Theoretical Frameworks for Testing Relativistic Gravity.IV. a Compendium of Metric Theories of Gravity and Their POST Newtonian Limits

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the Bergmann-Wagoner scalar-tensor theory and its special cases (Nordtvedt; Brans-Dicke-Jordan, (iii) recent, (as yet unpublished ) vector-tensor theory by Nordtvedt, Hellings, and Will, and (iv) a new stratified theory by the author, which is presented for the first time in this thesis.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.

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.

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).

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.

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.

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.

Laser ranging, both Lunar (LLR) and Satellite Laser Ranging (SLR), is one of the most accurate techniques to test gravitational physics and Einstein’s theory of General Relativity. Lunar Laser Ranging has provided very accurate tests of both the strong equivalence principle, at the foundations of General Relativity, and of the weak equivalence principle, at the basis of any metric theory of gravity; it has provided strong limits to the values of the so-called PPN (Parametrized Post-Newtonian) parameters, that are used to test the post-Newtonian limit of General Relativity, strong limits to conceivable deviations to the inverse square law for very weak gravity and accurate measurements of the geodetic precession, an effect predicted by General Relativity. Satellite laser ranging has provided strong limits to deviations to the inverse square gravity law, at a different range with respect to LLR, and in particular has given the first direct test of the gravitomagnetic field by measuring the gravitomagnetic shift of the node of a satellite, a frame-dragging effect also called Lense-Thirring effect. Here, after an introduction to gravitomagnetism and frame-dragging, we describe the latest results in measuring the Lense-Thirring effect using the LAGEOS satellites and the latest gravity field models obtained by the space mission GRACE. Finally, we describe an update of the LARES (LAser RElativity Satellite) mission. LARES is planned for launch in 2011 to further improve the accuracy in the measurement of frame-dragging.

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).

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.

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 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.   

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 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.

The confrontation between general relativity and experiment: An update