Simultaneity question – is sequence absolute?

[rede96]

There are two space stations A and B which are at rest wrt to each other and separated by some distance x. Each space station has a light that randomly flashes.

I am in a third space station, somewhere in the middle of A and B and at rest wrt to the other two stations.

I see Station A’s light flash then some short time later station B’s light flash.

As I understand it, as the flashes are not causally connected the order in which the lights flashes are dependant on the observer.

So some other observer may see B flash then A.

However, let’s say that I put an another set of lights C and D on each space station. I put C next to A and D next to B

These are special lights and can only flash in one set sequence, C then D and C will only flash if I send it a signal.

So I try to time my lights to flash at the same time as the random lights A and B. Tall task, but let's say I get lucky!

I send my signal and C flashes at the same time as A. C sends a signal to D and D then coincidently flashes at the same time as B.

This time, as there is a causal connection between C and D, all observers must agree that C flashed before D and therefore must also agree that A flashed before B even though A and B are not casually connected.

So although someone might see D before C, if they could work out the distance and time the light traveled correctly, they would find that it was actually C before D

So my point is, (albeit a bit long winded!) when ever any observer sees two events occurring, there is only one order that these really occurred in.

 

[pervect]

If C is in the lightcone of D for one observer, it will be in the lightcone of D for all observers. You should be able to see this if you draw a space-time diagram. The position of C will vary a bit, depending on your frame choice, and if you choose the right frame you can make the separation between C and D purely "timelike", but no matter what your choice, C will remain in the lightcone of D.

You can order events within a light-cone into "cause" and "effect" sequence.

So the short of it is that if I understand you correctly, you appear to understand correctly, , though the language you use to describe it is a bit informal.

 

[rede96]

If C is in the lightcone of D for one observer, it will be in the lightcone of D for all observers. You should be able to see this if you draw a space-time diagram. The position of C will vary a bit, depending on your frame choice, and if you choose the right frame you can make the separation between C and D purely "timelike", but no matter what your choice, C will remain in the lightcone of D.

You can order events within a light-cone into "cause" and "effect" sequence.

So the short of it is that if I understand you correctly, you appear to understand correctly, , though the language you use to describe it is a bit informal.

Thanks for the reply!

Sorry if my language is a bit informal. I am only an ‘amateur enthusiast’ and am still fairly new to all this.

I base most of my understanding on logic and thought experiments, I have not so far ventured into the math, let alone space-time diagrams. I simply want to get a good grasp of the principles first.

As for as your formal language, when you say C is in the light cone of D, I don’t quite understand. Could you explain a bit more please?

 

[Matterwave]

You don't need two events A and B to actually be "causally connected" (in the sense that A caused B) for all observers to agree which one came first. You just need B to be in the future light cone of A, or equivalently, that A and B must be time-like separated.

In easier to understand words. A just must in principle be able to send a light signal to B, and all observers will agree A occurred before B. Whether such a light signal was actually sent to trigger B doesn't matter.

 

[pervect]

Thanks for the reply!

Sorry if my language is a bit informal. I am only an ‘amateur enthusiast’ and am still fairly new to all this.

I base most of my understanding on logic and thought experiments, I have not so far ventured into the math, let alone space-time diagrams. I simply want to get a good grasp of the principles first.

As for as your formal language, when you say C is in the light cone of D, I don’t quite understand. Could you explain a bit more please?

Welcome aboard~ I don't want to be stuffy, just that sometimes I misread things if I'm not carefu :(. But you seem to have the basic idea, IMO.

 

[rede96]

You don't need two events A and B to actually be "causally connected" (in the sense that A caused B) for all observers to agree which one came first. You just need B to be in the future light cone of A, or equivalently, that A and B must be time-like separated.

In easier to understand words. A just must in principle be able to send a light signal to B, and all observers will agree A occurred before B. Whether such a light signal was actually sent to trigger B doesn't matter.

Thanks. So in layman’s terms, different observers may disagree on the timing of two events but not the sequence?

This makes sense to me but contradicts some of the other posts I've read here.

 

[rede96]

Welcome aboard~ I don't want to be stuffy, just that sometimes I misread things if I'm not carefu :(. But you seem to have the basic idea, IMO.

Thanks and no problem :)

 

[Dale]

So in layman’s terms, different observers may disagree on the timing of two events but not the sequence?

This is true if and only if the distance between the events is less than or equal to c times the time between the events. This is what Matterwave and perfect meant by the lightcone.

 

[rede96]

This is true if and only if the distance between the events is less than or equal to c times the time between the events. This is what Matterwave and perfect meant by the lightcone.

Thanks DaleSpam. Is the time & distance between the events relative? i.e. time / distance in the frame of the observer?

Or is this the proper time and proper length?

 

[Matterwave]

The time and distance separately is relative; however, the interval is the same for all observers:

$$S^2=c(t_B-t_A)^2-(x_B-x_A)^2=c(t_B'-t_A')^2-(x_B'-x_A')^2=S^2'$$

Therefore, if the distance between two events is less than c times the time between two events in 1 frames, the distance between those 2 events is ALWAYS less than c times the time between those 2 events in EVERY frame.

 

[rede96]

The time and distance separately is relative; however, the interval is the same for all observers:

$$S^2=c(t_B-t_A)^2-(x_B-x_A)^2=c(t_B'-t_A')^2-(x_B'-x_A')^2=S^2'$$

Therefore, if the distance between two events is less than c times the time between two events in 1 frames, the distance between those 2 events is ALWAYS less than c times the time between those 2 events in EVERY frame.

I suppose I need to get my head around the math at some point :)

So, assuming 300,000 km/sec for c, if the distance between the 2 events is 300,000 km and the interval between the events is 1.5 seconds, then all observers agree on sequence.

But if the interval is 0.5 secs, then sequence is observer dependant, is that right?

Also, could you demonstrate how the formula works using those numbers please :)

 

[Dale]

Thanks DaleSpam. Is the time & distance between the events relative? i.e. time / distance in the frame of the observer?

Yes.

 

[Matterwave]

I suppose I need to get my head around the math at some point :)

So, assuming 300,000 km/sec for c, if the distance between the 2 events is 300,000 km and the interval between the events is 1.5 seconds, then all observers agree on sequence.

But if the interval is 0.5 secs, then sequence is observer dependant, is that right?

Also, could you demonstrate how the formula works using those numbers please :)

Ok, so if in one frame you have 2 events separated by distance 300,000km, and 1.5 seconds, then you have that (c*1.5s)^2-(300,000km)^2>0 right. In another frame, these two events are separated by a different distance (let's call it d), and different time (let's call it t). Although d is not 300,000km, and t is not 1.5s, it is STILL true that (c*t)^2-(d)^2>0. In fact, it is true that: (c*t)^2-(d)^2=(c*1.5s)^2-(300,000km)^2.

If (c*t)^2-(d)^2>0 in SOME frame, EVERY observer will say A came before B. If not, then the sequence may be reversed in some frame.

 

[rede96]

Ok, so if in one frame you have 2 events separated by distance 300,000km, and 1.5 seconds, then you have that (c*1.5s)^2-(300,000km)^2>0 right. In another frame, these two events are separated by a different distance (let's call it d), and different time (let's call it t). Although d is not 300,000km, and t is not 1.5s, it is STILL true that (c*t)^2-(d)^2>0. In fact, it is true that: (c*t)^2-(d)^2=(c*1.5s)^2-(300,000km)^2.

If (c*t)^2-(d)^2>0 in SOME frame, EVERY observer will say A came before B. If not, then the sequence may be reversed in some frame.

Well that seems straight forward enough. :)

Just a couple of questions...

(c*1.5s)^2-(300,000km)^2 = 112,500,000,000, what does that answer represent?

Also if (c*t)^2-(d)^2=(c*1.5s)^2-(300,000km)^2 is true then why does the formula break down for any time t that is >5016?

EDIT: Ignor that last question! Excel doesn't seem to calculate large numbers correctly!

 

[Dale]

(c*1.5s)^2-(300,000km)^2 = 112,500,000,000, what does that answer represent?

The square of the spacetime interval.

 

[rede96]

The square of the spacetime interval.

At the risk of asking another stupid question, what is the spacetime intrval?

 

[Dale]

It is the "distance" between two events in spacetime. In Euclidean geometry you have distance between points and in Minkowski geometry you have the spacetime interval between events.

 

[rede96]

It is the "distance" between two events in spacetime. In Euclidean geometry you have distance between points and in Minkowski geometry you have the spacetime interval between events.

Thanks for that. I was just doing some reading to understand this a bit more and came across a different formula:

The space-time interval ds is defined by

ds^2 = -(cdt)^2 + dl^2

This obviously gives the negative interval. Is there any significance to this?

 

[PAllen]

Thanks for that. I was just doing some reading to understand this a bit more and came across a different formula:

The space-time interval ds is defined by

ds^2 = -(cdt)^2 + dl^2

This obviously gives the negative interval. Is there any significance to this?

Yes, the sign if ds^2 determines whether a path is spacelike or timelike. Using your conventionn for ds^2 (there is another), for a timelike path you integrate -ds^2, for a spacelike path integrate ds^2. The resulting intervals are called proper time and proper length.