Diameter of the earth with GRT

No. The proper diameter (the diameter we would measure with rulers) is longer than the circumference divided by ##\pi##. A brief calculation follows.

Looking back at the thread A. T. linked to, I realized that, while Jonathan Scott's formula for the potential (of a uniform density sphere) is correct, his assumption that the potential is what you need to calculate the proper diameter is wrong. What you need is the metric coefficient ##g_{rr}##; the potential is ##g_{tt}## .

Fortunately, the metric coefficient ##g_{rr}## is actually simpler to work with, particularly if we are willing to make a rough estimate. This metric coefficient is given by

$$
g_{rr} = \frac{1}{1 - \frac{2 G m(r)}{c^2 r}}
$$

where ##m(r)## is the "mass inside radius ##r## ". The square root of this value gives the ratio of proper distance to coordinate distance at radius ##r##. Since we always have ##g_{rr} \ge 1## for ##r \ge 0## , we can already see that proper distance will always be at least as large as coordinate distance, and will be strictly larger for any ##r## greater than zero and less than infinity. But the order of magnitude of my estimate should be correct.

Since ##m(r)## is zero at ##r = 0## , and is just ##M## , the total mass of the Earth, at ##r = r_e##, we can do a quick rough estimate by just averaging the two values of ##\sqrt{g_{rr}}##. One value is just 1; the other value, plugging in numbers, comes out to about ##1 + 2 \times 10^{-9}##. So averaging tells us that the proper diameter of the Earth is larger than the coordinate diameter (i.e., the circumference divided by ##\pi## ) by about ##10^{-9}## times the coordinate diameter, or about 1 centimeter. Note that this is a rough estimate; a more accurate calculation would have to first obtain the function ##m(r)## by integrating the density from ##r = 0## outward, and then evaluate the integral ##\int_0^{r_e} \sqrt{g_{rr}} dr## using the function ##m(r)## that was obtained.

 

Not just the exterior. Note that I put in ##m(r)##, the "mass inside radius ##r## " function. With that included, the ##g_{rr}## I wrote down is valid everywhere in a spherically symmetric spacetime. MTW shows this in one of their sections on the Schwarzschild metric; that's the derivation I'm most familiar with.

This gives the same ##g_{rr}## I wrote down, but with the appropriate form of ##m(r)## for a sphere of uniform density: ##m(r) = M r^3 / R^3##, where ##M## is the total mass and ##R## is the surface radius. You could certainly integrate this from ##r = 0## to ##r = r_e## to get a more accurate result; I was just too lazy to write down the complicated expression for the integral. ;)