Inconsistency in Length contraction

[KallaNikhil]

Let there be 2 astronauts A and B in uniform relative motion and they are moving towards each other. Let A be moving in a spaceship of length L and there are 2 clocks ca1 and ca2 attached to the front and back of the spaceship respectively and let clock of B be cb. The moment when the front of the A's spaceship crosses B then B sets it's time cb equal to ca1. Now from A's perspective when the back of the spaceship crosses B , ca2 must have elapsed more time than cb. But from B's perspective ca2 must have elapsed less time than cb (after including length contraction of spaceship from B's perspective). How can there be an inconsistency in the time measurements of ca2 ? (time elapsed is the time taken for the spaceship to cross B)

 

[Dale]

But from B's perspective ca2 must have elapsed less time than cb (after including length contraction of spaceship from B's perspective).

First, whenever there is one of these introductory paradoxes, the solution is almost always the relativity of simultaneity. In Bs frame clock cb is offset by an amount that explains this discrepancy.

Second, as a general piece of advice, don't use the length contraction and time dilation formulas. Use instead the full Lorentz transform. It will automatically simplify as appropriate, but it will not simplify in cases where the time dilation or length contraction formulas are inappropriate.

 

[Ibix]

The moment when the front of the A's spaceship crosses B then B sets it's time cb equal to ca1.

As Dale says, your problem is the relativity of simultaneity. The quoted sentence is where you went wrong. "The moment when..." according to which frame? The two astronauts won't agree what that moment is, except at the front of the rocket.

 

[PAllen]

First, whenever there is one of these introductory paradoxes, the solution is almost always the relativity of simultaneity. In Bs frame clock cb is offset by an amount that explains this discrepancy.

Second, as a general piece of advice, don't use the length contraction and time dilation formulas. Use instead the full Lorentz transform. It will automatically simplify as appropriate, but it will not simplify in cases where the time dilation or length contraction formulas are inappropriate.

I think you mean clock ca2 is offset compared to ca1 in B's frame.

 

[Dale]

I think you mean clock ca2 is offset compared to ca1 in B's frame.

oops, yes. Thank you

 

[Vitro]

The moment when the front of the A's spaceship crosses B then B sets it's time cb equal to ca1.

Exercise: what time does ca2 show according to astronaut B when the above takes place?

 

[KallaNikhil]

As Dale says, your problem is the relativity of simultaneity. The quoted sentence is where you went wrong. "The moment when..." according to which frame? The two astronauts won't agree what that moment is, except at the front of the rocket.

I am talking about the time when the back of the spaceship just crosses B. Shouldn't they both agree on the time shown by the clock ca2 ? (I am not talking about the time shown by their respective clocks but the time shown by ca2 only from the 2 frames, Assume that a physical clock is present)

 

[Ibix]

I am talking about the time when the back of the spaceship just crosses B. Shouldn't they both agree on the time shown by the clock ca2 ? (I am not talking about the time shown by their respective clocks but the time shown by ca2 only from the 2 frames, Assume that a physical clock is present)

No. A and B don't agree what "at the same time as ca1 and cb passed each other" means except at the location where ca1 and cb passed each other. So they don't agree what ca2 read "at that same time". They'll agree what ca2 reads when it passes cb.

 

[KallaNikhil]

I think you mean clock ca2 is offset compared to ca1 in B's frame.

No, I meant regarding the different values of ca2 as observed from the 2 different frames at the instant when the back of the spaceship just crosses B.

 

[KallaNikhil]

No. A and B don't agree what "at the same time as ca1 and cb passed each other" means except at the location where ca1 and cb passed each other. So they don't agree what ca2 read "at that same time". They'll agree what ca2 reads when it passes cb.

is it not that ca1 and ca2 are synchronous ?

 

[jbriggs444]

is it not that ca1 and ca2 are synchronous ?

They are syntonous according to all inertial frames (i.e. they tick at the same rate as each other, no matter what frame they are measured from)
They are synchronous according only according to A's frame (in that frame they show the same reading at the same time. In other frames they do not).

 

[Ibix]

is it not that ca1 and ca2 are synchronous ?

As jbriggs444 says, only in one frame. That's what "relativity of simultaneity" means - whether two events are regarded as simultaneous is a frame-dependent thing.

 

[PAllen]

No, I meant regarding the different values of ca2 as observed from the 2 different frames at the instant when the back of the spaceship just crosses B.

I was replying to Dale. He knows what I meant, and he was describing the same thing, just typed too quickly.

 

[Dale]

is it not that ca1 and ca2 are synchronous ?

They are synchronized only in A's frame. In B's frame they are not synchronized.

 

[Dale]

I was replying to Dale. He knows what I meant, and he was describing the same thing, just typed too quickly.

To confirm for the OP, yes @PAllen and I agree, I just made a small typo which he caught and corrected.

 

[SiennaTheGr8]

@KallaNikhil

You might find David Morin's "rear clock ahead" exercise helpful (it's on page 11): http://www.people.fas.harvard.edu/~djmorin/chap11.pdf