General Relativity — worked problems

(a) $ds^2 = a^2 d\lambda^2 - da^2$; $\Gamma^\lambda_{\lambda a} = 1/a$, $\Gamma^a_{\lambda\lambda} = a$. Lorentzian analogue of polar coordinates.

(b) $\nabla_\mu V^\mu = \partial_\lambda V^\lambda + (1/a)\partial_a(aV^a)$. $\Box\phi = (1/a^2)\partial_\lambda^2\phi - \partial_a^2\phi - (1/a)\partial_a\phi$.

(c) On the wedge $x^2 - t^2 = a^2$. Jacobian-substitution gives $T'^\lambda{}_\lambda = a^2\cosh(2\lambda) = x^2 + t^2$. Trace-check: $T'^\lambda{}_\lambda + T'^a{}_a = 0$ ✓ (matches Cartesian $T^\mu{}_\mu = 0$).

Christoffel: $\Gamma^r_{\phi\phi} = -r$, $\Gamma^\phi_{r\phi} = 1/r$. Non-trivial covariant-derivative entries: $\nabla_r V^\phi = \partial_r V^\phi + V^\phi/r$, $\nabla_\phi V^r = \partial_\phi V^r - r V^\phi$, $\nabla_\phi V^\phi = \partial_\phi V^\phi + V^r/r$.

$$\boxed{\;\nabla_\mu V^\mu = \partial_t V^t + \partial_r V^r + V^r/r + \partial_\phi V^\phi.\;}$$

(a) $t = \sinh\chi$, $x = \cosh\chi\cos\phi$, $y = \cosh\chi\sin\phi$.

(b) $ds^2 = d\chi^2 - \cosh^2\chi\,d\phi^2$ — global slicing of 2D de Sitter; circular spatial slice contracts from $\infty$ to $2\pi$ at $\chi = 0$, then re-expands.

(c) $\Gamma^\chi_{\phi\phi} = \sinh(2\chi)/2$, $\Gamma^\phi_{\chi\phi} = \tanh\chi$ — standard FRW form with $a(\chi) = \cosh\chi$. $R = -2$ (constant negative curvature in $(+,-)$ signature; equivalently dS$_2$ with $\ell = 1$).

(a) Standard: $R^\omega{}_{\mu\nu\lambda} = \partial_\nu\Gamma^\omega_{\lambda\mu} - \partial_\lambda\Gamma^\omega_{\nu\mu} + \Gamma\Gamma$ terms. First Bianchi: cyclic sum vanishes by symmetry $\Gamma^\sigma_{\mu\nu} = \Gamma^\sigma_{\nu\mu}$ (torsion-free).

(b) Riemann normal coords at $p$: $\Gamma|_p = 0$, $R_{\alpha\beta\mu\nu;\lambda}|_p$ reduces to second derivatives of $\Gamma$. Cyclic sum vanishes by commutativity of partials. Tensorially true at every $p$.

Contract second Bianchi twice with $g$: $\nabla_\mu G^{\mu\nu} = 0$ identically. Einstein's equation then forces $\nabla_\mu T^{\mu\nu} = 0$. In flat space, reduces to $\partial_\mu T^{\mu\nu} = 0$ — local energy-momentum conservation.

Historical: Einstein insisted matter-conservation $\nabla_\mu T^{\mu\nu} = 0$; sought a geometric tensor with identically-vanishing divergence. The Bianchi identities supply exactly one: $G^{\mu\nu}$ (up to a metric multiple — cosmological constant). The field equations $G^{\mu\nu} = \kappa T^{\mu\nu}$ are forced by Bianchi + equivalence principle.

(a) $\Gamma^r_{rr} = 1/r$, $\Gamma^r_{\phi\phi} = -2r$, $\Gamma^\phi_{r\phi} = 2/r$. $R^r{}_{\phi r\phi} = -2 - (-2) + (-2) - (-4) = 0$. In 2D this is the entire Riemann tensor — the manifold is flat where the metric is smooth.

Substitution $u = r^2/2$ gives $ds^2 = du^2 + u^2 d\psi^2$ with $\psi = 2\phi$ — flat polar coordinates but with period $4\pi$. Double cover of the plane, branched at the origin — cone with $2\pi$ excess angle.

(b) $C = 2\pi r_0^2$, $A = \pi r_0^4/2$. So $C^2 = 4\pi^2 r_0^4 = 8\pi A$: $$\boxed{\;C^2 = 8\pi A\;\text{(twice the Euclidean }4\pi A\text{).}\;}$$ Flatness ≠ Euclidean global geometry; topological defects shift the area-circumference relation while leaving $R = 0$ in the smooth bulk. The curvature concentrates as a delta at the origin (Gauss–Bonnet recovers the deficit angle).

Koszul formula with constant-metric simplification: $2g(\nabla_A B, C) = -g(A,[B,C]) + g(B,[C,A]) + g(C,[A,B])$.

$\nabla_X X = \nabla_Y Y = \nabla_Z Z = 0$. Off-diagonal: $\nabla_X Y = -Z/2$, $\nabla_Y Z = X/2$, $\nabla_Z X = Y/2$ (and reflected $\nabla_Y X = +Z/2$ etc. via torsion-freeness).

Riemann: $R(X,Y)X = Y/4$ etc. Recognise as maximally symmetric form: $$\boxed{\;R(A,B)C = (1/4)[g(B,C)A - g(A,C)B],\;}$$ i.e. $R_{ABCD} = (g_{AC}g_{BD} - g_{AD}g_{BC})/4$. Constant sectional curvature $K = 1/4$, scalar $R = 3/2$.

Identification. The frame is $\mathfrak{sl}(2,\mathbb{R}) \cong \mathfrak{so}(2,1)$ with metric proportional to the Killing form. The manifold is (the universal cover of) $SL(2,\mathbb{R})$ — a maximally symmetric 3D Lorentzian space, dS$_3$ or AdS$_3$ depending on overall sign, curvature radius $\ell = 2$.

Constraint: $(x^1)^2 + (x^2)^2 - (x^0)^2 = a^2$ — unit hyperboloid of "radius" $a$.

(a) $ds^2|_\Sigma = a^2 d\lambda^2 - a^2\cosh^2\lambda\,d\theta^2$ — 2D de Sitter with curvature radius $a$.

(b) $\Gamma^\lambda_{\theta\theta} = \sinh(2\lambda)/2$, $\Gamma^\theta_{\lambda\theta} = \tanh\lambda$.

(c) $\dot\lambda = 0$ in geodesic equation $\ddot\lambda + \cosh\lambda\sinh\lambda\,\dot\theta^2 = 0$ forces $\sinh\lambda_0 = 0$, i.e. $\lambda_0 = 0$ (the "throat"). Then $\ddot\theta = 0$, so $\theta = \theta_0 + c\tau$ — a closed spacelike geodesic circle of radius $a$ in the equatorial plane $\{x^0 = 0, x^3 = 0\}$.

Cross-check via embedded geometry: ambient acceleration $\ddot{\vec X} = -c^2 a(0,\sin\theta,\cos\theta,0)$ is anti-parallel to the outward normal — purely normal to $\Sigma$, hence geodesic. ✓. Just like the equator of a sphere is a great circle; only the throat (the minimum-expansion slice in cosmological language) hosts a closed spacelike geodesic.