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.