Equations having infinities are avoided in physics when possible. When that is not possible, we call it a singularity. Those singularities are not welcomed because they nullify one of the goals of science that is the application of equations to understand the system under study.

One well-known singularity appears in Schwarzschild’s metric [1] applied to the gravity created by a spherically symmetric, non-rotating body, of mass M and radius R. Strong conclusions coming from math deductions are risky as in [1], where it is claiming “The uniqueness of the solution resulted spontaneously through the present calculation”. That solution appears on page 20 on Carson Blinn [2] pdf paper as,

d s 2 = ( 1 − 2 G M c 2 r ) c 2 d t 2 − d r 2 1 − 2 G M c 2 r − r 2 d θ 2 − r 2 sin 2 θ d ϕ 2 (1)

In Equation (1) s is the space-time interval, G is the universal gravitational constant, c is the speed of light, t is time, and θ and ϕ are the spherical angles. If the gravitational fraction becomes bigger than one, we are forced to deal with imaginary time and imaginary radial distances.

As soon as 1935 [3] it was proposed to deal with those singularities by applying a change of variables (others propositions in [4] - [12] ). That move was a mathematical solution that required some extra physical consideration, further complicating the problem. Besides previous note, the same author concluded in 1939 [13] that “…the ‘Schwarzschild singularities’ do not exist in physical reality”. The authors proposed to take those singularities as mathematical concepts and not as real entities.

In [14] it was called to attention about the stability of any of the solutions proposed and clarifying that unstable solutions should not be taken into consideration. It was concluded in [15] that stars approaching the singular condition become unstable and must collapse to a point. In agreement with that idea, in [16] another surprising idea is mentioned by indicating mathematically that the centrifugal force is reversed bellow the Schwarzschild’s radius. The intention of removing the Schwarzschild singularity by a change in the coordinates system does not eliminate all the rare physical consequences. Now the authors are pointing to the non-sense of the dynamical inside those singularities.

The conclusion of [17] mentioned “…we are led to the unexpected conclusion that a particle which exerts a gravitational field must travel faster than light”. Another call back to the physics was mentioned in [18] by saying “It may be seen that de Sitter’s singularity like Schwarzschild’ singularity is an artificial singularity, not of the field but of the coordinates introduced to describe this field”. It looks as they have given up in [19] after concluding “This perhaps can be attributed to the fact, as remarked by Einstein [20] that the general theory of relativity would break down under such stringent conditions”. It is mentioned in [21] that there should be some physics because the Schwarzschild’s solution is mathematically stable under small non-spherical perturbations. In [22] appears a strong remark “…the singular character of the surface r = 1 is a feature intimately connected with the non-definiteness of the space-time metric”. At this time, it looks like the authors are trapped between the attractiveness and incoherence of the hall picture.

Other’s problems arise by connecting gravity with others disciplines of the physics. In [23] was introduced the idea that Quantum Mechanics affect the metric derivations as “Firstly, the event horizon obtained by setting g~ = 0, which occurs at r = L, in the classical theory, should now be modified”. Negating the power of theoretical physics, in [24] was claimed “In a real Universe an isolated body cannot be described strictly by the Schwarzschild solution because the body does not exist all alone in the Universe”. Professors teaching this material are not happy with its consequences, even in modern times, as in [25] where it was declared “In concluding, it is important to emphasize that the interior structure of realistic black holes has not been satisfactorily determined and is still open to considerable debate”. Another metric [26] is introduced with mathematical consequences related to the existence of the superluminal particles called tachyons.

Interesting propositions were made in [27] [28] [29]. Those ideas are pointing in the direction followed here in this paper.

Einstein’s General Theory of Relativity (GTR) [30] has been tested many times and its condition of General is accepted without questioning. This notion does not imply that any solution coming from applying GTR is going to also be general because some internal assumption can reduce its domain. This statement seems trivial, but it is the reason of this paper where the question Could exist another solution that include the Schwarzschild’s solution as a particular one? will be answer.

The differential equation on page 19 [2], where ℝ appears as the Ricci temporal tensor and the Ricci scalar is

ℝ 00 − 1 2 g 00 ℝ = 1 r V ′ V 2 + 1 r 2 ( 1 − 1 V ) = 8 π G c 2 T 00 . (2)

Normally Equation (2) is solved by nulling the temporal energy-stress tensor T₀₀. Let’s apply a fundamental variation. Here it will be assumed that the GTR is about the general properties of any field in the space-temporal reality and gravity should be treated as any other field. Then, the tensor T₀₀ could not be zero. Because the intensity of the Newtonian’s gravity decreases with the squared of the radius and the energy density of other fields are proportional to its intensity squared, it will be assumed that T₀₀ decreases inversely with the quartic of the distance. In that way, it is close to zero for regular gravitational fields. Mathematically the constrain 8 π G c − 4 T μ ν ≈ A r − 4 will be used because in weak fields that constrain includes the Schwarzschild’s solution as a particular case. Equation (2) becomes Equation (2a)

ℝ 00 − 1 2 g 00 ℝ = 1 r V ′ V 2 + 1 r 2 ( 1 − 1 V ) = A ( r ) r 4 . (2a)

The reader can check out that Equation (3) is solution of Equation (2a)

V ( r ) = 1 + 2 G M C 2 r , (3)

if A_(r) satisfices

A ( r ) = 4 G 2 M 2 C 4 ( 1 + 2 G M C 2 r ) 2 . (4)

Realize that for weak gravitational fields V ( r ) = 1 + 2 G M c − 2 r − 1 ≈ ( 1 − 2 G M c − 2 r − 1 ) − 1 , and A ≈ 0 as is in the Schwarzschild’s solution.

Similarly, the solution of the equivalent differential equation on the same page 19 [2], where now the tensor is pure radial,

ℝ 11 − 1 2 g 11 ℝ = − 1 r U ′ U V + 1 r 2 ( 1 − 1 V ) = A ( r ) r 4 (5)

becomes identical to Equation (2a) as expected if U = V⁻¹ holds.

Because on page 12 [2] the squared space-time interval was characterized as

d s 2 = ∑ μ ν g μ ν d x μ d x ν = U ( r ) c 2 d t 2 − V ( r ) d r 2 − r 2 d θ 2 − r 2 sin 2 θ d ϕ 2 ,

the substitution of Equation (3) and its inverse into the equation above output,

d s r ≥ R 2 = ( 1 + 2 G M C 2 r ) − 1 C 2 d t 2 − ( 1 + 2 G M C 2 r ) d r 2 − r 2 d θ 2 − r 2 sin 2 θ d ϕ 2 . (6)

The math done pays off because Equation (6) does not carry on any singularity, not even at the center of the gravitational body. This can be noticed in Figure 1 where it was assumed a constant average density defined as the ratio between the mass contained in the spherical volume of radius r, and that volume.

The metric described by Equation (6) becomes important for strong gravitational fields. In no circumstance does that metric show any skyrocket tendency, not even for the maximum gravitational potential at the surface of the body. Also, the temporal part of the metric has no negative values as expected. Additionally, the gravitational potentials approach flat values faster than in the Schwarzschild’s metric. That means that in any experimental observation of Kerr’s dragging or Yukawa’s potential, effects should be intended as close as possible to the source. The work in [31] is an example of negative observations attempted too far away from the center of our galaxy. In principle, it is possible that the idea proposed in this paper could explain that negative result.

Einstein understood that the GTR metric of space included Newtonian gravity because gravity produces the curvature of space-time. This is an example of the interplay between mathematics and physics that results in progress in both disciplines. However, it is important to realize that mathematical solutions can correspond to regions that predict exotic physical behavior of space and time.

The idea mentioned above was split into two ideas. Here, it was assumed that the GTR is about the relativistic connections in the space-time coordinates and the fields filling those coordinates. Then it became natural to manipulate the gravitational field as is done with any other field. It looks like a simple idea, but it allows to find an exact solution of the GTR equations that included the Schwarzschild’s solution as an approximate solution for weak gravitational fields. Our approach was based on the idea that gravitational field has energy and that energy can behave as a gravitational mass.

In [35] a sacrifice was made assuming that photons have rest mass and that makes the existence of exotic Dark Energy unnecessary. Here, applying the same logic to the possible particles of gravity, the idea of Dark Matter becomes unnecessary to explain what happens in regions where gravity is strong. If the model proposed here is correct, exotic, and yet unobserved phenomena may be unnecessary.

The author is grateful to FIU’s Center for Excellence in Writing for their valuable suggestions regarding the understanding of this paper.

The author declares no conflicts of interest regarding the publication of this paper.

Parra, J.L. (2021) D’addio to Blackness. International Journal of Astronomy and Astrophysics, 11, 509-516. https://doi.org/10.4236/ijaa.2021.114027