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

\begin{equation} \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} \end{equation}

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

$$\mathrm ds^2 = - \gamma'(r',t)c^2\mathrm dt^2 + \beta'(r',t)c\mathrm dr\mathrm dt + \alpha(r,t)[\mathrm dr^2 + r^2 (\mathrm d\theta^2 + \sin^2\theta \mathrm d\phi^2)]$$

Then compose the integral multiplier

$$c\mathrm dt'= \eta(r',t) [ - \gamma'(r',t) c \mathrm dt + \frac{1}{2} \beta'(r',t)\mathrm dr']$$

And finally,

$$\mathrm ds^2 = -\eta^{-2}(r',t) \gamma'^{-1}(r',t)c^2\mathrm dt'^2 + [\alpha'(r',t) + \frac{\beta'^2(r',t)}{4r'} ]\mathrm dr'^2 + r'^2(\mathrm d\theta^2 + \sin^2\theta\mathrm d\phi^2)$$

In general

\begin{equation} \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} \end{equation}

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.

Frame and Coordinates