Linearized gravity

In the theory of general relativity, linearized gravity is the application of perturbation theory to the metric tensor that describes the geometry of spacetime. As a consequence, linearized gravity is an effective method for modeling the effects of gravity when the gravitational field is weak. The usage of linearized gravity is integral to the study of gravitational waves and weak-field gravitational lensing.

In the theory of general relativity, linearized gravity is the application of perturbation theory to the metric tensor that describes the geometry of spacetime. As a consequence, linearized gravity is an effective method for modeling the effects of gravity when the gravitational field is weak. The usage of linearized gravity is integral to the study of gravitational waves and weak-field gravitational lensing. R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = \kappa T_{\mu\nu} where R_{\mu\nu} is the Ricci tensor, R is the Ricci scalar, T_{\mu\nu} is the energy–momentum tensor, \kappa is the Einstein gravitational constant, and g_{\mu\nu} is the spacetime metric tensor that represents the solutions of the equation. Although succinct when written out using Einstein notation, hidden within the Ricci tensor and Ricci scalar are exceptionally nonlinear dependencies on the metric tensor that render the prospect of finding exact solutions impractical in most systems. However, when describing systems for which the curvature of spacetime is small (meaning that terms in the EFE that are quadratic in g_{\mu\nu} do not significantly contribute to the equations of motion), one can model the solution of the field equations as being the Minkowski metric \eta_{\mu\nu} plus a small perturbation term h_{\mu\nu} . In other words: g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu},\qquad |h_{\mu\nu}| \ll 1. In this regime, substituting the general metric g_{\mu\nu} for this perturbative approximation results in a simplified expression for the Ricci tensor: R_{\mu\nu} = \frac{1}{2}(\partial_\sigma\partial_\mu h^\sigma_\nu + \partial_\sigma\partial_\nu h^\sigma_\mu - \partial_\mu\partial_\nu h - \square h_{\mu\nu}), where h = \eta^{\mu\nu}h_{\mu\nu} is the trace of the perturbation, \partial_\mu denotes the partial derivative with respect to the x^\mu coordinate of spacetime, and \square = \eta^{\mu\nu} \partial_\mu \partial_\nu is the d'Alembert operator. Together with the Ricci scalar, R = \eta_{\mu\nu}R^{\mu\nu} = \partial_\mu\partial_\nu h^{\mu\nu} - \square h, the left side of the field equation reduces to R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = \frac{1}{2}(\partial_\sigma\partial_\mu h^\sigma_\nu + \partial_\sigma\partial_\nu h^\sigma_\mu - \partial_\mu\partial_\nu h - \square h_{\mu\nu} - \eta_{\mu\nu}\partial_\rho\partial_\lambda h^{\rho\lambda} + \eta_{\mu\nu}\square h). and thus the EFE is reduced to a linear second order partial differential equation in terms of h_{\mu\nu} . h_{\mu\nu} = (\phi^*g)_{\mu\nu} - \eta_{\mu\nu}. The diffeomorphisms \phi may thus be chosen such that . Given then a vector field \xi^\mu defined on the flat background spacetime, an additional family of diffeomorphisms \psi_\epsilon may be defined as those generated by \xi^\mu and parameterized by . These new diffeomorphisms will be used to represent the coordinate transformations for "infinitesimal shifts" as discussed above. Together with , a family of perturbations is given by \begin{align} h^{(\epsilon)}_{\mu\nu} &= [(\phi\circ\psi_\epsilon)^*g]_{\mu\nu} - \eta_{\mu\nu} \\ &= [\psi^*_\epsilon(\phi^*g)]_{\mu\nu} - \eta_{\mu\nu} \\ &= \psi^*_\epsilon(h + \eta)_{\mu\nu} - \eta_{\mu\nu} \\ &= (\psi^*_\epsilon h)_{\mu\nu} + \epsilon\left[\frac{(\psi^*_\epsilon\eta)_{\mu\nu} - \eta_{\mu\nu}}{\epsilon}\right]. \end{align} Therefore, in the limit , h^{(\epsilon)}_{\mu\nu} = h_{\mu\nu} + \epsilon\mathcal{L}_\xi\eta_{\mu\nu} where \mathcal{L}_\xi is the Lie derivative along the vector field . The Lie derivative works out to yield the final gauge transformation of the perturbation metric : h^{(\epsilon)}_{\mu\nu} = h_{\mu\nu} + \epsilon(\partial_\mu\xi_\nu + \partial_\nu\xi_\mu), which precisely define the set of perturbation metrics that describe the same physical system. In other words, it characterizes the gauge symmetry of the linearized field equations. s_{ij} = h_{ij} - \frac{1}{3}\delta^{kl}h_{kl}\delta_{ij} (Note that the indices span only spatial components: ). Thus, by using , the spatial components of the perturbation can be decomposed as h_{ij} = s_{ij} - \Psi\delta_{ij} where \Psi = \frac{1}{3}\delta^{kl}h_{kl} . The tensor s_{ij} is, by construction, traceless and is referred to as the strain since it represents the amount by which the perturbation stretches and contracts measurements of space. In the context of studying gravitational radiation, the strain is particularly useful when utilized with the transverse gauge. This gauge is defined by choosing the spatial components of \xi^\mu to satisfy the relation \nabla^2\xi^j + \frac{1}{3}\partial_j\partial_i\xi^i = -\partial_i s^{ij}, then choosing the time component \xi^0 to satisfy \nabla^2\xi^0 = \partial_i h_{0i} + \partial_0\partial_i\xi^i. After performing the gauge transformation using the formula in the previous section, the strain becomes spatially transverse: \partial_i s^{ij}_{(\epsilon)} = 0, with the additional property: \partial_i h^{0i}_{(\epsilon)} = 0. h^{(\epsilon)}_{0\nu} = 0. This can be achieved by requiring the time component of \xi^\mu to satisfy \partial_0\xi^0 = -h_{00} and requiring the spatial components to satisfy \partial_0\xi^i = \partial_i\xi^0 - h_{0i}. \partial_\mu h^\mu_\nu = \frac{1}{2}\partial_\nu h is true. To achieve this, \xi_\mu is required to satisfy the relation \square\xi_\mu = -\partial_\nu h^\nu_\mu + \frac{1}{2}\partial_\mu h. Consequently, by using the harmonic gauge, the Einstein tensor G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} reduces to G_{\mu\nu} = -\frac{1}{2}\square\left(h^{(\epsilon)}_{\mu\nu} - \frac{1}{2}h^{(\epsilon)}\eta_{\mu\nu}\right). Therefore, by writing it in terms of a "trace-reversed" metric, , the linearized field equations reduce to \square \bar{h}^{(\epsilon)}_{\mu\nu} = -2\kappa T_{\mu\nu}. This can be solved exactly, to produce the wave solutions that define gravitational radiation.

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