Exploring Notable Solutions of Einstein's Field Equations: A Survey of Exact Models for Gravitating Systems

About the Study

In general relativity, the effects of gravitation are ascribed to space-time curvature instead of to a force [1]. The starting point for general relativity is the equivalence principle, which equates free fall with inertial motion. The issue that this creates is that free-falling objects can accelerate with respect to each other [2]. To deal with this difficulty, Einstein proposed that space-time is curved by matter, and that free-falling objects are moving along locally straight paths in curved space-time. More specifically, Einstein and David Hilbert discovered the field equations of general relativity, which relate the presence of matter and the curvature of space-time. These field equations are a set of 10 simultaneous, non-linear, differential equations. The solutions of the field equations are the components of the metric tensor of space-time, which describes its geometry. The geodesic paths of space-time are calculated from the metric tensor [3].

Notable solutions of the Einstein field equations include;

The Schwarzschild solution, which describes space-time surrounding a spherically symmetrical non-rotating uncharged massive object. For objects with radii smaller than the Schwarzschild radius, this solution generates a black hole with a central singularity. The Reissner–Nordström solution, in which the central object has an electrical charge. For charges with a geometrized length less than the geometrized length of the mass of the object, this solution produces black holes with an event horizon surrounding a Cauchy horizon [4]. The Kerr solution for rotating massive objects. This solution also produces black holes with multiple horizons [5]. The cosmological Robertson–Walker solution, which predicts the expansion of the universe. General relativity has enjoyed much success because its predictions not called for by older theories of gravity have been regularly confirmed [6]. General relativity accounts for the anomalous perihelion precession of Mercury. Gravitational lensing was first confirmed in 1919, and has more recently been strongly confirmed through the use of a quasar which passes behind the Sun as seen from the Earth. The expansion of the universe predicted by the Robertson–Walker metric was confirmed by Edwin Hubble in 1929 [7]. The prediction that time runs slower at lower potentials has been confirmed by the Pound–Rebka experiment, the Hafele–Keating experiment, and the GPS [8]. The time delay of light passing close to a massive object was first identified by Irwin Shapiro in 1964 in interplanetary spacecraft signals. Gravitational radiation has been indirectly confirmed through studies of binary pulsars such as PSR 1913+16 [9]. In 2015, the LIGO experiments directly detected gravitational radiation from two colliding black holes, making this the first direct observation of both gravitational waves and black holes. It is believed that neutron star mergers and black hole formation may also create detectable amounts of gravitational radiation [10].