1 Introduction

The following is a brief presentation cited from a PM entry reference on Einstein’s Field Equations, together with two original alternative formulations of GR Field Equations and the fate and current status of black hole conjectures. The interested reader may click on the above link to see the complete reference cited here which provides additional mathematical data/information on Einstein’s field Equations in GR.

1.1 Einstein’s Field Equations in General Relativity

“Then, the Einstein equations read as follows: “ ¹

Here, G_μυ = R_μυ −g_μυR is the Einstein tensor, R_μυ is the Ricci tensor, and R = g^μνR _μν is the Ricci scalar, and g^μν is the inverse metric tensor.

One possibility is that the tensor field T_μν is specified and that these equations are then solved to obtain g_μν. A noteworthy case of this is the vacuum Einstein equations, in which

Another possibility is that T_μν is given in terms of some other fields on the manifold and that the Einstein equations are augmented by differential equations which describe those fields. In that case, one speaks of Einstein-Maxwell equations, Einstein-Yang-Mills equations, and the like depending on what these other fields may happen to be. It should be noted that, on account of the Bianchi identity, there is an integrability condition ∇_μ(g)T^μν = 0. (Here, ∇(g) denotes covariant differentiation with respect to the Levi-Civita connection of the metric tensor g_μν).

When choosing T_μν, these conditions must be taken into account in order to guarantee that a solution is possible.”

2 Alternative Formulations of GR Field Equations and General Relativity theories

An alternative, more general formulation would involve a categorical framework such as the category of pseudo-Riemannian manifolds, and/or the category of Riemannian manifolds, with, or without, a Riemannian metric. Expanding universes and black hole singularities, with or without hair, either with an event horizon, or ‘naked’ can be treated within such an unified categorical framework of Riemannian/ pseudo-Riemanian manifolds and their transformations represented either as morphisms or by functors and natural transformations between functors. Quantized versions in quantum gravity may also be available based on spin foams represented by time-dependent/ parameterized functors between spin networks including extremely intense, but finite, gravitational fields. A quantum Riemannian geometry, that is, a quantized ‘Riemannian–like’ manifold has also been reported in attempts to formulate a quantum gravity theory based on a quantized (or deformed)non-commutative ‘Riemannian manifold’.

An alternative approach has already been reported recently as ‘Local Quantum Physics’ by Haag and others, or in a more general setting as “Algebraic (or ‘Axiomatic’) Quantum Field Theory” (AQFT).

3 Conjectures

3.1 The Penrose Conjecture

Sir Roger Penrose formulated sometime ago an important conjecture regarding physical black holes:

“All physical black holes have an event horizon; naked black holes are physically prohibited or forbidden even though they may be mathematically possible.”

John Wheeler also formulated a conjecture related to the above: “All black holes are ‘without hair’ (are completely invisible as no radiation escapes any black hole).”

Stephen William Hawking and others seem to have proven theoretically and decisively that black holes have ‘hair’, that is, that they can radiate what is now called ‘Hawking’ radiation. Thus, the J. Wheeler conjecture seems to be incorrect. Furthermore, recent astrophysical theories and observations seem to disprove also the ‘Penrose conjecture’, and suggest the existence of naked nlack holes without an event horizon, thus going much further than the Hawking’s model of black holes ‘with hair’.

4 Hyperbolic Formulations

5 Variational Principles

6 Global Structure

7 Initial Value Formulation

8 Special Solutions

8.0.1 Spatially Homogeneous Solutions

8.0.2 Solutions with Symmetries

8.0.3 Algebraically Special Solutions

8.0.4 Linearization

8.0.5 Singularities

8.0.6 Asymptotically Flat Solutions

8.0.7 Existence Theorems