It is said that if two events are causally related, then there exists a moving observer which perceives both of them happening at the same place, which I sort of understand but how can two events which are not causally related be perceived to be simultaneous by a moving observer? It is just not clicking in my mind.
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1$\begingroup$ Do you understand this diagram i.sstatic.net/AtqPQ.gif from en.wikipedia.org/wiki/Relativity_of_simultaneity ? $\endgroup$PM 2Ring– PM 2Ring2025-03-08 17:49:04 +00:00Commented Mar 8, 2025 at 17:49
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$\begingroup$ Yes I do but I can't see the connection, how does the two events not being causally related allow a moving observer to observe them to be simultaneous? $\endgroup$Soham Pine Std 9 A Roll no 31– Soham Pine Std 9 A Roll no 312025-03-08 17:56:53 +00:00Commented Mar 8, 2025 at 17:56
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1$\begingroup$ Based on the length of the 3 answers so far, I wonder if I am missing a nuance of your question. Say you are traveling in a car at a constant speed on a straight, level road (so it's an inertial frame). On your right, a person turns on their porch light. On your left, a cow is chewing her cud and her jaw is at a particular angle in her chewing cycle. The light from both events reach your left eye at the same time. Are you asking how those two events can be causally unrelated? $\endgroup$Syntax Junkie– Syntax Junkie2025-03-09 02:26:35 +00:00Commented Mar 9, 2025 at 2:26
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$\begingroup$ I have completely understood it now, should I close the question or what should I do? $\endgroup$Soham Pine Std 9 A Roll no 31– Soham Pine Std 9 A Roll no 312025-03-09 04:10:16 +00:00Commented Mar 9, 2025 at 4:10
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3$\begingroup$ You should select the check mark next to the answer that (best) resolved your problem. $\endgroup$Daniel R. Collins– Daniel R. Collins2025-03-09 04:40:10 +00:00Commented Mar 9, 2025 at 4:40
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3 Answers
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For the sake of intuitive understanding, it would be better to start the other way around: take two events that aren't causally related but that appear to occur at the same time for an inertial observer, and then see how simple it is to manipulate their apparent time ordering.
The following is based on an explanation given by Feynman, in his 4th Messenger lecture on Symmetry.
Imagine you're in the middle of a long spaceship. The spaceship is inertial, and two flashlights are adjusted at each end of it, so they will emit a flash of light simultaneously in this frame. Clearly, these flashes aren't causally related, neither flash occurred due to the other, the flashlights and their clocks are completely independent and don't need to communicate to carry out their function.
Also clearly, since you're standing exactly in the middle of the spaceship, you'll perceive both flashes as simultaneous.
But now, imagine that before perceiving the flashes, you begin moving with constant speed in the $+x$ direction rather than standing still. You'll perceive the "right" flash to occur before the left. The opposite will be true had you moved in the $-x$ direction. So the order of the flashes depends on your state of motion, exactly as expected for spacelike separated events.
I think this is a very simple way to see how the order of non-causally related events can be changed by switching between inertial reference frames.
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$\begingroup$ It is great point but why only non causally related events, why can't two causally related events be perceived in a similar fashion. $\endgroup$Soham Pine Std 9 A Roll no 31– Soham Pine Std 9 A Roll no 312025-03-08 18:18:29 +00:00Commented Mar 8, 2025 at 18:18
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2$\begingroup$ @SohamPineStd9ARollno31 imagine instead of one of the flashlights we had a mirror. Can you perceive the mirror to be reflecting the light coming from the flashlight, before the flashlight emitted this light, by moving to some other inertial frame? $\endgroup$Amit– Amit2025-03-08 18:20:29 +00:00Commented Mar 8, 2025 at 18:20
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$\begingroup$ Brother this "aha" moment was gorgeous, thanks a lot, but one more slightly unrelated question, can you give me an intuitive reason for why the x'-axis seems to be tilted at an angle from the x-axis, most explanations include that a light ray's world line has to be symmetric and at a 45 degree angle but it doesn't feel intuitive enough, like the ct or time axis being tilted feels perfectly valid as it represents the world line of the moving observer because it is always at t'=0, but the x' axis being tilted seems to be a little weird. $\endgroup$Soham Pine Std 9 A Roll no 31– Soham Pine Std 9 A Roll no 312025-03-08 18:28:29 +00:00Commented Mar 8, 2025 at 18:28
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1$\begingroup$ @SohamPineStd9ARollno31 The $x'$ axis in what you're mentioning is a hyperplane of simultaneity, meaning that lines parallel to it define what is "now" for the inertial observer, they are lines of constant $t'$. Just like in my answer above, when the observer started moving left/right, his "now" evidently changed relative to the spaceship (recall, relative to the spaceship the lights were emitted at the same time, but not for the moving observer), so his $x'$ is tilted if we choose to draw the spaceship's axes as the "stationary frame" (though there is no preferred stationary frame). $\endgroup$Amit– Amit2025-03-08 18:31:04 +00:00Commented Mar 8, 2025 at 18:31
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1$\begingroup$ @SohamPineStd9ARollno31 consider accepting & upvoting if you've found this answer helpful, cheers. $\endgroup$Amit– Amit2025-03-08 18:44:19 +00:00Commented Mar 8, 2025 at 18:44
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Here's a diagram of spacetime axes rotation from Wikipedia's article on the relativity of simultaneity:
Events A, B, and C occur in different order depending on the motion of the observer. The white line represents a plane of simultaneity being moved from the past to the future.
There are three types of spacetime interval. Light-like (or null) intervals are at 45° in the diagram. Time-like intervals are more vertical than 45°. Space-like intervals are more horizontal than 45°.
Changing to a frame with a different velocity rotates the time-like and space-like intervals, but the light-like intervals remain at 45°, since the speed of light is invariant. Also, a boost to a different frame cannot change the quality of an interval. Time-like intervals remain time-like, and space-like intervals remain space-like.
So given two events which are on a time-like interval we can find a velocity that rotates the interval to be purely time-like, i.e., perfectly vertical. In that frame, the two events have zero spatial separation. That is, they occur at the same place.
Similarly, given two events which are on a space-like interval we can find a velocity that rotates the interval to be purely space-like, i.e., perfectly horizontal. In that frame, the two events have zero temporal separation. That is, they are simultaneous.
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Consider 1+1 Minkowski spacetime.
Consider two events A and Z which are not causally related.
Their future light-cones intersect at an event (call it F)
and
their past-light cones intersect at an event (call it P).
https://www.desmos.com/calculator/mveyzcilol
The inertial observer Bob through P and F observes that A and Z are simultaneous since Bob performs a "radar experiment" to assign a time-coordinate to event A by sending (when Bob's clock reads $t_P$) a signal to A and waiting for the arrival of its echo (when Bob's clock reads $t_F$). Bob assigns a time-coordinate to A using his clock readings $t_P$ and $t_F$: $$t_{A, \rm accd\ to\ Bob}=\frac{t_F+t_P}{2}.$$ Similarly, $$t_{Z, \rm accd\ to\ Bob}=\frac{t_F+t_P}{2},$$ which is equal to $t_{A, \rm accd\ to\ Bob}$.
(If you draw any other inertial worldline that is not parallel to PF in this spacetime diagram, that observer will assign unequal time-coordinates to A and Z.)
[Note: in Minkowski spacetime geometry, the segment PF is Minkowski-orthogonal to segment AZ.]
