- #1

TheMan112

- 43

- 1

Given we have a spherically symmetric gravitational field around a spherically symmetric body of mass M.

How can I calculate the actual (geodesic) distance between two points with the same angle but at different distances from the center of the body (and field).

I came immediately to think of the Schwarzschild solution which reduces to:

[tex]ds^2 = -(1-r_s/r)^{-1} dr^2 = \frac{1}{(\frac{r_s}{r}-1)} dr^2[/tex]

The geodesic distance would then be:

[tex]\int ^{r_1}_{r_2} ds = \int ^{r_1}_{r_2} \frac{1}{\sqrt{r_s/r-1}}dr[/tex]

Which can't be used outside [tex]r_s[/tex], and creates an insanely complex primitive function inside [tex]r_s[/tex].

What should I do? I need help urgently.

How can I calculate the actual (geodesic) distance between two points with the same angle but at different distances from the center of the body (and field).

I came immediately to think of the Schwarzschild solution which reduces to:

[tex]ds^2 = -(1-r_s/r)^{-1} dr^2 = \frac{1}{(\frac{r_s}{r}-1)} dr^2[/tex]

The geodesic distance would then be:

[tex]\int ^{r_1}_{r_2} ds = \int ^{r_1}_{r_2} \frac{1}{\sqrt{r_s/r-1}}dr[/tex]

Which can't be used outside [tex]r_s[/tex], and creates an insanely complex primitive function inside [tex]r_s[/tex].

What should I do? I need help urgently.

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