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A beginner's question:

I have always understood that (four-dimensional) Minkowski spacetime can be recovered up to a constant factor—i.e. 'up to a dilation' or 'up to global scale'—from its causal structure. That is to say, I understand that whereas the symmetry group of Minkowski spacetime is the Poincaré group, the symmetries of its causal structure are the Poincaré group + dilations.

Recently, I was told that Minkowski geometry can only be recovered up to local scale from its causal structure. And elsewhere, I have now read that Minkowski geometry can be recovered up to a conformal factor from its causal structure.

Which is it, and how significant is the difference between a global constant factor, a local constant factor, and a conformal factor?

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  • $\begingroup$ Thanks to Danu and @Slereah for helpful replies. I can see now how a conformal factor (a strictly positive function) should preserve the causal structure of Minkowski spacetime. Do the conformal symmetries of Minkowski spacetime form a particular supergroup of the Poincaré group? $\endgroup$ – user61367 Apr 17 '14 at 10:20
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Partial answer:

In the context of general relativity, it is conventional to make use of conformal diagrams (a.k.a. Penrose diagrams to enable us to visualize a spacetime. These bring the boundaries of the spacetime (at infinitey) to a finite coordinate value, and keep lightlike worldlines at 45 degrees. As such, they're the generalization of Minkowski diagrams to arbitrary spacetimes. This is discussed in any textbook on general relativity.

Although one needs to make a coordinate change to construct the conformal diagram of a spacetime, the use of conformal transformations only ensures that the transformations preserve the causal structure and the 'local characteristics' of the spacetime: Conformal transformations leave angles and the shape of infinitesimally small figures invariant (a better mathematician than me is more than welcome to make this statement more precise/accessible). Spacelike and timelike intervals don't 'mix' under a conformal transformation, and therefore the conformal diagram conveys the causal structure of the spacetime.

In light of the above, I think that the statements that that the spacetime can recovered from its causal structure up to a conformal factor or up to local scale are really just the same statement, hinting at the above-mentioned conformal diagrams. Possibly, your first statements also means the same thing, and it's just a matter of semantics. Anyone who knows how to make these things more accurate/correct is welcome to contribute by improving my answer.

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The causal structure of spacetime is determined by the lightcones at every point of it, basically. From this you can construct such things as the chronological past (all events that influenced this point), the chronological future (all events that will be influenced by this point), the Cauchy horizon (the limits of what you can predict in the future from a given slice of space), and a lot of other related concepts.

This lightcone is determined by the metric tensor, $g_{\mu\nu}$, which defines the scalar product. The inside of the cone is composed of all points that can be reached from the center such that their tangent vectors are of positive norm (timelike vectors)

$g_{\mu\nu} k^\mu k^\nu > 0$

The edge of the cone has vectors of 0 norm (lightlike vectors)

$g_{\mu\nu} k^\mu k^\nu = 0$

And the outside is vectors of negative norm (spacelike vectors)

$g_{\mu\nu} k^\mu k^\nu < 0$

It is rather easy to see that, since the edge of the light cone has such a relation, a conformal transformation,

$g_{\mu\nu} \rightarrow \Omega(x) g_{\mu\nu}$

where $\Omega$ is some (always strictly positive) function, will not change the causal structure :

$\Omega(x) g_{\mu\nu} k^\mu k^\nu = 0 * \Omega(x) = 0$

The times and distances of those two spacetimes will be different, but not its causal structure, as defined by what will be in the future of some point.

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