The Material Connection between Classical Mechanics and Relativity

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.

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 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.

In this paper, we have calculated the angle of refraction that light travels approaching to the strong gravitational field like a black hole by combining the general relativity and the classical Snell’s law, assuming that the gravitational field can act as a non-vacuum filled with medium of some coefficients. We have found that the value of refracted angle exactly coincides with the value from the Einstein’s relativity theory in a weak gravitational field. From this optical interpretation of the traveling of light near a black hole, we have suggested that there might have the reflection phenomenon and investigated that the total reflection occurs at the surface of a black hole. Regardless this might cause controversy, we can explain the recent observation that light reflects from a black hole.

By using an alternative method valid in weak and strong gravitational fields, we study beam collimation, light bending and time delays in the equatorial Kerr spacetime. Three applications are carried out. First, it is shown that the interaction “spin-orbit” could collimate a beam only if the particle or photon moves in a opposite direction to the compact object's spin. Secondly, the deflection angles of light rays grazing a quasar, a pulsar (PSR 1937+214), and a rotating black hole are computed and compared with approximate solutions. Finally, we calculate the photon's time delay traveling from SN1987A to Earth. The result of this undertaking is compared with a similar calculation using the PPN formalism.

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.

It has been demonstrated that dark matter (DM) can theoretically be completely explained by a natural effect of General Relativity (GR) without exotic matter, the Lense-Thirring effect that exists exclusively in GR and that would be due to the clusters of galaxies.In this study, we show that this explanation of DM leads to a modelization that can be interpreted as MOND-based theories.More concretely, we retrieve from GR the value of MOND parameter 8 2 0 10 cm s a -- and deep MOND and AQUAL parameters 1.37 G G . It means that MOND-based theories could be interpreted as an approximation of the linearized GR (i.e.GR in a weak gravitational field or small acceleration) in a particular physical case of a uniform gravitic field (2 nd component of GR in its linearized form, similar to magnetic field of Electromagnetism).A publication has recently observed deviations from Newtonian acceleration with a 10 significance for wide binary stars at weak gravitational acceleration.The author demonstrates that these deviations can be explained by MOND theory with the previous parameters' values.This situation leads to a difficulty.On one hand, the traditional DM hypothesis can't explain these deviations and on the other hand, empirical MOND theories are difficult to justify compared to the success of GR.With our result, no more difficulty, these deviations do not need to be explained by MOND theory but by linearized GR with the uniform gravitic field explaining the DM component (Lense-Thirring effect of the clusters of galaxies).

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.

This paper studies two perfectly conducting parallel plates in the weak gravitational field on the surface of the Earth. Since the appropriate line element, to first order in the constant gravity acceleration $g$, is precisely of the Rindler type, we can exploit the formalism for studying Feynman Green functions in Rindler spacetime. Our analysis does not reduce the electromagnetic potential to the transverse part before quantization. It is instead fully covariant and well suited for obtaining all components of the regularized and renormalized energy-momentum tensor to arbitrary order in the gravity acceleration $g$. The general structure of the calculation is therefore elucidated, and the components of the Maxwell energy-momentum tensor are evaluated up to second order in $g$, improving a previous analysis by the authors and correcting their old first-order formula for the Casimir energy.

On covariant scalar green functions in a weak gravitational field

Motion of a thin layer of the Williamson fluid over an inclined solid surface in the gravitational field is considered. Expressions are obtained for the stationary motion of the film in the gravitation field. The influence of the latter and tangential vibrations of the solid surface on the flow in the layer is analyzed. It is shown that the vibrations generate a pronounced averaged fluid flow even in the weak gravitational field in which the film is at rest in the absence of vibrations. The effect of the nonmonotonic dependence of the mean flow velocity of the film on the vibration frequency of the surface is found.

In this paper the ‘‘free fall’’ motion of a test tetrad in a weak gravitational field is studied. This is done with the intention of exploring the geometry locally through geodesics and parallel transport. By use of the so-called ‘‘radial gauge’’ the action of the field on the test tetrad just in terms of the Riemann tensor is expressed. The average effect of small quantized fluctuations of the field can thus be computed using the Feynman–De Witt propagator. Since the dimension L of the test particle must be finite, only fluctuations of wavelength λ≥L have to be considered. The corresponding corrections to the geometry are proportional to lPlanck/L. Such dependence on L spoils, in fact, a genuine geometrical description.

We compute the gravitational effective action by integrating out quantum matter fields in a weak gravitational field, using the Schwinger-Keldysh (in-in) formalism. We pay particular attention to the role of the initial quantum state in the structure of the nonlocal terms in the effective action, with an eye to nonlinear completions of the theory that may be relevant in astrophysics and cosmology. In this first paper we consider a quantum scalar field in thermal equilibrium, in a stationary gravitational field. We obtain a covariant expression for the nonlocal effective action, that can be expressed in terms of the curvature tensor, the four-velocity of the thermal bath and the local Tolman temperature. We discuss the connection between the results for ultrastatic and static metrics through conformal transformations, and the main features of the thermal corrections to the semiclassical Einstein equations.

Given the extreme accuracy of modern space astrometry, a precise relativistic modeling of observations is required. Concerning light propagation, the standard procedure is the solution of the null-geodesic equations. However, another approach based on the time transfer functions (TTF) has demonstrated its capability to give access to key quantities such as the time of flight of a light signal between two point-events and the tangent vector to its null-geodesic in a weak gravitational field using an integral-based method. The availability of several models, formulated in different and independent ways, must not be considered like an oversized relativistic toolbox. Quite the contrary, they are needed as validation to put future experimental results on solid ground. The objective of this work is then twofold. First, we build the time of flight and tangent vectors in a closed form within the TTF formalism giving the case of a time-dependent metric. Second, we show how to use this new approach to obtain a comparison of the TTF with two existing modelings, namely the Gaia RElativistic Model (GREM) and the Relativistic Astrometric MODel (RAMOD). In this way, we highlight the mutual consistency of the three models, opening the basis for further links between all the approaches, which is mandatory for the interpretation of future space missions data. This will be illustrated through two recognized cases: a static gravitational field and a system of gravitational mass monopoles in uniform motion.

Starting from the equations of general relativity, equations similar to those of electromagnetic theory are derived. It is assumed that the particles are slowly moving (v≪c), and the gravitational field is sufficiently weak that nonlinear terms in Einstein’s field equations can be neglected. For static fields, the analogy to electrostatics and magnetostatics is very close. Results are compared with those of a previous derivation by Braginsky, Caves, and Thorne [Phys. Rev. D 15, 2047–2068 (1977)]. These results lead to very simple derivations of the Lense–Thirring precession [Phys. Z. 19, 156–163 (1918)] and the spin-curvature force of Papepetrou [Proc. R. Soc. London Ser. A 209, 248–258 (1951)] and Pirani [Acta Phys. Pol. 15, 389–405 (1956)].