Quantum Effects in General Relativity: Investigating Repulsive Gravity of Black Holes at Large Distances
Round 1
Reviewer 1 Report
While reading this ms many questions appear.
Say, if eq.2.5 is quatum potential then which theory induces it? Nothing is said about. Why there is no contribution from potential to scalar field eq. ?
If Gr is quantum, why author does not take into account quantum effects of metric?
The presentation of grav field for BH is written again in very implicit and not understandable form.
It looks the author mixes classical gravity by quantum effects, changing situations in classical gravity theory by ad hoc expressions saying that these are due to quantum effects.
So I would not recommend this ms for publication unless above points will be clarified and presentation will be given in logical way: theory then eqs of motion then solutions then approximations.
IT is OK..
Author Response
Response to Referee 1
While reading this ms many questions appear.
Say, if eq.2.5 is quantum potential then which theory induces it?
The quantum potential is firstly introduced by Madelung (and then reworked by Bhom)
The quantum hydrodynamic representation of the quantum field equation for the complex variable is reorganized as a function two real variables, namely: the mass densities , and the momentum . The result obtained by Madelung is the classical-like equation of motion with the additional presence of the theory-defined quantum potential.
In the paper the Madelung approach is generalized to the relativistic case of the Klein-Gordon equation ( for scalar uncharged fields). This can be easily checked in the low velocity limit where the Klein-Gordon equation converges to the Schrodinger one and the quantum potential in equation 2.5 converges to the Madelung quantum potential.
In the relativistic case, the Madelung procedure leads to the relativistic Hamilton-Jacobi like equation (where the quantum effects are introduced by the relativistic quantum potential) plus the current conservation equation.
Equation (2.1) has been changed in a more suitable form in order to make evident the quantum potential contribution into the Gr equation.
As observed since Madelung, the quantum effects are introduced by the quantum potential that is clearly non local thanks to its derivative structure where the mass density in a point depends by the values nearby.
Nothing is said about. Why there is no contribution from potential to scalar field eq. ?
The quantum potential gives a clear contribution that can be noticed both into the hydrodynamic Hamilton-Jacobi Equation (2.9) and in the motion equation (2.17) where it generates a force acting on the mass density even in absence of external potential.
If Gr is quantum, why author does not take into account quantum effects of metric?
Since the quantum hydrodynamic equations and the Klein-Gordon equation are equivalent, the quantum potential effect in the Klein-Gordon scalar field is implicit. Nevertheless, the quantum contribution in the Klein-Gordon equation in curved spacetime is anyway introduced by the metric tensor of the gravity equation (2.1). through the energy impulse tensor that is a function of the quantum potential (see eq. (2.3-2.8).
The presentation of grav field for BH is written again in very implicit and not understandable form.
More mathematical passages are added into the text to explain the derivation of the motion equation (2.17).
It looks the author mixes classical gravity by quantum effects, changing situations in classical gravity theory by ad hoc expressions saying that these are due to quantum effects.
Actually, being the mass quite null beyond the gravitational radius of the black hole, the approximated solution of the general relativity coincides with the quantum Gr. On the basis of this (zero order) approximated gravitational field, the mass density distribution whose quantum potential generates a force able to exactly counterbalance the gravitational force and leading to stationary configuration is derived
Below the link of published paper (ref. [11] that the referee can utilize for checking the formulas:
Symmetry 2019, 11(3), 322; https://doi.org/10.3390/sym11030322
So I would not recommend this ms for publication unless above points will be clarified and presentation will be given in logical way: theory then eqs of motion then solutions then approximations.
To better explain the content of the paper in the context of the current research it can be useful to compare the proposed theory with the non-commutative geometry approach. One of the main problems in the General Relativity is that the energy-impulse tensor density for classical bodies has a point-dependence on mass density so that the most general form remains undefined.
About this point, the non-commutative geometry shows that the general form of energy-impulse tensor density can introduces the quantum phenomenon of uncertainty and, hence, to reproduce the properties distinctive of quantum mechanics. Nevertheless, these uncertainty relations are not generally equal to those given by the quantum mechanics, but there is a degree of freedom that must be fixed.
In order to overcome this problem, the form of the energy tensor for the matter (such as, for instance, the Tolman-Oppenheimer- Volkov equation) is semiempirical assumed.
In the present work the energy impulse tensor (2.3) plays the same role of the Tolman-Oppenheimer- Volkov equation.
The the energy impulse tensor (2.3) basically fixes the undefined parameters and on the basis of quantum mechanics through the quantum potential (whose action against the mass concentration are at the basis of the quantum uncertainty relations). In this way, the resulting uncertainty relations are those of quantum mechanics.
The connection with the quantum mechanics is maintained since the spacetime geometrization is obtained by imposing the covariance of the quantum mechanical field equations and by utilizing the canonical method of imposing the least action condition.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The author analyzes some results by using the Einstein equations with the starting point being some metric with spherical symmetry. The paper could be interesting if it is improved considerably. Then I suggest the author to work over the following points:
1). It is not clear what is Quantum in this paper. I assume that the author means that the metric he uses has somehow some quantum properties or that the mass source term entering the Einstein equations is a Quantum field. This is a semiclassical approach and the author should specify this and then compare with other models offering similar analysis like the ones in J.Math.Phys. 51 (2010) 022503; Int. J. Mod. Phys. A24: 1229, 2009, among others. For my eyes, the model looks like a non-commutative model but with a different starting point for the matter source content.
2). The references should be enumerated.
3). Can the model proposed by the author develop interesting phenomena at large scales? It seems yes but the author needs to explain more about this.
4). Does the model solves the singularity problems of Black-Holes? Please mention a little bit more.
5). Please explain the content of the paper in detail at the end of the introduction. For example: In Sec. (2), we explain.... and so on.
6). The author should specify more details about the origin of the potential in eq. (2.5).
7). Please extend the Conclusions and increase the number of text explanations in the paper.
After the author revise the paper in agreement with these comments, I will happily revise the paper again.
I find the English understandable. just revise potential typos and grammar.
Author Response
Response to Referee 2
The author analyzes some results by using the Einstein equations with the starting point being some metric with spherical symmetry. The paper could be interesting if it is improved considerably. Then I suggest the author to work over the following points:
1). It is not clear what is Quantum in this paper. I assume that the author means that the metric he uses has somehow some quantum properties or that the mass source term entering the Einstein equations is a Quantum field.
The quantum potential is firstly introduced by Madelung (and then reworked by Bhom)
The quantum hydrodynamic representation of the quantum field equation for the complex variable is reorganized as a function two real variables, namely: the mass densities , and the momentum . The result obtained by Madelung is the classical-like equation of motion with the additional presence of the theory-defined quantum potential.
In the paper the Madelung approach is generalized to the relativistic case of the Klein-Gordon equation (for scalar uncharged fields). The convergence to the Madelung theory can be easily checked by doing the low velocity limit where the Klein-Gordon equation converges to the Schrodinger one and the quantum potential in equation 2.5 converges to the Madelung quantum potential.
In the relativistic case, the Madelung procedure leads to the relativistic Hamilton-Jacobi like equation (where the quantum effects are introduced by the relativistic quantum potential) plus the current conservation equation.
Equation (2.1) has been changed in a more suitable form in order to make evident the quantum potential contribution into the Gr equation.
As observed since Madelung, the quantum effects are introduced by the quantum potential that is clearly non local thanks to its derivative structure where the mass density in a point depends by the values nearby.
In this work, the quantum contribution is contained into the metric tensor of the gravity equation, through the energy impulse tensor (2.3) that is function of the quantum potential. Thence, the metric tensor, implicitly introduces the quantum effects into the field equation (2.2) (and thence in the mass density equation (2.17)) in curved spacetime.
This is a semiclassical approach
The classical-like form of the gravity equation is consequence (and the advantage) of the Madelung quantum hydrodynamic approach since the motion equation owns a classical-like form where the quantum effects are contained into the quantum potential.
The semiclassical approach is obtained in the case when for the scale of the problem , but the quantum non-locality survives (that’s why it is widely recognized that the semiclassical approximation does not lead to the macroscopic classical dynamics).
The treatment is fully quantum since no hypothesis on the smallness of is introduced (for more detail see: Symmetry 2019, 11(3), 322; https://doi.org/10.3390/sym11030322
and the author should specify this and then compare with other models offering similar analysis like the ones in J.Math.Phys. 51 (2010) 022503; Int. J. Mod. Phys. A24: 1229, 2009, among others. For my eyes, the model looks like a non-commutative model but with a different starting point for the matter source content.
One of the main problems in the General Relativity is that the energy-impulse tensor density for classical bodies has a point-dependence on mass density so that the most general form remains undefined. About this point, the the non-commutative geometry shows that the general form of energy-impulse tensor density introduces the physical phenomenon of uncertainty relations and, hence, can reproduce the properties distinctive of quantum mechanics, is correct. Nevetheless, these uncertainty relations are not generally equal to those given by the quantum mechanics, but there is a degree of freedom that must be fixed.
In the first cited paper the form of the energy tensor for the matter is given by the Tolman-Oppenheimer- Volkov equation, that is of semi-empirical derivation, which introduces the quantum properties by an ad hoc equation.
On the other hand, the form of the energy impulse tensor (i.e, the parameters and) in the present work is uniquely defined by the quantum potential (whose action against the mass concentration are at the basis of the uncertainty relations) and the resulting uncertainty relations are exactly those of quantum mechanics. This basically because the spacetime geometrization is obtained by assuming the covariance of the quantum mechanical field equations and by imposing imposing the least action condition. In the present paper the quantum mechanical properties fixes both the pressure term as well as the out-diagonal shear terms of the mass energy tensor leading, so, exactly to the uncertainty relations of quantum mechanics.
The second cited paper is on the same approach. Comparing the non-commutative geometry approach with the present work, it can be observed that the quantum potential fixes some properties of the spacetime non-commutativity and (for the unidimensional central symmetric black hole) leads to the minimum square area (see line 284-288) equal to the Compton’s length squared (which is the minimum area that can be explored in quantum mechanics with the energy ).
2). The references should be enumerated.
(re)done, it is an editorial typos .
3). Can the model proposed by the author develop interesting phenomena at large scales? It seems yes but the author needs to explain more about this.
The main large-scale phenomenon of the quantum effects on spacetime curvature are: the generation of the repulsive gravity at cosmological distance: the generation of a quintessence-like pressure density tensor that is able to give a mean value in the universe (i.e., the cosmological constant) of order of magnitude of the observed values (as shown in ref. (18)); The definition of a minimum mass for the formation of black holes that is really safe for our universe since elementary particles cannot form black holes and the quantum instability of the vacuum cannot leads to massive generation of micro black holes.
Explanations are added.
4). Does the model solves the singularity problems of Black-Holes? Please mention a little bit more.
This is the novelty of the introduction of quantum effects into the general relativity. The basis is the repulsive force of the quantum potential that grows when the mass density tends to be more and more concentrated and forbids the point singularity of the mass density.
An additional comment on this point is added into the revised text.
5). Please explain the content of the paper in detail at the end of the introduction. For example: In Sec. (2), we explain.... and so on.
This important note is introduced at the end of the introduction.
6). The author should specify more details about the origin of the potential in eq. (2.5).
Specific text is introduced about this point.
7). Please extend the Conclusions and increase the number of text explanations in the paper.
The advice, that increase the readability of the paper, has been followed
After the author revise the paper in agreement with these comments, I will happily revise the paper again.
Thanks for the useful suggestions.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The author actually avoided to answer clearly to my questions and make corresponding changes in the ms. It remains to be messy and badly understandable for readers.
Hence I suggest to do much more serious major revision of ms but not to send readers to some other papers in order to understand what is quantum potential etc. I still cannot recommend this ms for publication in present form.
ok
Author Response
The final objection concerns the clarity of the manuscript. While the subject matter of the work itself is inherently complex and possesses intrinsic intricacies (which I cannot alter), I have made revisions to simplify the presentation of concepts and mathematical developments, making them easier to comprehend.
However, it is not feasible to include all calculations (which have already been verified) from other published papers, primarily due to the inherent complexity of the subject matter.
Reviewer 2 Report
The authors have addressed the comments of the referee. The paper can be accepted after the author check the format of the journal. Some letters and typing appear distorted, perhaps because the author was using an editing version.
No comment
Author Response
There are no further objections. Nevertheless, since the subject matter of the work itself is inherently complex and possesses intrinsic intricacies (which I cannot alter), I have made revisions to simplify the presentation of concepts and mathematical developments, making them easier to comprehend.
Round 3
Reviewer 1 Report
It is still that ms is aimed to very few people including author themselves. It maybe accepted.
ok
