Time-Like Events: Spatial Order Not Relevant?

[BLevine1985]

Hi, I'm having trouble understanding that for time-like events where the order of time is absolute, these events can be in different spatial orders depending on reference frame. Can someone provide an example please?

Thank you!

 

[Mister T]

What do you mean by "spatial order"?

If you are standing on the platform watching a train go by, you will observe that the front of the train passes the platform before the back of the train passes the platform. Moreover, you will see the front of the train before you see the back of the train.

If another train is passing this train, they will see the back of the train before they see the front of the train. If the people in that passing train consider themselves at rest, they will conclude that the other train is moving backwards. Yet, they will still agree that the front of the first train passed the platform before the back.

 

[BLevine1985]

Well in spacelike intervals the time order doesn't matter, since the events can't be causally linked. So I was wondering about the converse. In the time-like interval where the time order is absolute, what does that imply about the space order? Is there such a thing?

 

[DrGreg]

Well in spacelike intervals the time order doesn't matter, since the events can't be causally linked. So I was wondering about the converse. In the time-like interval where the time order is absolute, what does that imply about the space order? Is there such a thing?

If you are driving along a straight road from south to north and pass through a town at 2pm, then "the town at 1pm" occurs to the north of you and "the town at 3pm" occurs to the south of you.

If someone else is driving along the same road in the opposite direction from north to south and also passes through the town at 2pm, then "the town at 1pm" occurs to the south of them and "the town at 3pm" occurs to the north of them.

So you and the other person disagree on the spatial order of the two timelike-separated events, "the town at 1pm" and "the town at 3pm", i.e. on which event is to the north of the other event.

You don't need relativity for this, everything I've said is true in Newtonian physics as well.

 

[BLevine1985]

Thanks. I had a feeling it was one of those things that is so simple that you miss it, and yeah it turned out to be that.

 

[Dale]

what does that imply about the space order? Is there such a thing?

Space isn't ordered anyway.

 

[PeterDonis]

Space isn't ordered anyway.

3D space isn't, but a 1D spacelike line is. Note that @DrGreg carefully gave a 1D example in post #4.

 

[Ibix]

time-like events

You seem to have used the terms correctly later, so perhaps this is just a typo, but be aware that "time-like events" doesn't make sense. Events can be time-like separated, but are not themselves time-like, space-like or otherwise.

Space isn't ordered anyway.

In general I agree, although you can impose an ordering by picking one direction and only comparing displacement in that direction, as @DrGreg did. It's probably worth noting that the fact you have to restrict yourself to one dimension is why you see talk of time ordering (naturally one dimensional) but not much of spatial ordering.

 

[Dale]

3D space isn't, but a 1D spacelike line is. Note that @DrGreg carefully gave a 1D example in post #4.

Even for a 1D line I would disagree. Or at least the ordering is not natural. There is no natural sense in which a point on the left is smaller than a point on the right. You can artificially impose an ordering but there is no natural "arrow of space" like there is a natural arrow of time.

 

[PeterDonis]

the ordering is not natural. There is no natural sense in which a point on the left is smaller than a point on the right

The timelike ordering is not "smaller than" and "larger than"; it is "before" and "after". These are symmetric in the same way that "north of" and "south of" are in @DrGreg's example. You don't have to have a concept of "size" of items to have an ordering. You just have to have a relation that satisfies the properties of an ordering relation.

 

[Dale]

The timelike ordering is not "smaller than" and "larger than"; it is "before" and "after". These are symmetric in the same way that "north of" and "south of" are

The second law of thermo disagrees about the symmetry of "before" and "after".

 

[PeterDonis]

The second law of thermo disagrees about the symmetry of "before" and "after".

But the second law of thermo is not something you can derive from Minkowski spacetime and its geometry. As far as the latter is concerned, "before" and "after" are symmetric.

 

[Dale]

But the second law of thermo is not something you can derive from Minkowski spacetime and its geometry. As far as the latter is concerned, "before" and "after" are symmetric.

Nevertheless there is a clear and unambiguous physical meaning for $t_a < t_b$ whereas there is not a clear and unambiguous physical meaning for $x_a<x_b$, even considering each restricted to a single 1D worldline (timelike and spacelike respectively). I don't think that we are required to turn off all of our other knowledge of physics when discussing relativity. In the physical world Minkowski geometry is not a standalone entity divorced from the other laws of physics.

 

[PeterDonis]

there is a clear and unambiguous physical meaning for $t_a < t_b$ whereas there is not a clear and unambiguous physical meaning for $x_a < x_b$, even considering each restricted to a single 1D worldline (timelike and spacelike respectively).

I disagree; @DrGreg described a clear and unambiguous physical meaning for $x_a < x_b$ along a single 1D spacelike line. The difference with space is that a single 1D spacelike line is only a restricted subset of space, whereas a 1D worldline is the full history of a timelike observer (at least an idealized pointlike one).

 

[Dale]

Sure, whatever. I stand by my statement that space isn't ordered.

 

[vanhees71]

The timelike ordering is not "smaller than" and "larger than"; it is "before" and "after". These are symmetric in the same way that "north of" and "south of" are in @DrGreg's example. You don't have to have a concept of "size" of items to have an ordering. You just have to have a relation that satisfies the properties of an ordering relation.

Sure, the "real line" can be ordered in the one or the other way, and if this "real line" models a spatial direction, there's not so much different in choosing the one or the other ordering. If the "real line" models a temporal direction, that's a different business in physics, because we assume that there is a "causal ordering", i.e., that if one event is the cause of another event these events must be ordered such that the cause is "before" its "effect". In my opinion that's a basic assumption behind all of physics since without causality it wouldn't make sense to look for "natural laws" to begin with.

That's also what's behind the math of relativistic space-time manifolds being not "Riemannian" but "pseudo-Riemannian" manifolds with a fundamental form of signature (1,3) (or equivalently (3,1)): It's because this enables to establish a "causal ordering" in the sense of "time ordering" in contradistinction to any possible "spatial ordering".

Take special relativity as an example and the derivation of the proper orthochonous Poincare group as the symmetry group of special relativistic spacetime this argument with "causal/temporal ordering" is important.

If you just start from the assumption of the special principle of relativity, i.e., the assumption that there are (global) inertial frames and that time is homogeneous and the space as observed by any inertial observer is a Euclidean 3D manifold, you get first 3 possible symmetry groups, the Galilei-group (leading to the spacetime model of Newtonian physics, which is a fiber bundle), the Poincare group (the spacetime model of special relativity, which is an affine pseudo-Euclidean space with a fundamental form of signature (1,3) or (3,1)), or the group ISO(4) (leading to a 4D affine Euclidean manifold), but this latter possibility is ruled out due to the fact that then, to have a full group as the symmetry transformations, you can't establish a causal/temporal order.

For details, see, e.g.

V. Berzi and V. Gorini, Reciprocity Principle and the Lorentz
Transformations, Jour. Math. Phys. 10, 1518 (1969),
https://doi.org/10.1063/1.1665000

 

[PeroK]

Sure, whatever. I stand by my statement that space isn't ordered.

Although, never eat shredded wheat!