Relativity of Simultaneity and Time

[Grimble]

Let me take this idea of causality a step further and ask what you think about it. This is not trying to rewrite anything but merely posing an interesting(?) logical line of thought...

Let us take another event P when the Apple and Orange are picked up from a basket on a table and then taken to where they will be dropped.
In the light cone from this event the dropping and landing of the apples will be causally connected and lie in Event P's future light cone - however far apart those later events might be; even in different galaxies if need be. Or a similar previous event for A,B and C in that thought experiment - let us say that A,B and C are guns firing and the Event P in this case is the loading of the three guns (or rather the act of picking up the three bullets.

Now in addition to our non causally connected events we have larger light cones that do connect all our events. A,B and C or the dropping and landing of the fruit, are in the light cone of the appropriate events P.

This means that the invariant Spacetime Interval between P and each of the others, being invariant has an absolute value. And must be the same for every other frame of reference (FoR), whether that interval is space-like, time-like or light-like in that frame.

The spatial distance between the events will be the same for each FoR as spacetime is stationary as mapped in any FoR.

If those subsequent events are fixed in time relative to event P, how can they then be reordered in another FoR - relative to another observer.

To take this a step further, every event in Spacetime is in the light cone of the Big Bang and must therefore be causally connected to the Big Bang and therefore have invariant spacetime intervals relative to the big bang and therefore be fixed in time.

 

[PeterDonis]

every event in Spacetime is in the light cone of the Big Bang and must therefore be causally connected to the Big Bang and therefore have invariant spacetime intervals relative to the big bang

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.

(This is a special case of the more general point that the "interval" between two events depends on which curve between them you pick. You can't talk about "the" unique interval between two events; you have to specify which curve the interval is evaluated on. In SR this is often glossed over because inertial frames pick out a unique curve between any pair of events, and the "interval" is assumed to be the interval along that curve. But even in SR that's not always the curve you want to focus on.)

In cosmology, it is common to pick one particular timelike curve from the big bang to a given event, which is the one that is the worldline of a "comoving" observer--an observer who always sees the universe as homogeneous and isotropic. The interval along the "comoving" worldline from the big bang to a given event is indeed unique.

and therefore be fixed in time.

This is not true. What is true is that we can choose coordinates in such a way that the "time" coordinate assigned to each event is the same as the interval along some chosen worldline--the usual choice in cosmology is the "comoving" worldline. But that choice is for convenience; nothing in the physics requires it. When cosmologists talk about "time", they are (in almost all cases) implicitly using the definition of "time" as the interval along comoving worldlines. They talk about "time" without qualification not because that is the unique physical definition of "time", but because it's the most convenient one.

 

[Dale]

The spatial distance between the events will be the same for each FoR as spacetime is stationary as mapped in any FoR.

This is not true.

If those subsequent events are fixed in time relative to event P

They are not, your premise is false.

 

[Grimble]

(This is a special case of the more general point that the "interval" between two events depends on which curve between them you pick. You can't talk about "the" unique interval between two events; you have to specify which curve the interval is evaluated on. In SR this is often glossed over because inertial frames pick out a unique curve between any pair of events, and the "interval" is assumed to be the interval along that curve. But even in SR that's not always the curve you want to focus on.)

So are you saying that the invariant spacetime interval between two events is not invariant?

 

[Grimble]

This is not true.
.

OK, it is the Spacetime interval, not the Spatial interval Every Frame of Reference maps spacetime as at rest relative to that spacemap - as Minkowski stated:
"The substance existing at any world point can always be conceived to be at rest, if time and space are interpreted suitably."

 

[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.