Topology outside vs inside black hole

[zankaon]

If topology of a manifold were invariant (or not), what specifically would topology of a patch of such manifold in neighborhood (outside) of such BH, suggest?

 

[Chris Hillman]

A quick sketch of the concept of "invariance" in mathematics

If topology of a manifold were invariant (or not), what specifically would topology of a patch of such manifold in neighborhood (outside) of such BH, suggest?

Not sure I understand the question, but let me try to provide some background which should be useful.

A topological invariant is a quantity $q(M)$ defined on topological manifolds, such that whenever $M_1,\, M_2$ are homeomorphic, then $q(M_1) = q(M_2)$. Well, that would be a numerical invariant like genus; more generally q might be something more elaborate, like a group, and then we'd say that whenever $M_1,\, M_2$ are homeomorphic, then $q(M_1), \, q(M_2)$ must be isomorphic as groups. In all cases, the invariant detects "structure" which is invariant under homemorphisms (continuous maps with continuous inverses; "no ripping, folding, crushing, or tearing").

Another common use of "invariant", particularly in this forum, is this: we say that a quantity defined on a Lorentzian manifold is Lorentz invariant when it is invariant under Lorentz transformations (or rather, Poincare transformations, which include translations). For example, spacetime interval is a Lorentz invariant of Minkowski vacuum.

As with topological invariance, the key idea is that we are assigning a "quantity" (a number, a group...) to things which classifies them into orbits under an action by a symmetry group (group of self-isometries on a Lorentzian-manifold, group of self-homeomorphisms of a topological manifold). (Well, not quite: the invariant can tell when two things belong to different orbits but not when they belong to the same orbit.) In the case of topological invariants defined on topological manifolds, we are attempting to (partially) classify all topological manifolds according to their topology. In the case of Lorentz invariants defined on a single Lorentzian manifold, we are attempting to (partially) classify geometric features on that manifold, e.g. classifying pairs of events in Minkowski vacuum by their pairwise spacetime intervals.

I hope this information will enable you to reformulate your question.

 

[zankaon]

A toplogical gedanken (thought simulation)

If topology of a manifold were invariant (or not), what specifically would topology of a patch of such manifold in neighborhood (outside) of such BH, suggest?

If one considered an idealized spherical expansion (of say neutrinos) for a patch of manifold outside a BH; then one could consider such expansion to be finite, bounded (in an orthogonal to the 2-surface) sense, and hence such expansion per se would be NOT closed. Now if one considered the same gedanken simulation, but for less than BH_h (BH horizon), then the experiment in principle, would seem to have the same topological description, but in an environment becoming more extreme; hence the same manifold (continuum i.e. same inbetweenness). But yet does one have a construct (BH_h) interspersed between such 2 environments, but with a different topology? Topology commentary is always appreciated.

 

[Chris Hillman]

Failed attempt to decipher the question

If one considered an idealized spherical expansion (of say neutrinos) for a patch of manifold outside a BH

Do you mean this (for photons instead of neutrinos): "Consider the beacon congruence, i.e. the null geodesic congruence consisting of all forward null geodesics issuing from an event in the exterior of the Schwarzschild vacuum, on some neighborhood N lying in the exterior region in which the congruence is nonsingular, and compute its optical expansion scalar"?

then one could consider such expansion to be finite, bounded (in an orthogonal to the 2-surface) sense,

Do you mean this: "choose a curve C from the congruence and choose a two-surface everywhere transverse to C"? Do you claim this: "the expansion scalar is bounded on N"?

and hence such expansion per se would be NOT closed. Now if one considered the same gedanken simulation, but for less than BH_h (BH horizon), then the experiment in principle, would seem to have the same topological description, but in an environment becoming more extreme; hence the same manifold (continuum i.e. same inbetweenness).

OK, I give up! I can't make sense of your question, so you'll have to clarify your terms.

What do you mean by "topological description"? What do you mean by "gedanken simulation"? What do you mean by the term "closed expansion"? For that matter, what do you mean by the term "expansion"? What do you mean by "finite"? What do you mean by "in the sense of an orthogonal two-surface"? (Orthogonal to what?) What do you mean by "patch of manifold"? What do you mean by "BH"? (Schwarzschild vacuum solution in gtr?)

Most of these terms are standard, but you do not appear to be using any of them in a standard way.

(If possible, please read the discussion in Poisson, A Relativist's Toolkit and/or Ludvigsen, General Relativity of expansion and vorticity for timelike congruences and for null geodesic congruences before answering.)

 

[zankaon]

If one considered an idealized spherical expansion (of say neutrinos) for a patch of manifold outside a BH; then one could consider such expansion to be finite, bounded (in an orthogonal to the 2-surface) sense, and hence such expansion per se would be NOT closed. Now if one considered the same gedanken simulation, but for less than BH_h (BH horizon), then the experiment in principle, would seem to have the same topological description, but in an environment becoming more extreme; hence the same manifold (continuum i.e. same inbetweenness). But yet does one have a construct (BH_h) interspersed between such 2 environments, but with a different topology? Topology commentary is always appreciated.

The ending answer would seem to be no. For example if a larger BH coalesced with a smaller BH, the former would enlarge. Hence an expansion orthogonal to BH_h. So therefore such expansion (orthogonal to surface) of BH_h would be finite, bounded and thus not closed.

 

[Chris Hillman]

Zankaon, you are confusing me!

The ending answer would seem to be no.

To what question?! I am alarmed that you might think I "confirmed" [sic] some guess you had (I did no such thing!).

For example if a larger BH coalesced with a smaller BH, the former would enlarge. Hence an expansion orthogonal to BH_h. So therefore such expansion (orthogonal to surface) of BH_h would be finite, bounded and thus not closed.

If you haven't read the popular book by Wald, Space, Time, and Gravity, University of Chicago Press, 1977, you would enjoy it!

I have no idea what you seem to be asserting about what happens to the horizon of a black hole when it coalesces with a smaller hole, but it doesn't sound right. To mention just one point, the horizon isn't a physical membrane but an abstract "teleologically defined" surface. At an event on the horizon, nothing in particular happens compared with nearby events, in terms of local physics.