The schwarzschild original 1916, metric equation and coordinates are chosen because they cover the entire maximally extended riemann manifold, without the problem of a coordinate singularity located at the black hole event horizon. Kruskal coordinates to uv 0, that is either t 0 or x 0. However, as emphasized long ago by ehlers 4, an explicit maximally extended and regular covering. Symmetry considerations that hold independent of the details of the. Sean carroll spacetime and geometry free ebooks sean carroll spacetime and geometry solutions 3. We show the maximal extension of schwarzschild spacetime in the penrosecarter diagram of fig. This leads to the notion of a maximally extended solution. Carroll, spacetime and geometry general relativity, schwarzschild solution chapter 6. Again, the point is not to be able to understand the details with extreme rigor, but to grasp the. Material can come out of a white hole, but nothing can fall in.
The possibility of the existence of white holes was put forward by russian cosmologist igor novikov in 1964. Almost always the covering is given by way of the implicit kruskalszekeres procedure. Almost always the covering is given by way of the implicit kruskal 1 szekeres 2 procedure 3. Institute for advanced study department of physics. Black holes in general relativity 5 2 which is now indeed also pastinextendible.
Schwarzschild spacetime and the deflection angle the schwarzschild spacetime is expressed by the line element ds2 1. Additionally the study of causal properties of spacetimes was making interesting headway zeeman 1964. This paper considers the effect of hawking radiation on an eternal black hole that is. According to birkhoffs theorem the schwarzschild solution is also a.
Symmetry considerations that hold independent of the details of the emission mechanism show there is an inconsistency in the claim that such a blackhole evaporates away in a finite time. On the paradox of hawking radiation in a maximally. Understanding maximally extended schwarzschild solution. White holes are predicted as part of a solution to the einstein field equations known as the maximally extended version of the schwarzschild metric clarification needed describing an eternal black hole with no charge and no rotation. In these new coordinates the diagram for the maximally extended schwarzschild solution.
However, there is another extension of region i, constructed as follows. A careful analysis of the maximally extended metrics of schwarzschild manifold shows that the original schwarzschilds solution 1916 and brillouins solution 1923 are the only ones that are adequate from the physical standpoint. In a maximally extended external schwarzschild solution, that photons frequency only stays regular if the mode is extended back into the past region where no observer can go. Gravitational waves and test particles, production of gw pages 144 156 of the notes. Request pdf on jan 1, 20, andrzej krasinski and others published gyorgy szekeres 1960, the maximal analytic extension of the schwarzschild solution. The schwarzschild solution describes spacetime under influence by a nonrotating massive sphericallysymmetric object.
Schwarzschild and kerr solutions of einsteins field equation arxiv. The lines u 0 and v 0 r 2m correspond to the event horizon. Pdf schwarzschild manifold and nonregular coordinate. The schwarzschild solution continued and interior solutionsstellar collapse section 5. Symmetry considerations that hold independent of the details. Really this is an extension of spacetime in two directions. Szekeres metric shows that all maximally extended geodesics which do not. The schwarzschild solution is spherically symmetric.
The spacetime solution to einsteins equation that describes a universe expanding from the big bang is very similar to the timereversal of a black hole, but you dont really learn much from making that statement, especially because there is no outside. The wavy horizontal lines are the future and past singularities in the schwarzschild spacetime. The mathematical analysis of black holes in general relativity. Mathematical relativity department of mathematics universidade. Embeddings and time evolution of the schwarzschild wormhole. A maximally extended and regular covering of the schwarzschild vacuum is now a fundamental part of any introduction to general relativity. This gives the socalled maximally extended schwarzschild solution m,g. Maximally extended schwarzschild solution, gravitational waves pages 1 143 of the notes week 10. Maximally extended, explicit and regular coverings of the. That region doesnt seem to be observable and is physically suspect, so hawking used a black hole solution without a past region which forms at a finite time in the past. Show that these coordinates cover both regions i and ii of the maximally extended schwarzschild solution, and that the metric of the constant time surfaces tp const is the. Gyorgy szekeres 1960, the maximal analytic extension of the. I have a question about ontological status and coordinate transformations in gr.
Simplest version is the maximally extended schwarzschild solution. Schwarzschild solution with the aim of probing the nonlinear stability of the black hole horizon. The globalstructure of schwarzschild isconveniently. If the submanifolds are maximally symmetric spaces as twospheres are, then there is the following powerful theorem. A mathematical derivation of the general relativistic schwarzschild metric an honors thesis presented to the faculty of the departments of physics and mathematics east tennessee state university in partial ful. The blue spatial slice contains the einsteinrosen bridge connecting the two regions. The physics of schwarzschilds original 1916 metric. Figure 2 shows the maximally extended schwarzschild spacetime in terms of kruskal coordinates, with the angular coordinates suppressed so that each point in the diagram represents a 2sphere. Schwarzschild solution quick recap to begin, lets recap what we learned from the previous lecture. Motion in the schwarzschild space, schwarzschild black hole, maximally extended schwarzschild solution pages 115 of the notes week 9. We consider now two events a and b with coordinates xa,ta and xb,tb in o and coordinates x.
The penrose diagram for the maximally extended schwarzschild solution. A physics rst approach to the schwarzschild metric 181 powers of the normally small quantity gmrc2, more than the three ingre dients sr, ep and nl are needed, which is su cient to falsify schi s claim. Note that this new manifold m,g does not admit r as a global. If a phenomemon p is predicted by one set of coordinates, but not by another, is that enough to say that p cannot be physical, but is observer dependent. Maximal extension of schwarzschild spacetime in kruskal coordi nates. A mathematical derivation of the east tennessee state.
Figure 7, which shows a maximum of v r at r 3gm for every l. There were a lot of abstract concepts and sophisticated mathematics displayed, so now would be a good time to summarize the main ideas. Birkhoffs theorem that states that the schwarzschild solution is the unique spherically. Why do we care about the maximally extended versions of. Pdf on the paradox of hawking radiation in a maximally. Draw a diagram of the maximally extended schwarzschild spacetime in kruskal coordinates, and. The analytic extension of the schwarzschild spacetime by kruskal coordinates. When this solution is maximally extended it has an event horizon at the schwarzschild radius r s 2m. This can easily seen from the penrose diagram, and is consistent with the fact that entanglement. This answer to my own question is the best example of which i know that seems to show that the maximally extended schwarzschild spacetime is of some interest other than as a purely mathematical object.
A physicsfirst approach to the schwarzschild metric. Schwarzschild solution is the unique spherically symmetric solution to einsteins equations in vacuum. I included it instead of the penrose diagram precisely because i struggle to understand the shape of maximally extended schwarzschild from the penrose diagram. The issue of the existence of spacetime singularities was a vexing one, as emphasized particularly by john wheeler wheeler 1963, because it meant an end to the. The singularity at r 0now corresponds to tx 1 and is plotted as hyperbola with 2 branches in the 2nd and 4th region fig.