_{1}

^{*}

The theory here developed, makes use of the decomposition of matter (mass) in different spatial frequencies k’s using spatial Fourier transforms, and the posterior use of modified inverse Fourier transforms to construct an accurate description of the classical Newtonian gravitational field. Introducing the concept of quantization of the spatial frequency
*k*, which means allowing only discrete values, such as
*k _{m}*, 2

*k*, 3

_{m}*k*, leads to the appearance of extra gravitational force regions that occur at distances equally spaced apart in 2π/

_{m}*k*. These areas of extra gravitational force decrease inscribed in an inverse of the distance envelope (1/

_{m}*r*). The value of 2π/

*k*can be adjusted to be of the order of kiloparsec (kpc), being this way a plausible explanation for the effect of the dark matter since this causes practically flat rotation curves for most of the galaxies. As these regions of extra gravitational force also have adjacent areas of negative values (repulsive gravitational force), it is possible to show that any mass placed in the gravitational field far from the galaxy center will acquire, on average, a null acceleration, thereby remains the “light push,” or in other words, the “mean luminosity density” between galaxies as an explanation for the accelerating expansion of the universe, today being considered mainly due to dark energy. Along with the article, it is showed that the effect of light push is sufficient to explain the expansion of the universe. The present work also explains the nonlinear behavior of gravitational fields near massive objects such as blackholes, not contradicting the theory of general relativity, instead giving a complementary description of how black holes work, even describing the gravitational field internally to it, which is not available in the GR theory.

_{m}The theory of general relativity GR [

Despite its immense success, GR theory lacks in explaining the behavior of the gravity inside black holes [

The present theory has the merit and the novelty of helping to explain or solve the three points above, besides offering a simple computational way to calculate gravity inside and outside black holes, or in any other massive object, using Fourier transforms, such as explained later.

As a start point for the theory of black holes, the author considers the existence of two kinds of energy sources for the gravitational field: The rest mass, which converts into energy by E = Mc^{2} and the energy due the motion of particles.

For calculations are used the Fourier transform, widely used in quantum mechanics for representing the wave function in the position or momentum space, among many other uses [

Regarding dark matter and dark energy, both have been the subject of discussion for a long time [

Based on the work discussed above, it is possible to consider, by hypothesis, the possibility of the quantization of the spatial frequency k, that is, only occurring at fixed intervals, such as k_{m}, 2k_{m}, 3k_{m}, where k_{m} stands for “minimum spatial frequency”.

The proposed quantization, when applied in the inverse Fourier transform, reveals areas of extra gravitational force spaced at 2π/k_{m}, which fall as inscribed in an inverse distance envelope (1/r). Therefore, choosing the correct value of 2π/k_{m} in the range of kiloparsecs (kpc) is sufficient to explain the practically flat rotation curves of most galaxies [

The null average effect of gravity between galaxies allows us to demonstrate that the mean luminosity density of the universe is a sufficient factor to explain the repulsive force responsible for the accelerating expansion of the universe.

By now, it is not clear what causes the universe to expand, puzzling the scientists even more since the discovery that the expansion is in fact accelerating [

The repulsive force of light as responsible for the accelerating expansion of the universe is mostly unconsidered due to its small magnitude compared to gravitational effects, being necessary the elaboration of alternative concepts such as negative vacuum pressure or dark energy, among others [

However, considering the average annulment of the force of gravity between galaxies, it is clear that luminosity could be a predominant factor and the only one necessary to explain the expansion of the universe. The physical model developed in the last part of the present paper has yielded results that agree very well with values that are within the current acceptable limits for the Hubble constant ≈ 67 - 73 km∙s^{−1}∙Mpc^{−1}, obtained respectively by [

The present model excludes the existence of non-baryonic matter, or the existence of dark matter or dark energy, further concludes that the expansion of the universe follows a coasting universe model, expanding forever, never contracting.

The k-space or reciprocal space is the spatial frequency domain of a spatial Fourier transform, as shown in Equation (1). It is possible to take a distribution of mass in position space, and through the use of a spatial Fourier transform to decompose it in different spatial frequencies.

D ( k ) = ∫ − ∞ ∞ D ( x ) ⋅ e − i x k d x (1)

As our object of study are mostly radially symmetric spherical bodies, such as stars, black holes, and so on, it is showed, using spherical coordinates, that the Fourier transform for such a case can be simplified as in Equation (2) [

D ( k ) = ∫ 0 R D ( r ) ⋅ 4 π r 2 ⋅ sin ( k r ) k r d r (2)

where r is the radial coordinate, and k is the spatial frequency in k-space, which unit is m^{−1}. For a spherical object of mass M, radius R, and volume V, the radial density D(r) is as shown in Equation (3).

D ( r ) = M V = 3 M 4 π R 3 (3)

Substituting Equation (3) in Equation (2) and resolving for D(k), we get

D ( k ) = 3 M R 3 sin ( R k ) − R k cos ( R k ) k 3 (4)

where R is the radius of the spherical object, k is the spatial frequency and M is the mass of the object. The function D(k) has unit kg and can be understood as being the decomposition of the mass M in different spatial frequencies k’s. Equation (4) is plotted in

From

of the object after the application of the inverse Fourier transform, as will be explained below.

Reconstruction of the Gravitational PotentialTo facilitate the understanding of topics to be discussed later, we will normalize the function D(k), that is, divide it by M and introduce the function F(k) which is simply D(k)/M.

The inverse Fourier transform associated with the Fourier transform of Equation (2) is defined as

D ( r ) = 1 ( 2 π ) 3 ∫ 0 ∞ M ⋅ F ( k ) ⋅ 4 π k 2 ⋅ sin ( k r ) k r d k (5)

Upon inspection, it is verified that the right transformation for the classical Newtonian gravitational potential is given by Equation (6).

U ( r ) = 2 π ∫ 0 ∞ G M ⋅ F ( k ) ⋅ sin ( k r ) k r d k (6)

Which gives the correct value for the classical Newtonian gravitational potential, of the form GM/r, where G is the gravitational constant.

Plotting the Equation (6) along with its negative derivative with respect to r ( a ( r ) = − d U ( r ) / d r ), which is the force per unit mass i.e., gravitational acceleration, we obtain as shown in

is plotted the value of the gravitational potential for a hollow sphere using the pair of direct and inverse Fourier transforms given respectively by the Equations (2) and (6) and using Equation (3) considering the volume V as the volume of the outer shell. Where the integration limits for the Equation (2) are taken between 0.9R and R. The respective value of its negative derivative (acceleration) is also showed in the dashed line.

Such previous examples prove the validity of the adopted method. Also, other validated cases are not shown here.

The same concept could be expanded to objects that do not necessarily have radial symmetry, for that it is enough to use the correct pairs of Fourier transforms for such cases. These results indicate that any radial distribution of matter can be decomposed in k-space, using Equation (2), and by using Equation (6) to reconstruct the corresponding classical Newtonian gravitational potential, which by the negative value of its differentiation in space results in force per unit mass (acceleration).

In the relativity theory [

E 2 = ( p c ) 2 + ( M c 2 ) 2 (7)

With Mc^{2} being the rest mass energy (from now on the mass M is also considered the rest mass) and its “motion energy” given by the product of its momentum and the speed of light c. Based on that, the theory presented here assumes that the use of Equation (6) only considers the rest mass M or in a similar way, the “rest energy” Mc^{2} to correctly calculate the classical Newtonian gravitational potential.

However, the motion energy of an object is not only due its translational motion, being also contained in the form of quantum motion of its constituent particles.

Such phenomenon is described in the well-known “Uncertainty Principle” [

Δ p ≥ ℏ 2 Δ x (8)

where ħ is the reduced Planck constant.

This principle can be used to estimate the average values for the motion energy of a quantum particle, which is inversely proportional to the dimension occupied by the particle ∆x.

It is assumed here that the increase in mass density will act in reducing the radius of the particles (more particles packed together) and consequently increasing the motion energy of its constituent particles. This can be clearly seen in

The spatial frequency k is directly related to the momentum of a particle through the De Broglie relation hk, where h is the Planck constant. As the Fourier transform indicated in Equation (2) is a 3D transform, the same is valid for the inverse transform. So, integrating the function all over the 3D space of the spatial frequency k, we obtain the expression in Equation (9), which indicates the gravitational potential due to the motion energy.

U M ( r ) ∝ ∫ 0 ∞ G 3 c 4 M 3 F ( k ) ⋅ k 2 sin ( k r ) k r d k (9)

where physical constants and extra terms were adjusted in order to give the same dimension as in Equation (6), since the unit for k is m^{−1}.

As the Equation (7) indicates that the rest mass energy and the momentum energy are perpendicular to each other we can suppose that the angle total energy E is shifted in arctan (Momentum energy/Total Energy) in relation to the rest mass. This all can be summarized in the Equation (10).

U T ( r ) = π 2 ∫ 0 ∞ F ( k ) ⋅ ( G M ) 2 + ( 2 G 3 c 4 M 3 ⋅ k 2 ) 2 ⋅ sin ( k r − arctan ( 2 G 2 c 4 M 2 ⋅ k 2 ) ) k r − arctan ( 2 G 2 c 4 M 2 ⋅ k 2 ) d k (10)

where U_{T} is the complete solution for the gravitational potential. The terms in the square root are respectively proportional to the rest mass energy and motion energy and the term inside the arctan function represents the ratio between them. The values 2 multiplying the motion energy were chosen in order to adjust the equation to the GR theory as seen later. The Equation (10) is reduced to the classical gravitational potential as in Equation (6) in the limit R ≫ R S , where R_{S} is the Schwarzschild radius.

Making use of the Equation (10), we are able to plot for the condition for the formation of a black hole R = R S , being R S = 2 G M / c 2 .

For comparison we use the relativistic gravitational potential as obtained by [

U T ( r ) = − c 2 2 ln ( 1 − 2 G M r c 2 ) (11)

By analyzing

Which gives a pretty good agreement and besides that offers an understanding of how the gravitational field behaves internally to the massive body, showing that the gravitational potential only tends to infinity in the border of the black hole.

Such procedure can be used for calculation of the gravitational field near massive bodies approaching the Schwarzschild limit.

An important question that can be raised throughout this topic is how the mass contained in an object can deflect the space-time fabric to exert a force at the distance on some other object? As an answer, we can infer that any object that has mass has constituent particles that are always in motion, and that movement would be transmitted to the space-time fabric in a way to combine with the other vibrations originated by other particles building up a resulting total deflection of the space-time fabric in all internal and external regions of such object.

The only difference between the deflection caused by rest mass and the motion energy of these particles would be the way they "move", causing different effects on the space-time fabric that would later be transmitted to all regions of the space.

Up to here, we have worked with gravitational effects more evident over small

distances on cosmological scales ≪ 1 kpc, contained in smaller dimensions than those of our solar system. Since the extra gravitational effect of black holes seems more obvious in their surroundings, it became asymptotically more similar to the classic Newtonian model for greater distances, since the extra effect due to the motion energy of the quantum particles decays more abruptly as distance increases.

However, for larger cosmological scales in the order of kpc, we verify the existence of the phenomenon known as dark matter, and on even larger scales (intergalactic medium) we verify the existence of the phenomenon of dark energy.

In the next topic, we will demonstrate that both dark matter and dark energy can be caused by the existence of the quantization of the spatial frequency k, which can be understood as a limitation of the matter moving at all values of spatial frequency k, or a limitation of the space-time field of being excited in all spatial frequencies, with such excitations only occurring in steps.

We must clarify here that the quantization of the spatial frequency does not affect the results obtained in the present topic, since it only has effects on a scale much larger than that of our solar system.

We here postulate the existence of a minimum value for the spatial frequency k, being the other values multiples of this minimum value. Therefore, the spatial frequency can only acquire discrete values, such as k_{m}, 2k_{m}, 3k_{m}, and so on.

This leads to a discrete representation of the integral in Equation (6), with an increment of k_{m}, as indicated in Equation (12).

U ( r ) = 2 π ∑ n = 1 ∞ G M ⋅ F ( n ⋅ k m ) sin ( n ⋅ k m ⋅ r ) n ⋅ k m ⋅ r k m a ( r ) = − d d r U ( r ) (12)

The value of k_{m} should be considered as very small, in such a way that the value of 2π/k_{m} is on the galaxies size scale (kpc). Therefore, the celestial objects considered in this work are galaxies in general, where the effects of the quantization of the spatial frequency are evident.

To demonstrate how the quantization of the spatial frequency k allows us to deduce the effect of dark matter, we will take the example of the Milky Way.

The distribution of mass in the bulge of the Milky Way D_{b}(r) can be modeled according to Equation (13) [

D b ( r ) = 3 ⋅ M ⋅ b 2 4 ⋅ π ⋅ ( r 2 + b 2 ) 5 2 (13)

With M = 9.5 × 10 9 M ⊙ and b = 1.9 kpc . This equation can be approximated using a bell curve, as indicated in Equation (14).

D b ( r ) = M ⋅ e − ( x b ) 2 π 3 2 ⋅ b 3 (14)

In

After the application of Equation (14) in Equation (2), with R → ∞ , we obtain the mass distribution in k-space, also a bell curve, in terms of the spatial frequency k, as shown in Equation (15).

D b ( k ) = M ⋅ e − ( k ⋅ b ) 2 4 F b ( k ) = D b ( k ) M = e − ( k ⋅ b ) 2 4 (15)

Applying this value in Equation 12, with k m = 1 kpc − 1 and comparing the effect with the classic Newtonian force, we get

Being therefore sufficient the occurrence of these regions to explain the fact that most spiral galaxies have flat rotational velocity curves, since such galaxies have a high concentration of mass in a volume in their center (bulge) as modeled in the present case. Besides, there are adjacent regions of negative gravitational force, which effect is to expel objects orbiting the galaxy towards regions of positive gravitational force, being therefore most likely responsible for the creation of voids in the galaxies, where there is a lower concentration of matter.

The present model is perfectly symmetrical, requiring more realistic models for the precise determination of where the matter will accumulate. More precisely, it is necessary and suggested by the author to run complex computational simulations to analyze the evolution of spiral galaxies, considering the quantization of the spatial frequencies.

Now applying the relationship v r = r ⋅ a ( r ) , where v r is the rotational velocity, a(r) is the acceleration, and r is the radial distance, we obtain

From

quantization of the spatial frequency k. The quantization of the spatial frequency used in

Knowing that the acceleration a_{cc}, in terms of the displacement variable s, can be expressed as indicated in Equation (16).

a c c = d v d t = d v d s d s d t = d v d s ⋅ v ↔ a c c ⋅ d s = v ⋅ d v (16)

Integrating both sides and solving for the velocity v, we obtain

∫ a c c ⋅ d s = ∫ v ⋅ d v ↔ v = 2 ∫ a c c ( s ) ⋅ d s (17)

Therefore, we have an expression that relates the variations of acceleration in space s, with velocity v acquired by a mass. Consequently, we notice that if the integral of the acceleration in space is limited ( ∫ a c c ( s ) ⋅ d s ), the velocity v will have a constant value, causing a zero acceleration of a mass placed in that field. Now considering a(r) as this acceleration, we have the necessary conditions to verify if the acceleration of gravity with the effect of the quantization of the spatial frequency causes a net acceleration or not. The force per unit mass a(r) (acceleration) due to the quantization varies positively and negatively inscribed in an envelope of 1/r (

According to what was discussed above, we must assume that, on average, the net gravitational force between galaxies is canceled. It hence requires a force of repulsion much smaller than what is currently used to explain the accelerating expansion of the universe.

The next topic goes on to explain that the average luminous density of the universe is a sufficient factor to explain the commonly accepted rates of acceleration of the universe.

In this topic, the influence of luminosity on the expansion of the universe is evaluated considering the net gravitational effect of the attraction between galaxies as null, as previously discussed.

Luminosity is defined as the total electromagnetic power emitted by a source, and for astronomy, such sources can be stars, galaxies, supernovae, or any other astronomical object [^{2}), commonly named “apparent brightness,” is proportional to the inverse of the square of the distance (1/r^{2}). This relation is quite useful for our study, since we can borrow some concepts used in classical gravity, which also follow an inverse square law.

Considering that the universe is homogeneous and isotropic when viewed at large scales, we can make use of the shell theorem, which allows significant simplifications for objects subjected to gravitational forces inside or outside a shell or sphere.

We realize, in

Since apparent brightness (W/m^{2}) also follows the inverse square law, just like the force of gravity, we can, by analogy, use the same simplification as for the luminosity.

Using again

This simplification, again, is only possible because, just like gravity, the apparent brightness also follows the inverse square law (1/r^{2}). However, this simplification has some constraints; among them, we can list:

1) The presence of intergalactic dust causes an attenuation of the order 0.01 mag∙h∙Gpc^{−1} [

2) The mean free path M.F.P for a photon traveling through the local universe is given by Equation (18), Where R_{g} is the mean galaxy radius, and n is mean density of galaxies per volume. n can be estimated as 0.014 h^{3}∙Mpc^{−3} [_{g} as 0.015 Mpc. Replacing these values into Equation (18) we obtain M.P.F = 101.05 h^{−1} Gpc.

M . F . P = 1 n ⋅ π ⋅ R g 2 (18)

So, assuming the two constraints above, we can fairly say the universe is transparent for our analysis, being this way, the simplification given by

Since the universe is expanding, the mean luminosity density given by ρ L ( t ) varies in time. It leads to a total luminosity L inside a sphere of a comoving radius R(t) given by:

L = ρ L ( t ) ⋅ 4 ⋅ π ⋅ R ( t ) 3 3 (19)

Generating a luminosity pressure P

P = L 4 π R 2 1 c (20)

According to [

B ( z ) = ( 1 + z ) − 4 B 0 (21)

where z is the redshift, and B_{0} is the surface’s bolometric luminosity at z = 0. The Tolman relation implies that galaxies at redshift z will appear fainter by a factor of (1 + z)^{4}. This relation is valid for an observer placed at z = 0. If the observer is moved to a position at the point A, at a redshift z, in relation to the center of a sphere, as indicated in

P = 1 a 4 L 4 π R 2 1 c (21)

A galaxy with a face-on surface S, as indicated in

A c c = 1 a 4 ⋅ L 4 π R 2 ⋅ 1 c ⋅ S ⋅ k M (22)

Replacing L of Equation (19) in Equation (22), produces

A c c = 1 a 4 ⋅ ρ L ( t ) ⋅ R ( t ) 3 ⋅ 1 c ⋅ S ⋅ k M (23)

Applying the scale factor a ( t ) ⋅ R 0 = R ( t ) , we get the following relationship for acceleration

A c c = d 2 a ( t ) d t 2 = a ¨ ⋅ R ( t 1 ) (24)

The Equation (25) is derived in [

ρ L ( t ) = ρ L ( t 1 ) a 3 (25)

Where t_{1} is the present time and ρ L ( t 1 ) is the luminosity density of the present time. Replacing Equations (24) and (25) in Equation (23) yield:

a ¨ ⋅ R ( t 1 ) = 1 a 4 ⋅ ρ L ( t 1 ) ⋅ a 3 ⋅ R ( t 1 ) a 3 ⋅ 3 ⋅ 1 c ⋅ S ⋅ k M = 1 a 4 ⋅ ρ L ( t 1 ) ⋅ R ( t 1 ) 3 ⋅ 1 c ⋅ S ⋅ k M a ¨ = 1 a 4 ⋅ ρ L ( t 1 ) 3 ⋅ 1 c ⋅ S ⋅ k M } (26)

The Equation (26) is correct for galaxies, which are partially opaque [

Studies conducted by [^{2}, produces

( a ˙ a ) 2 = − 2 ⋅ ρ L ( t 1 ) ⋅ S ⋅ k a 5 ⋅ 9 ⋅ c ⋅ M + 2 ⋅ B a 2 (27)

Which is an expression for the Hubble constant since H = a ˙ / a , where B is a term to be determined. The term B can be determined using initial conditions, assuming that the gravitational force between galaxies is on average, nullified. This means that for galaxies, placed, as shown in _{g}, with R_{g} being the average galactic radius. We can call the process, where the galaxies start to move away as “luminosity dominance,” and the time of its occurrence as t_{D}, during this time we have

a ( t D ) = R ( t D ) R ( t 1 ) (28)

where R ( t D ) is the mean distance between galaxies at the “luminosity dominance” time t_{d}, and R ( t 1 ) is the mean distance between galaxies at the present

time t_{1}.

The mean distance between galaxies at time t_{D}, as seen in

R ( t 1 ) = ( 3 4 π n ) 1 3 (29)

where n is the present time density number of galaxies. Another important consideration to be made is assuming that at the time of the “luminosity dominance” t_{D} the recessional velocity is zero since it is the time when the luminosity repulsive force starts to overcome the gravitational force. So, consequently

a ˙ ( t D ) = 0 (30)

Applying Equations (28) and (30) in Equation (27), we obtain

( a ˙ ( t D ) a ( t D ) ) 2 = 0 = − 2 ⋅ ρ L ( t 1 ) ⋅ S ⋅ k a ( t D ) 5 ⋅ 9 ⋅ c ⋅ M + 2 ⋅ B a ( t D ) 2 (31)

Solving for B yields

B = ρ L ( t 1 ) ⋅ S ⋅ k a ( t D ) 3 ⋅ 9 ⋅ c ⋅ M (32)

Replacing this in Equation (27) we obtain

( a ˙ a ) 2 = − 2 ⋅ ρ L ( t 1 ) ⋅ S ⋅ k a 5 ⋅ 9 ⋅ c ⋅ M + 2 ⋅ ρ L ( t 1 ) ⋅ S ⋅ k a ( t D ) 3 ⋅ a 2 ⋅ 9 ⋅ c ⋅ M (33)

Rearranging for the Hubble constant

H ( t ) = − 2 ⋅ ρ L ( t 1 ) ⋅ S ⋅ k a ( t ) 5 ⋅ 9 ⋅ c ⋅ M + 2 ⋅ ρ L ( t 1 ) ⋅ S ⋅ k a ( t D ) 3 ⋅ a ( t ) 2 ⋅ 9 ⋅ c ⋅ M (34)

OBS: The factor scale factor a here presented is not the acceleration.

Considering the Equation derived for the Hubble constant (Equation (34)), we can verify that the second term inside the square root is equal to the first term when t =t_{D}. For t >t_{D} the first term increasingly dominates as the time increases. So, for t ≫ t D we can approximate the Equation (34) as

H ( t ) = a ˙ a = 2 ⋅ ρ L ( t 1 ) ⋅ S ⋅ k a ( t D ) 3 ⋅ a ( t ) 2 ⋅ 9 ⋅ c ⋅ M (35)

Rearranging we obtain

a ˙ 2 = 2 ⋅ ρ L ( t 1 ) ⋅ S ⋅ k a ( t D ) 3 ⋅ 9 ⋅ c ⋅ M (36)

Equation (36) can be solved for the scale factor a. We can “guess” a solution form for the scale factor a as being

a ( t ) = D ⋅ t U (37)

where t is time, D and U are constants to be determined, replacing Equation (37) in Equation (36) and rearranging we get

D 2 ⋅ U 2 ⋅ t 2 U − 2 = 2 ⋅ ρ L ( t 1 ) ⋅ S ⋅ k a ( t D ) 3 ⋅ 9 ⋅ c ⋅ M (38)

Since the right term is a constant, we need to set U to force the left term to be also a constant, this is done doing

2 U − 2 = 0 → U = 1 (39)

Replacing U = 1 in Equation (38) and solving for D yields

D = 2 ⋅ ρ L ( t 1 ) ⋅ S ⋅ k a ( t D ) 3 ⋅ 9 ⋅ c ⋅ M (40)

Therefore, the approximated scale factor for t ≫ t D is

a ( t ) ≈ 2 ⋅ ρ L ( t 1 ) ⋅ S ⋅ k a ( t D ) 3 ⋅ 9 ⋅ c ⋅ M ⋅ t (41)

The Equation (41) is a good approximation for values of t ≫ t D but is not the whole answer for a(t), since if we derive twice the value present in Equation (41) it will become zero, what is not valid, since the acceleration value for a ¨ ( t ) is given in Equation (37). Therefore, we can refine the value of a(t) doing the same procedure as before for the acceleration term in Equation (37). Calling the acceleration term as a_{acc}(t), and guessing the answer to be in the form a a c c ( t ) = D ⋅ t U , we get

D 5 ⋅ U ⋅ ( U − 1 ) ⋅ t 5 ⋅ U − 2 = ρ L ( t 1 ) 3 ⋅ 1 c ⋅ S ⋅ k M (42)

Using the same procedure as in Equation (39) leads to U = 5/2, and solving for D produces

a a c c ( t ) = − ( 25 ⋅ ρ L ( t 1 ) ⋅ S ⋅ k 18 ⋅ c ⋅ M ) 1 5 ⋅ t 2 5 (43)

Adding this term to Equation (41), yields

a ( t ) ≈ − ( 25 ⋅ ρ L ( t 1 ) ⋅ S ⋅ k 18 ⋅ c ⋅ M ) 1 5 ⋅ t 2 5 + 2 ⋅ ρ L ( t 1 ) ⋅ S ⋅ k a ( t D ) 3 ⋅ 9 ⋅ c ⋅ M ⋅ t (44)

Considering our initial time as t_{D}= 0, we have the value of a(t_{D}) in such a case. Adding this to Equation (44) we obtain the complete solution for the scale factor as a function of time t expressed in Equation (45).

a ( t ) = − ( 25 ⋅ ρ L ( t 1 ) ⋅ S ⋅ k 18 ⋅ c ⋅ M ) 1 5 ⋅ t 2 5 + 2 ⋅ ρ L ( t 1 ) ⋅ S ⋅ k a ( t D ) 3 ⋅ 9 ⋅ c ⋅ M ⋅ t + a ( t D ) (45)

According to [

a ( t D ) = R ( t D ) R ( t 1 ) = 20 kpc 2.538 Mpc = 7.88 × 10 − 3 (46)

[

[^{2}. For the opacity, k is considered the values obtained by [

ρ L ( t 1 ) = 3.387 × 10 − 33 W ⋅ m − 3

S = 6.734 × 10 41 m 2

k = 0.48

M = 3 × 10 41 kg

c = 3 × 10 8 m ⋅ s − 1

a ( t D ) = 7.88 × 10 − 3

Applying those values for Equation (34), for the present time t_{1}, with a(t_{1}) = 1, we obtain

H ( t 1 ) = 2.35 × 10 − 18 s = 72.54 km ⋅ s − 1 ⋅ Mpc − 1 (47)

which agrees very well within the interval of acceptable values for the Hubble constant ≈ 67 - 73 km∙s^{−1}∙Mpc^{−1}, obtained respectively by [

Using Equation 34, we can also plot the normalized value of the Hubble parameter (H(a)/H(t_{1})) against the scale factor a, with the result indicated in

Using the Equations (34) and (45) we can also plot the normalized value of the Hubble parameter (H(t)/H(t_{1})) and scale factor a against time in years (Yr), with the result indicated in

Looking at

The developed theory proved to be entirely satisfactory in developing a model of the gravitational force for massive objects, such as non-rotating black holes, in line with Einstein’s theory of relativity (GR). As merit, the present theory is able to determine the behavior of the acceleration of gravity within a black hole. Besides, the theory offers a straightforward and simple method to numerically calculate the effect of gravity near massive objects. It also provides a “bridge” between gravity and quantum physics, since the deflection of the space-time fabric can be understood as originated in the quantum scale.

The developed theory also proved to be quite satisfactory in the development of a model for the dark matter, besides demonstrating the average absence of gravitational force between galaxies. From the absence of average force between galaxies, it was possible to demonstrate that the average luminosity density of the universe is enough to explain the currently accepted rates of expansion of the universe. As a suggestion for future studies, the author recommends simulations of evolution of the galaxies, taking into consideration the effect of quantization of the spatial frequency k.

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

Holdefer, A. (2021) Non-Rotating Black Holes, Dark Matter and Dark Energy in a Unifying Theory. Journal of Applied Mathematics and Physics, 9, 1560-1582. https://doi.org/10.4236/jamp.2021.97107