• Christoffel Symbols \begin{equation} \Gamma^{\lambda}_{\,\mu\nu} = \frac{1}{2} g^{\lambda \rho}\left(\partial_{\mu} g_{\rho \nu} + \partial_{\nu} g_{\mu \rho} - \partial_{\rho} g_{\mu \nu}\right) \end{equation}
• Riemann Tensor \begin{equation} R^{\lambda}{}_{\mu\sigma\nu} = \partial_{\sigma} \Gamma^{\lambda}_{\, \mu\nu} - \partial_{\nu} \Gamma^{\lambda}_{\, \mu\sigma} + \Gamma^{\kappa}_{\,\mu\nu} \Gamma^{\lambda}_{\, \kappa \sigma} - \Gamma^{\kappa}_{\,\mu\sigma} \Gamma^{\lambda}_{\, \kappa \nu} \end{equation}
• Ricci Tensor \begin{equation} R_{\mu\nu} = \delta^{\sigma}{}_{\lambda} R^{\lambda}{}_{\mu \sigma \nu} \end{equation}
• Schouten Tensor \begin{equation} S_{\mu\nu} = \frac{1}{d-1} \, \left(R_{\mu\nu} - \frac{1}{2\,d}\,g_{\mu\nu} R \right) \end{equation} \begin{equation} \nabla^{\nu} S_{\mu\nu} = \nabla_{\mu} S^{\nu}_{\dub \nu} \end{equation}
• Weyl Tensor \begin{equation} C^{\lambda}{}_{\mu \sigma \nu} = R^{\lambda}{}_{\mu \sigma \nu} + g^{\lambda}{}_{\nu} \, S_{\mu \sigma} - g^{\lambda}{}_{\sigma} \, S_{\mu\nu} + g_{\mu\sigma} \, S^{\lambda}{}_{\nu} - g_{\mu\nu} \, S^{\lambda}{}_{\sigma} \end{equation}
• Commutators of Covariant Derivatives \begin{equation} \left[ \nabla_{\mu}, \nabla_{\nu} \right] A_{\lambda} = R_{\lambda \sigma \mu \nu} A^{\sigma} \end{equation} \begin{equation} \left[ \nabla_{\mu}, \nabla_{\nu} \right] A^{\lambda} = R^{\lambda}{}_{\sigma \mu \nu} A^{\sigma} \end{equation}
• Bianchi Identities \begin{equation} \nabla_{\kappa} R_{\lambda \mu \sigma \nu} - \nabla_{\lambda} R_{\kappa \mu \sigma \nu} + \nabla_{\mu} R_{\kappa \lambda \sigma \nu} = 0 \end{equation} \begin{equation} \nabla^{\nu} R_{\lambda \mu \sigma \nu} = \nabla_{\mu} R_{\lambda \sigma} - \nabla_{\lambda} R_{\mu\sigma} \end{equation} \begin{equation} \nabla^{\nu} R_{\mu\nu} = \frac{1}{2} \, \nabla_{\mu} R \end{equation}
• Bianchi Identities for the Weyl Tensor \begin{align} \nabla^{\nu} C_{\lambda \mu \sigma \nu} = & \, (d-2) \, \left( \nabla_{\mu} S_{\lambda \sigma} - \nabla_{\lambda} S_{\mu \sigma} \right) \end{align} \begin{align} \nabla^{\lambda} \nabla^{\sigma} C_{\lambda \mu \sigma \nu} = \frac{d-2}{d-1} & \,\left[ \, \nabla^{2} R_{\mu\nu} - \frac{1}{2d} \, g_{\mu\nu} \nabla^{2} R - \frac{d-1}{2d} \, \nabla_{\mu} \nabla_{\nu} R \right. \\ \nonumber & \dub - \left. \left(\frac{d+1}{d-1}\right) \, R_{\mu}{}^{\lambda} R_{\nu \lambda} + C_{\lambda \mu \sigma \nu} R^{\lambda \sigma} + \frac{(d+1)}{d(d-1)} \, R \, R_{\mu\nu} \right. \\ \nonumber & \dub \left. + \frac{1}{d-1} \, g_{\mu\nu} \left(R^{\lambda\sigma} R_{\lambda \sigma} - \frac{1}{d} \, R^2 \right) \, \right] \end{align}

• p-Form Components \begin{align} {\bf A}_{(p)} = \frac{1}{p!}\,A_{\mu_1 \ldots \mu_p}\,dx^{\mu_1} \land \ldots \land dx^{\mu_p} \end{align}
• Exterior Derivative \begin{align} \Big(d{\bf A}_{(p)}\Big)_{\mu_1 \ldots \mu_{p+1}} = (p+1)\,\partial_{\,[\,\mu_1} A_{\mu_2 \ldots \mu_{p+1}\,]\,} \end{align} \begin{align} B_{\,[\, \mu_1 \ldots \mu_n ]\,} \defeq \frac{1}{n!}\,\Big(\,B_{\mu_1 \ldots \mu_n} + \textrm{permutations}\,\Big) \end{align}
• Hodge-Star \begin{align} \Big( \star {\bf A}_{(p)}\Big)_{\mu_1 \ldots \mu_{d+1-p}} = \frac{1}{p!}\,\eps_{\mu_1 \ldots \mu_{d+1-p}}{}^{\nu_1 \ldots \nu_p}\,A_{\nu_1 \ldots \nu_p} \end{align} \begin{align} \star \,\star = \big(-1 \,\big)^{p\,(d+1-p)+1} \end{align}
• Wedge Product \begin{align} \Big( {\bf A}_{(p)} \land {\bf B}_{(q)} \Big)_{\mu_1 \ldots \mu_{p+q}} = \frac{(p+q)!}{p!\,q!}\,A_{\,[\,\mu_1 \ldots \mu_p} \, B_{\mu_{p+1} \ldots \mu_{p+q} \,]} \end{align}

• Curvature Two-Form \begin{equation} {\bf R}^{a}_{\dub b} = \frac{1}{2} \, R^{a}_{\dub b \, c\,d} \,{\bf e}^c \land {\bf e}^d \end{equation}
• Euler Density \begin{equation} {\bf e}_{_{2n}} = \frac{1}{(4\pi)^n \, \Gamma(n+1)}\,\eps_{a_{_1} \ldots a_{_{2n}}} {\bf R}^{a_{_1} a_{_2}}\land \ldots \land{\bf R}^{a_{_{2n-1}} a_{_{2n}}} \end{equation} \begin{equation*} \EE_{2n} = \frac{1}{(8\pi)^n\,\Gamma(n+1)} \, \eps_{\mu_{_1} \ldots \mu_{_{2n}}} \, \eps_{\nu_{_1} \ldots \nu_{_{2n}}} \, R^{\mu_{_1}\mu_{_2}\nu_{_1}\nu_{_2}} \ldots R^{\mu_{_{2n-1}}\mu_{_{2n}}\nu_{_{2n-1}}\nu_{_{2n}}} \end{equation*}
• Euler Number \begin{align} \chi(\MM) & \, = \int_{\MM} \nts d^{2n}x \,\sqrt{g} \, \EE_{2n} \\ \nonumber & \, = \int_{\MM} \nts {\bf e}_{_{2n}} \end{align}
• Examples \begin{align} \EE_{2} & \, = \frac{1}{8\pi} \, \eps_{\mu\nu} \eps_{\lambda\rho} \, R^{\mu\nu\lambda\rho} \\ \nonumber & \, = \frac{1}{4\pi}\,R \\ \EE_{4} & \, = \frac{1}{128 \pi^2} \, \eps_{\mu\nu\lambda\rho} \, \eps_{\alpha\beta\gamma\delta} \, R^{\mu\nu\alpha\beta} \, R^{\lambda\rho\gamma\delta} \\ \nonumber & \, = \frac{1}{32 \pi^2} \, \left(R^{\mu\nu\lambda\rho} R_{\mu\nu\lambda\rho} - 4 \, R^{\mu\nu} R_{\mu\nu} + R^2 \right) \\ \nonumber & \, = \frac{1}{32 \pi^2} \, C^{\mu\nu\lambda\rho} C_{\mu\nu\lambda\rho} - \frac{1}{8\pi^2} \, \left(\frac{d-2}{d-1}\right)\,\left(R^{\mu\nu} R_{\mu\nu} - \frac{d+1}{4\,d} \, R^2 \right) \end{align}

• First Fundamental Form / Induced Metric on \(\Sigma\) \begin{equation} h_{\mu\nu} = g_{\mu\nu} - e\,n_{\mu} n_{\nu} \end{equation}
• Projection of a tensor along \(\Sigma\) \begin{equation} \bot \,T^{\mu \,\ldots}{}_{\nu \, \ldots} = h^{\mu}{}_{\lambda} \ldots h^{\sigma}{}_{\nu} \ldots T^{\lambda \, \ldots}{}_{\sigma \, \ldots} \end{equation}
• Covariant Derivative on \(\Sigma\) compatible with \(h_{\mu\nu}\) \begin{equation} \DD_{\mu} T^{\alpha \, \ldots}{}_{\beta \, \ldots} = \bot \, \nabla_{\mu} T^{\alpha \, \ldots}{}_{\beta \, \ldots} \quad \forall \quad T = \bot\,T \end{equation}
• Intrinsic Curvature of \((\Sigma,h)\) \begin{equation} \left[ \DD_{\mu}, \DD_{\nu} \right] A^{\lambda} = \RR^{\lambda}{}_{\sigma \mu \nu} A^{\sigma} \quad \forall \quad A^{\lambda} = \bot \, A^{\lambda} %= h^{\lambda}{}_{\sigma} \, A^{\sigma} \end{equation}
• “Acceleration” Vector \begin{equation} a^{\mu} = n^{\nu} \nabla_{\nu} n^{\mu} \end{equation}
• Second Fundamental Form / Extrinsic Curvature of \(\Sigma\) \begin{align} K_{\mu\nu} & \,\, = \frac{1}{2}\, \bot \, (\nabla_{\mu} n_{\nu} + \nabla_{\nu} n_{\mu} ) = \frac{1}{2}\,h_{\mu}{}^{\lambda} h_{\nu}{}^{\sigma}\,( \nabla_{\lambda} n_{\sigma} + \nabla_{\sigma} n_{\lambda} ) \\ &\,\, = \frac{1}{2} \, \pounds_{n} h_{\mu\nu} \\ &\,\, = \frac{1}{2}\,\big(\nabla_{\mu} n_{\nu} + \nabla_{\nu} n_{\mu} - e\,n_{\mu} a_{\nu} - e\,n_{\nu} a_{\mu} \big) \end{align}
• Trace of Extrinsic Curvature \begin{equation} K = \nabla_{\mu} n^{\mu} \end{equation}
• Given a local description of the surface by the condition \(f(x) = 0\), for some sufficiently differentiable function \(f\), the unit vector can be defined as a normalized gradient \begin{equation}\label{SFNV} n_{\mu} = \sqrt{\frac{e\vphantom{\big|}}{g^{\alpha\beta}\partial_{\alpha}f \partial_{\beta}f}}\,\partial_{\mu}f \end{equation}
• Useful results for surface-forming \(n_{\mu}\): \begin{gather} \bot (\nabla_{\mu} n_{\nu} - \nabla_{\nu} n_{\mu} ) = 0 \\ \bot (\nabla_{\mu} a_{\nu} - \nabla_{\nu} a_{\mu} ) = \DD_{\mu} a_{\nu} - \DD_{\nu} a_{\mu} = 0 \vphantom{\Big|} \\ \nabla_{\mu} n_{\nu} = K_{\mu\nu} + e\,n_{\mu} a_{\nu} \end{gather}
• Gauss-Codazzi \begin{align} \bot R_{\lambda \mu \sigma \nu} \, = & \dub \RR_{\lambda \mu \sigma \nu} + e \, (K_{\mu \sigma} K_{\nu \lambda} - K_{\lambda \sigma} K_{\mu \nu} ) \\ \bot \left( R_{\lambda \mu \sigma \nu} \,n^{\lambda} \right) \, = & \dub \DD_{\nu} K_{\mu\sigma} - \DD_{\sigma} K_{\mu\nu} \vphantom{\Big|} \\ \bot \left(R_{\lambda \mu \sigma \nu} \,n^{\lambda} n^{\sigma}\right) \, = & \, - \pounds_{n} \, K_{\mu\nu} + K_{\mu}{}^{\lambda} \, K_{\lambda \nu} + \DD_{\mu} a_{\nu} - e\,a_{\mu} a_{\nu} \end{align}
• Projections of the Ricci tensor \begin{align} \bot \left( R_{\mu\nu} \right) \, = & \,\, \RR_{\mu\nu} + e\,\big(\DD_{\mu} a_{\nu} - e\,a_{\mu} a_{\nu}\big) - e\,\pounds_{n} \, K_{\mu\nu} - e\,K \, K_{\mu\nu} \\ \nonumber & \quad + 2 e\,K_{\mu}{}^{\lambda} \, K_{\nu \, \lambda} \\ \bot \left( R_{\mu\nu} \,n^{\mu} \right) \, = & \,\, \DD^{\mu} K_{\mu\nu} - \DD_{\nu} K \vphantom{\Big|} \\ R_{\mu\nu} n^{\mu} n^{\nu} \, = & - \pounds_{n} \, K - K^{\mu\nu} \, K_{\mu\nu} + \DD_{\mu} a^{\mu} -e\,a_{\mu} a^{\mu} \end{align}
• Decomposition of the Ricci scalar \begin{align} R \, = & \dub \RR - e\,K^2 - e\,K^{\mu\nu}\,K_{\mu\nu} - 2e\,\pounds_{n} \, K + 2 e\,\big( \DD_{\mu} a^{\mu} - e \, a_{\mu} a^{\mu}\big) \end{align}
• Lie Derivatives along \(n^{\mu}\) for any \(\FF = \bot \FF\) \begin{gather} \bot \left( \pounds_{n} \, \FF^{\mu \,\ldots}{}_{\nu \, \ldots}\right) \, = \, \pounds_{n} \, \FF^{\mu \,\ldots}{}_{\nu \, \ldots} \\ h_{\mu}{}^{\nu}\,\pounds_{n}\,\FF_{\nu \ldots} \, = \, \pounds_{n}\,\FF_{\mu \ldots} \vphantom{\Big|} \\ h_{\mu}{}^{\nu}\,\pounds_{n}\,\FF^{\mu \ldots} \, = \, \pounds_{n}\,\FF^{\nu \ldots} \\ \pounds_{n} K_{\mu\nu} \, = \, n^{\lambda} \nabla_{\lambda} K_{\mu\nu} + K_{\lambda \nu} \nabla_{\mu} n^{\lambda} + K_{\mu \lambda} \nabla_{\nu} n^{\lambda} \vphantom{\Big|} \\ h^{\mu\nu}\,\pounds_{n} K_{\mu\nu} \, = \, \pounds_{n} K + 2\,K^{\mu\nu} K_{\mu\nu} \end{gather}

• Gravitational Action \begin{align} I_G = & \dub \frac{1}{2\,\kappa^2} \int_{\MM} \nts d^{d+1}x \sqrt{g} \left( R - 2 \,\Lambda \right) + \frac{1}{\kappa^2} \int_{\dM} \nts d^{d} x \sqrt{h} \, K \\ = & \dub \frac{1}{2\,\kappa^2} \int_{\MM} \nts d^{d+1}x \sqrt{g} \left( \RR + K^2 - K^{\mu\nu}\, K_{\mu\nu} - 2 \,\Lambda \right) \end{align}
• Gauge Field Coupled to Particle \begin{align} I_{Maxwell} + I_{Matter} = & \dub -\frac{1}{4} \int_{\MM} \nts d^{d+1}x \sqrt{g} \,F^{\mu\nu} F_{\mu\nu} \\ & \dub - \sum_{n} m_{n} \int dp \, \sqrt{ -g_{\mu\nu}(x_{n}(p))\,\frac{dx_{n}{}^{\mu}(p)}{dp} \frac{dx_{n}{}^{\nu}(p)}{dp} } \\ & \dub + \sum_{n} e_{n} \int dp \, \frac{dx_{n}{}^{\mu}(p)}{dp}\,A_{\mu}(x_{n}(p)) \end{align}
• Gravity Minimally Coupled to a Gauge Field \begin{align} I_G + I_{Maxwell} = & \dub \int_{\MM} \nts d^{d+1}x \sqrt{g} \left[ \frac{1}{2\,\kappa^2} \left( R - 2 \,\Lambda \right) - \frac{1}{4} F^{\mu\nu} F_{\mu\nu} \right] + \frac{1}{\kappa^2} \int_{\dM} \nts d^{d} x \sqrt{h} \, K \end{align}

• Bulk Lagrangian Density \begin{gather} \Lag_{\MM} = \frac{1}{2\,\kappa^2} \left( K^2 - K^{\mu\nu} \, K_{\mu\nu} + \RR - 2\,\Lambda \right) \end{gather}
• Momentum Conjugate to \(h_{\mu\nu}\) \begin{equation} \pi^{\mu\nu} = \frac{\partial \Lag_{\MM}}{\partial \left( \pounds_{n} h_{\mu\nu} \right)} = \frac{1}{2\,\kappa^2} \left(h^{\mu\nu}\,K - K^{\mu\nu} \right) \end{equation} \begin{equation} \pi = \pi^{\mu}{}_{\mu} = \frac{d-1}{2\,\kappa^2} K \end{equation}
• Momentum Constraint \begin{gather} \HH_{\mu} = \frac{1}{\kappa^2} \bot \left(n^{\nu} G_{\mu\nu} \right) = 2\,\DD^{\nu} \pi_{\mu\nu} = 0 \end{gather}
• Hamiltonian Constraint \begin{gather} \HH = - \frac{1}{\kappa^2} n^{\mu} n^{\nu} G_{\mu\nu} = 2 \, \kappa^2 \left(\pi^{\mu\nu} \, \pi_{\mu\nu} - \frac{1}{d-1} \, \pi^2 \right) + \frac{1}{2\,\kappa^2} \left( \RR - 2 \, \Lambda \right) = 0 \end{gather}

• Metric \begin{gather} \hg_{\mu\nu} = e^{\,2\,\sigma} \, g_{\mu\nu} \end{gather}
• Christoffel \begin{gather} \hat{\Gamma}^{\,\lambda}_{\,\mu\nu} = \Gamma^{\lambda}_{\,\mu\nu} + \Theta^{\lambda}_{\,\mu\nu} \\ \Theta^{\lambda}_{\,\mu\nu} = \delta^{\lambda}{}_{\mu} \nabla_{\nu} \sigma + \delta^{\lambda}{}_{\nu} \nabla_{\mu} \sigma - g_{\mu\nu} \nabla^{\lambda} \sigma \end{gather}
• Riemann Tensor \begin{align} \hat{R}^{\,\lambda}{}_{\mu\rho\nu} = & \dub R^{\lambda}{}_{\mu\rho\nu} + \delta^{\lambda}{}_{\nu} \, \nabla_{\mu} \nabla_{\rho} \sigma - \delta^{\lambda}{}_{\rho} \, \nabla_{\mu} \nabla_{\nu} \sigma + g_{\mu\rho} \, \nabla_{\nu} \nabla^{\lambda} \sigma - g_{\mu\nu} \, \nabla_{\rho} \nabla^{\lambda} \sigma \\ & \dub + \delta^{\lambda}{}_{\rho} \, \nabla_{\mu} \sigma \, \nabla_{\nu} \sigma - \delta^{\lambda}{}_{\nu} \, \nabla_{\mu} \sigma \, \nabla_{\rho} \sigma + g_{\mu\nu}\,\nabla_{\rho} \sigma \, \nabla^{\lambda} \sigma - g_{\mu\rho}\,\nabla_{\nu} \sigma \, \nabla^{\lambda} \sigma \\ & \dub + \left( g_{\mu\rho} \, \delta^{\lambda}{}_{\nu} - g_{\mu\nu} \, \delta^{\lambda}{}_{\rho} \right) \nabla^{\alpha} \sigma \, \nabla_{\alpha} \sigma \end{align}
• Ricci Tensor \begin{align} \hat{R}_{\mu\nu} = & \dub R_{\mu\nu} - g_{\mu\nu} \, \nabla^{2} \sigma - (d-1)\,\nabla_{\mu} \nabla_{\nu} \sigma + (d-1) \, \nabla_{\mu} \sigma \, \nabla_{\nu} \sigma \\ & \dub - (d-1) \, g_{\mu\nu} \nabla^{\lambda} \sigma \, \nabla_{\lambda} \sigma \end{align}
• Ricci Scalar \begin{align} \hat{R} = e^{-2\,\sigma} \left( R -2\,d\,\nabla^{2} \sigma - d\,(d-1)\,\nabla^{\mu} \sigma \, \nabla_{\mu} \sigma \right) \end{align}
• Schouten Tensor \begin{align} \hat{S}_{\mu\nu} = & \dub S_{\mu\nu} - \nabla_{\mu} \nabla_{\nu} \sigma + \nabla_{\mu} \sigma \, \nabla_{\nu} \sigma - \frac{1}{2} \, g_{\mu\nu} \nabla^{\lambda} \sigma \, \nabla_{\lambda} \sigma \end{align}
• Weyl Tensor \begin{gather} \hat{C}^{\,\lambda}{}_{\mu \rho \nu} = C^{\lambda}_{\dub \mu \rho \nu} \end{gather}
• Normal Vector \begin{gather} \widehat{n}^{\mu} = e^{-\sigma} \, n^{\mu} \quad \quad \widehat{n}_{\mu} = e^{\,\sigma} \, n_{\mu} \end{gather}
• Extrinsic Curvature \begin{gather} \hat{K}_{\mu\nu} = e^{\,\sigma} \left( K_{\mu\nu} + h_{\mu\nu} \, n^{\lambda} \nabla_{\lambda} \sigma \right) \\ \hat{K} = e^{-\sigma} \left( K + d \, n^{\lambda} \nabla_{\lambda} \sigma \right) \end{gather}

• Inverse Metric \begin{equation} g^{\mu\nu} \to g^{\mu\nu} - g^{\mu\alpha}\,g^{\nu\beta} \,\dg_{\alpha\beta} + g^{\mu\alpha}\,g^{\nu\beta}\,g^{\lambda\rho}\,\dg_{\alpha\lambda}\,\dg_{\beta\rho} + \ldots \end{equation}
• Square Root of Determinant of Metric \begin{equation} \sqrt{g} \to \sqrt{g} \, \left( 1 + \frac{1}{2}\,g^{\mu\nu} \dg_{\mu\nu} + \ldots \right) \end{equation}
• Variational Operator \begin{gather} \delta (g_{\mu\nu}) = \dg_{\mu\nu} \quad \quad \delta^{\,2} (g_{\mu\nu}) = \delta (\dg_{\mu\nu}) = 0 \\ \delta(g^{\mu\nu}) = - g^{\mu\alpha}\,g^{\nu\beta} \,\dg_{\alpha\beta} \\ \delta^{\,2} (g^{\mu\nu}) = \delta \left( -g^{\mu\lambda} \, g^{\nu\rho} \, \dg_{\lambda \rho}\right) = 2 \, g^{\mu\alpha}\,g^{\nu\beta}\,g^{\lambda\rho}\,\dg_{\alpha\lambda}\,\dg_{\beta\rho} \\ \FF(g+\dg) = \FF(g) + \delta \FF(g) + \frac{1}{2} \delta^{\,2} \FF(g) + \ldots + \frac{1}{n!} \delta^{\,n} \FF(g) + \ldots \end{gather}
• Christoffel \begin{gather} \delta \,\Gamma^{\lambda}{}_{\mu \nu} = \frac{1}{2} g^{\lambda \rho} \left( \nabla_{\mu} \,\dg_{\rho \nu} + \nabla_{\nu} \,\dg_{\mu\rho} - \nabla_{\rho} \,\dg_{\mu\nu} \right) \\ \delta^{\,2} \Gamma^{\lambda}{}_{\mu \nu} = - g^{\lambda \alpha}\,g^{\rho \beta}\,\dg_{\alpha\beta} \left( \nabla_{\mu} \,\dg_{\rho \nu} + \nabla_{\nu} \,\dg_{\mu\rho} - \nabla_{\rho} \,\dg_{\mu\nu} \right) \\ \delta^{\,n} \Gamma^{\lambda}{}_{\mu \nu} = \frac{n}{2} \delta^{\,n-1}\left(g^{\lambda \rho} \right) \, \left( \nabla_{\mu} \,\dg_{\rho \nu} + \nabla_{\nu} \,\dg_{\mu\rho} - \nabla_{\rho} \,\dg_{\mu\nu} \right) \end{gather}
• Riemann Tensor \begin{gather} \delta R^{\lambda}{}_{\mu \sigma \nu} = \nabla_{\sigma} \delta \Gamma^{\lambda}{}_{\mu\nu} - \nabla_{\nu} \delta \Gamma^{\lambda}{}_{\mu\sigma} \end{gather}
• Ricci Tensor \begin{align} \delta R_{\mu\nu} = & \dub \nabla_{\lambda} \delta \Gamma^{\lambda}{}_{\mu\nu} - \nabla_{\nu} \delta \Gamma^{\lambda}{}_{\mu\lambda} \\ = & \dub \frac{1}{2} \left( \nabla^{\lambda} \nabla_{\mu} \dg_{\lambda \nu} + \nabla^{\lambda} \nabla_{\nu} \dg_{\mu \lambda} - g^{\lambda \rho} \nabla_{\mu} \nabla_{\nu} \dg_{\lambda \rho} - \nabla^{2} \dg_{\mu\nu} \right) \end{align}
• Ricci Scalar \begin{align} \delta R = - R^{\mu\nu} \dg_{\mu\nu} + \nabla^{\mu} \left( \nabla^{\nu} \dg_{\mu\nu} - g^{\lambda \rho} \nabla_{\mu} \dg_{\lambda \rho} \right) \end{align}
• Surface Forming Unit Vector \begin{gather} \delta n_{\mu} = \frac{e}{2} n_{\mu} n^{\nu} n^{\lambda} \dg_{\nu\lambda} = \frac{1}{2} \dg_{\mu\nu} n^{\nu} + \frac{1}{2} c_{\mu} \end{gather} \begin{gather} c_{\mu} = e\,n_{\mu} n^{\nu} n^{\lambda} \dg_{\nu\lambda} - \dg_{\mu\nu} n^{\nu} = - h_{\mu}{}^{\lambda} \dg_{\lambda \nu} n^{\nu} \end{gather}
• Acceleration vector \begin{gather} \delta a^{\mu} = -\frac{1}{2}\,\DD^{\mu}\big(n^{\nu} n^{\lambda} \delta g_{\nu\lambda} \big) + c^{\mu} \end{gather}
• Extrinsic Curvatures \begin{align} \delta K_{\mu\nu} = & \dub \frac{e}{2} n^{\alpha} n^{\beta} \dg_{\alpha \beta} K_{\mu\nu} + e\,\dg_{\lambda \rho} n^{\rho} \left(n_{\mu} K^{\lambda}{}_{\nu} + n_{\nu} K_{\mu}{}^{\lambda} \right) \\ \nonumber & \dub - \frac{1}{2} h_{\mu}{}^{\lambda} h_{\nu}{}^{\rho} n^{\alpha} \left(\nabla_{\lambda} \dg_{\alpha \rho} + \nabla_{\rho} \dg_{\lambda \alpha} - \nabla_{\alpha} \dg_{\lambda \rho} \right) \end{align} \begin{gather} \delta \, K = - \frac{1}{2} K^{\mu\nu} \dg_{\mu\nu} - \frac{1}{2} n^{\mu} \left(\nabla^{\nu} \dg_{\mu\nu} - g^{\nu\lambda} \nabla_{\mu} \dg_{\nu \lambda} \right) + \frac{1}{2} \DD_{\mu} c^{\mu} \end{gather}
• Gibbons-Hawking-York Lagrangian ( \(e = \pm 1\) ) \begin{align} \delta \big(e\,\sqrt{h}\, K \big) = & \,\, e\,\sqrt{h} \,\left[ \, \frac{1}{2} \, \big( h^{\mu\nu}\,K - K^{\mu\nu} \big) \dg_{\mu\nu} + \frac{1}{2} \DD_{\mu} c^{\mu} \right. \\ \nonumber & \quad\quad\quad \left. - \frac{1}{2} n^{\mu} \big(\nabla^{\nu} \dg_{\mu\nu} - g^{\nu\lambda} \nabla_{\mu} \dg_{\nu \lambda} \big) \right] \end{align}

We start by identifying a scalar field \(t\) whose isosurfaces \(\Sigma_t\) are normal to the timelike unit vector given by \begin{gather} u_{a} = - \alpha \, \partial_{a} t ~, \end{gather} where the lapse function \(\alpha\) is \begin{gather} \label{Lapse} \alpha \defeq \frac{1}{\sqrt{-h^{ab}\,\partial_a t \, \partial_b t}} ~. \end{gather} An observer whose worldline is tangent to \(u_{a}\) experiences an acceleration given by the vector \begin{gather} a_{b} = u^{c} \cdot \dnab_{c} u_{b} ~, \end{gather} which is orthogonal to \(u_{a}\). The (spatial) metric on the \(d-1\) dimensional surface \(\Sigma_t\) is given by \begin{gather} \sigma_{ab} = h_{ab} + u_a u_b ~. \end{gather} The intrinsic Ricci tensor built from this metric is denoted by \(\RR_{ab}\), and its Ricci scalar is \(\RR\). The covariant derivative on \(\Sigma_t\) is defined in terms of the \(d\) dimensional covariant derivative as \begin{gather} D_{a} V_{b} \defeq \sigma_{a}{}^{c} \sigma_{b}{}^{e} \big(\dnab_{c} V_{e} \big) \quad \text{for any} \quad V_{b} = \sigma_{b}{}^{c} V_{c} ~. \end{gather} The extrinsic curvature of \(\Sigma_t\) embedded in the ambient \(d\) dimensional spacetime is \begin{gather}\label{tExtrinsic} \theta_{ab} \defeq - \sigma_{a}{}^{c} \sigma_{b}{}^{d} \big( \dnab_{c} u_{d} \big) = - \dnab_{a} u_{b} - u_{a} a_{b} = - \frac{1}{2}\,\pounds_{u} \sigma_{ab} ~. \end{gather} This definition has an overall minus sign compared to the extrinsic curvature in section 4, where we considered surfaces normal to a spacelike vector.

Now we consider a ‘time flow’ vector field \(t^{a}\), which satisfies the condition \begin{gather} t^{a} \, \partial_{a} t = 1 ~. \end{gather} The vector \(t^{a}\) can be decomposed into parts normal and along \(\Sigma_t\) as \begin{gather} t^{a} = \alpha \, u^{a} + \beta^{a} ~, \end{gather} where \(\alpha\) is the lapse function \eqref{Lapse} and \(\beta^{a} \defeq \,\sigma^{a}{}_{b}\, t^{b}\) is the shift vector. An important result in the derivations that follow relates the Lie derivative of a scalar or spatial tensor (one that is orthogonal to \(u^{a}\) in all of its indices) along the time flow vector field, to Lie derivatives along \(u^{a}\) and \(\beta^{a}\). Let \(S\) be a scalar. Then \begin{gather} \pounds_{t} S = \pounds_{\alpha \, u} S + \pounds_{\beta} S = \alpha \,\pounds_{u} S + \pounds_{\beta} S ~. \end{gather} Rearranging this expression then gives \begin{gather} \pounds_{u} S = \frac{1}{\alpha}\, \big(\pounds_{t} S - \pounds_{\beta} S \big) ~. \end{gather} Similarly, for a spatial tensor with all lower indices we have \begin{gather} \pounds_{t} W_{a\ldots} = \alpha \,\pounds_{u} W_{a\ldots} + \pounds_{\beta} W_{a\ldots} ~. \end{gather} This is not the case when the tensor has any of its indices raised. In a moment, these identities will allow us to express certain Lie derivatives along \(u^{a}\) in terms of regular time derivatives and Lie derivatives along the shift vector \(\beta^{a}\).

Next, we construct the coordinate system that we will use for the decomposition of the equations of motion. The adapted coordinates \((t,x^i)\) are defined by \begin{gather} \partial_{t} x^{a} \defeq t^{a} ~. \end{gather} The \(x^i\) are coordinates along the \(d-1\) dimensional hypersurface \(\Sigma_t\). If we define \begin{gather} P_{i}{}^{a} \defeq \frac{\partial x^{a}}{\partial x^{i}} ~, \end{gather} then it follows from the definition of the coordinates that \(P_{i}{}^{a} \partial_{a} t = 0\) and we can use \(P_{i}{}^{a}\) to project tensors onto \(\Sigma_t\). For example, in the adapted coordinates the spatial metric, extrinsic curvature, and acceleration and shift vectors are \begin{gather} \sigma_{ij} = P_{i}{}^{a} P_{j}{}^{b} \sigma_{ab} \\ \theta_{ij} = P_{i}{}^{a} P_{j}{}^{b} \theta_{ab} \\ a_{j} = P_{j}{}^{b} a_{b} \\ \beta_{i} = P_{i}{}^{a} \beta_{a} = P_{i}{}^{a} t_{a}~. \end{gather} The line element in the adapted coordinates takes a familiar form: \begin{align} h_{ab} dx^a dx^b = & \,\, h_{ab} \bigg(\frac{\partial x^a}{\partial t}\,dt + \frac{\partial x^a}{\partial x^i}\,dx^i\bigg) \bigg(\frac{\partial x^b}{\partial t}\,dt + \frac{\partial x^b}{\partial x^j}\,dx^j\bigg) \\ = & \,\, h_{ab} \big(t^{a} dt + P_{i}{}^{a} dx^i \big)\big(t^{b} dt + P_{j}{}^{b} dx^j \big) \\ = & \,\, t^{a} t_{a} dt^2 + 2 t_{a} dt P_{i}{}^{a} dx^i + h_{ab} P_{i}{}^{a} P_{j}{}^{b} dx^i dx^j \\ = & \,\, \big( - \alpha^{2} + \beta^i \beta_i \big) dt^2 + 2 \beta_i dt dx^i + \sigma_{ij} dx^i dx^j \\ \label{ADMds} \Rightarrow h_{ab} dx^a dx^b = & \,\, -\alpha^2 dt^2 + \sigma_{ij} \big(dx^i + \beta^i dt \big) \big(dx^j + \beta^j dt \big) ~. \end{align} Thus, in the adapted coordinate system we can express the components of the spacetime metric \(h_{ab}\) and its inverse \(h^{ab}\) as \begin{gather} h_{ab} = \left( \begin{array}{c|c} -\alpha^{2} + \beta^i \beta_i & \hspace{1em} \sigma_{ij} \beta^{j} \hspace{1em} \\ \hline \sigma_{ij} \beta^{j} & \sigma_{ij} \end{array} \right) \\ \label{ADMInverseMetric} h^{ab} = \left( \begin{array}{c|c} \hspace{1em} -\frac{1}{\alpha^{2}} \hspace{1em} & \frac{1}{\alpha^2} \, \beta^{i} \\ \hline \frac{1}{\alpha^2} \, \beta^{i} & \sigma^{ij} - \frac{1}{\alpha^2} \, \beta^{i} \beta^{j} \end{array} \right) \\ \det(h_{ab}) = -\alpha^{2} \det(\sigma_{ij}) \vphantom{\Big|} \end{gather} Obtaining the components of the inverse is a short algebraic calculation. Note that the spatial indices \(i,j,\ldots\) in the adapted coordinates are lowered and raised using the spatial metric \(\sigma_{ij}\) and its inverse \(\sigma^{ij}\).

In adapted coordinates there are several results concerning the projections of Lie derivatives of scalars and tensors which will be important in what follows. The first, which is trivial, is that the Lie derivative of a scalar \(S\) along the time-flow vector \(t^a\) is just the regular time-derivative \begin{gather} \pounds_{t} S = t^{a} \partial_{a} S = \frac{\partial x^{a}}{\partial t}\,\frac{\partial S}{\partial x^{a}} = \partial_{t} S ~. \end{gather} Next, we consider the projector \(P_{i}{}^{a}\) applied to the Lie derivative along \(t^{a}\) of a general vector \(W_{a}\), which gives \begin{gather} P_{i}{}^{a} \pounds_{t} W_{a} = \partial_{t} W_{a} \quad \forall \quad W_{a} ~. \end{gather} The important point is that this applies not just to spatial vectors but to any vector \(W_{a}\), as a consequence of the result \begin{gather} P_{i}{}^{a} \pounds_{t} u_{a} = 0 ~. \end{gather} Finally, we can show that the Lie derivative along \(t^{a}\) of any contravariant spatial vector satisfies \begin{gather} P^{i}{}_{a} \pounds_{t} V^{a} = \partial_{t} V^{i} \quad \forall \quad V^{i} = P^{i}{}_{a} V^{a} ~. \end{gather} This follows from a lengthier calculation than what is required for the first two results.

Given these results, we can express various geometric quantities and their projections normal to and along \(\Sigma_t\) in terms of quantities intrinsic to \(\Sigma_t\) and simple time derivatives. First, the extrinsic curvature is \begin{align} \theta_{ij} = & \,\, - \frac{1}{2}\, P_{i}{}^{a} P_{j}{}^{b} \pounds_{u} \sigma_{ab} \\ = & \,\, - \frac{1}{2}\, P_{i}{}^{a} P_{j}{}^{b} \bigg(\frac{1}{\alpha}\,\big( \pounds_{t} \sigma_{ab} - \pounds_{\beta} \sigma_{ab} \big)\bigg) \\ \Rightarrow \, \theta_{ij} = & \,\, - \frac{1}{2 \alpha} \, \bigg(\partial_{t} \sigma_{ij} - \big( D_{i} \beta_{j} + D_{j} \beta_{i} \big) \bigg) ~. \end{align} Since \(\theta_{ab}\) is a spatial tensor, projections of its Lie derivative along \(u^{a}\) can be expressed in a similar manner \begin{gather} P_{i}{}^{a} P_{j}{}^{b} \pounds_{u} \theta_{ab} = \frac{1}{\alpha}\,\big(\partial_t \theta_{ij} - \pounds_{\beta} \theta_{ij} \big) ~. \end{gather} Now we present the Gauss-Codazzi and related equations in adapted coordinates: \begin{align} P_{i}{}^{a} P_{j}{}^{b} \big( \dR_{ab} \big) = & \,\, \RR_{ij} + \theta \theta_{ij} - 2 \theta_{i}{}^{k} \theta_{jk} - \frac{1}{\alpha}\,\big( \partial_t \theta_{ij} - \pounds_{\beta} \theta_{ij} \big) - \frac{1}{\alpha}\,D_i D_j \alpha \\ P_{i}{}^{a}\big( \dR_{ab} u^{b} \big) = & \,\, D_{i} \theta - D^{j} \theta_{ij} \\ \dR_{ab} u^a u^b = & \,\, \frac{1}{\alpha}\,\big(\partial_t \theta - \beta^{i}\partial_i \theta \big) - \theta^{ij} \theta_{ij} + \frac{1}{\alpha}\,D_i D^i \alpha \\ \dR = & \,\, \RR + \theta^{2} + \theta^{ij} \theta_{ij} - \frac{2}{\alpha}\,\big(\partial_t \theta - \beta^i \partial_i \theta \big) - \frac{2}{\alpha}\,D_i D^i \alpha ~. \end{align} These allow us to write out the various projections of the Einstein equations \(G_{ab} = \kappa^{2} T_{ab}\).

• Hamiltonian Constraint \begin{gather} \RR + \theta^{2} - \theta^{ij}\,\theta_{ij} = 2\kappa^{2}\,\rho \\ \rho \defeq T_{ab} u^{a} u^{b} \end{gather}
• Momentum Constraint \begin{gather} D^{j}\theta_{ij} - D_{i}\theta = \kappa^{2} j_{i} \\ j_{i} \defeq - P_{i}{}^{a}(T_{ab} u^{b}) \end{gather}
• ADM Evolution Equations \begin{align} \RR_{ij} - \frac{1}{2} \sigma_{ij} \RR - \frac{1}{\alpha}\left(\partial_{t} \theta_{ij} - \frac{1}{2} \sigma_{ij} \partial_{t} \theta \right) + \frac{1}{\alpha} \left(\pounds_{\beta} \theta_{ij} - \frac{1}{2} \sigma_{ij} \pounds_{\beta} \theta \right) & \\ \nonumber + \,\theta\,\theta_{ij} -2 \theta_{i}{}^{k} \theta_{jk} -\frac{1}{2}\sigma_{ij}\left(\theta^{2} + \theta^{kl} \theta_{kl} \right) - \frac{1}{\alpha} \left(D_i D_j \alpha - \sigma_{ij} D^{k} D_{k} \alpha \right) & \\ \nonumber = \kappa^{2} P_{i}{}^{a} P_{j}{}^{b} T_{ab} & \end{align} Or, if we use the trace of this equation to rewrite \(\RR\) and define the spatial stress tensor as \(\SS_{ij} \defeq P_{i}{}^{a} P_{j}{}^{b} T_{ab}\), this can be written as \begin{align} \partial_{t} \theta_{ij} = & \,\,\pounds_{\beta} \theta_{ij} + \alpha \left(\RR + \theta \, \theta_{ij} - 2 \,\theta_{i}{}^{k} \theta_{jk} \right) - D_{i} D_{j} \alpha \\ \nonumber & \, - \kappa^{2}\,\alpha\,\left(\SS_{ij} - \frac{1}{d-2} \sigma_{ij} \SS^{k}{}_{k}\right) - \kappa^{2}\,\frac{1}{d-2} \sigma_{ij} \alpha \,\rho \end{align}

The line element is often presented in the form \begin{gather} h_{ab} dx^a dx^b = h_{tt} dt^2 + 2 h_{ti} dt dx^i + h_{ij} dx^i dx^j ~. \end{gather} We would like to relate the components of the metric to the ADM variables: the lapse function \(\alpha\), the shift vector \(\beta_i\), and the spatial metric \(\sigma_{ij}\). This is a straightforward exercise in linear algebra. Comparing with \eqref{ADMds}, we first note that \begin{gather} \sigma_{ij} = h_{ij} ~. \end{gather} The inverse spatial metric, \(\sigma^{ij}\), is literally the inverse of \(h_{ij}\), which is not in general the same thing as \(h^{ij}\) (see, for instance, equation \eqref{ADMInverseMetric}) \begin{gather} \sigma^{ij} = (\sigma_{ij})^{-1} = (h_{ij})^{-1} \neq h^{ij} ~. \end{gather} For the shift vector we have \begin{gather} h_{ti} = \sigma_{ij} \beta^{j} \quad \rightarrow \quad \sigma^{ik} h_{tk} = \sigma^{ik} \sigma_{kl} \beta^{l} = \beta^{i} \\ \Rightarrow \beta^i = \sigma^{ij} h_{tj} ~. \end{gather} Finally, for the lapse we obtain \begin{gather} \alpha^{2} = \sigma^{ij} h_{ti} h_{tj} - h_{tt} ~. \end{gather}

August 31, 2023

• Updated hypersurface results to include both spacelike and timelike unit normals.
• Added useful simplifications associated with surface-forming (normalized gradient) unit normals.
• Reorganized the order of results in the section on hypersurfaces.
• Changed definition of the vector \(c^{\mu}\) in the section on small variations of the metric.
• Updated variations of \(n^{\mu}\), \(K_{\mu\nu}\), and \(K\) to accommodate both spacelike and timelike hypersurfaces.
• Added explicit variation of the Gibbons-Hawking-York Lagrangian, including the (usually neglected) corner term.
• Added a note about the different conventions used for the extrinsic curvature in the ADM and hypersurface sections.

• Added a short section with the explicit form of the Lie derivative of an arbitrary tensor.