I Supernova 100,000 Light Years Away: Agree on Time?

[Vampyr]

If on Earth we detect the light from a supernova 100,000 light years away, we can say that the supernova happened 100,000 years ago (ignoring any dust etc. that might slow down the light). However, would all observers agree that the event happened 100,000 years ago? If a spaceship was traveling in the direction of the supernova at 0.99c at the same distance as the Earth, what would they see?

By my thinking, they see light still traveling at c, and they observe no difference in their own clock. However, they would see the distance to the supernova Lorentz contracted. Since the light traveled a shorter distance from the spaceship's perspecitive, the supernova happened sooner from the spaceship's perspective than the Earth's perspective. I.e. the spaceship sees the supernova less than 100,000 light years away but no other changes that offset this.

Is my thinking correct?

 

[haushofer]

Well, draw a spacetime diagram {t,x} indicating the rest frame of the supernova and earth, neglecting the expansion of space. Put the Earth in its origin. Now introduce a second observer moving with v w.r.t. earth, having a frame {t',x'} with t=t'=0 coinciding. Let the supernova explode at t=-T. What is T'?

 

[Ibix]

Depends what you mean. Your reasoning is correct if you want to ask where the supernova remnant is now using the spaceship's frame. But the explosion itself happened in the past and the star/remnant is moving very fast in this frame, so was a lot further away when it went off.

It's much safer to use the Lorentz transforms than the special cases of length contraction and time dilation.

 

[PeroK]

If on Earth we detect the light from a supernova 100,000 light years away, we can say that the supernova happened 100,000 years ago (ignoring any dust etc. that might slow down the light). However, would all observers agree that the event happened 100,000 years ago? If a spaceship was traveling in the direction of the supernova at 0.99c at the same distance as the Earth, what would they see?

By my thinking, they see light still traveling at c, and they observe no difference in their own clock. However, they would see the distance to the supernova Lorentz contracted. Since the light traveled a shorter distance from the spaceship's perspecitive, the supernova happened sooner from the spaceship's perspective than the Earth's perspective. I.e. the spaceship sees the supernova less than 100,000 light years away but no other changes that offset this.

Is my thinking correct?

Relativity is not about the delay in light (or other) signals reaching an observer. That has no bearing on the time of an event in a reference frame. To obtain the time of the event in the spaceship frame you must use the Lorentz Transformation.

 

[PeterDonis]

Is my thinking correct?

No, because you're leaving out relativity of simultaneity. You cannot correctly analyze any relativity problem using length contraction and time dilation if you ignore relativity of simultaneity. Or, you could take the advice @Ibix gave in post #3 and just use the Lorentz transformations instead, since that automatically takes everything into account.