Topology of Minkowski spacetime

[ConradDJ]

I recently Googled "spacetime topology" and found that the topology of Minkowski spacetime is generally described as that of an R⁴ manifold.

This is not my field, but I'm surprised. Perhaps mathematically the (---+) "Lorentz signature" can be taken as a secondary characteristic of the manifold... but physically, it seems very basic to the topology of spacetime that any two points on a "light-like interval" are directly connected.

I understand that they are not the same point in spacetime, and I understand that there is a time-direction in the connection between them. That is, when I look at a star, there is a "null" spacetime distance between the place and time the photon was emitted and the place and time where it reaches my eye -- but this is a one-way "causal" connection from the star to my eye.

Does anyone know of a treatment of spacetime topology that discusses this kind of directed connection across a "null interval"?

 

[dextercioby]

Please, see this ongoing discussion in the topology subforum.

https://www.physicsforums.com/showthread.php?t=495816

 

[atyy]

The ability to define light cones is due to the (-+++) signature of the spacetime metric. R^4 itself does not inherently have a metric, and we can place on it metrics of different signature, and different metrics of the same signatue.

Spacetimes with (++++) signature are usually called "Euclidean", and spacetimes with (-+++) signature are usually called "Lorentzian","semi-Riemanninan", or "pseudo_Riemannian". The (-+++) spacetime with flat metric is usually called "Minkowski spacetime".

Minkowski spacetime can have different topologies. These topologies may induce "preferred frame" effects. http://arxiv.org/abs/gr-qc/0101014

A general reference: http://books.google.com/books?id=d6q8LAGPBecC&dq=joshi+global+structure&source=gbs_navlinks_s, especially chapter 4 "Causality and spacetime topology".

 

[ConradDJ]

Please, see this ongoing discussion in the topology subforum.

https://www.physicsforums.com/showthread.php?t=495816

Thank you – both these replies were very helpful. The thread referenced above was started by dslowik, who posted on the same issue https://www.physicsforums.com/showthread.php?t=495486" –

Dslowik seems to be concerned about the same issue I am, though he’s coming at it from a mathematical perspective. He says, for example –

Lorentz metric is not really a metric in the sense of metric spaces of a topology course since it admits negative values. If I use it to define the usual open sphere about a point, that sphere includes the entire light-cone through that point...

It seems that ST is a topological manifold with a locally Euclidean metric. This describes its topological structure as a metric space. We than add further structure to this metric/topological space by adding the non-Riemanian Lorentz metric. Thus we are using one metric and corresponding open balls to describe the topology, and another metric to describe the 'physical' distance between points. The physical distance between some points is 0, which is a very different topology than the locally Euclidean one; but the Lorentz metric can't be used to describe open balls for a topology?

... it seems odd to me that we impose a locally euclidean topology, then use a quite different metric to describe physical separation of points.

If I understand this correctly, it seems that describing spacetime as a manifold with a Lorentz metric is a mathematical kludge – something that works for purposes of calculations, but doesn’t really show us what’s going on, physically.

Dslowik seems to think that the zero absolute spacetime distance between points on the light-cone could and maybe should be treated as a basic topological feature of spacetime, not as something secondary, added on with the metric.

It seems that maybe Hawking et al did something like this in a 1976 paper that I’m not able to access. Does anyone know where I can find out more about their approach?
http://link.aip.org/link/doi/10.1063/1.522874"


My issue here is a philosophical one. In order to understand the underlying connection between Quantum theory and Relativity, I think we need to be able to describe the world of physical interaction that can actually be observed, from a point of view inside it.

Instead, both Quantum theory and Relativity have so far been formulated within an essentially classical framework, where every effort is made to eliminate “the observer” so we can continue imagining the universe from a “God’s-eye” viewpoint. Clearly this approach works very well for most purposes – but I think, among other things, it prevents us from seeing how these two theories are fundamentally related.

Specifically – when we start by describing spacetime as an R^4 manifold that’s locally Euclidean, we’re already rooting ourselves in the classical framework. Dslowik is pointing out that physical spacetime – as opposed to the convenient mathematical structure we use to describe it – is not Euclidean even locally. The main issue is not that it’s “curved” but that the topology of physical connections – i.e. the actual interactions taking place at a given place and time – is entirely different from R^4.

In the spacetime of physical connections, for example, there is no such thing as a “spacelike interval” or a “spacelike hypersurface”. All physical interaction operates either extremely locally – on the scale of an atomic nucleus – or on the light-cone, across a “null interval”... depending on whether the interaction is mediated by massless or massive particles.

I’m not saying there’s anything wrong with the usual mathematics of Relativity, wherever it works. Because light-speed is so fast as compared with the processes we normally deal with, it makes lots of sense to describe the world as if it were locally Euclidean. And of course, adding a (+++–) metric to the R^4 manifold gives us an effective way of describing spacetime curvature. But maybe we shouldn’t expect our mathematical conveniences to give us insight into fundamental physics.

I’ve been trying to envision the topology of the “physical spacetime” we actually observe, as something like a web of communications among different observers, on their respective light-cones. So far the closest analogy I can think of is the “topology” of distributed parallel processing in a computer system with many CPUs. http://en.wikipedia.org/wiki/Spacetime_topology" , there are alternative spacetime topologies (“Zeeman” and “Alexandrov”) that may relate to this picture.