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The equivalence principle allows Riemann normal coordinates in which the first derivatives of the metric tensor vanish at a chosen point. A purported local gravitational stress-energy tensor built from those first derivatives would vanish there in that chart and therefore, if tensorial, in all charts. Such an expression cannot encode the gravitational energy assigned by a coordinate-dependent field description. This rules out an ordinary universal local covariant gravitational energy density, not the construction of local curvature tensors or boundary and averaged notions of gravitational energy. The Riemann curvature tensor still describes invariant tidal gravity.

Radiating material close to a compact object probes relativistic orbital motion, gravitational redshift and light propagation. A thin disk's inferred inner edge can be compared with the predicted innermost stable circular orbit, while iron-line transfer and thermal-continuum fitting probe complementary aspects of the same geometry. Agreement tests a combined spacetime-and-disk model. A black-hole spin measurement made after imposing a Kerr metric is not alone an independent verification of that metric tensor; geometry tests must vary gravitational predictions and astrophysical nuisance parameters together.

Relative to a nondegenerate metric tensor, any affine connection is uniquely the Levi-Civita connection plus a contorsion tensor contribution and a disformation tensor contribution. In derivative-last notation, with $T_{ab}{}^c={\Gamma }^c{}_{ab}-{\Gamma }^c{}_{ba}$ and $N_{abc}={\nabla }_c{g}_{ab}$, the lowered difference is $A_{abc}=\frac{1}{2}(T_{abc}+T_{bca}-T_{cab}+N_{abc}-N_{bca}-N_{cab})$. This follows by solving $T_{abc}=A_{abc}-A_{bac}$ and $N_{abc}=-A_{acb}-A_{bca}$. The difference of affine connections is a tensor.

An analytic extension embeds a spacetime isometrically as a proper open region in a larger analytic Lorentzian manifold with a nondegenerate metric tensor. A divergence of coordinate components does not obstruct extension if a new chart makes all metric coefficients analytic and the determinant nonzero. For a static spherical metric with an analytic lapse $f$ and a positive simple horizon radius, ingoing coordinates replace $-fd{t}^2+d{r}^2/f$ by $-fd{v}^2+2dvdr$. This removes the coordinate singularity at $f=0$; transverse metric coefficients must also remain regular. An extension across such a horizon is distinct from extending through a curvature singularity.

Geometric or field data held fixed when a theory's dynamical degrees of freedom vary. The Minkowski metric is background structure in ordinary special relativity; the spacetime metric tensor is dynamical in general relativity. A fixed timelike vector field can introduce a preferred direction even when an equation is written using tensor notation.

The simply connected Bertotti-Robinson geometry is the product of the universal cover of two-dimensional anti-de Sitter spacetime and a two-sphere with equal radius $m>0$. In metric signature $(-,+,+,+)$,Its radial Penrose diagram is an infinite conformal strip of two-dimensional anti-de Sitter spacetime. The metric tensor $-{\lambda }^{2}d{t}^2+m^2{\lambda }^{-2}d{\lambda }^2+m^{2}d{\Omega }_2^2$ is a patch of it, with $\lambda =m(\sin T-\sin X)/\cos X$. Its Killing vector field ${\partial }_t$ extends across alternating degenerate Killing horizons $\lambda =0$. Both factors are geodesically complete, giving a maximal analytic extension of this patch without periodic time identification.

The Brans-Dicke theory adds a dynamical scalar field $\Phi$ to a metric tensor theory of gravity, with constant dimensionless coupling $\omega$. In a $(+---)$ signature and the reversed-curvature convention used here, the displayed Lagrangian is supplemented by matter and integrated against the metric tensor volume density with prefactor $1/(16\pi G)$. Its scalar multiplier of the Ricci scalar lets the curvature coupling vary in spacetime. Both the metric tensor and scalar are varied; the scalar equation of Brans-Dicke theory follows by eliminating curvature between the scalar equation and the trace of the metric tensor equation. The original construction is described in Brans and Dicke's 1961 article.

For the displayed metric tensor, its inverse has components $g^{rr}=e^{-2\chi }$ and $g^{\theta \theta }=e^{-2\chi }/r^2$. The nonzero second-kind Christoffel symbols are ${\Gamma }^r{}_{rr}={\chi }^{\prime }$, ${\Gamma }^r{}_{\theta \theta }=-r-r^2{\chi }^{\prime }$ and ${\Gamma }^{\theta }{}_{r\theta }={\Gamma }^{\theta }{}_{\theta r}=1/r+{\chi }^{\prime }$. They follow by differentiating the two diagonal metric entries and raising the first index of the first-kind connection coefficients. Polar coordinates require treating $r=0$ separately.

For a nondegenerate metric tensor and its Levi-Civita connection, contracting the Christoffel symbol formula gives ${\Gamma }^{\rho }{}_{\rho \mu }=\frac{1}{2}{g}^{ab}{\partial }_{\mu }{g}_{ab}$. The Jacobi determinant derivative formula identifies this with ${\partial }_{\mu }\log \sqrt{|\det g|}$. The volume density, rather than the determinant without an absolute value, works in every fixed metric signature.

The prescription replaces partial derivatives in flat-spacetime matter equations by the Levi-Civita connection and the Minkowski metric by the curved metric tensor. It implements minimal gravitational coupling and reproduces flat-spacetime physics at a point in a local inertial frame. It is not a unique rule for rearranged higher-derivative equations: commuting partial derivatives before replacement can conceal a Riemann curvature tensor term afterwards. The derivative-order ambiguity in covariant Maxwell potential equations is an explicit example. The equivalence principle removes uniform gravitational acceleration locally but does not erase tidal curvature over a finite experiment.

For metric tensors related by $\tilde{g}=e^{2\omega }g$, differentiating the rescaled metric tensor in the Christoffel symbol formula gives the displayed difference, where ${\omega }_a={\partial }_a\omega$ and the index on ${\omega }^a$ is raised with $g$. Consequently ${\tilde{\nabla }}_{V}V={\nabla }_{V}V+2V(\omega )V-g(V,V){\mathrm{grad}}_g\omega$. The last term vanishes on a null vector, proving conformal preservation of null geodesic paths; for a non-null geodesic its orthogonal component generally changes the path.

Conformal geometry studies a metric tensor only up to multiplication by a nowhere-zero positive scalar function on a real slice. A conformal class determines angles in Riemannian signature and null directions in Lorentzian signature. The complexified flat four-dimensional conformal geometry is encoded by the Klein quadric, whose incidence relation describes null separation.

For a metric-compatible affine connection, its difference from the Levi-Civita connection is the contorsion tensor. It is determined by the torsion tensor. In derivative-last notation let $T_{ab}{}^c={\Gamma }^c{}_{ab}-{\Gamma }^c{}_{ba}$, the negative of geometric torsion, and lower the last slot with the metric tensor. Then $C_{abc}=\frac{1}{2}(T_{abc}+T_{bca}-T_{cab})$. This is the actual correction added to the Levi-Civita connection; a convention writing $\Gamma =S-K$ has $K=-C$.

For positive coordinate charts put $J=\partial y/\partial x$. The metric tensor transforms by $g_x=J^T{g}_{y}J$, giving $\det g_x=(\det J{)}^2\det g_y$. The top wedge product of differential forms transforms by $d{y}^1\wedge \cdots \wedge d{y}^n=(\det J)d{x}^1\wedge \cdots \wedge d{x}^n$. Since $\det J>0$, the two factors match. Thus the local expressions define one global Riemannian volume form.

Cosmological perturbation theory studies departures of the metric tensor and matter fields from a homogeneous and isotropic FRW metric, order by order. Linear cosmological perturbation theory keeps only first-order terms. Higher orders describe nonlinear coupling between perturbations. Nonlinear cosmological perturbation theory also uses expansions that allow finite amplitudes, such as the long-wavelength approximation in cosmology.

A space with nonzero constant sectional curvature $K$ has curvature radius of a constant-curvature space $R_c=1/\sqrt{|K|}$. Scaling a metric tensor by $a^2$ divides its sectional curvature by $a^2$. In an FLRW metric whose spatial normalization is $k=\pm 1$, the spatial curvature of an FLRW universe is $K=k/a^2$, so its scale factor equals $R_c$. For $k=0$ there is no finite intrinsic curvature length.

Let $C^a{}_{bcd}={\partial }_d{\Gamma }_{bc}^a-{\partial }_c{\Gamma }_{bd}^a-{\Gamma }_{ec}^a{\Gamma }_{bd}^e+{\Gamma }_{ed}^a{\Gamma }_{bc}^e$, with all other index positions obtained by the metric tensor. This convention is the negative of the convention $[{\nabla }_c,{\nabla }_d]V^a=R^a{}_{bcd}{V}^b$. The Ricci identity is $k_{a;bc}-k_{a;cb}=C_a{}^d{}_{bc}{k}_d$, with $k_{a;bc}={\nabla }_c{\nabla }_b{k}_a$. For a Killing vector, the Killing equation makes this second derivative antisymmetric in $a,b$. Applying the commutator three times gives $2k_{a;bc}=(C_a{}^d{}_{bc}+C_c{}^d{}_{ba}-C_b{}^d{}_{ac})k_d=-2C_{abc}{}^d{k}_d$, where the last step uses the first Bianchi identity. Switching the sign of the Riemann curvature tensor switches the sign in this identity; moving a raised index by ordinary metric contraction does not switch it.

At $\omega =-3/2$, the combined scalar equation of Brans-Dicke theory becomes the matter consistency condition $T=0$. It does not determine $\square \Phi$ by division, since the wave coefficient has vanished. The original metric tensor and scalar Euler-Lagrange equations must be retained. Introducing scalar-dependent matter or a scalar potential changes the consistency condition.

A symmetric bilinear form on a tangent space is degenerate when some nonzero vector pairs to zero with every vector; equivalently its matrix has zero determinant. At such a point there is no inverse metric tensor, and the usual metric tensor construction of a unique Levi-Civita connection does not apply. A coordinate expression that degenerates must be checked to distinguish a removable coordinate failure from a genuinely excluded point of the geometric domain.

Transforming a metric tensor and all matter fields together by the pullback of a diffeomorphism preserves the Einstein field equations and covariant matter equations. Passive coordinate changes only change the description of the fields; active pullbacks produce another representative of the geometry. The treatment of transformations acting on a boundary depends on the boundary conditions and possible charges. A diffeomorphism is not necessarily an isometry of a fixed metric.