Talk:Penrose–Hawking singularity theorems
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Maybe we could add a sentence about the meaning of this? Even I as physicist (however never having properly studied general raltivity) am unsure about whether this is to mean: There can be black holes, e.g. region in spacetime from which escape is impossible. Sanders muc 21:57, 16 May 2004 (UTC)
The Gromov theorem as it is presented is wrong. For exemple take a paboloid with induced metric of R^3. It as a scalar curvature everywhere positiv (and even a ricci tensor positiv) but isn't compact. The true hypothesis is that the spectrum of the ricci tensor is uniformaly minored by a strictly positiv number on the manifold. The scalar curvature it self give no information: The Yamabé problem says that there exists a conformably diffeomorphic manifold with constant scalar cuvature. —Preceding unsigned comment added by 220.127.116.11 (talk) 17:35, 24 June 2008 (UTC)
- Right-- the correct constraint is that the curvature is bounded away from zero, I'll fix it. The Ricci curvature condition is correctly stated in the article, as the convergence property of geodesics. This is the way physicists state it, and it is directly related to the proof.Likebox (talk) 01:30, 25 June 2008 (UTC)
- Actually, on rereading, it is correct as stated--- the condition of positive curvature is defined in the article by the convergence properties of geodesics, and this is the same as the statement that the Ricci tensor is "uniformly minored" by a strictly positive number. There is no statement about the scalar curvature.Likebox (talk) 01:35, 25 June 2008 (UTC)
Energy-mass limit in given finite space
Need a little more clarification for laymen: there's a bound for the amount of information in a finite region of space (Bekenstein bound, which depends on the amount of energy-mass in that region. Does it mean if there's an infinite amount of energy in a region of space then there's an infinite amount of information? Should there be an upper bound for energy-mass if Pauli exclusion principle really holds true? Also, if that's the case then singularity is ruled out?Mastertek (talk) 14:53, 23 October 2011 (UTC)
Instead of citing the sources, I wanted to prove the theorems. They are easy. The best source is "Large Scale Structure of Space Time", but it wastes your time by reviewing everything and the kitchen sink, and then in the interesting new parts, it is written densely and unclearly. But that's the source I read.
You were complaining about Gromov's theorem. I stated the theorem and sketched the proof--- it says that if a manifold has the property that when you extend any geodesic it will collide with a neighbor after a finite time, then the manifold is compact. I don't know the math literature well enough to know the actual cite. I heard this theorem while auditing a class.Likebox (talk) 22:32, 1 January 2009 (UTC)
Attribution of mathematical results
I have a very crappy knowledge of the mathematics literature, so I was attributing results without knowing the whole story. Sorry about that--- somebody please fix this. The important thing is to make sure that the analogous Riemannian geometry results remain--- the focusing theorems in Riemannian geometry are naturally parallel to Penrose's method, which uses the same type of ideas to prove something more surprising.Likebox (talk) 01:29, 27 October 2009 (UTC)
Einstein Maxwell Dirac
Somebody added this sentence to the lead:
(This doesn't hold for the Einstein-Maxwell-Dirac system, which is the (super)classical limit of quantum electrodynamics in a curved space-time.)
And the "this" refers to the statement that light rays are always focused by gravity. The weak energy condition is just that the projection of the stress energy tensor onto null vectors is nonnegative. This is certainly true in all models of matter, including Einstein Maxwell Dirac, or even with scalars. The dominant energy condition, on the other hand, is violated by scalars with a VEV. Maybe what the person meant here is that with Einstein Maxwell Dirac including positrons and electrons of arbitrary charge and mass, one could imagine forming a scalar positronium condensate which violates the dominant energy condition?
Another possibility is that the person just meant that classical Dirac spinors have solutions of positive and negative energy. This is true, but is physically spurious--- this type of negative energy is well understood to go away in second quantization.Likebox (talk) 00:26, 9 December 2009 (UTC)
- Metric Completeness and Singularity
Understanding the definition of the singularity is very important, since our universe has started from singularity and, and all black holes contain singularity. It is not reasonable to define a space-time singularity as a point where the metric tensor was undefined or was not suitably differentiable. However the trouble with this is that one could simply cut out such points and say that the remaining manifold represented the whole of space-time, which would then be non-singular. Although we define the singularity of any black hole as a place which space time is undefined in many occasions, such as the singularity of Kerr black hole, note that in reality this is not true. Indeed, it would seem inappropriate to regard such singular points as being part of spacetime, for the normal equations of physics would not hold at them and it would be impossible to make any measurements. The most important mathematical definition in the singularity theorem is the geodesic completeness. In the case of a manifold (M,g) with a positive definite metric g one can define a distance function ρ(x,y) which is the greatest lower bound of the length of curves from x to y. The distance function is a metric in the topological sense; that is, a basis for the open sets of M is provided by the sets B(x,r) consisting of all points y which are element to M such that ρ(x,y)<r. The pair (M,g) is said to be metrically complete (m-complete) if every Cauchy sequence with respect to the distance function ρ converges to a point in M. (A Cauchy sequence is an infinite sequence of points x(n)such that for any ε>0 there is a number N such that ρ(x(n),x(m))<ε whenever n and m are greater than N. Therefore we define m-completeness implies geodesic completeness (g-completeness), that is every geodesic can be extended to arbitrary values of its affine parameter. This is because if (M,g) is m-complete if every differentiable(at least once) of finite length has an endpoint(a definition from the causal structure) Timelike geodesic incompleteness has an important property; there could be freely moving observers or particles whose histories did not exist after a finite interval of proper time. So it is appropriate to regard such a space as singular. Even though the affine parameter on a null geodesic does not have quite the same physical significance as proper time does on timelike geodesics, one should regard that a null geodesically incomplete space time as singular both because null geodesics are the histories of zero rest mass particles and because there are some examples(such as RN solution) which one would think of as singular but which are timelike but not null geodesically complete(meaning that there are freely-falling particles whose motion cannot be determined at a finite time at the point of reaching the singularity). Therefore if a space time is timelike or null geodesically complete, we shall say that it has a singularity. A theorem that were provided by Penrose in 1965 is very important. Before he provided the theorem to be proven, it was possible to hope that collapse to a Schwarzschild singularity was an artifact of spherical symmetry, and typical geometries would remain nonsingular. But they provided a fact that once a collapse reaches a certain point, evolution to a singularity is inevitable.(For more detailed disscusion see http://cafe.daum.net/grelativitycosmology)
- This just repeats many of the points made earlier in an opaque way, although perhaps the statement that the singularities are not an artifact of spherical symmetry could be emphasized more in the article. I cleaned it up below:
In special exact solutions, the coordinate values where the metric diverges in an essential way describe a singularity. But it seems inappropriate to regard the actual points described by these coordinate values as a part of spacetime, because the normal laws of physics do not hold there, in particular, the spacetime no longer looks like Minkowski space locally. One could cut out such points and say that the remaining manifold represented the whole of space-time.
For a Euclidean manifold with a positive definite metric g one can define a distance function ρ(x,y) which is the greatest lower bound of the length of curves from x to y. This distance function makes the manifold into a topological metric space. The manifold is metrically complete if every Cauchy sequence using ρ as a metric converges to a point in M. Metric completeness implies geodesic completeness, that is every geodesic can be extended to arbitrary values of its arclength. The reason is that if a geodesic g is inextendible past a certain arclength p, points on the geodesic which approach the inextensible limit form a Cauchy sequence which cannot have a limit on the manifold, else the geodesic could be extended a little bit further.
For Minkowski manifolds, where the metric is not positive definite, timelike geodesic incompleteness has a physical interpretation. Timelike geodesics are the paths of freely moving particles or observers whose history ends after a finite interval of proper time. The physical interpretation shows that it is appropriate to regard a geodesically incomplete space as singular.
Even though the affine parameter on a null geodesic does not quite have the same physical significance as proper time, a null geodesically incomplete space time is considered singular too. This is because null geodesics are the histories of zero rest mass particles and can be approximated by specially accelerated observers. The singular exact Reissner-Nordström solution and Kerr solution are timelike but not null geodesically complete.
Penrose's 1965 singularity theorem established that any spacetime with a closed trapped surface is singular. Before Penrose proved his result, it was conceivable that typical geometries could remain nonsingular for all time. But since a small perturbation of a closed trapped surface is still closed and trapped, geodesic incompleteness is a general property of any spacetime which develops a gravitational horizon which vaguely resembles the Schwartschild horizon.Likebox (talk) 01:00, 9 December 2009 (UTC)
deleted incorrect statement
I've removed the following sentence from the lead: "It is still an open question whether time-like singularities ever occur in the interior of real charged or rotating black holes, or whether they are artifacts of high symmetry and disappear when realistic perturbations are added." Actually one of the most important applications of the theorems is to show that the development of a singularity is robust with respect to perturbations and asymmetry.--18.104.22.168 (talk) 01:38, 27 December 2009 (UTC)
another incorrect statement
The last part of the last sentence in the introduction is wrong:
"inflationary cosmologies avoid the initial big-bang singularity, rounding them out to a smooth beginning."
Inflation is just a phase of accelerated expansion and it has to begin at some time (there is no past eternal inflation). Hence without invoking some sort of Quantum Gravity inflation can in no way remove the initial Big Bang singularity. Someone should confirm this and remove the statement. —Preceding unsigned comment added by 22.214.171.124 (talk) 20:37, 22 August 2010 (UTC)
Arguements against a point singularity and infinite density
(1) The contents of a black hole would be expected to be relativistic. If the pressure P of relativistic material is given as (rho)(c^2)/3, the total thermal energy of this star would be ∫PdV = (Mc^2)/3, meaning a whopping 1/3 of the mass energy of the star would be used just to oppose the force of gravity. Interestingly, using the viral equation, if (Mc^2)/3 is equal to 1/2 the (newtonian) gravitational potential energy, or G(M^2)/2R, the radius R of this star equals 1.5GM/(c^2), or 3/4 of the Schwarzchild radius. Because of relativistic gravity the gravitational binding energy and radius would be somewhat larger. (2) Black holes have spin. If a 5 solar mass object of 10 kilometers radius spins at 500 revolutions per second, the surface velocity is 10% of the speed of light. If a 5 solar mass object of 1 centimeter radius spins at 500 revolutions per second, angular momentum isn't conserved. 126.96.36.199 (talk) 21:34, 1 January 2013 (UTC)BG