Stress-energy Tensor
Tags: #physics #stress-energy tensorEquation
$$T_{\mu v}=(\varrho c^{2}+p)u_{p}u_{v}+pg_{\mu v}+\frac{1}{c^{2}}(F^{\mu}_{\alpha}F^{\alpha v} + \frac{1}{4}g^{\mu v}F^{\alpha\beta}F_{\alpha\beta}) \\ \triangledown_{v} T_{\mu v}=0 \\ F_{\alpha\beta}=\frac{\partial{A_{\beta}}}{\partial{x^{\alpha}}} - \frac{\partial{A_{\alpha}}}{\partial{x^{\beta}}} \\ \frac{d p_{\alpha}}{d \tau}=qF_{\alpha\beta}u^{\beta}$$Latex Code
T_{\mu v}=(\varrho c^{2}+p)u_{p}u_{v}+pg_{\mu v}+\frac{1}{c^{2}}(F^{\mu}_{\alpha}F^{\alpha v} + \frac{1}{4}g^{\mu v}F^{\alpha\beta}F_{\alpha\beta}) \\
\triangledown_{v} T_{\mu v}=0 \\
F_{\alpha\beta}=\frac{\partial{A_{\beta}}}{\partial{x^{\alpha}}} - \frac{\partial{A_{\alpha}}}{\partial{x^{\beta}}} \\
\frac{d p_{\alpha}}{d \tau}=qF_{\alpha\beta}u^{\beta}
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Introduction
Latex code for the stress-energy tensor and the field tensor. I will briefly introduce the notations in this formulation.
- The stress-energy tensor is given by:
- The conservation laws can than be written as:
. And the electromagnetic field tensor is given by
.
- The equations of motion for a charged particle in an EM field become with the field tensor:
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