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Definition/Summary
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
* 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!
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
* 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!