How are Coordinates Defined in General Relativity?

[jdstokes]

In SR, coordinates are unambiguously defined with reference to a lattice of rigid steel rulers with synchronized clocks at each vertex.

How does the situation change in GR? Consider e.g. the Scharzschild metric

$ds^2 = g(r) c^2 dt^2 - \frac{dr^2}{g(r)} - r^2(d\theta^2 + \sin^2\theta d\varphi^2), \quad g(r) \stackrel{\mathrm{def}}{=} 1 - \frac{2GM}{c^2r}$.

This is the metric pertaining to an accelerated observer located at coordinates $P= (r,\theta,\varphi)$ with respect to the source. The radius r is not the radius that an observer at P measures using a lattice of rigid steel rulers. How then is r to be defined?

I find the coordinate distance in the FRW metric easier to understand than in Scharzschild geometry since it is defined so that two freely falling observers hold a constant coordinate separation.

Here is a related question. Consider an ant sitting on the surface of a sphere. If the ant tries to establish a lattice of rigid rulers two things can happen. If the ant is aware of its surroundings it will find that the lattice extends tangentially off the sphere. If however, the ant is unaware of the surrounding space, then the lattice must curve with the surface of the sphere.

Doesn't one face a similar problem in constructing a coordinate lattice in curved spacetime? Is this how coordinates are defined in the Scharzschild metric with respect to a curved lattice of rigid steel rulers?

 

[Mentz114]

From Wiki ( easier to copy it than to rewrite )

... the Schwarzschild radial coordinate does not accurately represent radial distances, i.e. distances taken along the spacelike geodesic congruence which arise as the integral curves of $\partial_r$. Rather, to find a suitable notion of 'spatial distance' between two of our nested spheres, we should integrate g(r)dr along some coordinate ray from the origin.

$$\Delta\rho = \int_{r_0}^{r_1} g(r) dr$$

The 'g(r)' is your g(r)^-1. If you're aware of this, apologies for wasting time.

I was glad to refresh my memory ...

There's also a goodish thread on rulers in Schwarzschild spacetime recently, but I can't track it down.

M

 

[snoopies622]

There's also a goodish thread on rulers in Schwarzschild spacetime recently, but I can't track it down.

Perhaps this one?

https://www.physicsforums.com/showthread.php?t=248015

Entry #5 seems like a good starting point.

 

[Mentz114]

There's no need to be sarcastic. If you find my reply inadequate or incorrect, you have the option of ignoring it, rather than lashing out.

M

 

[snoopies622]

That was not an attempt at sarcasm; I was just trying to be helpful.

I'm sorry it came across otherwise.

 

[Mentz114]

Mea culpa. I thought you'd posted the OP ! Stupid error. Apologies all round. The heat from my red face is melting the screen ...

M