Are The Field Equations of General Relativity Compatible with Fluid Dynamics?
The evolution of the universe is enshrined in the Friedmann model of a homogeneous evolving universe. But, are the Einstein field equations compatible with the elementary equations of fluid dynamics.
What stands out as a sore thumb is the expression for the hydrostatic pressure that general relativity assigns in terms of a sum of sectional curvatures. We know from the Euler equations that the gradient of pressure appears together with gravitational acceleration, yet the Einstein equations would have us assign one sectional curvature to gravitational acceleration, and another to the pressure. Moreover, it turns out that the pressure is always negative, and this has led to the unfounded belief that negative pressure creates accelerated expansion. In addition, one must distinguish between a constant mass and a constant mass density from the outset for, otherwise, the sum of sectional curvatures will be different. As a result, critical values of the density of the universe, and the seeming necessity to introduce 'dark' matter are nonsensical and violate the basic laws of physics.
The spatial part of the metric is
g=dR^2/(1-kR^2) + R^2 d\theta^2,
where k is a constant and R=R(r), r the radial coordinate. Defining an angle, B=sin^-1(k^1/2 R), the metric can be expressed as
g= dB^2+sin^2B d\theta^2.
With the angle as the independent variable it is easy to see that the sectional curvature K=+1.
Now add the time part to -g after multiplying it by the space part by the square of the radius of the universe, A^2(t),
G =-c^2dt^2-A^2(t) g
which is a function only of time. The sectional curvature of the time part is -d^2A/dt^2/A, which is indeterminate. That is, we don't know if it is constant or not, and what it has to do with the positive, constant curvature of the spatial part of the metric, which is k, in its original form.
If we combine sectional curvatures, it is like adding apples and pears. But, for the sake of argument, let us consider the velocity
v=dR/dt= (1-k R^2)^1/2.
Differentiating again--in time--yields
This says the sum of the sectional curvature vanish.
Where does gravity fit in? Frankly, it doesn't! However, this is precisely what the Einstein field equations are supposed to do. Fortunately, we don't need them; consider the velocity
Now if we convert from constant mass M to constant density k, we come out with the same result as above. However, differentiating the last expression in time, keeping the mass constant, results in
The factor of 1/2 means that the difference between twice the translational curvature plus the rotational curvature vanishes. But, this is precisely the expression that the Einstein field equations give for the negative of the hydrostatic pressure: If the eigenvalues of the Ricci tensor don't vanish that must mean the universe is not empty and should be proportional to the sources--density and pressure. The density M/R^3 is what is proportional to the acceleration in the above expression.
However, if we wanted to consider a system of constant density, like in the inner Schwarzschild solution, we should have started out with it, and then differentiated. We would have come out with (*) and not (**)! We have absolutely no right to convert (**) to a mass density. Yet, if we do precisely that the second equality in (**) looks like the expression for the square of the Hubble parameter, H=dR/dt/R. But this entails interpreting the left-hand side as a constant density. Taking the time derivative would give a positive acceleration instead of a negative one! The moral of the story is that one cannot convert from constant mass to constant density at will!
The so-called critical density is determined by the condition E=0, that the total energy vanish.
Is this what a 'flat' universe means? And what happened to the pressure? By the equality of the sectional curvatures in (**), the pressure would vanish. Not so if (*) were valid.
Pressure and density are not separate entities. Just look at the Euler equation for an inviscid liquid:
acceleration=-the ratio of the gradient of the pressure divided by the density and R divided by the square of the Newtonian free-fall time, 1/(G\rho)^1/2, where \rho is the constant density.
The equation of motion contains BOTH the gradient in pressure and the constant gravitational acceleration at the surface of the earth, or G\rho R. This is not borne out by the field equations of general relativity, which, frankly speaking, are meaningless. And so, too, are a critical density and 'dark' matter.