Geometric Formulation of Gravity

Contents

Geodesics as the trajectories of free fall
Curvature and tidal forces
The Einstein–Hilbert action

The equivalence principle (The Equivalence Principle and Classical Tests) motivates, but does not by itself prove, a specific mathematical statement: that gravity is the geometry of the spacetime manifold itself. This chapter states that geometric formulation precisely, reusing throughout the machinery already built, in full generality, in Differentiable Manifolds, Tensors, and Curvature and Lie Groups, Lie Algebras, and Fibre Bundles. Physical spacetime instantiates that general framework at \(p+q=3+1\), torsion set to zero: no experiment to date requires \(T^{a}\neq0\) (Section 13.13.2), and this treatise's scope rule (Epistemology and the Scientific Method) accordingly restricts the physical theory to the Levi-Civita connection.

Geodesics as the trajectories of free fall

Postulate 43.1 (Geodesic motion).

A test particle subject to no force other than gravity follows a timelike geodesic of \((M,g)\): an autoparallel curve of the Levi-Civita connection (Definition 13.146), parametrized by its own proper time \(\tau\),

\begin{equation}\tag{43.1} \frac{\dd^{2}x^{\lambda}}{\dd\tau^{2}} + \mathring{\Gamma}^{\lambda}{}_{\mu\nu}\, \dv{x^{\mu}}{\tau}\,\dv{x^{\nu}}{\tau} = 0\ec \end{equation}

the specialization of Equation (13.297) to the torsion-free connection of Theorem 13.150.

This is the direct generalization of Newton's first law (Newtonian Dynamics): in the absence of gravity, \(g_{\mu\nu}=\eta_{\mu\nu}\), \(\mathring{\Gamma}=0\), and Equation (43.1) reduces to \(\dd^{2}x^{\lambda}/\dd\tau^{2}=0\), uniform motion. Equation (43.1) also follows from extremizing the proper-time functional \(\int\dd\tau=\int\sqrt{-g_{\mu\nu}\dot x^{\mu}\dot x^{\nu}}/c\,\dd\lambda\) (Remark 13.151), the relativistic analogue of Hamilton's principle (Lagrangian Mechanics).

Derivation. Derives Equation (43.1). Take the action of a free massive particle proportional to its proper time,

\begin{equation}\tag{43.2} S[x] = -mc^{2}\!\int\dd\tau = -mc\!\int\dd\lambda\, \sqrt{-g_{\mu\nu}(x)\,\dot x^{\mu}\dot x^{\nu}}\ec \end{equation}

with \(\dot x^{\mu} = \dd x^{\mu}/\dd\lambda\) and \(\lambda\) an arbitrary parameter. The Euler–Lagrange equations of Calculus of Variations applied to \(L = \sqrt{-g_{\mu\nu}\dot x^{\mu}\dot x^{\nu}}\) simplify upon choosing \(\lambda = \tau\) after the variation, so that \(L = c\) is constant along the curve:

\begin{equation*} \dv{}{\tau}\bigl(g_{\lambda\nu}\dot x^{\nu}\bigr) - \frac{1}{2}\,\pp_{\lambda}g_{\mu\nu}\, \dot x^{\mu}\dot x^{\nu} = 0\ec \end{equation*}

i.e. \(g_{\lambda\nu}\ddot x^{\nu} + \bigl(\pp_{\mu}g_{\lambda\nu} - \tfrac{1}{2}\pp_{\lambda}g_{\mu\nu}\bigr) \dot x^{\mu}\dot x^{\nu} = 0\). Symmetrizing \(\pp_{\mu}g_{\lambda\nu} \to \tfrac{1}{2}(\pp_{\mu}g_{\lambda\nu} + \pp_{\nu}g_{\lambda\mu})\) (legitimate under the symmetric contraction with \(\dot x^{\mu}\dot x^{\nu}\)) and raising with \(g^{\sigma\lambda}\) produces exactly the Levi-Civita coefficients of Equation (13.301) with \(T = 0\), which is Equation (43.1).

For the Newtonian limit take a static weak field, \(g_{00} = -(1 + 2\Phi/c^{2})\) with \(\abs{\Phi}/c^{2} \ll 1\), and slow motion, \(\abs{\dd x^{i}/\dd t} \ll c\), so that with \(x^{0} = ct\) the four-velocity is dominated by \(\dd x^{0}/\dd\tau \approx c\). The only Christoffel symbol surviving at leading order in Equation (43.1) is \(\mathring{\Gamma}^{i}{}_{00} = -\tfrac{1}{2}\,\delta^{ij}\pp_{j}g_{00} = \pp_{i}\Phi/c^{2}\), and the spatial components collapse to

\begin{equation*} \dv[2]{x^{i}}{t} = -c^{2}\,\mathring{\Gamma}^{i}{}_{00} = -\pp_{i}\Phi\ec \end{equation*}

which is Newton's law of gravitation \(\ddot{\vect x} = -\nabla\Phi\) (Newtonian Dynamics): the geodesic postulate contains Newtonian free fall as its slow-motion, weak-field limit.

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Curvature and tidal forces

Two neighbouring free-falling geodesics, separated by a small connecting vector \(\xi^{\mu}\), obey the geodesic deviation equation

\begin{equation}\tag{43.3} \frac{\mathrm{D}^{2}\xi^{\lambda}}{\dd\tau^{2}} = -R^{\lambda}{}_{\mu\nu\rho}\, \dv{x^{\mu}}{\tau}\,\xi^{\nu}\,\dv{x^{\rho}}{\tau}\ec \end{equation}

with \(\mathrm{D}/\dd\tau\) the covariant derivative along the geodesic and \(R^{\lambda}{}_{\mu\nu\rho}\) the Riemann tensor of Theorem 13.152 (torsion-free). Curvature, not the connection, is therefore the geometric object with direct physical content: it is observed as the relative (tidal) acceleration of nearby free-falling bodies, exactly as Newtonian tidal forces are governed by the second derivatives of the potential, \(-\pp_{i}\pp_{j}\Phi\).

Derivation. Derives Equation (43.3). This is the Jacobi equation Equation (13.309) (Proposition 13.155), instantiated on spacetime with the Levi-Civita connection: in the index order of Equation (13.309), \(\mathrm{D}^{2}\xi^{\lambda}/\dd\tau^{2} = R^{\lambda}{}_{\rho\mu\nu}u^{\rho}u^{\mu}\xi^{\nu}\), and the antisymmetry of the Riemann tensor in its last index pair (Proposition 13.154) rewrites this as \(-R^{\lambda}{}_{\mu\nu\rho}\,u^{\mu}\xi^{\nu}u^{\rho}\), the form quoted in Equation (43.3).

For the Newtonian identification take the static weak field of the previous derivation, \(u^{\mu} \approx (c,\vect 0)\), and compute from Equation (13.306): at linear order in \(\Phi/c^{2}\) and for time-independent fields,

\begin{equation*} R^{i}{}_{0j0} = \pp_{j}\mathring{\Gamma}^{i}{}_{00} - \pp_{0}\mathring{\Gamma}^{i}{}_{j0} = \frac{\pp_{i}\pp_{j}\Phi}{c^{2}}\ec \end{equation*}

so Equation (43.3) reduces to

\begin{equation*} \dv[2]{\xi^{i}}{t} = -c^{2}R^{i}{}_{0j0}\,\xi^{j} = -\pp_{i}\pp_{j}\Phi\;\xi^{j}\ec \end{equation*}

the Newtonian tidal equation: the Riemann component \(R^{i}{}_{0j0}\) is the tidal tensor \(\pp_{i}\pp_{j}\Phi/c^{2}\), which is the sense of the correspondence asserted above.

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The Einstein–Hilbert action

The field equations governing \(g_{\mu\nu}\) itself (The Einstein Field Equations) follow from an action principle, stated here in both the variables of this treatise's double geometric language (Section 13.1): the metric \(g_{\mu\nu}\), and the vielbein \(e^{a}\) together with the curvature two-form \(R^{a}{}_{b}\) of Section 13.12.

Definition 43.2 (Einstein–Hilbert action).

In \(D=p+q\) dimensions,

\begin{equation}\tag{43.4} S_{\mathrm{EH}}[g] = \frac{1}{2\kappa}\int_{M}\dd^{D}x\, \sqrt{\abs{g}}\;R\ec \end{equation}

with \(\kappa\) a gravitational coupling constant, fixed in \(D=4\) by the Newtonian limit to \(\kappa=8\pi G/c^{4}\) (The Einstein Field Equations). Equivalently, in vielbein-form variables,

\begin{equation}\tag{43.5} S_{\mathrm{EH}}[e,\omega] = \frac{1}{2\kappa\,(D-2)!}\int_{M} \epsilon_{a_{1}\ldots a_{D}}\, R^{a_{1}a_{2}}\wedge e^{a_{3}}\wedge\cdots\wedge e^{a_{D}}\ec \end{equation}

with \(\omega^{a}{}_{b}=\omega^{a}{}_{b}(e)\) the torsion-free connection determined by \(e\) (Proposition 13.158).

Theorem 43.3 (Equivalence of the two variable sets).

\(S_{\mathrm{EH}}[g]=S_{\mathrm{EH}}[e,\omega]\) for \(g_{\mu\nu}= \eta_{ab}e^{a}{}_{\mu}e^{b}{}_{\nu}\) (Equation (13.264)). Rests on Definition 43.2, Definition 13.122 and Proposition 13.158.

Proof.

This is Equation (A.150), proven in Appendix A.8.

∎

Remark 43.4 (Why the vielbein form matters).

The equivalence of Theorem 43.3 shows that Equation (43.5) is not a mere notational variant: it is the same physical action, written in the language in which gravity appears as an \(\SO(p,q)\) gauge theory (Remark 13.159) — the language required the moment matter includes spinor fields (The Dirac Equation), which possess no tensorial transformation law under diffeomorphisms and must instead be coupled through the frame and the spin connection [Ortin:2015]. Varying Equation (43.5) independently with respect to \(e^{a}\) (with \(\omega\) eliminated via Proposition 13.158, first-order/Palatini formalism) reproduces the same field equations obtained from varying Equation (43.4) with respect to \(g_{\mu\nu}\); that variation, and the resulting field equations, are carried out in full in Section 44.1.3: the metric route is Theorem 44.9, and the independent first-order (vielbein–connection) route is Proposition 44.13.