How Does Light Reach Bob in Special Relativity?

[Ned Stark]

Homework Statement

Andrew is at rest in frame S and observes Bob moving to the right (frame S0 is
bobs rest frame) with speed v. On the ground, a distance L to the right of Andrew, is a lamp.
At the instant that bob passes Andrew (as observed in frame S), the lamp emits a flash of light

At what time (in frame S0) does the light reach bob?

Homework Equations

Δt'=γΔt
Δx'=(1/γ)Δx
γ=1/sqrt(1-(v/c)^2)

The Attempt at a Solution

Ive tried this problem using two methods, though I am not sure which one is correct, if any.
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attempt 1)

let the instant that bob passes Andrew occur at t=0 and x=0

in frame S, bob receives the light at t = L/(v + c), but bobs clock runs slower by a factor of gamma

so in S0, bob receives the light at t'=γ*L/(v + c).

But wouldn't this imply that in bobs frame, the flash moves at velocity v+c? which violates the postulate that the speed of light is the same in any frame.

attempt 2)

in bobs frame, the lamp is moving towards him at speed v, so the distance to the lamp in frame S0 is contracted to L'=(1/γ)*L.

and in S0 the flash travels at speed c, so in S0, the flash reaches bob at time t'= L/(c*γ)


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i am leaning towards attempt 2, however in bobs frame, would the flash of light occur before bob passes andrew?

 

[Doc Al]

attempt 1)

let the instant that bob passes Andrew occur at t=0 and x=0

in frame S, bob receives the light at t = L/(v + c), but bobs clock runs slower by a factor of gamma

so in S0, bob receives the light at t'=γ*L/(v + c).

Sounds good to me.

But wouldn't this imply that in bobs frame, the flash moves at velocity v+c? which violates the postulate that the speed of light is the same in any frame.

No. Why would you think that?

attempt 2)

in bobs frame, the lamp is moving towards him at speed v, so the distance to the lamp in frame S0 is contracted to L'=(1/γ)*L.

and in S0 the flash travels at speed c, so in S0, the flash reaches bob at time t'= L/(c*γ)

This attempt fails because in Bob's frame the light does not flash when Andrew passes Bob. You neglected the relativity of simultaneity.

Attempt 1 wins!

 

[Ned Stark]

Is it correct to say that in andrews frame, the relative velocity between the flash and bob is c+v?

That seems kinda funky.

But i do get what your saying about relativity of simultinaity. I drew a Minkowski diagram for bobs frame, and sure enough, in bobs frame the lamp does not flash at the same time as when bob passes andrew.

 

[Ned Stark]

Is it correct to say that in andrews frame, the relative velocity between the flash and bob is c+v?

That seems kinda funky

actually It doesn't seem to funky now that I've thought about it some more. the important thing is that is bobs inertial frame, the speed of light relative to him is c

 

[Doc Al]

actually It doesn't seem to funky now that I've thought about it some more. the important thing is that is bobs inertial frame, the speed of light relative to him is c

Good! Looks like you figured it out.

The rate at which Bob and the flash approach each other as seen in Andrew's frame is often called the closing speed to distinguish it from relative velocity, which means something different. The closing speed of two objects can be greater than c, since it's not really the speed of anything. But the relative velocity of two objects can never exceed c.