Hindawi Advances in High Physics Volume 2018, Article ID 4537058, 7 pages https://doi.org/10.1155/2018/4537058

Research Article A Covariant Canonical Quantization of

Stuart Marongwe

Department of Physics and Astronomy, Botswana International University of Science and Technology, P. Bag , Palapye, Botswana

Correspondence should be addressed to Stuart Marongwe; [email protected]

Received 13 June 2018; Accepted 8 November 2018; Published 19 December 2018

Guest Editor: Farook Rahaman

Copyright ©  Stuart Marongwe. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e publication of this article was funded by SCOAP.

A Hamiltonian formulation of General Relativity within the context of the Nexus Paradigm of quantum gravity is presented. We show that the Ricci flow in a compact matter free manifold serves as the Hamiltonian density of the vacuum as well as a evolution operator for the vacuum energy density. e metric tensor of GR is expressed in terms of the Bloch energy eigenstate functions of the quantum vacuum allowing an interpretation of GR in terms of the fundamental concepts of quantum mechanics.

1. Introduction the geometry and matter content of the universe of each time slice. However, the Hamiltonian no longer determines e gravitational field which is elegantly described by Ein- the evolution of the system and leads to the problem of stein’s field equations has so far eluded a quantum descrip- timelessness [, ]. Hawking rightly points out that the very tion.MuchefforthasbeenplacedintoformulatingGeneral act of splitting -time into space and time destroys the Relativity (GR) in terms of Hamilton’s equations since a spirit of GR (general covariance) and therefore not much Hamiltonian formulation of a classical field theory leads can be gained from this approach to quantization of gravity. naturally to its quantization. e earliest such attempt is the Perturbative covariant approaches to the problem of quantum ADM formalism [], named for its authors Richard Arnowitt, gravity have an inherent weakness in that they depend on , and Charles W. Misner first published in a fine classical background. It is therefore difficult to obtain  . is formalism starts from the assumption that space a self-consistent quantum theory of gravity with a classical Σ is foliated into a family of time slices �,labeledbytheir background space-time. Steven Carlip [ , ] and Claus time coordinate t and with space coordinates on each slice � Kiefer [, ] have made excellent and extensive reviews of given by � . e dynamic variables of this theory are then the problems faced by current approaches to the problem taken to be the metric tensor of three-dimensional spatial of quantum gravity. Carlip [ ] in particular singles out the � �� � slices ��� (�.� ) andtheirconjugatemomenta� (�.� ).Using lack of a firm conceptual understanding of the foundational these variables it is possible to define a Hamiltonian and concepts of quantum gravity as the source of much of the thereby write the equations of motion for GR in Hamilton's difficulty in understanding quantum gravity. ese deep form.etimeslicesarethenweldedtogetherusingfour conceptual issues result in technical problems in the attempt Lagrange multipliers and components of a shi vector field. to develop a consistent quantum theory of gravity. An extensive review of this formalism can be found in the In this paper we report a successful covariant canonical literature notably in [– ]. quantization of the gravitational field which preserves the e ADM formalism was first applied by Bryce De Witt in success of GR while simultaneously explaining Dark Energy   [] to quantize gravity which resulted in the Wheeler–De (DE) and (DM). is approach to quantization Witt equation of quantum gravity. It is a functional differen- takes place in -space of metric signature (-,,,) in which tial equation in which the three-dimensional spatial metrics the quanta are excitations of the quantum vacuum called havetheformofanoperatoractingonawavefunction. Nexus gravitons. ough the Nexus Paradigm has been is wave function contains all of the information about introduced in the following papers [– ], the aim of this  Advances in High Energy Physics study is to explicitly express the Hamiltonian formulation where � is the amplitude of a displacement vector in of the theory using the Bloch eigenstate functions of the Minkowski space. If we consider the Hubble diameter as quantum vacuum. ese wave functions contain information the maximum dimension of the local patch of space then about the energy state of the quantum vacuum which in turn �=��� where ��� istheHubbleradius.ischoiceofa dictates the geometry of space-time. maximum amplitude is justified from the fact that we cannot physically interact with objects beyond the Hubble -radius. 2. Methods Itisimportanttonotethatthelineelementisthesquare of the amplitude of the displacement -vector. us if we Our first step towards a covariant canonical quantization are to express GR in terms of the language of QM we must begins with defining a quantized space-time and its quanta. make the radical assumption that the displacement vectors We then modify Einstein’s vacuum equations to be consistent in Minkowski space are pulses of -space which can be with the quantized space-time followed by the defining of expressed in terms of Fourier functions as follows: Hamilton’s equations of the quantized space-time. is step ∞ is then followed by the Poisson brackets which provide the � 2��� � ��� Δ�� = � ∫ sinc (��) � �� bridge between classical and quantum mechanics (QM). e �� −∞ ∞ ( ) covariant canonical quantization procedure is carried out � =� ∫ ����(���)�� within the context of the Nexus Paradigm of quantum gravity. −∞ 2� �=+∞ .. Quantization of -Space and the Nexus Graviton. e pri- �� = ∑ � Where �� �� () mary objective of physics is the study of functional relation- �=−∞ ships amongst measurable physical quantities. In particular, a unifying paradigm of physical phenomena should reveal ��� Here �(���) = sinc(��)� are Bloch energy eigenstate the functional relationship between the fundamental physical functions. e Bloch functions can only allow the four wave quantities of -space and -momentum. Currently GR and vector to assume the following quantized values: QM offer the best predictions of the results of measurement �� of physical phenomena in their respective domains using �� = �=±1,±2...1060 � () different languages. GR describes gravitation in the language ��� of geometry and thus far, it has been difficult to apply thethelanguageofwavefunctionsusedinQMtogive e minimum -radius in Minkowski space is the Planck - it a QM description. e problem of quantum gravity is length since it is impossible to measure this length without 60 therefore to interpret GR in terms of the wavefunctions of forming a black hole. e  statesarisefromtheratioof QM.TranslatingthelanguageofmeasurementofGRintothat Hubble -radius to the Planck -length. e displacement - of QM becomes the primary objective of this present attempt vectors in each eigenstate of space-time generate an infinite to resolve the problem of quantum gravity. Bravais -lattice. Also, condition () transforms ( ) to Measurements in GR take place in a local patch of a ��1 Reimannian manifold. is local patch can be considered � � Δ�� =� ∫ ����(�,�,�)�� () asaflatMinkowskispace.elineelementinMinkowski −��1 space which is the subject of measurement can be computed through the inner product of the local coordinates as e second assumption we make is that each displacement - � 2 2 2 2 2 vector is associated with a conjugate pulse of four-momentum Δ� Δ�� =Δ� +Δ� +Δ� −� Δ� which can also be expressed as a Fourier integral

=(�Δ�+�Δ�+�Δ�+���Δ�) () �� 2��1 1 ⋅ (�Δ� + �Δ� + �Δ� + ���Δ�) Δ�(�)� = �� ∫ �(�,�,�)�� � −�� 1 ( ) On multiplying the right hand side we see that to get all the ��1 cross terms such as Δ�Δ� to cancel out we must assume =�� ∫ ����(�,�,�)�� −��1 �� + �� = 0 () where �(1)� is the four-momentum of the ground state. �2 =�2 =⋅⋅⋅=1 A displacement -vector and its conjugate -momentum e above conditions therefore imply that the coefficients satisfy the Heisenberg uncertainty relation (�,�,�,�) generate a Clifford algebra and therefore must ℎ be matrices. We rewrite these coefficients in the -tuple Δ� Δ� ≥ () 1 2 3 0 � � 2 form as (� ,� ,� ,� ) which may be summarized using the Minkowski metric on space-time as follows: e Uncertainty Principle plays the important role of gener- � ] �] {� ,� }=2� () ating a vector bundle, out of the total uncertainty space E of trivial displacement -vectors from which a closed compact egammasareofcoursetheDiracmatrices.usinorder manifold X is formed, i.e., (� : � �→ �). Each point on the to satisfy () we can express a displacement -vector as manifold is associated with a vector which is along a normal � � Δ� =�� () to the manifold. Advances in High Energy Physics 

2 2 −1 e wave packet described by () is essentially a particle of ��2 =−(1−( )) �2��2 +(1−( )) ��2 2 2 four-space. e spin of this particle can be determined from � � () 2 2 2 2 the fact that each component of the four-displacement vector +� (�� + sin ��� ) transforms according to the law ere are no singularities in (). At high , character- �� 1 � ized by microcosmic scale wavelengths of the Nexus graviton Δ� = exp ( ��� [��,��]) Δ� () � 8 � and high values of n,space-timeisflatandhighlycompact. � is implies a continuous length contraction of the local where �� is an antisymmetric x matrix parameterizing the coordinates via the addition of more waves resulting in an transformation. increase in localization. Also, from a world line perspective of erefore, each component of the -vector has a spin half. a test particle, it implies that the deviation from a rectilinear A summation of all the four half spins yields a total spin of trajectory due to uncertainities in its location in -space is .WegivethenametheNexusgravitontothisparticleof- small. us at microcosmic scales, in the realm of subatomic spacesincetheprimaryobjectiveofquantumgravityistofind particles space-time is extremely flat. e world line begins the nexus between the concepts of GR and QM. to substantially deviate from a well defined rectilinear path at From () the norm squared of the -momentum of the low energies as the uncertainities in its location are increased. n-th state graviton is atis,atlowvaluesofn, the world line becomes degenerate �2 3(�ℎ� )2 allowing a multiplicity of trajectories which in GR are averaged (ℎ)2 ��� = � − 0 =0 () by the Ricci curvature tensor. us gravity is a low energy � 2 2 � � phenomenon which vanishes asymptotically at high energies. −18 −1 e gravitational effects of the Nexus graviton manifest where H0 is the Hubble constant (. x  s )andcanbe Λ at large scales and for galaxies, these effects begin to manifest expressed in terms of the cosmological constant, ,as when the density of DE is equal to the density of baryonic �2 3�2 matter as described by () or when the acceleration due to Λ = � = � =�2Λ � 2 2 () baryonic matter is equal and opposite to the acceleration due (ℎ�) (2�) to the emission of the ground state graviton as described in []. A solution to eqn () from [] in the weak field at We infer from () that the Nexus graviton (or displacement galactic scale radii is -vector) in the n-th quantum state forms a trivial vector 2 bundle via the Uncertainty Principle which generates a � � �� (�) = +�0V� −�0� () compact Riemannian manifold of positive Ricci curvature ��2 �2 that can be expressed in the form Here c is the speed of light. e first term on the right is the Newtonian gravitational � =�2Λ� (��)�� (�,�)�� () acceleration, the second term is a radial acceleration due to space-time in the n-th quantum state, and the final term is where �(��)�� is the Einstein tensor of space-time in the acceleration due to DE. e dynamics becomes strongly non- n-th state. Equation () depicts a contracting geodesic Newtonian when ball and as explained in [– ] this is DM which is an �� (�) V2 intrinsic compactification of the elements space-time in the =��= � 2 0 ( ) n-th quantum state. is compactification is a result of the � � superposition of several plane waves as described by () to ese are conditions in which the acceleration due to bary- form an increasingly localized wave packet as more waves are onic matter is annulled by that due to the DE. Under such added. Similary the converse is also true. e loss of harmonic conditions waves expands the elements of space-time which gives rise to V2 �= � () DE. us the DE arises from the emission of a ground state � � graviton such that () becomes 0

2 Substituting for r in ( ) yields �(��)�� =(� −1)Λ�(�,�)�� ( ) 4 V� =��(�) �0� () ese are Einstein’s vacuum field equations in the quantized is is the Baryonic Tully–Fisher relation. e conditions space-time. If the graviton field is perturbed by the presence permitting the DE to cancel out the acceleration due to of baryonic matter then ( ) becomes baryonic matter leave quantum gravity as the unique source

2 of gravity. us condition ( ) reduces () to �(��)�� =���] +(� −1)Λ�(�,�)�� 2 � � �V� () = =�0V� () 2 ��2 �� =���] +(� −1)�����(�,�)�� from which we obtain the following equations of galactic and where ��� is the density of DE. cosmic evolution: 1 From [] the static solution to () for a spherically (�0�) 1/4 �� = � (�� (�) �0�) () symmetric Nexus graviton is computed as �0  Advances in High Energy Physics

(�0�) 1/4 V� =� (�� (�) �0�) () expand or contract and does not execute translational motion (� �) 1/4 implying that the Hamiltonian density of the system is equal � =�� 0 (�� (�) � �) � 0 0 ( ) to the Lagrangian density. �=� () Here �� is the radius of curvature of space-time in the n- th quantum state (which is also the radius of the n-th state GR is a metric field in which the energy density in four- V nexus graviton), � the radial velocity of objects embedded in space determines its value. Since the Bloch energy eigenstate � that space-time, and � their radial acceleration within it. e functions determine the energy of space-time, we must amplification of the radius of curvature with time explains seek to express the metric in terms of the Bloch wave the existence of ultra-diffuse galaxies and the spiral shapes functions. To this end we shall express the eigenstate four- ofmostgalaxies.eincreaseinradialvelocitywithtime space components of the Nexus graviton in the k-th band as explains why early type galaxies composed of population II Δ�� =�� =� �� (��) ���� stars are fast rotators. Equation ( ) explains late time cosmic �,� �,� �� sinc () acceleration which began once condition ( ) was satisfied or such that an infintesimal four-radius within the k-th band is equivalently from (), when the density of baryonic matter computed as wasatthesamevalueasthatofDE.uscondition( )also ��� explains the Coincidence Problem. �� = �,� ��� =��� �� (��) ������ () �,� ��� �� sinc .. Canonical Transformations in the Nexus Paradigm. In In () the first-order derivative of the periodic sinc function classical mechanics, a system is described by n indepen- is equal to zero for all integral values of n.eintervalwithin dent coordinates (�1,�2,...��) together with their conjugate thebandisthencomputedas momenta (�1,�2,...��). In the Nexus Paradigm, the labeling ��� ��� �� refers to a creation of a Nexus graviton in the n-th 2 2 �,� �,� � ] �� =���,� = �� �� quantum state associated with a conjugate momentum ��. ��� ��] () e Hamiltonian equation � ] � ] =��� � �(�,�,�)�(�,�,�)�� �� �� ��̇ = () Here the interval is described in terms of the reciprocal lattice ��� �=��� �=��� with � �� and ] ��. e metric tensor of refers to the rate of expansion or contraction of space-time four-space in the k-th band is therefore associated with the generated by the addition or emission of harmonic waves to Bloch energy eigenstate functions of the quantum vacuum as the Nexus graviton and follows: ��] =���]� � =��]� � �� (�,�) (�,�,�) (�,�,�) (�,�,�) (�,�,�) ( ) ̇ �� =− () ��� Equation ( ) translates the geometric language of GR into thewavefunctionlanguageofQM. refers to the force field associated with the addition or We initiate the translation procedure of GR into QM by emission of harmonic waves to the Nexus graviton. It is first finding the Lagrange density for the quantized vacuum important to note that this force field generates an isotropic from ( ) which following Einstein and Hilbert is found to be expansion or contraction of space-time within the spatiotem- � = � (� − 2 (�2 −1)Λ) poral dimensions of the graviton. �� () We can also rewrite the Hamiltonian equations in terms Given that the Einstein tensor in a compact manifold is equal of Poisson brackets which are invariant under canonical to the Ricci flow transformations as 1 −����] =Δ��] =��] − ���] =��] () ��̇ ={��,�}, 2 ̇ () �� ={��,�} therefore the equations of motion of the quantum vacuum obtained from () yield the following quantized field equa- e Poisson brackets provide the bridge between classical and tions: � ] QMandinQM,thesebracketsarewrittenas −�� (� �(�,�,�)� �(�.�.�)) ̇ ̂ 2 � ] () �̂� =[�̂�, �] , =(� −1)Λ� � � � ( ) (�,�,�) (�,�,�) �̂̇ =[�̂ , �]̂ � � which can be written as � (��� �]� ) and obey the following commutation rules � (�−1,�,�) (�+1,�,�) 2 −� � ] [�̂�, �̂�]=0, = ∇ ∇ � � � � (2�)2 � ] (�−1,�,�) (�+1,�,�) ( ) [�̂�, �̂�]=0, () 1 � ] [�̂ , �̂ ]=� = ∇ ∇ � � � � � � �� 4�2 � ] (�−1,�,�) (�+1,�,�) where .. e Hamiltonian Formulation for the Quantum Vacuum. �(�−1)�1� e Nexus graviton is a pulse of -space which can only �(�−1,�,�) = sinc ((�−1) �1�) � () Advances in High Energy Physics

�(�+1)�1� �(�+1,�,�) = sinc ((�+1) �1�) � () a heat sink. e second term on the R.H.S. of () is an -cell 3�2 or -cube that operates as a sinc filter with a four-wave cut-off 1 =Λ () of (2�)2 Λ �2� � =�� =2�√ ∙ () For large values of n the Bloch functions statisfy the condition � 1 3 �� �(�−1,�,�) ≈�(�,�,�) ≈�(�+1,�,�) () e filtration of high frequencies from the vacuum lowers e quantum vacuum can therefore be interpreted as a system the quantum vacuum state and generates a gravitational field in which there are a constant annihilation and creation of inmuchthesamewayastheCasimirEffectisgenerated. quanta as implied by () and () which causes the Nexus e physical process of filtration occurs as follows: A -cube graviton to either expand or contract. with baryonic matter embedded with in it acquires inertia. e inertia gives the cell inductive impedance and becomes .. e Hamiltonian Formulation in the Presence of Matter less reponsive to high frequency vibrations of the quantum Fields. We now seek to introduce matter fields into the vacuum. us the heavier the cell, the lower the cut-off fre- quantum vacuum. If we compare the quantized metric of quency. Equation () shows that when the radius of the cell () describing the gravity within a Nexus graviton with isequaltohalftheSchwarzschildradius,theonlypermited the Schwarzschild metric describing the gravitational field frequencyisthatofthegroundstate.ustheinsideofablack around baryonic matter, we notice that we can describe the hole is in the ground state having a negative metric signature. gravitational field around baryonic matter in terms of the We now introduce a test particle of m,intothe quantum state of space-time through the relation quantum vacuum perturbed by matter fields. e particle will 2 2�� flow along with the Ricci flow and the Hamiltonian of the = �2 �2� () system becomes �(�̂ �� �]� ) is yields a relationship between the quantum state of space- (�−1,�,�,�) (�+1,�,�,�) time and the amount of baryonic matter embedded within it ̂ ̂ P�P] as follows: = (��� �]� ) ( ) 2� (�−1,�,�) ( �+1,�,�) �2� �2 �2 = = 2 ( ) � ] �� V −�(� �(�,�,�)� �(�,�,�)) Equation ( ) reveals a family of concentric stable circular 2 2 Here orbits �� =���/� with corresponding orbital speeds of V =�/� ℎ2 ��2 3ℎ2�2 � . us in the Nexus Paradigm, unlike in GR, the �=�2 Λ= . 1 ( ) innermost stable circular orbit ocurrs at �=1or at half 2� �� 2� the Schwarzchild radius which implies that the event horizon �̂ = −�ℎ∇ predicted by the Nexus Paradigm is half the size predicted � � ( ) in GR. Also ( ) reveals how the Nexus graviton in the n- �=−�ℎ�̂ th quantum state imitates DM if M is considered as the � ( ) apparent mass of the DM. rough this comparision, we can Equation ( ) is equivalent to () in which the Hamiltonian also deduce that the deflection of light through gravitational is equal to the Lagrangian. lensing by space-time in the n-th quantum state is ̂ � ] 4 �(� �(�−1,�,�,�)� �(�+1,�,�,�)) �= 2 () � ] ( ) � = L (� �(�−1,�,�,�)� �(�+1,�,�,�))

us gravitational lensing can be used to constrain the value e Lagrangian is not relativistic because the energy reference of the quantum state n of space-time within a lensing system. frames or space-time states involved in the transition dynam- Having obtained the relationship between the quantum ics �−1, �,and�+1are separated by a small energy gap, stateofspace-timeandtheamountofbaryonicmatter �=√3∙ℎ�0. e exchanged quantum of energy between embedded within it, the result of ( ) is then added to ( ) states is responsible for “welding” them together. to yield the time evolution of the quantum vacuum in the Equation ( ) is the gravitational interaction in reciprocal presence of baryonic matter as space-time. e weakness of the gravitational interaction is � ] �� (� �(�−1,�,�,�)� �(�+1,�,�,�)) duetothesmallvalueofthecosmologicalconstant. 1 � ] = ∇ ∇ � � � � () 4�2 � ] (�−1,�,�) (�+1,�,�) 3. Line Elements and Information in K-Space 2 � ] −� Λ� �(�,�.�)� �(�,�,�) A close inspection of () reveals the relationship between a line element in -space and its conjugate line in K-space us the time evolution of the quantum vacuum in the ��2 =����2 presence of matter resembles thermal flow in the presence of ( )  Advances in High Energy Physics

Shadow of Sagittarius A∗ Shadow diameter = 26 micro 52 n =2 arcseconds . At the event horizon the quantum state of space-time is n =1 39

26 5.2n2GM r = n 2c2 13 n =1

5.2GM  arcseconds r = 0 1 2c2

10.4GM 13 r = 2 c2

26

39

52 39 26 13 0 13 26 39 52

F : Shadow of Sagittarius A∗. where pixel slots available in the n-th quantum state or the number 2 � ] � ] of degenerate energy levels. �� =� � � � �� �� ( ) (�,�,�) (�,�,�) � � � � ��⟩ =�1 ��1⟩ +�2 ��2⟩ ...�� ���⟩ . � ] � � � � � � � � ( ) e term � �(�,�,�)� �(�,�,�) refers to the normalized infor- mation flux density passing through an elementary surface e entropy of the n-state therefore becomes �����] in K-space. e gradient of the flux density yields the 2 3 � �p � � � baseband bandwidth (�)�. �=� �=� =� � =� � � ln � ln �2 � ln �2 � ln ℎ ( ) � ] � ] n p G ��� �(�,�,�)� �(�,�,�) =�(�)�� �(�,�,�)� �(�,�,�) ( ) Here we observe that objects fall into a gravitational field is gradient is also a measure of the acutance or sharpness because it leads to an increase in entropy. In strong fields the of the information. A large bandwidth provides detailed information is delocalized. A black hole does not annihilate information while a small bandwidth provides diffuse infor- information; it simply diffuses or dilutes it by increasing the mation. Equation ( ) describes the diffusion of information entropy. It can also be interpreted as a low pass filter that as it flows into a gravitational well. Unitarity is preserved if permitsonebitofinformationtopassperHubbletime. the flux is summed over the total diffusion surface. Since information is delocalized close to a black hole, then Σ we do not expect the Event Horizon Telescope to observe a � ^ ∬ � �(�,�,�)� �(�,�,�)�����] =1 ( ) silhouette surrounded by a well defined accretion disc but 0 a circular silhouette surrounded by a cloud of degenerate Here the integral implies that one bit of information is found (delocalized) matter (grey area) as depicted in Figure . 2 e radii are calculated from ( ) and the magnification on a surface Σ=�� in K-space which corresponds to an 2 2 factor, ./, arises from gravitational lensing. e silhouette area �� =4�/� in -space. e �1 bandwidth permitted � is half the size predicted in GR. within a black hole generates the smallest elementary surface �2 or pixel in K-space of 1 suggesting that information inside 4. Discussion a blackhole is diluted to one bit per area equivalent to the square of the Hubble -radius in -space or from () the A successful covariant canonical quantization of the gravita- �2 =4�2Λ/3 information surface density is 1 .usthe tional field has been presented in which we find that gravity cosmological constant can also be interpreted as a unit of is akin to a thermal flow of space-time and that space-time information surface density. e ratio of the Planck surface in can also be described in terms of a reciprocal lattice. e K-space to the smallest elementary surface in K-space yields 120 presence of matter creates an impure lattice and a potential  . is huge number represents the maximum number of well arises in the region of perturbation through filtration of pixels that can fit on the largest surface in K-space. Hence high frequencies from the quantum vacuum. A test particle 2 2 the expression �=�p/�n represents the number of empty flows along with the quantum vacuum and the presence of Advances in High Energy Physics 

a potential well will cause it to flow towards the sink. is [] C. Kiefer, “Conceptual problems in quantum gravity and quan- formulation will find important applications in high energy tum cosmology,” ISRN Mathematical Physics,vol.,Article lattice gauge field theories where it may help expand the ID  ,pages,. standard model of by eliminating divergent [] S. Marongwe, “e Nexus graviton: A quantum of Dark Energy terms in the current theory and predicting hitherto unknown and Dark Matter,” International Journal of Geometric Methods in phenomena. Modern Physics, vol. , no. , Article ID   , . [] S. Marongwe, “e Schwarzschild solution to the Nexus gravi- ton field,” International Journal of Geometric Methods in Modern Data Availability Physics, vol. , no. , Article ID  ,  pages,  . eresultsofthistheoreticalresearchwillbeconfirmedby [ ] S. Marongwe, “e electromagnetic signature of gravitational data from the Event Horizon Collaboration on the size and wave interaction with the quantum vacuum,” International Jour- nalofModernPhysicsD:Gravitation,Astrophysics,Cosmology, shape of the shadow of Sgr A∗. vol.,no.,ArticleID , pages,. Conflicts of Interest e author declares no conflicts of interest.

Acknowledgments e author gratefully appreciates the funding and support from the Department of Physics and Astronomy at the Botswana International University of Science and Technol- ogy (BIUST).

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