2016-07-02

# Time as a Derived Quantity

Some time ago I wrote down an interesting derivation of the local time in a box filled with photons to be the same as the average delta velocity between the photons in the box. In particular, local time $d\tau$ is related to the velocity $\v{B}$ of the box and the average velocity $\av$ of the photons as follows: \begin{equation} \left(\frac{c\,d\tau(t)}{dt} \right)^2 = c^2- |\Vec{v}_B|^2 \approx c^2 - \av^2 = \frac{1}{n^2} \sum_{i<j} (\v{i}-\v{j})^2 , \label{eq:start} \end{equation} where the $\v{i}$ are the velocities of the photons in the box, of course with $|\v{i}|=c$, the speed of light.

While I find this already quite remarkable, I think it is possible to go even one step further and, at least formally, get rid of the coordinate time $t$ alltogether. If we call $\vx{i}$ the position of a photon in the box, then clearly $\v{i}=d\vx{i}/dt$. Inserting this into equation \eqref{eq:start}, we get \begin{equation*} \left(\frac{c\,d\tau(t)}{dt} \right)^2 \approx \frac{1}{n^2} \sum_{i<j} \left( \frac{d\vx{i}-d\vx{j}}{dt} \right)^2 \end{equation*} Lets define $\vdel_{ij}=\vx{i}-\vx{j}$. If we invoke the notion of the differential of a function, we can further simplify this last equation to: \begin{equation} c^2 d\tau^2 \approx \frac{1}{n^2} \sum_{i<j} (d\vdel_{ij})^2 \end{equation} This can be interpreted as saying that (the square of) small increments of the local time, $d\tau$, in a box of photons, is equal to the average of (the square of) small changes in the mutual distance of the photons in the box.

I wonder if this has anything to do with the timeless physics that Barbour has in mind. The formula seems to indicate that the flow of local time is just a secondary effect of a bunch of photons not all having identical velocity vectors. Because if they have, $d\tau$ becomes zero.