General Relativity and the curvature of space: more space or less than flat?

[jbriggs444]

Thank you. I think I'm understanding better now. Could you please clarify: invariants are e.g. causal relationships, that is clear. Are paths that a hypothetical flash of light would follow through spacetime invariants (I say paths, because light can go many directions, correct?)?

I would say that the path that a flash of light follows is invariant, yes. The coordinates used to describe the set of events on that path will vary from one coordinate system to the next. But the set of events on the path is the same.

However, the angle that a particular light pulse takes from its launching point or the angle at which it arrives at its detection point can vary depending on one's choice of reference frame (e.g. stellar abberation). The path is still an invariant. The angle that it takes when projected onto a spacelike snapshot is not.

 

[PeroK]

Thank you. I think I'm understanding better now. Could you please clarify: invariants are e.g. causal relationships, that is clear. Are paths that a hypothetical flash of light would follow through spacetime invariants (I say paths, because light can go many directions, correct?)?

A path through spacetime is not really what we mean by invariant. That's a defined set of points in the spacetime manifold. It's not easy to describe that path until you have chosen a coordinate system, but (and this is the key point), the path exists and is well-defined without being given a coordinate description.

Generally there are two types of path: timelike (followed by massive particles) and null (followed by light). And, there are general timelike and null paths and geodesic timelike and null paths, which are the natural paths that particles and light follow through spacetime. Massive particles can, of course, be forced off geodesic paths, but the path remains timelike. I'm not sure there's any way to force a light onto a null non-geodesic path(?)

There are clearly an infinitude of possible paths, but each particle or light ray can only take one path through spacetime (its worldline).

An invariant is something you calculate, like the length of a spacetime path between two events. Null paths have zero length in all coordinate systems and timelike paths have the same non-zero length in all coordinate systems. So, it's the length of the spacetime path that is invariant.

We don't really talk about the path itself being invariant.

 

[PeterDonis]

A path through spacetime is not really what we mean by invariant.

It can be. You say:

the path exists and is well-defined without being given a coordinate description

That's what "invariant" means, so yes, a path through spacetime would be an invariant.

What it would not be is what I would call a "local" invariant, i.e., an invariant defined at a single spacetime point. As you say, it's a set of spacetime points. But that set of points is the same no matter what coordinates you choose.

I'm not sure there's any way to force a light onto a null non-geodesic path(?)

There is: a waveguide or fiber optic cable are examples of things that can do this.

it's the length of the spacetime path that is invariant.

We don't really talk about the path itself being invariant.

It's true that the term "invariant" is more likely to be used to describe the arc length along the path than the path itself. However, I don't think that means it's wrong to describe the path itself as invariant; it's just a less common use of the term.