A Brief Review of Internal Inconsistencies in General Relativity Theory (GRT) and Comparisons to Metric Theory of Gravity (MTG)

Einstein’s General theory of relativity (GR) successfully describes gravity. Although GR has been accurately tested in weak gravitational fields, it remains largely untested in the general strong field cases. One of the most fundamental predictions of GR is the existence of black holes (BHs). After the recent direct detection of gravitational waves by LIGO, there is now near conclusive evidence for the existence of stellar-mass BHs. In spite of this exciting discovery, there is not yet direct evidence of the existence of BHs using astronomical observations in the electromagnetic spectrum. Are BHs observable astrophysical objects? Does GR hold in its most extreme limit or are alternatives needed? The prime target to address these fundamental questions is in the center of our own Milky Way, which hosts the closest and best-constrained supermassive BH candidate in the universe, Sagittarius A* (Sgr A*). Three different types of experiments hold the promise to test GR in a strong-field regime using observations of Sgr A* with new-generation instruments. The first experiment consists of making a standard astronomical image of the synchrotron emission from the relativistic plasma accreting onto Sgr A*. This emission forms a “shadow” around the event horizon cast against the background, whose predicted size ([Formula: see text]as) can now be resolved by upcoming very long baseline radio interferometry experiments at mm-waves such as the event horizon telescope (EHT). The second experiment aims to monitor stars orbiting Sgr A* with the next-generation near-infrared (NIR) interferometer GRAVITY at the very large telescope (VLT). The third experiment aims to detect and study a radio pulsar in tight orbit about Sgr A* using radio telescopes (including the Atacama large millimeter array or ALMA). The BlackHoleCam project exploits the synergy between these three different techniques and contributes directly to them at different levels. These efforts will eventually enable us to measure fundamental BH parameters (mass, spin, and quadrupole moment) with sufficiently high precision to provide fundamental tests of GR (e.g. testing the no-hair theorem) and probe the spacetime around a BH in any metric theory of gravity. Here, we review our current knowledge of the physical properties of Sgr A* as well as the current status of such experimental efforts towards imaging the event horizon, measuring stellar orbits, and timing pulsars around Sgr A*. We conclude that the Galactic center provides a unique fundamental-physics laboratory for experimental tests of BH accretion and theories of gravity in their most extreme limits.

Collaborative international efforts under the name of the Event Horizon Telescope project, using sub- mm very long baseline interferometry, are soon expected to provide the first images of the shadow cast by the candidate supermassive black hole in our Galactic center, Sagittarius A*. Observations of this shadow would provide direct evidence of the existence of astrophysical black holes. Although it is expected that astrophysical black holes are described by the axisymmetric Kerr solution, there also exist many other black hole solutions, both in general relativity and in other theories of gravity, which cannot presently be ruled out. To this end, we present calculations of black hole shadow images from various metric theories of gravity as described by our recent work on a general parameterisation of axisymmetric black holes [R. Konoplya, L. Rezzolla and A. Zhidenko, Phys. Rev. D 93, 064015 (2016)]. An algorithm to perform general ray-tracing calculations for any metric theory of gravity is first outlined and then employed to demonstrate that even for extremal metric deformation parameters of various black hole spacetimes, this parameterisation is both robust and rapidly convergent to the correct solution.

Accretion of magnetized gas on compact astrophysical objects such as black holes (BHs) has been successfully modeled using general relativistic magnetohydrodynamic (GRMHD) simulations. These simulations have largely been performed in the Kerr metric, which describes the spacetime of a vacuum and stationary spinning BH in general relativity (GR). The simulations have revealed important clues to the physics of accretion flows and jets near the BH event horizon and have been used to interpret recent Event Horizon Telescope images of the supermassive BHs M87* and Sgr A*. The GRMHD simulations require the spacetime metric to be given in horizon-penetrating coordinates such that all metric coefficients are regular at the event horizon. Only a few metrics, notably the Kerr metric and its electrically charged spinning analog, the Kerr–Newman metric, are currently available in such coordinates. We report here horizon-penetrating forms of a large class of stationary, axisymmetric, spinning metrics. These can be used to carry out GRMHD simulations of accretion on spinning, nonvacuum BHs and non-BHs within GR, as well as accretion on spinning objects described by non-GR metric theories of gravity.

We propose the almost-geodesic motion of self-gravitating test bodies as a possible selection rule among metric theories of gravity. Starting from a heuristic statement, the ``gravitational weak equivalence principle,'' we build a formal operative test able to probe the validity of the principle for any metric theory of gravity in an arbitrary number of spacetime dimensions. We show that, if the theory admits a well-posed variational formulation, this test singles out only the purely metric theories of gravity. This conclusion reproduces known results in the cases of general relativity (as well as with a cosmological constant term) and scalar-tensor theories, but extends also to debated or unknown scenarios, such as the $f(R)$ and Lanczos-Lovelock theories. We thus provide new tools going beyond the standard methods, where the latter turn out to be inconclusive or inapplicable.

The violation of energy conservation law is a death sentence for the General Relativity Theory (GRT). This paper investigates the correctness of the General Relativity Theory by studying the energy conservation during the relativistic free fall of a small test body in a uniform gravitational field. The paper compares predictions of energy conservation obtained from the GRT and from the Metric Theory of Gravity (MTG). It is found that the gravitational mass dependence on velocity in the GRT is not correct, because this dependency leads to a prediction of violation of energy conservation while the MTG having a different gravitational mass dependency on velocity predicts correctly the energy conservation.

The increasing importance of relativistic gravity in astrophysics has led to the need for a detailed analysis of theories of gravity and their viability. Accordingly, in this thesis, metric theories of gravity are compiled, and are classified into four groups: (i) general relativity (ii) scalar-tensor theories (iii) conformally flat theories and (iv) stratified theories. The post-Newtonian limit of each theory is constructed and its Parametrized Post-Newtonian (PPN) values are obtained. These results, when combined with experimental data and with recent work by Nordtvedt and Will, show that, of all theories thus far examined by our group, the only currently viable ones are (i) general relativity, (ii) the Bergmann-Wagoner scalar-tensor theory and its special cases (Nordtvedt; Brans-Dicke-Jordan, (iii) recent, (as yet unpublished ) vector-tensor theory by Nordtvedt, Hellings, and Will, and (iv) a new stratified theory by the author, which is presented for the first time in this thesis. The PPN formalism is used to analyze stellar stability in any metric theory of gravity. This analysis enables one to infer, for any given gravitation theory, the extent to which post-Newtonian effects induce instabilities in white dwarfs, in neutron stars, and in supermassive stars. It also reveals the extent to which our current empirical knowledge of post-Newtonian gravity (based on solar-system experiments) actually guarantees that relativistic instabilities exist. In particular, it shows that for conservative theories of gravity, current solar-system experiments gua­rantee that relativistic corrections do induce dynamical instabilities in stars with adiabatic indices slightly greater than 4/3, while for non-conservative theories, current experiments do not permit any firm conclusion.

I show that several observable properties of bursting neutron stars in metric theories of gravity can be calculated using only conservation laws, Killing symmetries, and the Einstein equivalence principle, without requiring the validity of the general relativistic field equations. I calculate, in particular, the gravitational redshift of a surface atomic line, the touchdown luminosity of a radius-expansion burst, which is believed to be equal to the Eddington critical luminosity, and the apparent surface area of a neutron star as measured during the cooling tails of bursts. I show that, for a general metric theory of gravity, the apparent surface area of a neutron star depends on the coordinate radius of the stellar surface and on its gravitational redshift in the exact same way as in general relativity. On the other hand, the Eddington critical luminosity depends also on an additional parameter that measures the degree to which the general relativistic field equations are satisfied. These results can be used in conjunction with current and future high-energy observations of bursting neutron stars to test general relativity in the strong-field regime.

Cosmological and astrophysical observations lead to the emerging picture of a universe that is spatially flat and presently undertaking an accelerated expansion. The observations supporting this picture come from a range of measurements encompassing estimates of galaxy cluster masses, the Hubble diagram derived from type-Ia supernovae observations, the measurements of Cosmic Microwave Background radiation anisotropies, etc. The present accelerated expansion of the universe can be explained by admitting the existence of a cosmic fluid, with negative pressure. In the simplest scenario, this unknown component of the universe, the Dark Energy, is represented by the cosmological constant ([Formula: see text]), and accounts for about 70% of the global energy budget of the universe. The remaining 30% consist of a small fraction of baryons (4%) with the rest being Cold Dark Matter (CDM). The Lambda Cold Dark Matter ([Formula: see text]CDM) model, i.e. General Relativity with cosmological constant, is in good agreement with observations. It can be assumed as the first step towards a new standard cosmological model. However, despite the satisfying agreement with observations, the [Formula: see text]CDM model presents lack of congruence and shortcomings and therefore theories beyond Einstein’s General Relativity are called for. Many extensions of Einstein’s theory of gravity have been studied and proposed with various motivations like the quest for a quantum theory of gravity to extensions of anomalies in observations at the solar system, galactic and cosmological scales. These extensions include adding higher powers of Ricci curvature [Formula: see text], coupling the Ricci curvature with scalar fields and generalized functions of [Formula: see text]. In addition, when viewed from the perspective of Supergravity (SUGRA), many of these theories may originate from the same SUGRA theory, but interpreted in different frames. SUGRA therefore serves as a good framework for organizing and generalizing theories of gravity beyond General Relativity. All these theories when applied to inflation (a rapid expansion of early universe in which primordial gravitational waves might be generated and might still be detectable by the imprint they left or by the ripples that persist today) can have distinct signatures in the Cosmic Microwave Background radiation temperature and polarization anisotropies. We give a review of [Formula: see text]CDM cosmology and survey the theories of gravity beyond Einstein’s General Relativity, specially which arise from SUGRA, and study the consequences of these theories in the context of inflation and put bounds on the theories and the parameters therein from the observational experiments like PLANCK, Keck/BICEP, etc. The possibility of testing these theories in the near future in CMB observations and new data coming from colliders like the LHC, provides an unique opportunity for constructing verifiable models of particle physics and General Relativity.

This paper investigates by simple means the relativistic accelerated motion of a small test body in a simulated uniform gravitational like field and compares the predictions of energy loss, perhaps by radiation, obtained from the General Relativity Theory (GRT) and from the Metric Theory of Gravity (MTG). The study is first conducted in a flat Minkowski space-time with simulated constant gravitational like force and later in a true curved space-time with a metric, which, however, is not derived from the GRT. It is found that the gravitational mass dependence on velocity in GRT is not correct, because it predicts a negative loss of energy while the MTG predicts correctly a positive loss. The energy is conserved in a curved space-time free fall where the gravitational mass does not depend on velocity. There can be no energy radiation during the test body free fall in a uniform gravitational field.

We propose a new parametric framework to describe in generic metric theories of gravity the spacetime of spherically symmetric and slowly rotating black holes. In contrast to similar approaches proposed so far, we do not use a Taylor expansion in powers of M/r, where M and r are the mass of the black hole and a generic radial coordinate, respectively. Rather, we use a continued-fraction expansion in terms of a compactified radial coordinate. This choice leads to superior convergence properties and allows us to approximate a number of known metric theories with a much smaller set of coefficients. The measure of these coefficients via observations of near-horizon processes can be used to effectively constrain and compare arbitrary metric theories of gravity. Although our attention is here focussed on spherically symmetric black holes, we also discuss how our approach could be extended to rotating black holes.

We present a simple method of deriving the semiclassical equations of motion for a spinning particle in a gravitational field. We investigate the cases of classical, rotating particles, i.e. the so-called pole–dipole particles, as well as particles with an additional intrinsic spin. We show that, starting with a simple Lagrangian, one can derive equations for the spin evolution and momentum propagation in the framework of metric theories of gravity (general relativity) and in theories based on a Riemann–Cartan geometry (Poincaré gauge theory), without explicitly referring to matter current densities (spin and stress energy). Our results agree with those derived from the multipole expansion of the current densities by the conventional Papapetrou method and from the WKB analysis for elementary particles.

This thesis presents theoretical frameworks for the analysis and testing of gravitation theories - both metric and non-metric. For non-metric theories, the high-precision Eotvos-Dicke-Braginskii (EDB) experiments are demonstrated to be powerful tests of their gravitational coupling to electromagnetic interactions. All known non-metric theories are ruled out to within the precision of the EDB experiments. We present a new metric theory of gravity that cannot be distinguished from general relativity in all current and planned solar system experiments. However, this theory has very different gravitational-wave properties. Hence, we point out the need for further tests of metric theories beyond the Parametrized Post-Newtonian formalism, and emphasize the importance of the observation of gravitational waves as a tool for testing relativistic gravity in the future. A theory-independent formalism delineating the properties of weak, plane gravitational waves in metric theories is set up. General conservation laws that follow from variational principles in metric theories of gravity are investigated.

The direct observation of gravitational waves will provide a unique tool for probing the dynamical properties of highly compact astrophysical objects, mapping ultra-relativistic regions of space-time, and testing Einstein's general theory of relativity. LISA (Laser Interferometer Space Antenna), a joint NASA-ESA mission to be launched in the next decade, will perform these scientific tasks by detecting and studying low-frequency cosmic gravitational waves through their influence on the phases of six modulated laser beams exchanged between three remote spacecraft. By directly measuring the polarization components of the waves LISA will detect, we will be able to test Einstein's theory of relativity with good sensitivity. Since a gravitational wave signal predicted by the most general relativistic metric theory of gravity accounts for {\it six} polarization modes (the usual two Einstein's tensor polarizations as well as two vector and two scalar wave components), we have derived the LISA Time-Delay Interferometric responses and estimated their sensitivities to vector- and scalar-type waves. We find that (i) at frequencies larger than roughly the inverse of the one-way light time ($\approx 6 \times 10^{-2} $ Hz.) LISA is more than ten times sensitive to scalar-longitudinal and vector signals than to tensor and scalar-transverse waves, and (ii) in the low part of its frequency band is equally sensitive to tensor and vector waves and somewhat less sensitive to scalar signals.

The first law of black hole mechanics (in the form derived by Wald) is expressed in terms of integrals over surfaces, at the horizon and spatial infinity, of a stationary, axisymmetric black hole, in a diffeomorphism-invariant Lagrangian theory of gravity. The original statement of the first law given by Bardeen, Carter, and Hawking for an Einstein-perfect fluid system contained, in addition, volume integrals of the fluid fields, over a spacelike slice stretching between these two surfaces. One would expect that Wald's methods, applied to a Lagrangian Einstein-perfect fluid formulation, would convert these terms to surface integrals. However, because the fields appearing in the Lagrangian of a gravitating perfect fluid are typically nonstationary (even in a stationary black-hole--perfect-fluid spacetime) a direct application of these methods generally yields restricted results. We therefore first approach the problem of incorporating general nonstationary matter fields into Wald's analysis, and derive a first-law-like relation for an arbitrary Lagrangian metric theory of gravity coupled to arbitrary Lagrangian matter fields, requiring only that the metric field be stationary. This relation includes a volume integral of matter fields over a spacelike slice between the black hole horizon and spatial infinity, and reduces to the first law originally derived by Bardeen, Carter, and Hawking when the theory is general relativity coupled to a perfect fluid. We then turn to consider a specific Lagrangian formulation for an isentropic perfect fluid given by Carter, and directly apply Wald's analysis, assuming that both the metric and fluid fields are stationary and axisymmetric in the black hole spacetime. The first law we derive contains only surface integrals at the black hole horizon and spatial infinity, but the assumptions of stationarity and axisymmetry of the fluid fields make this relation much more restrictive in its allowed fluid configurations and perturbations than that given by Bardeen, Carter, and Hawking. In the Appendix, we use the symplectic structure of the Einstein-perfect fluid system to derive a conserved current for perturbations of this system: this current reduces to one derived ab initio for this system by Chandrasekhar and Ferrari.

A certain vector-tensor (VT) theory of gravitation was tested in previous papers. In the background universe, the vector field of the theory has a certain energy density, which is appropriate to play the role of vacuum energy (cosmological constant). Moreover, this background and its perturbations may explain the temperature angular power spectrum of the cosmic microwave background (CMB) obtained with WMAP (Wilkinson Map Anisotropy Probe), and other observations, as e.g., the Ia supernova luminosities. The parametrized post-Newtonian limit of the VT theory has been proved to be identical to that of general relativity (GR), and there are no quantum ghosts and classical instabilities. Here, the stationary spherically symmetric solution, in the absence of any matter content, is derived and studied. The metric of this solution is formally identical to that of the Reissner-Nordstr\"om-de Sitter solution of GR, but the role of the electrical charge is played by a certain quantity $\Gamma $ depending on both the vector field and the parameters of the VT theory. The black hole and cosmological horizons are discussed. The radius of the VT black hole horizon deviates with respect to that of the Kottler-Schwarzschild-de Sitter radius. Realistic relative deviations depend on $\Gamma $ and reach maximum values close to 30 per cent. For large enough $\Gamma $ values, there is no any black hole horizon, but only a cosmological horizon. The radius of this last horizon is almost independent of the mass source, the vector field components, and the VT parameters. It essentially depends on the cosmological constant value, which has been fixed by using cosmological observational data (CMB anisotropy, galaxy correlations and so on).