I Physical meaning of null spacetime interval

[jbriggs444]

There is no frame where two null separated events are in either the same location or the same position. There is also no frame where the time order of two null separated events swaps.

If I interpret this correctly, you are saying that boosts and rotations preserve time orientation [and chirality] and that you are restricting your attention to future-directed, right-handed frames?

 

[PeterDonis]

boosts and rotations preserve time orientation [and chirality]

This is the usual definition, yes. The more technical term is that boosts and rotations are elements of the proper orthochronous Lorentz group. This is the component of the full Lorentz group that is connected to the identity.

 

[Dale]

If I interpret this correctly, you are saying that boosts and rotations preserve time orientation [and chirality] and that you are restricting your attention to future-directed, right-handed frames?

Yes, as @PeterDonis confirmed.

This goes back to the idea that surfaces of constant spacetime interval from some origin form hyperboloids. For timelike intervals it is a hyperboloid of two sheets, one future and one past. There is no way to smoothly move from one sheet to the other, they are disconnected.

This is in contrast to spacelike intervals which form hyperboloid a of one sheet. So any fixed spacelike imterval can be smoothly connected.

So spacetime intervals split spacetime into four distinct regions about any event: future timelike, past timelike, lightlike, and spacelike. There is no arrow of time, but there is a clear separation between future and past.

 

[vanhees71]

If I interpret this correctly, you are saying that boosts and rotations preserve time orientation [and chirality] and that you are restricting your attention to future-directed, right-handed frames?

The orthochonous Lorentz group preserves time-ordering between timelike or light-like separated events. That's why only such events can be causally connected, because this group doesn't preserve the time-ordering of space-like separated events, i.e., you can always transform to another frame, where the temporal order is switched or to a frame, where the events are simultaneous.

 

[pervect]

TL;DR Summary: Physical meaning of null spacetime interval

Hi all,

I've been trying to wrap my mind around the physical meaning of null spacetime intervals. I understand the metric of Minkowski geometry. And I understand that photons have no proper time and no reference frame: the spacetime interval between the emission and absorption of a photon is zero. They are the same event.

My question is: What does this mean? Is the fabric of spacetime warped to infinity by every photon passing through it? If I look up at Polaris at night, from my reference frame it appears to be 433 light years away in space, and 433 years away in time. But in spacetime, the emission of that photon and its absorption by my eye are the same event.

This is a bit of a necro post, but I have a different take. As others have mentioned, the emission event and reception events are different events, it's incorrect to assume that because the space-time interval is zero, that that means that the events are the same.

What I think one needs to know about null intervals is that mathematically, they have an affine geometry, not a Euclidean geometry.

In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting"[1][2]) the metric notions of distance and angle.

What can do with an affine geometry? Affine geometries have a concept of "in the same direction", which allows one to talk about geodesics. These would be "null geodoesics". And affine geometry have a concept of marking regular intervals along any particular null geodesic, which is an "affine parameterization" of the null geodesic.

But there isn't any meaningful notion of "distance" along a null interval, even though there is a notion of "equal" intervals along any one particular geodesic. A distance would require the ability to compare regular intervals from one null geodesic to another - but there's no unique way of doing this.

 

[vanhees71]

That's what preach all the time: Hammer into your students the fact that the indefinite fundamental form of Minkowski space does NOT induce a metric on $\mathbb{R}^4$ as a Euclidean scalar product does. Light-like separated events thus are, by definition, never the same event. As you say, it describes the emission and absorption event of light signals, as described by the retarded propagator of the electromagnetic field in classical electrodynamics.