# General Relativity Primer

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## General Relativity Primer

Lorentz invariance dictates that two inertial frame ${\displaystyle (x,y,z,t)}$ and ${\displaystyle (x^{\prime },y^{\prime },z^{\prime },t^{\prime })}$, with one moving with respect to the other at velocity ${\displaystyle {\hat {v}}=v{\hat {x}}}$, are related by:

${\displaystyle {\begin{matrix}x^{\prime }=\gamma (x-vt),&y^{\prime }=y,&z^{\prime }=z,&t^{\prime }=\gamma \left(t-{v \over c^{2}}x\right)\end{matrix}}\,\!}$

where ${\displaystyle \gamma \equiv {1 \over {\sqrt {1-{v^{2} \over c^{2}}}}}}$. Note, to give a taste of tensor forms, this all may be written as ${\displaystyle {x^{\prime }}^{\alpha }=\Lambda _{\beta }^{\alpha }x^{\beta }+I_{0}^{\alpha }}$.

Remember the Lorentz invariant interval, which is conserved between frames:

${\displaystyle ds^{2}=c^{2}dt^{2}-(dx^{2}+dy^{2}+dz^{2})\,\!}$

Light travels a ${\displaystyle ds^{2}=0}$ path. In tensor form, this equation looks like:

${\displaystyle ds^{2}=\eta _{\alpha \beta }dx^{\alpha }dx^{\beta }\,\!}$

where ${\displaystyle \eta _{\alpha \beta }}$, the metric tensor, is given by:

${\displaystyle \eta _{\alpha \beta }\equiv {\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\\\end{pmatrix}}\,\!}$