Relativity of Simultaneity and Time

[PeterDonis]

Every Frame of Reference maps spacetime as at rest relative to that spacemap

That's not what Minkowski said. He didn't say "spacetime" was at rest; he said "the substance existing at any world point" was at rest (with an appropriate choice of frame). Spacetime is not something that can be "at rest" or "moving"; the concept doesn't make sense.

 

[Dale]

OK, it is the Spacetime interval, not the Spatial interval

Then the rest of your reasoning doesn't follow. Fixing the spacetime interval doesn't fix the time in Minkowski geometry any more than fixing the distance to the origin fixes the y coordinate in Euclidean geometry.

 

[Ibix]

Grimble - your basic idea is flawed. Proper time along a worldline is analogous to distance along a curve in Euclidean geometry. So your idea is analogous to saying that we were all together at one point and have all traveled one mile since then, so therefore we must all be at the same distance north (say) of our start point. And that's only one frame. If you claim this to be true in all frames then this is analogous to defining all possible rotated Cartesian reference frames and asserting that we have the same "north" coordinate as each other in every one of those systems.

You could also draw a Minkowski diagram and plot a selection of straight line paths starting at the origin and with fixed path length. You'll find that the ends define a hyperbola ##ct=\sqrt {s^2-x^2}##. The hyperbola is invariant under Lorentz transform, but the endpoints of the straight lines move along it. The ##\Delta t## and ##\Delta x## values for any pair change as they do so.

As PeterDonis noted this analysis only works for straight lines in flat spacetime. Arbitrary curves with equal proper time don't necessarily terminate on the same hyperbola.

 

[Grimble]

This is sort of true. The part you left out is that the invariant interval between the big bang and a particular event depends on which curve between the two you pick. There are an infinite number of possible timelike curves between the big bang and a given event, which can have different intervals between the two events.

I am sorry but I understood that the Spacetime interval between two events was invariant, however it was measured - that although the time component and space components can vary according to the Frame they are measured in - which I guess is what you mean by the different curves - the sum remains invariant?

 

[Dale]

I am sorry but I understood that the Spacetime interval between two events was invariant, however it was measured

The spacetime interval is invariant. The time is not fixed, contrary to your conclusion.

 

[PeterDonis]

I understood that the Spacetime interval between two events was invariant

Not as you are interpreting that statement. The correct statement is that the spacetime interval along a particular curve between two events is invariant. But "spacetime interval" has no meaning if you don't specify a curve between the two points. SR textbooks often gloss over this by implicitly assuming that the curve along which the interval is computed is the "straight line" between the two events, i.e., the Minkowski straight line. But glossing over the choice does not mean such a choice is not being made.

 

[Grimble]

OK. Accepting what you say, that time is not fixed however one looks at it, and that invariant intervals are not invariant; What I believe was implied above is that for timelike intervals one event must lie within the lightcone of the other event while for spacelike intervals neither event can lie within the lightcone of the other event?

But that brings me back to the thought I didn't express clearly enough above.

If we take an event in the past - say the decision to perform the experiment and measure the falling of the fruit or the three events, A, B and C; and plot the light cone of that event, then all the other events follow as a consequence of making that decision. All those events are the separated from the decision event by timelike intervals, not spacelike intervals as they each have a causal connection to that decision.

Therefore being timelike intervals rather than spacelike intervals they cannot be rearranged in sequence - what other factors are present here that haven't been raised yet?

 

[Ibix]

In your example, what happens at event P causes what happens at events A, B and C which are simultaneous in some frame. That means that A, B and C are timelike separated from P, but spacelike separated from each other. You can rearrange the order of A, B and C by a different choice of frame, but no one will say they happen before P.

 

[Grimble]

Thank you, Got It!

 

[Dale]

invariant intervals are not invariant;

Invariant intervals are invariant. Peter Donis' point was not that they are not Invariant. His point was that they are defined along curves in spacetime.

All those events are the separated from the decision event by timelike intervals, not spacelike intervals as they each have a causal connection to that decision.

Therefore being timelike intervals rather than spacelike intervals they cannot be rearranged in sequence

Yes.

 

[sydneybself]

There is a simpler way to look at which was originated by Korzybski about 80 years ago. He originated the aphorism “The map is not the territory”. To Korzybski, the events, which happened physically, were ‘territory’. What the observers observed were ‘maps’ - each observer having his own map. Assume that the events consisted of three objects falling and hitting the ground, making a noise. Assume the observers couldn’t see the events because it was dark and determined the sequence of events by what they heard. If they were at different distances from the events and from each other, the sounds would reach them at different times. That is called time delay. The maps of each would be different but each would be equally valid.