Understanding the Metric Tensor: Definition, Equations, and Properties

[Greg Bernhardt]

Definition/Summary

The metric tensor $$g_{\mu\nu}$$ is a 4x4 matrix that is determined by the curvature and coordinate system of the spacetime

Equations

The proper time is given by the equation
$$d\tau^2=dx^{\mu}dx^{\nu}g_{\mu\nu}$$
using the Einstein summation convention

It is a symmetric tensor meaning that
$$g_{\mu\nu}=g_{\nu\mu}$$

The contravariant version of the metric is the inverse of the covariant metric
$$g_{\mu\nu}g^{\nu\lambda}=\delta_{\mu}^{\lambda}$$
where
$$\delta_{\mu}^{\nu}=\left\{\begin{array}{cc}0,&\mbox{ if }<br /> \mu\neq\nu\\1, & \mbox{ if } \mu=\nu\end{array}\right.$$
In Cartesian coordinates and flat space-time
$$g_{\mu\nu}=\eta_{\mu\nu}$$
where
$$\eta_{\mu\nu}=\left\{\begin{array}{cc}0, & \mbox{ if }<br /> \mu\neq\nu\\-1, & \mbox{ if } \mu=\nu & \mu,\nu=1,2,3\\1, & \mbox{ if } \mu=\nu & \mu,\nu=0 \end{array}\right.$$
The Christoffel symbols are defined by
$$\Gamma^{\lambda}_{\mu\nu}=\frac{1}{2}g^{\lambda\rho}\displaystyle{(}\frac{\partial g_{\rho\mu}}{\partial x^{\nu}}+\frac{\partial g_{\rho\nu}}{\partial x^{\mu}}-\frac{\partial g_{\mu\nu}}{\partial x^{\rho}}\displaystyle{)}$$

Extended explanation



* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[hideelo]

Is it correct to say that the metric tensor is the choice of isomorphism between the tangent and cotangent space? This is equivalent to saying that it is our tensor of choice for raising and lowering indices. If I were to start with this definition, would everything else follow?

 

[andrewkirk]

Just a minor point: 'a metric tensor' is a more general concept than the 4 x 4 item described above. Every Riemannian or pseudo-Riemannian manifold has a metric tensor, and the dimension will be n x n, where n is the dimension of the manifold. To avoid confusing students into thinking that 'metric tensor' is a term specific to GR, or that metric tensors have to be 4 x 4, it might be better to re-title the article as 'What is the metric tensor of space-time?' or 'What is Einstein's metric tensor?'