Metric

A metric is a symmetric non-degenerate (0,2) tensor field:

• Symmetric: The indices can be interchanged since $g_{\mu \nu }=g_{\nu \mu }$
• Non-degenerate: The determinant of a metric is non-zero; the inverse exists. So, the eigenvalues of a metric (as an nxn matrix) are non-zero.

The pattern of +ve and -ve eigenvalues is called a metric’s signature.

• (++++) - Riemannian Signature
• (-+++) - Lorentzian Signature

Although the values of the eigenvalues depend on the choice of coordinate system, Sylvester’s Law of Inertia says that the signature is coordinate-independent.

Uses of a metric:

Defines the inner product on Tₚ

$g:T_p\times T_p\to \mathbb{R}$

$(V,W)\equiv g(V,W)=g_{\mu \nu }{V}^{\mu }{W}^{\nu }$

Maps vectors to co-vectors

$g:T_p\to T_p^{\ast },{\omega }_{\mu }=g_{\mu \nu }{V}^{\nu }$

This map has an inverse too, with a (2,0) tensor called the inverse metric defined such that $g^{\mu \nu }{g}_{\nu \lambda }={\delta }_{\lambda }^{\mu }$ More generally, from tensors of type (r, s) to (r+1, s-1) and (r-1, s+1), called raising and lowering indices.

Defines a notion of length

We can define the line element as $d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$and hence the proper length and proper time are

$l=\int \sqrt{g_{\mu \nu }{\mathring{x}}^{\mu }{\mathring{x}}^{\nu }}d\lambda$

$\tau =\int \sqrt{-g_{\mu \nu }{\mathring{x}}^{\mu }{\mathring{x}}^{\nu }}d\lambda$

Examples:

1. TheMinkowski metric - Although this is not a tensor equation since it is not a tensor under general coordinate transformations.
2. Unit Sphere with the “round” metric - Obtained by restricting the flat metric on the 3-d real space to a unit sphere, i.e.,

$x=sin\theta cos\varphi ,y=cos\theta sin\varphi ,z=cos\theta$

$d{s}^2=d{\theta }^2+si{n}^2\theta d{\varphi }^2$ with $g_{\mu \nu }=\left(\begin{array}{cc}1& 0\\ 0& si{n}^2\theta \end{array}\right)$ 3. Poincare Disc - $d{s}^2=\frac{4}{(1-r^2{)}^2}(d{r}^2+r^{2}d{\varphi }^2)$

This arises as a hyperbolic surface $-t^2+x^2+y^2=-1$ in 3dMinkowski space.

A simple coordinate system on this surface is

$t=cosh\rho ,x=sinh\rho cos\varphi ,y=sinh\rho sin\varphi$

$d{s}^2=d{\rho }^2+sin{h}^2\rho d{\varphi }^2$