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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.

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.

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.

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 $K=-C$.

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.

Simultaneously varying a metric tensor and a scalar field by the Lie derivative along a compactly supported vector field leaves a covariant matter action invariant. Integration by parts gives the displayed Noether identity. It implies stress-energy conservation when the scalar equation holds without needing to impose the metric field equation.

Geometrized units set $G=c=1$, allowing mass, length, and time to be expressed in common units. Restore $GM/c^2$ for a mass parameter in a radial metric tensor.

With the Riemann curvature tensor convention $R^{\mu }{}_{\nu \rho \sigma }={\partial }_{\rho }{\Gamma }^{\mu }{}_{\nu \sigma }-{\partial }_{\sigma }{\Gamma }^{\mu }{}_{\nu \rho }+\cdots$, harmonic coordinates remove the mixed second derivatives from the Ricci tensor. The remainder $Q_{ab}$ contains at most first derivatives of the metric tensor. The vacuum Einstein field equations become a system of quasilinear partial differential equations with a Lorentzian wave equation principal part. Harmonic coordinate constraints must be maintained when using the reduced system.

The induced worldvolume metric is the pullback of the target metric tensor by a brane embedding. For a nondegenerate timelike embedding, $\sqrt{-\det h}$ is its invariant worldvolume volume density. The $p=1$ specialization is the induced worldsheet metric.

For a nondegenerate metric tensor, differentiating $g_{ac}{g}^{cb}={\delta }_a{}^b$ gives $\delta g_{ab}=-g_{ac}{g}_{bd}\delta g^{cd}$. Combining this with the Jacobi determinant derivative formula proves the displayed volume-density variation. It is the determinant contribution when defining a stress-energy tensor by varying the inverse metric tensor.

An isometric extension of a connected metric manifold is a proper embedding onto an open subset of a larger connected manifold of the same dimension, preserving the metric tensor. Smooth complete connected Riemannian manifolds admit no such proper extension; Lorentzian geodesic completeness has different subtleties.

The isometry group of a metric space consists of its bijective isometries, with composition as the group operation. For a Riemannian metric, these are the diffeomorphisms preserving the metric tensor. Right translations of a right-invariant Riemannian metric supply a faithful subgroup isomorphic to the underlying Lie group.

For a differentiable invertible matrix $A(t)$, $\frac{d}{dt}\log \det A=\mathrm{tr}(A^{-1}\dot{A})$ wherever a consistent logarithm is defined. In particular a positive spatial metric tensor satisfies $h^{ij}{\dot{h}}_{ij}=\dot{h}/h$. This separates its local volume change from shape change.

Where a Killing vector field is ${\partial }_x$ and its squared norm $V=h_{xx}$ is nonzero, an invariant metric tensor decomposes as $h=V(dx+A{)}^2+\gamma$. The quotient one-form has $A_i=h_{xi}/V$ and the quotient metric has ${\gamma }_{ij}=h_{ij}-h_{xi}{h}_{xj}/V$. All coefficients are independent of $x$. Changing the fibre coordinate to $x^{\prime }=x+\lambda$ sends $A$ to $A-d\lambda$. A null Killing direction requires a different decomposition.

A local inertial frame is a coordinate frame in which, at one chosen spacetime event, the metric tensor is Minkowskian and the Levi-Civita connection vanishes. Freely falling experiments confined to a sufficiently small neighbourhood therefore obey the laws of special relativity up to tidal corrections.

A Lorentzian metric is a smooth nondegenerate symmetric metric tensor with one negative and all other positive directions, or its opposite-sign convention. At each point, a nonzero tangent is timelike, null or spacelike according as $g(v,v)$ is negative, zero or positive in the first convention. Unlike a Riemannian metric, it gives a light cone rather than a positive norm. Minkowski spacetime is its constant flat model, and proper time along a timelike curve is $\int \sqrt{-g(\dot{\gamma },\dot{\gamma })}du$.

Metric compatibility is ${\nabla }_{\alpha }{g}_{\mu \nu }=0$, allowing the metric to pass through covariant derivatives and index operations.

The metric signature records the signs of the diagonalized nondegenerate metric tensor, equivalently its numbers of negative and positive directions. For a four-dimensional Lorentzian manifold, both $(-,+,+,+)$ and its overall negative convention are common. The numerical signs of contractions and component Hodge star operator identities must be consistent with that choice and the orientation.

A real scalar field is minimally coupled to a metric tensor when its kinetic term uses the metric contraction of first derivatives and its scalar potential depends only on the field, without an explicit scalar curvature coupling. For metric signature $(-,+,+,+)$ the displayed Lagrangian and volume density $\sqrt{-g}$ give $\square \varphi -V^{\prime }(\varphi )=0$. Its stress-energy tensor is $T_{ab}={\nabla }_a\varphi {\nabla }_b\varphi -g_{ab}[\frac{1}{2}(\nabla \varphi {)}^2+V]$. The scalar stress-energy divergence identity explains its conservation on the field equation.