Geodesics in Some Exact Rotating Solutions of Einstein's Equations
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Gravitational Lensing from a Spacetime Perspective
Gravitational Lensing from a Spacetime Perspective Volker Perlick Physics Department Lancaster University Lancaster LA1 4YB United Kingdom email: [email protected] Abstract The theory of gravitational lensing is reviewed from a spacetime perspective, without quasi-Newtonian approximations. More precisely, the review covers all aspects of gravita- tional lensing where light propagation is described in terms of lightlike geodesics of a metric of Lorentzian signature. It includes the basic equations and the relevant techniques for calcu- lating the position, the shape, and the brightness of images in an arbitrary general-relativistic spacetime. It also includes general theorems on the classification of caustics, on criteria for multiple imaging, and on the possible number of images. The general results are illustrated with examples of spacetimes where the lensing features can be explicitly calculated, including the Schwarzschild spacetime, the Kerr spacetime, the spacetime of a straight string, plane gravitational waves, and others. arXiv:1010.3416v1 [gr-qc] 17 Oct 2010 1 1 Introduction In its most general sense, gravitational lensing is a collective term for all effects of a gravitational field on the propagation of electromagnetic radiation, with the latter usually described in terms of rays. According to general relativity, the gravitational field is coded in a metric of Lorentzian signature on the 4-dimensional spacetime manifold, and the light rays are the lightlike geodesics of this spacetime metric. From a mathematical point of view, the theory of gravitational lensing is thus the theory of lightlike geodesics in a 4-dimensional manifold with a Lorentzian metric. The first observation of a ‘gravitational lensing’ effect was made when the deflection of star light by our Sun was verified during a Solar eclipse in 1919. -
Spacetimes with Singularities Ovidiu Cristinel Stoica
Spacetimes with Singularities Ovidiu Cristinel Stoica To cite this version: Ovidiu Cristinel Stoica. Spacetimes with Singularities. An. Stiint. Univ. Ovidius Constanta, Ser. Mat., 2012, pp.26. hal-00617027v2 HAL Id: hal-00617027 https://hal.archives-ouvertes.fr/hal-00617027v2 Submitted on 30 Nov 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Spacetimes with Singularities ∗†‡ Ovidiu-Cristinel Stoica Abstract We report on some advances made in the problem of singularities in general relativity. First is introduced the singular semi-Riemannian geometry for met- rics which can change their signature (in particular be degenerate). The standard operations like covariant contraction, covariant derivative, and constructions like the Riemann curvature are usually prohibited by the fact that the metric is not invertible. The things become even worse at the points where the signature changes. We show that we can still do many of these operations, in a different framework which we propose. This allows the writing of an equivalent form of Einstein’s equation, which works for degenerate metric too. Once we make the singularities manageable from mathematical view- point, we can extend analytically the black hole solutions and then choose from the maximal extensions globally hyperbolic regions. -
Generalizations of the Kerr-Newman Solution
Generalizations of the Kerr-Newman solution Contents 1 Topics 1035 1.1 ICRANetParticipants. 1035 1.2 Ongoingcollaborations. 1035 1.3 Students ............................... 1035 2 Brief description 1037 3 Introduction 1039 4 Thegeneralstaticvacuumsolution 1041 4.1 Line element and field equations . 1041 4.2 Staticsolution ............................ 1043 5 Stationary generalization 1045 5.1 Ernst representation . 1045 5.2 Representation as a nonlinear sigma model . 1046 5.3 Representation as a generalized harmonic map . 1048 5.4 Dimensional extension . 1052 5.5 Thegeneralsolution . 1055 6 Tidal indicators in the spacetime of a rotating deformed mass 1059 6.1 Introduction ............................. 1059 6.2 The gravitational field of a rotating deformed mass . 1060 6.2.1 Limitingcases. 1062 6.3 Circularorbitsonthesymmetryplane . 1064 6.4 Tidalindicators ........................... 1065 6.4.1 Super-energy density and super-Poynting vector . 1067 6.4.2 Discussion.......................... 1068 6.4.3 Limit of slow rotation and small deformation . 1069 6.5 Multipole moments, tidal Love numbers and Post-Newtonian theory................................. 1076 6.6 Concludingremarks . 1077 7 Neutrino oscillations in the field of a rotating deformed mass 1081 7.1 Introduction ............................. 1081 7.2 Stationary axisymmetric spacetimes and neutrino oscillation . 1082 7.2.1 Geodesics .......................... 1083 1033 Contents 7.2.2 Neutrinooscillations . 1084 7.3 Neutrino oscillations in the Hartle-Thorne metric . 1085 7.4 Concludingremarks . 1088 8 Gravitational field of compact objects in general relativity 1091 8.1 Introduction ............................. 1091 8.2 The Hartle-Thorne metrics . 1094 8.2.1 The interior solution . 1094 8.2.2 The Exterior Solution . 1096 8.3 The Fock’s approach . 1097 8.3.1 The interior solution . -
Singularities, Black Holes, and Cosmic Censorship: a Tribute to Roger Penrose
Foundations of Physics (2021) 51:42 https://doi.org/10.1007/s10701-021-00432-1 INVITED REVIEW Singularities, Black Holes, and Cosmic Censorship: A Tribute to Roger Penrose Klaas Landsman1 Received: 8 January 2021 / Accepted: 25 January 2021 © The Author(s) 2021 Abstract In the light of his recent (and fully deserved) Nobel Prize, this pedagogical paper draws attention to a fundamental tension that drove Penrose’s work on general rela- tivity. His 1965 singularity theorem (for which he got the prize) does not in fact imply the existence of black holes (even if its assumptions are met). Similarly, his versatile defnition of a singular space–time does not match the generally accepted defnition of a black hole (derived from his concept of null infnity). To overcome this, Penrose launched his cosmic censorship conjecture(s), whose evolution we discuss. In particular, we review both his own (mature) formulation and its later, inequivalent reformulation in the PDE literature. As a compromise, one might say that in “generic” or “physically reasonable” space–times, weak cosmic censorship postulates the appearance and stability of event horizons, whereas strong cosmic censorship asks for the instability and ensuing disappearance of Cauchy horizons. As an encore, an “Appendix” by Erik Curiel reviews the early history of the defni- tion of a black hole. Keywords General relativity · Roger Penrose · Black holes · Ccosmic censorship * Klaas Landsman [email protected] 1 Department of Mathematics, Radboud University, Nijmegen, The Netherlands Vol.:(0123456789)1 3 42 Page 2 of 38 Foundations of Physics (2021) 51:42 Conformal diagram [146, p. 208, Fig. -
Generalizations of the Kerr-Newman Solution
Generalizations of the Kerr-Newman solution Contents 1 Topics 663 1.1 ICRANetParticipants. 663 1.2 Ongoingcollaborations. 663 1.3 Students ............................... 663 2 Brief description 665 3 Introduction 667 4 Thegeneralstaticvacuumsolution 669 4.1 Line element and field equations . 669 4.2 Staticsolution ............................ 671 5 Stationary generalization 673 5.1 Ernst representation . 673 5.2 Representation as a nonlinear sigma model . 674 5.3 Representation as a generalized harmonic map . 676 5.4 Dimensional extension . 680 5.5 Thegeneralsolution ........................ 683 6 Static and slowly rotating stars in the weak-field approximation 687 6.1 Introduction ............................. 687 6.2 Slowly rotating stars in Newtonian gravity . 689 6.2.1 Coordinates ......................... 690 6.2.2 Spherical harmonics . 692 6.3 Physical properties of the model . 694 6.3.1 Mass and Central Density . 695 6.3.2 The Shape of the Star and Numerical Integration . 697 6.3.3 Ellipticity .......................... 699 6.3.4 Quadrupole Moment . 700 6.3.5 MomentofInertia . 700 6.4 Summary............................... 701 6.4.1 Thestaticcase........................ 702 6.4.2 The rotating case: l = 0Equations ............ 702 6.4.3 The rotating case: l = 2Equations ............ 703 6.5 Anexample:Whitedwarfs. 704 6.6 Conclusions ............................. 708 659 Contents 7 PropertiesoftheergoregionintheKerrspacetime 711 7.1 Introduction ............................. 711 7.2 Generalproperties ......................... 712 7.2.1 The black hole case (0 < a < M) ............. 713 7.2.2 The extreme black hole case (a = M) .......... 714 7.2.3 The naked singularity case (a > M) ........... 714 7.2.4 The equatorial plane . 714 7.2.5 Symmetries and Killing vectors . 715 7.2.6 The energetic inside the Kerr ergoregion . -
The Physics of Schwarzschild's Original 1916 Metric Solution And
The Physics of Schwarzschild’s Original 1916 Metric Solution and Elimination of the Coordinate Singularity Anomaly By Robert Louis Kemp Super Principia Mathematica The Rage to Master Conceptual & Mathematical Physics www.SuperPrincipia.com www.Blog.Superprincipia.com Flying Car Publishing Company P.O Box 91861 Long Beach, CA 90809 Abstract This paper is a mathematical treatise and historical perspective of Karl Schwarzschild’s original 1916 solution and description of a Non-Euclidean Spherically Symmetric Metric equation. In the modern literature of the Schwarzschild Metric equation, it is described as predicting an output of a “Coordinate Singularity” anomaly located at the surface of the Black Hole Event Horizon. However, in Schwarzschild’s original 1916 metric solution there is not a “Coordinate Singularity” located at the Black Hole Event Horizon, but there is an actual quantitative value for space and time, that is predicted there. This paper and work compares the Schwarzschild Spherically Symmetric Metric equation original 1916 results with the results predicted by the modern literature on the Schwarzschild Metric. This work also describes conceptually the physics behind the Gaussian Distance Curvature and Reduction Density equation that Schwarzschild used to eliminate and avoid a “Coordinate Singularity”. This work reveals that Schwarzschild’s original 1916 solution, predicts that the Inertial Mass Volume Density, Escape Velocity, and gravitational field acceleration is reduced at all points in the inhomogeneous gradient gravitational -
PHY390, the Kerr Metric and Black Holes
PHY390, The Kerr Metric and Black Holes James Lattimer Department of Physics & Astronomy 449 ESS Bldg. Stony Brook University April 1, 2021 Black Holes, Neutron Stars and Gravitational Radiation [email protected] James Lattimer PHY390, The Kerr Metric and Black Holes What Exactly is a Black Hole? Standard definition: A region of space from which nothing, not even light, can escape. I Where does the escape velocity equal the speed of light? r 2GMBH vesc = = c RSch This defines the Schwarzschild radius RSch to be 2GMBH M RSch = 2 ' 3 km c M I The event horizon marks the point of no return for any object. I A black hole is black because it absorbs everything incident on the event horizon and reflects nothing. I Black holes are hypothesized to form in three ways: I Gravitational collapse of a star I A high energy collision I Density fluctuations in the early universe I In general relativity, the black hole's mass is concentrated at the center in a singularity of infinite density. James Lattimer PHY390, The Kerr Metric and Black Holes John Michell and Black Holes The first reference is by the Anglican priest, John Michell (1724-1793), in a letter written to Henry Cavendish, of the Royal Society, in 1783. He reasoned, from observations of radiation pressure, that light, like mass, has inertia. If gravity affects light like its mass equivalent, light would be weakened. He argued that a Sun with 500 times its radius and the same density would be so massive that it's escape velocity would exceed light speed. -
Complex Structures in the Dynamics of Kerr Black Holes
Complex structures in the dynamics of Kerr black holes Ben Maybee [email protected] Higgs Centre for Theoretical Physics, The University of Edinburgh YTF 20, e-Durham In a certain film… Help! I need to precisely scatter off a spinning black hole and reach a life supporting planet… Spinning black holes are very special…maybe some insight will simplify things? Christopher Nolan and Warner Bros Pictures, 2013 In a certain film… Okay, you’re good with spin; what should I try? Black holes are easy; the spin vector 푎휇 determines all multipoles. Try using the Newman- Janis shift… Hansen, 1974 Outline 1. The Newman-Janis shift 2. Kerr black holes as elementary particles? 3. Effective actions and complex worldsheets 4. Spinor equations of motion 5. Discussion The Newman-Janis shift Simple statement: Kerr metric, with spin parameter 푎, can be Newman & obtained from Schwarzschild by complex Janis, 1965 transformation 푧 → 푧 + 푖푎 Kerr and Schwarzschild are both exact Kerr-Schild solutions to GR: Kerr & Schild, 1965 Background metric Null vector, Scalar function Classical double copy: ∃ gauge theory solution Monteiro, O’Connell & White, 2014 → EM analogue of Kerr: Kerr! The Newman-Janis shift Metrics: Under 푧 → 푧 + 푖푎, 푟2 → 푟ǁ + 푖푎 cos 휃 2. Why did that just happen??? Scattering amplitudes and spin No hair theorem → all black hole multipoles specified by Hansen, 1974 mass and spin vector. Wigner classification → single particle Wigner, states specified by mass and spin s. 1939 Little group irreps Any state in a little group irrep can be represented using chiral spinor- helicity variables: Distinct spinor reps Arkani-Hamed, Huang & Huang, 2017 Little group metric Massless: 푈(1) → 훿푖푗 2-spinors Massive: 푆푈 2 → 휖푗푖 3-point exponentiation Use to calculate amplitudes for any spin s: Chiral Spin = parameter e.g. -
Problems in the Schwarzschild Geometry
Problems in the Schwarzschild geometry April 4, 2015 Start work on the problems below. See the notes on Schwarzschild geodesics to get started. Help is available. Summary of geodesics: We have, for timelike curves: ! dt 1 − 2M = u0 r0 dτ 0 2M 1 − r d' L = dτ r2 r dr 2M L2 2ML2 = 2E + − + dτ r r2 r3 where 2 d' L = r0 dτ 0 " 2 # 1 2M 02 E = − 1 − 1 − u0 2 r0 Problems: dr 1. Find the form of dτ for spacelike and null geodesics. 2. Find the proper time required for a particle to fall from radius r0 > 2M to r = 2M. Evaluate the time numerically for a solar mass black hole, and a galactic black hole with mass 107 times the mass of the sun. 3. Find the proper time required for a particle to fall from radius r0 = 2M to r = 0. Evaluate for solar mass and 107 solar mass black holes. Find predictions for the three classical tests of general relativity. For all three problems, 2M assume r 1. dr 1. Perihelion advance of Mercury. From the orbital equation derived by integrating d' (given in the Notes), find the angle between adjacent minima of r ('). The amount by which this exceeds 2π is the perihelion advance. 1 2. Gravitational red shift. Use your expression for null geodesics, restricted to outward radial motion !0 (L = 0) to find the fractional change in frequency ! (or wavelength) of the light. Remember that the α E ! momentum 4-vector, p = c ; p = ~ c ; k is tangent to the null curve. -
Lense-Thirring Precession in Strong Gravitational Fields
1 Lense-Thirring Precession in Strong Gravitational Fields Chandrachur Chakraborty Tata Institute of Fundamental Research, Mumbai 400005, India Abstract The exact frame-dragging (or Lense-Thirring (LT) precession) rates for Kerr, Kerr-Taub-NUT (KTN) and Taub-NUT spacetimes have been derived. Remarkably, in the case of the ‘zero an- gular momentum’ Taub-NUT spacetime, the frame-dragging effect is shown not to vanish, when considered for spinning test gyroscope. In the case of the interior of the pulsars, the exact frame- dragging rate monotonically decreases from the center to the surface along the pole and but it shows an ‘anomaly’ along the equator. Moving from the equator to the pole, it is observed that this ‘anomaly’ disappears after crossing a critical angle. The ‘same’ anomaly can also be found in the KTN spacetime. The resemblance of the anomalous LT precessions in the KTN spacetimes and the spacetime of the pulsars could be used to identify a role of Taub-NUT solutions in the astrophysical observations or equivalently, a signature of the existence of NUT charge in the pulsars. 1 Introduction Stationary spacetimes with angular momentum (rotation) are known to exhibit an effect called Lense- Thirring (LT) precession whereby locally inertial frames are dragged along the rotating spacetime, making any test gyroscope in such spacetimes precess with a certain frequency called the LT precession frequency [1]. This frequency has been shown to decay as the inverse cube of the distance of the test gyroscope from the source for large enough distances where curvature effects are small, and known to be proportional to the angular momentum of the source. -
Kerr Black Hole and Rotating Wormhole
Kerr Fest (Christchurch, August 26-28, 2004) Kerr black hole and rotating wormhole Sung-Won Kim(Ewha Womans Univ.) August 27, 2004 • INTRODUCTION • STATIC WORMHOLE • ROTATING WORMHOLE • KERR METRIC • SUMMARY AND DISCUSSION 1 Introduction The wormhole structure: two asymptotically flat regions + a bridge • To be traversable: exotic matter which violates the known energy conditions • Exotic matter is also an important issue on dark energy which accelerates our universe. • Requirement of the more general wormhole model - ‘rotating worm- hole’ • Two-dimensional model of transition between black hole and worm- hole ⇒ Interest on the general relation between black hole and wormhole 2 Static Wormhole(Morris and Thorne, 1988) The spacetime metric for static wormhole dr2 ds2 = −e2Λ(r)dt2 + + r2(dθ2 + sin2 θdφ2) 1 − b(r)/r Λ(r): the lapse function b(r): wormhole shape function At t =const. and θ = π/2, the 2-curved surface is embedded into 3-dimensional Euclidean space dr2 d˜s2 = + r2dφ2 = dz2 + dr2 + r2dφ2 1 − b(r)/r Flare-out condition d2r b − b0r = > 0 dz2 2b2 With new radial coordinate l ∈ (−∞, ∞) (proper distance), while r > b ds2 = −e2Λ(l)dt2 + dl2 + r(l)2(dθ2 + sin2 θdφ2) where dl b−1/2 = ± 1 − dr r 3 Rotating Wormhole(Teo, 1998) The spacetime in which we are interested will be stationary and axially symmetric. The most general stationary and axisymmetric metric can be written as 2 2 2 i j ds = gttdt + 2gtφdtdφ + gφφdφ + gijdx dx , where the indices i, j = 1, 2. Freedom to cast the metric into spherical polar coordinates by setting 2 2 g22 = gφφ/ sin x → The metric of the rotating wormhole as: ds2 = −N 2dt2 + eµdr2 + r2K2dθ2 + r2K2 sin2 θ[dφ2 − Ωdt]2 dr2 = −N 2dt2 + + r2K2dθ2 + r2K2 sin2 θ[dφ2 − Ωdt]2, 1 − b(r)/r where Ω is the angular velocity dφ/dt acquired by a particle that falls freely from infinity to the point (r, θ), and which gives rise to the well- known dragging of inertial frames or Lense-Thirring effect in general relativity. -
The Newman Janis Algorithm: a Review of Some Results
Thirteenth International Conference on Geometry, Integrability and Quantization June 3–8, 2011, Varna, Bulgaria Ivaïlo M. Mladenov, Andrei Ludu and Akira Yoshioka, Editors Avangard Prima, Sofia 2012, pp 159–169 doi: 10.7546/giq-12-2011-159-169 THE NEWMAN JANIS ALGORITHM: A REVIEW OF SOME RESULTS ROSANGELA CANONICO, LUCA PARISI and GAETANO VILASI Dipartimento di Fisica “E.R. Caianiello”, Università di Salerno, I-84084 Fisciano (SA), INFN, Sezione di Napoli, GC di Salerno, Italy Abstract. In this paper we review some interesting results obtained through the Newman-Janis algorithm, a solution generating technique employed in General Relativity. We also describe the use of this algorithm in different theories, namely f(R), Einstein-Maxwell-Dilaton gravity, Braneworld, Born- Infeld Monopole and we focus on the validity of the results. 1. Introduction In order to find new solutions of Einstein field equations, several methods were introduced, such as the Newman-Penrose formalism and a technique founded by Newman and Janis [15]. In the recent years, many papers have appeared focusing on the Newman-Janis algorithm, considered as a solution generating technique which provides metrics of reduced symmetries from symmetric ones. Our aim is to give a summary of the use of this method and to review the most interesting applications in different gravity theories. The outline of this paper is as follows: in Section 2, we present the general pro- cedure of the Newman-Janis algorithm (henceforth NJA). In Section 3, we review the interesting attempts made in General Relativity, focusing on the most intrigu- ing results. In Section 4, we have analyzed how to apply this technique to the f(R) modified theories of gravity, by following the procedure used in GR.