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# Measurements in special relativity

## The photon bounce

Let's consider two observers, along curves $x^\mu(\tau)$ and $y^\mu(\sigma)$. We also define the function of the spacetime interval between any two points of those curves by

$$d(\sigma, \tau) = \eta_{\mu\nu} (x^\mu(\tau) - y^\mu(\sigma)) (x^\nu(\tau) - y^\nu(\sigma))$$

The derivatives of our distance function are

\begin{eqnarray} \frac{d}{d\sigma} d(\sigma, \tau) &=& -2\eta_{\mu\nu} \dot{y}^\mu(\sigma) (x^\nu(\tau) - y^\nu(\sigma))\\ \frac{d}{d\tau} d(\sigma, \tau) &=& 2\eta_{\mu\nu} \dot{x}^\mu(\sigma) (x^\nu(\tau) - y^\nu(\sigma))\\ \end{eqnarray}
Last updated : 2020-03-24 13:52:04
Tags : physics , special-relativity