Newtonian Limit Of General Relativity Research Articles

Tidal disruption of white dwarfs in a modified gravity theory with SPH

Abstract Low energy imprints of modifications to general relativity are often found in pressure balance equations inside stars. These modifications are then amenable to tests via astrophysical phenomena, using observational effects in stellar astrophysics that crucially depend on such equations. One such effect is tidal disruption of stars in the vicinity of black holes. In this paper, using a numerical scheme modelled with smoothed particle hydrodynamics, we study real time tidal disruption of a class of white dwarfs by intermediate-mass black holes, in the low energy limit of a theory of modified gravity that alters the internal physics of white dwarfs, namely the Eddington inspired Born-Infeld theory. In this single parameter extension of general relativity, the mass-radius relation of white dwarfs as well as their tidal disruption radius depend on the modified gravity parameter, and these capture the effect of modifications to general relativity. Our numerical simulations incorporating these show that departure from general relativity in these scenarios might be observationally significant, and should therefore be contrasted with data. In particular, we study observationally relevant physical quantities, i.e., tidal kick velocity and trajectory deviation of the remnant core and fallback rates of the tidal debris in this theory and compare them to the Newtonian limit of general relativity. We also comment on the qualitative differences between the modified gravity theory and one with stellar rotation.

Teleparallel Newton–Cartan gravity

We discuss a teleparallel version of Newton–Cartan gravity. This theory arises as a formal large-speed-of-light limit of the teleparallel equivalent of general relativity (TEGR). Thus, it provides a geometric formulation of the Newtonian limit of TEGR, similar to standard Newton–Cartan gravity being the Newtonian limit of general relativity. We show how by a certain gauge-fixing the standard formulation of Newtonian gravity can be recovered.

Simultaneous determination of mass parameter and radial marker in Schwarzschild geometry

We show that mass parameter and radial marker values can be indirectly measured in thought experiments performed in Schwarzschild spacetime, without using the Newtonian limit of general relativity or approximations based on Euclidean geometry. Our approach involves different proper time quantifications as well as solutions to systems of algebraic equations, and aims to strengthen the conceptual independence of general relativity from Newtonian gravity.

Long-Range Scalar Forces in Five-Dimensional General Relativity

We present new results regarding the long-range scalar field that emerges from the classical Kaluza unification of general relativity and electromagnetism. The Kaluza framework reproduces known physics exactly when the scalar field goes to one, so we studied perturbations of the scalar field around unity, as is done for gravity in the Newtonian limit of general relativity. A suite of interesting phenomena unknown to the Kaluza literature is revealed: planetary masses are clothed in scalar field, which contributes 25% of the mass-energy of the clothed mass; the scalar potential around a planet is positive, compared with the negative gravitational potential; at laboratory scales, the scalar charge which couples to the scalar field is quadratic in electric charge; a new length scale of physics is encountered for the static scalar field around an electrically-charged mass, L s = μ 0 Q 2 / M ; the scalar charge of elementary particles is proportional to the electric charge, making the scalar force indistinguishable from the atomic electric force. An unduly strong electrogravitic buoyancy force is predicted for electrically-charged objects in the planetary scalar field, and this calculation appears to be the first quantitative falsification of the Kaluza unification. Since the simplest classical field, a long-range scalar field, is expected in nature, and since the Kaluza scalar field is as weak as gravity, we suggest that if there is an error in this calculation, it is likely to be in the magnitude of the coupling to the scalar field, not in the existence or magnitude of the scalar field itself.

Oscillating Cosmological Force Modifies Newtonian Dynamics

In the Newtonian limit of general relativity a force acting on a test mass in a central gravitational field is conventionally defined by the attractive Newtonian gravity (inverse square) term plus a small repulsive cosmological force, which is proportional to the slow acceleration of the universe expansion. In this paper we considered the cosmological-force correction due to fast quantum oscillations of the universe scale factor as a potential solution of the cosmological constant problem. These fast fluctuations of the cosmological scale factor violate Lorentz invariance at the Planck scale, and they induce strong changes to the current sign and magnitude of the average cosmological force, thus making it one of the potential probable causes for the modification of Newtonian dynamics in galaxy-scale systems. The modified cosmological force may be responsible for the recently discovered “cosmic-clock” behavior of disk galaxies in the low-redshift universe. The obtained results have strong implications for astroparticle physics since they demonstrate that typical galaxy rotation curves may be obtained without (or almost without) dark-matter particles.

More problems for Newtonian cosmology

I point out a radical indeterminism in potential-based formulations of Newtonian gravity once we drop the condition that the potential vanishes at infinity (as is necessary, and indeed celebrated, in cosmological applications). This indeterminism, which is well known in theoretical cosmology but has received little attention in foundational discussions, can be removed only by specifying boundary conditions at all instants of time, which undermines the theory's claim to be fully cosmological, i.e., to apply to the Universe as a whole. A recent alternative formulation of Newtonian gravity due to Saunders (Philosophy of Science 80 (2013) pp. 22-48) provides a conceptually satisfactory cosmology but fails to reproduce the Newtonian limit of general relativity in homogenous but anisotropic universes. I conclude that Newtonian gravity lacks a fully satisfactory cosmological formulation.

Constraining alternative theories of gravity using GW150914 and GW151226

The recently reported gravitational wave events GW$150914$ and GW$151226$ caused by the mergers of binary black holes [arXiv:1602.03841],[arXiv:1606.04855],[arXiv:1606.04856] provide a formidable way to set constraints on alternative metric theories of gravity in the strong field regime. In this paper, we develop an approach where an arbitrary theory of gravity can be parametrised by an effective coupling $G_{eff}$ and an effective gravitational potential $\Phi(r)$. The standard Newtonian limit of General Relativity is recovered as soon as $G_{eff}\rightarrow G_N$ and $\Phi(r)\rightarrow \Phi_{N}$. The upper bound on the graviton mass and the gravitational interaction length, reported by the LIGO-VIRGO collaboration, can be directly recast in terms of the parameters of the theory which allows an analysis where the gravitational wave frequency modulation sets constraints on the range of possible alternative models of gravity. Numerical results based on published parameters for the binary black hole mergers are also reported. Comparison of the observed phase of the GW$150914$ and GW$151226$ with the modulated phase in alternative theories of gravity does not give reasonable constraints due the large uncertainties in the estimated parameters for the coalescing black holes. In addition to these general considerations, we obtain limits for the frequency dependence of the $\alpha$ parameter in scalar tensor theories of gravity.

Satellite Test of the Equivalence Principle as a Probe of Modified Newtonian Dynamics.

The proposed satellite test of the equivalence principle (STEP) will detect possible violations of the weak equivalence principle by measuring relative accelerations between test masses of different composition with a precision of one part in 10^{18}. A serendipitous by-product of the experimental design is that the absolute or common-mode acceleration of the test masses is also measured to high precision as they oscillate along a common axis under the influence of restoring forces produced by the position sensor currents, which in drag-free mode lead to Newtonian accelerations as small as 10^{-14} g. This is deep inside the low-acceleration regime where modified Newtonian dynamics (MOND) diverges strongly from the Newtonian limit of general relativity. We show that MOND theories (including those based on the widely used "n family" of interpolating functions as well as the covariant tensor-vector-scalar formulation) predict an easily detectable increase in the frequency of oscillations of the STEP test masses if the strong equivalence principle holds. If it does not hold, MOND predicts a cumulative increase in oscillation amplitude which is also detectable. STEP thus provides a new and potentially decisive test of Newton's law of inertia, as well as the equivalence principle in both its strong and weak forms.

Lattice Universe: examples and problems.

We consider lattice Universes with spatial topologies Ttimes Ttimes T, Ttimes Ttimes R , and Ttimes Rtimes R. In the Newtonian limit of General Relativity, we solve the Poisson equation for the gravitational potential in the enumerated models. In the case of point-like massive sources in the Ttimes Ttimes T model, we demonstrate that the gravitational potential has no definite values on the straight lines joining identical masses in neighboring cells, i.e. at points where masses are absent. Clearly, this is a nonphysical result, since the dynamics of cosmic bodies is not determined in such a case. The only way to avoid this problem and get a regular solution at any point of the cell is the smearing of these masses over some region. Therefore, the smearing of gravitating bodies in N-body simulations is not only a technical method but also a physically substantiated procedure. In the cases of Ttimes Ttimes R and Ttimes Rtimes R topologies, there is no way to get any physically reasonable and nontrivial solution. The only solutions we can get here are the ones which reduce these topologies to the Ttimes Ttimes T one.

Gravitational cusp anomalous dimension from AdS space

Recently a new picture has been developed for examining Wilson lines, and the corresponding anomalous dimensions which govern their renormalization properties. By making a particular coordinate transform, the calculation of the cusp anomalous dimension in QED or QCD can be related to the energy of a pair of static charges in Euclidean Anti-de Sitter space. This paper shows how the same picture can be used to describe Wilson lines in quantum gravity. We show how the relevant cusp anomalous dimension (which has recently been shown to be one-loop exact) can be obtained using the Newtonian limit of general relativity. We also show how both the QED and gravity cases emerge as special cases of a general formulation, and that a continuous parameter exists which interpolates between them. The results may be useful in examining the relations between gauge and gravity theories.

Probing the dark matter issue in f(R)-gravity via gravitational lensing

For a general class of analytic f(R)-gravity theories, we discuss the weak field limit in view of gravitational lensing. Though an additional Yukawa term in the gravitational potential modifies dynamics with respect to the standard Newtonian limit of General Relativity, the motion of massless particles results unaffected thanks to suitable cancellations in the post-Newtonian limit. Thus, all the lensing observables are equal to the ones known from General Relativity. Since f(R)-gravity is claimed, among other things, to be a possible solution to overcome for the need of dark matter in virialized systems, we discuss the impact of our results on the dynamical and gravitational lensing analyses. In this framework, dynamics could, in principle, be able to reproduce the astrophysical observations without recurring to dark matter, but in the case of gravitational lensing we find that dark matter is an unavoidable ingredient. Another important implication is that gravitational lensing, in the post-Newtonian limit, is not able to constrain these extended theories, since their predictions do not differ from General Relativity.

A canonical dynamics view of the Newtonian limit of general relativity

The Newtonian limit of general relativity was Jurgen Ehlers favourite model for limit relations between theories of physics. In this contribution, for the case of isolated systems, the Newtonian limit of general relativity will be illuminated from a canonical dynamics point of view. The canonical dynamics approach naturally supplies a post-Newtonian expansion of general relativity.

Peter Havas

Peter Havas

On the Newtonian limit and cut-off scales of isothermal dark matter haloes with cosmological constant

We examine isothermal dark matter haloes in hydrostatic equilibrium with a 'A-field', or cosmological constant A = Ω Λ ρ critc 2 , where Ω Λ ≃ 0.7 and ρ c r i t is the present value of the critical density with h ≃ 0.65. Modelling cold dark matter (CDM) as a self-gravitating Maxwell-Boltzmann gas, the Newtonian limit of general relativity yields equilibrium equations that are different from those arising by merely coupling an 'isothermal sphere' to the A-field within a Newtonian framework. Using the conditions for the existence and stability of circular geodesic orbits, the numerical solutions of the equilibrium equations (Newtonian and Newtonian limit) show the existence of: (i) an 'isothermal region' (0 ≤ r r 1 ) dominated by the A-field, where the Newtonian potential oscillates and circular orbits only exist in disconnected patches of the domain of r; (iii) a 'transition region' (r 2 ≤ r 0.008 M O . pc - 3 , in interesting agreement with rotation curve studies of dwarf galaxies. Because r 2 marks the largest radius of a stable circular orbit, it provides a characteristic boundary or 'cut-off' maximal radius for isothermal spheres in equilibrium with a A-field. For current estimates of ρ c and velocity dispersion of virialized galactic structures, this cut-off scale ranges from 90 kpc for dwarf galaxies, up to 3 Mpc for large galaxies and 22 Mpc for clusters. In a purely Newtonian framework, these length-scales are about 10 per cent smaller, although in either case r 2 is between five and seven times larger than the physical cut-off scales of isothermal haloes, such as the virialization radius or the critical radius for the onset of Antonov instability. These results indicate that the effects of the A-field can be safely ignored in studies of virialized structures, but could be significant in the study of structure formation models and the dynamics of superclusters still in the linear regime or of gravitational clustering at large scales (r 30 Mpc).

THE CONSISTENT NEWTONIAN LIMIT OF EINSTEIN'S GRAVITY WITH A COSMOLOGICAL CONSTANT

We derive the "exact" Newtonian limit of general relativity with a positive cosmological constant Λ. We point out that in contrast to the case with Λ=0, the presence of a positive Λ in Einsteins's equations enforces, via the condition |Φ|≪1 on the potential Φ, a range ℛ max (Λ)≫r≫ℛ min (Λ), within which the Newtonian limit is valid. It also leads to the existence of a maximum mass, ℳ max (Λ). As a consequence we cannot put the boundary condition for the solution of the Poisson equation at infinity. A boundary condition suitably chosen now at a finite range will then get reflected in the solution of Φ provided the mass distribution is not spherically symmetric.

Newtonian and Post-Newtonian Limits of Relativistic Cosmology

The Newtonian limit of general relativity is by no means as straightforward as is commonly assumed. In particular, the correct limit of the Bianchi identities must be taken to the second (non-linear) order. Furthermore Newtonian cosmology does not have a well-posed initial value formulation, while relativistic cosmology does. We show in this paper that the c-4 approximation of general relativity, although non-linear, provides a non-standard version of Newtonian theory which is in fact completely equivalent to the Heckmann–Schucking version of Newtonian cosmology. The next approximation (order c-6), when the limit is taken in a particular way, gives rise to a closed and self-consistent post-Newtonian cosmological theory which has a well-posed initial value problem. This seems to be a suitable, if somewhat complicated, theory for cosmological and astrophysical problems

On the Newtonian limit of general relativity

We find a choice of variables for the 3+1 formulation of general relativity which casts the evolution equations into (flux-conservative) symmetric-hyperbolic first order form for arbitrary lapse and shift, for the first time. We redefine the lapse function in terms of the determinant of the 3-metric and a free function U which embodies the lapse freedom. By rescaling the variables with appropriate factors of 1/c, the system is shown to have a smooth Newtonian limit when the redefined lapse U and the shift are fixed by means of elliptic equations to be satisfied on each time slice. We give a prescription for the choice of appropriate initial data with controlled extra-radiation content, based on the theory of problems with different time-scales. Our results are local, in the sense that we are not concerned with the treatment of asymptotic regions. On the other hand, this local theory is all what is needed for most problems of practical numerical computation.

On the definition of post-newtonian approximations

A definition of post-newtonian approximations is presented where the whole formalism is derived from a minimal set of axioms. This establishes a link between the existing precise formulation of the newtonian limit of general relativity and the post-newtonian equations which are used in practical calculations. The breakdown of higher post-newtonian approximations is examined within this framework. It is shown that the choice of harmonic gauge leads to equations which do not admit asymptotically flat solutions at the second post-newtonian level if one starts with a generic newtonian solution. The most simple choice of gauge gives equations which are solvable at the 2PN level but which in general have no solutions in the case of the third post-newtonian approximation.

The Newtonian limit of Penrose's quasi-local mass

Penrose's definition, (1982), of quasi-local mass is applied in the Newtonian limit of general relativity. The calculations are for 2-surfaces within a hypersurface of constant Newtonian time t. The leading-order terms in both the mass within the 2-surface and the difference between the mass and a conserved mass within the 2-surface are calculated. The former is just the Newtonian mass contained within the 2-surface and the latter, for a 2-surface surrounding an isolated system, gives the standard expression for the energy of the system. When applied to 2-surfaces containing parts of a system results interpretable in terms of binding energies appear.

Gravitational radiation and the validity of the far-zone quadrupole formula in the Newtonian limit of general relativity

We examine the gravitational radiation emitted by a sequence of spacetimes whose near-zone Newtonian limit we have previously studied. The spacetimes are defined by initial data which scale in a Newtonian fashion: the density as ${\ensuremath{\epsilon}}^{2}$, velocity as \ensuremath{\epsilon}, pressure as ${\ensuremath{\epsilon}}^{4}$, where \ensuremath{\epsilon} is the sequence parameter. We asymptotically approximate the metric at an event which, as \ensuremath{\epsilon}\ensuremath{\rightarrow}0, remains a fixed number of gravitational wavelengths distant from the system and a fixed number of wave periods to the future of the initial hypersurface. We show that the radiation behaves like that of linearized theory in a Minkowski spacetime, since the mass of the metric vanishes as \ensuremath{\epsilon}\ensuremath{\rightarrow}0. We call this Minkowskian far-zone limiting manifold FM; it is a boundary of the sequence of spacetimes, in which the radiation carries an energy flux given asymptotically by the usual far-zone quadrupole formula (the Landau-Lifshitz formula), as measured both by the Isaacson average stress-energy tensor in FM or by the Bondi flux on ${\mathit{I}}_{\mathrm{FM}}^{+}$. This proves that the quadrupole formula is an asymptotic approximation to general relativity. We study the relation between ${\mathit{I}}_{\mathrm{\ensuremath{\epsilon}}}^{+}$, the sequence of null infinities of the individual manifolds, and ${\mathit{I}}_{\mathrm{FM}}^{+}$; and we examine the gauge-invariance of FM under certain gauge transformations. We also discuss the relation of this calculation with similar ones in the frame-work of matched asymptotic expansions and others based on the characteristic initial-value problem.