# What is the formal difference between the light cone and a black hole?

A black hole can be loosely defined as a spatial closed surface from which nothing, not even light, can leave.

The light cone of special relativity is in some sense similar to a black hole because by definition it cannot be left by any physical object (light-like or time-like curves cannot leave it).

So I was wandering which is the difference between these two objects.

For instance if we choose a set of coordinates in which the radius that measures the distance from the light cone source is rescaled at every instant so to have a constant value, then, wouldn't the light cone look exactly like a black hole in these coordinates?

I would say that the difference is that the light cone becomes non compact as time-infinity is approached, so the change of coordinates above is not good anymore.. but I am far from sure and I am a bit confused.

In a general spacetime the distinction between the insides of a black hole and “insides” of a light cone (an absolute future of some event) could be quite subtle and even subjective. For example, among cosmological spacetimes with the Big Crunch there is a continuum of solutions smoothly interpolating from a black hole cosmology (where the “matter” consists of many black holes) to a uniform FLRW cosmology. The intermediate solutions could be subjectively interpreted as either growing black holes coalescing together or the cosmological density fluctuations slowing the expansion of lightcones.

So a formal definition of black hole exists only for certain classes of spacetimes with “nice” asymptotic behavior, such as asymptotically flat spacetimes, to which we restrict our further attention.
The key here is the future null infinity (denoted by $$\mathscr{I}^+$$). This is an important concept, and one should look for details in a book such as [1, 2], but informally $$\mathscr{I}^+$$ consists of all future endpoints of null geodesics “escaping to infinity”.

For spacetime points inside a black hole no causal trajectory (such as null geodesic) could reach $$\mathscr{I}^+$$, while for the points “inside” a normal light cone there are null geodesics reaching this null infinity. So the definition of a black hole region in [1 ] is: $$B= M - J^-(\mathscr{I}^+)$$, or “all the points of a manifold $$M$$ that do not lie in the past of $$\mathscr{I}^+$$” (in other words, all the events from which no signal could ever escape to infinity). In contrast, for an asymptotically flat spacetime without an event horizon all points of spacetime are in the past of $$\mathscr{I}^+$$, in other words $$M=J^-(\mathscr{I}^+)$$.

Note, that the existence of singularities inside of a black hole is not a necessity, but an artefact of an “ordinary” general relativity with a particular matter content. One could consider modified theories of relativity which do not have singularities but have black holes in the sense outlined above.

1. R.M. Wald, General Relativity, University of Chicago Press (Chicago, 1984).

2. S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Spacetime, Cambridge University Press (Cambridge, 1973).

• Nice, this is what I was looking for! Here I get that with $J^-$ you mean on operator that maps endpoints to geodesics, right? – AoZora May 13 at 6:34
• @france95: For a set $S$ of spacetime points $J^-(S)$ is a causal past of the set. It contains all points $q$ of the manifold such that there exist a future directed causal curve $\lambda(t)$ (with tangent vector future null or timelike everywhere) connecting $q$ and some point $p\in S$ such that $\lambda(0)=q$ and $\lambda(1)=p$. See section 8.1 of Wald. – A.V.S. May 13 at 16:10

Basically, a black hole is a consequence of general relativity, a deformed spacetime region caused by a sufficiently compact mass-energy. A light cone is a set of events, a surface in spacetime describing the temporal evolution of photons with respect to one given event (the source of those photons). These concepts are very different.

• sure but in which way do these objects differ mathematically? what puzzles me is that from what I grasp of their formal definitions these objects looks equivalent. As you say they obviously are not.. – AoZora May 12 at 16:09
• The mathematical difference between the two concepts is that a black hole (as a region of spacetime that is not in the causal past of the infinite future) implies a region of spacetime in which the curvature is infinite in a way that does not depend on the coordinate system, while the definition of cone of light does not. – NarcosisGF May 12 at 16:32
• mmm I am not convinced: I would say that a real black hole will not have the singularity because it will be smoothed by the presence of matter. At least a black hole created by the collapse of a star. It should be like a neutron star but smaller than Schwartzschild radius, shouldn't it? And the inside of the matter distribution should not have a singularity as in the case of a distribution larger than the Schwartzschild radius.. In any case I guess there should be a clearer way to tell the difference between the light cone and BHs, maybe it is a matter of characterizing them as hypersurfaces? – AoZora May 12 at 20:12
• @france95 In a purely classical GR black hole (i.e., no QM), the matter gets completely crushed. Now a quantum gravity theory may prevent a mathematical singularity from forming, but it's very likely that the predictions of GR & quantum gravity agree down to at least atomic scale, if not smaller. That is, the core of a BH is unlikely to be larger than an atom. – PM 2Ring May 13 at 1:36