# What Frame Are We In

Synge once said, use space and time, and define them.

This post is aimed to make clear what frame are we in.

## Intro

In general relativity, we often transform coordinates. Here is an example.

The general form of metric with spherical space component is

$$\mathrm ds^2 = - \gamma(r,t)c^2\mathrm dt^2 + \beta(r,t)c\mathrm dr\mathrm dt + \alpha(r,t)[\mathrm dr^2 + r^2 (\mathrm d\theta^2 + \sin^2\theta \mathrm d\phi^2)]\label{MetricForm1}$$

With a transformation $\alpha(r,t)r^2 = r’^2$,

Then compose the integral multiplier

And finally,

In general

$$\mathrm ds^2 = -b(r,t)c^2\mathrm dt^2 + a(r,t)\mathrm dr^2 + r^2(\mathrm d\theta^2 + \sin^2\theta\mathrm d\phi^2) \label{MetricForm2}$$

Then what? The two forms of metric demonstrate different properties. Take Birkhoff theorem as an example. The results could be very different startting from the form (\ref{MetricForm1}) and (\ref{MetricForm2}).

It is obviously very important to show what the coordinate transformation means and what frame are the observers in indicated by the coordinates.