In this section we will describe how the nearhorizon limit of extremal reissnernordstrom is ads2. It is wellknown that there are two horizons for the reissnernordstromde sitter spacetime, namely, the black hole horizon and the cosmological one. Mit lectures on the geometry and analysis of black hole. Use your diagram to explain why radiation becomes in nitely blueshifted at the inner horizon of the rn spacetime. Friedmanlemaitrerobertsonwalker flrw and reissner nordstrom metrics have true curvature singularities.
There are basically six possibilities as depicted in figure 2. Entering a black hole reissnernordstrom you would still be broken apart by the tidal forces within the outer horizon the singularity would appear as a place in time within the inner horizon time and space change roles again and so the. Both horizons can usually seem to be two independent thermodynamic systems. Spherical symmetry the line element in coordinates t,r. Mathematical tripos part iii lent term 2020 black holes. The most astrophysically realistic case is that of a rotating, neutral black hole.
The same way as mapping schwarzschild coordinates onto kruskalszekeres coordinate reveals two seperate external regions for the schwarzschild black hole, such a. Mit lectures on the geometry and analysis of black hole spacetimes in general relativity mihalis dafermos. Conformal isometry of the lukewarm reissner nordstr\omde sitter spacetime. Schwarzschild and kerr solutions of einsteins field equation. The analogous solution for a charged, rotating body is given by the kerrnewman metric the metric was discovered between 1916 and. By doing a conformal transforma tion, as explained. The penrose diagram for the minkowski space explains all of its causal structure. Although they have been studied for years,1 they still attract tremendous attention in the physics and astrophysics literature. Could the maximally extended rn solution describe a physical black hole. Picture of a black hole that forms from a collapsing shell of matter. Roger penrose, who invented this kind of diagram in the early 1950s, himself calls them conformal diagrams.
Reissnernordstrom metric derivation however, we cannot set the stressenergy tensor to zero as in the schwarzschild derivation there is electric charge present, and hence a nonzero maxwell tensor by equating the ricci tensor with the nonzero maxwell tensor, and after working. All this information is from this great site for penrose diagrams by andrew hamilton, who seems to do a lot of work with black holes based on his site. Carterpenrose diagram of a reissnernordstrom black hole in the non extreme. Gunnar nordstrom later independently of reissner arrived at the same solution 3 now known as the reissnernordstrom metric. The conformal rescaling does not change causal relations. Removing a coordinate singularity at the schwarzschild radius.
Examples of penrose diagrams for some types of black holes will be presented schwarzschild, reissnernordstrom and kerrnewman, beyond the. Schwarzschild and kerr solutions of einsteins field. The white hole is actually a completely separate piece of spacetime, as is apparent in the penrose diagram of the reissnernordstrom geometry. Introduction to black hole evaporation pierrehenry lambert contents introduction 3 1. Explicit global coordinates for schwarzschild and reissner. Gq2 0 so there is a minimal mass mmin q2 g below this mass, we. Preface these are lecture notes for the course on black holes in part iii of the cambridge mathematical tripos. What is the penrose diagram for a reissnernordstrom. Penrose diagramfor themaximally extendedreissnernordstri. The white hole continuation of the reissnernordstrom geometry is an exact mathematical solution, but it is no more realistic than a waterfall that falls down one side of a ravine, and then climbs. Since the schwarzschild \time coordinate t goes to in. Cosmic strings, white holes and blackholes are deduced from the reissner nordstrom singularities. We employ a late time expansion and argue that decaying fluxes of radiation through the event horizon imply that the black hole must contain a null singularity on the cauchy horizon and a central spacelike singularity.
The cp diagram of ds2 is rotated by 90 relative to ads2. The future and past null infinities of the positive mass case are denoted, respectively, by, the horizons by, the future and past timelike infinities. Rotating black holes in penrose diagrams physics forums. Interestingly, the fourdimensional reissnernordstrom black hole in fact, all of elec. Rotating black holes in penrose diagrams thread starter relativityfan. The analogous solution for a charged, rotating body is given by the kerrnewman metric. Gravitational collapse and singularities in some non.
Cosmic strings, white holes and blackholes are deduced from the reissnernordstrom singularities. Properties of classical black holes and both classical and quantum black hole thermodynamics are treated. Penrosecarter diagram for the naked reissnernordstrom black hole q 2 m 2, analytically extended beyond the singularity. Consider the einsteinmaxwell action setting units g n 1,10 s 1 16.
Angel, oregon, was a german aeronautical engineer whose avocation was mathematical physics. The reissnernordstrom metric is a famous solution to the einstein. We study the collision of two reissnernordstrom gravitational shock waves in ads and show that the charge completely prevents the formation of marginally trapped surfaces of the penrose type with topology s d. In higher dimensions the celestial circle is replaced by a celestial sphere of dimension d. Sep 30, 2010 hello, in penrose diagrams it says that once you have crossed a first cauchy horizonof a rotating black hole, then, with the repulsive singularity, it is possible to cross another cauchy horizon, and then another event horizon to escape the black hole and go into another universe. Pdf conformal isometry of the lukewarm reissnernordstr\om. The selection and focus is determined by my idiosyncracies, time limitations, and an e. Extending the reissnernordstrom spacetime at the singularity 2 2. The line element continues smoothly through the horizon. In particular, we obtain conformal representations of the neighbourhoods of spatial infinity, timelike infinity and the cylindrical endthe socalled cylinders at spatial infinity and at the horizon, respectivelywhich are regular with respect to the conformal einstein field equations and their. On the conformal structure of the extremal reissnernordstr\ om. Carterpenrose diagram of a reissnernordstrom black hole in the nonextreme.
An embedding diagram in minkowski space of a timelike schwarzschild spacetime hypersurface described by a vertical line in the kruskalszekerescoordinates. Real black holes probably spin, but probably have almost no electric charge, because our universe appears to be electrically neutral, and a charged black hole would quickly neutralize by attracting charge. The schwarzschild metric 1916 is a solution to the vacuum einstein equations. A note on the collision of reissnernordstrom gravitational. It is indeed only in 2011 that light was shed on a subtle instability, occuring in an extreme reissner nordstr om background, by stephanos aretakis in 3 and 4. As such, they contain some sections which are entirely absent from the audio recordings. Friedmanlemaitrerobertsonwalker flrw and reissnernordstrom metrics have true curvature singularities. Gravitational collapse in the flrw metric yields a curvature singularity which shows the universe started a finite time ago. Although they have been studied for years,1 they still attract.
These sections are in italics, allowing readers already familiar with the material to easily bypass them. The black holes of a general twodimensional dilaton gravity theory. In a penrose diagram, light cones are drawn at 45 degrees just as in minkowski spacetime, so these causal relations are easy to see. Plate analysis using classical or reissner mindlin theories. Remo garattini facolt a di ingegneria, universit a degli studi di bergamo, viale marconi, 5, 24044 dalmine bergamo italy. The definitions of the regions, and are given in the main text. In physics and astronomy, the reissner nordstrom metric is a static solution to the einsteinmaxwell field equations, which corresponds to the gravitational field of a charged, nonrotating, spherically symmetric body of mass m. Overview the schwarzschild black hole is the simplest black hole solution.
On the conformal structure of the extremal reissner. To this end, we show that the reissner nordstrom metric charge denoted by. Glue together all efpatches on overlap regions to get full cpdiagram 7. The constant time hypersurface we will consider will be denoted by. Finally we present the penrose diagram illustrating the region covered by the coordinates. On the conformal structure of the extremal reissnernordstrom. The penrose diagram of the schwarzschild geometry clarifies this unexpected. The critical phenomena and thermodynamics of the reissner. Penrose diagram of the reissner nordstrom geometry. Above we see a penrose diagram representing a coordinate extension 1 for a charged or rotating black hole. Hans jacob reissner, also known as jacob johannes reissner 18 january 1874, berlin 2 october 1967, mt. Extending the reissner nordstrom spacetime at the singularity 2 2. What is the penrose diagram for a reissnernordstrom metric.
Compare this with a penrose diagram or kruskalszekeres diagram for an eternal nonrotating. Consider the reissnernordstrom solution with m eusing advanced eddingtonfinkelstein coordinates. The penrose diagram for the subextremal reissnernordstrom solution. Faculty of civil engineering, state universiv at campinas, brazil abstract plates can be solved with classical or reissnermindlin plate theory in the same computer code with an appropriate treatment of the direct boundary element formulation. Vast amounts of interesting and important work on the subject are not mentioned.
Definition of a penrose diagram a penrose diagram is a kind of spacetime diagram arranged to make clear the complete causal structure of any given geometry. We analyse various conformal properties of the extremal reissnernordstrom spacetime. A bisognano wichmannlike theorem in a certain case of a. Two dimensions are special, since the 0sphere consists of two disjoint points.
It will serve as an example many times in this course. The reissner nordstrom solution and its classification. We study black holes produced via collapse of a spherically symmetric charged scalar field in asymptotically flat space. Schwarzschild black holes, nonextreme reissnernordstrom and very. Conformal diagram of the extremal reissnernordstrom spacetime in the positive mass. Introduction to black hole evaporation pierrehenry lambert introduction there are several reasons for which the general relativity theory cannot be the. From one side the theory predicts the existence of. Instability of extreme reissnernordstrom black holes imperial. Chapter 1 an introduction to black holes black holes belong to the most fascinating objects predicted by einsteins theory of gravitation.
Hans jacob reissner, 1 also known as jacob johannes reissner 2 18 january 1874, berlin 2 october 1967, mt. Penrose carter diagram for the naked reissnernordstrom black hole q 2 m 2, analytically extended beyond the singularity. Note that in the latter case, the normal is contained in the tangent space. Flatten the cylinder to get the carterpenrose diagram of minkowski space time. Journey into and through a reissnernordstrom black hole. The known charged rotating black hole solution is the kerrnewman metric. Algorithms for the explicit computation of penrose diagrams. However, for the reissner nordstrom and kerr backgrounds, the stability in the extreme case was surprisingly assumed for many years. During world war i he was awarded the iron cross second class for civilians for his pioneering work on aircraft design biography. Diagram of the positive mass ef spacetime, suppressing the angular coordinates, with constant r surfaces vertical and constant v surfaces at 45. The penrose diagram is identical to the 4dimensional reissnernordstrom. However, for the reissnernordstrom and kerr backgrounds, the stability in the extreme case was surprisingly assumed for many years.
W e can visualize these results in a carter penrose diagram for the conformally. The lecture notes will be updated as the course progresses. They are an indispensable map for navigating inside a black hole. Copy three mirror images of the cpdiagram of the basic efpatch 6. We construct the complete penrose diagram by enforcing timereversal symmetry. Here is the extended conformal penrose diagram for the reissnernordstrom metric. For a more detailed account in two spacetime dimensions see section 3. The two pdf files i linked to both seem to say that a penroseconformal diagram should have two basic features1 that the worldlines of light beams null geodesics are represented as straight lines, usually at 45 degrees, so that the causal structure is obvious, and 2 that these geodesics all have finite length in the diagram, even if a. In physics and astronomy, the reissnernordstrom metric is a static solution to the einsteinmaxwell field equations, which corresponds to the gravitational field of a charged, nonrotating, spherically symmetric body of mass m. Lecture 1 these lecture notes provide somewhat more detailed explanations than those given in the audio recordings. Plate analysis using classical or reissner mindlin theories l. Hans reissner generalized schwarzschilds solution to include an electrically charged object 2. Use the info above to guess the cpdiagram of a basic efpatch 5. The meaning of the analytic extension at the singularity 14 references 15 introduction 0.
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