Conventions

The equations on this page are written

Manifold calculus (?)

Metric and inverse metric

The inverse metric \(g^{\mu\nu}\) of a metric \(g_{\mu\nu}\) is defined by \[ g^{\mu\rho} g_{\nu\rho} = \delta^\mu_\nu. \]

Covariant derivative

The covariant derivative for a rank \((M,N)\) tensor \(T\) is \[ \begin{align} \nabla_\mu T^{\alpha_1 \ldots \alpha_M}_{\beta_1 \ldots \beta_N} &= \partial_\mu T^{\alpha_1 \ldots \alpha_M}_{\beta_1 \ldots \beta_N} \\ &+ \Gamma^{\alpha_1}_{\sigma \mu} T^{\sigma \alpha_2 \ldots \alpha_M}_{\beta_1 \ldots \beta_N} + \ldots + \Gamma^{\alpha_M}_{\sigma\mu} T^{\alpha_1 \ldots \alpha_{M-1} \sigma}_{\beta_1 \ldots \beta_N} \\ &- \Gamma^\sigma_{\beta_1 \mu} T^{}_{\sigma \beta_2 \ldots \beta_N} - \ldots - \Gamma^\sigma_{\beta_N \mu} T^{}_{\beta_1 \ldots \beta_{M-1} \sigma}. \end{align} \] That is, for each ….

Christoffel symbols

\[ \Gamma^\lambda_{\mu\nu} = \ldots \] Assuming torsion-free \(\Gamma^\lambda_{\mu\nu} = \Gamma^\lambda_{\nu\mu}\) (symmetric in lower indices) and metric compatibility \(\nabla_\lambda g_{\mu\nu} = 0\)): \[ \Gamma^\lambda_{\mu\nu} = \frac12 g^{\lambda \rho} (\partial_\mu g_{\nu\rho} + \partial_\nu g_{\mu\rho} - \partial_\rho g_{\mu\nu}) \tag{123} \] (123)

Riemann tensor

Ricci tensor

Ricci scalar

Einstein tensor (common in physics)

\[ G_{\mu\nu} = R_{\mu\nu} - \frac12 g_{\mu\nu} R \]

Variational manifold calculus (?)

Variation of Christoffel symbols

Variation with respect to metric (inspiration) \[ \begin{align} \delta \Gamma^\lambda_{\mu\nu} &= \frac12 \delta g^{\lambda\rho} \overbrace{(\partial_\mu g_{\nu\rho} + \partial_\nu g_{\mu\rho} - \partial_\rho g_{\mu\nu})}^{2 g_{\sigma\rho} \Gamma^\sigma_{\mu\nu}} \\ &+ \frac12 g^{\lambda\rho} (\partial_\mu \delta g_{\nu\rho} + \partial_\nu \delta g_{\mu\rho} - \partial_\rho \delta g_{\mu\nu}) & (\text{product rule}) \\ &= \frac12 g^{\lambda\rho} (\partial_\mu \delta g_{\nu\rho} + \partial_\nu \delta g_{\mu\rho} - \partial_\rho \delta g_{\mu\nu} - 2 \Gamma^\sigma_{\mu\nu} \delta g_{\rho\sigma}) & (\delta g^{\lambda\rho} = -g^{\alpha\lambda} g^{\beta\rho} \delta g_{\alpha\beta}) \\ &= \frac12 g^{\lambda\rho} (\partial_\mu \delta g_{\nu\rho} - \Gamma^\sigma_{\nu\mu} \delta g_{\sigma\rho} - \textcolor{red}{\Gamma^\sigma_{\rho\mu} \delta g_{\nu\sigma}} +{} \\ &\phantom{{}={}} \phantom{\frac12 g^{\lambda\rho} (} \partial_\nu \delta g_{\mu\rho} - \Gamma^\sigma_{\mu\nu} \delta g_{\sigma\rho} - \textcolor{blue}{\Gamma^\sigma_{\rho\nu} \delta g_{\mu\sigma}} -{} \\ &\phantom{{}={}} \phantom{\frac12 g^{\lambda\rho} (} \partial_\rho \delta g_{\mu\nu} + \textcolor{red}{\Gamma^\sigma_{\mu\rho} \delta g_{\sigma\nu}} + \textcolor{blue}{\Gamma^\sigma_{\nu\rho} \delta g_{\mu\sigma}}) & (\text{add $0 = \textcolor{red}{A} - \textcolor{red}{A} + \textcolor{blue}{B} - \textcolor{blue}{B}$}) \\ &= \frac12 g^{\lambda\rho} (\nabla_\mu g_{\nu\rho} + \nabla_\nu g_{\mu\rho} - \nabla_\rho g_{\mu\nu}) \end{align} \]

Variation of Riemann tensor

Variation of Ricci tensor

Variation of Ricci scalar (Palatini identity)

Principle of least action

\[ \delta S = 0 \]

Euler-Lagrange equation

The variaction of a general action \(S[\phi, \nabla_\mu \phi] = \int d^4 x \sqrt{-g(x)} \, \mathcal{L}(\phi, \nabla_\mu \phi)\) with a scalar field \(\phi(x)\) is \[ \begin{align} \delta S &= \int d^4 x \sqrt{-g} \bigg[ \frac{\delta \mathcal{L}}{\delta \phi} \delta\phi + \frac{\delta \mathcal{L}}{\delta (\nabla_\mu \phi)} \delta (\nabla_\mu \phi) \bigg] & (\text{chain rule}) \\ &= \int d^4 x \sqrt{-g} \bigg[ \frac{\delta \mathcal{L}}{\delta \phi} \delta\phi + \frac{\delta \mathcal{L}}{\delta (\nabla_\mu \phi)} \nabla_\mu (\delta \phi) \bigg] & (\delta \nabla \phi = \nabla \delta \phi) \\ &= \int d^4 x \sqrt{-g} \bigg[ \frac{\delta \mathcal{L}}{\delta \phi} - \nabla_\mu \bigg( \frac{\delta \mathcal{L}}{\delta (\nabla_\mu \phi)} \bigg) \bigg] \delta\phi & (\text{by parts, no variation on boundary}) \\ &= \int d^4 x \sqrt{-g} \bigg[ \frac{\partial \mathcal{L}}{\partial \phi} - \nabla_\mu \bigg( \frac{\partial \mathcal{L}}{\partial (\nabla_\mu \phi)} \bigg) \bigg] \delta\phi & (\delta/\delta \phi = \partial/\partial \phi) , \\ \end{align} \] so for the principle of least action to hold for any field variation \(\delta\phi\), the field must satisfy the Euler-Lagrange equation \[ \frac{\partial \mathcal{L}}{\partial \phi} - \nabla_\mu \bigg( \frac{\partial \mathcal{L}}{\partial (\nabla_\mu \phi)} \bigg) = 0 . \]

For even more details, see correct Euler-Lagrange equation in curved spacetime.

Jordan-Brans-Dicke gravity

The Jordan-Brans-Dicke theory of gravity is a scalar-tensor modified gravity theory. Intuition: Newton’s constant replaced by a time-varying field, allowing it to vary from time to time (and place to place?).

Action

The total action of Brans-Dicke theory coupled to matter is \[ S = S_{BD} + S_M = \frac{1}{16\pi} \int d^4 x \sqrt{-g} \, \Big(\phi R - \frac{\omega}{\phi} g^{\mu\nu} \nabla_\mu \phi \, \nabla_\nu \phi\Big) + \int d^4x \sqrt{-g} \, \mathcal{L}_M , \] where \(g_{\mu\nu}(x)\) is the metric tensor, \(g(x) < 0\) is its determinant, \(\phi(x)\) is a scalar field, \(\omega\) is a constant parameter and \(\nabla_\mu \phi = \partial_\mu \phi\) coincides for the scalar field.

Equations of motion

(inspiration)

The (classical) equations of motion for the metric and scalar field follows from the principle of least action.

First, the variation of the Brans-Dicke action (variaction) with respect to the metric is \[ \begin{align} \delta S_{BD} &= \int d^4 x \frac{\delta S_{BD}}{\delta g^{\mu\nu}} \delta g^{\mu\nu} \\ &= \frac{1}{16\pi} \int d^4 x \Big[\frac{\delta\sqrt{-g}}{\delta g^{\mu\nu}} \Big(\phi R - \frac{\omega}{\phi} g^{\mu\nu} \partial_\mu \phi \, \partial_\nu \phi\Big) + \sqrt{-g} \Big(\phi \frac{\delta R}{\delta g^{\mu\nu}} - \frac{\omega}{\phi} \frac{\delta g^{\alpha\beta}}{\delta g^{\mu\nu}} \partial_\alpha \phi \, \partial_\beta \phi\Big)\Big] \delta g^{\mu\nu} \\ &= \frac{1}{16\pi} \int d^4 x \sqrt{-g} \Big[ \phi \Big(\frac{\delta R}{\delta g^{\mu\nu}} - \frac{R}{2} g_{\mu\nu} \delta g^{\mu\nu}\Big) + \frac{\omega}{\phi} \Big(\frac12 g_{\alpha\beta} \delta g^{\alpha\beta} g^{\mu\nu} - \delta g^{\mu\nu}\Big) \partial_\mu \phi \, \partial_\nu \phi \Big] \\ &= \frac{1}{16\pi} \int d^4 x \sqrt{-g} \Big[ \phi \Big(R_{\mu\nu} - \frac{R}{2} g_{\mu\nu} - \nabla_\mu \nabla_\nu + g_{\mu\nu} \nabla^2 \Big) + \frac{\omega}{\phi} \Big(\frac12 g_{\mu\nu} (\partial \phi)^2 - \partial_\mu \phi \, \partial_\nu \phi \Big) \Big] \delta g^{\mu\nu} \\ &= \frac{1}{16\pi} \int d^4 x \sqrt{-g} \, \delta g^{\mu\nu} \Big[ \phi \, G_{\mu\nu} - \nabla_\mu \nabla_\nu \phi + g_{\mu\nu} \nabla^2 \phi + \frac{\omega}{\phi} \Big(\frac12 g_{\mu\nu} (\partial \phi)^2 - \partial_\mu \phi \, \partial_\nu \phi \Big) \Big] . \end{align} \]

Second, the energy-momentum tensor is \(T_{\mu\nu} = -(\sqrt{-g}/2) \, (\delta S_M / \delta g^{\mu\nu})\), so the matter variaction is \[ \delta S_M = \int d^4 x \frac{\delta S_M}{\delta g^{\mu\nu}} \delta g^{\mu\nu} = - \frac12 \int d^4 x \sqrt{-g} \, T_{\mu\nu}. \]

Thus, the full variaction with respect to the metric is \[ \delta S = \frac{1}{16\pi} \int d^4 x \sqrt{-g} \, \delta g^{\mu\nu} \Big[ \phi \, G_{\mu\nu} - \nabla_\mu \nabla_\nu \phi + g_{\mu\nu} \nabla^2 \phi + \frac{\omega}{\phi} \Big(\frac12 g_{\mu\nu} (\partial \phi)^2 - \partial_\mu \phi \, \partial_\nu \phi \Big) - 8 \pi \, T_{\mu\nu} \Big]. \]

For this to hold for any variation \(\delta g^{\mu\nu}\) of the metric, it must satisfy the metric field equation \[ G_{\mu\nu} = \frac{8 \pi}{\phi} T_{\mu\nu} + \frac{\omega}{\phi^2} \Big(\partial_\mu \phi \, \partial_\nu \phi - \frac12 g_{\mu\nu} (\partial \phi)^2 \Big) + \frac{1}{\phi} \Big( \nabla_\mu \nabla_\nu \phi - g_{\mu\nu} \nabla^2 \phi \Big). \]

The equation of motion for the scalar field \(\phi\) is easier to get from the Euler-Lagrange equation with Lagrangian density \(\mathcal{L} = \mathcal{L}_{BD} = \phi R - (\omega / \phi) g^{\mu\nu} \nabla_\mu \phi \, \nabla_\nu \phi\) (the matter Lagrangian \(\mathcal{L}_M\) is independent of \(\phi\) and does not contribute): \[ \begin{align} 0 &= R + \frac{\omega}{\phi^2} g^{\mu\nu} \nabla_\mu \phi \, \nabla_\nu \phi - \nabla_\mu \Big(\frac{\omega}{\phi} g^{\mu\nu} \nabla_\nu \phi\Big) \\ &= R + \frac{\omega}{\phi^2} (\nabla \phi)^2 - \Big(-\frac{\omega}{\phi^2} \nabla_\mu \phi \, \nabla^\mu \phi + \frac{\omega}{\phi} \nabla^2 \phi \Big) \\ &= R + 2 \frac{\omega}{\phi^2} (\nabla \phi)^2 - \frac{\omega}{\phi} \nabla^2 \phi . \end{align} \] We can eliminate \(R\) by using that \(G^\mu_{\phantom{\mu}\mu} = R^\mu_{\phantom{\mu}\mu} - R \, g^\mu_{\phantom{\mu}\mu}/2 = R - 2R = -R\), so the trace of the metric field equations is \[ -R = \frac{8\pi}{\phi} T - \frac{\omega}{\phi^2} (\nabla \phi)^2 - \frac{3}{\phi} \nabla^2 \phi . \]

Eliminating \(R\) and using that \((\nabla \phi)^2 = -\phi \, \nabla^2 \phi\) by parts under the action integral with no variation at the boundary, we find the scalar field equation \[ \nabla^2 \phi = \frac{8\pi}{3+2\omega} T . \]

Background evolution (0th order perturbation theory solution)

At the background level, the universe is assumed to be homogeneous and isotropic, as described by the Friedmann-Lemaitre-Robertson-Walker metric \[ ds^2 = -dt^2 + a^2(t) \bigg[ \frac{dr^2}{1-k r^2} + r^2 ( d\theta^2 + \sin^2\theta \, d\phi^2 ) \bigg], \] and filled with a perfect fluid with energy density \(\rho\), pressure \(P\) and rest-frame energy-momentum \[ T^\mu_{\phantom{\mu}\nu} = \mathrm{diag} [-\rho, P, P, P] . \]

The \(\vphantom{G}_{00}\)-component of the metric field equations then gives \[ 3H^2 + 3 \frac{k}{a^2} = \frac{8 \pi}{\phi} \rho + \frac{\omega}{2} \frac{\dot{\phi}^2}{\phi^2} - 3H \frac{\dot\phi}{\phi} \] Reexpressing \(\dot{\phi} = H \phi \, d \log \phi / d \log a\), we can collect all terms with \(H\) on one side to find \[ H^2 \Bigg[ 1 + \frac{d \log \phi}{d \log a} - \frac{\omega}{6} \bigg( \frac{d \log \phi}{d \log a}\bigg)^2 \Bigg] = \frac{8 \pi}{3 \phi} \rho - \frac{k}{a^2} . \] Defining the usual (effective) density parameters \(\Omega_{r0} = \rho_{r0} / (3 H_0^2 / 8 \pi)\), \(\Omega_{m0} = \rho_{m0} / (3 H_0^2 / 8 \pi)\), \(\Omega_{\Lambda 0} = \rho_{\Lambda 0} / (3 H_0^2 / 8 \pi)\) and \(\Omega_{k0} = -k/H_0^2\), we find the Friedmann equation \[ E^2(a) = \bigg(\frac{H(a)}{H_0(a)}\bigg)^2 = \frac{1}{\phi} \cdot \frac{\Omega_{r0} a^{-4} + \Omega_{m0} a^{-3} + \phi \Omega_{k0} a^{-2} + \Omega_{\Lambda 0}}{1 + \frac{d \log \phi}{d \log a} - \frac{\omega}{6} \big( \frac{d \log \phi}{d \log a}\big)^2} . \] If we define the “phinalized” parameters \(\tilde \Omega_{i0} = \Omega_{i0} / \phi_0\), except for curvature \(\tilde \Omega_{k0} = \Omega_{k0}\), and introduce \[ \tilde{\Omega}_{\phi0} = -\frac{d \log \phi}{d \log a} + \frac{\omega}{6} \bigg(\frac{d \log \phi}{d \log a}\bigg)^2 \, \Bigg|_{a=1} \] we find that \(E^2(a=1)=1\) constrains the parameters by the modified closure condition \[ 1 = \tilde{\Omega}_{r0} + \tilde{\Omega}_{m0} + \tilde{\Omega}_{k0} + \tilde{\Omega}_{\Lambda 0} + \tilde{\Omega}_{\phi 0} . \]

Weak-field limit

Trace-reversed form of the metric field equation: $$

$$