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Linearized gravity

Linearized gravity is a perturbative approximation to Einstein's general theory of relativity, valid in the weak-field regime where the spacetime metric g_{\mu\nu} is expressed as a small deviation h_{\mu\nu} from the flat Minkowski metric \eta_{\mu\nu}, with |h_{\mu\nu}| \ll 1.^[1] This framework linearizes the Einstein field equations to first order in the perturbation, simplifying the nonlinear partial differential equations of full general relativity into a set of linear equations that resemble those of electromagnetism or other classical field theories.^[2] It serves as a foundational tool for analyzing phenomena such as the Newtonian limit of gravity, the propagation of gravitational waves, and weak-field tests of general relativity in astrophysical and cosmological contexts.^[3] In linearized gravity, the metric perturbation h_{\mu\nu} has 10 components in four-dimensional spacetime, but coordinate gauge freedom—arising from infinitesimal transformations x^\mu \to x^\mu - \xi^\mu—reduces the independent physical degrees of freedom to six: two scalar modes, two vector modes, and two tensor modes.^[3] A common choice is the Lorentz (or harmonic) gauge, where \partial^\mu \bar{h}_{\mu\nu} = 0 with the trace-reversed perturbation \bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2} \eta_{\mu\nu} h, which further simplifies the vacuum field equations to the wave equation \Box \bar{h}_{\mu\nu} = 0, indicating that gravitational disturbances propagate at the speed of light.^[1] For non-vacuum cases, the sourced equation becomes \Box \bar{h}_{\mu\nu} = -16\pi G T_{\mu\nu}, where T_{\mu\nu} is the stress-energy tensor, linking the perturbation directly to matter distributions.^[2] The theory's tensor modes correspond to gravitational waves, which are transverse and traceless in the appropriate gauge, exhibiting two independent polarization states (plus and cross) that cause tidal distortions in test masses without net displacement.^[1] In the static weak-field limit, the scalar modes recover the Newtonian Poisson equation \nabla^2 \Phi = 4\pi [G](/page/G) \rho, where \Phi relates to the time-time component of the metric, thus bridging classical gravity with relativistic corrections.^[3] Linearized gravity underpins the detection and analysis of gravitational waves from sources like binary black hole mergers, as observed by detectors such as LIGO, and informs precision tests of general relativity in the solar system.^[2]

Fundamentals

Definition and Scope

Linearized gravity refers to the first-order perturbative approximation of Einstein's field equations in the regime of weak gravitational fields, where the spacetime metric is expanded around the flat Minkowski background as g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, with the perturbation satisfying |h_{\mu\nu}| \ll 1.^[1] This approach linearizes the nonlinear Einstein equations by retaining only terms up to first order in h_{\mu\nu}, effectively treating gravity as a small deviation from special relativity.^[1] It provides a simplified framework for analyzing phenomena where the curvature is mild, such as distant regions from massive sources.^[4] The key assumptions underlying linearized gravity include a flat background metric \eta_{\mu\nu} of Minkowski spacetime, small metric perturbations h_{\mu\nu} that do not significantly alter the geometry, and coordinate systems chosen to maintain the linearity of the approximation.^[1] These assumptions hold in scenarios with weak fields and low matter densities, often in vacuum or with collisionless matter distributions where higher-order nonlinear effects are negligible.^[4] The scope of linearized gravity encompasses applications in weak-field environments, such as far from compact objects like black holes or neutron stars, and situations involving slow motions where velocities are much less than the speed of light (v \ll c).^[4] It is particularly useful for studying large-scale cosmological structures under the \LambdaCDM model and for weak-field tests of general relativity, but its validity breaks down in strong-field regimes or dense matter configurations where nonlinear corrections become significant, potentially introducing percent-level errors.^[4] Historically, linearized gravity was developed in the 1910s and 1920s by Albert Einstein and contemporaries, initially to explore gravitational waves as solutions to the approximated field equations.^[5] Einstein's 1916 paper demonstrated that these waves propagate at the speed of light in the weak-field limit, laying the groundwork for later refinements and experimental verifications.^[5]

Relation to General Relativity

Linearized gravity emerges from general relativity as the lowest-order approximation in the perturbative expansion of the nonlinear Einstein field equations G_{\mu\nu} = 8\pi G T_{\mu\nu} around a flat Minkowski spacetime background.^[1] This approach, first explored by Einstein in his derivation of gravitational waves, treats deviations from flatness as small perturbations, allowing the complex nonlinear structure of full general relativity to be simplified for analysis.^[6] The expansion is a Taylor series in the perturbation amplitude, retaining only terms up to first order in h, where higher-order contributions capture the nonlinear interactions inherent to strong gravitational fields.^[7] The perturbative hierarchy is formalized by decomposing the full metric as g_{\mu\nu} = \eta_{\mu\nu} + \epsilon h_{\mu\nu} + O(\epsilon^2), with \eta_{\mu\nu} the Minkowski metric, h_{\mu\nu} the small perturbation satisfying |h_{\mu\nu}| \ll 1, and \epsilon a bookkeeping parameter set to unity after linearization.^[8] This ansatz linearizes the Ricci tensor and scalar curvature, transforming the Einstein tensor into a form that resembles the field equations of a massless spin-2 field propagating on flat spacetime.^[1] Higher-order terms in \epsilon, such as O(\epsilon^2) and beyond, encode nonlinear effects like gravitational self-interaction, which are essential for phenomena involving intense curvature, such as black hole formation or mergers.^[7] The validity of this linearization rests on the weak-field assumption, where spacetime curvature is mild enough that nonlinear terms in the field equations can be neglected, yielding a wave equation for h_{\mu\nu} analogous to the linearized Maxwell equations for electromagnetism.^[8] In such regimes—exemplified by the solar system's gravitational field or distant astronomical sources—the perturbation h_{\mu\nu} remains small compared to unity, enabling exact solutions in vacuum or simple matter distributions without resorting to the full theory's computational complexity.^[1] However, linearized gravity has inherent limitations, breaking down in regions of strong gravitational fields, such as near black holes, or in scenarios involving relativistic velocities where higher-order corrections become significant.^[7] It also fails to account for the backreaction of gravitational waves on the source or the conservation of gravitational energy-momentum in a local sense, issues resolved only in the nonlinear framework.^[8] Consequently, it serves primarily as the foundational step for more advanced approximations, including post-Newtonian expansions that systematically incorporate nonlinear terms for precision tests of general relativity in weakly curved but dynamically rich environments.^[1]

Formulation

Metric Perturbation

In the weak-field approximation of general relativity, linearized gravity describes spacetime geometry by perturbing the flat Minkowski metric. The standard ansatz for the metric tensor is

g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu},

where \eta_{\mu\nu} denotes the Minkowski metric with signature (-,+,+,+), and h_{\mu\nu} represents a small symmetric perturbation tensor satisfying |h_{\mu\nu}| \ll 1. This expansion is valid when gravitational fields are weak, such as far from massive sources or for small-amplitude gravitational waves. To first order, all indices on the perturbation are raised and lowered using the background Minkowski metric, ensuring the approximation remains consistent without contributions from higher-order terms in the inverse metric g^{\mu\nu} \approx \eta^{\mu\nu} - h^{\mu\nu}.^[1] The contravariant form of the perturbation is defined as h^{\mu\nu} = \eta^{\mu\alpha} \eta^{\nu\beta} h_{\alpha\beta}, which preserves the symmetry h^{\mu\nu} = h^{\nu\mu}. This convention for index placement aligns with the linear order, where the difference between raising/lowering indices with g^{\mu\nu} or \eta^{\mu\nu} introduces negligible corrections of order h^2. The trace of the perturbation, h = \eta^{\mu\nu} h_{\mu\nu}, provides a scalar measure of the overall deviation from flatness, with the mostly-plus signature yielding h = -h_{00} + h_{ii} in coordinate components.^[9] A useful reformulation involves the trace-reversed perturbation \bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2} \eta_{\mu\nu} h, which has trace \bar{h} = \eta^{\mu\nu} \bar{h}_{\mu\nu} = -h in four dimensions. This definition simplifies subsequent calculations by decoupling the trace from certain tensor components, though it does not alter the physical content of the theory. The corresponding contravariant version follows analogously from raising indices with \eta^{\mu\nu}.^[1] For the approximation to hold in isolated systems, boundary conditions require the perturbation to decay at infinity: h_{\mu\nu} \to 0 as the spatial distance r \to \infty. This condition is well-suited to asymptotically flat spacetimes, where the geometry approaches Minkowski far from sources, enabling a well-defined perturbative expansion without divergent contributions.^[1]

Linearized Field Equations

The linearized Einstein field equations are obtained by expanding the full nonlinear equations of general relativity to first order in the metric perturbation h_{\mu\nu}, assuming a background Minkowski metric \eta_{\mu\nu} with |h_{\mu\nu}| \ll 1.^[10] The process begins with the linearization of the Christoffel symbols, which appear in the Riemann tensor and subsequently in the Ricci tensor. At linear order, the Christoffel symbols simplify to \Gamma^\lambda_{\mu\nu} = \frac{1}{2} \eta^{\lambda\sigma} (\partial_\mu h_{\sigma\nu} + \partial_\nu h_{\sigma\mu} - \partial_\sigma h_{\mu\nu}), where partial derivatives are taken with respect to Minkowski coordinates and indices are raised/lowered using \eta^{\mu\nu}.^[1] From these, the linearized Ricci tensor R_{\mu\nu}^{(1)} is derived as

R_{\mu\nu}^{(1)} = \frac{1}{2} \left( \partial^\sigma \partial_\mu h_{\sigma\nu} + \partial^\sigma \partial_\nu h_{\sigma\mu} - \partial_\mu \partial_\nu h - \Box h_{\mu\nu} \right),

where h = \eta^{\mu\nu} h_{\mu\nu} is the trace of the perturbation and \Box = \eta^{\mu\nu} \partial_\mu \partial_\nu is the flat-space d'Alembertian operator.^[3] The linearized scalar curvature follows as R^{(1)} = \eta^{\mu\nu} R_{\mu\nu}^{(1)} = \partial^\sigma \partial^\rho h_{\sigma\rho} - \Box h. The Einstein tensor at linear order is then G_{\mu\nu}^{(1)} = R_{\mu\nu}^{(1)} - \frac{1}{2} \eta_{\mu\nu} R^{(1)}, yielding the full sourced linearized field equations

\frac{1}{2} \left( \partial^\sigma \partial_\mu \bar{h}_{\sigma\nu} + \partial^\sigma \partial_\nu \bar{h}_{\sigma\mu} - \Box \bar{h}_{\mu\nu} - \eta_{\mu\nu} \partial^\sigma \partial^\rho \bar{h}_{\sigma\rho} \right) = 8\pi G T_{\mu\nu},

where \bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2} \eta_{\mu\nu} h is the trace-reversed perturbation and T_{\mu\nu} is the stress-energy tensor (raised to first order in the matter fields).^[1] These equations hold without imposing a specific gauge and capture the dynamics of weak gravitational fields coupled to matter.^[1] In the vacuum case, where T_{\mu\nu} = 0, the equations simplify to R_{\mu\nu}^{(1)} = 0. In the harmonic (Lorenz-de Donder) gauge, defined by \partial^\mu \bar{h}_{\mu\nu} = 0, this further reduces to the wave equation \Box \bar{h}_{\mu\nu} = 0, describing freely propagating gravitational disturbances at the speed of light.^[3] The linearization preserves the contracted Bianchi identity \partial^\mu G_{\mu\nu}^{(1)} = 0, which implies the conservation law for the stress-energy tensor at linear order: \partial^\mu T_{\mu\nu} = 0. This consistency ensures that the matter dynamics remain compatible with the gravitational field equations without introducing additional constraints.^[1]

Gauge Invariance

Gauge Transformations

In general relativity, the full theory is invariant under general coordinate transformations, reflecting the diffeomorphism invariance of the theory. In the linearized approximation, this invariance manifests as a residual gauge freedom, where infinitesimal coordinate transformations generated by a small vector field \xi^\mu (with |\xi^\mu| \ll 1) leave the physical content unchanged at linear order. These transformations correspond to diffeomorphisms of the flat background spacetime, preserving the structure of the perturbation theory without mixing higher-order terms.^[1] Under such an infinitesimal gauge transformation, the metric perturbation h_{\mu\nu} transforms as

h'_{\mu\nu}(x) = h_{\mu\nu}(x) - \partial_\mu \xi_\nu(x) - \partial_\nu \xi_\mu(x),

where the derivatives are with respect to the flat Minkowski coordinates, and the transformation is exact to first order in \xi^\mu. This law arises directly from the Lie derivative of the metric along \xi^\mu, truncated at linear order. For the commonly used trace-reversed perturbation \bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2} \eta_{\mu\nu} h (with h = \eta^{\alpha\beta} h_{\alpha\beta}), the transformation becomes

\bar{h}'_{\mu\nu}(x) = \bar{h}_{\mu\nu}(x) - \partial_\mu \xi_\nu(x) - \partial_\nu \xi_\mu(x) + \eta_{\mu\nu} \partial^\sigma \xi_\sigma(x),

reflecting the additional contribution from the trace shift under the gauge change. These transformations ensure that the linearized Einstein equations remain form-invariant, as the gauge part satisfies the vacuum equations trivially.^[1]^[11] The vector \xi^\mu(x) introduces four arbitrary functions per spacetime point, corresponding to the four coordinates in four dimensions. Since h_{\mu\nu} is a symmetric tensor with 10 independent components, this gauge freedom removes four degrees of freedom, leaving six physical components that describe the propagating gravitational degrees of freedom, such as the two polarizations of gravitational waves. The smallness of \xi^\mu guarantees that the transformations do not generate nonlinear corrections, maintaining the validity of the linear approximation throughout.^[12]

Gauge-Invariant Quantities

In linearized gravity, gauge-invariant quantities are physical observables that remain unchanged under infinitesimal coordinate transformations, ensuring that predictions depend only on the intrinsic geometry rather than the choice of coordinates. These quantities are constructed by forming combinations of the metric perturbation h_{\mu\nu} (or its trace-reversed form \bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2} \eta_{\mu\nu} h) that eliminate the gauge-dependent parts. Under a gauge transformation parameterized by an infinitesimal vector \xi^\mu, the change in the trace-reversed perturbation is given by

\bar{h}_{\mu\nu} - \bar{h}'_{\mu\nu} = - \left( \partial_\mu \xi_\nu + \partial_\nu \xi_\mu - \eta_{\mu\nu} \partial \cdot \xi \right),

where \partial \cdot \xi = \partial^\rho \xi_\rho. Gauge-invariant quantities can thus be built as differences or projections that are orthogonal to these transformation terms, such as subtracting the gauge contribution from \bar{h}_{\mu\nu} to isolate the physical part.^[13] A common method involves projecting the perturbation onto transverse and traceless (TT) components, which are inherently gauge-invariant because gauge transformations cannot generate TT modes. In Fourier space, this projection removes longitudinal and trace parts, yielding two independent polarization degrees of freedom for the tensor sector. For example, the linearized Weyl tensor components, such as the Newman-Penrose scalars \Psi_0 and \Psi_4, are gauge-invariant at first order and capture the tidal field effects without coordinate artifacts. The metric perturbation h_{\mu\nu} can be decomposed into scalar, vector, and tensor modes based on their transformation properties under spatial rotations, a technique particularly useful in cosmological contexts. The tensor modes, consisting of the TT part of the spatial perturbation h_{ij}, are fully gauge-invariant with two degrees of freedom. In contrast, the four scalar modes (e.g., involving h_{00}, the trace of h_{ij}, and longitudinal parts) yield two gauge-invariant combinations, such as the Bardeen potentials \Phi and \Psi, which represent curvature and Newtonian-like potentials, respectively; the two vector modes reduce to one gauge-invariant vorticity-like quantity.^[3] These gauge-invariant quantities are crucial for deriving coordinate-independent physical predictions, such as the geodesic deviation equation, which measures relative accelerations of test particles due to tidal forces encoded in the Weyl tensor. In cosmological perturbation theory, they facilitate the study of structure formation by ensuring that observables like density contrasts evolve independently of gauge choices.^[13]

Gauge Choices

Harmonic Gauge

The harmonic gauge, also known as the de Donder gauge, imposes the condition \partial^\mu \bar{h}_{\mu\nu} = 0 on the trace-reversed perturbation \bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2} \eta_{\mu\nu} h, where h_{\mu\nu} is the metric perturbation from the flat background \eta_{\mu\nu} and h = h^\lambda_\lambda is its trace.^[14]^[2] This condition is analogous to the Lorentz gauge in electromagnetism and was originally introduced in the context of general relativity to facilitate the treatment of the gravitational field. Imposing the harmonic gauge simplifies the linearized Einstein field equations to the sourced wave equation

\Box \bar{h}_{\mu\nu} = -16\pi G T_{\mu\nu},

where \Box = \partial^\mu \partial_\mu is the d'Alembertian operator and T_{\mu\nu} is the stress-energy tensor (in units where c=1).^[14]^[2] This form decouples the equations for each component of \bar{h}_{\mu\nu}, allowing solutions that propagate disturbances at the speed of light, much like electromagnetic waves.^[14] The advantages include enhanced solvability for problems involving propagating gravitational fields, as the equation resembles the inhomogeneous wave equation familiar from classical field theories.^[2] In the far field, away from sources, the solution can be expressed using retarded potentials:

\bar{h}_{\mu\nu}(t, \mathbf{x}) = 4G \int \frac{T_{\mu\nu}(t - |\mathbf{x} - \mathbf{y}|, \mathbf{y})}{|\mathbf{x} - \mathbf{y}|} d^3 y,

which ensures causality by evaluating the source at the retarded time.^[14]^[2] This integral form is particularly useful for analyzing radiation from localized sources, such as binary systems, where the leading-order far-field behavior scales as $1/r.^[2] The harmonic gauge does not exhaust all gauge freedom; residual transformations of the form \bar{h}_{\mu\nu} \to \bar{h}_{\mu\nu} + \partial_\mu \xi_\nu + \partial_\nu \xi_\mu, where \xi^\mu satisfies the homogeneous wave equation \Box \xi^\mu = 0, remain possible.^[14]^[2] These four constraints eliminate unphysical degrees of freedom, reducing the 10 components of the symmetric h_{\mu\nu} to 6 physical ones, consistent with the propagation properties of a massless spin-2 field in the presence of sources.^[14]

Synchronous Gauge

The synchronous gauge is a coordinate choice in linearized gravity where the metric perturbation satisfies h_{0\mu} = 0, eliminating time-space mixing terms in the perturbed metric.^[15] This condition, originally introduced by Lifshitz in the context of cosmological perturbations, sets the off-diagonal components h_{0i} = 0 and often includes h_{00} = 0 for comoving coordinates, resulting in a line element of the form

ds^2 = -dt^2 + (\delta_{ij} + h_{ij}) dx^i dx^j

in flat background spacetimes or, more generally in expanding universes,

ds^2 = a^2(\eta) \left[ -d\eta^2 + (\delta_{ij} + h_{ij}) dx^i dx^j \right],

where \eta is conformal time and a(\eta) is the scale factor.^[16]^[17] This gauge choice ensures that the worldlines of comoving observers—fundamental matter particles at rest in the coordinate system—coincide with the coordinate time lines, preserving matter geodesics as straight lines in spatial coordinates without acceleration.^[15] In the linearized Einstein field equations (EFE), the $00-component becomes a constraint equation relating the spatial perturbation h_{ij} directly to the energy density perturbation \delta T_{00}, such as k^2 \eta - \frac{1}{2} \mathcal{H} \dot{h} = 4\pi G a^2 \delta \rho in Fourier space for scalar modes, where \mathcal{H} = \dot{a}/a and h_{ij} decomposes into scalar, vector, and tensor modes, with the scalar contribution given by h_{ij}^{(s)} = \frac{h}{3} \delta_{ij} + \left( \partial_i \partial_j - \frac{1}{3} \delta_{ij} \partial^2 \right) \eta, where h = h^k_k is the trace.^[16] The remaining equations govern the evolution of h_{ij}, decoupling scalar, vector, and tensor modes while incorporating matter sources like \delta T^i_i.^[16] The synchronous gauge offers significant advantages in spacetimes filled with matter, particularly in the Newtonian limit, where it aligns the metric perturbation with the gravitational potential, simplifying the recovery of Poisson's equation from the EFE constraints.^[16] It is extensively used in cosmological perturbation theory to model density perturbations, where the matter overdensity \delta evolves as \dot{\delta} = -\frac{1}{2} \dot{h} for pressureless dust (cold dark matter), directly linking metric variables to observable quantities like galaxy clustering.^[16]^[15] However, the gauge retains residual freedom under spatial coordinate transformations x^i \to x^i + \xi^i(t, \mathbf{x}) with time-independent \dot{\xi}^i = 0, allowing shifts that mix physical modes with gauge artifacts, which must be carefully subtracted to isolate true perturbations.^[16] Additionally, it can obscure the hyperbolic wave nature of gravitational propagation, as the coordinate system ties observers to geodesics that may intersect, leading to singularities in highly nonlinear regimes.^[15]

Transverse-Traceless Gauge

The transverse-traceless (TT) gauge is a specialized coordinate choice in linearized general relativity, particularly suited for describing gravitational waves propagating in vacuum. It imposes strict conditions on the metric perturbation h_{\mu\nu}: the trace vanishes (h = h^i_i = 0), the components involving time are zero (h_{0\mu} = 0), and the spatial part is transverse (\partial^i h_{ij} = 0), ensuring that the perturbation is purely spatial and divergence-free.^[1]^[18] These conditions reduce the ten components of h_{\mu\nu} to just two independent degrees of freedom, corresponding to the plus (h_+) and cross (h_\times) tensor polarizations for a wave propagating in the z-direction.^[19]^[1] In this gauge, the linearized vacuum Einstein field equations simplify dramatically to the wave equation \Box \bar{h}_{ij}^{\rm TT} = 0, where \bar{h}_{ij}^{\rm TT} = h_{ij}^{\rm TT} (since the trace vanishes) and \Box = -\partial_t^2 + \nabla^2 is the flat-space d'Alembertian. This form highlights the propagation of pure tensor modes at the speed of light, free from scalar or vector contributions that could arise in other gauges. The TT gauge thus captures the physical content of gravitational radiation as transverse quadrupolar distortions, aligning directly with the gauge-invariant tensor modes of the theory.^[18]^[19] The TT gauge is typically imposed starting from the more general harmonic (Lorenz) gauge, where \partial^\mu \bar{h}_{\mu\nu} = 0, by exploiting the remaining gauge freedom through infinitesimal coordinate transformations x^\mu \to x^\mu + \xi^\mu satisfying \Box \xi^\mu = 0. Specifically, one chooses \xi^0 and \xi^i to eliminate the trace and longitudinal components of the spatial perturbation, projecting it onto the transverse plane perpendicular to the propagation direction. This process yields the TT form without altering the physical propagation, as the gauge conditions are adapted to the plane-wave ansatz.^[1]^[19] A key advantage of the TT gauge is its direct connection to observable effects: the perturbation h_{ij}^{\rm TT} represents the physical tidal strain on test masses, with no gauge artifacts contaminating the signal, making it ideal for gravitational wave detection experiments. For instance, the plus polarization stretches space along the x-direction while compressing it along y, and vice versa, whereas the cross polarization shears space in the xy-plane. This purity of tensor modes, with exactly two degrees of freedom, facilitates precise modeling of detector responses and energy flux calculations in vacuum radiative scenarios.^[18]^[1]

Applications

Gravitational Waves

In linearized general relativity, gravitational waves emerge as propagating disturbances in the metric perturbation h_{\mu\nu} that satisfy the vacuum field equations \square \bar{h}_{\mu\nu} = 0, where \bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2} \eta_{\mu\nu} h is the trace-reversed perturbation and \square is the flat-space d'Alembertian operator. These waves describe weak, far-field radiation from accelerating sources, analogous to electromagnetic waves but tensorial in nature.^[20] The simplest vacuum solutions are monochromatic plane waves of the form

h_{\mu\nu}(x) = \mathrm{Re} \left[ A_{\mu\nu} \, e^{i k^\lambda x_\lambda} \right],

where A_{\mu\nu} is a constant amplitude tensor and the wave vector k^\mu satisfies the null condition k^\mu k_\mu = 0, ensuring propagation along lightlike geodesics. In the transverse-traceless (TT) gauge, where the waves propagate in the z-direction, the perturbation simplifies to h_{\mu\nu}^{\mathrm{TT}} = h_{ij}^{\mathrm{TT}} with only spatial components transverse to the propagation direction (i,j = x,y), traceless (h_{ii}^{\mathrm{TT}} = 0), and divergence-free (\partial_i h_{ij}^{\mathrm{TT}} = 0). This gauge reveals two independent polarization states: the plus polarization h_+, which stretches and compresses spacetime alternately along the x- and y-axes, and the cross polarization h_\times, which shears spacetime at 45 degrees to these axes. These polarizations are orthogonal and carry the physical degrees of freedom of the massless spin-2 graviton field.^[20]^[21] Gravitational waves propagate at the speed of light c without dispersion, as dictated by the null wave vector condition, allowing them to travel vast cosmic distances coherently from their sources. This null propagation implies that the waves are causal, with no superluminal signaling, and their phase velocity equals the group velocity.^[20] For generation by weak, isolated sources, the dominant far-field contribution arises from the time-varying mass quadrupole moment Q_{ij}, projected into the TT gauge. In the post-Newtonian approximation (with c = 1 units), the TT component of the metric perturbation at a distance r from the source is given by

h_{ij}^{\mathrm{TT}}(t, \mathbf{x}) = \frac{2G}{r} \ddot{Q}_{ij}^{\mathrm{TT}}(t - r),

where \ddot{Q}_{ij}^{\mathrm{TT}} is the second time derivative of the transverse-traceless quadrupole moment, evaluated at retarded time t - r. This formula captures the leading-order (v^2/c^2) radiation from non-spherical accelerations, such as in binary systems, with higher multipoles contributing at smaller orders.^[22] The energy and momentum carried by gravitational waves are described by an effective stress-energy pseudotensor, which in the linearized regime averages to t_{\mu\nu} \sim \langle \partial_\mu h_{ij} \partial_\nu h^{ij} \rangle, where the angular brackets denote spatial and temporal averaging over several wavelengths. For plane waves in the TT gauge, the time-averaged energy flux (Poynting vector) is \langle t_{0i} \rangle \approx \frac{c^3}{32\pi G} \langle \dot{h}_{ij}^{\mathrm{TT}} \dot{h}^{ij\mathrm{TT}} \rangle \hat{n}_i, indicating that the waves transport positive energy density and momentum along the propagation direction, consistent with their null character. This pseudotensor, while coordinate-dependent, provides a gauge-invariant measure of the backreaction on the background spacetime when averaged.^[23]

Newtonian Limit and Matter Coupling

In the Newtonian limit of linearized gravity, the theory is applicable to weak gravitational fields and non-relativistic motions, where velocities are much smaller than the speed of light (v \ll c) and the field strength satisfies |\Phi| \ll c^2, with \Phi denoting the Newtonian gravitational potential.^[2] In this regime, the metric perturbation takes the form h_{00} = -2\Phi/c^2, h_{0i} = 0, and h_{ij} = -2\Phi \delta_{ij}/c^2 in the isotropic gauge, which ensures spatial isotropy and simplifies the spatial components to a conformal factor times the flat metric.^[24] This ansatz aligns the linearized Einstein field equations with the classical description of gravity, recovering Newtonian behavior while preserving the relativistic structure to first order. The recovery of Poisson's equation emerges directly from the 00-component of the linearized Einstein field equations in this limit. Substituting the metric perturbation into the Ricci tensor yields R_{00} \approx -\frac{1}{2} \nabla^2 h_{00}, and with the stress-energy tensor dominated by the rest mass energy, the equation simplifies to \nabla^2 \Phi = 4\pi G \rho, where \rho is the mass density, matching the Newtonian gravitational potential equation.^[25] This derivation confirms that linearized gravity encompasses classical gravity as a low-energy approximation, with the coupling constant G emerging from the relativistic framework via \kappa = 8\pi G / c^4.^[24] Coupling to matter in the Newtonian limit involves the stress-energy tensor T^{\mu\nu}, which for non-relativistic sources (e.g., dust or pressureless matter) approximates as T_{00} \approx \rho c^2 and T_{0i} \approx -\rho v_i c, with spatial components T_{ij} negligible to leading order.^[2] These components source the metric perturbations through the linearized field equations, \square \bar{h}_{\mu\nu} = -16\pi G T_{\mu\nu}/c^4, where the bar denotes the trace-reversed perturbation. Post-Newtonian corrections arise from the shear in h_{ij}, which introduces small deviations from the scalar potential but remains linear in the weak-field expansion.^[25] The motion of test particles follows geodesics in the perturbed metric, with the linearized Christoffel symbols yielding the equation \ddot{x}^\mu \approx -\Gamma^\mu_{\alpha\beta} \dot{x}^\alpha \dot{x}^\beta. For non-relativistic particles (\dot{x}^0 \approx c, \dot{x}^i \ll c), this reduces to \ddot{\vec{x}} = -\nabla \Phi, reproducing Newton's second law under gravity.^[24] This geodesic approximation validates the equivalence principle at linear order, linking the curved spacetime geometry to inertial forces in the Newtonian framework.^[25]

References

[1]
[PDF] Lecture VIII: Linearized gravity - Caltech (Tapir)
Nov 5, 2012 · In the most general case, gravitational waves can be elliptically polarized. An alternative and frequently useful description of gravitational ...
[2]
6. Weak Fields and Gravitational Radiation
In fact, we can think of the linearized version of general relativity (where effects of higher than first order in h ... gravity should predict their existence.
[3]
[PDF] General Relativity Fall 2017 Lecture 12: Linearized gravity, Part II
While a gauge transformation in electromagnetism amounts to providing one scalar function, a gauge transformation in linearized GR amounts to providing 4 ...
[4]
[PDF] arXiv:1810.05203v1 [astro-ph.CO] 11 Oct 2018
Oct 11, 2018 · In this work we examine the accuracy of linearized gravity in the presence of collisionless matter and a cosmological constant utilizing fully ...
[5]
https://arxiv.org/abs/1602.04040
[6]
[PDF] Einstein's Discovery of Gravitational Waves 1916-1918 - arXiv
Dec 2, 2016 · In June 1916, Einstein presented to the Prussian Academy of. Sciences a paper on gravitational waves, published later under the title, " ...
[7]
[PDF] 5. When Gravity is Weak - DAMTP
Because gravitational waves are so weak, the linearised metric is entirely adequate for any properties of gravitational waves that we wish to calcuate.<|control11|><|separator|>
[8]
[PDF] 14.1 Solving the Einstein field equation: General considerations - MIT
We begin by studying the linearized field equations. 14.2 The “weak-field” limit: Linearized gravity. The weak-field limit is defined by the idea that ...
[9]
[PDF] Linearized Gravity
Jun 4, 2024 · In this chapter we will derive an approximation of Einstein's equations in a scenario with weak gravity. Additionally we will assume the ...Missing: scope | Show results with:scope<|control11|><|separator|>
[10]
[PDF] Lecture 15: Linearized Einstein field equations
Oct 17, 2019 · Since we will linearize the relevant equations, we may work in Fourier space: each Fourier mode satisfies an independent equation. We denote by ...
[11]
[PDF] General Relativity Fall 2019 Lecture 11: Linearized gravity, Part I
Oct 15, 2019 · Such a gauge transformation does not affect any measurable quantity at linear order. ... To summarize, the parallels between E&M and linearized ...
[12]
[PDF] 4 Linearized gravity - It works!
We can use the linearization not only at the level of field equations but also at the level of the action. As a first task we fill in a gap that was left open ...
[13]
[PDF] 1 The metric, stress-energy tensor, and gauge transformations.
These are gauge-invariant at first order. These parts will correspond to gravitational waves, and they cannot be gauged away. • There are 4 scalar parts of ...
[14]
None
Below is a merged response that consolidates all the information from the provided summaries into a single, comprehensive summary. To maximize detail and clarity, I will use a table in CSV format to organize the key points across the different sections and sources, followed by a narrative summary that ties everything together. Since the system has a "no thinking token" limit, I’ll focus on directly compiling and presenting the data without additional interpretation beyond what’s provided.
[15]
Synchronous gauge - Cosmological Dynamics - E. Bertschinger
Synchronous gauge has the property that there exists a set of comoving observers who fall freely without changing their spatial coordinates.
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[astro-ph/9506072] Cosmological Perturbation Theory in the ... - arXiv
Jun 11, 1995 · This paper presents a systematic treatment of the linear theory of scalar gravitational perturbations in the synchronous gauge and the conformal Newtonian (or ...Missing: gravity | Show results with:gravity
[17]
E. Lifshitz, On the gravitational stability of the expanding universe
Jan 18, 2017 · The coordinates are not uniquely determined by the synchronous gauge choice (4), hence some solutions to these equations are gauge modes (they ...
[18]
[PDF] Gravitational Waves - Theory - Theoretical Astrophysics Group
Feb 5, 2020 · This is a gauge transformation and not a coordinate one. • We are still working on the same set of coordinates xµ and have defined a new tensor ...
[19]
[PDF] General Relativity and Gravitational Waves
5.2 The Transverse-Traceless Gauge. We begin where we left off last time. We seek to solve the weak-field Einstein equation in nearly cartesian coordinates, ...
[20]
[PDF] Sitzungsberichte der Königlich Preussischen Akademie der ...
833. Page 2. Einstein: Näherungsweise Integration der Feldgleichungen der Gravitation. 689. § i. Integration der Näherungsgleichungen des. Gravitationsfeldes.
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Gravitational Radiation in the Limit of High Frequency. I. The Linear ...
This paper develops a formalism for high-frequency gravitational waves, using a linearized equation, and shows they travel on null geodesics like light waves.
[22]
[PDF] Über Gravitationswellen.
Februar 1918. – Mitteilung vom 31. Januar. Über Gravitationswellen. Von A. ENsri. Vorgelegt am 31 Januar 1918 s. oben S. 79). | ). wichtige Frage, wie die ...
[23]
Gravitational Radiation in the Limit of High Frequency. II. Nonlinear ...
This paper considers nonlinearities in high-frequency gravitational waves, finding a stress tensor for energy and momentum, and that waves propagate along null ...
[24]
[PDF] Einstein's Theory of Gravity
Dec 4, 2019 · We will show that Einstein's equations in the Newtonian limit reduce to the well known Poisson equation of Newtonian gravity. For small ...
[25]
[PDF] Newtonian limit
Oct 28, 2020 · Poisson equation for Newtonian gravitational potential; and ii) the geodesic equation reduces to Newton's equation of motion. Newtonian ...