# I Are Special Relativity effects valid for angular velocity?

#### Lohan

I like a to ask a very simple question. I have read about time-dilation when it comes to SR.

So, like:
(1.) A and B are are rest and close to each other.
(2.) B starts travelling at .5C relative to A, away from A, in a straight line.
(3.) B comes back at .5C towards A.

Now, according to SR, A has aged, and B has not (or aged less).

My question is, will this work, even for angular velocity? i.e.
(1.) A and B are millions of miles apart. Say, A is on Earth, and B is at a distance 100 times that of the moon. i.e. B is at a distance, 100 times the distance between the earth and moon.
(b.) Now, B starts travelling at .5C, but with an angular velocity relative to A. i.e. A is the center, B is rotating around A an angular velocity of .5C.

Now, will the time dilation effects of SR happen in this instance also?? Will the clock in B, tick slower, than the clock A has?

Or, does SR apply only if B travels away or towards A in a straight line?

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#### A.T.

B is rotating around A an angular velocity of .5C.
Usually A & B have to meet twice, to compare their elapsed proper times in non-ambiguous way. Also, angular velocity isn't measured in c. You probably mean tangential velocity.

If A is inertial, and B is circling around A, then B will age less. Ignoring gravity etc.

#### Lohan

A & B have to meet twice, to compare their elapsed proper times in non-ambiguous way. Also, angular velocity isn't measured in c. You probably mean tangential velocity.
Thanks for the very quick reply.

Yes, I mean B is rotating around A at a speed of .5C.

I have a small experiment, like a thought experiment as such: Instead of meeting each other, what if B has a very powerful telescope that can see the clock of A from his spaceship.

Similarly, if A has a very powerful telescope that can see the clock inside B's spaceship. Just to make it interesting, suppose A is not on Earth, but inside small ball shaped spaceship. The spaceship rotates constantly to track the spaceship of B. So both can see each other through the 2 powerful telescopes. Since B is rotating around A, the light which shows A's clock, will travel to B always at the same time, and vice-versa as well.

They coordinate the experiment in such a fashion that both start at at the same time. I think you can start at the same time. Like, both spaceships are close together, they know each others times. Then B starts moving at a slow speed towards the rotating "orbit". I think you can use some calculations, used in the GPS satellites to coordinate this. I have read that the GPS systems have to make a "relativity" correction to tell the correct coordinates of a place.

So, what do you think both of them will see when they look at each other and the clocks??

So, basically, they are looking at each other in real-time using 2 very powerful telescopes. Will B see A age, like a movie-being fast forwarded???

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#### A.T.

Will B see A age, like a movie-being fast forwarded???
Yes, and A will see B age slower.

#### Lohan

So, does this mean that B will see A age faster like what we see when we fast-forward a VCR tape???

Will they be able talk to each other???

Say, there is a long fiber optics communication cable fixed between A and B which rotates with A and B. Now, will they be able to speak to each other, if both are ageing differently??

#### Ibix

So, does this mean that B will see A age faster like what we see when we fast-forward a VCR tape???

Will they be able talk to each other???

Say, there is a long fiber optics communication cable fixed between A and B which rotates with A and B. Now, will they be able to speak to each other, if both are ageing differently??
Yes. Why bother with the fiber optics? Radio communication would be easier.

GR and SR to a lesser extent affects even geostationary satellites, we communicate with them daily and already have a very good understanding of how to correct for the effect. So this isn't really a thought experiment, it's something that happens every day.

#### Lohan

So, they use radio communication. Fine.

So, B is rotating around A at .5c and talking to A. Both are talking to each other.

But doesn't it cause a problem when they start communicating???

If A is aging much faster than B, then how can they talk?

Geostationary satellites travel at very low speeds compared to C. So, I think the effects of SR is negligible, right?

What I have read is that GPS systems use GR to make corrections.

Never heard of communications systems using GR & SR for corrections in communications. Can you point to some sources for this??

#### Nugatory

Mentor
So, B is rotating around A at .5c and talking to A. Both are talking to each other.
But doesn't it cause a problem when they start communicating?
If A is aging much faster than B, then how can they talk?
One of them thinks the other one is talking slow, and vice versa. But they can still communicate.

Geostationary satellites travel at very low speeds compared to C. So, I think the effects of SR is negligible, right?
What I have read is that GPS systems use GR to make corrections.
They need to make relativistic corrections, and that includes the effects of both SR and GR. You are right that SR effects are generally negligible for satellites, but GPS calculations are more sensitive.
Never heard of communications systems using GR & SR for corrections in communications. Can you point to some sources for this??
No one is saying that (with the exception of the GPS system) we routinely make relativistic corrections in communications. We don't need to, because the relativistic effects are small compared with other effects (I expect that signal travel time and ordinary classical Doppler are the most important). However, we can and do routinely observe them even if we don't bother correcting for them, and this is the point Jadin Andrews was making.

Geostationary satellites travel at very low speeds compared to C. So, I think the effects of SR is negligible, right?
What I have read is that GPS systems use GR to make corrections.
Never heard of communications systems using GR & SR for corrections in communications. Can you point to some sources for this??
It's not necessary to correct for SR or GR on communications satellites, the effect is rather insignificant. However, we have to account for it in GPS systems. Also, the effect of GR is much stronger than SR, especially for the relatively slow velocity of satellites. SR basically predicts that the faster we travel the slower time will pass for us, however GR predicts that time will pass faster if we are higher up in a gravity well. So our satellites are not as deep in earths gravity as what we are, and therefore time travels faster for them, even though they are travelling much faster than we are.

In fact, GPS systems are so sensitive to time dilation that even their elliptical orbits need to be accounted for when correcting their clocks. IE, at some points in their orbit (periapsis) they are closer and at some points (apoapsis) they are further from the earth, and this is significant enough to affect the accuracy of GPS.

But I think you seem to have a problem with the idea of communicating with someone for whom time is travelling slower/faster. My point is that it wont be a problem, since we communicate with satellites every day, and it's not impossible. However, it would just be 'cool' to see time dilation in such a tangible way as in your thought experiment. Basically, apart from the fact that our radio channels for communication would be red/blue shifted, we still wouldn't notice the other person talking slower, there will be a delay, but if we use digital encoding then communication fidelity would be preserved. The information won't be altered.
But why just imagine talking to someone face to face for whom time is passing slower? I think the only thing we would notice is their sloth like responses and slow speech, but that person would see us running around like chipmunks.

#### pervect

Staff Emeritus
Thinking of it in terms of time dilation can miss some points, due to issues involving the relativity of simultaneity. If the term "relativity of simultaneity" itself is unfamiliar, the probability of it being a roadblock to understanding is very high, though unfortunately that doesn't help explain what the problem is because the term that describes the problem is unfamiliar. Rather than spending a lot of time trying to explain the relativity of simultaneity, I'll take a different course and try to explain the actual observed signals.

If you imagine an observer O sitting out in space, and a radio source S doing a relativistic flyby, initially the frequency received by O will be blueshifted via the relativistic doppler shift formula. The frequency will continuously decrease as time progresses, eventually it will turn into a blueshift when the radio source S moves from "in front" of O to "behind" O.

Referring to the ASCII diagram below, the signal sent at S1 is received by O as blueshifted, as the object S moves through S2, S3, and S4, it gets progressively redshifted, the net effect at S4 being a redshift.

S1------->S2---------->S3------------S4

.............................O

See https://en.wikipedia.org/wiki/Relativistic_Doppler_effect for the detailed formula for doppler shift vs angle.

The thing I want to stress is that the frequency received by O depends on the transmetted frequency, the speed of S,AND the angle at which O receives the signal.

You can imagine that O has a little tube sticking out at some angle, so as to receive the light emitted when the source S is "in line" with the tube. Then the received frequency depends on the angle of the tube. The case where the tube on O points at right angles to the path of S gives you what's called the Transverse doppler effect, which has the formula given in the wiki, a redshift by a factor $\sqrt{(1+\beta)/(1-\beta)}$. This formula for the transverse doppler effect is a special case of the more complicated formula for the doppler shift "for arbitrary motion" described in the Wiki article.

For the orbiting observer, let A be the observer in the center, and B be the observer in a circular orbit. Then in the frame of A, the wiki formulas for the transverse doppler effect applies, and the frequency received by A according to A's clock will be lower than the frequency B receives according to B's clock. Note B is accelerating, so it doesn't stay in the same frame of reference, so we need to talk about an "instantaneous frame of reference" at some specific instant of time when we talk about B. In this instantaneous frame of reference, while the general formula for doppler shift given in Wiki still applies, it's no longer the transverse doppler effect, because the apparent angle of A shifts due to what's called relativistic abberation. See for instance https://en.wikipedia.org/wiki/Relativistic_aberration. So the frequency received by B according to B's clock is higher than the frequency sent by A acccording to A's clock.

If you draw a diagram you can see why the angle changes. If you imagine a tube attached to B that always points at A, you can see that light emitted from A will hit the front part of the tube, but will be blocked by the rear of the tube as B continues its orbit and the light continues in it's straight line path. ,Therfore, while we do use a relativisitic dopper formua to find the doppler shift inthis case, it's not the transverse doppler formula, becuase the signal isn't transverse anymore, it is at an angle, due to the effects of relativisitc abberation.

#### Lohan

pervect, why would the communication signals get red or blue shifted in this special case.

B is rotating around A. So, when B sends a signal to A, isn't it just coming at A in a straight line? Because B moving at .5C with respect to A in a circle.

It's not that B is moving away or towards A right?? B, in fact has the same distance to A all the time, because B is moving around A in a circle. So, how can this red, blue shifting occur?

That is why I ask how can they communicate with each other?

#### Nugatory

Mentor
It's not that B is moving away or towards A right?? B, in fact has the same distance to A all the time, because B is moving around A in a circle. So, how can this red, blue shifting occur?
Imagine that B is carrying a strobe light that flashes once every second according to B's clock. What is the frequency of the flashes according to A's clock?

#### jartsa

Let's say Bob and Alice are standing on a race track side by side. Bob starts running straight ahead, while Alice stays still. Let's assume an abrupt acceleration of Bob.

Right after Bob has done accelerating, he sees Alice a little bit ahead of himself, and himself moving towards Alice, and he sees a blueshift of the light emitted by Alice.

If Bob was running around Alice, it would be like that all the time according to Bob. (A blueshifted Alice a little bit ahead of Bob)

#### pervect

Staff Emeritus
pervect, why would the communication signals get red or blue shifted in this special case.

B is rotating around A. So, when B sends a signal to A, isn't it just coming at A in a straight line? Because B moving at .5C with respect to A in a circle.

It's not that B is moving away or towards A right?? B, in fact has the same distance to A all the time, because B is moving around A in a circle. So, how can this red, blue shifting occur?

That is why I ask how can they communicate with each other?
WIthout the effects of special relativity, you'd expect no transverse doppler effect, i.e. if an object is moving in a circle keeping a constant distance from a central object, you would expect the frequency received by the orbiting object to be the same as the frequency transmitted by the central object.

In special relativity, however, you do get a transverse doppler shift, and the received frequency is different from the transmitted frequency.

As to the "why" of it. That's harder to answer, I suppose the short answer without a lot of math is "time dilation". I actually don't care much for that answer, but it's probably the best I can do at this point, since I don't really want to get into a lot of math. A better answer might be "it's due to the Lorentz transform", but that's probably over-technical. I'l additionally note that the orbiting body is a special case of the twin paradox (which you mentioned in your original post as I recall, you were curious about the circular case of this paradox).

I'd also like to mention experimental tests. If you look at the FAQ for experimental tests of special relativity, you'll find several experiments that have been carried out to confirm the existence of the transverse doppler effect.

"Are Special Relativity effects valid for angular velocity?"

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