Can Two Events Be Simultaneous in Multiple Reference Frames?

[itsthemac]

Simple question, but I don't know exactly where to find the answer so I'll just ask. I'm just starting out learning SR, so I apologize if this is kind of trivial. If you have two inertial reference frames, and one is in motion relative to the other, I know it's possible for two events to be simultaneous in one reference frame, and not in the other one. But, I'm wondering if it's ever possible for two events to be simultaneous in both reference frames. Is there ANY circumstance, given the above specifications, in which two events can be simultaneous in BOTH reference frames?

My guess is no, but I want to understand why it can never be that way.

 

[JesseM]

They can be simultaneous in both frames, but only if the axis between the two events is exactly perpendicular to the axis of relative motion between the two frames. You can show this using the simplest form of the Lorentz transformation, where the axis of relative motion between the frames is defined to be the x-axis, and the transformation looks like this:

x' = gamma*(x - vt)
y' = y
z' = z
t' = gamma*(t - vx/c^2)

with gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

So, suppose in the unprimed coordinate system you have two events that have the same x and t coordinates but different y coordinates, you can show that in the primed frame both events still have identical t' coordinates.

 

[itsthemac]

They can be simultaneous in both frames, but only if the axis between the two events is exactly perpendicular to the axis of relative motion between the two frames. You can show this using the simplest form of the Lorentz transformation, where the axis of relative motion between the frames is defined to be the x-axis, and the transformation looks like this:

x' = gamma*(x - vt)
y' = y
z' = z
t' = gamma*(t - vx/c^2)

with gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

So, suppose in the unprimed coordinate system you have two events that have the same x and t coordinates but different y coordinates, you can show that in the primed frame both events still have identical t' coordinates.

Thanks! I did not expect that. Now I have a few follow-up questions.

First of all, does this imply that two events are ALWAYS simultaneous in two different inertial frames, whenever they're simultaneous in one of the frames, if they occur along that perpendicular axis that you mentioned?

Secondly, (again, this is all new to me), is an "event" limited to something that's localized at a literally dimensionless point in space? Meaning for example if we have the event of a firecracker exploding, we can really only look at it as something that occurred at some exact point (I'm guessing you would take the center of mass of the firecracker to be that point)?

Also, just to make sure I understand this, please tell me if this example is correct. There's a rocket flying through space at a constant velocity, moving relative to and toward the sun on its way to fly directly past it. So we have two reference frames; the rocket's and the sun's. Does this mean that at the instant the center of mass of the rocket passes the center of mass of the sun, that a clock stationary in the sun's frame and a clock mounted to the rocket could both be zeroed simultaneously within both frames? If the answer to this is yes, I have another follow-up question, if you don't mind.

 

[JesseM]

First of all, does this imply that two events are ALWAYS simultaneous in two different inertial frames, whenever they're simultaneous in one of the frames, if they occur along that perpendicular axis that you mentioned?

Yes.

Secondly, (again, this is all new to me), is an "event" limited to something that's localized at a literally dimensionless point in space? Meaning for example if we have the event of a firecracker exploding, we can really only look at it as something that occurred at some exact point (I'm guessing you would take the center of mass of the firecracker to be that point)?

Yes, an event is understood as a dimensionless point, though often in problems physicists will idealize processes that take place in a very small region of spacetime as "events" (small compared to the distances between different 'events' being considered in the problem, that is), such as the "event" of two clocks passing right next to each other.

Also, just to make sure I understand this, please tell me if this example is correct. There's a rocket flying through space at a constant velocity, moving relative to and toward the sun on its way to fly directly past it. So we have two reference frames; the rocket's and the sun's. Does this mean that at the instant the center of mass of the rocket passes the center of mass of the sun, that a clock stationary in the sun's frame and a clock mounted to the rocket could both be zeroed simultaneously within both frames? If the answer to this is yes, I have another follow-up question, if you don't mind.

Yes, if two physical processes happen at the same position and time coordinate in one frame, they also happen at the same position and time coordinate in any other frame. This is an important principle of relativity, since if different frames disagreed about such localized occurrences, they could disagree about basic physical questions like whether a small bomb went off next to a space traveler and killed him or whether it went off sometime after it passed him and he survived!

 

[LBrandt]

I don't usually "chime in" on these posts, but I thought that I'd chime in here on this one. The OP's question involved the rocket passing "close" to the sun, but the center of mass of the rocket won't be in the same "place" as the center of mass of the sun. In every other thread on the subject of relativity of simultaneity, it has been stated that only when two events happen at the same place can observers in all frames determine whether they happened at the same time. In the above case, the rocket will not be in the same place as the center of mass of the sun, so how can you say that the two events can be considered simulaneous in all frames of reference?

 

[itsthemac]

Yes, if two physical processes happen at the same position and time coordinate in one frame, they also happen at the same position and time coordinate in any other frame.

Dumb question, but when you say same position coordinates in one frame, you mean only along one axis, right? They don't have to actually be physically at the same point, right? (otherwise I misunderstood your point before).

And here's my follow-up:

I saw an example that dealt with time dilation in my physics book in which a rocket traveling at a constant speed (0.9c relative to the solar system) goes from the sun to saturn (never accelerating). In the example, they show how you can calculate how long the journey took as measured by someone on earth, and as measured by someone on the rocket. Taking a proper sun-saturn distance of 1.43*10^12 meters, they show that the journey takes 5300 seconds as measured by someone on earth, and 2310 seconds as measured by someone in the rocket.

And from what we just discussed, as the rocket passes directly past the sun (coming from the side opposite saturn), we can start both clocks (one on the sun and one in the rocket) simultaneously. By this I mean both clocks will start ticking simultaneously from the point of view of the sun, AND both clocks will start sticking simultaneously from the point of view of the rocket.

What I'm having a problem understanding is this: We can imagine a huge digital clock on saturn that is in sync with the sun's and Earth's clocks (since they aren't moving relative to each other in this problem, and let's ignore the effects of gravity) and timing the journey. So, if you were standing on saturn, at the instant that the center of mass of the rocket passed by the center of mass of saturn, you would see "5300 sec" displayed on the clock. But what would someone on the rocket see looking down at saturn on the big digital clock at the instant they passed by the center of mass of saturn?

The reason I'm confused is similar to the twin paradox. From the rocket's frame of reference, the clock on saturn ticks slower than the clock on the rocket. But since the clock on the rocket shows "2310 sec" (from the rocket's frame of reference) at the instant they pass the center of mass of saturn, this would mean that the clock on saturn would have to read a value *less* than 2310 seconds (from the rocket's reference frame) since they were both zeroed at the same time (from the perspective of each reference frame), and thus much less than 5300 seconds. So what do the people on the rocket actually see on saturn's clock as they pass?

I could imagine a situation where the rocket physically made contact with the clock on saturn, so it's really just one event. So at that one event, what does the clock read? 5300 seconds or something less? Surely one event can't be different from two different reference frames.

Part of me doesn't even know if I'm asking the right questions to clear up my confusion.

 

[LBrandt]

I don't consider myself to be an expert on relativity, but my understanding is that for two events to be able to be viewed as simultaneous in all frames of reference, they must occur at the same point, not merely on a given axis. You can always draw a straight line between two points in space, and those two points will lie along a given axis.

 

[JesseM]

I don't consider myself to be an expert on relativity, but my understanding is that for two events to be able to be viewed as simultaneous in all frames of reference, they must occur at the same point, not merely on a given axis. You can always draw a straight line between two points in space, and those two points will lie along a given axis.

You're right, but the original question wasn't about what was needed for two events to be simultaneous in all frames of reference, just in two different frames of reference. In this case they can be at different positions as long as their separation is perpendicular to the axis of motion between the two frames (but of course you can find a new frame whose axis of relative motion is not perpendicular to the axis joining the two events, and in this frame the events are non-simultaneous).

 

[JesseM]

Dumb question, but when you say same position coordinates in one frame, you mean only along one axis, right? They don't have to actually be physically at the same point, right? (otherwise I misunderstood your point before).

In that quote I was talking about events at the same position on all frames, since that's what I thought you were talking about with the rocket/sun example (an idealization where the rocket passes right through the Sun). But it's true that we could also just imagine a rocket that is at the same position coordinate as the center of the Sun on the axis of relative motion between the Sun's frame and the rocket's frame, and then both frames would agree the clocks were zeroed simultaneously even if the rocket was far from the Sun on other axes.

And here's my follow-up:

I saw an example that dealt with time dilation in my physics book in which a rocket traveling at a constant speed (0.9c relative to the solar system) goes from the sun to saturn (never accelerating). In the example, they show how you can calculate how long the journey took as measured by someone on earth, and as measured by someone on the rocket. Taking a proper sun-saturn distance of 1.43*10^12 meters, they show that the journey takes 5300 seconds as measured by someone on earth, and 2310 seconds as measured by someone in the rocket.

And from what we just discussed, as the rocket passes directly past the sun (coming from the side opposite saturn), we can start both clocks (one on the sun and one in the rocket) simultaneously. By this I mean both clocks will start ticking simultaneously from the point of view of the sun, AND both clocks will start sticking simultaneously from the point of view of the rocket.

What I'm having a problem understanding is this: We can imagine a huge digital clock on saturn that is in sync with the sun's and Earth's clocks (since they aren't moving relative to each other in this problem, and let's ignore the effects of gravity) and timing the journey. So, if you were standing on saturn, at the instant that the center of mass of the rocket passed by the center of mass of saturn, you would see "5300 sec" displayed on the clock. But what would someone on the rocket see looking down at saturn on the big digital clock at the instant they passed by the center of mass of saturn?

The reason I'm confused is similar to the twin paradox. From the rocket's frame of reference, the clock on saturn ticks slower than the clock on the rocket. But since the clock on the rocket shows "2310 sec" (from the rocket's frame of reference) at the instant they pass the center of mass of saturn, this would mean that the clock on saturn would have to read a value *less* than 2310 seconds (from the rocket's reference frame) since they were both zeroed at the same time (from the perspective of each reference frame), and thus much less than 5300 seconds. So what do the people on the rocket actually see on saturn's clock as they pass?

I could imagine a situation where the rocket physically made contact with the clock on saturn, so it's really just one event. So at that one event, what does the clock read? 5300 seconds or something less? Surely one event can't be different from two different reference frames.

Part of me doesn't even know if I'm asking the right questions to clear up my confusion.

In the rocket traveler's frame, it would still be true that the rocket clock only showed a time of 2309.2 seconds when it passed by Saturn (you had some roundoff error with the 2310 figure), and that Saturn's clock showed 5300 seconds. What I think you're missing here is that even if the Saturn clock and the Sun clock were synchronized in the Sun/rest frame, those two clocks were not synchronized in the rocket's frame since the axis between Sun & Saturn is not perpendicular to the axis of relative motion between the rocket & Sun. So in the rocket frame, when both the rocket and Sun clocks read 0 seconds, the Saturn clock already read 4293.88 seconds. So as the rocket clock ticked forward by 2309.2 seconds in this frame during its journey from Sun to Saturn, the Saturn clock was running slow by a factor of 1/sqrt(1 - 0.9^2) = 2.29515734, so it only ticked forward by 2309.2/2.29515734 = 1006.12...but since it had already read 4293.88 at the start, by the time the rocket clock reached it it now read 4293.88 + 1006.12 = 5300.

 

[LBrandt]

You're right, but the original question wasn't about what was needed for two events to be simultaneous in all frames of reference, just in two different frames of reference. In this case they can be at different positions as long as their separation is perpendicular to the axis of motion between the two frames (but of course you can find a new frame whose axis of relative motion is not perpendicular to the axis joining the two events, and in this frame the events are non-simultaneous).

Again, I'm not representing myself as a relativity expert, but I've never heard any discussion or seen any textbook on relativity that uses the concept of an axis of relative motion to determine whether two events can be simultaneous in more than one frame of reference. Here is a quote from Wikipedia on the subject, and it's pretty clear to me that it states that unless the two events occur at the same place, they cannot be considered to be simultaneous by more than one set of observers.

In physics, the relativity of simultaneity is the concept that simultaneity—whether two events occur at the same time—is not absolute, but depends on the observer's reference frame. According to the special theory of relativity, it is impossible to say in an absolute sense whether two events occur at the same time if those events are separated in space. Where an event occurs in a single place—for example, a car crash—all observers will agree that both cars arrived at the point of impact at the same time. But where the events are separated in space, such as one car crash in London and another in New Delhi, the question of whether the events are simultaneous is relative: in some reference frames the two accidents may happen at the same time, in others (in a different state of motion relative to the events) the crash in London may occur first, and in still others the New Delhi crash may occur first.

 

[itsthemac]

In the rocket traveler's frame, it would still be true that the rocket clock only showed a time of 2309.2 seconds when it passed by Saturn (you had some roundoff error with the 2310 figure), and that Saturn's clock showed 5300 seconds. What I think you're missing here is that even if the Saturn clock and the Sun clock were synchronized in the Sun/rest frame, those two clocks were not synchronized in the rocket's frame since the axis between Sun & Saturn is not perpendicular to the axis of relative motion between the rocket & Sun. So in the rocket frame, when both the rocket and Sun clocks read 0 seconds, the Saturn clock already read 4293.88 seconds. So as the rocket clock ticked forward by 2309.2 seconds in this frame during its journey from Sun to Saturn, the Saturn clock was running slow by a factor of 1/sqrt(1 - 0.9^2) = 2.29515734, so it only ticked forward by 2309.2/2.29515734 = 1006.12...but since it had already read 4293.88 at the start, by the time the rocket clock reached it it now read 4293.88 + 1006.12 = 5300.

Wow, ok. This is a lot to take in for me. I guess this all just stems from the basic premise that two things that are simultaneous in one reference frame (in this case, a clock on the sun and a clock on saturn that are both in sync in the sun-saturn frame) aren't necessarily simultaneous in any other reference frame (and thus the sun and saturn clocks aren't in sync in the rocket's frame). Seems somewhat simple now, but obviously I still haven't internalized and fully accepted a lot that follows from the basic premises of SR. Thanks a lot for taking the time to answer my questions, you've been a big help.

Just a side question, and this is for anyone, are there any good computer programs/simulation software (online apps even) that deal with relativity that can allow you to set up your own experiments and view them from different reference frames, etc (or anything like that)? I think this would really help me out, but I couldn't find anything online.

 

[JesseM]

Again, I'm not representing myself as a relativity expert, but I've never heard any discussion or seen any textbook on relativity that uses the concept of an axis of relative motion to determine whether two events can be simultaneous in more than one frame of reference. Here is a quote from Wikipedia on the subject, and it's pretty clear to me that it states that unless the two events occur at the same place, they cannot be considered to be simultaneous by more than one set of observers.

In physics, the relativity of simultaneity is the concept that simultaneity—whether two events occur at the same time—is not absolute, but depends on the observer's reference frame. According to the special theory of relativity, it is impossible to say in an absolute sense whether two events occur at the same time if those events are separated in space. Where an event occurs in a single place—for example, a car crash—all observers will agree that both cars arrived at the point of impact at the same time. But where the events are separated in space, such as one car crash in London and another in New Delhi, the question of whether the events are simultaneous is relative: in some reference frames the two accidents may happen at the same time, in others (in a different state of motion relative to the events) the crash in London may occur first, and in still others the New Delhi crash may occur first.

The quote says you can't tell whether two events are simultaneous "in an absolute sense", which is the same as simultaneous in all frames (or simultaneous in a preferred frame, but relativity says there are no physically preferred frames). Nowhere in that quote do they say it's impossible to find two specific frames where such that a pair of events at different locations are simultaneous in both of those specific frames. Are you familiar with the use of the Lorentz transformation? It's easy to use it to demonstrate what I'm saying. Suppose the frames are moving relative to each other parallel to their x-axes as is assumed in the Lorentz transformation for frames in standard configuration, and assume v=0.6c so that gamma=1.25. Then the Lorentz transformation is:

t' = 1.25*(t - vx/c^2)
x' = 1.25(x - vt)
y' = y
z' = z

So suppose our two events A and B have coordinates (t=0, x=0, y=0, z=0) and (t=0, x=0, y=10, z=0). If you plug the first event into the transformation above you get (t'=0, x'=0, y'=0, z'=0); if you plug the second event into the transformation you get (t'=0, x'=0, y'=10, and z'=0). So, in both frames the two events each have a time-coordinate of 0.