# What is gravitating ?

1. Sep 7, 2013

### Mentz114

There is a static spherically symmetric perfect fluid solution of the EFE where the energy-momentum tensor is $diag(\rho,p,p,p)$ with $\rho=b\,\left( 2\,b\,{r}^{2}+3\,a\right) /{\left( 2\,b\,{r}^{2}+a\right) }^{2}$ and $p={b}/({2\,b\,{r}^{2}+a})$. a and b are parameters with b>0 and 0<a<1. On the surface $r=\sqrt{(1-a)/(2b)}\equiv r_{max}$ the PF metric coincides with the Schwarzschild exterior, as long as the Schwarzschild parameter m has the value $m_s= {\sqrt{1-a}\,\left( 1-a\right) }/( {4\,\sqrt{2b}})$.

Calculating $M_s$ the mass/energy total of the PF
\begin{align*} M_s &=\ 4\pi\int_0^{r_{max}} r^2\rho\ dr = 4\pi\left[ \frac{\,b\,{r}^{3}}{2\,b\,{r}^{2}+a} \right]_0^{r_{max}}\\ &= \frac{\sqrt{2}\,\pi\,\sqrt{1-a}\,\left( 1-a\right) }{\sqrt{b}}\\ &= 8\pi\ m_s \end{align*}
This seems most satisfactory but raises the question - what happened to the pressure terms in the EMT ? It appears that the integral of the energy density accounts for all the exterior vacuum curvature. Is this an anomaly or am I right to be surprised ?

(Actually I was very glad when the integral turned out like this - until the question of the pressure appeared).

2. Sep 7, 2013

### Staff: Mentor

Heuristically, the positive contribution of the pressure to the mass is exactly canceled by the negative contribution of gravitational binding energy to the mass. The easiest way I know of to see how that works is to look at the Komar mass integral, in which $\rho + 3 p$ appears in the integrand, but also the "redshift factor" $\sqrt{1 - 2m(r) / r}$ appears, and the two contributions cancel each other when the integral is computed over the entire volume of the object.

3. Sep 8, 2013

### Mentz114

Thanks. That could account for it. I'll check out the KM integral.

This PF is more realistic than I first thought because the radius of the ball can be set to any multiple of 2m by a suitable choice of parameter a. As a-> 0 so rmax -> 4m, and as a->1 so rmax increases without bound.

Last edited: Sep 8, 2013