# Proper Distance and Coordinate Distance in General Relativity

When I started studying general relativity,I was told that the coordinate distance is not covariant and it does not have any physical meaning.I realized that only the proper length is the physical-measured length.

However,I found that it is not always true when I study more.I am learning the Post-Newtonian Theory of General Relativity by the book Gravity:Newtonian, Post-Newtonian, Relativistic.The book hardly ever mentions the proper distance.On page 509,it says:

In our development of the foundations of special and general relativity, we have emphasized that only physically measurable quantities are relevant, and that the selection of coordinates is completely arbitrary, devoid of physical meaning. On the other hand, to calculate such things as equations of motion or the Shapiro time delay, it is essential to have a suitable coordinate system. No one should consider doing such calculations using a proper time variable or using physically-measured lengths, because this would introduce unnecessary complications into the calculations.

However,it doesn't explain why the calculations are unnecessary.During the whole book,I can only find these words about the proper distance.

For example,it proves the pericenter advance on page 485: $$\Delta\omega=6\pi\frac{Gm}{c^2p}$$ where $p$ is the Keplerian semi-latus rectum,$\omega$ denoting the Keplerian longitude of pericenter.These are obviously depending on the coordinate.But it just says "Substituting the values for Mercury, $e = 0.2056$ and $P = 87.97\ \text{days}$, we obtain $42.98\ \text{arcseconds per century}$."

Even though the differnce between the proper distance and the coordinate distance for this example may be able to be ignored,I do not think it should be ignored when calculating the orbital evolution under radiation reaction,it's $O(c^{-5})$ (see page 692).How to explain this?

• When I've seen the calculation done it's used the weak field approximation and done using the Schwarzschild coordinates, so the precession is in the $\phi$ coordinate. I guess using harmonic coordinates would be a lot more elegant but I must confess I don't know how this would be done. Jul 5, 2016 at 15:45