Aberration of Light in Circular Motion: Does Distance Change?

[Phylosopher]

TL;DR Summary

What happens to distance?

Simple as it sounds!

Usually people derive aberration of light using linear motion, not circular motion. When aberration happens in linear motion, one would expect distance between the source and the observer to change. But, in circular motion, the path light takes in the circular motion, in the source frame of reference is $r$.

Thus, I would expect distance in the frame of reference of the moving object to be $r$ as well!

The motion is perpendicular to $r$, speed of light is fixed, yet we have time dilation! therefore traveled distance is elongated!

What am I missing? Does distance change?

 

[PeroK]

I don't understand your scenario. Is this about time dilation for uniform circular motion?

 

[Dale]

Summary: What happens to distance?

Thus, I would expect distance in the frame of reference of the moving object to be rrr as well!

There are many different rotating reference frames, which one are you using?

 

[Phylosopher]

I don't understand your scenario. Is this about time dilation for uniform circular motion?

Let me clarify.

Problem:
Take a reference frame S, it represents the reference frame of a point source (say a star). In this reference frame, the distance between the orbiting object and the source is $r$. Further, the object have a perfect circular orbit and so $r$ is constant and the speed of the object $v$ is perpendicular to $r$.

In the reference frame of the circulating object $S'$, the star have a velocity $v$ and is titled with an angle $\theta$ due to aberration of light.

In the reference frame of the object $S'$, what is the distance $r'$ between the source and the object?

My attempt:
My intuition says that $r=r'$ since the velocity $v$ is perpendicular to $r$ in the reference frame of the source $S$. Also, if I want to define r in this reference frame, then I would write $r=c \Delta t$, where $Delta t$ is the time it takes light to travel from the center to the rim of the circle.

From another point of view, in the reference frame $S'$ time dilation must occur no matter the direction of velocity. Therefore, $\Delta t'=\gamma \Delta t$. Speed of light is fixed $c$. If I want to know how long it takes light to travel from the source to the object in $S'$, then I would find it as follows $r'= c\Delta t'=c \gamma \Delta t= \gamma r$.

 

[PeroK]

Let me clarify.

Problem:
Take a reference frame S, it represents the reference frame of a point source (say a star). In this reference frame, the distance between the orbiting object and the source is $r$. Further, the object have a perfect circular orbit and so $r$ is constant and the speed of the object $v$ is perpendicular to $r$.

In the reference frame of the circulating object $S'$, the star have a velocity $v$ and is titled with an angle $\theta$ due to aberration of light.

In the reference frame of the object $S'$, what is the distance $r'$ between the source and the object?

My attempt:
My intuition says that $r=r'$ since the velocity $v$ is perpendicular to $r$ in the reference frame of the source $S$. Also, if I want to define r in this reference frame, then I would write $r=c \Delta t$, where $Delta t$ is the time it takes light to travel from the center to the rim of the circle.

From another point of view, in the reference frame $S'$ time dilation must occur no matter the direction of velocity. Therefore, $\Delta t'=\gamma \Delta t$. Speed of light is fixed $c$. If I want to know how long it takes light to travel from the source to the object in $S'$, then I would find it as follows $r'= c\Delta t'=c \gamma \Delta t= \gamma r$.

S' is not an inertial frame. The basic equations of time dilation and length contraction (and the Lorentz transformation in general) apply only to inertial reference frames.

You need a more complex analysis for S'.

 

[Dale]

In the reference frame of the object S′S′S', what is the distance r′r′r' between the source and the object?

Which version of the reference frame of the object? There are several.

You need to either provide the metric in the rotating frame or the transformation from the inertial frame to the rotating frame (or both) before your question can be answered. The answer depends on your arbitrary choice of coordinates

 

[Phylosopher]

S' is not an inertial frame. The basic equations of time dilation and length contraction (and the Lorentz transformation in general) apply only to inertial reference frames.

You need a more complex analysis for S'.

Don't say that LOL.

I am actually aware of this! I was hoping that someone can lead me to a simpler answer.

Which version of the reference frame of the object? There are several.

A momentarily co-moving frame.

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Months ago, I was reading research papers on Ehrenfest Paradox, and I was seriously annoyed with it. Not because it is hard, but because it made a clear distinction for me between linear and rotational motion.

Anyway, in the vicinity of $\beta <<1$, can I assume that $r'\approx r$?

I would really appreciate it if anyone can suggest for me papers/books in this specific subject. I am also interested in rotation around extended sources (a sphere instead of a point), but I couldn't find anything on it.

 

[Dale]

A momentarily co-moving frame.

Ok, which moment do you want the momentarily co-moving frame to be co-moving?

Note that the momentarily comoving frame is an inertial frame, not a rotating frame. So I am not sure why it is relevant, but we certainly can use it to answer.

 

[Phylosopher]

Ok, which moment do you want the momentarily co-moving frame to be co-moving?

at $t$ I guess! Since I should be an observer in $S$.

 

[Dale]

at $t$ I guess! Since I should be an observer in $S$.

Hmm, looking at your posts above $t$ appears to be a coordinate, not a moment. I may be misunderstanding your $t$ above.

Look, this is getting irritating. You have asked an ambiguous question and I have told you what we need to resolve the ambiguity and 9 posts later the same ambiguity remains.

You need to define the reference frame clearly. Actually write down the transformation you are interested in or provide enough information for us to do so. This back and forth is annoying.

If you don’t care about your question enough to clarify it as repeatedly requested then we will just close the thread. If it is not important to you then it doesn’t need to clutter up the site.

 

[vanhees71]

Doesn't the question simply refer to the transverse Doppler effect observed by an observer rotating around a light source placed at the origin. It's explained in all detail in Wikipedia:

https://en.wikipedia.org/wiki/Relat...ne_object_in_circular_motion_around_the_other

 

[PAllen]

My first reaction to the thread title is that aberration in SR has nothing whatsoever to do with distance. It has only to do with how angles between velocities transform from one frame to another.

 

[pervect]

If you are interested in distance in rotating frames, I'd suggest forgetting for the time being about aberration, and doing some research on the concept of distance in rotating frames first.

The cliff notes version is that radial distances are fairly intuitive in rotating frames, but the transverse distances (for instance, computing the circumference of a circle in a rotating frame) are not.

For more specifics - in general, the topic is called the Ehrenfest pardox. There's a vast and confusing literature on the topic, unfortunately. I don't have a guide as to which particular papers have high impact, and which are nonsense, unfortunately.

I do like Ruggieo's "The Relative space: space measurements on a rotating platform". https://arxiv.org/abs/gr-qc/0309020, though I can't necessarily claim it has a high impact factor, but I like the way it applies the fundamental SI defintion of distance via round-trip travel times to rotating frames.

The basic idea is that space-time is 4-dimenstional, and that distances are 3-dimensional, so we need to project the 4-dimensional space-time onto a 3-dimensional space to measure distances. Ruggiero paper basically addresses the nature of the "space" defined by such a projection operation.