New parametrization for spherically symmetric black holes in metric theories of gravity

Following previous work of ours in spherical symmetry, we here propose a new\nparametric framework to describe the spacetime of axisymmetric black holes in\ngeneric metric theories of gravity. In this case, the metric components are\nfunctions of both the radial and the polar angular coordinates, forcing a\ndouble expansion to obtain a generic axisymmetric metric expression. In\nparticular, we use a continued-fraction expansion in terms of a compactified\nradial coordinate to express the radial dependence, while we exploit a Taylor\nexpansion in terms of the cosine of the polar angle for the polar dependence.\nThese choices lead to a superior convergence in the radial direction and to an\nexact limit on the equatorial plane. As a validation of our approach, we build\nparametrized representations of Kerr, rotating dilaton, and\nEinstein-dilaton-Gauss-Bonnet black holes. The match is already very good at\nlowest order in the expansion and improves as new orders are added. We expect a\nsimilar behavior for any stationary and axisymmetric black-hole metric.\n

The general parametrization of spherically symmetric and asymptotically flat black-hole spacetimes in arbitrary metric theories of gravity was suggested in [3]. The parametrization is based on the continued fraction expansion in terms of the compact radial coordinate and has superior convergence and strict hierarchy of parameters. It is known that some observable quantities, related to particle motion around the black hole, such as the eikonal quasinormal modes, radius of the shadow, frequency at the innermost stable circular orbit, and others, depend mostly on only a few of the lowest coefficients of the parametrization. Here we continue this approach by studying the dominant (low-lying) quasinormal modes for such generally parametrized black holes. We show that, due to the hierarchy of parameters, the dominant quasinormal frequencies are also well determined by only the first few coefficients of the expansion for the so-called moderate black-hole geometries. The latter are characterized by a relatively slow change of the metric functions in the radiation zone near the black hole. The nonmoderate metrics, which change strongly between the event horizon and the innermost stable circular orbit are usually characterized by echoes or by the distinctive (from the Einstein case) quasinormal ringing which does not match the current observational data. Therefore, the compact description of a black-hole spacetime in terms of the truncated general parametrization is an effective formalism for testing strong gravity and imposing constraints on allowed black-hole geometries.

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.

Increasing sophistication and precision of experimental tests of relativistic gravitation theories has led to the need for a detailed theoretical framework for analysing and interpreting these experiments. Such a framework is the Parametrized Post-Newtonian (PPN) formalism, which treats the post-Newtonian limit of arbitrary metric theories of gravity in terms of nine metric parameters, whose values vary from theory to theory. The theoretical and experimental foundations of the PPN formalism are laid out and discussed, and the detailed definitions and equations for the formalism are given. It is shown that some metric theories of gravity predict that a massive, self-gravitating body's passive gravitational mass should not be equal to its inertial mass, but should be an anisotropic tensor which depends on the body's self-gravitational energy (violation of the principle of equivalence). Two theorems are presented which probe the theoretical structure of the PPN formalism. They state that (i) a metric theory of gravity possesses post-Newtonian integral conservation laws if and only if its nine PP parameters have values which satisfy a set of seven constraint equations, and (ii) a metric theory of gravity is invariant under asymptotic Lorentz transformations if and only if its PPN parameters satisfy a set of three constraint equations. Some theories of gravity (including Whitehead's theory and theories which violate one of the Lorentz-invariance parameter constraints) are shown to predict an anisotropy in the Newtonian gravitational constant. Gravimeter data on the tides of the solid Earth are used to put an upper limit on the magnitude of the predicted anisotropy, and thence to rule out such theories.

This paper is written mostly in an overview style in its nature thus avoiding many equations and computations, which casual readers do not necessarily understand. Paper investigates and compares side by side in detail assumptions with their logical consequences and resulting internal inconsistencies in both; the General Relativity Theory and the Metric Theory of Gravity. It is found that the GRT has many such internal inconsistencies, which have to be corrected by unusual and difficult to believe assumptions that are not backed up by a typical experience one encounters in a real life, while the MTG avoids such problems. For the readers who are interested in proofs of discussed findings the paper provides internet links to papers where such proofs are available. The key differences between the GRT and MTG theories are: the gravitational mass dependence on velocity, nature of the “empty” space, the finite or infinite size of the Universe, the existence of Black Holes (BH) with their Event Horizons (EH), the creation of Universe by the Big Bang (BB), and the relation between the Cosmic Microwave Background Radiation (CMBR) temperature, and the Hubble constant that characterizes the velocity of receding Galaxies.   

The general parametrization for spacetimes of spherically symmetric Lorentzian, traversable wormholes in an arbitrary metric theory of gravity is presented. The parametrization is similar in spirit to the post-Newtonian parametrized formalism, but with validity that extends beyond the weak field region and covers the whole space. Our method is based on a continued-fraction expansion in terms of a compactified radial coordinate. Calculations of shadows and quasinormal modes for various examples of parametrization of known wormhole metrics that we have performed show that, for most cases, the parametrization provides excellent accuracy already at the first order. Therefore, only a few parameters are dominant and important for finding potentially observable quantities in a wormhole background. We have also extended the analysis to the regime of slow rotation.

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.

Here we have developed the general parametrization for spherically symmetric and asymptotically flat black-hole spacetimes in an arbitrary metric theory of gravity. The parametrization is similar in spirit to the parametrized post-Newtonian (PPN) approximation, but valid in the whole space outside the event horizon, including the near horizon region. This generalizes the continued-fraction expansion method in terms of a compact radial coordinate suggested by Rezzolla and Zhidenko [Phys.Rev.D 90 8, 084009 (2014)] for the four-dimensional case. As the first application of our higher-dimensional parametrization we have approximated black-hole solutions of the Einstein-Lovelock theory in various dimensions. This allows one to write down the black-hole solution which depends on many parameters (coupling constants in front of higher curvature terms) in a very compact analytic form, which depends only upon a few parameters of the parametrization. The approximate metric deviates from the exact (but extremely cumbersome) expressions by fractions of one percent even at the first order of the continued-fraction expansion, which is confirmed here by computation of observable quantities, such as quasinormal modes of the black hole.

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.

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.

Metric of axially symmetric asymptotically flat black holes in an arbitrary metric theory of gravity can be represented in the general form which depends on infinite number of parameters. We constrain this general class of metrics by requiring the existence of additional symmetries, which lead to the separation of variables in the Hamilton-Jacobi and Klein-Gordon equations, and show that once the metric functions change sufficiently moderately in some region near the black hole, the black-hole shadow depends on a few deformation parameters only. We analyze the influence of these parameters on the black-hole shadow. We also show that the shadow of the rotating black hole in the Einstein-dilaton-Gauss-Bonnet theory is well approximated if the terms violating the separation of variables are neglected in the metric.

Hod's proposal claims that the least damped quasinormal mode of a black hole must have the imaginary part smaller than half of the surface gravity at the event horizon. The Strong Cosmic Censorship in General Relativity implies that this bound must be even weaker: half of the surface gravity at the Cauchy horizon. The appealing question is whether these bounds are limited by the Einstein theory only? Here we will present numerical evidence that once the black hole size is much smaller than then the radius of the cosmological horizon, both the Hod's proposal and the strong cosmic censorship bound for quasinormal modes are satisfied for general spherically symmetric black holes in an arbitrary metric theory of gravity. The low-lying quasinormal frequencies have the universal behavior in this regime and do not depend on the near-horizon geometry, but only on the asymptotic parameters: the value of the cosmological constant and black hole mass.

Hod's proposal claims that the least damped quasinormal mode of a black hole must have the imaginary part smaller than half of the surface gravity at the event horizon.The Strong Cosmic Censorship in General Relativity implies that this bound must be even weaker: half of the surface gravity at the Cauchy horizon. The appealing question is whether these bounds are limited by the Einstein theory only?Here we will present numerical evidence that once the black hole size is much smaller than then the radius of the cosmological horizon, both the Hod's proposal and the strong cosmic censorship bound for quasinormal modes are satisfied for general spherically symmetric black holes in an arbitrary metric theory of gravity. The low-lying quasinormal frequencies have the universal behavior in this regime and do not depend on the near-horizon geometry, but only on the asymptotic parameters: the value of the cosmological constant and black hole mass.

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.

Conservation of energy, momentum, and angular momentum in metric theories of gravity is studied extensively both in Lagrangian formulations (using generalized Bianchi identities) and in the post-Newtonian limit of general metric theories. Our most important results are the following: (i) The matter response equations $T_{}^{\ensuremath{\mu}\ensuremath{\nu}}{}_{;\ensuremath{\nu}}{}^{}=0$ of any Lagrangian-based, generally covariant metric theory (LBGCM theory) are a consequence of the gravitational-field equations if and only if the theory contains no absolute variables. (ii) Almost all LBGCM theories possess conservation laws of the form $\ensuremath{\theta}_{\ensuremath{\mu}}^{}{}_{}{}^{\ensuremath{\nu}}{}_{,\ensuremath{\nu}}{}^{}{}_{}{}^{}=0$ (where $\ensuremath{\theta}_{\ensuremath{\mu}}^{}{}_{}{}^{\ensuremath{\nu}}$ reduces to $T_{\ensuremath{\mu}}^{}{}_{}{}^{\ensuremath{\nu}}$ in the absence of gravity). (iii) $\ensuremath{\theta}_{\ensuremath{\mu}}^{}{}_{}{}^{\ensuremath{\nu}}$ is always expressible in terms of a superpotential, $\ensuremath{\theta}_{\ensuremath{\mu}}^{}{}_{}{}^{\ensuremath{\nu}}=\ensuremath{\Lambda}_{\ensuremath{\mu}}^{}{}_{}{}^{[\ensuremath{\nu}\ensuremath{\alpha}]}{}_{,\ensuremath{\alpha}}{}^{}{}_{}{}^{}$, If the superpotential $\ensuremath{\Lambda}_{\ensuremath{\mu}}^{}{}_{}{}^{[\ensuremath{\nu}\ensuremath{\alpha}]}$ can be expressed in terms of asymptotic values of field quantities, then the conserved integral ${P}_{\ensuremath{\mu}}=\ensuremath{\int}\ensuremath{\theta}_{\ensuremath{\mu}}^{}{}_{}{}^{\ensuremath{\nu}}{d}^{3}{\ensuremath{\Sigma}}_{\ensuremath{\nu}}$ can be measured by experiments confined to the asymptotically flat region outside the source. (iv) In the Will-Nordtvedt ten-parameter post-Newtonian (PPN) formalism there exists a conserved ${P}_{\ensuremath{\mu}}$ if and only if the parameters obey five specific constraints; two additional constraints are needed for the existence of a conserved angular momentum ${J}_{\ensuremath{\mu}\ensuremath{\nu}}$ (This modifies and extends a previous result due to Will.) (v) We conjecture that for metric theories of gravity, the conservation of energy-momentum is equivalent to the existence of a Lagrangian formulation; and using the PPN formalism, we prove the post-Newtonian limit of this conjecture. (vi) We present "stress-energy-momentum complexes" $\ensuremath{\theta}_{\ensuremath{\mu}}^{}{}_{}{}^{\ensuremath{\nu}}$ for all currently viable metric theories known to us.