Lemaitre-Tolman-Bondi (LTB) solutions are used frequently to describe the collapse or expansion of spherically symmetric inhomogeneous mass distributions in the Universe. These exact solutions are obtained in the synchronous gauge where nonlinear dynamics (with respect to the FLRW background) induce large deviations from the FLRW metric. In this paper we show explicitly that this is a gauge artefact (for realistic sub-horizon inhomogeneities). We write down the nonlinear gauge transformation from synchronous to Newtonian gauge for a general LTB solution using the fact that the peculiar velocities are small. In the latter gauge we recover the solution in the form of a weakly perturbed FLRW metric that is assumed in standard cosmology. Furthermore we show how to obtain the LTB solutions directly in Newtonian gauge and illustrate how the Newtonian approximation remains valid in the nonlinear regime where cosmological perturbation theory breaks down. Finally we discuss the implications of our results for the backreaction scenario.
The construction of Dirac observables, that is gauge invariant objects, in General Relativity is technically more complicated than in other gauge theories such as the standard model due to its more complicated gauge group which is closely related to the group of spacetime diffeomorphisms. However, the explicit and usually cumbersome expression of Dirac observables in terms of gauge non invariant quantities is irrelevant if their Poisson algebra is sufficiently simple. Precisely that can be achieved by employing the relational formalism and a specific type of matter proposed originally by Brown…
A numerical approach is considered for spherically symmetric spacetimes that generalize Lemaitre-Tolman-Bondi dust solutions to nonzero pressure ("LTB spacetimes"). We introduce quasi-local (QL) variables that are covariant LTB objects satisfying evolution equations of Friedman-Lemaitre-Robertson-Walker (FLRW) cosmologies. We prove rigorously that relative deviations of the local covariant scalars from the QL scalars are non-linear, gauge invariant and covariant perturbations on a FLRW formal "background" given by the QL scalars. The dynamics of LTB spacetimes is completely determined by the Q…
We study some consequences of the loop quantization of the outermost dust shell in the Lemaître-Tolman-Bondi spacetime with a homogeneous dust density using different quantization strategies motivated by loop quantum gravity. Prior work has dealt with loop quantizing this model by employing holonomies and the triads, following the procedure in standard loop quantum cosmology. In this work we compare this quantization with the one in which holonomies and gauge-covariant fluxes are used. While both of the quantization schemes resolve the central singularity, they lead to different mass gaps at w…
We present in this paper a new application of the geodesic light-cone (GLC) gauge for weak lensing calculations. Using interesting properties of this gauge, we derive an exact expression of the amplification matrix - involving convergence, magnification and shear - and of the deformation matrix - involving the optical scalars. These expressions are simple and non-perturbative as long as no caustics are created on the past light-cone and are, by construction, free from the thin lens approximation. We apply these general expressions on the example of an Lemaître-Tolman-Bondi (LTB) model with an …
The "Newtonian" or non-relativistic decomposition of Einstein's gravitational field is useful in the post-Newtonian approximation. We obtain the full non-quadratic Einstein-Hilbert action in terms of these fields as well as the harmonic gauge fixing term and find fairly simple expressions. We discuss alternatives to the harmonic gauge.
We examine the relation between the dynamics of Lema\^ıtre-Tolman-Bondi (LTB) dust models (with and without $Λ$) and the dynamics of dust perturbations in two of the more familiar formalisms used in cosmology: the metric based Cosmological Perturbation Theory (CPT) and the Covariant Gauge Invariant (GIC) perturbations. For this purpose we recast the evolution of LTB models in terms of a covariant and gauge invariant formalism of local and non-local "exact fluctuations " on a Friedmann-Lema\^ıtre-Robertson-Walker (FLRW) background defined by suitable averages of covariant scalars. We examine th…
Relativistic cosmological perturbation analyses can be made based on several different fundamental gauge conditions. In the pressureless limit the variables in certain gauge conditions show the correct Newtonian behaviors. We consider the general curvature and the cosmological constant in the background medium. The perturbed density in the comoving gauge, and the perturbed velocity and the perturbed potential in the zero-shear gauge show the same behavior as the Newtonian ones in a general scale. Far inside horizon, except for the uniform-density gauge, density perturbations in all the fundame…
We discuss the gravitational self-force on a particle in a black hole space-time. For a point particle, the full (bare) self-force diverges. It is known that the metric perturbation induced by a particle can be divided into two parts, the direct part (or the S part) and the tail part (or the R part), in the harmonic gauge, and the regularized self-force is derived from the R part which is regular and satisfies the source-free perturbed Einstein equations. In this paper, we consider a gauge transformation from the harmonic gauge to the Regge-Wheeler gauge in which the full metric perturbation c…
We derive the equilibrium hydrostatic equation of a spherical star for any gravitational Lagrangian density of the form $L=\sqrt{-g}f(R)$. The Palatini variational principle for the Helmholtz Lagrangian in the Einstein gauge is used to obtain the field equations in this gauge. The equilibrium hydrostatic equation is obtained and is used to study the Newtonian limit for $f(R)=R-\frac{a^{2}}{3R}$. The same procedure is carried out for the more generally case $f(R)=R-\frac{1}{n+2}\frac{a^{n+1}}{R^{n}}$ giving a good Newtonian limit.
Recently, a model for an emergent gravity based on $SO(5)$ Yang-Mills action in Euclidian 4-dimensional spacetime was proposed. In this work we provide some 1 and 2-loops computations and show that the model can accommodate suitable predicting values for the Newtonian constant. Moreover, it is shown that the typical scale of the expected transition between the quantum and the geometrodynamical theory is consistent with Planck scale. We also provide a discussion on the cosmological constant problem.
Continuing our analytic computation of the first-order self-force contribution to Detweiler's redshift variable we provide the exact expressions of the ninth and ninth-and-a-half post-Newtonian terms.
As an alternative to dark energy it has been suggested that we may be at the center of an inhomogeneous isotropic universe described by a Lemaitre-Tolman-Bondi (LTB) solution of Einstein's field equations. In order to test this hypothesis we calculate the general analytical formula to fifth order for the redshift spherical shell mass. Using the same analytical method we write the metric in the light-cone by introducing a gauge invariant quantity $G(z)$ which together with the luminosity distance $D_L(z)$ completely determine the light-cone geometry of a LTB model.
A new kind of gauge theory is introduced, where the minimal coupling and corresponding covariant derivatives are defined in the space of functions pertaining to the functional Schroedinger picture of a given field theory. While, for simplicity, we study the example of a U(1) symmetry, this kind of gauge theory can accommodate other symmetries as well. We consider the resulting relativistic nonlinear extension of quantum mechanics and show that it incorporates gravity in the (0+1)-dimensional limit, similar to recently studied Schroedinger-Newton equations. Gravity is encoded here into a univer…
Newton's standard theory of gravitation is reformulated as a {\it gauge} theory of the {\it extended} Galilei Group. The Action principle is obtained by matching the {\it gauge} technique and a suitable limiting procedure from the ADM-De Witt action of general relativity coupled to a relativistic mass-point.
We study the evolution of linear perturbations in a Lemaître-Tolman-Bondi (LTB) void model with realistic cosmological initial conditions. Linear perturbation theory in LTB models is substantially more complicated than in standard Friedmann universes as the inhomogeneous background causes gauge-invariant perturbations to couple at first order. As shown by Clarkson et al. (2009), the evolution is constrained by a system of linear partial differential equations which need to be integrated numerically. We present a new numerical scheme using finite element methods to solve this equation system an…
We prove that in the infinite speed-of-light limit (i.e., non-relativistic and subhorizon limits), the relativistic fully nonlinear cosmological perturbation equations in two gauge conditions, the zero-shear gauge and the uniform-expansion gauge, exactly reproduce the Newtonian hydrodynamic perturbation equations in the cosmological background; as a consequence, in the same two gauge conditions, the Newtonian hydrodynamic equations are exactly recovered in the Minkowsky background.
We present an approach to the parametrized post-Newtonian (PPN) formalism which is based on gauge-invariant higher order perturbation theory. This approach divides the components of the metric perturbations into gauge-invariant quantities, which carry information about the physical system under consideration, and pure gauge quantities, which describe the choice of the coordinate system. This separation generally leads to a simplification of the PPN procedure, since only the gauge-invariant quantities appear in the field equations and must be determined by solving them. Another simplification a…
We compare the cosmological first-order post-Newtonian (1PN) approximation with the relativistic cosmological linear perturbation theory in a zero-pressure medium with the cosmological constant. We compare equations and solutions in several different gauge conditions available in both methods. In the PN method we have perturbation equations for density, velocity and gravitational potential independently of the gauge condition to 1PN order. However, correspondences with these 1PN equations are available only in certain gauge conditions in the perturbation theory. Equations of perturbed velocity…
We present an extension to the gauge-invariant formulation of the parameterized post-Newtonian (PPN) formalism, which allows its application to symmetric teleparallel gravity theories. In its original formulation, the gauge-invariant PPN formalism makes use of a gauge-invariant perturbative expansion of tensor fields; however, one of the fundamental gravitational field variables in symmetric teleparallel gravity theories is a flat, torsion-free connection, hence not a tensor field. Since connections transform differently from tensor fields under diffeomorphisms, we introduce here an adapted no…
Using the fully nonlinear and exact perturbation formulation with magnetohydrodynamics (MHD) in Minkowski background we derive first-order post-Newtonian (1PN) equations without imposing the slicing (temporal gauge) condition. The 1PN MHD formulation is complementary to our recently presented fully relativistic MHD combined with 0PN gravity available only in the maximal slicing. We present the 1PN MHD equations in two gauge conditions previously used in the literature and provide gauge transformation relations between different gauges. We derive the PN effects on MHD waves in a static homogene…
Relativistic cosmological perturbation analyses can be made based on several different fundamental gauge conditions. In the pressureless limit the variables in certain gauge conditions show the correct Newtonian behaviors. Considering the general curvature ($K$) and the cosmological constant ($Λ$) in the background medium, the perturbed density in the comoving gauge, and the perturbed velocity and the perturbed potential in the zero-shear gauge show the same behavior as the Newtonian ones in general scales. In the first part, we elaborate these Newtonian correspondences. In the second part, us…
We present a systematic treatment of the linear theory of scalar gravitational perturbations in the synchronous gauge and the conformal Newtonian (or longitudinal) gauge. We first derive the transformation law relating the two gauges. We then write down in parallel in both gauges the coupled, linearized Boltzmann, Einstein and fluid equations that govern the evolution of the metric perturbations and the density fluctuations of the particle species. The particle species considered include cold dark matter (CDM), baryons, photons, massless neutrinos, and massive neutrinos (a hot dark matter or H…
In a preceding paper we developed a reformulation of Newtonian gravitation as a {\it gauge} theory of the extended Galilei group. In the present one we derive two true generalizations of Newton's theory (a {\it ten-fields} and an {\it eleven-fields} theory), in terms of an explicit Lagrangian realization of the {\it absolute time} dynamics of a Riemannian three-space. They turn out to be {\it gauge invariant} theories of the extended Galilei group in the same sense in which general relativity is said to be a {\it gauge} theory of the Poincaré group. The {\it ten-fields} theory provides a dynam…
After reviewing how Albert Einstein's general relativity (GR) can be viewed as a gauge theory of the Poincaré algebra, we show how Élie Cartan's geometric formulation of Newtonian gravity (Newton-Cartan gravity) can be viewed as a gauge theory of the Bargmann algebra following the construction of [R. Andringa, E. Bergshoeff, S. Panda, and M. de Roo, "Newtonian Gravity and the Bargmann Algebra," Class. Quant. Grav. 28 (2011) 105011, arXiv:1011.1145 [hep-th]]. In doing so, we will touch on the following auxiliary topics: the extension of Yang-Mills gauge theory to a more generic formalism of gau…
We revisit and clarify the gauge dependence of gravitational waves generated at second order from scalar perturbations. In a universe dominated by a perfect fluid with a constant equation-of-state parameter $w$, we compute the energy density of such induced gravitational waves in the Newtonian, comoving, and uniform curvature gauges. Huge differences are found between the Newtonian and comoving gauge results for any $w \,(\ge 0)$. This is always caused by the perturbation of the shift vector. Interestingly, the Newtonian and uniform curvature gauge calculations give the same energy density for…
We parameterize the second post-Newtonian (2PN) metric for a gravitating system of fluids in the generalized harmonic gauge, and find that there are only three independent 2PN parameters (i.e. $ω$, $δ$ and $δ_2$) for satisfying some conservation laws including the conservations of energy, momentum, angular-momentum and the uniform motion of the center-of-mass.
A general analytic procedure is developed for the post-Newtonian limit of $f(R)$-gravity with metric approach in the Jordan frame by using the harmonic gauge condition. In a pure perturbative framework and by using the Green function method a general scheme of solutions up to $(v/c)^4$ order is shown. Considering the Taylor expansion of a generic function $f$ it is possible to parameterize the solutions by derivatives of $f$. At Newtonian order, $(v/c)^2$, all more important topics about the Gauss and Birkhoff theorem are discussed. The corrections to "standard" gravitational potential ($tt$-c…
In classical gravity deviations from the predictions of the Einstein theory are often discussed within the framework of the conformal Newtonian gauge, where scalar perturbations are described by two potentials $ϕ$ and $ψ$. In this paper we use the above gauge to explore possible cosmological consequences of a running Newton's constant $ G (\Box) $, as suggested by the nontrivial ultraviolet fixed point scenario arising from the quantum field-theoretic treatment of Einstein gravity with a cosmological constant term. Here we focus on the effects of a scale-dependent coupling on the so-called gra…
Charged Black-Hole Binary Evolution at Second Post-Newtonian Order
Università di Perugia（佩鲁贾大学）；Niels Bohr Institute, Copenhagen University（哥本哈根大学尼尔斯·玻尔研究所）；Università degli Studi di Padova（帕多瓦大学）；INFN, Sezione di Padova（意大利国家核物理研究所帕多瓦分部）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理学研究所（阿尔伯特·爱因斯坦研究所））；INAF - Osservatorio Astronomico di Padova（意大利国家天体物理研究所帕多瓦天文台）
We clarify the subtle issue of finding the correct mapping of Newtonian simulations to light-cone observables at very large distance scales. A faithful general-relativistic interpretation specifies a gauge, i.e. a chart that relates the simulation data to points of the space-time manifold. It has already been pointed out that the implicit gauge choice of Newtonian simulations is indeed different from the Poisson gauge that is commonly adopted for relativistic calculations, the difference being most significant at large scales. It is therefore inconsistent, for example, to predict weak-lensing …
Applying the recently developed dynamical perturbation formalism on cosmological background to scalar-tensor theory, we provide a solid theoretical basis and a rigorous justification for phenomenological models of orbital dynamics that are currently used to interpret experimental measurements of the time-dependent gravitational constant. We derive the field equations for the scalar-tensor perturbations and study their gauge freedom associated with the cosmological expansion. We find a new gauge eliminating a prohibitive number of gauge modes in the field equations and significantly simplifying…
The asymptotic scheme of post-Newtonian approximation defined for general relativity (GR) in the harmonic gauge by Futamase & Schutz (1983) is based on a family of initial data for the matter fields of a perfect fluid and for the initial metric, defining a family of weakly self-gravitating systems. We show that Weinberg's (1972) expansion of the metric and his general expansion of the energy-momentum tensor ${\bf T}$, as well as his expanded equations for the gravitational field and his general form of the expanded dynamical equations, apply naturally to this family. Then, following the asympt…
There are well-known problems associated with the idea of (local) gravitational energy in general relativity. We offer a new perspective on those problems by comparison with Newtonian gravitation, and particularly geometrized Newtonian gravitation (i.e., Newton-Cartan theory). We show that there is a natural candidate for the energy density of a Newtonian gravitational field. But we observe that this quantity is gauge dependent, and that it cannot be defined in the geometrized (gauge-free) theory without introducing further structure. We then address a potential response by showing that there …
We review and extend the Gauge Vectors-Tensor gravity: a covariant theory of gravity composed of a metric and gauge fields, leading to simple second order partial differential equations of motion, whose Newtonian and strong limits coincide to those of the Einsten-Hilbert action but the physics of its very weak fields should be identified through observation. We show that GVT is at least as dynamically stable as the Einstein-Hilbert gravity. It accommodates the MOND paradigm. We study its gravitational light deflection. We show that the post Newtonian parameter of $γ$ vanishes in the MOND regim…
In this paper, we derive the post-Newtonian equations of the ideal Magnetohydrodynamics. To do so, we use the modern approach to post-Newtonian theory, where the harmonic gauge is used instead of the standard post-Newtonian gauge, and find the post-Newtonian metric in the presence of the electromagnetic fields. We show that although the electric field does not contribute in the metric and curvature of the spacetime, the magnetic field appears in the time-time component of the metric. The appearance of the magnetic field, in principle, leads to new relativistic contributions to the magnetohydro…
We present a tetrad-based method for solving the Einstein field equations for spherically-symmetric systems and compare it with the widely-used Lemaître-Tolman-Bondi (LTB) model. In particular, we focus on the issues of gauge ambiguity and the use of comoving versus 'physical' coordinate systems. We also clarify the correspondences between the two approaches, and illustrate their differences by applying them to the classic examples of the Schwarzschild and Friedmann-Robertson-Walker spacetimes. We demonstrate that the tetrad-based method does not suffer from the gauge freedoms inherent to the …
In a 4-dimensional (4D) weak field geometry governed by the linearized Einstein's equations and sourced primarily by a static, spatially localized, but otherwise arbitrary mass density $T_{00}$, it is known that the leading order tail part of the Green's functions of the minimally coupled massless scalar, Lorenz gauge photon, and de Donder gauge graviton at late times are not only time translation symmetric, but also space-translation and spherically symmetric. Only the monopole moment of the matter source is responsible for the late-time tail of the Green's functions. We provide evidence, in …
We define a new paradigm --- postrelativity --- based on the hypothesis of a preferred hidden Newtonian frame in relativistic theories. It leads to a modification of general relativity with ether interpretation, without topological problems, black hole and big bang singularities. Semiclassical theory predicts Hawking radiation with evaporation before horizon formation. In quantum gravity there is no problem of time and topology. Configuration space and quasiclassical predictions are different from canonical quantization of general relativity. Uncertainty of the light cone or an atomic structur…
Newtonian cosmological perturbation equations valid to full nonlinear order are well known in the literature. Assuming the absence of the transverse-tracefree part of the metric, we present the general relativistic counterpart valid to full nonlinear order. The relativistic equations are presented without taking the slicing (temporal gauge) condition. The equations do have the proper Newtonian and first post-Newtonian limits. We also present the relativistic pressure correction terms in the Newtonian hydrodynamic equations.
In this work a study of the gravity is made using Einstein's equation in the post-Newtonian approach. This is a method to linearise the General Relativity indicated to treat non-relativistic objects. It enables us to construct, from metric-independent elements, fields that are governed by equations similar to the Maxwell's ones in Lorentz gauge. We promediate these equations for matter distributed in local systems, like solar systems or galaxies. Finally we define the large scale fields for this distribution, which includes terms analogous to electromagnetic case, like polarization, magnetizat…
The present paper outlines theoretical principles of the post-Newtonian mechanics in the expanding universe. It is based upon the gauge-invariant theory of the Lagrangian perturbations of cosmological manifold caused by an isolated astronomical N-body system. We postulate that the background manifold is described by Friedman-Lemaitre-Robertson-Walker (FLRW) metric governed by two primary components - the dark matter and the dark energy. The dark matter is treated as an ideal fluid. The dark energy is described by a single scalar field with a potential which is hold unspecified as long as the t…
As a crucial step towards the completion of the fourth post-Newtonian (4PN) gravitational-wave generation from compact binary systems, we obtain the expressions of the so-called "canonical" multipole moments of the source in terms of the "source" and "gauge" moments. The canonical moments describe the propagation of gravitational waves outside the source's near zone, while the source and gauge moments encode explicit information about the matter source. Those two descriptions, in terms of two sets of canonical moments or in terms of six sets of source and gauge moments, are isometric. We thus …
We derive the basic equations of the cosmological first-order post-Newtonian approximation from the recently formulated fully nonlinear and exact cosmological perturbation theory in Einstein's gravity. Apparently the latter, being exact, should include the former, and here we use this fact as a new derivation of the former. The complete sets of equations in both approaches are presented without fixing the temporal gauge conditions so that we can use the gauge choice as an advantage. Comparisons between the two approaches are made. Both are potentially important in handling relativistic aspects…
Gauging of space translations for nonrelativistic point particles in one dimension leads to general coordinate transformations with fixed Newtonian time. The minimal gauge invariant extension of the particle velocity requires the introduction of two gauge fields whose minimal self interaction leads to a Maxwellian term in the Lagrangean. No dilaton field is introduced. We fix the gauge such that the residual symmetry group is the Galilei group. In case of a line the two-particle reduced Lagrangean describes the motion in a Newtonian gravitational potential with strength proportional to the ene…
We implement the effective field theory for gravitating spinning objects in the post-Newtonian scheme at the next-to-next-to-leading order level to derive the gravitational spin-orbit interaction potential at the third and a half post-Newtonian order for rapidly rotating compact objects. From the next-to-next-to-leading order interaction potential, which we obtain here in a Lagrangian form for the first time, we derive straightforwardly the corresponding Hamiltonian. The spin-orbit sector constitutes the most elaborate spin dependent sector at each order, and accordingly we encounter a prolife…
We use the effective field theory for gravitational bound states, proposed by Goldberger and Rothstein, to compute the interaction Lagrangian of a binary system at the second Post-Newtonian order. Throughout the calculation, we use a metric parametrization based on a temporal Kaluza-Klein decomposition and test the claim by Kol and Smolkin that this parametrization provides important calculational advantages. We demonstrate how to use the effective field theory method efficiently in precision calculations, and we reproduce known results for the second Post-Newtonian order equations of motion i…
We present the general relativistic pressure correction terms in Newtonian hydrodynamic equations to the nonlinear order: these are equations (\ref{mass-conservation-Mink})-(\ref{Poisson-eq-Mink}). The derivation is made in the zero-shear gauge based on the fully nonlinear formulation of cosmological perturbation in Einstein's gravity. The correction terms {\it differ} from many of the previously suggested forms in the literature based on hand-waving manners. We confirm our results by comparing with (i) the nonlinear perturbation theory, (ii) the first order post-Newtonian approximation, and (…
On large-scales, comparable to the horizon, the observable clustering properties of galaxies are affected by various general relativistic effects. To calculate these effects one needs to consistently solve for the metric, densities and velocities in a specific coordinate system or gauge. The method of choice for simulating large-scale structure is numerical N-body simulations which are performed in the Newtonian limit. Even though one might worry that the use of the Newtonian approximation would make it impossible to use these simulations to compute properties on very large-scales, we show tha…
Radiation-reaction driven dynamics at the third-and-a-half post-Newtonian order in different gauges
Istituto per le Applicazioni del Calcolo “M. Picone,” CNR（意大利国家研究委员会计算应用研究所）；Scuola Superiore Meridionale（南方高等学院）；INFN, Sezione di Napoli（意大利国家核物理研究所那不勒斯分部）
Using the (3+1) formalism in general relativity, we perform the post-Newtonian(PN) approximation to clarify what sort of gauge condition is suitable for numerical analysis of coalescing compact binary neutron stars and gravitational waves from them. We adopt a kind of transverse gauge condition to determine the shift vector. On the other hand, for determination of the time slice, we adopt three slice conditions(conformal slice, maximal slice and harmonic slice) and discuss their properties. Using these conditions, the PN hydrodynamic equations are obtained up through the 2.5PN order including …
We consider a general relativistic zero-pressure irrotational cosmological medium perturbed to the third order. We assume a flat Friedmann background but include the cosmological constant. We ignore the rotational perturbation which decays in expanding phase. In our previous studies we discovered that, to the second-order perturbation, except for the gravitational wave contributions, the relativistic equations coincide exactly with the previously known Newtonian ones. Since the Newtonian second-order equations are fully nonlinear, any nonvanishing third and higher order terms in the relativist…
We introduce the Gauge Vector-Tensor (GVT) theory by extending the AQUAL's approach to the GravitoElectroMagnetism (GEM) approximation of gravity. GVT is a generally covariant theory of gravity composed of a pseudo Riemannian metric and two U(1) gauge connections that reproduces MOND in the limit of very weak gravitational fields while remains consistent with the Einstein-Hilbert gravity in the limit of strong and Newtonian gravitational fields. GVT also provides a simple framework to study the GEM approximation to gravity. We illustrate that the gravitomagnetic force at the edge of a galaxy c…
The initial conditions for Newtonian $N$-body simulations are usually generated by applying the Zel'dovich approximation to the initial displacements of the particles using an initial power spectrum of density fluctuations generated by an Einstein-Boltzmann solver. We show that in most gauges the initial displacements generated in this way receive a first-order relativistic correction. We define a new gauge, the $N$-body gauge, in which this relativistic correction vanishes and show that a conventional Newtonian $N$-body simulation includes all first-order relativistic contributions (in the ab…
