# A “local clock time” coordinate system outside spherical mass

With an appropriate coordinate change from Schwarzschild coordinates, is it possible to get the metric in the following form?

$$ds^2 = -dt^2 + A(r) dr^2 + B(r)\ r^2 \left( d\theta^2 + \sin^2 \theta \ d\phi^2 \right) + dt \left[ D(x^\mu) dr + E(x^\mu) d\theta + F(x^\mu) d\phi \right]$$

The idea being that the time coordinate would agree with the time measured by a clock sitting at constant spatial coordinates. Hopefully by pushing some of the issues with clock synchronization over into time's off diagonal components.

The Schwarzschild coordinate chart doesn't cover the whole space-time, so if this only works for some finite range or something, that would be fine.

I have a feeling if such a coordinate system was possible, it would already be known and named after someone. So please let me know if there is a standard name for this, or if it turns out to be not possible for some reason, please help me understand why this is not possible.

Here is the best answer I can work out at the moment. No, it does not appear possible to choose a coordinate system with all the desired properties.

Starting with the Schwarzschild coordinates $(t,r,\phi,\theta)$, the coordinate components of the metric are:

$$g_{\mu\nu} = \begin{bmatrix} 1-2M/r & 0 & 0 & 0 \\ 0 & \frac{-1}{1-2M/r} & 0 & 0 \\ 0 & 0 & -r^2 & 0 \\ 0 & 0 & 0 & -r^2 \sin(\theta) \end{bmatrix}$$

Conceptually, I can imagine using the spatial coordinates from the Schwarzschild chart, using the initial $t=0$ time slice, and then using clocks at constant spatial coordinates to define the time coordinate for events. This would give us our new coordinate system. And that would make $g_{00}$ have to be -1.

This change in time coordinates would only agree on simultaneity at $t=0$, so measuring a path along $d\bar{r}$ with $d\bar{t}=0$, is likely to now have a length depending on $\bar{t}$ unless fortuitous cancellation can be arranged.

To allow more freedom to hopefully help that cancellation, we can generalize to consider a different slice for time $\bar{t}=0$, which would correspond to offsetting each "sphere" of clocks by some amount depending only on $\bar{r}$. So the coordinate transformation is:

$$t = f(\bar{r})\bar{t} + h(\bar{r}), \ r = \bar{r}, \phi = \bar{\phi}, \theta = \bar{\theta} \\ \text{where } f(\bar{r}) = \frac{1}{\sqrt{ 1 - 2 M / \bar{r}}}$$

The coordinate change has only two interesting components of the Jacobian:

$$\frac{\partial t}{\partial \bar{t}} = f(\bar{r})$$ $$\frac{\partial t}{\partial \bar{r}} = \bar{t} \frac{\partial f(\bar{r})}{\partial \bar{r}} + \frac{\partial h(\bar{r})}{\partial \bar{r}} = - \bar{t} \frac{M}{\bar{r}^2} f^3(\bar{r}) + \frac{\partial h(\bar{r})}{\partial \bar{r}}$$

The coordinate components of the metric are:

$$g_{\bar{\mu}\bar{\nu}} = \frac{\partial x^\mu}{\partial x^\bar{\mu}} \frac{\partial x^\nu}{\partial x^\bar{\nu}} g_{\mu\nu}$$

This is block diagonal, with the $\phi$ and $\theta$ terms unaffected. So only the $\bar{r}$, $\bar{t}$, and cross terms need to be inspected.

Evaluating the $(d\bar{t})^2$ term shows the local clock time requirement is met:

$$g_{\bar{0}\bar{0}} = \frac{\partial x^\mu}{\partial x^\bar{0}} \frac{\partial x^\nu}{\partial x^\bar{0}} g_{\mu\nu} = \left( \frac{\partial t}{\partial \bar{t}} \right)^2 g_{00} = 1$$

The remaining requirement is to get the $(d\bar{r})^2$ term to only depend on the radial coordinate.

$$g_{\bar{1}\bar{1}} = C(\bar{r})$$

$$g_{\bar{1}\bar{1}} = \frac{\partial x^\mu}{\partial x^\bar{1}} \frac{\partial x^\nu}{\partial x^\bar{1}} g_{\mu\nu} = \left( \frac{\partial t}{\partial \bar{r}} \right)^2 g_{00} + \left( \frac{\partial r}{\partial \bar{r}} \right)^2 g_{11} =$$

$$\left(- \bar{t} \frac{M}{\bar{r}^2} f^3(\bar{r}) + \frac{\partial h(\bar{r})}{\partial \bar{r}}\right)^2 \frac{1}{f^2(\bar{r})} - f^2(\bar{r})$$

But no choice of $h(\bar{r})$ can counter the time dependence. No coordinate transformation can prevent a similar fate if it only redefines the time coordinate and has the spatial coordinates independent of the new time coordinate.

All non-zero components of the Schwarzschild metric depend on the radial coordinate, so considering a coordinate transformation with time dependence in the radial coordinate would result in time dependence in all of the non-zero metric components.

So it is not possible to put the metric in the desired form.