Light Refraction and Reflection near a Black Hole

Einstein's General Relativity theory has been tested in many ways during the last hundred years as reviewed in this chapter. Two tests are discussed in detail in this article: the concept of a zero gravity surface, the roots of which go back to Jarnefelt, Einstein and Straus, and the no-hair theorem of black holes, first proposed by Israel, Carter and Hawking. The former tests the necessity of the cosmological constant Lambda, the latter the concept of a spinning black hole. The zero gravity surface is manifested most prominently in the motions of dwarf galaxies around the Local Group of galaxies. The no-hair theorem is testable for the first time in the binary black hole system OJ287. These represent stringent tests at the limit of weak and strong gravitational fields, respectively. In this article we discuss the current observational situation and future possibilities.

This chapter presents pedagogical approaches to light deflection by gravitational fields. Karl-Heinz Lotze and Silvia Simionato put particular emphasis on historical perspectives to contextualize gravitational lensing in light of the larger development of Einsteinian physics. It is intended for readers who look for instructional models of the gravitational-lens effect in general relativity suitable at the secondary school level. The chapter treats the deflection, or curvature, of the trajectories of light propagating in weak gravitational fields. In Einstein's general theory of relativity, gravitation is no longer a force in the sense of Newtonian dynamics. Gravitational lenses are a powerful educational tool. Light deflection by gravitational fields is the key mechanism behind the gravitational lens effect. It discusses the gravitational lens effect. The idea of light deflection in strong gravitational fields has a long history as well and can be traced back to the late 18th century.

As soon as the theory of general relativity was propounded, astronomers and physicists made ready to test its observational predictions. There was of course the excess motion of the perihelion of the orbit of Mercury, which the theory had been designed in part to explain. There was the gravitational bending of light beams (verified at the eclipse of 1919), the gravitational redshifting of light and the gravitational retarding of light (both verified in recent years). Light rays and the planet Mercury would exist without the theory of general relativity. Its effects on their behavior are small. But there is a class of objects whose existence is predicted by general relativity and which do not exist if the theory in some form is not true. These are the black holes, objects that have collapsed under the influence of their own gravitation until they are so dense and have such strong gravitational fields that neither matter nor radiation can escape from their surfaces. To a general relativist a black hole is an example of extremely curved space. In and around it the effects of general relativity are not minor; they are dominant, and they determine the nature of the phenomena that occur. The discovery of an actual black hole would provide an astrophysical working place for general relativity, a region where the effects of space curvature could be studied readily rather than having to be sorted out as tiny corrections to the nearly flat space that surrounds ordinary objects. As Remo J. Ruffini of Princeton University puts it, the discovery of a black hole would be for general relativity what the discovery of the photon (also, by the way, associated with Einstein) was for quantum theory and the physics of the microcosm. Ruffini believes, and has believed for more than a year, that at least one black hole is currently under observation (SN: 1/13/73, p. 28). Controversy rages, and to the discussion Ruffini now brings theoretical arguments and observational evidence, some of which was barely analyzed in time for him to use it at the meeting of the American Physical Society two weeks ago in Chicago. Ruffini's candidate for a black hole is Cygnus X-1, one of a newly discovered class of celestial X-ray sources, X-ray binaries. Up to, a point there is general agreement as to what these binary sources are: They are systems in which a more or less normal star is gravitationally bound to a highly condensed dark companion. The usual explanation as to how the companion got condensed is that it is the last stage of stellar evolution, what is left over after a supernova explosion. Matter streams out from the normal star and falls onto the dark companion. The streaming matter forms a rotating disk around the dark companion in which pieces of matter gradually spiral inward until they fall on the dark companion. As they do so they convert energy they gain through their fall to X-rays. (The matter in the disks is electrically charged ionized plasma, not neutral gas, so it has means to do this.) The argument starts over the nature of the condensed body. The main candidates are black holes and neutron stars-though a few observers may hold out for white dwarfs. To make the case for black holes or for a black hole one needs a way of distinguishing observationally between the two. Ruffini provides a way. It has to do with magnetic fields. Stars have magnetic fields. When they collapse, even through the violence of a supernova explosion, they should retain those fields. Thus neutron stars and black holes can have magnetic fields. But in the configuration of those fields lies the difference. A neutron star

A small body moving in the field of a much larger black hole and subjected to its own gravity moves on an accelerated world line in the background spacetime of the large black hole. The acceleration is produced by the body's gravitational self-force, which is constructed from the body's retarded gravitational field. The adiabatic approximation to the gravitational self-force is obtained instead from the half-retarded minus half-advanced field; it is known to produce the same dissipative effects as the true self-force. We argue that the adiabatic approximation is limited, because it discards important conservative terms which lead to the secular evolution of some orbital elements. We argue further that this secular evolution has measurable consequences; in particular, it affects the phasing of the orbit and the phasing of the associated gravitational wave. Our argument rests on a simple toy model involving a point electric charge moving slowly in the weak gravitational field of a central mass; the charge is also subjected to its electromagnetic self-force. In this simple context the true self-force is known explicitly and it can cleanly be separated into conservative and radiation-reaction pieces. We observe that the conservative part of the self-force produces a secular regression of the orbit's periapsis. We explain how the conclusions reached on the basis of the toy model can be extended to the gravitational self-force, and to fast motions and strong fields. While the limitations of the adiabatic approximation are quite severe in a post-Newtonian context in which the motion is slow and the gravitational field weak, they appear to be less so for rapid motions and strong fields.

It is assumed here that the energy of a strong gravitational field creates non-linear effects over enclosed masses. This idea and the rigorous rules of the General Theory of Relativity output a metric that covers strong and weak gravitational fields. The proposed metric could be correct because it included the Schwarzschild’s metric as a particular case and has no singularities. Additionally, it appears here that the massive condition of the gravitational fields has properties like the so-called Dark Matter.

We develop a method for finding the exact equations of structure and motion of multipole test particles in Einstein's unified field theory---the theory of the nonsymmetric field. The method is also applicable to Einstein's gravitational theory. Particles are represented by singularities in the field. The method is covariant at each step of the analysis. We also apply the method and find both in Einstein's unified field theory and in Einstein's gravitational theory the equations of structure and motion of neutral pole-dipole test particles possessing no electromagnetic multipole moments. In the case of Einstein's gravitational theory the results are the well-known equations of structure and motion of a neutral pole-dipole test particle in a given background gravitational field. In the case of Einstein's unified field theory the results are the same, providing we identify a certain symmetric second-rank tensor field appearing in Einstein's theory with the metric and gravitational field. We therefore discover not only the equations of structure and motion of a neutral test particle in Einstein's unified field theory, but we also discover what field in Einstein's theory plays the role of metric and gravitational field.

This paper is concerned with the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat space–time and weak radiation at a very late time. We evaluate quantum amplitudes (not just probabilities) for transitions from initial to final states. This quantum description shows that no information is lost in collapse to a black hole. Boundary data for the gravitational field and (in this paper) a scalar field are posed on an initial space-like hypersurface ΣIand a final surface ΣF. These asymptotically flat three-surfaces are separated by a Lorentzian proper-time interval T (typically very large), as measured at spatial infinity. The boundary-value problem is made well-posed, both classically and quantum-mechanically, by a rotation of T into the lower-half complex plane: T → |T| exp (- iθ), with 0 < θ ≤ π/2. This corresponds to Feynman's +iϵ prescription. We consider the classical boundary-value problem and calculate the second-variation classical Lorentzian action [Formula: see text] as a functional of the boundary data. Following Feynman, the Lorentzian quantum amplitude is recovered in the limit θ → 0+from the well-defined complex-T amplitude. Dirac's canonical approach to the quantisation of constrained systems shows that, for locally supersymmetric theories of gravity, the amplitude is exactly semi-classical, namely [Formula: see text] for weak perturbations, apart from delta functionals of the supersymmetry constraints. We treat such quantum amplitudes for weak scalar-field configurations on ΣF, taking (for simplicity) the weak final gravitational field to be spherically symmetric. The treatment involves adiabatic solutions to the scalar wave equation. This considerably extends work reported in previous papers, by giving explicit expressions for the real and imaginary parts of such quantum amplitudes.

We introduce the Gauge Vector-Tensor (GVT) theory by extending the AQUAL's approach to the GravitoElectroMagnetism (GEM) approximation of gravity. GVT is a generally covariant theory of gravity composed of a pseudo Riemannian metric and two U(1) gauge connections that reproduces MOND in the limit of very weak gravitational fields while remains consistent with the Einstein-Hilbert gravity in the limit of strong and Newtonian gravitational fields. GVT also provides a simple framework to study the GEM approximation to gravity. We illustrate that the gravitomagnetic force at the edge of a galaxy can be in accord with either GVT or ΛCDM but not both. We also study the physics of the GVT theory around the gravitational saddle point of the Sun and Jupiter system. We notice that the conclusive refusal of the GVT theory demands measuring either both of the gravitoelectric and gravitomagnetic fields inside the Sun-Jupiter MOND window, or the gravitoelectric field inside two different solar GVT MOND windows. The GVT theory, however, will be favored by observing an anomaly in the gravitoelectric field inside a single MOND window.

We show that rotating black holes do not experience any tidal deformation when they are perturbed by a weak and adiabatic gravitational field. The tidal deformability of an object is quantified by the so-called "Love numbers", which describe the object's linear response to its external tidal field. In this work, we compute the Love numbers of Kerr black holes and find that they vanish identically. We also compute the dissipative part of the black hole's tidal response, which is non-vanishing due to the absorptive nature of the event horizon. Our results hold for arbitrary values of black hole spin, for both the electric-type and magnetic-type perturbations, and to all orders in the multipole expansion of the tidal field. The boundary conditions at the event horizon and at asymptotic infinity are incorporated in our study, as they are crucial for understanding the way in which these tidal effects are mapped onto gravitational-wave observables. In closing, we address the ambiguity issue of Love numbers in General Relativity, which we argue is resolved when those boundary conditions are taken into account. Our findings provide essential inputs for current efforts to probe the nature of compact objects through the gravitational waves emitted by binary systems.

QUANTUM gravitational effects are usually ignored in calculations of the formation and evolution of black holes. The justification for this is that the radius of curvature of space-time outside the event horizon is very large compared to the Planck length (Għ/c3)1/2 ≈ 10−33 cm, the length scale on which quantum fluctuations of the metric are expected to be of order unity. This means that the energy density of particles created by the gravitational field is small compared to the space-time curvature. Even though quantum effects may be small locally, they may still, however, add up to produce a significant effect over the lifetime of the Universe ≈ 1017 s which is very long compared to the Planck time ≈ 10−43 s. The purpose of this letter is to show that this indeed may be the case: it seems that any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole was a body with a temperature of (κ/2π) (ħ/2k) ≈ 10−6 (M/M)K where κ is the surface gravity of the black hole1. As a black hole emits this thermal radiation one would expect it to lose mass. This in turn would increase the surface gravity and so increase the rate of emission. The black hole would therefore have a finite life of the order of 1071 (M/M)−3 s. For a black hole of solar mass this is much longer than the age of the Universe. There might, however, be much smaller black holes which were formed by fluctuations in the early Universe2. Any such black hole of mass less than 1015 g would have evaporated by now. Near the end of its life the rate of emission would be very high and about 1030 erg would be released in the last 0.1 s. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs. It is often said that nothing can escape from a black hole. But in 1974, Stephen Hawking realized that, owing to quantum effects, black holes should emit particles with a thermal distribution of energies — as if the black hole had a temperature inversely proportional to its mass. In addition to putting black-hole thermodynamics on a firmer footing, this discovery led Hawking to postulate 'black hole explosions', as primordial black holes end their lives in an accelerating release of energy.

This study analyzes the effect of multipole values (\( \ell\)) on the distribution of gravitational or electromagnetic fields in black holes, using the Regge-Wheeler potential approach. Various masses (5, 10, 20, 50) and varying \( \ell\) values are analyzed to explore the impact of multipole on space-time geometry. The results show that increasing the value of \( \ell\) reduces the values of \( r_{\text{positive}}\) and \( r_{\text{negative}}\), which indicates the field distribution becomes more focused at higher multipole values. The sensitivity to changes in the \( \ell\) value is especially noticeable in the colinear \( \ell\) interval between 5 to 17, where small changes in the \( \ell\) value result in large changes in the \( r\) value. In the framework of black hole theory, two solutions of the quadratic equation \( \ell = \frac{-1 \pm\sqrt{13}}{2}\) yield two values \( \ell_+\) and \( \ell_-\), which affect the distribution of the gravitational field. Logarithmic analysis of the \( r \) value reveals variations in the field distribution in the multipole range, and shows a strong dependence on the black hole mass, with more significant changes at small masses. This study provides a deeper understanding of the dependence of the field distribution on multipole and opens up opportunities for further research into the complexity of multipolar interactions in black hole theory.

In accordance with the special theory of relativity all forms of energy, including heat, have inertia and hence in accordance with the equivalence principle also have weight. The purpose of the present article is to investigate the thermodynamic implications of the idea that heat has weight. In particular an investigation is made to see if a temperature gradient is a necessary accompaniment of thermal equilibrium in a gravitational field, in order to prevent the flow of heat from regions of higher to those of lower gravitational potential.A preliminary non-rigorous treatment of this problem is first given by attempting to modify the classical thermodynamics only to the extent of associating with each intrinsic quantity of energy an additional amount of potential gravitational energy. In this way an expression is obtained for increase in equilibrium temperature with decrease in gravitational potential which, however, could in any case only be correct as a first approximation in a weak gravitational field. A discussion of the uncertainties and lack of rigor of this preliminary treatment is then given and the necessity pointed out for a rigorous treatment based on the principles of general relativity.A rigorous relativistic treatment is then undertaken using the extension of thermodynamics to general relativity previously presented by the author. The system to be treated is taken as a static spherical distribution of perfect fluid which has come to gravitational and thermodynamic equilibrium. The principles of relativistic mechanics are first applied to such a system in order to obtain results needed in the later work. And it is then shown that these mechanical principles themselves are sufficient to determine the temperature distribution as a function of potential in the simple case of black-body radiation. The principles of relativistic thermodynamics are then applied to this same case of pure black-body radiation and the same expression for temperature as a function of potential obtained by the thermodynamic as by the mechanical treatment. This may be regarded as giving some measure of check on the validity of the proposed relativistic thermodynamics.Following this, a thermodynamic treatment is given for the temperature distribution in the more general case of matter and radiation and a result found which harmonizes with that for radiation alone. A treatment is then given to the distribution of a perfect monatomic gas in a gravitational field both on the assumption that the total number of atoms must remain constant and on the assumption of the ready interconvertibility of matter and radiation. In the latter case the same dependence of concentration on temperature is obtained as was found by Stern and by the author for the case of flat space-time.Using a system of coordinates such that the line element for the sphere of fluid takes the form$d{s}^{2}=\ensuremath{-}{e}^{u}(d{r}^{2}+{r}^{2}d{\ensuremath{\theta}}^{2}+{r}^{2}{sin}^{2}\ensuremath{\theta}d{\ensuremath{\varphi}}^{2})+{e}^{\ensuremath{\nu}}d{t}^{2}$ the general result for the relation between gravitational potential and equilibrium temperature ${T}_{0}$ as measured by a local observer in proper coordinates can be given by the equation $\frac{d \mathrm{ln}{T}_{0}}{\mathrm{dr}}=\ensuremath{-}\frac{1}{2}\frac{d\ensuremath{\nu}}{\mathrm{dr}}$This equation reduces in the case of a weak field to that obtained by the preliminary non-rigorous treatment, and gives a very small change of temperature with position in fields of ordinary intensity. The result, however, is one of great theoretical interest, since constant temperature throughout any system which has come to thermal equilibrium has hitherto been regarded as an inescapable thermodynamic conclusion. It is also not out of the question that the effect might sometime be of experimental or observational importance.

We present a first-principles derivation of the main results of the Kerr/CFT correspondence and its extensions using only tools from gravity and quantum field theory. Firstly, we review properties of extremal black holes with in particular the construction of an asymptotic Virasoro symmetry in the near-horizon limit. The entropy of extremal spinning or charged black holes is shown to match with a chiral half of Cardy’s formula. Secondly, we show how a thermal 2-dimensional conformal field theory (CFT) is relevant to reproduce the dynamics of near-superradiant probes around near-extremal black holes in the semi-classical limit. Thirdly, we review the hidden conformal symmetries of asymptotically-flat black holes away from extremality and present how the non-extremal entropy can be matched with Cardy’s formula. We follow an effective field theory approach and consider the Kerr–Newman black hole and its generalizations in various supergravity theories. The interpretation of these results by deformed dual conformal field theories is discussed and contrasted with properties of standard 2-dimensional CFTs. We conclude with a list of open problems.

Essentially, in this paper we propose a new description of the quantum dynamics by two relativistic propagation wave packets, in the two conjugated spaces, of the coordinates and of the momentum. Compared to the Schrödinger-Dirac equation, which describes a free particle by a wave function continuously expanding in time, considered as the amplitude of a probabilistic distribution of this particle, the new equations describe a free particle as an invariant distribution of matter propagating in the two spaces, as it should be. Matter quantization arises from the equality of the integral of the matter density with the mass describing the dynamics of this density in the phases of the wave packets. In this description, the classical Lagrange and Hamilton equations are obtained as the group velocities of the two wave packets in the coordinate and momentum spaces. When to the relativistic Lagrangian we add terms with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and a scalar potential conjugated to time, we obtain the Lorentz force and the Maxwell equations as characteristics of the quantum dynamics. In this framework, the conventional Schrödinger-Dirac equations of a quantum particle in an electromagnetic field obtain additional terms explicitly depending on velocity, as is expected in the framework of relativistic theory. Such a particle wave function takes the form of a rapidly varying wave, with the frequency corresponding to the rest energy, modulated by the electric potential  where this particle is confined, the vector potential   of the electromagnetic radiation, and the gravitational potential. In the gravitational field of a black hole, the velocity of a quantum particle is directed to the Schwarzschild boundary from the outside and from the inside, without passing this boundary. When the very strong gravitational field of a large agglomeration of matter has spherical symmetry with a radius smaller than the Schwarzschild radius, the quantum particles in the central region are crushed up to quasi-uniform distributions of matter, exploding with velocities much larger than the light velocity, towards the Schwarzschild boundary, but decelerating in the gravitational field and reaching this boundary only in an infinite time. The matter of a black hole has the tendency to concentrate around the Schwarzschild boundary, with a radius proportional to its mass. In this framework, we believe that our universe is a huge black hole, with the explosion of its central matter as the Big Bang-inflation process, and an increasing radius on account of an absorption of matter from the Total Universe, where our universe has been formed. In a gravitational wave, the time-space interval in the first-order approximation is the product of a gravitational invariant including the amplitude tensor, and a coordinate invariant including the polarization vector. The gravitational invariant describes a rotation of the metric tensor around the direction of the propagation of this wave, with spin 2, as a gravitational quantum called graviton. The coordinate invariant describes a matter rotation with the spins ½ for Fermions, and 1 for Bosons. From the new dynamic equations, for a free particle in the coordinate and momentum spaces, we reobtain the two basic equations of the quantum field theory, but with a change of sign, and an additional term depending on momentum, to the rest mass as the eigenvalue of these equations. However, when these eigenvalues are eliminated, the wave function takes the form of a wave packet of spinors of the same form as in the conventional quantum field theory, with a normalization volume as the integral of the ratio of the energy to the rest energy, over the momentum domain which gives finite dimensions to the quantum particle, as a finite distribution of matter in the coordinate space.

Black holes are one of nature's greatest mysteries. For years, these celestial bodies have evaded scientists due to their intense gravity and extreme density. A specific area of black hole research that is overlooked is how these cosmic bodies affect the space-time continuum. Developing a complete understanding of space-time in and around black holes is essential for future research relating to black holes and combining the two great theories of this generation: general relativity and quantum mechanics. This paper aims to analyze how the space-time continuum is distorted by black holes and explain the applications that space-time around black holes has to modern physics. It first develops an understanding of space-time linkage and curvature through Einstein’s theories of relativity and presents the geometry of space-time as predicted by Minkowski’s equation. When inside a black hole, space and time essentially trade places so that the flow of time causes matter to be drawn within a black hole. As one explores further toward the centre of a black hole, curvature increases until it is predicted by general relativity to reach a gravitational singularity where density becomes infinite, and the laws of space-time break down. In the paper, the discrepancy between general relativity and quantum mechanics is explored through Hawking Radiation and the theoretical singularity at a black hole's centre. A combination of written explanations, math models, and diagrams are used to communicate how the space-time continuum is affected by black holes.