Relativity and a Mirror

[saddlestone-man]

TL;DR Summary

How does my image change as I move away from a mirror?

If I park my space ship close to a highly-reflective mirror in space and then accelerate away from it at exactly 90 degress to its surface, how will the image I see of myself change as I approach the speed of light?

best regards ... Stef

 

[PeroK]

TL;DR Summary: How does my image change as I move away from a mirror?

If I park my space ship close to a highly-reflective mirror in space and then accelerate away from it at exactly 90 degress to its surface, how will the image I see of myself change as I approach the speed of light?

best regards ... Stef

Is that so difficult to calculate?

 

[Nugatory]

If I park my space ship close to a highly-reflective mirror in space and then accelerate away from it at exactly 90 degress to its surface, how will the image I see of myself change as I approach the speed of light?

(You mean that your speed relative to the mirror is approaching $c$, I presume? That seems the only natural interpretation here, but it is good to always be careful about what velocities are relative to).

This problem will be easier to understand if you start with a somewhat simpler problem: You had been accelerating away from the mirror but have stopped accelerating and now are just coasting, moving at a constant speed relative to the mirror. Now you can draw a simple Minkowski diagram using the frame in which the ship is at rest and the mirror is moving away from it. Draw the worldlines of successive flashes of light leaving the ship at a fixed frequency according to ship-board clock (say one flash emitted per second), reaching the mirror and being reflected back to the ship. What is the frequency with which the returning flashes return to the ship?
Hint 1: The motion of the mirror relative to the ship at the moment that a light flash reaches it is not relevant.
Hint 2: The distance traveled by the light is relevant.

Work this out and you'll be able see how it works in the case in which the ship is accelerating and we're considering the image of the ship during acceleration.

 

[Orodruin]

Hint 1: The motion of the mirror relative to the ship at the moment that a light flash reaches it is not relevant.

Huh? Maybe you are meaning something else, but the motion of the mirror is highly relevant to the frequency of the returning flash.

 

[Nugatory]

Huh? Maybe you are meaning something else, but the motion of the mirror is highly relevant to the frequency of the returning flash.

I'm thinking the time interval between successive returning flashes considered as zero-width pulses (similar to how the Doppler effect is treated in the Twin Paradox FAQ). Understand this, then apply the same considerations to successive wave crests, and the red shift will naturally appear.

 

[saddlestone-man]

Many thanks for the replies so far.

If I may ask a supplementary question: would the red shift I see in my reflection in the mirror be the same seen by an observer standing on the mirror? Or would the fact that the light I see has travelled a round-trip distance twice that of the light observed on the mirror make a difference?

 

[Nugatory]

If I may ask a supplementary question: would the red shift I see in my reflection in the mirror be the same seen by an observer standing on the mirror? Or would the fact that the light I see has travelled a round-trip distance twice that of the light observed on the mirror make a difference?

Now this you really can calculate for yourself, if you’ve understood the previous answers. Try it for yourself, you’ll be glad that you did.

(And if you really can’t work it out after making a good-faith effort someone here will give you the answer).

 

[Ibix]

If I may ask a supplementary question: would the red shift I see in my reflection in the mirror be the same seen by an observer standing on the mirror?

There are several ways to interpret this question. Are you considering your original accelerating case or Nugatory's simplified inertial version? And what redshift are you referring to here? Are you comparing light before or after the reflection for each observer?

Or would the fact that the light I see has travelled a round-trip distance twice that of the light observed on the mirror make a difference?

Distance has no effect at all. Unless you're talking about cosmological redshift, in which case there's a lot more detail needed in your problem specification.

 

[Nugatory]

Distance has no effect at all. Unless you're talking about cosmological redshift, in which case there's a lot more detail needed in your problem specification

The distance between the ship and the mirror has no effect, but differences in the distance travelled by successive flashes of light does - indeed that’s the basis of the Doppler effect.

 

[pervect]

TL;DR Summary: How does my image change as I move away from a mirror?

If I park my space ship close to a highly-reflective mirror in space and then accelerate away from it at exactly 90 degress to its surface, how will the image I see of myself change as I approach the speed of light?

best regards ... Stef

The problem I'm familiar with is accelerating away from a beacon at a constant proper acceleration. The short summary for this problem is that if you accelerate at 1 light year/year^2 away from a beacon (which is approximately 1G in standard units), you'll see a signal that gets more and more redshifted, to the point where you will _never_ receive a signal dated later than 1 year after the date of your departure. So if you departed jan 1 2024, you'd see timestamped signals from the beacon getting slower and slower, gradually approaching but never qutie reaching jan 1 2025, but never anything past that.

Adding a mirror adds a bit of complexity which I haven't worked out, but the phenomenon of a "last image" remains.

Wiki discusses the motion of an observer with a constant proper acceleration, which is known as "hyperbolic motion", but it doesn't work out this particular problem.

The "last image" from the beacon is a light ray which is the asymptote of the hyperbola - it never arrives, because the accelerating spaceship never reaches the asymptote.

Your problem will be more complex, but should have similar features of a "last image".

 

[Orodruin]

which is approximately 1G in standard units

It is approximately 1G regardless of units.

Adding a mirror adds a bit of complexity which I haven't worked out, but the phenomenon of a "last image" remains.

It really isn’t that hard. You just add a mirror image of the observer in the mirror’s rest frame.

 

[pervect]

The hardest part of the problem, in my mind, is figuring out what question the OP is asking. Adittionally, I find myself making errors nowadays even on fairly simple calculations nowadays, so I mostly stick to general observations and familiar problems :(. Getting old is better than the alternative, but it still has it's downsides :(.

Hopefully the observation that that the observer in the accelerating spaceship will never see his image age pass some particular instant in time is useful to the OP.