Special Relativity-Lorentz Transformation

[Qyzren]

Can you guys please verify/help me with some questions.

Consider 2 events, A & B. In frame S they are separated by Δx and Δt. It is reasonable to say that the events are causally connected if it is possible for a signal (such as a light pulse) to travel between them. We mean that one might have caused the other. Write down a relation (an inequality) between Δx and Δt that expresses this. Make sure your relation is still valid when Δx and/or Δt are negative (hint: use absolute values)

I was thinking, since I = -(cΔt)² + (Δx)² + (Δy)² + (Δz)², with Δy and Δz = 0. We have I = -(cΔt)² + (Δx)²

Since this is a time like interval or at most light like interval, I <= 0.
so -(cΔt)² + (Δx)² <= 0.

I believe this should be correct but i didn't use any absolute value signs... or should i express it in terms of Δt & Δx instead of its square?


Show that the concept of being causally connected is invarient, i.e, that all inertial observers will agree on whether A & B are causally connected.
We have -(cΔt)² + (Δx)²
I define a new frame S', with interval (cΔt', Δx', Δy', Δz').
Δy' & Δz' is 0.
so using lorentz transformations:
x = γ(x'+vt), t = γ(t'+vx'/c²)
-(cγt' + γvx'/c)² + (γx' + γvt')² <= 0
some expanding & simplifying gives
γ²t'²(v²-c²) + γ²x'²(1-v²/c²) <= 0
subbing in γ² for c²/(c²-v²) for the first product and 1/(1-v²/c²) for the second gives
-(ct')² + (x')² <= 0.
Therefore all inertial observers will agree on whether A & B are causally connected.

Does this make sense? Or am i doing the question wrong?

Suppose that A & B are causally connected, and that A occurs before B in frame S, Show that all observers will agree that A occurs before B.

I am not so sure about this one, but i thinking...
Suppose there is a time t > 0 between events A and B (and a distance x),
i have to show that t' in any reference frame is > 0
that is to say, the time between the events is always positive, meaning A happened first.

Something along those lines, if someone could help me and check my answers, that would be great.
Thanks everyone for helping.

 

[javierR]

Hey,
Looks fine to me. Their reference to absolute values means that you can write your condition as
|x|^2 <= c^2*|t|^2
then taking square roots
|x|<= (+-) c*|t|.
Drawing a t vs. x diagram with event A at the origin, this condition means that event B is *within* the forward or backward lightcone of event A. A non-causally connected event B would be outside of the light-cone of A. (You can exchange A with B everywhere in those sentences).

Your thinking on the last question is also fine, keeping in mind the causality condition you found is satisfied (i.e., you may use the condition in your chain of reasoning).