Updated: Sep 22, 2019
Numerical relativity pawns itself as offering numerical solutions to the set of nonlinear equations that Einstein wrote down in which he "equated" geometry with its source energy-stress.
While general relativity makes no claims to have solved the two-body problem, and cannot even reproduced Newton's law of superposition of gravitational fields, numerical relativity vants itself to have described the merger of binary black holes and neutron stars. Einstein was very clear that the motion he intended to describe was geodesic in origin; that is, paths of shortest connection where the mobile travels at constant velocity. Whereas its velocity can change direction, it can't change its magnitude. And the acceleration vector has a component perpendicular to the surface, but no tangential component to a given surface. Given this condition, the equation of a geodesic follows: a second-order differential equation that parametrically describes the curve.
Non-euclidean metrics of constant curvature have been studied since the middle of the nineteenth century. Those of non-constant curvature are less well-known. In this respect, Schwarzschild is to be credited with the discovery of a non-Euclidean metric of non-constant curvature, which could be transformed into one of constant curvature simply by replacing the mass by a constant mass density. Commonly, they are referred to his "outer" and "inner" solutions, respectively.
With the passage of time, things tend to get garbled by those who choose not to read the original publications. General relativity, in this respect, is no acception. Schwarzschild's inner solution was conveniently "swept under the carpet", and the outer solution was extended beyond its domain of validity. Thus was born the "black hole" with its "event" horizon which served as the boundary of the non-Euclidean metric.
The great mathematicians of the nineteenth century, that include Beltrami, Klein, and Poincare, would never have dreamed of extending their "disc" models of the hyperbolic plane beyond the rim of the disc. As an inhabitant of the disc makes his journey to the rim of the disc, it would appear to him to take an infinite amount of time because his clocks and rulers shrink in size along with him. However, inventive physicists with no background in non-Euclidean geometry have found transforms that make the rim disappear making the singularity at the origin the only non-removable one.
Once inside the rim, we are in the domain of a black hole which sucks us into its singularity. It is commonly believed, although there is proof to the contrary, that as the voyager approaches the event horizon his velocity increases to that of the speed of light, thereby making the black hole's binding energy the same order as its rest energy. This has lead to confusing the internal energy of a black hole with its rest energy, a temperature that is inversely proportional to its internal energy, and an "entropy" that is quadratic in the internal energy! If that is not enough, the heat capacity turns out to be negative, which is impossible for a system comprising solely of a single phase. This has lead to the nonsensical conclusion that black holes of smaller mass have higher energies.
All this could have been avoided had the general relativists had consulted the original works of Schwarzschild, Droste, Hilbert, and (even) Einstein. In fact, Einstein and his younger colleague Rosen set themselves out to "building bridges" to avoid any and all notions of the hideous singularity. In truth, as the test particle approaches the horizon his velocity and acceleration both tend to zero, just like our inhabitant of the two dimensional disc that serves as a model of the hyperbolic plane.
Then what is all the hype of having "observed" black holes---even a "super" one at the center of our galaxy? We share the view of Wolfgang Kundt that the stellar-mass black hole candidates are really neutron stars inside massive accretion discs. Moreover, the "bars" seen at the center of spiral galaxies are central engines to which "active galactic nuclei" (AGN) owe their extreme properties to. Because of their axial symmetry the central bars may even be the progenitors of binary stars.
Another approximation that is used is the ppN or the parametrized post-Newton approximation in which unknown constants are introduced and bounds are obtained using the results of general relativity. Two of its central results in which ppN has been applied to is the filling of the "gap" in the advance of the perihelion of the planet Mercury, and Shapiro's time delay. The latter has been related to gravitational lensing. In both cases, there is the nagging problem of integrating over an unperturbed trajectory. A comparison of Einstein's derivation of the classical deflection of light in 1911 with Soldner's 1801 derivation shows clearly that in Soldner's case there is no need for integration. The deflection angle is simply the angle of the asymptotes of the hyperbolic trajectory. This clearly shows that the two phenomena that will set bounds on the parameters in ppN are related to gravitational phenomena in two---not three---spatial dimensions.
The logarithmic potential in the Shapiro time delay is the hallmark of two dimensional gravity.
The numerical equivalence with the missing number of arcseconds in the perihelion of Mercury was heralded as a monumental achievement of general relativity. It consists of adding a non-linear term to an otherwise Keplerian orbit that Einstein obtained using the (outer) Schwarzschild metric. That seemingly innocuous addition leads to the introduction of a rest mass which is completely foreign to Keplerian mechanics. We need only recall the success that the Ptolemaic cycles and epicycles were in predicting the motion of the planets, not to be fooled into believing that a physical mechanism has been found to explain the "gap" in the perihelion of Mercury. Rather, all that is necessary is to add a cubic term to the gravitational force that Newton did in order to obtain the same quadratic term to Kepler's equation. All this is explained in my book Seeing Gravity.
Finally, the attentive reader should have asked himself or herself of why develop a theory of 3 (space) + 1 (time) dimensionality when cosmological structures like spiral galaxies, AGNs, QSOs (Quasi-Stellar Objects) use reduced spatial dimensionalities. Here, general relativity has been less successful in providing a proper Newtonian limit to spacetimes of dimensionality 2+1,