Metric Signature and Curvature Signs

The site uses one fixed metric-and-curvature package. This record collects that package, shows how every contraction changes under the two sign choices most often encountered in relativity sources, and gives checks that can distinguish a real convention mismatch from a harmless change of notation. Full geometric treatment is given in Curvature, Holonomy, and Geometric Invariants. Specific derivations belong to Pseudo-Riemannian Metrics and Signature, The Riemann Curvature Tensor, and Ricci, Scalar, and Einstein Tensors.

The site metric and curvature package

The metric signature is $(-,+,+,+)$: timelike vectors have negative squared norm, spacelike vectors have positive squared norm, and null vectors have zero norm. Reversing the overall metric sign preserves the null cone but reverses the timelike and spacelike inequalities used to describe it. For the torsion-free, metric-compatible Levi-Civita connection,

$$\begin{aligned} \Gamma^\rho{}_{\mu\nu} &=\frac12 g^{\rho\lambda} \bigl( \partial_\mu g_{\lambda\nu} +\partial_\nu g_{\lambda\mu} -\partial_\lambda g_{\mu\nu} \bigr). \end{aligned}$$

This formula follows from $\nabla_\lambda g_{\mu\nu}=0$ and $\Gamma^\rho{}_{\mu\nu}=\Gamma^\rho{}_{\nu\mu}$; it is not a convention for a general affine connection. The corresponding construction and uniqueness argument are given in Wendl 2021–2022, §§ 22–24, pp. 184–211.

The Riemann tensor is fixed first as an operator on vectors:

$$[\nabla_\mu,\nabla_\nu]V^\rho =R^\rho{}_{\sigma\mu\nu}V^\sigma.$$

Duality then fixes the sign on a covector; it is not another independent choice:

$$[\nabla_\mu,\nabla_\nu]\omega_\rho =-R^\sigma{}_{\rho\mu\nu}\omega_\sigma.$$

The minus sign follows by applying the commutator to the scalar $\omega_\rho V^\rho$: the vector and covector curvature terms must cancel. This is a useful diagnostic because two sources can display opposite-looking commutators while using the same Riemann tensor—one may simply act on a vector and the other on a covector.

In a coordinate basis, the exact index order is

$$\begin{aligned} R^\rho{}_{\sigma\mu\nu} ={}&\partial_\mu\Gamma^\rho{}_{\nu\sigma} -\partial_\nu\Gamma^\rho{}_{\mu\sigma} \\ &+\Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma} -\Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma}. \end{aligned}$$

These operator and component definitions agree with Tong 2019, § 3.2, Eqs. (3.15)–(3.18), pp. 107–110 and Carroll 1997, § 3, Eqs. (3.63)–(3.71), pp. 74–77. The site lowers the first index with the metric and contracts the first and third positions:

$$R_{\rho\sigma\mu\nu} =g_{\rho\lambda}R^\lambda{}_{\sigma\mu\nu}, \qquad R_{\sigma\nu}=R^\rho{}_{\sigma\rho\nu}.$$

Finally,

$$R=g^{\mu\nu}R_{\mu\nu}, \qquad G_{\mu\nu}=R_{\mu\nu}-\frac12 Rg_{\mu\nu}.$$

Writing only a symbol such as $R^\rho{}_{\sigma\mu\nu}$ is therefore insufficient metadata: the commutator, component order, and Ricci contraction must agree as one package.

Translating two independent sign switches

Let a primed source use the same coordinates and index order but

$$g'_{\mu\nu}=\sigma g_{\mu\nu}, \qquad R'^\rho{}_{\sigma\mu\nu} =\kappa R^\rho{}_{\sigma\mu\nu}, \qquad \sigma,\kappa\in\{+1,-1\}.$$

Here $\sigma=-1$ reverses the metric signature, whereas $\kappa=-1$ reverses the Riemann definition. Because $\sigma$ is constant, the inverse metric obeys $g'^{\mu\nu}=\sigma g^{\mu\nu}$. Substitution into the Levi-Civita formula gives the complete translation:

Quantity	Primed expression	What controls the sign
Levi-Civita coefficients	$\Gamma'^\rho{}_{\mu\nu}=\Gamma^\rho{}_{\mu\nu}$	A constant metric reversal cancels between $g^{-1}$ and $\partial g$.
Mixed Riemann tensor	$R'^\rho{}_{\sigma\mu\nu}=\kappa R^\rho{}_{\sigma\mu\nu}$	Only the curvature-definition switch.
All-down Riemann tensor	$R'_{\rho\sigma\mu\nu}=\sigma\kappa R_{\rho\sigma\mu\nu}$	Lowering the first index adds $\sigma$.
Ricci tensor	$R'_{\mu\nu}=\kappa R_{\mu\nu}$	The declared first–third mixed contraction adds no metric factor.
Scalar curvature	$R'=\sigma\kappa R$	The inverse-metric trace adds $\sigma$.
Einstein tensor	$G'_{\mu\nu}=\kappa G_{\mu\nu}$	The two metric-reversal factors in $R'g'_{\mu\nu}$ cancel.

Thus an opposite signature does not by itself reverse $R^\rho{}_{\sigma\mu\nu}$, $R_{\mu\nu}$, or $G_{\mu\nu}$. Conversely, swapping the last two Riemann indices is a $\kappa=-1$ operation even if the metric signature is unchanged. A source using another Ricci contraction or another index order must first be rewritten into the definitions above; the two-switch table cannot repair an undeclared permutation.

A safe comparison therefore proceeds in a fixed order. First identify the source’s vector commutator and the positions represented by each Riemann index. Second rewrite its Ricci tensor as an explicit contraction of that mixed Riemann tensor. Only then determine $\sigma$ and $\kappa$ and apply the table to the whole equation. Comparing a single all-down component before these steps is ambiguous: its sign mixes the curvature definition with the metric used to lower the first index. Likewise, comparing only a quadratic invariant such as $R_{\rho\sigma\mu\nu}R^{\rho\sigma\mu\nu}$ cannot detect an overall curvature sign.

This table translates the geometric side only. A field equation also contains the cosmological term and a stress tensor whose sign depends on the matter action, the variable varied, and the source’s definition of $T_{\mu\nu}$. Translate the Einstein–Hilbert and matter actions, the volume density, $\Lambda g_{\mu\nu}$, and $T_{\mu\nu}$ as a complete package before comparing field equations. Flipping signs only on the left-hand side can turn two equivalent conventions into inequivalent dynamics.

Checks that expose a wrong sign

Flat polar coordinates. For $ds^2=dr^2+r^2d\phi^2$, the nonzero coefficients include $\Gamma^r{}_{\phi\phi}=-r$ and $\Gamma^\phi{}_{r\phi}=1/r$. The derivative contribution to $R^r{}_{\phi r\phi}$ is $-1$, while the quadratic contribution is $+1$. Their sum vanishes. Nonzero connection coefficients therefore do not imply curvature, and a component formula that misses either term fails immediately.

A round sphere and constant curvature. For a two-sphere of radius $a$, the site convention gives positive Gaussian curvature $K=1/a^2$ and $R=2/a^2$. More generally, in $n$ dimensions,

$$R_{\rho\sigma\mu\nu} =K\bigl( g_{\rho\mu}g_{\sigma\nu} -g_{\rho\nu}g_{\sigma\mu} \bigr),$$

so

$$R_{\mu\nu}=(n-1)Kg_{\mu\nu}, \qquad R=n(n-1)K.$$

This test distinguishes $\kappa$ because reversing the curvature definition reverses the displayed sign of $K$ and $R$. Constant-curvature and infinitesimal-loop checks are developed in Wendl 2021–2022, §§ 25–31, pp. 212–278. The benchmark is local: it fixes the sign and contractions but does not identify global topology. Distinct constant-curvature quotients can share the same local tensor formula.

Contracted Bianchi identity. With the definitions above,

$$\nabla^\mu G_{\mu\nu}=0.$$

The contraction follows from the differential Bianchi identity; see Tong 2019, § 3.4, Eqs. (3.40)–(3.41), pp. 131–132 and Carroll 1997, § 3, Eqs. (3.90)–(3.96), pp. 81–82. It checks that contractions and coefficients are internally consistent, but it cannot by itself determine $\kappa$: both $G_{\mu\nu}$ and $-G_{\mu\nu}$ are divergence free. Pair it with the signed sphere test.

Where this map stops

The table assumes a Levi-Civita connection, constant $\sigma$ and $\kappa$, and unchanged coordinates and tensor index order. A position-dependent conformal rescaling is not a signature switch; its derivatives change the connection. With torsion, common commutator conventions can contain an additional torsion-dependent derivative term. With nonmetricity, raising and lowering need not commute with covariant differentiation, and the connection is not fixed by the metric formula above. In either setting, state the connection, torsion, nonmetricity, curvature operator, and contraction rules explicitly rather than extending this table by analogy.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” arXiv:gr-qc/9712019v1, 1997. Open PDF.
• Tong, David. General Relativity. Cambridge Part III Mathematical Tripos lecture notes, 2019. Official PDF.
• Wendl, Chris. Differential Geometry I and II. Humboldt-Universität zu Berlin lecture notes, 2021–2022. Open PDF.