Why Pressure does not fit into General Relativity

In 1934 Milne and McCrea [Quart. J. Math.  5 (1934) 73-80] made a startling discovery that still is not fully resolved. In Harrison's [Cosmology (Cambridge U. P., 2000), 2nd ed. p.326] words: "Why should Newtonian theory and general relativity, when applied to a uniform universe, yield identical results? At best, Newtonian theory is only approximately true, and yet in this most unlikely of all applications it gives the correct result." Milne and McCrea derived the Friedmann-Lemaitre equations for the expansion rate of the universe at zero pressure. This was later generalized to non-zero pressure by Harrison himself [Annals of Phys.  35 (1965) 437]. Gravity causes acceleration, but uniform pressure not. It is the non-uniformness of the pressure that causes motion whereas even uniform gravity can and does cause motion.

General relativity has confused a material velocity $v$ with the time rate of change of a scale factor, $\frac{dr}{dt}$, which can even be the radius of the universe. Moreover,  the mass density density  is confused with the internal energy density. It is the latter that is related to the pressure through a thermal equation of state. All this is made transparent in the Milne-McCrea analysis.

The starting point for the Milne-McCrea analysis is Newton's equation \[ [1]                           \frac{dv}{dt}=-\frac{GM(r)}{r^2},\] which they observed may be written as \[[2]                                  \frac{\partial v}{\partial t}+v\frac{\partial v}{\partial r}=-\frac{4\pi}{3} G\rho r,\] where $\rho$ \is the density. We can introduce the pressure $p$ by writing Newton's equation [1] as Euler's equation \[ [3]                                            \frac{dv}{dt}=-\frac{1}{\rho}\frac{dp}{dr}.\] Then, equating the right-hand sides of [1] and [2] gives \[ [4]                     dp=\frac{GM}{r^2}\rho dr,\] which is \emph{not} the equation of hydrostatic equilibrium.


Euler's equation in the presence of a gravitational field is \[ [5]                \frac{dv}{dt}=-\frac{1}{\rho}\frac{dp}{dr}-\frac{GM}{r^2}.\] If the fluid is at rest, $v=0$, then [5] reduces to the equation of hydrostatic equilibrium \[ [4']                    dp=-\frac{GM}{r^2}\rho dr,\] which differs from [4] by a sign.


Now, let us see what happens when we try to use the Friedmann-Lemaitre equation in the presence of a pressure, \[ [6]                 \frac{d^2r}{dt^2}=-\frac{4\pi G}{3}\left(\rho+3\frac{p}{c^2}\right)r.\] This equation can be used to eliminate the last term in Euler's equation \[ [5'] \frac{dv}{dt}=-\frac{1}{\rho}\frac{dp}{dr}-\frac{4\pi}{3}G\rho r,\] to get \[[7]                          \frac{dv}{dt}=-\frac{1}{\rho}\frac{dp}{dr}+\ddot{r}+\frac{4\pi G}{c^2}p r.\] If $v=\dot{r}$, then the sum of the first and third terms on the right-hand side of [8] must vanish, requiring that \[ [8]             4\pi G\rho=\frac{c^2}{r}\frac{d}{dr}\ln p.\] This is not the well-known condition for spherically symmetric body in equilibrium [Landau & Lifshitz, Fluid Mechanics (Pergamon Press, Oxford, 1959) p. 8 Eq (3.7)] \[ [9]                                                  4\pi G\rho=-\frac{1}{r^2}\frac{d}{dr}\left(\frac{r^2}{\rho}\frac{dp}{dr}\right). \]


Consequently, $v\neq\dot{r}$ even when the pressure is the same at every point in the fluid because it enters Einstein's equation  as $p$ itself and not as its gradient as in Euler's equation [6']. So it is not as Harrison claims that the "Newtonian picture of a sphere expanding in flat Euclidean space and Einstein's picture of it expanding in a curved space  "despite their basic differences, yield identical equations." It is not that the "Newtonian picture breaks down when the pressure is not small compared with the energy density $\rho c^2$," for $\rho c^2$ is the rest energy density, and it is not comparable to the pressure. The pressure is, however, comparable to the internal energy density which does not appear in Einstein's equations. Gravity causes both a solid and fluid particles to accelerate, but, by Euler's authority, it is the gradient of the pressure that causes the rate of change of a given fluid particle as it moves about in space. Gravity is a force, and so too is the gradient of the pressure. In other words, pressure is stress, but it is the change in the stress that causes motion. And this is unlike gravity which causes motion even when it is uniform.


In the weak (Newtonian) gravitational field, the time component of the metric tensor is $g_{00}=-1-2\phi/c^2$. The Newtonian gravitational potential $\phi$, in this limit, satisfies Poisson's equation \[[10]                                      \Delta\phi=4\pi G \rho,\] where $\Delta$ is the Laplacian. But, if Einstein's equation, [6], is true then we would have to generalize [10] to \[ [11]                                                              \Delta\phi=4\pi G\left(\rho+3\frac{p}{c^2}\right),\] making pressure a source of gravity even in a weak gravitational field!

This is precisely the Oppenheimer-Volkoff [Phys. Rev. 55 (1939) 374-381] for quasi-hydrostatic equilibrium:

\[[12]                                     \frac{dp}{dr}=-\frac{G(M+4\pi r^3 p/c^2)\rho}{r^2}=\frac{G(M+3pV/c^2)\rho}{r^2},\]

used in their study of gravitational collapse in the classical region $c^2>>G(M+3 pV/c^2)/r$. It is clearly evident from [12] that general relativity considers the compressional work $pV$ as contributing to the inertial mass.