Academic literature on the topic 'Wave equations on black hole spacetimes'
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Journal articles on the topic "Wave equations on black hole spacetimes"
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Pailas, Theodoros. "“Time”-Covariant Schrödinger Equation and the Canonical Quantization of the Reissner–Nordström Black Hole." Quantum Reports 2, no. 3 (August 7, 2020): 414–41. http://dx.doi.org/10.3390/quantum2030029.
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A “time”-covariant Schrödinger equation is defined for the minisuperspace model of the Reissner–Nordström (RN) black hole, as a “hybrid” between the “intrinsic time” Schrödinger and Wheeler–DeWitt (WDW) equations. To do so, a reduced, regular, and “time(r)”-dependent Hamiltonian density was constructed, without “breaking” the re-parametrization covariance r→f(r˜). As a result, the evolution of states with respect to the parameter r and the probabilistic interpretation of the resulting quantum description is possible, while quantum schemes for different gauge choices are equivalent by construction. The solutions are found for Dirac’s delta and Gaussian initial states. A geometrical interpretation of the wavefunctions is presented via Bohm analysis. Alongside this, a criterion is presented to adjudicate which, between two singular spacetimes, is “more” or “less” singular. Two ways to adjudicate the existence of singularities are compared (vanishing of the probability density at the classical singularity and semi-classical spacetime singularity). Finally, an equivalence of the reduced equations with those of a 3D electromagnetic pp-wave spacetime is revealed.
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BINI, DONATO, CHRISTIAN CHERUBINI, ROBERT T. JANTZEN, and REMO RUFFINI. "DE RHAM WAVE EQUATION FOR TENSOR VALUED p-FORMS." International Journal of Modern Physics D 12, no. 08 (September 2003): 1363–84. http://dx.doi.org/10.1142/s0218271803003785.
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The de Rham Laplacian Δ (dR) for differential forms is a geometric generalization of the usual covariant Laplacian Δ, and it may be extended naturally to tensor-valued p-forms using the exterior covariant derivative associated with a metric connection. Using it the wave equation satisfied by the curvature tensors in general relativity takes its most compact form. This wave equation leads to the Teukolsky equations describing integral spin perturbations of black hole spacetimes.
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ATIQUR RAHMAN, M. "NUMERICAL SOLUTIONS OF IDEAL TWO-FLUID TRANSVERSE EQUATIONS VERY CLOSED TO THE EVENT HORIZON OF REISSNER–NORDSTRÖM–ANTI-DE SITTER BLACK HOLE." International Journal of Modern Physics D 22, no. 09 (June 26, 2013): 1350063. http://dx.doi.org/10.1142/s0218271813500636.
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The Alfvén and high frequency electromagnetic waves propagating in a relativistic two-fluid plasma influenced by the gravitational field of the Reissner–Nordström–anti-de Sitter (RNAdS) black hole have been investigated applying 3+1 split of spacetime. The extremal cases also discussed here based on the simple observation that the near-horizon geometry of a static extremal black hole contains two-dimensional anti-de Sitter factor even in the presence of positive cosmological constant. We reformulate the relativistic two-fluid equations with the set of simultaneous linear equations for the perturbations. We derive the dispersion relation for these waves and solve numerically for the wave number k.
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Momeni, Davood. "Covariant quantum-mechanical scattering via Stueckelberg–Horwitz–Piron theory." International Journal of Modern Physics D 29, no. 01 (January 2020): 2050008. http://dx.doi.org/10.1142/s021827182050008x.
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Based on the Stueckelberg–Horwitz–Piron theory of covariant quantum mechanics on curved spacetime, we solved the wave equations for a charged covariant harmonic oscillator in the background of a charged static spherically symmetric black hole. Using Green’s functions, we found an asymptotic form for the wave function in the lowest mode ([Formula: see text]-mode) and in higher moments. It has been proven that for [Formula: see text]-wave, in a definite range of solid angles, the differential cross-section depends effectively to on the magnetic and electric charges of the black hole.
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Al-Badawi, A., and I. sakalli. "Dirac and Klein–Gordon–Fock equations in Grumiller’s spacetime." International Journal of Geometric Methods in Modern Physics 15, no. 04 (March 13, 2018): 1850051. http://dx.doi.org/10.1142/s0219887818500512.
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We study the Dirac and the chargeless Klein–Gordon–Fock equations in the geometry of Grumiller’s spacetime that describes a model for gravity of a central object at large distances. The Dirac equation is separated into radial and angular equations by adopting the Newman–Penrose formalism. The angular part of the both wave equations are analytically solved. For the radial equations, we managed to reduce them to one dimensional Schrödinger-type wave equations with their corresponding effective potentials. Fermions’s potentials are numerically analyzed by serving their some characteristic plots. We also compute the quasinormal frequencies of the chargeless and massive scalar waves. With the aid of those quasinormal frequencies, Bekenstein’s area conjecture is tested for the Grumiller black hole. Thus, the effects of the Rindler acceleration on the waves of fermions and scalars are thoroughly analyzed.
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Rodríguez, J. F., J. A. Rueda, and R. Ruffini. "Strong-field gravitational-wave emission in Schwarzschild and Kerr geometries: some general considerations." EPJ Web of Conferences 168 (2018): 02006. http://dx.doi.org/10.1051/epjconf/201816802006.
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We have used the perturbations of the exact solutions of the Einstein equations to estimate the relativistic wave emission of a test particle orbiting around a black hole. We show how the hamiltonian equations of motion of a test particle augmented with the radiation-reaction force can establish a priori constraints on the possible phenomena occurring in the merger of compact objects. The dynamical evolution consists of a helicoidal sequence of quasi-circular orbits, induced by the radiation-reaction and the background spacetime. Near the innermost stable circular orbit the evolution is followed by a smooth transition and finally plunges geodesically into the black hole horizon. This analysis gives physical insight of the merger of two equal masses objects.
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Allahyari, Alireza, Javad T. Firouzjaee, and Reza Mansouri. "Gravitational collapse in the AdS background and the black hole formation." International Journal of Modern Physics D 25, no. 01 (January 2016): 1650005. http://dx.doi.org/10.1142/s021827181650005x.
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We study the time evolution of the Misner-Sharp mass and the apparent horizon for gravitational collapse of a massless scalar field in the [Formula: see text] spacetime for both cases of narrow and broad waves by numerically solving the Einstein’s equations coupled to a massless scalar field. This is done by relying on the full dynamics of the collapse including the concept of the dynamical horizon. It turns out that the Misner-Sharp mass is everywhere constant except for a rapid change across a thin shell defined by the density profile of the collapsing wave. By studying the evolution of the apparent horizon, indicating the formation of a black hole at different times we see how asymptotically an event horizon forms. The dependence of the thermalization time on the radius of the initial black hole event horizon is also studied.
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Bokhari, Ashfaque H., A. H. Kara, B. B. I. Gadjagboui, and Ghulam Shabbir. "Symmetries and conservation laws of some asymptotically symmetric spacetimes of interest in gravitational waves." International Journal of Geometric Methods in Modern Physics 16, no. 10 (October 2019): 1950152. http://dx.doi.org/10.1142/s0219887819501524.
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In this paper, we discuss symmetries and the corresponding conservation laws of certain exact solutions of the Einstein field equations (EFEs) representing a Schwarzschild black hole and gravitational waves in asymptotically flat space times. Of particular interest are symmetries of asymptotically flat spacetimes because they admit a property that identifies them for the existence of gravitational waves there. In the light of this fact, we discuss symmetry algebras of a few recently published solutions of Einstein equations in asymptotically flat metrics. Given the fact that gravitational waves are of great interest in relativity, we focus in this paper on finding the type of symmetries they admit and their corresponding conservation laws. We also show how these symmetries are radically different from the other well-known symmetries and present necessary condition that distinguishes them.
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Tokgöz, Gülni̇hal, and İzzet Sakallı. "Fermion clouds around z = 0 Lifshitz black holes." International Journal of Geometric Methods in Modern Physics 17, no. 09 (August 2020): 2050143. http://dx.doi.org/10.1142/s0219887820501431.
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In this work, the Dirac equation is studied in the [Formula: see text] Lifshitz black hole ([Formula: see text]LBH) spacetime. The set of equations representing the Dirac equation in the Newman–Penrose (NP) formalism is decoupled into a radial set and an angular set. The separation constant is obtained with the aid of the spin weighted spheroidal harmonics. The radial set of equations, which are independent of mass, is reduced to Zerilli equations (ZEs) with their associated potentials. In the near horizon (NH) region, these equations are solved in terms of the Bessel functions of the first and second kinds arising from the fermionic perturbation on the background geometry. For computing the boxed quasinormal modes (BQNMs) instead of the ordinary quasinormal modes (QNMs), we first impose the purely ingoing wave condition at the event horizon. Then, Dirichlet boundary condition (DBC) and Newmann boundary condition (NBC) are applied in order to get the resonance conditions. For solving the resonance conditions, we follow the Hod’s iteration method. Finally, Maggiore’s method (MM) is employed to derive the entropy/area spectra of the [Formula: see text]LBH which are shown to be equidistant.
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Most, Elias R., L. Jens Papenfort, Samuel D. Tootle, and Luciano Rezzolla. "On accretion discs formed in MHD simulations of black hole–neutron star mergers with accurate microphysics." Monthly Notices of the Royal Astronomical Society 506, no. 3 (July 1, 2021): 3511–26. http://dx.doi.org/10.1093/mnras/stab1824.
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ABSTRACT Remnant accretion discs formed in compact object mergers are an important ingredient in the understanding of electromagnetic afterglows of multimessenger gravitational-wave events. Due to magnetically and neutrino-driven winds, a significant fraction of the disc mass will eventually become unbound and undergo r-process nucleosynthesis. While this process has been studied in some detail, previous studies have typically used approximate initial conditions for the accretion discs, or started from purely hydrodynamical simulations. In this work, we analyse the properties of accretion discs formed from near equal-mass black hole–neutron star mergers simulated in general-relativistic magnetohydrodynamics in dynamical spacetimes with an accurate microphysical description. The post-merger systems were evolved until $120\, {\rm ms}$ for different finite-temperature equations of state and black hole spins. We present a detailed analysis of the fluid properties and of the magnetic-field topology. In particular, we provide analytic fits of the magnetic-field strength and specific entropy as a function of the rest-mass density, which can be used for the construction of equilibrium disc models. Finally, we evolve one of the systems for a total of $350\, \rm ms$ after merger and study the prospect for eventual jet launching. While our simulations do not reach this stage, we find clear evidence of continued funnel magnetization and clearing, a prerequisite for any jet-launching mechanism.
Dissertations / Theses on the topic "Wave equations on black hole spacetimes"
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Schlue, Volker. "Linear waves on higher dimensional Schwarzschild black holes and Schwarzschild de Sitter spacetimes." Thesis, University of Cambridge, 2012. https://www.repository.cam.ac.uk/handle/1810/243640.
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I study linear waves on higher dimensional Schwarzschild black holes and Schwarzschild de Sitter spacetimes. In the first part of this thesis two decay results are proven for general finite energy solutions to the linear wave equation on higher dimensional Schwarzschild black holes. I establish uniform energy decay and improved interior first order energy decay in all dimensions with rates in accordance with the 3 + 1-dimensional case. The method of proof departs from earlier work on this problem. I apply and extend the new physical space approach to decay of Dafermos and Rodnianski. An integrated local energy decay estimate for the wave equation on higher dimensional Schwarzschild black holes is proven. In the second part of this thesis the global study of solutions to the linear wave equation on expanding de Sitter and Schwarzschild de Sitter spacetimes is initiated. I show that finite energy solutions to the initial value problem are globally bounded and have a limit on the future boundary that can be viewed as a function on the standard cylinder. Both problems are related to the Cauchy problem in General Relativity.
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Dempsey, David. "Wave propagation on black hole spacetimes." Thesis, University of Sheffield, 2017. http://etheses.whiterose.ac.uk/18023/.
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This thesis studies the propagation of fundamental fields on black hole and black hole analogue spacetimes. We consider the scalar, electromagnetic, gravitational and Dirac fields, and their governing equations, in various scenarios. We initially consider an analogue gravity model, the draining bathtub vortex, that shares features with the Kerr black hole, such as a horizon and an ergoregion. We solve the wave equation approximately, via the eikonal approximation, and numerically, using the method of lines, and show that a point-like disturbance maps out the lightcone of the effective spacetime. The Schwarzschild and Kerr black hole spacetimes are then introduced and we discuss their key features. We solve the scalar wave equation for the black hole spacetimes and compare with the analogue spacetime. We then introduce the self force, the back reaction of a body's own field on its motion. The scalar self force on Kerr spacetime is calculated using the worldline integration method. This involves solving the scalar wave equation to find the Green function via the Kirchhoff representation and integrating over the entire past history of the worldline. The electromagnetic (EM) self force is calculated via the mode sum method. We use both analytical and numerical techniques to calculate EM self force for a particle held static outside of a Schwarzschild black hole. The gauge freedom of the gravitational self force is also discussed. We construct for eccentric orbits on Schwarzschild the spin precession invariant, a gauge invariant quantity. We compare the spin precession invariant calculated using numerical self force data with a post-Newtonian calculation. Finally we investigate the Dirac (fermionic) field in searching for the existence of bound states. We find that the solutions which satisfy the boundary conditions, obey a three-term recurrence-relation. Using continued-fraction methods we find a spectrum of quasi-bound states of the Dirac field exists.
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Sbierski, Jan. "On the initial value problem in general relativity and wave propagation in black-hole spacetimes." Thesis, University of Cambridge, 2014. https://www.repository.cam.ac.uk/handle/1810/248837.
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The first part of this thesis is concerned with the question of global uniqueness of solutions to the initial value problem in general relativity. In 1969, Choquet-Bruhat and Geroch proved, that in the class of globally hyperbolic Cauchy developments, there is a unique maximal Cauchy development. The original proof, however, has the peculiar feature that it appeals to Zorn’s lemma in order to guarantee the existence of this maximal development; in particular, the proof is not constructive. In the first part of this thesis we give a proof of the above mentioned theorem that avoids the use of Zorn’s lemma. The second part of this thesis investigates the behaviour of so-called Gaussian beam solutions of the wave equation - highly oscillatory and localised solutions which travel, for some time, along null geodesics. The main result of this part of the thesis is a characterisation of the temporal behaviour of the energy of such Gaussian beams in terms of the underlying null geodesic. We conclude by giving applications of this result to black hole spacetimes. Recalling that the wave equation can be considered a “poor man’s” linearisation of the Einstein equations, these applications are of interest for a better understanding of the black hole stability conjecture, which states that the exterior of our explicit black hole solutions is stable to small perturbations, while the interior is expected to be unstable. The last part of the thesis is concerned with the wave equation in the interior of a black hole. In particular, we show that under certain conditions on the black hole parameters, waves that are compactly supported on the event horizon, have finite energy near the Cauchy horizon. This result is again motivated by the investigation of the conjectured instability of the interior of our explicit black hole solutions.
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Civin, Damon. "Stability of charged rotating black holes for linear scalar perturbations." Thesis, University of Cambridge, 2015. https://www.repository.cam.ac.uk/handle/1810/247397.
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In this thesis, the stability of the family of subextremal Kerr-Newman space- times is studied in the case of linear scalar perturbations. That is, nondegenerate energy bounds (NEB) and integrated local energy decay (ILED) results are proved for solutions of the wave equation on the domain of outer communications. The main obstacles to the proof of these results are superradiance, trapping and their interaction. These difficulties are surmounted by localising solutions of the wave equation in phase space and applying the vector field method. Miraculously, as in the Kerr case, superradiance and trapping occur in disjoint regions of phase space and can be dealt with individually. Trapping is a high frequency obstruction to the proof whereas superradiance occurs at both high and low frequencies. The construction of energy currents for superradiant frequencies gives rise to an unfavourable boundary term. In the high frequency regime, this boundary term is controlled by exploiting the presence of a large parameter. For low superradiant frequencies, no such parameter is available. This difficulty is overcome by proving quantitative versions of mode stability type results. The mode stability result on the real axis is then applied to prove integrated local energy decay for solutions of the wave equation restricted to a bounded frequency regime. The (ILED) statement is necessarily degenerate due to the trapping effect. This implies that a nondegenerate (ILED) statement must lose differentiability. If one uses an (ILED) result that loses differentiability to prove (NEB), this loss is passed onto the (NEB) statement as well. Here, the geometry of the subextremal Kerr-Newman background is exploited to obtain the (NEB) statement directly from the degenerate (ILED) with no loss of differentiability.
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Babb, James Patrick. "The derivation and quasinormal mode spectrum of acoustic anti-de sitter black hole analogues." Thesis, 2013. http://hdl.handle.net/1828/4484.
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Dumb holes (also known as acoustic black holes) are fluid flows which include an "acoustic horizon:" a surface, analogous to a gravitational horizon, beyond which sound may pass but never classically return. Soundwaves in these flows will therefore experience "effective geometries" which are identical to black hole spacetimes up to a conformal factor. By adjusting the parameters of the fluid flow, it is possible to create an effective geometry which is conformal to the Anti-de Sitter black hole spacetime- a geometry which has recieved a great deal of attention in recent years due to its conjectured holographic duality to Conformal Field Theories. While we would not expect an acoustic analogue of the AdS-CFT correspondence to exist, this dumb hole provides a means, at least in principle, of experimentally testing the theoretical properties of the AdS spacetime. In particular, I have calculated the quasinormal mode spectrum of this acoustic geometry.Graduate
0986
0753
jpbabb@yahoo.ca
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Bonning, Erin Wells Matzner Richard A. "Computational and astrophysical studies of black hole spacetimes." 2004. http://wwwlib.umi.com/cr/utexas/fullcit?p3139194.
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Bonning, Erin Wells. "Computational and astrophysical studies of black hole spacetimes." Thesis, 2004. http://hdl.handle.net/2152/1212.
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"study of the continuous spectrum for wave propagation on Schwarzschild spacetime =: 史瓦兹西爾德時空中波動傳播之連續頻譜." 2002. http://library.cuhk.edu.hk/record=b5895981.
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Mak Ka Wai Charles.Thesis submitted in: October 2001.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2002.
Includes bibliographical references (leaves 89-91).
Text in English; abstracts in English and Chinese.
Mak Ka Wai Charles.
Chapter 1 --- Introduction --- p.1
Chapter 1.1 --- Overview of the Mathematical Framework --- p.2
Chapter 1.2 --- System of Interest --- p.7
Chapter 1.2.1 --- Klein-Gordon equation --- p.7
Chapter 1.2.2 --- QNM boundary conditions --- p.12
Chapter 1.3 --- Outline of This Thesis --- p.14
Chapter 2 --- Green's Function --- p.15
Chapter 2.1 --- "Formal Expression for G(x,y，w)" --- p.16
Chapter 2.2 --- "Leaver's Series Solution: An Analytic Expression for g(r, w)" --- p.17
Chapter 2.3 --- Location of the Cut --- p.22
Chapter 2.4 --- "Jaffe's Series Solution: An Analytic Expression for f(r,w)" --- p.23
Chapter 2.5 --- QNMs and Their Locations --- p.26
Chapter 2.5.1 --- Alternative definitions of QNM --- p.26
Chapter 2.5.2 --- Methods of searching for QNMs --- p.28
Chapter 2.5.3 --- Locations of QNMs --- p.29
Chapter 2.6 --- Green's Function and Eigenspectra --- p.30
Chapter 3 --- Normalization Function: Analytical Treatment --- p.34
Chapter 3.1 --- Definition and Properties --- p.34
Chapter 3.2 --- Analytic Approximations for --- p.36
Chapter 3.3 --- Polar Perturbations --- p.39
Chapter 4 --- Normalization Function: Numerical Treatment --- p.42
Chapter 4.1 --- "Numerical Algorithm for g(x,w)" --- p.42
Chapter 4.1.1 --- Method --- p.42
Chapter 4.1.2 --- Equation governing R(z) --- p.45
Chapter 4.1.3 --- "Equations governing A(x, z) and B(x, z)" --- p.45
Chapter 4.2 --- "Numerical Algorithm for g(x, ´ؤw)" --- p.49
Chapter 4.3 --- Numerical Result of q(γ) --- p.50
Chapter 4.4 --- Comparison of Numerical Result with Analytic Approximations --- p.56
Chapter 5 --- "Branch Cut Strength of G(x, y, w)" --- p.58
Chapter 5.1 --- "Relation between q(γ) and ΔG(x,y, ´ؤiγ)" --- p.58
Chapter 5.2 --- Proof of the Power Law --- p.60
Chapter 5.3 --- "Numerical Results for ΔG(x, y, ´ؤiγ)" --- p.63
Chapter 5.4 --- Study of a Physically Important Limit --- p.65
Chapter 5.4.1 --- Limiting x and y --- p.65
Chapter 5.4.2 --- Poles on the unphysical sheet --- p.69
Chapter 5.4.3 --- Zerilli potential --- p.77
Chapter 6 --- Conclusion --- p.81
Chapter A --- Tortoise Coordinate --- p.84
Chapter B --- Solution of the Generalized Coulomb Wave Equation --- p.86
Chapter C --- Derivation of (5.1) --- p.88
Bibliography --- p.89
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Ryzner, Jiří. "Fyzikální interpretace speciálních řešení Einsteinových-Maxwellových rovnic." Master's thesis, 2016. http://www.nusl.cz/ntk/nusl-389282.
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V klasické fyzice m·že být ustavena statická rovnováha v soustavě, která obsahuje extrémně nabité zdroje gravitačního a elektromagnetického pole. Udivujícím faktem je, že tato situace m·že nastat i pro černé díry v relativis- tické fyzice. Tato práce vyšetřuje speciální případ nekonečně dlouhé, extrémně nabité struny, zkoumá geometrii prostoročasu, elektrogeodetiky, vlastnosti zdroje a srovnává řešení se situací v klasické fyzice. Dále se zabýváme analogickou situací v dynamickém prostoročase s kosmologickou konstantou, a řešení porovnáváme s jeho statickou verzí. Nakonec zkoumáme periodické řešení Laplaceovy rovnice, které odpovídá nekonečně mnoha extremálním bodovým zdroj·m rozloženým v pravidelném rozestupu podél přímky. Vyšetřujeme vlastnosti elektrostatického potenciálu a ukazujeme, že v limitě velké vzdálenosti od osy tvořené zdroji pře- chází toto řešení v nabitou strunu. 1
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Μυλωνάς, Διονύσιος. "Δυισμοί στη γραμμικοποιημένη βαρύτητα." Thesis, 2010. http://nemertes.lis.upatras.gr/jspui/handle/10889/3288.
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Στη γραμμική εκδοχή της γενικής θεωρίας της σχετικότητας, θεωρεί κανείς τις διαταραχές κάποιας μετρικής γύρω από κάποιο χωροχρονικό υπόβαθρο. Κρατώντας όρους διαταραχών μέχρι και πρώτης τάξεως, οδηγείται κανείς στις γραμμικές εξισώσεις Einstein. Σε αυτό το πλαίσιο αποδεικνύεται μια σχέση δυισμού ανάμεσα στα διάφορα στοιχεία του τανυστή Weyl, αντίστοιχη με το δυισμό ανάμεσα στην ηλεκτρική και τη μαγνητική ροή της ηλεκτρομαγνητικής θεωρίας του Maxwell.
Στην εργασία αυτή κάνουμε μία ανασκόπηση της έρευνας που έχει γίνει μέχρι τώρα αναφορικά με αυτές τις σχέσεις δυισμού. Πιο συγκεκριμένα, εξετάζουμε την ισχύ των σχέσεων στον Anti-de Sitter χωρόχρονο και επισημαίνουμε το τρόπο με τον οποίο κατασκευάζει κανείς δυικές δομές από τις εκφράσεις για τις διαταραχές. Επίσης, χρησιμοποιώντας τη τεχνική της ολογραφικής επανακανονικοποίησης, εξετάζουμε το σύμμορφο σύνορο του χωροχρόνου. Βρίσκουμε εκεί μια σχέση δυισμού ανάμεσα στα στοιχεία του τανυστή ενέργειας-ορμής και του τανυστή Cotton της αντίστοιχης Chern - Simons θεωρίας, η οποία αποδεικνύεται ότι είναι άμεση συνέπεια του δυισμού στο AdS υπόβαθρο.
Τέλος, εφαρμόζουμε την ίδια συλλογιστική στο Schwarzschild - Anti-de Sitter υπόβαθρο, όπου η παρουσία της μελανής οπής διαφοροποιεί τις συνοριακές συνθήκες του προβλήματος. Λόγω αυτού του γεγονός δεν μπορεί να πει κανείς με σιγουριά εάν μπορούν να διατυπωθούν σχέσεις δυισμού σε αυτή τη περίπτωση. Παρόλα αυτά βρίσκουμε ότι ισχύουν σχέσεις δυισμού στο σύμμορφο σύνορο παρόμοιες με αυτές του AdS υποβάθρου, πράγμα που σημαίνει ότι στο σύστημα παραμένει κάποια συμμετρία από τη γραμμική θεωρία. Η εργασία καταλήγει σε σχόλια και μία εκτενή συζήτηση για τις πιθανές μελλοντικές κατευθύνσεις.In the linear version of the general theory of relativity, one considers metric perturbations around a fixed background. Keeping terms up to first order of perturbation leads to the linearized Einstein equations. In this framework it has been proved that a duality between the various elements of the Weyl tensor holds. This duality is similar to the one between the electric and magnetic fluxes of Maxwell's electromagnetism. In the present work we review the status of these dualities for non trivial backgrounds. We examine the Anti-de Sitter background, where we point out the way to explicitly construct dual configurations using the metric perturbation expressions. Using the holographic renormalization technique, we examine the conformal boundary where a duality between the components of the energy-momentum tensor and the Cotton tensor of the corresponding Chern - Simons theory holds. It is then proved that this duality is a direct consequence of the electric/magnetic duality in the bulk, in the case of the AdS background. Finally, we apply same procedure to the Schwarzschild - Anti-de Sitter background, where the presence of the black hole changes the boundary conditions of the problem. This simple fact makes it impossible say whether such a duality exists in this case. Nevertheless, we find that a duality similar to that of the AdS background still holds for the conformal boundary, which means that there is a remnant of symmetry from the linear theory. We conclude with comments and a extensive discussion on possible future directions.
Books on the topic "Wave equations on black hole spacetimes"
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Deruelle, Nathalie, and Jean-Philippe Uzan. The two-body problem: an effective-one-body approach. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0056.
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This chapter presents the basics of the ‘effective-one-body’ approach to the two-body problem in general relativity. It also shows that the 2PN equations of motion can be mapped. This can be done by means of an appropriate canonical transformation, to a geodesic motion in a static, spherically symmetric spacetime, thus considerably simplifying the dynamics. Then, including the 2.5PN radiation reaction force in the (resummed) equations of motion, this chapter provides the waveform during the inspiral, merger, and ringdown phases of the coalescence of two non-spinning black holes into a final Kerr black hole. The chapter also comments on the current developments of this approach, which is instrumental in building the libraries of waveform templates that are needed to analyze the data collected by the current gravitational wave detectors.
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Chruściel, Piotr T. Geometry of Black Holes. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198855415.001.0001.
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There exists a large scientific literature on black holes, including many excellent textbooks of various levels of difficulty. However, most of these prefer physical intuition to mathematical rigour. The object of this book is to fill this gap and present a detailed, mathematically oriented, extended introduction to the subject. The first part of the book starts with a presentation, in Chapter 1, of some basic facts about Lorentzian manifolds. Chapter 2 develops those elements of Lorentzian causality theory which are key to the understanding of black-hole spacetimes. We present some applications of the causality theory in Chapter 3, as relevant for the study of black holes. Chapter 4, which opens the second part of the book, constitutes an introduction to the theory of black holes, including a review of experimental evidence, a presentation of the basic notions, and a study of the flagship black holes: the Schwarzschild, Reissner–Nordström, Kerr, and Majumdar–Papapetrou solutions of the Einstein, or Einstein–Maxwell, equations. Chapter 5 presents some further important solutions: the Kerr–Newman–(anti-)de Sitter black holes, the Emperan–Reall black rings, the Kaluza–Klein solutions of Rasheed, and the Birmingham family of metrics. Chapters 6 and 7 present the construction of conformal and projective diagrams, which play a key role in understanding the global structure of spacetimes obtained by piecing together metrics which, initially, are expressed in local coordinates. Chapter 8 presents an overview of known dynamical black-hole solutions of the vacuum Einstein equations.
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Wittman, David M. The Elements of Relativity. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780199658633.001.0001.
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Relativity is a set of remarkable insights into the way space and time work. The basic notion of relativity, first articulated by Galileo, explains why we do not feel Earth moving as it orbits the Sun and was successful for hundreds of years. We present thinking tools that elucidate Galilean relativity and prepare us for the more modern understanding. We then show how Galilean relativity breaks down at speeds near the speed of light, and follow Einstein’s steps in working out the unexpected relationships between space and time that we now call special relativity. These relationships give rise to time dilation, length contraction, and the twin “paradox” which we explain in detail. Throughout, we emphasize how these effects are tightly interwoven logically and graphically. Our graphical understanding leads to viewing space and time as a unified entity called spacetime whose geometry differs from that of space alone, giving rise to these remarkable effects. The same geometry gives rise to the energy?momentum relation that yields the famous equation E = mc2, which we explore in detail. We then show that this geometric model can explain gravity better than traditional models of the “force” of gravity. This gives rise to general relativity, which unites relativity and gravity in a coherent whole that spawns new insights into the dynamic nature of spacetime. We examine experimental tests and startling predictions of general relativity, from everyday applications (GPS) to exotic phenomena such as gravitomagnetism, gravitational waves, Big Bang cosmology, and especially black holes.
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Deruelle, Nathalie, and Jean-Philippe Uzan. The physics of black holes I. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0049.
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This chapter describes two physical processes related to the Schwarzschild and Kerr solutions which can be induced by the gravitational field of a black hole. The first is the Penrose process, which suggests that rotating black holes are large energy reservoirs. Next is superradiance, which is the first step in the study of black-hole stability. The study of the stability of black holes involves the linearization of the Einstein equations about the Schwarzschild or Kerr solution. As this chapter shows, the equations of motion for perturbations of the metric are wave equations. The problem then is to determine whether or not these solutions are bounded.
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Szeftel, Jérémie, and Sergiu Klainerman. Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations. Princeton University Press, 2020. http://dx.doi.org/10.23943/princeton/9780691212425.001.0001.
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One of the major outstanding questions about black holes is whether they remain stable when subject to small perturbations. An affirmative answer to this question would provide strong theoretical support for the physical reality of black holes. This book takes an important step toward solving the fundamental black hole stability problem in general relativity by establishing the stability of nonrotating black holes — or Schwarzschild spacetimes — under so-called polarized perturbations. This restriction ensures that the final state of evolution is itself a Schwarzschild space. Building on the remarkable advances made in the past fifteen years in establishing quantitative linear stability, the book introduces a series of new ideas to deal with the strongly nonlinear, covariant features of the Einstein equations. Most preeminent among them is the general covariant modulation (GCM) procedure that allows them to determine the center of mass frame and the mass of the final black hole state. Essential reading for mathematicians and physicists alike, the book introduces a rich theoretical framework relevant to situations such as the full setting of the Kerr stability conjecture.
Book chapters on the topic "Wave equations on black hole spacetimes"
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Moffat, John W. "The Biggest Ears in the Sky: LIGO." In The Shadow of the Black Hole, 108–36. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780190650728.003.0007.
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At a press conference on February 11, 2016, David Reitz, LIGO Executive Director, announced, “We did it!” They detected gravitational waves for the first time. Both LIGO sites, in Washington state and Louisiana, registered the incoming gravitational waves from two black holes colliding and merging far away. Over the following months, more mergers were detected. Gravitational waves are caused by the acceleration of a massive object, which stretches and compresses spacetime in a wave-like motion that is incredibly small and difficult to detect. Numerical relativity research over decades has produced over a quarter of a million template solutions of Einstein’s equations. The best template fit to the wave form data identifies the masses and spins of the two merging black holes. Much of this chapter describes the technology of the LIGO apparatus. On October 3, 2017, Barish, Thorne, and Weiss, the founders of LIGO, received the Nobel Prize for Physics.
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Chruściel, Piotr T. "Some applications." In Geometry of Black Holes, 85–114. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198855415.003.0003.
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The aim of this chapter is to present key applications of causality theory, as relevant to black-hole spacetimes. For this we need to introduce the concept of conformal completions, which is done in Section 3.1. We continue, in Section 3.2, with a review of the null splitting theorem of Galloway. Section 3.3 contains complete proofs of a few versions of the topological censorship theorems, which are otherwise scattered across the literature, and which play a basic role in understanding the topology of black holes. In Section 3.4 we review some key incompleteness theorems, also known under the name of singularity theorems. Section 3.5 is devoted to the presentation of a few versions of the area theorem, which is a cornerstones of ‘black-hole thermodynamics’. We close this chapter with a short discussion of the role played by causality theory when studying the wave equation.
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Manton, Nicholas, and Nicholas Mee. "General Relativity." In The Physical World. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198795933.003.0007.
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This chapter presents the physical motivation for general relativity, derives the Einstein field equation and gives concise derivations of the main results of the theory. It begins with the equivalence principle, tidal forces in Newtonian gravity and their connection to curved spacetime geometry. This leads to a derivation of the field equation. Tests of general relativity are considered: Mercury’s perihelion advance, gravitational redshift, the deflection of starlight and gravitational lenses. The exterior and interior Schwarzschild solutions are discussed. Eddington–Finkelstein coordinates are used to describe objects falling into non-rotating black holes. The Kerr metric is used to describe rotating black holes and their astrophysical consequences. Gravitational waves are described and used to explain the orbital decay of binary neutron stars. Their recent detection by LIGO and the beginning of a new era of gravitational wave astronomy is discussed. Finally, the gravitational field equations are derived from the Einstein–Hilbert action.
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Chruściel, Piotr T. "Dynamical black holes." In Geometry of Black Holes, 312–36. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198855415.003.0008.
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In this chapter we review what is known about dynamical black hole-solutions of Einstein equations. We discuss the Robinson–Trautman black holes, with or without a cosmological constant. We review the Cauchy-data approach to the construction of black-hole spacetimes. We propose some alternative approaches to a meaningful definition of black hole in a dynamical spacetime, and we review the nonlinear stability results for black-hole solutions of vacuum Einstein equations.
Conference papers on the topic "Wave equations on black hole spacetimes"
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DAFERMOS, MIHALIS, and IGOR RODNIANSKI. "A NEW PHYSICAL-SPACE APPROACH TO DECAY FOR THE WAVE EQUATION WITH APPLICATIONS TO BLACK HOLE SPACETIMES." In XVIth International Congress on Mathematical Physics. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/9789814304634_0032.
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