Why does gravity affect time?

Question

So Special Relativity states that for all non-accelerating objects of matter the laws of physics are the same. I'm confused on why this law of physic applies to objects in acceleration and gravity since I presumed it only applied to objects in a constant speed. Have I been miss-enlightened ? Also I am aware of that the change who deserves attention is at the fraction of the speed of light because it's more notable at that point. But, again. Why does time dilation and length contraction occur for objects in acceleration(or gravity).

Answer 1

So Special Relativity states that for all non-accelerating objects of matter the laws of physics are the same.

I think the point is just that the constants and the time and space derivatives that appear in a law of physics should not have to change the form of the equation if you measure the time and the space in two frames that move relative to each other a fixed velocity). You don't have to have objects per set.

I'm confused on why this law of physic applies to objects in acceleration and gravity since I presumed it only applied to objects in a constant speed.

When one person is moving relative to another they might disagree about what constitutes a change in the x coordinate so they might disagree about what a derivative with respect to x looks like. But they should each be able to write the equation for the laws of physics the same and have the laws look the same. They might disagree about the values of the electric field about whether there even is a magnetic field at a point and what momentum some object has. But they will agree what the laws look like.

The principle is about the laws. Think of it as something that tells us how to make our laws. It tells us to make our laws more like maxwell than like Newton. And thus we learned to improve Newton to something that also works to high accuracy with regular speeds and regular accuracy with high speed.

Have I been miss-enlightened ? Also I am aware of that the change who deserves attention is at the fraction of the speed of light because it's more notable at that point.

Yes, Newtonian Physics is more obviously wrong at higher speeds. You can see it as wrong at low speeds too if you measure very precisely.

But, again. Why does time dilation and length contraction occur for objects in acceleration(or gravity).

Acceleration and gravity are different. This might be where you misunderstand. Special Relativity handles acceleration just fine. And you don't need General Relativity to handle acceleration.

This is why it is important to realize that special relativity was never about objects it was about observers.

For instance if you are your friend move away from each other and send signs to each at the rate as measured by your wristwatches that you designed and built the same way and you move at 80%c then signals from your friend arrive to you at 3 times the rate you are sending them. And your friend notices the same thing. It isn't about objects its about how nature measures time.

For instance if you accelerate then we can approximate your motion with a bunch of lines that matches the curve pretty well and then the laws of physics are the same for each of them so we can figure out how a clock measures time on each of those segments and add them up. If the segments were small that sum is very very close to what the clocks measures and we can get as accurate as we want by taking more line segments that are each smaller.

So acceleration is not a problem. And we can compute things by using small lines in small regions of space and time. In General Relativity you take the fact that you only ever need to compute things in small regions and then you say that locally gravity doesn't accelerate things towards the earth (which is far away) it brings things together (when you drop two things and they fall towards the center of the earth they getting closer together) that is a local thing. So in General Relativity you work with local regions about you allow gravity to change which motions are natural so the nearby things can be brought together. And gravity becomes about things like that and about sewing up different regions together.

Time dilation is about comparing different things and it is better to just accept that we need to know how many times a clock ticks from one place-time to another place-time by a particular route (including how fast it went).

That route can be broken into small pieces and then the local Physics takes over. But it is just about the geometry of spacetime. Rods measure distances and clocks measure times and it is based on the actual place-time and the other actual place-time and the collection of place-times in between.

Jumping straight to how different people would describe it is a bit backwards (but has a long history) think of it as measuring the way spacetime is and that that is what you want to learn.

Answer 2

SR is based upon two postulates:

1)The laws of physics are the same in all inertial frames of reference.

2)Speed of light $c$ is constant for all inertial frames of reference.

To arrive at some results in relativity, one only needs to assume one of the postulates, other results are logical conclusions that require assuming the two postulates together.

For instance, postulate 1 alone(without postulate 2), implies the relativity of constant velocity motion.

You're correct, The laws of physics are indeed the same only for observers with constant velocity(postulate 1), but not for accelerated frames.

However Time dilation(the fact that relative motion affects time) is not a consequence of postulate 1, but it's only a consequence of the constancy of $c$ (postulate 2). To see how the constancy of $c$ leads to time dilation you should read about Simple inference of time dilation due to relative velocity section in this article on Wikipedia.

If you combined postulate 1 and 2 together you arrive at the fact that, if two inertial frames in relative motion then each of them claim other's clock is slower.

The difference between accelerated and non-accelerated frames with respect to time dilation is this:

If we have two frames, one that is at rest(an inertial frame) and another that is moving with respect to it with constant velocity, The observer at rest claims that time is slower on the moving clock(using postulate 2). However since the moving observer moves with constant velocity, He can use the principle of relativity(postulate 1) with postulate 2, to claim that he's in fact at rest and not moving and it's the other observer that is moving and experiences time dilation.

However, if we have two observers, one who's at rest and another another that is moving with respect to it in an accelerated motion. Then the accelerated frame cannot use postulate 1 to claim he's at rest and it's the other guy who has time dilation, since he's accelerating and thus is truly moving. Only the observer at rest will claim the accelerated frame has time dilation(using postulate 2).

So the constancy of $c$ implies that, if you're an observer at rest, and there's another frame of reference that is moving with respect to you(it does not matter if it's accelerated motion or constant velocity) then you'll measure the time on moving clock to be slowed down relative to your clock.

Now according to the equivalence principle between acceleration and gravity, since they're equivalent, therefore gravity should affect time as well.

Also I am aware of that the change who deserves attention is at the fraction of the speed of light because it's more notable at that point

This is a plot of the lorentz factor: $gamma=\dfrac{1}{\sqrt{1-(\dfrac{v}{c}})^2}$.

To give you a feeling for $\gamma$, if for example, you calculated $\gamma$ for a frame that is moving relative to be $\gamma=2$ , then you'd observe that the time on the moving clock ticks as half as slow as your clock at rest.

$\dfrac{v}{c}$ is the ratio between the velocity $v$ of a certain frame with $c$ and it's on the x-axis, and $\gamma$ on the Y-axis, as $\gamma$ increases, relativistic effects become manifest. You'll notice that $\gamma$ is pretty steady through out the graph, it only starts to increase notably as $\dfrac{v}{c}$ approaches 1. So relativistic effects are only manifest at near the speed of light.