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

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

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.

For a nondegenerate metric tensor whose signature is fixed during the variation, the Jacobi formula gives $\delta \det g=(\det g)g^{ab}\delta g_{ab}$. Differentiating the square root of the absolute determinant proves the displayed identity. Expressed through the variation of inverse metric, the formula is $-\sqrt{|\det g|}{g}_{ab}\delta g^{ab}/2$. Absolute values accommodate either Riemannian or Lorentzian signature.

Differentiating $g^{ac}{g}_{cb}={\delta }_b^a$ and multiplying by the inverse gives the displayed variation. It is the matrix inverse derivative, expressed with the covariant and contravariant components of a metric tensor.

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 $\mu =-\beta$.

For focal distance $a>0$, $\xi \ge 0$ and $0\le \theta \le \pi$, the displayed prolate spheroidal coordinates are orthogonal coordinates. The three coordinate tangent vectors have lengths $h_{\xi }=h_{\theta }=a\sqrt{{\sinh }^2\xi +{\sin }^2\theta }$ and $h_{\varphi }=a\sinh \xi \sin \theta$. Their product gives the Jacobian determinant $a^3({\sinh }^2\xi +{\sin }^2\theta )\sinh \xi \sin \theta$. A positive constant $\xi$ gives a prolate spheroid with equatorial semiaxis $a\sinh \xi$, polar semiaxis $a\cosh \xi$, and foci $(0,0,\pm a)$. The surface degenerates to the focal segment at $\xi =0$; this coordinate degeneracy is distinct from a singularity of the surrounding Euclidean space.

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 spacetime metric is the Lorentzian metric on a spacetime. It determines proper time along timelike paths and the null directions that describe local light propagation.

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.

In the exterior of Schwarzschild spacetime, $d{r}_{\ast }/dr=(1-2M/r{)}^{-1}$. Outgoing radial null geodesics therefore have constant $u=t-r_{\ast }$. At fixed receiver radius $R$, their arrival Schwarzschild time is $u+r_{\ast }(R)$, so differences in $u$ give arrival-time differences. A static receiver's proper time includes the factor $\sqrt{1-2M/R}$.

// Target: optics.bigb

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.

The timelike cone is the open set of nonzero tangent vectors with timelike squared norm. It has future and past components once a time orientation is chosen. Homogeneity extends an identity for unit timelike vectors to this whole open set. In particular, if a symmetric quadratic form vanishes on every timelike vector, its polynomial vanishes on an open set and therefore vanishes identically. This observation proves the converse from geodesic first integrals to the Killing equation.

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

Set $x=a\cosh \rho \cos u\cos v$, $y=a\cosh \rho \cos u\sin v$, $z=a\sinh \rho \sin u$, with $a>0$, $\rho \ge 0$ and $|u|\le \pi /2$. Differentiating the position vector gives pairwise orthogonal coordinate tangents and the displayed scale factors of orthogonal coordinates. Thus the positive volume density is $a^3\cosh \rho \cos u({\sinh }^2\rho +{\sin }^{2}u)$. The ordered Jacobian determinant is negative for the order $(\rho ,u,v)$; volume uses its absolute value. Positive constant $\rho$ surfaces are oblate spheroids. For $0<|u|<\pi /2$, constant $u$ gives the half of the one-sheet hyperboloid $(x^2+y^2)/(a^2{\cos }^{2}u)-z^2/(a^2{\sin }^{2}u)=1$ with the sign of $z$ fixed by $u$. At $\rho =0$ the coordinates cover the equatorial disk, while $u=0$ covers the exterior equatorial plane.

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.