Defining Spacetime Coordinates

[PeroK]

I was looking at the Static Weak Field Metric, which Hartle gives as:

$ds^2 = (1- \frac{2\Phi(x^i)}{c^2})(dx^2 + dy^2 + dz^2)$

For a fixed time.

Where, for example, $\Phi(r) = \frac{-GM}{r}$

I was trying to figure out how the coordinates (x, y, z) could be defined. Clearly, they can't be defined by measurements of length. Hartle says nothing about this.

I suspect that the $r$ in the second equation is probably the measurable distance, and not $(x^2 + y^2 + x^2)^{1/2}$

The best explanation I could come up with myself is that if you measured $r$ and $\Phi(r)$ at every point and knew $G$ and $M$ then you could define $x, y, z$ precisely so that the equation for $ds^2$ holds!

Does that sound right and/or can anyone shed any light on this?

 

[PeterDonis]

I was trying to figure out how the coordinates (x, y, z) could be defined.

In the usual convention, they are, heuristically, the Cartesian coordinates that would be assigned if space were flat.

I suspect that the $r$ in the second equation is probably the measurable distance

In the usual convention, it isn't, it's $\left( x^2 + y^2 + z^2 \right)^{1/2}$. This basically corresponds to the Schwarzschild coordinate definition of $r$.

I don't know for sure that Hartle is using this convention, but I suspect he is.

 

[PeroK]

In the usual convention, they are, heuristically, the Cartesian coordinates that would be assigned if space were flat.
In the usual convention, it isn't, it's $\left( x^2 + y^2 + z^2 \right)^{1/2}$. This basically corresponds to the Schwarzschild coordinate definition of $r$.

I don't know for sure that Hartle is using this convention, but I suspect he is.

In terms of coordinate time, I can imagine a remote clock in flat spacetime keeping coordinate time. Is there an equivalent for understanding how you can assign Cartesian coordinates to a curved region of spacetime as if it were flat? Projecting onto a flat coordinate map?

That would explain a lot.

 

[PeterDonis]

Projecting onto a flat coordinate map?

More or less, yes. It helps if there is spatial symmetry present; for example, in the spherically symmetric case, you can calculate the "areal radius" $r$ (i.e., the quantity $\sqrt{A / 4 \pi}$, where $A$ is the area of a 2-sphere containing a given point) and assign x, y, z coordinates using the standard conversion from spherical to Cartesian coordinates in flat space. That's what I think Hartle was doing.

 

[Ibix]

My understanding (at least for Schwarzschild coordinates) is that the rubber sheet analogy is actually helpful here. You can imagine a flat sheet with a circular polar coordinate grid drawn on it. Below it you imagine another sheet stretched as in the rubber sheet analogy (I believe the relevant surface for Schwarzschild coordinates is called Flamm's hyperboloid). Then you project the circular coordinate grid vertically downwards. Tangential distances are preserved, but radial distances are not.

 

[PeroK]

More or less, yes. It helps if there is spatial symmetry present; for example, in the spherically symmetric case, you can calculate the "areal radius" $r$ (i.e., the quantity $\sqrt{A / 4 \pi}$, where $A$ is the area of a 2-sphere containing a given point) and assign x, y, z coordinates using the standard conversion from spherical to Cartesian coordinates in flat space. That's what I think Hartle was doing.

Thanks, Peter. That's exactly what Hartle is doing.