## General Relativity Primer

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

${\begin{matrix}x^{\prime }=\gamma (x-vt),&y^{\prime }=y,&z^{\prime }=z,&t^{\prime }=\gamma \left(t-{v \over c^{2}}x\right)\end{matrix}}\,\!$
where $\gamma \equiv {1 \over {\sqrt {1-{v^{2} \over c^{2}}}}}$. Note, to give a taste of tensor forms, this all may be written as ${x^{\prime }}^{\alpha }=\Lambda _{\beta }^{\alpha }x^{\beta }+I_{0}^{\alpha }$.

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

$ds^{2}=-c^{2}dt^{2}+(dx^{2}+dy^{2}+dz^{2})\,\!$
Light travels a $ds^{2}=0$ path. In tensor form, this equation looks like:

$ds^{2}=\eta _{\alpha \beta }dx^{\alpha }dx^{\beta }\,\!$
where $\eta _{\alpha \beta }$, the metric tensor, is given by:

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