In the case of the basic static model for compact objects, in the theory up to date the collapse is ruled by a specific solution (called Schwarzschild solution
but not given explicitly by Schwarzschild, coming from the Hilbert's interpretation instead) that contains mathematical and thus physical singularities leading to a mass limit for ordinary compact objects and to the consequent black hole hypothesis (generalization to rotating or charged objects contains as well the features of singularity and horizon surface and it is not necessary in this context).
In a maximally extended external Schwarzschild solution
, that photon's frequency only stays regular if the mode is extended back into the past region where no observer can go.
His topics include LaGrangian and Hamiltonian formalism, and the relation between relativity and essential tensor calculus, along with Einstein's equation in special cases with explicit presentations of calculations for all steps, with coverage of Newtonian mechanics, symmetries, bodies' central forces, rigid body dynamics, small oscillations and stability, phenomenological consequences, aspects of special relativity, the equation of motion of the particle in a gravitational field, tensor calculus for Reimann spaces, Einstein's equation of the gravitational field, and the Schwarzschild solution
For example, the Schwarzschild solution
in quasiMinkowskian coordinates [11Weinberg 1972; p.
CONFORMAL FLUCTUATIONS OF THE INTERIOR SCHWARZSCHILD SOLUTION
The Schwarzschild solution
of the general relativity for a static spherically symmetric body predicts the perihelion precession of planets, the deflection of distant star light by the Sun, the gravitational redshift of Sun's light, and the time delay of radar echoes, which have been well tested by the measurements [1-4].
According to the Einsteinian general theory of relativity  and its Schwarzschild solution
, the gravitational field (or acceleration) at the surface of a black hole is inversely proportional to its mass or radius.
Formally, the problem we are considering is a generalization of the Schwarzschild solution
produced for an analogous case (a sphere of incompressible liquid).
The "Schwarzschild" solution, as espoused by Hilbert and others is different to the Schwarzschild solution
obtained originally by Schwarzschild ;
According to the Schwarzschild solution
of the Einstein general theory of relativity , the radius of a black hole with mass [M.
As a substitute, the line elements for the generalized Schwarzschild solution
of a point mass will be used to address the luminosity variability.
Also known analytic solutions of Einstein's equations like the isotropic Schwarzschild solution
,  indicate this.