Black holes and fundamental fields II C. Herdeiro U. Aveiro and - PowerPoint PPT Presentation

Black holes and fundamental fields II C. Herdeiro U. Aveiro and CENTRA Portugal

Lecture plan: a) Introduction: the simplicity of black holes b) Story I: Linear analysis and new dof (“hair”) c) Story II: Non-linear analysis - new black holes and solitons d) Discussion

REVIEW LETTERS PHYSICAL VOLUME 11, NUMBER 5 1 SEPTEMBER 196$ Professor Robert Kenney, Mr. Howard rate for different masses of the intermediate Goldberg, and Mr. James Vale and the crew of the cyclotron, boson. spectrum The end point of the neutrino from the 184-in. cyclotron is -250 MeV, and whose full cooperation the run would without not We are also grateful with a have been possible. with this energy in collision to neutrinos Mr. Philip Beilin, Mr. Ned Dairiki, and Mr. Rob- proton would produce a boson of mass stationary ert Shafer for their help in running the experiment. equal to 2270m~. However, with the momentum boson masses distribution in the nucleus, higher but only a small fraction of the may be attained, so the rate of events can participate, protons *This work was done under the auspices of the U. S. falls off rapidly. Atomic Energy Commission pro- Because of the low energy of the neutrinos 'Clyde L. Cowan, Bull. Am. Phys. Soc. 8, 383 (1963); only a rather con- duced at the 1S4-in. cyclotron, and (private communication). Phys. Bev. Letters 4, 378 (1960). 2Toichino Kinoshita, limit of 2130m can be placed on the servative and S. Watanabe, ST. Tanikawa Phys. Bev. 113, 1344 mass of the intermediate boson. 1963: Kerr’s solution (19593. We would like to thank Professor Luis Alvarez 4Hugo B. Rugge, Lawrence Radiation Laboratory for suggesting and showing a this measurement Report UCBL-10252, 20 May 1962 (unpublished). and also Profes- Phys. Rev. Lett. 11 (1963) 237-238 in its progress, ~Richard J. Kurz, keen interest Lawrence Laboratory Radiation sor Clyde Cowan for communicating his results Report UCBL-10564, 15 November 1962 (unpublished). Our thanks are due before their publication. 6Howard Goldberg (private communication). FIELD OF A SPINNING MASS AS AN EXAMPLE GRAVITATIONAL GF ALGEBRAICALLY SPECIAL METRICS Roy P. Kerr* of Texas, Austin, Texas and Aerospace Research Laboratories, Air Force Base, Ohio University Wright-Patterson 26 July 1963) (Received and Sachs' have proved g is a complex coordinate, a dot denotes that the alge- where Goldberg of Einstein's with respect to g, and the operator braically special solutions differentiation empty- D is defined are characterized space field equations by the by existence of a geodesic and shear-free ray con- D = 8/st; - Qs/su. Among these spaces are the plane- gruence, A&. P is real, whereas metrics' fronted waves and the Robinson- Trautman Q and m (which is defined to for which the congruence diver- be m, +im, ) are complex. They are all independ- has nonvanishing but is hypersurface ent of the coordinate ~. L is defined gence, orthogonal. by the class of solu- In this note we shall present 6 =Im(P 'D~Q). is diverging, tions for which the congruence and is not necessarily There are two natural hypersurface orthogonal. The choices that can be made (A) P can be for the coordinate only previously known example of the general system. Either ds 2 = − ( ∆ − a 2 sin 2 θ ) dt 2 − 2 a sin 2 θ ( r 2 + a 2 − ∆ ) in which case 0 is complex, case is the Newman, met- chosen to be unity, Unti, and Tamburino dtd φ rics, 'which is of Petrov Type D, and possesses or (B) Q can be taken pure imaginary, with P dif- Σ Σ a four-dimensional group of isometrics. In case (A), the field e(luations Σ = r 2 + a 2 cos 2 θ ferent from unity. ✓ ( r 2 + a 2 ) 2 − ∆ a 2 sin 2 θ If we introduce a complex null tetrad (t~ is the are ◆ sin 2 θ d φ 2 + Σ ∆ dr 2 + Σ d θ 2 of t), with complex conjugate + ∆ = r 2 − 2 GMr + a 2 (m -D*D*DQ) = Is DQI', Σ = 2tt*+ 2m'', Q ds Im(m -D*D*DQ) =0, then the coordinate may be chosen so that system (in the coordinates introduced by Robert H. Boyer and Richard W. Lindquist, in 1967, D*m = 3mb. (4) t =P(r+f~)dg, system is probably J. Math. Phys. 8 (1967) 265 ) The second coordinate better, )t =du+2Re(Qdg), but it gives more complicated I dr — 2 Re[[(r — ie))) ~ ())ii]d([=+(rPi')' field equations. It will be observed that if m is zero then the field equations are integrable. These spaces '+, +6 +Re[P 'D(o*lnP correspond to the Type-III and null spaces with h*)+]

1967: Israel’s theorem REVIEW P H YS ICAL 164, NUM BER VOLUME 25 DECEMBER f 967 5 Event Horizons in Static Vacuum Space-Times WERNER ISRAEL mathematics Department, of Atberta, Alberta, Canada Unzoerszty QSd DubLin Instztute for Adoanced Studzes, Dublin, Ireland (Received 27 April 1967) The following all static, asymptotically theorem is established. Among Rat vacuum space-times with connected surfaces g00=constant, closed simply equipotential the Schwarzschild is the only one solution gpp = 0. Thus there exists no static asymmetric which has a nonsingular infinite-red-shift surface perturbation of the Schwarzschild due to internal (e. g. , a quadrupole manifold sources moment) which will preserve a regular event horizon. Possible implications of this result for asymmetric gravitational collapse are briefly discussed. horizon. It is the aim of this paper to give a precise 1. INTRODUCTION (see Sec. 4) and proof of this conjecture. formulation HK peculiar of the infinite-red-shift properties surface g00= 0 (r = 2trt) in Schwarzschild's spheri- 2. IMBEDDING FORMULAS cally vacuum field, and the question analo- qf whether space-times' ' have gous surfaces exist in asymmetric Ke begin by collecting some general formulas Israel’s theorem: for the become a focus of attention in connection with recent immersion of hypersurfaces in an (st+1)-dimensional space. ' An asymptotically flat static vacuum spacetime that is non-singular on and outside an event horizon, interest in gravitational collapse. Riemannian For static fields (to which we confine ourselves in this Let the equations must be isometric to the Schwarzschild spacetime. the history of an infinite-red-shift surface can paper) x =x'(e', as a 3-space S on which , e", V), V=const be de6ned the Killing vector (2) becomes null. Then S itself is null, and acts as a station- hypersurf ace 2 an for causal inQuence. ' represent orientable unit with ary unidirectional membrane normal n; the effect on S In the special case of axial symmetry, +1 (spacelike n) out explicitly. ' " A fundamental of static perturbations n n=e(n)= of the Schwarzschild metric can n e&;&=0, (3) — 1 (timelike n) be worked diEerence according to whether the source of the pertur- emerges The e holonomic base vectors to Z, e(;) tangent bation is external or internal. If the perturbation is due with components solely to the presence of exterior bodies, and if it is not — e&;& — ex /Be' (4) too strong (e. g. , if the spherically symmetric particle is by a ring of mass some distance encircled the away), are such that an infinitesimal displacement in Z has effect is merely to distort S while preserving its essential the form e(;~d8'. event horizon. ' On features as a nonsingular qualitative The Gauss-%eingarten relations a quadrupole the other hand, superimposing moment be&, &o/M'= — q, no matter how small, causes S to become singular. ' (The e(n)E, srto+1' (5) 0'e&, &" square of the four-dimensional Riemann tensor diverges decompose the absolute derivative b/bee Lreferred to the according to (rt+1)-dimensional metric( of the vector with e&, & g00 ~ 0) respect to the (st+1)-dimensional fe&, &, n). They basis as RABCDR &I /g00 ~ be regarded as defining the extrinsic may curvature tensor E ~ and the intrinsic one connection I', ~' of Z. static A study of small (linearized) perturbations From (3) and (5). to of the Schwarzschild manifold4 points similar conclusions. erto/be'= E;e&. &o. Partial results of this that suggest strongly type The Ricci commutation Schwarzschild's relations solution is uniquely distinguished all static, asymptotically Bat, vacuum 6elds by among — ( e expel& e e e the fact that it alone a nonsingular possesses event e& &"= -R"-ere&. & (7) (be ee' ee'ee be'be 'A. G. Doroshkevich, and I. D. Xovikov, Ya. B. Zel'dovich, Zh. Eksperim. i Teor. Fiz. 49, 170 (1965) (English transl. : Soviet Phys. — the aid of (5) and (6), to the equations JETP 22, 122 (1966) j. lead, with of 'C. V. Vishveshwara, University of Maryland 1966 Report, 1 to I+1. Italic indices (unpublished). Greek indices run from distinguish ' L. A. Mysak and G. Szekeres, Can. J. Phys. 44, 617 (1966); on the imbedded quantities defined manifold (e. g. , E~f„g is the W. Israel and K. A. Khan, Nuovo Cimento 33, 331 (1964). tensor of Z) and have the range 1 — intrinsic curvature sz. Covariant " T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 (1957). differentiation with respect to the (n+1)-dimensional or n-dimen- " Q. Krez and N. Rosen, Bull. Res. Council Israel Fs, 47 (1959). bv a stroke or a semicolon, sional metric is denoted respectively. 164 1776

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1967-...: The electro-vacuum uniqueness theorems Phys. Rev. Lett. 26 (1971) 331-333 Vacuum: Kerr Kerr 1963 Uniqueness Israel 1967; Carter 1971; 1 Z d 4 x √− gR D.C. Robinson, Phys. Rev. Lett. 34, 905 (1975). S = 16 π

1971: Wheeler and Ruffini coin the expression “a black hole has no hair” R. Ruffini and John Wheeler, “Introducing the black hole”, Physics Today, January 1971, Pages 30-41