Gravitational lensing with f (χ) = χ3/2 gravity in accordance with astrophysical observations

In this article, we present the results of a fourth order perturbation analysis of the metric theory of gravity f(chi) = chi^(3/2) , with chi a suitable dimensionless Ricci scalar. Such a model corresponds to a specific f(R) metric theory of gravity, where the mass of the system is included in the gravitational field's action. In previous works we have shown that, up to the second order in perturbations, this theory reproduces the flat rotation curves of galaxies and the details of the gravitational lensing in individual, groups, and clusters of galaxies. Here, leaving fixed the results from our previous works, we show that the theory reproduces the dynamical masses of 12 Chandra X-ray galaxy clusters, without the need of dark matter, through the metric coefficients up to the fourth order of approximation. In this sense, we calculate the first relativistic correction of the f(chi) metric theory and apply it to fit the dynamical masses of clusters of galaxies.

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.

Gravitational lensing in metric theories of gravity is discussed. I introduce a generalized approximate metric element, inclusive of both post-post-Newtonian contributions and a gravitomagnetic field. Following Fermat's principle and standard hypotheses, I derive the time delay function and deflection angle caused by an isolated mass distribution. Several astrophysical systems are considered. In most of the cases, the gravitomagnetic correction offers the best perspectives for an observational detection. Actual measurements distinguish only marginally different metric theories from each other.

From a parametrized post-Newtonian (PPN) perspective, we address the question of whether or not the new degrees of freedom represented by the PPN potentials can lead to significant modifications in the dynamics of galaxies in the direction of rendering dark matter obsolete. Here, we focus on the study of rotation curves associated with spherically symmetric configurations. The values for the post-Newtonian parameters, which help us to classify the different metric theories of gravity, are tightly constrained, mainly by Solar System experiments. Such restrictions render the modifications of gravitational effects, with respect to general relativity (GR), to be insignificant, making attempts to find alternative metrical theories rather fruitless. However, in recent years, metric theories characterized by screening mechanisms have become popular, due to the fact that they lead to the possibility of modifications in larger scales than the Solar System while retaining the success of GR on it, allowing for violations of the constraints of the post-Newtonian parameters. In such a context, we consider here two kinds of solutions for field equations: (i) Vacuum solutions (i.e., when no matter fields are present) and (ii) fields in the presence of a polytropic distribution of matter. For case (i), we find that the post-Newtonian corrections do not lead to modifications significant enough to be considered an alternative to the dark matter hypothesis. In case (ii), we find that for a wide range of values for the PPN parameters $\ensuremath{\gamma}$, $\ensuremath{\beta}\ensuremath{\le}1$, $\ensuremath{\xi}\ensuremath{\ge}0$, ${\ensuremath{\alpha}}_{3}$, ${\ensuremath{\zeta}}_{1}$, and ${\ensuremath{\zeta}}_{2}$, the need for dark matter is unavoidable in order to find flat rotation curves. It is only for theories in which ${\ensuremath{\zeta}}_{3}g0$, or $\ensuremath{\beta}g1$, or $\ensuremath{\xi}l0$ that some resemblance of flat rotation curves is found. But the latter two require some direct fine-tuning of the screening radius ${r}_{c}$, while ${\ensuremath{\zeta}}_{3}g0$ implies the most sound modifications. The latter suggests, at least for the models considered, that these are the only theories (consistent with the usual PPN approach) capable of replacing dark matter as a possible explanation for the dynamics of galaxies.

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 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.

Pulsar timing experiments aimed at the detection of gravitational radiation have been performed for decades now. With the forthcoming construction of large arrays capable of tracking multiple millisecond pulsars, it is very likely we will be able to make the first detection of gravitational radiation in the nano-Hertz band, and test Einstein's theory of relativity by measuring the polarization components of the detected signals. Since a gravitational wave 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 estimated the single-antenna sensitivities to these six polarizations. We find pulsar timing experiments to be significantly more sensitive, over their entire observational frequency band ($\approx 10^{-9} - 10^{-6}$ Hz), to scalar-longitudinal and vector waves than to scalar-transverse and tensor waves. At $10^{-7}$ Hz and with pulsars at a distance of $1$ kpc, for instance, we estimate an average sensitivity to scalar-longitudinal waves that is more than two orders of magnitude better than the sensitivity to tensor waves. Our results imply that a direct detection of gravitational radiation by pulsar timing will result into a test of the theory of general relativity that is more stringent than that based on monitoring the decay of the orbital period of a binary system.

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.

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.

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)

view Abstract Citations (52) References (19) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Testing Einstein's Theory of Gravity by Analyzing Lunar Laser Ranging Data Mueller, J. ; Schneider, M. ; Soffel, M. ; Ruder, H. Abstract Lunar Laser Ranging data of more than 20 years have been used to accurately determine parameters of the earth-moon system (e.g., the mass of the system, the lunar tidal acceleration etc.), the station-reflector geometry, and other physical parameters like the solar quadrupole moment. In addition to these, parameters suitable for testing metric theories of gravity in the first post-Newtonian approximation have been determined with great accuracy. These are the parameterized post-Newtonian parameters gamma and beta, the Nordtvedt parameter, the geodetic precession of the lunar orbit, as well as a possible time variation of the gravitational constant. Publication: The Astrophysical Journal Pub Date: December 1991 DOI: 10.1086/186222 Bibcode: 1991ApJ...382L.101M Keywords: Earth-Moon System; Einstein Equations; Gravitation Theory; Laser Range Finders; Lunar Rangefinding; Ephemeris Time; Geodesy; Precession; Relativistic Theory; Tides; Astronomy; EARTH: GENERAL; FUNDAMENTAL CONSTANTS; MOON; RELATIVITY; SOLAR SYSTEM: GENERAL; SUN: INTERIOR full text sources ADS |