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PeterDonis submitted a new PF Insights post
The Schwarzschild Geometry: Part 4
Continue reading the Original PF Insights Post.
The Schwarzschild Geometry: Part 4
Continue reading the Original PF Insights Post.
Haelfix said:There is a serious Cauchy problem with having a past singularity that is allowed to communicate information off to infinity.
Haelfix said:Since nothing is allowed to get in, that means that 'test' particles traveling in orbits around the white hole horizon (more precisely the particle horizon) will accumulate there
Haelfix said:there will be a severe blue shift when viewed from infinity
Haelfix said:See:
Death of White Holes in the Early Universe - Eardley, Douglas M. Phys.Rev.Lett. 33 (1974) 442-444
Haelfix said:Quantum mechanically, if you believe in Hawking radiation/evaporation, and black hole thermodynamics, in some sense black hole and white hole microstates have to be the same thing!
PeterDonis said:If "Cauchy problem" is intended to mean that the spacetime has a Cauchy horizon, this is not true. The Schwarzschild spacetime is globally hyperbolic.
PeterDonis said:Since it is dealing with the early universe, it obviously is not using a vacuum geometry, and the Schwarzschild spacetime I am discussing in this series is a vacuum solution (except for the Oppenheimer-Snyder model, which has a non-vacuum region, but that model also has no region III or IV so it's not relevant here). In short, I'm not sure the term "white hole" in that paper means the same thing as I mean by "white hole" in these articles.
PeterDonis said:I'm aware of this hypothesis by Hawking, but I don't know if it has led to anything in the field of quantum gravity.
Haelfix said:A Cauchy problem ('initial value problem') in GR is a statement about taking surfaces of initial data (in GR-- spacelike surfaces but they could in principle also involve data from other matter fields) and developing them forward in some regular way subject to the relevant partial differential equations
Haelfix said:Here, the initial data surface is singular as there is geodesic incompleteness
Haelfix said:The geometry I'm referring to is not vacuum, but it is somewhat similar to Oppenheimer Snyder which you were discussing. It is the *perturbed* extended Schwarzschild solution with an infalling sheet of spherically symmetric null dust.
Haelfix said:A Cauchy problem ('initial value problem') in GR is a statement about taking surfaces of initial data (in GR-- spacelike surfaces but they could in principle also involve data from other matter fields) and developing them forward in some regular way subject to the relevant partial differential equations such that the process satisfies certain constraints (basically you want reversibility, avoiding many to one mappings, etc). Here, the initial data surface is singular as there is geodesic incompleteness, and physically this manifests itself as a loss of predictability between any 'two' distinct states in the theory, provided the singular surface was is in at least ones past lightcone. Basically you are taking an infinite amount of information (states) and allowing that to propagate throughout spacetime. This language is often used when discussing formulations of cosmic censorship, but for some reason that I don't understand the FRW singularity and the White hole singularity seem to be excluded from theorems about cosmic censorship (probably b/c they are trivial).
Haelfix said:So there are certainly spacelike Cauchy surfaces that one can construct that will have finite values for all physical quantities arbitrarily 'near' the singularity, but I don't believe this is sufficient condition for being a well posed surface (regular is I agree an incorrect word choice). There are other technical restrictions on the form of the initial data and I'd have to consult a textbook (im currently away) for the exact statements. Clearly having arbitrarily large(but finite) tidal forces is not what one would want for well behaved data.
Haelfix said:2) The white hole horizon is conceptually really bizarre...
Since nothing is allowed to get in, that means that 'test' particles traveling in orbits around the white hole horizon (more precisely the particle horizon) will accumulate there, and there will be a severe blue shift when viewed from infinity. This blue sheet is a sort of classical instability, and it is argued that it leads to gravitational collapse, and thus there is likely a singularity in the future as well!
stevendaryl said:It's true by definition that:
- If the test particle is below the event horizon and ##\frac{dr}{d\tau} > 0##, then the particle is in the black hole interior.
- If the test particle is below the event horizon and ##\frac{dr}{d\tau} < 0##, then the particle is in the white hole interior.
stevendaryl said:Why was entropy lower in the far past? General Relativity doesn't answer this question. (I'm not sure what does
stevendaryl said:Another complication is to include test particles that don't move on geodesics, because of non-gravitational forces. How does that affect the picture?
Haelfix said:Surfaces that include data with arbitrarily large curvature invariants are thus being evolved forward with Einsteins equations
Haelfix said:when they likely don't even obey the equation to begin with.
Haelfix said:Yes but think about it, any such line has access to the singularity region in its causal past. Surfaces that include data with arbitrarily large curvature invariants are thus being evolved forward with Einsteins equations, when they likely don't even obey the equation to begin with. The entire future spacetime is thus built out of that dubious development.
When people formulate statements about cosmic censorship they are trying to formalize that notion somehow (and I know there are difficulties with making the statement precise). I'll look into it when I get the chance
martinbn said:Well, it's not how it works. The initial data doesn't include anything from the past of the Cauchy surface. In fact until you solve the equations, there is no past nor future. The initial data consists of fields defined on the surface. Whatever the values of the past and future evolution may be, say arbitrary large, they are not part of the initial conditions. So there is nothing dubious here and by construction you get solutions to the Einstein equation.
PeterDonis said:You have these backwards.
Haelfix said:Sure, you can formally do this. Butt then I can formally take a line in the middle of the diagram, evolve it arbitrarily far backwards to the singularity region, then evolve it forward again back to the start. The two resulting hypersurfaces won't necessarily agree anymore depending upon details of what takes place near the singularity. This is why it's often said that naked singularities yield problems for determinism. So I would say the propriety of those sorts of manipulations are basically equivalent to whether you accept (weak) cosmic censorship or not.
PeterDonis said:You don't have to evolve them forward. You can evolve the initial data on the hypersurface ##T = 0## in Kruskal-Szekeres coordinates both forwards and backwards. Doing so will give you the complete globally hyperbolic region, all the way back to the past singularity and forward to the future singularity. Since the equations are time symmetric, this is perfectly well-defined and justified.
I don't know what you're basing this on. The subject under discussion is a well-defined solution of the classical Einstein Field Equation. Any event with finite spacetime curvature invariants, including arbitrarily large ones, can occur in such a solution. The solution might not end up describing anything physically relevant, but that doesn't mean the points with large spacetime curvature values "don't obey the equation"; it just means physics, unlike this particular mathematical model, chooses some other equation at that point.
stevendaryl said:What is it that prevents having two nearby test particles with opposite signs of ##\frac{dr}{d\tau}##?
Applying an orientation to an orientable spacetime involves choosing a consistent labeling of past/future of all light cones. Then, for any world line, a tangent directed one way (one sign, in your case) is future directed, while the other sign is past directed.stevendaryl said:Right. In the black hole interior, [itex]\frac{dr}{d\tau}< 0[/itex] and in the white hole interior, [itex]\frac{dr}{d\tau} > 0[/itex].
So now I'm a little confused: What is it that prevents having two nearby test particles with opposite signs of [itex]\frac{dr}{d\tau}[/itex]?
Ben Niehoff said:I disagree with the terminology "globally hyperbolic" here.
Ben Niehoff said:The equations of motion fail at the singularities, and the singularities are reachable in finite proper time.
Ben Niehoff said:This means you cannot just excise the singularities
Ben Niehoff said:The issue is that the singularities don't obey the equation.
Ben Niehoff said:What, then, stops me from saying the Reissner-Nordstrom geometry is globally hyperbolic? Can't I just excise the Cauchy horizons (and whatever lies beyond them) and call it a day?
In a similar fashion, it seems I can call anything "globally hyperbolic", if I just cut out the bad parts and redefine what I'm talking about.
Ben Niehoff said:Most (open patches of) manifolds can be extended in more than one way.
Ben Niehoff said:If you can point to a mathematician's definition of "global"
Ben Niehoff said:I still feel that an incomplete spacetime is physically unhealthy in some way, as it means that there are some observers who can reach a region of "Here be dragons" in finite proper time.
Ben Niehoff said:Perhaps that is simply the best that GR can say about such spacetimes, and an obvious indication that the theory cannot be fundamental
Ben Niehoff said:But I still feel that an incomplete spacetime is physically unhealthy in some way, as it means that there are some observers who can reach a region of "Here be dragons" in finite proper time. Perhaps that is simply the best that GR can say about such spacetimes, and an obvious indication that the theory cannot be fundamental (since it cannot answer physically reasonable questions about what happens to some observers).
The cosmic censorship I'm familiar with only requires the BH be surrounded by a horizon. It doesn't say anything about the a-causality of the manifold inside the inner horizon. Does the strong version go further and say something like no singularity in the past light cone of any observer? This would require rejection of the inner horizon region due to CTCs. It also would reject the full Kruskal geometry.martinbn said:My personal view is that incompleteness as in the Schwartzschild solution is OK. The observer is torn apart by infinite curvature and ceases to exists, but everyone is accounted for. Incompleteness as in the Kerr solution, where the observer reaches the Cauchy horizon in finite proper time and there is no unique extension beyond it, is not OK. The theory loses its predictability. But this is where the strong cosmic censorship conjecture comes in. If true, these situations are non generic and therefore the theory is still as good as ever.
PeterDonis said:Yes. The problem is that we don't (yet) have a more fundamental theory that covers the regime that GR does not.
The Schwarzschild geometry is a solution to the equations of Einstein's theory of general relativity. It describes the gravitational field around a spherically symmetric mass, such as a star or black hole.
In the Schwarzschild geometry, time is affected by the presence of a massive object. As an object gets closer to the massive body, time appears to slow down for an outside observer. This is known as gravitational time dilation and is a consequence of the curvature of spacetime caused by the massive object.
The Schwarzschild radius is the distance from the center of a massive object at which the escape velocity is equal to the speed of light. This is also known as the event horizon of a black hole, as anything that crosses this radius cannot escape the gravitational pull of the black hole.
The singularity in the Schwarzschild geometry is a point of infinite density and zero volume. It is where the curvature of spacetime becomes infinite, and the laws of physics as we know them break down. This singularity is thought to exist at the center of a black hole.
Gravitational lensing occurs when the path of light is bent by the curvature of spacetime caused by a massive object. In the Schwarzschild geometry, the curvature of spacetime is described by the Schwarzschild metric, which can be used to calculate the bending of light around a massive object. This phenomenon has been observed and confirmed by astronomers, providing evidence for the validity of the Schwarzschild geometry and general relativity.