Publisher Correction: The Newtonian limit of orthonormal frames in metric theories of gravity

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.

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 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.

We discuss the advantages of using metric theories of gravity with curvature–matter couplings in order to construct a relativistic generalization of the simplest version of Modified Newtonian Dynamics (MOND), where Tully–Fisher scalings are valid for a wide variety of astrophysical objects. We show that these proposals are valid at the weakest perturbation order for trajectories of massive and massless particles (photons). These constructions can be divided into local and non-local metric theories of gravity with curvature–matter couplings. Using the simplest two local constructions in an FLRW universe for dust, we show that there is no need for the introduction of dark matter and dark energy components into the Friedmann equation in order to account for type Ia supernovae observations of an accelerated universe at the present epoch.

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.

In this article we perform a second order perturbation analysis of the gravitational metric theory of gravity $ f(\chi) = \chi^{3/2} $ developed by Bernal et al. (2011). We show that the theory accounts in detail for two observational facts: (1) the phenomenology of flattened rotation curves associated to the Tully-Fisher relation observed in spiral galaxies, and (2) the details of observations of gravitational lensing in galaxies and groups of galaxies, without the need of any dark matter. We show how all dynamical observations on flat rotation curves and gravitational lensing can be synthesised in terms of the empirically required metric coefficients of any metric theory of gravity. We construct the corresponding metric components for the theory presented at second order in perturbation, which are shown to be perfectly compatible with the empirically derived ones. It is also shown that under the theory being presented, in order to obtain a complete full agreement with the observational results, a specific signature of Riemann's tensor has to be chosen. This signature corresponds to the one most widely used nowadays in relativity theory. Also, a computational program, the MEXICAS (Metric EXtended-gravity Incorporated through a Computer Algebraic System) code, developed for its usage in the Computer Algebraic System (CAS) Maxima for working out perturbations on a metric theory of gravity, is presented and made publicly available.

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.

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.

view Abstract Citations (5) References (8) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Relativistic tidal forces. Nordtvedt, K. Abstract The post-Newtonian, relativistic (1/c-squared) tidal force is calculated within the general PPN framework of all metric theories of gravity. There are no relativistic tides in general relativity, but they are generally nonzero in other metric theories of gravity, including scalar-tensor theories. In close binaries, such as the binary pulsar system PSR 1913+16, the relativistic tides can be orders of magnitude larger than Newtonian tides. In 'preferred frame' theories of gravity in which the PPN coefficient alpha(1) is nonzero, the relativistic tidal field is not the gradient of a scalar potential, but includes also a circulating, nonconservative field in a body. Publication: The Astrophysical Journal Pub Date: January 1983 DOI: 10.1086/160633 Bibcode: 1983ApJ...264..620N Keywords: Binary Stars; Free Fall; Gravitation Theory; Relativity; Tides; Equations Of Motion; Pulsars; Scalars; Space-Time Functions; Astrophysics full text sources ADS | data products SIMBAD (1)

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.

We develop the parametrized post-Newtonian (PPN) formalism, which encompasses the weak-field, slow-motion regime, known as the post-Newtonian limit, of a wide range of metric theories of gravity. Ten PPN parameters are introduced, whose values depend upon the theory of gravity under study. We show that general properties of metric theories of gravity may be reflected in specific values of the PPN parameters, including the presence or absence of a preferred universal frame of reference, and the presence or absence of global conservation laws for energy, momentum and angular momentum.

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.

Part I. The theory of time-independent accretion disks around compact objects is developed, generalizing the stationary models of various authors to allow time dependence on the radial-flow time scale. Equations are derived for the time evolution of matter surface density [Sigma] and for implicit expressions of relevant disk variables in terms of [Sigma]. Analytic and numerical studies of these equations yield numerical models of mass accretion from a disk onto a compact object and a discovery of the unstable nature of the inner region of the disk, causing a breakdown of current accretion disk models. Part II. Theoretical frameworks for analyzing and testing gravitation theories are developed for both nonmetric and metric theories. Highly precise experimental confirmation of the Weak Equivalence Principle is shown to be deadly if not fatal evidence for ruling out all nonmetric theories of gravity. For the class of metric theories we demonstrate the necessity for going beyond current frameworks of analysis (e.g.,the PPN framework) by constructing a new theory of gravity identical to GRT in the Post-Newtonian limit. As a first step in transcending current frameworks, we develop a formalism for delineating and testing all metric theories of gravity on the basis of their gravitational-wave properties and thereby emphasize gravitational-wave observations as a future tool for testing gravitation theories. We also investigate conservation laws and some common properties of Lagrangian-based metric theories of gravity.

Metric theories of gravity are compiled and classified according to the types of gravitational fields they contain, and the modes of interaction among those fields. The gravitation theories considered are classified as (1) general relativity, (2) scalar-tensor theories, (3) conformally flat theories, and (4) stratified theories with conformally flat space slices. The post-Newtonian limit of each theory is constructed and its Parametrized Post-Newtonian (PPN) values are obtained by comparing it with Will's version of the formalism. Results obtained here, when combined with experimental data and with recent work by Nordtvedt and Will and by Ni, show that, of all theories thus far examined by our group, the only currently viable ones are general relativity, the Bergmann-Wagoner scalar-tensor theory and its special cases (Nordtvedt; Brans-Dicke-Jordan), and a recent, new vector-tensor theory by Nordtvedt, Hellings, and Will.