Metric tensor ($g_{\mu \nu }$)

A metric tensor is a nondegenerate symmetric covariant tensor that defines squared line elements by $d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$.

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.

For $0<\alpha <\beta$, define $-\beta \le \mu \le -\alpha \le \lambda$ by $R^2/(\tau +\alpha )+z^2/(\tau +\beta )=1$. Then $R^2=(\lambda +\alpha )(\mu +\alpha )/(\alpha -\beta )$ and $z^2=(\lambda +\beta )(\mu +\beta )/(\beta -\alpha )$. Constant $\lambda$ surfaces are confocal prolate spheroids and constant interior $\mu$ surfaces are two-sheeted hyperboloids, with foci $z=\pm \sqrt{\beta -\alpha }$. The coordinates are orthogonal away from their degeneracies. The squared scale factors are $h_{\lambda }^2=(\lambda -\mu )/[4(\lambda +\alpha )(\lambda +\beta )]$, $h_{\mu }^2=(\mu -\lambda )/[4(\mu +\alpha )(\mu +\beta )]$ and $h_{\varphi }^2=R^2$. The two signs of $z$ are coordinate branches, joined at the equatorial fold $\mu =-\beta$.

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 $\int \sqrt{-g(\dot{\gamma },\dot{\gamma })}du$.

A null coordinate is a coordinate function $u$ whose gradient is null, $g^{ab}{\partial }_{a}u{\partial }_{b}u=0$. Its level sets are null hypersurfaces. Pairs of such coordinates straighten radial null directions in two-dimensional orbit metrics; a monotone reparametrization of either function preserves its null level sets. Examples include the retarded and advanced null coordinates and the radial Kruskal–Szekeres coordinates.

With metric signature $(-,+,+,+)$, a nonzero vector $v$ is spacelike when its Lorentzian inner product with itself satisfies $g(v,v)>0$. In Minkowski spacetime, this means its spatial components have larger squared Euclidean length than its time component. The classification is preserved by Lorentz transformations. A timelike vector has negative squared Lorentzian inner product, and a null vector has zero squared Lorentzian inner product.

A Lorentzian inner product is a nondegenerate symmetric bilinear form with one negative and the remaining positive eigenvalues in the mostly-plus convention. Two nonzero future causal vectors have nonpositive inner product; if either is timelike, it is strictly negative.

A nonzero covector $\ell$ is null when its squared dual Lorentzian inner product is zero: $g^{\alpha \beta }{\ell }_{\alpha }{\ell }_{\beta }=0$. Raising its index gives a null vector. Null covectors describe normals to characteristic light cones and are the test directions in the classical null condition for wave equations.

A nonzero causal vector is timelike or null: $g(v,v)\le 0$ with metric signature $(-,+,+,+)$. The zero vector is commonly included when stating the dominant energy condition.

A timelike vector $v$ obeys $g(v,v)<0$ in metric signature $(-,+,+,+)$. Its orthogonal complement has positive-definite metric; the two timelike cones give the local future and past choices.

A timelike vector field assigns a timelike vector continuously or smoothly to every event in its domain. A globally defined such field gives a time orientation, but it need not be a Killing vector field or a timelike geodesic vector field.

With parameter $a\ge 0$, oblate spheroidal coordinates on Euclidean three-space areWriting $\Sigma =r^2+a^2{\cos }^2\theta$ and $\Delta =r^2+a^2$, their spatial line element is

Starting from the Minkowski metric in oblate spheroidal coordinates, set $u=t-r$, $R=r$, and choose an azimuth $\phi$ satisfyingThenso the coordinate vector ${\partial }_R$ is null.

The inverse metric satisfies $g^{\mu \rho }{g}_{\rho \nu }={\delta }_{\nu }^{\mu }$ and raises covariant tensor indices.