Quantum Effects in General Relativity: Investigating Repulsive Gravity of Black Holes at Large Distances

: This paper proposes a theoretical study that investigates quantum effects on the gravity of black holes. This study utilizes a gravitational model that incorporates quantum mechanics derived from the classical-like quantum hydrodynamic representation. This research calculates the mass density distribution of quantum black holes, speciﬁcally in the case of central symmetry. The gravity of a quantum black hole shows contributions coming from quantum potential energy, which is also sensitive to the presence of a background of gravitational noise. The additional energy, stored in quantum potential ﬂuctuations and constituting a form of dark energy, leads to a repulsive gravity in the weak gravity limit. This repulsive gravity overcomes the attractive classical Newtonian force at large distances of order of the intergalactic length.


Introduction
One of the biggest challenges in physics today is the unification of general relativity and quantum theory.General relativity provides a description of gravity as the curvature of spacetime caused by the presence of matter and energy, while quantum theory describes the behavior of matter and energy at the smallest scales.
The problem is that these two theories are fundamentally different in their approaches, and attempts to merge them have been unsuccessful so far.One of the key challenges is the existence of singularities, such as those found in black holes, which arise from the application of general relativity at very small scales.
Another issue is the conflict between the principles of quantum mechanics and those of general relativity, such as the principle of locality, which states that information cannot travel faster than the speed of light, and the principle of unitarity, which requires that the total probability of all possible outcomes of an experiment add up to one.
There have been various proposals for reconciling general relativity and quantum theory, such as string theory [1][2][3], loop quantum gravity [4][5][6], and causal dynamical triangulation [7-9], but none have yet been confirmed by experimental evidence, and the search for a unified theory remains an active area of research in theoretical physics.
Since the discovery of the universe's accelerated expansion, scientists have been trying to determine what is driving this acceleration.However, despite many attempts to explain it, the current observational data cannot conclusively identify a source.
In 1926, Schrödinger presented his findings on wave mechanics [1,2] by formulating a linear differential equation for a complex wave function ψ(r, t) = n(r, t)exp iS (r, t) .However, in the same year, Madelung [3] discovered an alternative formulation that consisted of two real equations, which exhibited striking similarities to equations commonly used in hydrodynamics.The first equation was a continuity equation for the squared amplitude, representing the probability density n(r, t) = ψ * (r, t)ψ(r, t), where the phase

Cosmological Scalar Boson Mass under Self-Gravity
During the collapse of a black hole, its mass distribution becomes highly concentrated, but the repulsive force of its quantum potential may become strong enough to counteract the gravitational force and prevent its collapse.This can result in the formation of stationary mass distributions.The uncertainty principle ensures that the repulsive quantum nonlocal potential grows sufficiently to overcome the gravitational force, thereby preventing a point-like collapse.
When the mass distribution of a scalar uncharged boson becomes extremely concentrated in space, its gravitational force can generate stable self-bonded states.These states are the quantum mechanical analogue of a black hole predicted by general relativity.
In this section, the author investigates whether quantum potential force can stop gravitational collapse when the mass distribution approaches the classical point singularity.
In order to obtain quantum mechanical stationary black hole configurations on a cosmological scale with large mass distributions, we make the assumption that the mass field can be represented as a scalar variable.This simplified model of a scalar black hole mass serves as a "macroscopic" viewpoint that is acceptable for studying the gravitational behavior of black holes on a cosmological scale.
The distribution of mass in space-time (ST) is attributed to the formation of vacuum states resulting from the quantization of spinor and massive boson fields.This mass distribution in ST is not only non-continuous, but also exhibits physical phenomena arising from the other three fundamental interactions that remain out of the description.

Stationary Scalar Mass Distribution
In the case of a scalar mass field obeying the Klein-Gordon equation, the gravitationally coupled system of motion equation reads [11] The quantum contribution in (1) is contained in the energy impulse tensor that reads [11] T where [11] T where ∆ λλ is given in [11], and where the quantum potential reads The KGE (2) in hydrodynamic notation, as a function of the real variables |ψ| 2 and S, with momentum ∂ µ S = −p µ , leads to a couple of real variable equations, Following the same method used by Landau e Lifšits in general relativity [12] from the generalized Hamilton-Jacobi Equation (9), it is possible to derive (see Appendix A) the motion equation in curved spacetime [11] that reads (see Equation (A6) in Appendix A) (see Appendix A) and cD q 0 is the curvilinear covariant total time derivative.Moreover, by utilizing the relation Since we are interested in the stationary mass density distribution of a black hole, we have to impose the stationary condition where the force generated by the gravitational force c γ Γ α µν u α u ν is balanced by the quantum potential force In the classical case where = 0, and thence V qu = 0, the counterbalancing expansive quantum force is null, and thus, the collapse proceeds with the generation of point singularity.

The Mass Distribution in a Central Symmetric Scalar Uncharged Black Hole
In classical general relativity, the collapse of a central gravitational field results in a final point-like mass density being approached with increasing velocity.However, in the quantum case, the quantum potential generates an expansive force that counteracts gravity, leading to deceleration and potentially halting the collapse.As a result, stable stationary configurations may exist at an equilibrium point, eliminating the classical point singularity.This suggests that the interplay between quantum effects and gravity can lead to different outcomes than those predicted by classical general relativity.From a general standpoint, the stationary mass distribution, as described by Equation ( 12), depends on the metric tensor defined by the quantum Einstein gravity (QGE) Equation ( 1) and vice versa.Although the general solution of these coupled equations is quite complex, the simplifying assumption of central symmetry can be introduced to extract useful information.This assumption leads to the quantum analogue of the Schwarzschild black hole, where the metric tensor satisfies a particular condition [12].
Assuming that, in stationary distributions, the mass is enclosed in a sphere of the radius R 0 , for r > R 0 , we can use the approximated gravitational relations [12] −e −λ ν r from which the quantum potential reads [13] By introducing the relations ( 26)-(30) into the motion equation, it follows that and, by the stationary condition in the BH system of reference at a large distance, and g 00 , 0, 0, 0 ∼ = (1, 0, 0, 0) ( We obtain where R g is the gravitational radius of BH and R c is the Compton's length, leading to the BH mass density at large distances (see Appendix B), which follows the law The constant, G 0 , is defined by the normalization condition.

The Mass Distribution near the Center of a Schwarzschild Black Hole
In the classical case, a BH mass collapses into a point, whereas in the quantum case, for the uncertainty principle (see (46)), the maximum concentration is inside a sphere whose radius is in the order of magnitude of the Compton's length R c .Thence, for a macroscopically massive BH with the condition R g R c (for a BH with a mass m ∼ 10 35 kg R g R c ∼ 10 85 ), we can assume with good approximation that, in the limit r R g → 0 (at least r R g 10 −85 ), there is no mass for R c r < R g .Thus, by observing that and that 8πG Equation (A14), in Appendix B, can be also retained for x = r R g → 0 .Thus, providing that at small distance from the BH center, it holds we obtain the differential equation whose solution, given in Appendix B, reads The output (43) shows that, for large mass BHs (e.g., m ∼ 10 35 kg) the mass density is practically null outside the sphere of the Compton's radius at the center of the BH (for instance, at a distance r = 10R c , the mass density |ψ| 2 ∼ G 0 e − 2 1−ς 2 e 10 86 ∼ G 0 e − 10 86 (see Appendix B).
This output, in agreement with the uncertainty principle, leads to a piece of information about the minimum mass for the formation of BHs.In fact, in cases where the BH energy does not exceed the value mc 2 due to its localization (otherwise a new BH is formed), by the uncertainty principle, it follows that where which leads to mc 2 > 2 4m∆x 2 and, finally, to Additionally, since in order to form a BH, all the mass must be inside the gravitational radius, we must have that and, thence, that leading to the following condition for the black hole mass m: where m p = c 8πG is the reduced Planck mass.For small masses when m → 0 (quantum case), the gravitational radius R g tends to be zero while the Compton's radius R c goes to infinity so that, in order to have all the black hole mass inside its gravitational radius, for R c = R g , we have the minimum mass (49) for the formation of a black hole (R g = R c 4 ).This condition is safe for our universe since low energy elementary particles cannot form BHs.
On the other hand, it is noteworthy to observe that, for a very large mass m → ∞ , V qu ∝ 1 m → 0 (classical limit), the BH Compton's radius R c goes to zero and the point singularity of the classical general relativity is asymptotically approached.
Additionally, as black holes with a Planck mass cannot be divided into two smaller black holes, they represent the lightest possible configuration of scalar uncharged mass density that can be achieved solely through gravitational interaction.Moreover, since the condition expressed in Equation (49) also applies to quantized fields, the fundamental lowest state of a quantum black hole is heavier than π 1/2 m p .

Gravitational Field of Black Holes at Large Distance in Spacetime with Background Fluctuations
In this section, we derive the weak gravitational force of black holes over long distances.The large distance approximation is used because the gravitational radius of a black hole is much smaller than the cosmological physical scale, allowing us to treat the mass distribution of the black hole as point-like.
In fact, the mass distribution of a BH (36) arranged in the form at a large distance reads where the normalization condition ´|ψ| Furthermore, as black holes are quantum objects with significant quantum potential energy (as described in Appendix B), we anticipate that their gravity over long distances may result in quantum effects contributing to Newtonian law.
The contribution coming from the quantum potential, contained in the energy density tensor of the QGE, reads where it has used the identity 1 Thence, the gravitational Equation (53) in a mixed form reads leading to the equation for the trace of the Ricci tensor and to Given that, at a large distance, we can use the approximations (61) reads (64) leading to the identity By integrating the flux of the gravitational force ∂φ ∂q α with a sphere with the radius r − r BH , it follows that ˚∂ ∂q α ∂φ ∂q α dV = By the Dirac δ-shape approximation of the BH mass distribution, it can be posed that V qu (r−r BH =0) = 0, so that the BH gravitational field at a large distance reads where the repulsive force −Gm 1 3 overcomes the attractive one when From ( 59) we can observe that the cosmological pressure density originating from a BH at a large distance is constant and reads and that the repulsive gravity is generated by the presence of the dark energy/mass density |δψ 0 | of the background fluctuations.From (70) it is also interesting to note that the large distance mass density of a BH (36) acquires the form Generally speaking, beyond the centrally symmetric case, the pressure density tensor, denoted as Λ Q , is a function of mass fields and exhibits point dependence similar to the quintessence model.The key distinction lies in its dependence on the quantum properties of spacetime rather than an obscure physical field.Additionally, Λ Q can yield a cosmological constant, representing the mean value in the universe, with magnitudes consistent with observed values (see reference [18]).Furthermore, the definition of the minimum radius of a black hole mass distribution, which solves the case of the classical general relativity point singularity and, consequently, the determination of a minimum mass for black hole formation, represents the primary large-scale manifestation of quantum effects on the space-time curvature within the theory.The establishment of a minimum mass for black hole formation holds significant importance, ensuring a secure condition for our universe, as elementary particles cannot generate black holes, and the quantum instability of vacuum does not lead to the massive production of micro black holes.

Quantum Potential Fluctuations Generated by Background Fluctuations
To determine the parameter |δψ 0 | 2 , we must move beyond the static vacuum solution and consider that the vacuum is filled with stochastic gravitational waves.These waves originate from various sources, including relic gravitational waves from the Big Bang and other sources [19].
Considering the vacuum fluctuations in the background, it becomes possible to define the stochastic generalization of the quantum hydrodynamic equations [20] so that the wave function ψ = |ψ|e − iS , in the low velocity limit, is given by the equations . .
where S (q,t) = − 2 ln ψ ψ * , H is the Hamiltonian of the system, and V qu is given by the low velocity limit of (8).
The ripples of the vacuum curvature are assumed to manifest themselves through an additional fluctuating mass density δn vac into the vacuum so that where n is linked to n by the relation lim δn vac →0 n = n, that, introduced into the quantum potential leads to the quantum fluctuating force [20] − ∂q i that we are going to determine.Given that the energy/mass density δn vac is defined as positive, the mean vacuum fluctuations give rise to an additional non-zero (dark) energy density in the vacuum.
Given that the energy/mass density δn vac is defined as positive, this paragraph describes the assumption that the mean vacuum fluctuations < δn vac > give rise to an additional dark energy density in the vacuum.The assumption is made that this vacuum of dark energy/matter does not interact with the physical system under consideration, and therefore, the gravity interaction is disregarded in the Hamiltonian H in (74).The evolution of the total dark energy is assumed to depend on cosmological dynamics and to have reached an equilibrium configuration.

Spectrum and Correlation Function of Mass Density Noise in Quantum Spacetime with Curvature Fluctuations
When determining the features of the fluctuations of quantum potential, which consequently produce force noise, we employ the postulate that the fluctuations of the vacuum curvature are described by the wave function ψ vac with the density δn vac = |ψ vac | 2 and that they do not have a Hamiltonian interaction with the physical system (gravitational interaction is disregarded).
In this case, the wave function of the overall system ψ tot reads Moreover, by assuming that the equivalent mass of dark energy is much smaller than the mass of the system (i.e., m tot = m dark + m ∼ = m), the overall quantum potential (8) reads Moreover, given the vacuum mass density noise of wave-length λ, associated with the fluctuation wave function it follows that the quantum potential energy fluctuations read where For V → ∞ , the unidimensional case leads to In (84), the normalization condition V n tot(q,t) dq = n tot V has been used, and for a large volume (V λ c 3 see (87)), the following approximation has been used: For the three-dimensional case, (84) leads to Equation ( 85) reveals that the energy arising from the mass density fluctuations of the vacuum becomes greater as the square of the inverse of λ.Thus, the corresponding fluctuations in quantum potential produce extremely large energy fluctuations δE qu , even for very small noise amplitudes (i.e., T → 0 when λ approaches zero) at very short distances.
A convergence to the deterministic limit of quantum mechanics (72-74) (for T → 0 ) is warranted by the fact that the higher the energy-to-noise amplitude ratio kT is, the smaller the probability is of a convergence happening.This brings a condition on the spatial correlation function of the quantum potential noise as λ → 0 or for T → 0 .
One way to obtain the shape of the spatial correlation function G(λ) is through a stochastic calculation, which can be quite complex [20].However, a simpler approach for obtaining G(λ) can be achieved by considering the spectrum of the fluctuations, as described in [16].
Since each component of spatial frequency, k = 2π λ , brings the quantum potential energy contribution (84), its probability of happening reads where is the De Broglie length.From (86), it comes out that the spectrum S(k) of the spatial frequency is not white, and the components with a wave-length, λ, smaller than λ c go quickly to zero.Additionally, from (88), the spatial shape G(λ) reads One can see from Equation (89) that the quantum potential progressively suppresses uncorrelated mass density fluctuations at shorter and shorter distances, which in turn allows for the realization of deterministic quantum mechanics in systems whose physical length is much smaller than the De Broglie length.

The (Dark) Energy Density of Quantum Potential Fluctuations
The energy associated with the quantum potential noise of a body with a mass m can be evaluated using the probability energy fluctuation function where where In this case, the energy density of the quantum potential fluctuation reads where, for (85) in the three-dimensional case λ =|k|, m|δψ 0 | 2 is the additional mass density in the vacuum that the black hole mass m acquires due to the background fluctuations, and where where l u is the diameter of the universe.Moreover, for SMBHs (in the order of Sagittarius A* with a mass of about of 10 38 kg) at and that leading to

Repulsive Gravity at Large Distance
By introducing (98) into (69), the repulsive Newtonian gravity overcomes the attractive one at the distance The gravitational force between galaxies becomes repulsive at intergalactic distances, which is on the same order of magnitude as the typical radius of galaxies (∼ 10 20 m).This may affect the external part of the galactic disc.However, since the energy density of BH quantum potential fluctuations decreases with the expansion of the radius l u of the universe according to lim t→∞ |δψ 0 | 2 (r) ∼ 1 l u 2mc πkT 2mc 2 → 0 ), the BH repulsive force asymptotically approaches zero, leading to a final static universe.This effect can furnish the empirical confirmation of the theory.
Since, as shown in [21], the quantum potential of macroscopic low-density mass bodies is practically null and does not contribute to the expansive gravity of the universe, the repulsive gravitational force in (100), causing the repulsion of the galaxies, is mainly attributed to black holes and supermassive black holes due to the quantum nature of space-time with fluctuating background metrics.
The correction to Newtonian gravity that arises from the fluctuations of quantum potential in massive bodies such as BHs and SMBHs complies with the concept of modified Newtonian dynamics (MOND) [22], which suggests a modification in Newtonian gravity for very low accelerations in order to account for the observed motion of the galaxies.

Conclusions
This work shows that quantum black holes with a central symmetry have a mass density distribution that is not point-like but is concentrated in a sphere with a radius in the order of its Compton wavelength.
Due to significant quantum potential energy, there exists a supplemental term in the gravity equation, resulting in an additional contribution to the gravity force at large distances.
The pressure density tensor is a function of mass fields and demonstrates a pointspecific behavior akin to the quintessence model.However, what sets it apart is its reliance on the quantum properties of space-time instead of an elusive physical field.
The present model shows that the alteration of Newtonian gravity over long distances is explained by the gravitational effect of the quantum potential of enormously massive entities, such as black holes and supermassive black holes, subject to background dark energy fluctuations.In the presence of fluctuations in the space-time background metric, the dark energy arising from fluctuations in the quantum potential energy of black holes results in a repulsive contribution to gravitational force.This repulsive force of SMBH dominates over the Newtonian force at distances characteristic of intergalactic space.On this basis, it has the capacity to generate a cosmological constant producing the acceleration of the universe, aligning reasonably well with the observed low value.
Furthermore, the establishment of a minimum radius for the mass distribution of black holes, which solves the issue of classical general relativity's point singularity, and subsequently, the determination of a minimum mass required for black hole formation, represents the foremost large-scale manifestation of quantum effects on the curvature of spacetime as posited by the theory.This concept holds important significance as it ensures the stability of our universe, preventing elementary particles from spontaneously generating black holes and averting the excessive production of microscopic black holes due to quantum vacuum instability.x (to be checked at the end), Equation (A14) simplifies to where x 1 and y (x) = u R g u , leading to the solution