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The metric tensor [tex]g_{\mu\nu}[/tex] 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

[tex]d\tau^2=dx^{\mu}dx^{\nu}g_{\mu\nu}[/tex]

using the Einstein summation convention

It is a symmetric tensor meaning that

[tex]g_{\mu\nu}=g_{\nu\mu}[/tex]

The contravariant version of the metric is the inverse of the covariant metric

[tex]g_{\mu\nu}g^{\nu\lambda}=\delta_{\mu}^{\lambda}[/tex]

where

[tex]\delta_{\mu}^{\nu}=\left\{\begin{array}{cc}0,&\mbox{ if }

\mu\neq\nu\\1, & \mbox{ if } \mu=\nu\end{array}\right.[/tex]

In Cartesian coordinates and flat space-time

[tex]g_{\mu\nu}=\eta_{\mu\nu}[/tex]

where

[tex]\eta_{\mu\nu}=\left\{\begin{array}{cc}0, & \mbox{ if }

\mu\neq\nu\\-1, & \mbox{ if } \mu=\nu & \mu,\nu=1,2,3\\1, & \mbox{ if } \mu=\nu & \mu,\nu=0 \end{array}\right.[/tex]

The Christoffel symbols are defined by

[tex]\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{)}[/tex]

Extended explanation

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# What is a Metric Tensor

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