Special relativity thought experiment

I'm trying to build some intuition to some basic concepts in Special Relativity. The following is a thought experiment from one of my lectures.

Here are the statements:

(1) According to Alice, her light signal reaches Charlie before the light signal sent by Bob.

(2) According to Alice, her light signal reaches Charlie at the same time as the light signal sent by Bob.

Here's my thought process:

• It's clear to me that one of the big points in Special Relativity is that light moves at the same speed in all inertial frames. So, both signals would hit Charlie at the same time.
• However, according to Alice it would perhaps be the case that her signal hits Bob first. In Alice's inertial frame, Charlie and Bob are moving at speed $-v$. However, due to their initial positioning, Bob is moving away from her and Charlie is moving towards her.
• However, what's immutable is that both light signals were emitted when Alice and Bob were at the same $x$ position. Therefore, according to Alice both signals hit Charlie at the same time. It doesn't matter if she's closer to Charlie by the time this happens.

Am I going wrong somewhere in my thinking?

The light signals travel as a pair. They are emitted at the same time and place in every frame of reference (even non-inertial). Therefore both Alice and Bob will think that both signals reach Charlie at the same time. What will differ between Alice and Bob is how long they think that it takes for the signal pair to reach Charlie.

Thus, statement (2) is correct.

Recall that special relativity introduces the notion of the spacetime interval, or the idea that the significant quantity relating the displacements in space and time between two events is invariant between reference frames. The way the problem is set up, we can immediately identify two points in spacetime: The point at which the signals are emitted (they occur at the same time and place for both observers, so they are the same spacetime point), and the point at which the signal is received.

If the second statement were true, then the time component of the spacetime interval for both observers would have to be the same, but since there is a relative velocity between their reference frames, this cannot be the case. Since Bob's spacetime interval has only a time component (since he stands still) but Alice moves through space and time, we can see that time component in Alice's frame will be smaller than in Bob's, since the space component has to account for some of the length of the spacetime interval, which is the same for both observers. This means that in her frame, the time separation between emission and reception will be smaller than that of Bob, and she will interpret her signal as being received before Bob's.

• Sorry, I'm extremely new to this topic so I have some trouble understanding you in some areas. Let me try and give a jist of what you're saying and you tell me where I'm wrong: both Alice and Bob emit a light signal at the exact same place and at the exact same time -- you cannot argue objectively that Charlie received the light beam first by any of them since that would imply one light beam was faster than another. However, once the light beam is emitted, I don't see how Alice's motion matters after this point, other than the fact that she will be closer to Charlie during reception. – sangstar Nov 14 '17 at 1:38
• Sort of. The whole idea of special relativity is that you can't argue anything objectively, except for the speed of light and the length of the spacetime interval between two events (also sometimes called the invariant interval, for that reason). Because Alice is moving with respect to Bob, she will perceive space and time differently than Bob will. It doesn't matter where she is during reception - just the very fact that she's moving means she is in a different reference frame, and therefore has different interpretations of the displacements in space and time between two events. – Billy Kalfus Nov 14 '17 at 1:54
• I'm afraid I haven't covered spacetime intervals before, but from what I do understand, are you saying that since Alice is moving, the distance between her and Bob at emission is closer for her than it is for Bob? Even though they look to be the same distance away objectively? And this is because of length contraction? – sangstar Nov 14 '17 at 1:56
• Ah - if you haven't covered the spacetime interval yet then it may be confusing. Not quite - in fact, in her reference frame the length is longer than in Bob's. You need to toss out this idea of the way something is "objectively" - in special relativity, you always identify which observer is making an observation, and therefore nothing (except the speed of light and the length of the invariant interval) is objective. – Billy Kalfus Nov 14 '17 at 2:01
• You are wrong. The light signals travel as a pair. Therefore both Alice and Bob will think that both signals reach Charlie at the same time. What will differ between Alice and Bob is how long they think that it takes for the signal pair to reach Charlie. – md2perpe Feb 14 at 17:51