The Schwarzschild metric is unique in general relativity, and its shows. The contradictions that it provides casts doubts on the validity of the whole theory. Depending on whether we reason from the geodesic equation, or its integrated form we obtain conflicting results. And if we further consider geodesic orbital equations in the space submanifold of the Schwarzschild meteric we come out with nonsensical results.
What is gravitational acceleration? In Newton's theory the acceleration of gravity is always negative, which supposedly indicates that gravity is attractive. If we consider the geodesic equations, it would appear that at high enough velocities gravity can turn around and become negative! Another sticking point is that how do we know how fact a "test" particle is traveling since Schwarzschild is a single mass metric? The "test" particle is introduced by us to measure the temperature, so to speak, but it does not enter into any relation. This can be attributed to Newton's second law where the relation between the force of gravity and acceleration depends only on the central mass. But, force does not enter into general relativity, or at least not if we don't take the asymptotic limit.
Schwarzschild solved Einstein's equations in "vacuum". The central mass enters only as an arbitrary constant of integration, and our desire that in the asymptotic limit of large distances, we regain Newton's theory. But why should Newton's theory of gravitation apply only to the asymptotic limit, and weak fields? Moreover, the expressions of the gravitational potentials, found by Schwarzschild, contain the ratio of the gravitational potential to the square of the speed of light, or the rest mass energy per unit mass. How can this ever lead to closed orbits and even those which precess, as in the precession of the line of apsides in the case of Mercury?
Why should we start from an indefinite metric that is good for the propagation of light, and by its generalization to where the coefficients depend on space call it a metric for gravitational waves? That already implicitly implies that gravity propagates at the same speed of light. But, what if gravity does not propagate at all, and its minute changes to Newtonian theory are carried by electromagnetic waves? Moreover, why should the time component of the metric influence what is essentially a static metric?
Schwarzschild found that the coefficients in his line element,
c^2 ds^2=B(r)c^2 dt^2- A(r)dr^2- r^2d(phi)^2,
in the azimuthal plane where (theta)=pi/2, were given by
B=1/A=(1-2m/c^2 r), (1)
where m=GM, the Newtonian gravitational parameter. Looking at (1), we see that the second term is a very small correction to what would otherwise be the line element for light curves were the local time increment ds were to vanish. Taken at face value, we would be hard pressed to explain how Keplerian orbits can arise from such minute perturbations. Yet, it seems to!
Curves in the 3D spatial manifold, with an affine parameter, p, such that the tangent vectors V^i=dx^i/dp, where x^i=(r, (theta), (phi)), have a positive definite norm
V^iV_i = g_(ij)V^iV^j=c^2>0 (2)
which, supposedly satisfy the geodesic equations
where the G's are the Christoffel symbols, or connection coefficients. Thus, in the equatorial plane,
where we used the conservation of angular momentum,
V_3=g_(33)V^3=r^2 d(phi)/dp (5)
where we set (phi)=3.
Writing (4) out we have
(dr/dp)^2=-h^2/r^2(1-2m/c^2r)+c^2(1-2m/c^2 r)=-h^2/r^2-2m/r+2h^2m/c^2 r^3 (6)
where the fact that the square root of the velocity norm travels the speed of light has cancelled the c^2 in the denominators of (1) to introduce the Newtonian gravitational potential, but with the wrong sign!!! The last term is usually interpreted as the correction to the Keplerian orbit necessary to give the correct numerical expression for the precession of the line of apsides in the case of Mercury.
In 4D, the equation analogous to (6) is
(dr/dp)^2+ h^2/r^2 A(r) +B'A/A'B = -E<0 (6)
where the last term on the left comes from the time component, (dt/dp)^2, and
E>0 for material particles and 0 for photons. This is eqn (8.4.13) in Weinberg's Gravitation and Cosmology. The last term on the left hand side is -1 while E is an arbitrary constant of integration. No longer does it compensate the c^2 in the Schwarzschild expressions thus relegating the Newtonian gravitational potential to a much less important role.
Combining 3D and 4D results gives
c^2= B(r)-E (7)
which is hardly feasible given that E>0, or at least =0 for photons. But, in that case, (7) becomes an impossibility. In our previous blog, we asserted that for velocities greater than c/3^1/2, the gravitational acceleration becomes positive, indicating gravitational repulsion. Why do we even have to go to the geodesic equation when we have the equation of the orbit already. Yet, even if we do why should it introduce extraneous information that is not forthcoming in the metric itself?
Moreover, we got gravitational repulsion merely by considering a submanifold of space dimensions! And there is no reference to the velocity of the "test" particle which is completely extraneous to the Schwarzschild metric which has a single mass.
How can you build a theory upon such inconsistencies?