Analyzing Time Dilation in a Spaceship-Mars Scenario

[dirac26]

Here is the simple problem from the book, but I have a hard time understanding how to solve it, or how to think about it.
A spaceship flies past Mars with a speed of relative 0.985 c to the surface of the planet. When the spaceship is directly overhead, a signal light on the Martian surface blinks on and then off. An observer on Mars measures that the signal light was on for 75 microseconds. (a) Does the observer on Mars or the pilot on the spaceship measure the proper time? (b) What is the duration of the light pulse measured by the pilot of the spaceship?

My first though is that proper time is 75 microseconds, since both of these events(start and the end of the signal) happened at the same location - on Mars. So, the Δt
₀ = 75μs.
The duration of the light pulse measured by the pilot of the spaceship should then be simply Δt' = γΔt
₀, right? That gives 435μs. It should be longer since for the observer on the spaceship, the Mars is moving, and (similar like the experiment with the mirror and light source) the light passes longer distance, so the time interval is larger. But, here is what I don't understand, observer on the spaceship is moving, and he should measure shorter time interval, right? Or, does he looks at the clock on the Mars, and that clock is slowed down??? I am kind of confused. What is wrong in my train of thought?

 

[Ibix]

Does the observer on Mars or the pilot on the spaceship measure the proper time?

The proper time of what clock?

My first though is that proper time is 75 microseconds,

Assuming the question wants the proper time of the clock controlling the signal light, I agree.

The duration of the light pulse measured by the pilot of the spaceship should then be simply Δt' = γΔt

Are you sure? Remember that the spaceship is moving and you need to take Doppler into account.

 

[Sagittarius A-Star]

Here is the simple problem from the book,

Which book?

but I have a hard time understanding how to solve it, or how to think about it.
A spaceship flies past Mars with a speed of relative 0.985 c to the surface of the planet. When the spaceship is directly overhead, a signal light on the Martian surface blinks on and then off. An observer on Mars measures that the signal light was on for 75 microseconds.

The spaceship flies past Mars over a distance of $22.2 km$ while the $75 \mu s$, with reference to the Mars-rest frame. It only can be approximately "directly overhead" of both blink-events, if you assume a "high enough" altitude. I assume, that you want to argue with the transverse Doppler effect.

My first though is that proper time is 75 microseconds, since both of these events(start and the end of the signal) happened at the same location - on Mars. So, the Δt
₀ = 75μs.
The duration of the light pulse measured by the pilot of the spaceship should then be simply Δt' = γΔt
₀, right? That gives 435μs. It should be longer since for the observer on the spaceship, the Mars is moving, and (similar like the experiment with the mirror and light source) the light passes longer distance, so the time interval is larger. But, here is what I don't understand, observer on the spaceship is moving, and he should measure shorter time interval, right? Or, does he looks at the clock on the Mars, and that clock is slowed down??? I am kind of confused. What is wrong in my train of thought?

As you can see at the example of a moving light clock, the angle between a light-pulse direction and the direction of the relative velocity between the frames is in general differently in the 2 frames. This effect is called "aberration".

If the angle is for example 90° in the receivers frame, then there is a Doppler redshift by $1/\gamma$.
But if the angle is 90° in the senders frame, then there is a Doppler blueshift by $\gamma$.

Source:
https://en.wikipedia.org/wiki/Relativistic_Doppler_effect#Transverse_Doppler_effect

So, that with reference to each frame the "moving" clock ticks slower than the clock "at rest" does not lead to a contradiction.

 

[Vanadium 50]

Which book?

THE Book:

Man, you guys need to study the classics more!

I am going to suggest you draw a spaectime diagram of the problem (and all non-trivial problems). Carefully label it (r.g. pulse emitted, pulse detected) While it may not directly provide the answer, it will force you to think about the problem and to identify what is meant by qords like "distance" and "duration" and 'frequency".

 

[Orodruin]

Are you sure? Remember that the spaceship is moving and you need to take Doppler into account.

The spaceship is directly overhead. This removes the classical part of the Doppler effect and the only remaining effect is due to time dilation.

 

[robphy]

Which book?

From a google book search, it looks like it's from Hugh Young's University Physics.

 

[jbriggs444]

The spaceship is directly overhead. This removes the classical part of the Doppler effect and the only remaining effect is due to time dilation.

Is the signal part of a broad beam or a narrow ray?

If it is a narrow ray, the pilot cannot measure its duration without a second observer and a simultaneity convention.

If it is a broad beam, then the top of the beam's wavefront will not be horizontal as assessed by the pilot and the duration of the signal may not be in accordance with a naive calculation.

 

[Ibix]

The spaceship is directly overhead. This removes the classical part of the Doppler effect and the only remaining effect is due to time dilation.

It moves 22km in 75 microseconds. Depending how close to Mars it is (we can assume it's not in the atmosphere at 0.985c, but that's only a few kilometres deep) it might or might not be plausible to describe it as "directly overhead" the whole time.

The question doesn't seem particularly well specified to me (note also the "who measures proper time" question). At least not as reproduced here.

 

[Sagittarius A-Star]

If it is a narrow ray, the pilot cannot measure its duration without a second observer and a simultaneity convention.

The pilot can measure it's duration without a second observer, i.e. if his Galilei-telescope has a diameter of $23 km$. The time-dilation factor depends on the simultaneity convention.

Source:
https://de.wikipedia.org/wiki/Fernrohrbrille#Optischer_Hintergrund

​

 

[jbriggs444]

The pilot can measure its duration without a second observer and with a simultaneity convention, i.e. if his Galilei-telescope has a diameter of $23 km$.

View attachment 338447​

Source:
https://de.wikipedia.org/wiki/Fernrohrbrille#Optischer_Hintergrund

​

So you are going for the "broad beam" interpretation. You also seem to assume that the signal front arrives at $A$ and $B$ simultaneously. In what frame is that assumption justifiable? Do you see that it must be false in every other frame?

 

[Sagittarius A-Star]

So you are going for the "broad beam" interpretation. You also seem to assume that the signal front arrives at $A$ and $B$ simultaneously. In what frame is that assumption justifiable?

No, I assume a narrow ray. The first signal reaches the lens at location A, the second later at location B.

 

[jbriggs444]

No, I assume a narrow ray. The first signal reaches the lens at location A, the second later at location B.

So you have a moving lens that sweeps across the narrow beam?

 

[Sagittarius A-Star]

So you have a moving lens that sweeps across the narrow beam?

Yes.

 

[jbriggs444]

Yes.

Or, equivalently, two mirrors arranged carefully so that the leading mirror catches the leading edge of the signal and reflects that leading edge to the pilot awhile the trailing mirror catches the trailing edge of the signal and reflects that trailing edge to the pilot.

Refraction or reflection, the result is the same. The physical situation will allow the realization of a simultaneity convention, as you had pointed out.

Yes, that could work.

Of course, there will be some disagreement about the geometry of the situation due to aberration.

 

[Sagittarius A-Star]

Or, equivalently, two mirrors arranged carefully so that the leading mirror catches the leading edge of the signal and reflects that leading edge to the pilot awhile the trailing mirror catches the trailing edge of the signal and reflects that trailing edge to the pilot.

Refraction or reflection, the result is the same. The physical situation will allow the realization of a simultaneity convention, as you had pointed out.

Yes, that could work.

Of course, there will be some disagreement about the geometry of the situation due to aberration.

It must work, because it was done in 1979:

Direct observation of the transversal Doppler-shift

D. Hasselkamp, E. Mondry & A. Scharmann​

Abstract
An experiment is reported in which the second order Doppler-shift has been determined by the observation of theHα-line emitted by linearly moving hydrogen atoms of velocities 2.53×108 cm/s−9.28×108 cm/s. In contrast to previous experiments the direct transversal observation has been used for the first time. The coefficient of the second order term in the relativistic approximation is found to be 0.52±0.03 which compares good with the theoretical value of 1/2.

Source:
https://link.springer.com/article/10.1007/BF01435932