Updated: Jun 18, 2021
Due to General Relativity's (GR) illogical foundations, many paradoxes have been underlined. No less important than others is the possibility that gravity can become repulsive for values of the velocity greater than the speed of light divided by the square root of 3 in the asymptotic regions of space far from the central mass. This was first proposed by Hilbert, and later confirmed by Treder. Hilbert could be excused because he was a mathematician, and the field equations, taken at face value, led him to such a conclusion.
But just pause a minute and try to rationalize why high speed celestial bodies should become repulsive of other bodies if the speed is great enough. For beyond the critical speed, any body at any point in the field would be repelled by other bodies in the field. That makes no sense.
Luckily, the Schwarzschild solution in particular, and GR in general treats only a single body. In other words, GR can't affront the two body problem let along the n-body one. The condition that the acceleration be positive is
v = dr/dt > (1-2GM/r)c/3^1/2.
The metric is
when the motion is constrained to like in the azimuthal plane. The coefficients, A and B, are unknowns to be found using the Einstein field equations valid in an "empty" universe. The solution
does, in fact, contradict the condition under which the Einstein equations are solved because the universe does not appear "empty" at all, but has a central mass M. The effect of this central mass on other (nonexistent) masses, is usually accounted for by introducing fictitious "test" masses.
The equation of motion is found to be
where the prime means differentiation with respect to r. If we now introduce "time-slicing" whereby 4D space is cut into 3+1 dimensions, it is possible to study the spatial evolution of a system at a given epoch. This method has been called the holy-grail of numerical relativity which attempts to analyze the propagation of gravitational waves in 3D.
For a Newtonian "slice of time" in GR is a "space-like hypersurface", which is a 3D surface embedded in 4D space-time. It is non-causal because any two points "space-like" hypersurface because is connected by a "space-like" path in the metric so that they are independent of one another since they can't be related by any time sequence of events. Notwithstanding this evident fact, numerical relativists claim that it "can form the basis of forward predictions" by patching together sequences of time slices.
In contrast to Newtonian theory with (x,y,z,t) as coordinates, once time t is fixed, at the origin of the coordinate system for example, the entire hypersurface is determined, GR has greater flexibility together with greater indeterminism. The time like vector will be normal to each of the hypersurfaces requiring 3 degrees of freedom allowing for a foliation. Curved hypersurfaces require the specification of a greater number of degrees of freedom.
Be that as it may, if we impose a time slicing, or introducing the weaker condition that B'=0, the above equation of motion will reduce to
d^2r/dt^2=-A'/2A (dr/dt)^2 (**)
in the case of radial motion. And when the value of A is introduced, there results
d^2r/dt^2=+ g >0,
where g=GM/r^2. This means that gravity is repulsive since the acceleration is everywhere positive. It also implies that the velocity is given by
v= dr/dt = (1-2GM/r)^1/2 c (***)
Squaring both sides of (***) allow it to be interpreted as an energy balance equation
v^2= 2E- 2GM/r,
where 2E is the rest energy per unit mass. It makes no difference what this energy is because it is a constant.
However interpreting (***) as an energy conservation equation displays the fact that the gravitational potential has entered with the wrong sign! This is the origin of the singularity in the Schwarzschild solution. Here, gravity is everywhere repulsive, not just for velocities greater than the critical value. If time-slicing in numerical relativity is valid, it is an inevitable conclusion that the Schwarzschild exterior solution is unphysical!