# Time dilation, length contraction and relative simultaneity on Earth

1. Mar 25, 2014

### analyst5

The three main effects of SR occur in inertial frames and change the description of space-time relative to a particular observer. My question here is how do these effects occur on Earth, since we know that motion on Earth is non-inertial. I know that we travel at small speeds and that we can't really perceive them, but if we could travel at greater speeds undergoing circular motion on the surface of Earth, what would happen, and how would it differ from the standard definition of these 3 effects in inertial frames?

2. Mar 25, 2014

### A.T.

It already differs significantly from inertial frames because of gravity. The rest frame of your house for example is highly non-inertial as it accelerates with 1g away from the the Earth's center. If the Earth was spinning faster, your house would be actually more inertial, than it is now.

3. Mar 25, 2014

### analyst5

If we don't take gravitational effects into account, just circular motion, what are the differences?

4. Mar 25, 2014

### A.T.

You mean just a rotating reference frame, or the restframe of a spinning disc. There are many threads on this here already. The key points:
- You cannot synchronize clocks in that frame.
- You can synchronize clocks at equal radius, but not with the standard Einstein convention
- Light propagation is not isotropic
- The geometry measured with rulers at rest to the disc is non-Euclidean

Maybe you should first look into linearly accelerated frames, to see the differences to inertial frames there.

Last edited: Mar 25, 2014
5. Mar 25, 2014

### Staff: Mentor

It's not "motion on Earth" that's not inertial, it's motion described using coordinates in which the surface of the earth is at rest that's non-inertial. Seal me and my lab equipment into a box, then drop the box from an airplane.... I can use coordinates in which my lab is at rest (until I hit the ground) and the experiments I do inside the box will register time dilation, length contraction, relativity of simultaneity, just as they do in the textbook thought experiments conducted in empty space. In this coordinate system the surface of the earth is both rushing towards at 9.8 $m/sec^2$ and rotating sideways underneath me.

Or I could choose to use the coordinates that come naturally to an observer on the surface of the earth; in these coordinates the surface of the earth is at rest and my lab is falling from the sky and moving sideways from Coriolis force. The calculation of the motions of objects in my lab will be much messier, but as long as I get the coordinate transformations right I'll get the exact same results - it has to come out that way because it's still the exact same experiment and the objects in my lab are all doing the exact same thing.

Many things are harder in the non-inertial coordinates: A.T. has provided a list of things that are hard or impossible using the coordinates in which the surface of the earth is at rest. However, as long as we're aware of these difficulties, we can do the necessary calculations to predict how relativistic effects will appear on earth, and perform experiments that verify these predictions.

In practice and for realistic measurements we seldom have to worry about the non-inertial nature of the earth's surface; the difference in relativistic results between a true inertial frame and a frame in which the surface of the earth is at rest are generally negligible. Gravity is totally irrelevant for the motion of particles in a collider; the rotation of the Earth is pretty much negligible during the hundred or so microseconds that it takes a cosmic ray particle to make from upper atmosphere to ground; and so forth.

6. Mar 25, 2014

### Staff: Mentor

This can't be correct as you state it. Whether or not a given state of motion is inertial is invariant; it doesn't depend on the coordinates you choose. Inertial motion means zero proper acceleration; non-inertial motion means non-zero proper acceleration. Proper acceleration is an invariant.

Which means, the box is in free fall (ignoring air resistance), therefore it has zero proper acceleration, therefore the motion is inertial. That's true regardless of the coordinates we use to describe it.

Agreed.

Right; and those experimental results will include a reading of zero on an accelerometer that is at rest in the free-falling lab, corresponding to a zero proper acceleration for the lab's worldline. So the lab's motion is inertial even though we are using a non-inertial frame to describe it.

On the other hand, if the lab was sitting on the surface of the Earth, an accelerometer at rest in the lab would *not* read zero; it would read 1 g, meaning the lab's motion would be non-inertial. This would change a number of other experimental results as well. And that would be true regardless of whether you chose coordinates in which the surface of the Earth was at rest, or coordinates in which an observer free-falling towards the Earth was at rest.

7. Mar 25, 2014

### Staff: Mentor

You're right. The point I was trying to make is that "on earth" (which is usually understood to mean "using coordinates in which the surface of the earth is at rest"), has nothing to do with whether the motion is inertial or not; the motion is what it it is and it can be inertial or not on earth as easily as anywhere else.

I'm hearing in OP's question, perhaps wrongly, some confusion between the invariant property (accelerometer does or does not read zero) and the coordinate accelerations that appear and complicate things when we choose coordinates in which the surface of the earth is at rest.

[And for analyst5's benefit - PeterDonis and I are not arguing here. He's right and I'm trying to clean up my answer to our question so that it better describes physics about which we are in complete agreement while still responding to your question. Maybe instead I should have asked you to go back and look very carefully at the first two sentences of your original post where you slid almost imperceptibly from "inertial frame" to "non-inertial motion"]

8. Mar 25, 2014

### analyst5

Thanks for the answers, but it seems that nobody mentioned what does happen with the three relativistic effects I mentioned in the title of my thread, viewed from a non-inertial frame on Earth? Let's say a car is moving with some speed relative to Earth's surface. Can anybody explain the relativistic effects from his perspective and how do they compare to the classical ones in inertial frames?

9. Mar 25, 2014

### A.T.

A frame moving fast in a circle has locally the same SR effects as its instantaneous inertial rest frame. But additionally it has radial proper acceleration and thus (pseudo)gravitational time dilation along this direction.