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,&amp;\mbox{ if }<br /> \mu\neq\nu\\1, &amp; \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, &amp; \mbox{ if }<br /> \mu\neq\nu\\-1, &amp; \mbox{ if } \mu=\nu &amp; \mu,\nu=1,2,3\\1, &amp; \mbox{ if } \mu=\nu &amp; \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



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[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?'