extra distance to 'catch up' to the upper mirror as it accelerates upwards, but as it travels downwards, the lower mirror will move towards it, reducing the required travel distance. Overall, the distance for a single up-down tick is larger in the linearly accelerating frame than in the inertial frame.

What does all of this have to do with gravity? The equivalence principle demands that someone standing in a gravitational field must experience the same time dilation and weight as if they were being spun in a circle at a certain speed/having rockets strapped to them to yield an equivalent acceleration. If our axioms are to hold, then time must run slower in (stronger) gravitational fields. In regards to linear motion, time dilation calculated from both of these regimes (kinematic and gravitational) just give us the same result. However, we must note that circular motion in a gravitational field is different to our rotating lab analogy. Then, both kinematic and gravitational time dilations play separate roles and must be treated individually. Is it some kind of coincidence that the time dilation kinematically leads to the same result as time dilation via artificial gravity? Well, not really - it's just telling us that the fundamental source of the time dilation is the same.

We've seen that gravitational time dilation is a thing if our axioms hold, but what about a gravitational field actually causes time to tick more slowly? Perhaps the photon does actually have to travel further between mirrors or the forces binding matter in gravitational fields... Or it could be a case of your sense of 'now' in a gravitational field being continually swept forward relative to regions outside the gravitational field. There are even more ways to think about this question - these explanations are ultimately just ways we map mathematics to our intuition. A more interesting question to ask may be "why does slowed time cause gravity?"

published: 09/01/23 by kaan evcimen