Prove that “leading clocks lag”

[alexmahone]

Homework Statement

A spaceship has two clocks: one in the front and one at the back. The clocks are synchronized in the spaceship's frame of reference.

The spaceship zooms past the earth's surface at a relativistic speed. Prove that in the earth's frame of reference, the clock at the front lags behind the clock at the back.

Relevant Equations

No equations

Homework Statement: A spaceship has two clocks: one in the front and one at the back. The clocks are synchronized in the spaceship's frame of reference.

The spaceship zooms past the Earth's surface at a relativistic speed. Prove that in the Earth's frame of reference, the clock at the front lags behind the clock at the back.
Homework Equations: No equations

Let us suppose two light pulses are simultaneously emitted (from the spaceship's frame of reference) from the center of the spaceship to the front and the back.

Refer to https://d3c33hcgiwev3.cloudfront.net/_d661836c6bcee877d087ef0a9e3cf564_Wk4_relativityofsimultaneityleadingclockslag_handout.pdf?Expires=1569369600&Signature=Tu3pd2pk6jS~BUA714JOMgZNt-nvwwPNxr66nx79KIA~QLb7GU-XtMG0qsukf2ojcp5v5BfsPAJzayNRclOTeCYEJdVUPt5ZeeW8dxqFqq1NSsVptx9Bg1ZcdrA6NMFI5sqN3FvoTyyI3hXT6ZnWWP~I~m8Z5DNmVU3FgNgfX50_&Key-Pair-Id=APKAJLTNE6QMUY6HBC5A for a diagram.

In the Earth frame of reference, the light pulse approaching the back of the spaceship has to travel a shorter distance. So, the pulse reaches the back of the spaceship before the other pulse reaches the front of the spaceship.

I'm not sure how to proceed.

 

[Janus]

What happens for the the light pulses according to the spaceship frame of reference?

 

[alexmahone]

What happens for the the light pulses according to the spaceship frame of reference?

In the spaceship frame of reference, the light pulses reach the front and the back of the spaceship simultaneously. So...?

 

[Janus]

So, they should arrive, when the clocks at the ends read the same.
What time should each clock read when its light pulse reaches it in the Earth frame?

 

[alexmahone]

Hey @Janus, sorry for my delay in replying. The truth is that I've been having a very hard time with this problem and in formulating my thoughts regarding the same. But I'll give it a go:

I have a conceptual misunderstanding here: One can very well calculate the time when each pulse reads the front and rear clocks in the Earth frame -- but why should this light pulse matter at all? IMO we should rather find the time that the front and rear clocks read at any given instant on the Earth frame, not at two different instances when the light pulses reach them.

I hope what I just said makes some sense!

 

[FactChecker]

It has been shown experimentally that the spaceship will think that the clocks will indicate identical times when the light pulses reach the spaceship clocks. The observers on the ground will think that the light pulses hit the spaceship ends at different times according to ground clocks at the immediately adjacent locations. What does that mean about the spaceship clocks compared to the immediately adjacent ground clocks?

 

[PeroK]

Hey @Janus, sorry for my delay in replying. The truth is that I've been having a very hard time with this problem and in formulating my thoughts regarding the same. But I'll give it a go:

I have a conceptual misunderstanding here: One can very well calculate the time when each pulse reads the front and rear clocks in the Earth frame -- but why should this light pulse matter at all?

I hope what I just said makes some sense!

It matters because you are trying to compare times in two different reference frames. If you never think beyond one single reference frame per experiment, then all the results of SR will elude you.

In particular it is the invariant speed of light (the same light pulse has the same "invariant" speed in reference frames moving with respect to each other) that disrupts the classical notions of absolute time, space and simultaneity.

 

[Janus]

So, jut to clarify what the last two two posters have said. If in the ship frame, the light pulses reaches a clock when it reads 12:00:01, then according to the Earth frame, this is exactly what that clock reads when the light pulse reaches it. In the ship frame, both clocks read the same when the light pulses reach them. Thus in the Earth frame, the reading for each clock when the light pulses reaches it must be identical. So if, in the Earth frame the front and rear clocks both read 12:00:01 when the light reaches them, and the light reaches them at different times( by the Earth clock), then the clocks on the ship, at any given moment, read different times.

 

[PeroK]

... you can take this a step further conceptually. Counting the arrival of the pulses constitutes two synchronized clocks in the ship frame; but two unsynchronized clocks in the Earth frame. The clocks may simply be counters, counting the number of pulses.