Expansion Of General Relativity Research Articles

Toward a covariant framework for post-Newtonian expansions for radiative sources

We consider the classic problem of a compact fluid source that behaves nonrelativistically and that radiates gravitational waves. The problem consists of determining the metric close to the source as well as far away from it. The nonrelativistic nature of the source leads to a separation of scales resulting in an overlap region where both the 1/c and (multipolar) G expansions are valid. Standard approaches to this problem (the Blanchet-Damour and the DIRE approach) use the harmonic gauge. We define a “post-Newtonian” class of gauges that admit a Newtonian regime in inertial coordinates. In this paper we set up a formalism to solve for the metric for any post-Newtonian gauge choice. Our methods are based on previous work on the covariant theory of nonrelativistic gravity (a 1/c expansion of general relativity that uses post-Newton-Cartan variables). At the order of interest in the 1/c and G expansions we split the variables into two sets: transverse and longitudinal. We show that for the transverse variables the problem can be reduced to inverting Laplacian and d’Alembertian operators on their respective domains subject to appropriate boundary conditions. The latter are regularity in the interior and asymptotic flatness with a Sommerfeld no-incoming radiation condition imposed at past null infinity. The longitudinal variables follow from the gauge choice. The full solution is then obtained by the method of matched asymptotic expansion. We show that our methods reproduce existing results in harmonic gauge to 2.5PN order. Published by the American Physical Society 2024

A 3+1 formulation of the 1/c expansion of General Relativity

Expanding General Relativity in the inverse speed of light, 1/c, leads to a nonrelativistic gravitational theory that extends the Post-Newtonian expansion by the inclusion of additional strong gravitational potentials. This theory has a fully covariant formulation in the language of Newton-Cartan geometry but we revisit it here in a 3+1 formulation. The appropriate 3+1 formulation of General Relativity is one first described by Kol and Smolkin (KS), rather than the better known Arnowitt-Deser-Misner (ADM) formalism. As we review, the KS formulation is dual to the ADM formulation in that the role of tangent and co-tangent spaces get interchanged. In this 3+1 formulation the 1/c expansion can be performed in a more systematic and efficient fashion, something we use to extend the computation of the effective Lagrangian beyond what was previously achieved and to make a number of new all order observations.

Gravitational radiation from inspiralling compact objects: Spin effects to the fourth post-Newtonian order

The linear- and quadratic-in-spin contributions to the binding potential and gravitational-wave flux from binary systems are derived to next-to-next-to-leading order in the Post-Newtonian (PN) expansion of general relativity, including finite-size and tail effects. The calculation is carried out through the worldline effective field theory framework. We find agreement in the overlap with the available PN literature and test-body limit. As a direct application, we complete the knowledge of spin effects in the evolution of the orbital phase for aligned-spin circular orbits to fourth PN order. We estimate the impact of the new results in the number of accumulated gravitational-wave cycles. We find they will play an important role in providing reliable physical interpretation of gravitational-wave signals from spinning binaries with future detectors such as LISA and the Einstein Telescope.

Torsion-induced gravitational theta term and gravitoelectromagnetism

Motivated by the analogy between a weak field expansion of general relativity and Maxwell’s laws of electrodynamics, we explore physical consequences of a parity violating theta term in gravitoelectromagnetism. This is distinct from the common gravitational theta term formed as a square of the Riemann tensor. Instead it appears as a product of the gravitoelectric and gravitomagnetic fields in the Lagrangian, similar to the Maxwellian theta term. We show that this sector can arise from a quadratic torsion term in nonlinear gravity. In analogy to the physics of topological insulators, the torsion-induced theta parameter can lead to excess mass density at the interface of regions where theta varies and consequently it generates a correction to Newton’s law of gravity. We discuss also an analogue of the Witten effect for gravitational dyons.

Note on scalar\u2013graviton and scalar\u2013photon\u2013graviton amplitudes

In this short note, we propose an algorithm based on the expansions of amplitudes, the dimensional reduction technique and the approach by differential operators, to calculate the tree level scalar–graviton amplitudes with two massive scalars and the tree level scalar–photon–graviton amplitudes with two massive scalars and one photon. While applying the unitarity method, these amplitudes are necessary inputs for the calculation of post-Newtonian and post-Minkowskian expansions in general relativity for two massive charged objects interacting with gravity and the electromagnetic field.

Asymptotic expansion of general relativity with Galilean covariance

The Galilean gravitation derives from a scalar potential and a vector one. Poisson’s equation to determine the scalar potential has not the expected Galilean covariance. Moreover, there are three missing equations to determine the potential vector. Besides, we require they have the Galilean covariance. These are the issues addressed in the paper. To avoid the drawbacks of the PPN approach and the NCT, we merge them into a new framework. The key idea is to take care that every term of the c expansion of the fields are Galilean covariants or invariants. The expected equations are deduced by variation of the Hilbert–Einstein functional. The contribution of the matter to the functional is derived from Souriau’s conformation tensor. We obtain a system of four non linear equations, solved by asymptotic expansion.

Oddity in nonrelativistic, strong gravity

We consider the presence of odd powers of the speed of light c in the covariant nonrelativistic expansion of General Relativity (GR). The term of order c in the relativistic metric is a vector potential that contributes at leading order in this expansion and describes strong gravitational effects outside the (post-)Newtonian regime. The nonrelativistic theory of the leading order potentials contains the full non-linear dynamics of the stationary sector of GR.

Action Principle for Newtonian Gravity.

We derive an action whose equations of motion contain the Poisson equation of Newtonian gravity. The construction requires a new notion of Newton-Cartan geometry based on an underlying symmetry algebra that differs from the usual Bargmann algebra. This geometry naturally arises in a covariant 1/c expansion of general relativity, with c being the speed of light. By truncating this expansion at subleading order, we obtain the field content and transformation rules of the fields that appear in the action of Newtonian gravity. The equations of motion generalize Newtonian gravity by allowing for the effect of gravitational time dilation due to strong gravitational fields.

Holographic non-computers

We introduce the notion of holographic non-computer as a system which exhibits parametrically large delays in the growth of complexity, as calculated within the Complexity-Action proposal. Some known examples of this behavior include extremal black holes and near-extremal hyperbolic black holes. Generic black holes in higher-dimensional gravity also show non-computing features. Within the 1/d expansion of General Relativity, we show that large-d scalings which capture the qualitative features of complexity, such as a linear growth regime and a plateau at exponentially long times, also exhibit an initial computational delay proportional to d. While consistent for large AdS black holes, the required ‘non-computing’ scalings are incompatible with thermodynamic stability for Schwarzschild black holes, unless they are tightly caged.

Computational Algorithm for Covariant Series Expansions in General Relativity

We present a new algorithm for computing covariant power expansions of tensor fields in generalized Riemannian normal coordinates, introduced in some neighborhood of a parallelized k-dimensional submanifold (k = 0, 1, . . .< n; the case k = 0 corresponds to a point), by transforming the expansions to the corresponding Taylor series. For an arbitrary real analytic tensor field, the coefficients of such series are expressed in terms of its covariant derivatives and covariant derivatives of the curvature and the torsion. The algorithm computes the corresponding Taylor polynomials of arbitrary orders for the field components and is applicable to connections that are, in general, nonmetric and not torsion-free. We show that this computational problem belongs to the complexity class LEXP.

Torsional Newton–Cartan gravity from the large c expansion of general relativity

We revisit the manifestly covariant large c expansion of general relativity, c being the speed of light. Assuming the relativistic connection has no pole in , this expansion is known to reproduce Newton–Cartan gravity and a covariant version of post-Newtonian corrections to it. We show that relaxing this assumption leads to the inclusion of twistless torsion in the effective non-relativistic theory. We argue that the resulting twistless torsion Newton–Cartan theory is an effective description of a non-relativistic regime of general relativity that extends Newtonian physics by including strong gravitational time dilation.

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.

Post-Minkowski expansion of general relativity

A post-Minkowski approximation of general relativity is described as a power series expansion in $G$, Newton's gravitational constant. Material sources are hidden behind boundaries, and only the vacuum Einstein equations are considered. An iterative procedure is outlined which, in one complete step, takes any approximate solution of the Einstein equations and produces a new approximation which has the error decreased by a factor of $G$. Each step in the procedure consists of three parts: first the equations of motion are used to update the trajectories of the boundaries; then the field equations are solved using a retarded Green function for Minkowski space; finally, a gauge transformation is performed which makes the geometry well behaved at future null infinity. Differences between this approach to the Einstein equations and similar ones are that we use a general (nonharmonic) gauge and formulate the procedure in a constructive manner which emphasizes its suitability for implementation on a computer.

Radiation reaction and energy loss in the post-newtonian approximation of general relativity

We calculate the radiation reaction force within the Anderson-Decanio scheme of a Post-Newtonian expansion of general relativity modified such that no infinite expressions appear up to the order considered. We show that for quasiperiodic motions of bodies in bound systems, i.e., motions which differ little from Keplerian motion, the dominant contribution to the change of the Newtonian energy of the system equals the power loss given by Einstein's “quadrupole formula.”

Causes and cures for the infinities in slow-motion expansions in general relativity

The causes of the divergent integrals arising in slow-motion expansions of the general relativistic field equations are studied and a remedy for them is suggested. This is done within the context of a model problem involving a coupled nonlinear scalar field and isotropic oscillator. The model is shown to give rise to divergent integrals directly attributable to the nonlinearity when the field is assumed to be analytic in a slowness parameter. Application of a nonregular perturbation approach which includes the method of matched asymptotic expansions is shown to eliminate the infinite contributions.