Silly question about Coordinates in General Relativity Why, we must take care when we talk about physical meaning of coordinates in General Relativity? Or, why coordinates in General Relativity don’t "mean anything"? (I'll appreciate if you give me a explicit example)
 A: The point of general relativity is that the laws of physics look the same regardless of the coordinates chosen (as long as s diffeomorphism relates the old coordinate system to the new one). This means so laws can be written a using true tensors. For example, $\nabla_\mu F^{\mu\nu}=\mu_0 j^\nu$ is true whether we work with Cartesian, cylindrical or spherical polar coordinates, or something else entirely. We can even change the choice of time coordinate. Which brings us to the second question: tortoise coordinates impose an $r$-dependent shift in the time coordinate to remove a coordinate singularity from the metric tensor. It's very important to distinguish coordinate singularities from physical ones. (For example, the point $r=0$ is a physical one, as can be seen from the divergence of the Kreschmann scalar )
A: What is meaningful is the geometry of spacetime which is described by the square of the infinitesimal line element $ds^2 = g_{\mu \nu} dx^\mu dx^\nu$, where $g_{\mu \nu}$ is the metric tensor and $dx^\mu$ is the infinitesimal coordinate displacement four-vector.  
The spacetime interval $ds^2$ is an invariant and provides information about the causal structure of spacetime:
$ds^2 \lt 0$ timelike interval; traversable by massive objects; the square root of the absolute value is the infinitesimal proper time
$ds^2 = 0$ lightlike interval; traversable only by massless objects, e.g. photons; null interval
$ds^2 \gt 0$ spacelike interval; not traversable by causally related objects; the square root is the infinitesimal proper length  
What is physical is the composition of the coordinates with the metric tensor, not the coordinates themselves.
