[Physics FAQ] - [Copyright]
Don Koks, 2022.
That question doesn't contain enough information to be answered. But we can always say one thing for certain: what you see involves light signals that travel at a finite speed, and you have to take that speed into account to ascertain when the events happened. If, once you've done that, you find that the events happened at the same time for you, then yes, they were simultaneous in your frame. If you find that they didn't happen at the same time for you, then no, they were not simultaneous in your frame.
Specifically, consider the following question, a slightly modified quote from the second reference below:
Two volcanoes, A and B, are a distance D apart in Earth's reference frame. Alice is standing at rest relative to the volcanoes, halfway between them. The two volcanoes erupt, and Alice sees the light from the eruptions at the same time. Bob is standing next to volcano B, and is at rest relative to Alice. What does Bob see, and are the eruptions simultaneous for him?
This question can be answered quickly. From the symmetry of Alice's measurements, she says the eruptions occurred simultaneously for her. Because Bob shares Alice's frame, he says that the eruptions were simultaneous for him too. Also, because he was closer to volcano B than to A, Bob saw B's eruption before he saw A's eruption.
Let's shed more light on that by analysing the scenario in detail. Alice is halfway between the volcanoes, and so she can trivially account for the travel times of the eruptions' light signals by tracing those signals back to their sources, in time. Suppose she saw the light from both eruptions at her time 12:00:
1. Alice sees the light from both eruptions (a.k.a. "Alice sees the eruptions happen") at her time 12:00.She knows the volcanoes are both at a distance of D/2 from her, and so she calculates that the volcanoes both erupted at her time 12:00 − D/(2c), where c is the speed of light:
2. Alice concludes that the volcanoes both erupted at her time 12:00 − D/(2c).Since they erupted at the same time for her, then by definition of the word "simultaneous", she says they erupted simultaneously for her.
What about Bob? He is at rest relative to Alice, and hence he shares her frame, which includes sharing her concept of time: that is, both Alice and Bob agree that their clocks are always synchronised. It follows that the eruptions must be simultaneous for him too. And if he says they are simultaneous but is closer to B and at rest relative to both volcanoes, then he must see B erupt before A. So that answers the original question.
But let's explicitly ask what Bob infers from what he sees, without our knowing anything about his standard of simultaneity. Bob's experience cannot be disagreed about by anyone; if someone disagreed with him, that would be as nonsensical as if they had disagreed about what Bob's name is. We are supposing that we know little about Bob, so let's ask Alice what Bob sees. She knows that the volcanoes erupted simultaneously, and she knows that Bob is at rest relative to these volcanoes, and right next to B. Hence she says that Bob sees the light from volcano B as soon as it erupts, meaning he sees it erupt at the time that Alice worked out in item 2 above (a time that he agrees on):
3. Bob sees the light from B's eruption (a.k.a. "he sees B erupt") at 12:00 − D/(2c).Alice says that Bob saw B erupt before he saw A erupt, since the light from A took some time to reach Bob. Because she and Bob cannot disagree about what he sees, that answers the question "What does Bob see?". Also, because Bob is right next to B, he concludes that B erupted precisely when he saw it:
4. Bob concludes that B erupted at 12:00 − D/(2c).Now, what about volcano A? The light from A's eruption takes a time D/c to reach Bob, and so he sees A's eruption at a later time:
5. Bob sees the light from A's eruption (a.k.a. "he sees A erupt") at 12:00 − D/(2c) + D/c = 12:00 + D/(2c).But he's not fooled by what he merely sees of a distant event; he knows that he has to account for light's finite speed that carried information of that event to him. He traces the light from A back in time: he saw it erupt at 12:00 + D/(2c), and knows he must subtract the time D/c that the light from the eruption took to reach him. So,
6. Bob concludes that A erupted at 12:00 + D/(2c) − D/c = 12:00 − D/(2c).This is the same time that he says B erupted, in item 4. So, he says the volcanoes erupted at the same time: they erupted simultaneously for him.
Note that when we discuss such scenarios, we have to be careful with our words. For example, "Bob sees A erupt at time T" is understood to mean "At Bob's time T, the light from A's eruption reaches his eyes". It is not understood to mean that Bob says "A erupted at time T". If Bob sees A erupt at time T, then he knows A erupted at either time T (if A is right next to him) or time T minus the light-transit time (if A is not right next to him).
Relativity is built on straightforward ideas. It assumes that people making measurements know all about signal-transit times, and that they account for these in their measurements.
The volcano scenario is a classic one. It was discussed in:
R. Scherr, P. Shaffer, S. Vokos, The challenge of changing deeply held student beliefs about the relativity of simultaneity, Am. J. Phys. 70 1238 (2002).
The scenario and the above reference were also discussed in:
J. Aslanides, Relativity Concept Inventory, Australian National University, 2012, page 116.
Unfortunately, the question was answered incorrectly there—presumably a simple mistake by its author.