Time-dependent accretion disks around compact objects. Theoretical frameworks for analyzing and testing gravitation theories

After an introduction to theories of gravity alternative to general relativity, metric theories (Sec. 1) and the parametrized post-Newtonian (PPN) formalism (Sec. 2), we define a new class of metric theories of gravity (Sec. 3). It turns out that the post-Newtonian approximation of these new theories is not described by the PPN formalism (Sec. 4); in fact, in the limit of weak field and slow motions, the post-Newtonian expression of the metric tensor contains an, a priori, infinite set of new terms and correspondingly an, a priori, infinite set of new PPN parameters. As a consequence, the parametrized post-Newtonian formulas describing the classical relativistic tests should include these new parameters, and therefore the experimental values of the classical relativistic effects should not be used to put limits only on the standard ten PPN parameters. Finally, we note that a subset of this new class of theories has the same post-Newtonian limit and value of the PPN parameters as general relativity, and therefore is automatically in agreement with the classical general-relativistic tests (Sec. 4, theory III).

If the Pioneer anomaly has a gravitational origin, it would, according to the equivalence principle, distort the motions of the planets in the Solar System. Since no anomalous motion of the planets has been detected, it is generally believed that the Pioneer anomaly can not originate from a gravitational source in the Solar System. However, this conclusion becomes less obvious when considering models that either imply modifications to gravity over long distances or gravitational sources localized to the outer Solar System, given the uncertainty in the orbital parameters of the outer planets. Following the general assumption that the Pioneer spacecraft move geodesically in a spherically symmetric space-time metric, we derive the metric disturbance that is needed in order to account for the Pioneer anomaly. We then analyze the residual effects on the astronomical observables of the three outer planets that would arise from this metric disturbance, given an arbitrary metric theory of gravity. Providing a method for comparing the computed residuals with actual residuals, our results imply that the presence of a perturbation to the gravitational field necessary to induce the Pioneer anomaly is in conflict with available data for the planets Uranus and Pluto, but not for Neptune. Wemore » therefore conclude that the motion of the Pioneer spacecraft must be nongeodesic. Since our results are model-independent within the class of metric theories of gravity, they can be applied to rule out any model of the Pioneer anomaly that implies that the Pioneer spacecraft move geodesically in a perturbed space-time metric, regardless of the origin of this metric disturbance.« less

Einstein’s theory of gravity has been extensively tested on solar system scales, and for isolated astrophysical systems, using the perturbative framework known as the parameterized post-Newtonian (PPN) formalism. This framework is designed for use in the weak-field and slow-motion limit of gravity, and can be used to constrain a large class of metric theories of gravity with data collected from the aforementioned systems. Given the potential of future surveys to probe cosmological scales to high precision, it is a topic of much contemporary interest to construct a similar framework to link Einstein’s theory of gravity and its alternatives to observations on cosmological scales. Our approach to this problem is to adapt and extend the existing PPN formalism for use in cosmology. We derive a set of equations that use the same parameters to consistently model both weak fields and cosmology. This allows us to parameterize a large class of modified theories of gravity and dark energy models on cosmological scales, using just four functions of time. These four functions can be directly linked to the background expansion of the universe, first-order cosmological perturbations, and the weak-field limit of the theory. They also reduce to the standard PPN parameters on solar system scales. We illustrate how dark energy models and scalar-tensor and vector-tensor theories of gravity fit into this framework, which we refer to as ‘parameterized post-Newtonian cosmology’ (PPNC).

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 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.

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.

The successful background-independent quantization of Loop Quantum Gravity relies on the key observation that classical General Relativity can be cast into the connection-dynamical formalism with the structure group of SU(2). Due to this particular formalism, Loop Quantum Gravity was generally considered as a quantization scheme that applies only to General Relativity. However, we will show that the nonperturbative quantization procedure of Loop Quantum Gravity can be extended to a rather general class of metric theories of gravity, which have received increased attention recently due to motivations coming form cosmology and astrophysics. In particular, we will first introduce how to reformulate the 4-dimensional metric f(R) theories of gravity, as well as Brans-Dicke theory, into connection-dynamical formalism with real SU(2) connections as configuration variables. Through these formalisms, we then outline the nonpertubative canonical quantization of the f(R) theories and Brans-Dicke theory by extending the loop quantization scheme of General Relativity.

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.

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.

We investigate the possibility of testing of the Einstein Equivalence\nPrinciple (EEP) using measurements of anomalous magnetic moments of elementary\nparticles. We compute the one loop correction for the $g-2$ anomaly within the\nclass of non metric theories of gravity described by the \\tmu formalism. We\nfind several novel mechanisms for breaking the EEP whose origin is due purely\nto radiative corrections. We discuss the possibilities of setting new empirical\nconstraints on these effects.\n

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.

view Abstract Citations (26) References (18) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Gravitational Stokes Parameters Anile, A. Marcello ; Breuer, Reinhard A. Abstract The electromagnetic and gravitational Stokes parameters are defined in the general theory of relativity. The general-relativistic equation of radiative transfer for polarized radiation is then derived in terms of the Stokes parameters, for both high-frequency electromagnetic and gravitational waves. We conclude by generalizing the concept of Stokes parameters for the most general class of metric theories of gravity, where six (instead of two) independent states of polarization are present. Sabject headings: polarization - radiative transfer - relativity Publication: The Astrophysical Journal Pub Date: April 1974 DOI: 10.1086/152766 Bibcode: 1974ApJ...189...39A full text sources ADS |

The angular momentum of the Earth produces gravitomagnetic components of the Riemann curvature tensor, which are of the order of ${10}^{\mathrm{\ensuremath{-}}10}$ of the Newtonian tidal terms arising from the mass of the Earth. These components could be detected in principle by sensitive superconducting gravity gradiometers currently under development. We lay out the theoretical principles of such an experiment by using the parametrized post-Newtonian formalism to derive the locally measured Riemann tensor in an orbiting proper reference frame, in a class of metric theories of gravity that includes general relativity. A gradiometer assembly consisting of three gradiometers with axes at mutually right angles measures three diagonal components of a 3\ifmmode\times\else\texttimes\fi{}3 ``tidal tensor,'' related to the Riemann tensor. We find that, by choosing a particular assembly orientation relative to the orbit and taking a sum and difference of two of the three gradiometer outputs, one can isolate the gravitomagnetic relativistic effect from the large Newtonian background.

A Lagrangian-based metric theory of gravity is developed with three adjustable constants and two tensor fields, one of which is a nondynamic 'flat space metric' eta. With a suitable cosmological model and a particular choice of the constants, the 'post-Newtonian limit' of the theory agrees, in the current epoch, with that of general relativity theory (GRT); consequently the theory is consistent with current gravitation experiments. Because of the role of eta, the gravitational 'constant' G is time-dependent and gravitational waves travel null geodesics of eta rather than the physical metric g. Gravitational waves possess six degrees of freedom. The general exact static spherically-symmetric solution is a four-parameter family. Future experimental tests of the theory are discussed.

view Abstract Citations (162) References (15) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Time-dependent accretion disks around compact objects. I. Theory and basic equations. Lightman, A. P. Abstract We consider accretion disks around compact stars and black holes, generalizing the stationary disk models of various authors to allow time dependence on the radial-flow time scale. The structure and evolution of the disk are governed by an 'evolution equation' for matter surface density, plus a set of implicit algebraic equations determining various thermodynamic and radiation variables in terms of surface density. The analytic structure of these equations is studied. It is shown that there exists a maximum permissible value of the surface density at which gas and radiation pressure are approximately equal, and beyond which viscosity generates more energy than radiative transport can remove. Publication: The Astrophysical Journal Pub Date: December 1974 DOI: 10.1086/153259 Bibcode: 1974ApJ...194..419L Keywords: Binary Stars; Black Holes (Astronomy); Magnetohydrodynamics; Rotating Disks; Stellar Gravitation; Time Dependence; Astronomical Models; Density Distribution; Deposition; Neutron Stars; Nonlinear Equations; Plasma Dynamics; Radiative Transfer; Stellar Evolution; Viscosity; X Ray Sources; Astrophysics full text sources ADS |