The Geometry of Gravity

The equivalence principle forces the conclusion that spacetime is locally Minkowskian but globally curved. This page makes that precise: spacetime is a four-dimensional Lorentzian manifold $(M,g)$, and the metric tensor $g_{\mu \nu }$ — the object that assigns intervals, times, and angles — is simultaneously the gravitational field. What Newton packaged in a single potential $\Phi$, general relativity carries in the ten independent components of $g_{\mu \nu }$.

We use $c=1$, keep $G$ explicit, and — following the standard GR literature — adopt the mostly-plus signature.

Sign convention (read once)

The SR pages use mostly-minus ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$, natural when the timelike interval $\Delta s^2>0$ is to be positive. The GR literature (Wald, MTW, Carroll) overwhelmingly uses mostly-plus:

$${\eta }_{\mu \nu }=\mathrm{diag}(-1,+1,+1,+1),$$

for which spacelike intervals are positive and proper time is $d{\tau }^2=-d{s}^2=-g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$. We adopt mostly-plus for the entire GR section. The physics is convention-independent; only signs of individual terms differ. Where a formula's sign flips relative to the SR pages, it is because of this choice, not a disagreement about physics. (Our other conventions: Riemann and Einstein-equation signs are fixed on Curvature; these are the "$(-,+,+)$" conventions in the MTW classification, matching Misner–Thorne–Wheeler and Carroll.)

Spacetime as a manifold

A spacetime is a pair $(M,g)$ where:

• $M$ is a smooth, connected, four-dimensional manifold — locally ${\mathbb{R}}^4$, coordinatized by charts $x^{\mu }:U\to {\mathbb{R}}^4$, but with no preferred global coordinates;
• $g$ is a Lorentzian metric: a smooth, symmetric, non-degenerate $(0,2)$-tensor field of signature $(-,+,+,+)$.

Dropping global coordinates is the mathematical face of general covariance: no coordinate system is physically preferred, and the laws must be written as tensor equations valid in every chart. Coordinates are labels; the geometry lives in $g$. (The deep consequence — that only coincidences of events, not coordinate values, are physical — is the content of the hole argument.)

At each point $p\in M$ the tangent space $T_{p}M$ is a four-dimensional vector space carrying the values $g_p$; it is the arena of the local inertial frame guaranteed by the EEP. Vectors, covectors, and tensors are defined pointwise on these tangent spaces exactly as the SR four-vector formalism does on flat space — the difference is that the metric now varies from point to point.

The metric tensor

In a coordinate chart the metric is a symmetric $4\times 4$ matrix field $g_{\mu \nu }(x)$, encoding the infinitesimal interval

$$d{s}^2=g_{\mu \nu }(x)d{x}^{\mu }d{x}^{\nu }.$$

Being symmetric, $g_{\mu \nu }$ has $10$ independent components — but $4$ of them are pure coordinate (gauge) freedom, leaving $6$ physical functions. (Deriving dynamics will further split these; see the initial-value formulation.)

The metric does all the geometric work:

• Intervals and proper time. Along a timelike worldline, $d{\tau }^2=-g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$; a clock reads $\tau =\int d\tau$, generalizing the SR proper time.
• Causal structure. The sign of $g_{\mu \nu }{V}^{\mu }{V}^{\nu }$ classifies a vector $V$ as timelike ($<0$), null ($=0$), or spacelike ($>0$); the null vectors at $p$ form the light cone in $T_{p}M$. Light cones exist at every event but tilt and open differently from place to place — this position-dependence of the causal structure is the geometric content of gravity.
• Raising and lowering. The inverse metric $g^{\mu \nu }$ (defined by $g^{\mu \alpha }{g}_{\alpha \nu }={\delta }^{\mu }{}_{\nu }$) raises indices; $g_{\mu \nu }$ lowers them, as on flat spacetime.
• Volume. The invariant volume element is $\sqrt{-g}{d}^{4}x$, with $g=\det g_{\mu \nu }<0$; the $\sqrt{-g}$ makes integrals coordinate-independent.

The metric is the gravitational potential

The connection to Newtonian gravity is direct. In the weak-field, slow-motion limit the metric is a small perturbation of flat spacetime,

$$g_{00}=-\left(1+\frac{2\Phi }{c^2}\right)+\cdots ,$$

where $\Phi$ is the Newtonian potential. Gravitational time dilation, $d\tau =\sqrt{-g_{00}}dt\approx (1+\Phi /c^2)dt$, is then just the statement that clocks deeper in the well ($\Phi <0$) run slow — the gravitational redshift of the previous page. So $g_{\mu \nu }$ is the relativistic gravitational potential, with $\Phi$ recovered as (essentially) one component in the appropriate limit. The remaining components encode effects with no Newtonian analog (frame dragging, gravitational waves).

Local flatness: the EEP as a theorem of geometry

The equivalence principle becomes a precise statement about the metric. At any event $p$ one can choose local inertial (Riemann normal) coordinates in which

$$g_{\mu \nu }(p)={\eta }_{\mu \nu },{\partial }_{\alpha }{g}_{\mu \nu }(p)=0.$$

That is, at a chosen point the metric can be brought to the Minkowski form and its first derivatives made to vanish — the local inertial frame of the freely falling observer. Counting confirms this is always possible: a coordinate change has enough freedom to set the $10$ values $g_{\mu \nu }(p)$ to ${\eta }_{\mu \nu }$ and the $40$ first derivatives ${\partial }_{\alpha }{g}_{\mu \nu }(p)$ to zero.

What one cannot generally remove is the second derivatives: of the $100$ components ${\partial }_{\alpha }{\partial }_{\beta }{g}_{\mu \nu }(p)$, coordinate freedom can kill only $80$, leaving $20$ that no choice of frame can eliminate. Those $20$ are exactly the independent components of the Riemann curvature tensor. This is the geometric statement of the tidal-force argument: a uniform field (first derivatives) is gauge, but curvature (second derivatives) is physical. Gravity, stripped of coordinate artifacts, is the non-removable second derivative of the metric.

What "gravity is geometry" buys

Recasting gravity as the metric resolves the incompatibilities of the opening crisis:

• No action at a distance. The metric is a dynamical field obeying local equations (the Einstein equations); disturbances propagate causally, at the speed of light, as gravitational waves.
• General covariance. Written as tensor equations on $(M,g)$, the laws take the same form in every coordinate system, satisfying the principle of relativity in its strongest form.
• Universality is automatic. Test bodies follow geodesics of $g$, which know nothing of mass or composition — the UFF is built in, not imposed.

Summary

• Spacetime is a Lorentzian manifold $(M,g)$; the metric $g_{\mu \nu }$ carries all geometry and is the gravitational field.
• We use mostly-plus signature $(-,+,+,+)$, unlike the SR pages.
• The metric fixes proper time, causal structure (position-dependent light cones), and volume; in the weak-field limit $g_{00}\approx -(1+2\Phi )$ recovers the Newtonian potential.
• At every event the metric can be made Minkowskian with vanishing first derivatives (local inertial frame = EEP); the irreducible $20$ second derivatives are the curvature.

Next

To do physics on $(M,g)$ we need to differentiate tensor fields covariantly and say what "straightest possible line" means. That is the covariant derivative and parallel transport, which defines geodesics — the worldlines of freely falling bodies.