Physical significance of 3rd TOV equation

Question

There are three TOV equations :

$$m(r)=\int_{0}^{r}4\pi \rho(r') r'^2 dr'$$

$$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{dP}{dr}=\frac{-G(\rho+P)(m+4\pi r^3\rho)}{r(r-2Gm)}$$

$$ \frac{d\Phi}{dr}=\frac{G(m+4\pi r^3\rho)}{r(r-2Gm)}$$

These equation comes after solving Einstein equation for the interior of star.

The $\Phi$ in the third eqution comes form the ansatz of the metric that is : $$ ds^2=e^{-2\Phi(r)}dt^2+e^{2\Lambda(r)}dr^2+r^2(d\theta^2+\sin^2(\theta)d\phi^2)$$

Now my confusion is: What is the physical significance of the third TOV equation, it seems that it signifies the modified potential in the case of Einstein gravity?

I would be very thankful if someone can provide a better insight!!

It plays the same role that $1 - (2M/r)$ plays in Schwarzschild. — Connor Behan, Dec 04 '21 at 11:12
If $\Phi$ were anything else, Einstein's equation would not be satisfied. Are you asking what coefficients of $dt^2$ mean physically? — Connor Behan, Dec 04 '21 at 12:03
Yes, actually what does it signify for a neutron star, Is it the potential energy, if so then how can the relationship be understood? — user235005, Dec 04 '21 at 12:11

JanG · Answer 1 · 2021-12-06T07:12:24.027

1

The three TOV equations are just reshaped EFE equations showed for example in the answer https://physics.stackexchange.com/a/679431/281096 (equations 1, 2 and 3). Please take into account that I am using there variable $u=(r/R)^{2}$ and not $r$. To express the equations in variable $r$ one has to replace $du$ by $2r dr$.

edited Dec 06 '21 at 07:12

answered Dec 05 '21 at 19:14

JanG

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Physical significance of 3rd TOV equation

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