Approximate Noether symmetries of the geodetic Lagrangian of spherically symmetric spacetimes

Abstract

Spherical symmetry of heavenly bodies is an important property in gravitational theories. All celestial bodies are approximately spherical shapes due to their gravitational pull in all directions. However, observation shows that the earth and other planets are not perfect spheres, they are flattened at poles and bulged at their equators. The black holes are considered to be spherical in shape. Their accelerated motion radiate mass in the form of gravitational waves. It is proved by Hawking that the black holes evaporate in radiation called Hawking radiation. It means that their mass-energy momentum tensor is a decreasing function of time. But all the known black holes solutions are static. To get the time dependent mass-energy momentum tensor we need a time dependent black hole solution of the Einstein field equations. To form the black hole spacetime time dependent, we perturbed the general spherically symmetric spacetime metric by a general time dependent conformal factor. Then use the Noether theorem and obtain a particular time dependent conformal factor without losing the spherical symmetry of the spacetimes produced by heavenly bodies. We define several theorems in this regard, which will provide a way to convert the static form of spherically symmetric static spacetimes (particularly black holes solutions) into non static spacetimes (non static black holes solutions). All spherically symmetric distributions that satisfy certain conditions will admit the time conformal approximation.