Notes on Relativity and Gravity

The principle of relativity: Physics works the same in every reference frame, and the speed of light in a vacuum is part of the laws of physics. Thus this speed is always measured to be the same thing no matter what the motion of the observer doing the measurement.

This seemingly simple principle is actually very profound, because it means that if a police car is approaching a stationary police car and both officers use their radar guns on each other, they must both measure the same Doppler shift regardless of which one is moving. In short, they could not tell which one is moving if they didn't have the ground as a reference point, so imagine they are both in spaceships moving through a complete vacuum-- it is a symmetric problem that only depends on the relative velocity between them. Note sound does not work that way-- the Doppler shift of sound is different depending on whether the source or the observer is moving relative to the air. But light requires no medium to carry it, unlike sound which needs air or ocean waves which need water, and so light can move right through vacuum. That's why there's no preferred reference frame from which to describe the motion of the light, all you can determine is the relative motion of the source and the observer.

So far, this may not be so surprising. But if you imagine that the officers are in spaceships in a complete vacuum, then you see by the principle of relativity that there is a symmetry to the situation, so that both must get the same result on their radar gun. The surprising ramification of this is that if you do a simple calculation of what Doppler shifts the officers should see, you will get that they should not get the same answer if one is moving and the other isn't, so there has to be something more to the equation. The only resolution must be that in order for the two police officers to get the same result on their radar guns, each one must conclude that time is running slowly for the other officer! It is inescapable aspect of reality that time must flow at different rates in different reference frames, if you want the principle of relativity to hold. This different rate of flow of time has been measured, and the principle does hold! Kudos to Einstein.

Since the speed of light must be the same in all frames, and since time is flowing differently, then distances must be measured differently also. To make a long story short (no pun intended), this requires that rulers that are in motion must shrink relative to rulers that are standing still. Again, this is the character of reality itself, and stems solely from the principle of relativity. It has nothing to do with errors in perception due to the lag in time for light to get to the observer, those factors have all been included and you still need to monkey with space and time to make the calculation work out.

What this means is, the passage of time is a little like leafing through the pages in someone's diary. Reading a single diary, we have the sense of normal passage of time, but let's say we have one diary in our right hand and another in our left-- how do we connect what is happening in one book at the same time as what is happening in the other? The natural way would be to assume that one day in one diary is the same as one day in the other, but what if these are diaries from beings at opposite ends of the galaxy, moving at high relative speed to each other? Do we know that the days match up like that? No, we know that they do not-- Einstein's relativity tells us how many days, or "pages", you have to leaf through in one of the diaries to equal one day, or "page", in the other, and still have the events being described happening "simultaneously". So with our new concept of the passage of time, we find we need a more sophisticated idea of what "happening at the same time" actually means. The official definition in relativity is that two events are simultaneous if an observer between them can at the same time send out two light signals, one which reaches each event, such that if the light signal was reflected back by those events, the two signals would return to the observer at the same time. This depends on the motion of the observer, and so does the concept of what is simultaneous!

A set of pages that explain the surprises of relativity can be found here.

These strange changes in space and time have been tested in many situations, are are exactly correct to the accuracy to which we are capable of measuring them. Thus if faced with either believing that space and time are absolute entities, or that the principle of relativity is correct, only the latter idea has stood the tests it has been subjected to.

An interesting exercise to show why light moving at the same speed regardless of the speed of the experimental apparatus (like the Michelson-Morley experiment) is a strange result, and has strange ramifications for time and distance, can be found here.

Now, given that space and time are not absolute, this opens an entirely new concept of gravity, conceived by Albert Einstein. The idea is that gravity is not actually a force at all. Rather, gravity stems from the fact that time flows more slowly when you get closer to a massive body. It slows a lot when you get close to a black hole, and only a very tiny amount when you get close to the Earth or the Sun, but nevertheless it slows. This slowing of time causes a "curvature" in the space-time coordinates, which creates the illusion that we are all sitting still in our seats, when in fact we are all being accelerated upward by the force we feel from our chairs. The reason we do not know we are accelerating is that we apply uncurved coordinates when we consider our motion, but gravity has curved the spacetime, so the natural coordinates we use unknowingly are that of an accelerating reference frame, like the perceptions of a person on an amusement park ride going round and round. We don't notice this because we are all accelerating together, and the curvature means that we never actually have to leave the surface of the Earth, or even get any farther from the center of the Earth. The irony is that we used to think that objects in a circular orbit were accelerating toward the center, so were accelerating without changing their distance, but in fact those objects are not accelerating at all, they are moving in a straight path in a curved spacetime. We are the ones who are accelerating, without changing our distance from the center. Curved coordinates are rather bizarre that way! An analogy often used is an ant crawling on a curved apple surface-- the ant thinks it's always marching in a straight line, but in fact it is "orbiting" the apple.

What this also means is, if you were born a twin but lived your life in an orbiting space station, while your twin stayed on Earth, if you were later reunited you would actually be older than your twin! For the weak gravity of Earth, the effect would be a small fraction of a second, not at all noticeable. Nevertheless, this tiny difference in time is enough to generate all the things that we observe gravity does, like making objects that we throw up into the sky fall back down on us, and keeping the Moon from flying off into space. All the things we used to think we needed a force to do, but now we just need a tiny difference in the rate that time is flowing, i.e., curved coordinates. Even for the gravity of the Sun the difference in the rate of flow of time is miniscule, but it gives rise to Kepler's laws anyway. Only for a neutron star or black hole would the age difference be noticeable, and you could actually be years older than your twin.