General Relativity Cheatsheet

Cheatsheet Content

Non-inertial Frames & Curved Spacetime Non-inertial Frames: Frames where Newton's laws do not hold without introducing fictitious forces (e.g., centrifugal, Coriolis). Non-Euclidean Geometry: Geometry where Euclid's fifth postulate (parallel lines) does not hold. Sum of angles in a triangle is not $180^\circ$. Curved Spacetime: In General Relativity (GR), gravity is not a force but a manifestation of the curvature of spacetime caused by mass and energy. Equivalence Principle Weak Equivalence Principle (WEP): The trajectory of a test particle in a gravitational field depends only on its initial position and velocity, not on its composition. (Gravitational mass = Inertial mass). Einstein Equivalence Principle (EEP): In a sufficiently small region of spacetime, the laws of physics are the same as those in an unaccelerated reference frame (local Lorentz frame). This implies WEP and local position/Lorentz invariance. Strong Equivalence Principle (SEP): EEP holds for self-gravitating bodies as well. Coordinate Transformations & Covariance General Coordinate Transformations: $x'^\mu = x'^\mu(x^0, x^1, x^2, x^3)$. General Covariance: Physical laws should be expressible in a form that is invariant under general coordinate transformations. This ensures laws are independent of the chosen coordinate system. Tensor Calculus Fundamentals Tangent Space ($T_p M$): The set of all possible tangent vectors at a point $p$ on a manifold $M$. Forms a vector space. Dual Space ($T_p^* M$): The space of linear maps from the tangent space to the real numbers (1-forms). Contravariant Vectors (Tangent Vectors): Components transform as $V'^\mu = \frac{\partial x'^\mu}{\partial x^\nu} V^\nu$. Represent direction. Covariant Vectors (1-forms): Components transform as $\omega'_\mu = \frac{\partial x^\nu}{\partial x'^\mu} \omega_\nu$. Represent surfaces/gradients. Metric Tensor ($g_{\mu\nu}$): Defines the inner product of vectors and measures distances/angles in spacetime. $ds^2 = g_{\mu\nu} dx^\mu dx^\nu$. Raises/lowers indices: $V_\mu = g_{\mu\nu} V^\nu$, $V^\mu = g^{\mu\nu} V_\nu$. Christoffel Symbols ($\Gamma^\alpha_{\mu\nu}$): Coefficients of the affine connection, describe how basis vectors change. First kind: $\Gamma_{\sigma\mu\nu} = \frac{1}{2} (\partial_\mu g_{\nu\sigma} + \partial_\nu g_{\sigma\mu} - \partial_\sigma g_{\mu\nu})$ Second kind: $\Gamma^\alpha_{\mu\nu} = g^{\alpha\sigma} \Gamma_{\sigma\mu\nu} = \frac{1}{2} g^{\alpha\sigma} (\partial_\mu g_{\nu\sigma} + \partial_\nu g_{\sigma\mu} - \partial_\sigma g_{\mu\nu})$ Covariant Derivative ($\nabla_\mu$): Generalizes partial derivative to curved spacetime, ensuring tensor properties are preserved. For scalar $\phi$: $\nabla_\mu \phi = \partial_\mu \phi$ For contravariant vector $V^\alpha$: $\nabla_\mu V^\alpha = \partial_\mu V^\alpha + \Gamma^\alpha_{\mu\nu} V^\nu$ For covariant vector $\omega_\alpha$: $\nabla_\mu \omega_\alpha = \partial_\mu \omega_\alpha - \Gamma^\nu_{\mu\alpha} \omega_\nu$ Parallel Transport: Moving a vector along a curve such that it "stays parallel" to itself. A vector $V^\mu$ is parallel transported along a curve $x^\nu(\lambda)$ if $\frac{dV^\mu}{d\lambda} + \Gamma^\mu_{\alpha\beta} V^\alpha \frac{dx^\beta}{d\lambda} = 0$. Intrinsic Derivative: For a vector $V^\mu$ along a curve $x^\nu(\lambda)$, the intrinsic derivative is $\frac{DV^\mu}{d\lambda} = \frac{dV^\mu}{d\lambda} + \Gamma^\mu_{\alpha\beta} V^\alpha \frac{dx^\beta}{d\lambda}$. Geodesic Equation: The path of a freely falling particle in curved spacetime. It is the path of "straightest possible" line. $\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0$, where $\tau$ is proper time. Gravitation as Spacetime Curvature Riemann Curvature Tensor ($R^\rho_{\sigma\mu\nu}$): Measures the curvature of spacetime. Defined by the non-commutativity of covariant derivatives: $[\nabla_\mu, \nabla_\nu] V^\sigma = R^\sigma_{\rho\mu\nu} V^\rho$. $R^\rho_{\sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda} \Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda} \Gamma^\lambda_{\mu\sigma}$ Properties of Riemann Tensor: Antisymmetric in last two indices: $R^\rho_{\sigma\mu\nu} = -R^\rho_{\sigma\nu\mu}$ Antisymmetric in first two (covariant) indices: $R_{\rho\sigma\mu\nu} = -R_{\sigma\rho\mu\nu}$ Symmetric under exchange of pairs of indices: $R_{\rho\sigma\mu\nu} = R_{\mu\nu\rho\sigma}$ Cyclic identity: $R^\rho_{\sigma[\mu\nu;\lambda]} = 0$ (or $R^\rho_{\sigma\mu\nu} + R^\rho_{\sigma\nu\lambda} + R^\rho_{\sigma\lambda\mu} = 0$) Ricci Tensor ($R_{\mu\nu}$): Contraction of Riemann tensor: $R_{\mu\nu} = R^\alpha_{\mu\alpha\nu}$. Represents average curvature. Ricci Scalar ($R$): Contraction of Ricci tensor: $R = g^{\mu\nu} R_{\mu\nu}$. Bianchi Identities: $\nabla_{[\alpha} R_{\beta\gamma]\delta\epsilon} = 0$. Implies $\nabla^\mu (R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R) = 0$. Geodesic Deviation Equation: Describes the relative acceleration of two nearby geodesics. $\frac{D^2 \xi^\alpha}{d\tau^2} = -R^\alpha_{\beta\gamma\delta} U^\beta U^\gamma \xi^\delta$, where $\xi^\alpha$ is the separation vector and $U^\alpha$ is the 4-velocity. Relativistic Tidal Forces: Manifestation of spacetime curvature; described by geodesic deviation. Energy-Momentum & Einstein's Equations Energy-Momentum Tensor ($T_{\mu\nu}$): Source of gravity, describes the density and flux of energy and momentum. Dust: $T^{\mu\nu} = \rho_0 U^\mu U^\nu$, where $\rho_0$ is rest mass density. Perfect Fluid: $T^{\mu\nu} = (\rho_0 + P) U^\mu U^\nu + P g^{\mu\nu}$, where $P$ is pressure. Conservation Laws: $\nabla_\mu T^{\mu\nu} = 0$ (covariant conservation of energy and momentum). Hilbert's Variational Principle: Action $S = \int \sqrt{-g} (R + \mathcal{L}_M) d^4x$. Varying with respect to $g^{\mu\nu}$ yields Einstein's equations. Einstein-Hilbert Action: $S_G = \frac{1}{16\pi G} \int R \sqrt{-g} d^4x$. Einstein's Equation: $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = \frac{8\pi G}{c^4} T_{\mu\nu}$. Relates spacetime curvature to energy-momentum. Newtonian Approximation: In weak gravitational fields and slow speeds, GR reduces to Newtonian gravity. $g_{00} \approx -(1 + 2\Phi/c^2)$ where $\Phi$ is Newtonian potential. Geodesic Equations from Variational Principle: Geodesics are paths that extremize the proper time interval $\int ds$. Weak Field & Gravitational Waves Weak Field Metric: $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$, where $\eta_{\mu\nu}$ is Minkowski metric and $|h_{\mu\nu}| \ll 1$. Linearized Field Equations: Approximated Einstein equations for weak fields. $\Box \bar{h}^{\mu\nu} = -16\pi G/c^4 T^{\mu\nu}$, where $\bar{h}^{\mu\nu} = h^{\mu\nu} - \frac{1}{2} \eta^{\mu\nu} h$. Gravitational Energy-Momentum Pseudotensor ($t_{\mu\nu}$): Describes the energy and momentum carried by the gravitational field itself (not a true tensor). Gravitational Waves: Ripples in spacetime that propagate at the speed of light. Wave Equation in Linearized Theory: Solutions to $\Box \bar{h}^{\mu\nu} = 0$ in vacuum. Plane Wave Solutions: $h_{\mu\nu} = A_{\mu\nu} e^{ik_\rho x^\rho}$. Transverse Traceless (TT) Gauge: A gauge choice where $h_{0\mu}=0$, $h^\mu_\mu=0$, $\partial^\mu h_{\mu\nu}=0$. In this gauge, only two independent polarizations exist ($h_+$ and $h_\times$). Effect on Test Particles: GWs cause a strain on test particles, changing their relative distances. Quadrupole Formula: Luminosity of GWs $P_{GW} = \frac{G}{5c^5} \frac{d^3 Q_{ij}}{dt^3} \frac{d^3 Q^{ij}}{dt^3}$, where $Q_{ij}$ is the quadrupole moment of the source. Schwarzschild Spacetime & Tests of GR Static, Spherically Symmetric Spacetime: Described by the Schwarzschild metric for vacuum outside a non-rotating, uncharged, spherical mass. Schwarzschild Exterior Solution: $ds^2 = -(1 - \frac{2M}{r}) dt^2 + (1 - \frac{2M}{r})^{-1} dr^2 + r^2 (d\theta^2 + \sin^2\theta d\phi^2)$ (using $G=c=1$) $M$ is the mass of the central object. Schwarzschild Geodesics: Paths of particles in this spacetime. Metric in Newtonian Limit: For large $r$, $g_{00} \approx -(1 - \frac{2M}{r}) \approx -(1 + \frac{2\Phi}{c^2})$, where $\Phi = -GM/r$. Effective Potential for Particle Orbits: $V_{eff}(r) = -\frac{M}{r} + \frac{L^2}{2r^2} - \frac{ML^2}{r^3}$ (for massive particles, $E^2/2$). This potential has an extra $1/r^3$ term compared to Newtonian gravity. $R=2M$ Surface (Schwarzschild Radius): Event Horizon of a non-rotating black hole. $g_{00}$ becomes zero, $g_{rr}$ becomes infinite. A singularity in coordinates, not physically. Innermost Stable Circular Orbit (ISCO): For massive particles, ISCO is at $r=6M$. Inside this, circular orbits are unstable. Classical Tests of GR: Precession of Perihelion of Mercury: Observed $43''$ per century, predicted by GR. Bending of Light: Light rays bend by $4GM/(rc^2)$ (for grazing ray, $1.75''$ at sun's limb). Confirmed by Eddington. Gravitational Redshift: Photons lose energy climbing out of a gravitational well. $\Delta\nu/\nu \approx -GM/(rc^2)$. Radar Echo Delay (Shapiro Delay): Time delay for radar signals passing near a massive object. Cosmological Dynamics Weyl's Postulate: Fundamental particles of the cosmic fluid are on spacelike geodesics that do not intersect. Cosmological Principle: On large scales, the universe is homogeneous and isotropic. Co-moving Coordinates: Coordinates that expand with the universe, so co-moving observers see constant spatial separations between galaxies. Maximally Symmetric Spaces: Spaces that admit the maximum number of Killing vectors. Homogeneous and isotropic. Robertson-Walker Metric: The most general metric consistent with the cosmological principle. $ds^2 = -c^2 dt^2 + a(t)^2 \left[ \frac{dr^2}{1 - kr^2} + r^2 (d\theta^2 + \sin^2\theta d\phi^2) \right]$ $a(t)$ is the scale factor, $k$ is the curvature parameter ($+1$ closed, $0$ flat, $-1$ open). Expanding Universe: $a(t)$ increases with time, indicating expansion. Anisotropies, Vorticity, Shear: Deviations from perfect homogeneity and isotropy. Anisotropies: Directional dependence. Vorticity: Rotation of matter. Shear: Distortion of shape without rotation or volume change. Raychaudhuri Equation: Describes the evolution of the expansion, shear, and vorticity of timelike congruences. $\frac{d\theta}{d\tau} + \frac{1}{3}\theta^2 + 2\sigma^2 - 2\omega^2 = -R_{\mu\nu}U^\mu U^\nu + \nabla_\mu a^\mu$ $\theta$ is expansion scalar, $\sigma^2$ is shear scalar, $\omega^2$ is vorticity scalar. Crucial for singularity theorems.