arXiv:2011.05328v1 [gr-qc] 10 Nov 2020 qain;Q,qatmmcais U,gnrlzduncert generalized GUP, mechanics; quantum QM, equations; 1 lmn,mti esr n geodesics. and tensor, metric line in element, modifications work possible present the namely The earlier, 2015). of the (2014, on Diab focuses modification & Tawfik a N. effects, A. mention gravitational GUP to i.e. we due latter, principle the uncertainty Heisenberg For into elements QM. new and add to GR conjectured (1990). is gravity Smolin quantum & Rovelli The quan- (1994); entire an Donoghue in framework relativity tum general of theory reconcile the to of attempts principles various were There physics. in questions is gravity quantum for se theory consistent a of Formulation Tawfik* Nasser Abdel Equation Geodesic Elem and Tensor Line Metric on Discretization Length Minimal of Consequences PAPER CONFERENCE xxx/xxxx DOI: xxxxx; Received Eal tawfi[email protected] *Email: Correspondence 5 4 3 2 1 n nomto,161Cio Egypt Cairo, 11671 Information, and Computers of Faculty (MTI), Information and Technology for University Modern Egypt Cairo, 11571 Department, Science Basic Engineering, for Academy Modern Egypt. Cairo, 11571 Engineering, of Faculty (MTI), Information and Technology for University Modern Germany. Main, am Frankfurt D-60438, Physics, Theoretical for Institute University, Goethe Egypt. Giza, 12588 (ECTP), Physics Theoretical for Center Egyptian nutmt olbtufruaeysiloeo h open the of one still unfortunately but goal ultimate an 0 Abbreviations: INTRODUCTION eie xxxxx; Revised R eea eaiiy gEFE, relativity; general GR, 1,2 cetdxxxxx Accepted be aidDiab Magied Abdel | ”gravitational atcei the in particle a s curved acceleration, and encompasses flat apparently latter in The equation modified. geodesic the and element to line GUP the confront we difficulties, technical the Despite state. relates gravity gEFE tivity hand, other the On etc. opera operators, momentum and (distance) length i of gravitational relations the mutation account into takes GUP gravity. quantum of properties manifesting metric in modifications quential edsceuto,Gnrlzducranyprinciple uncertainty Generalized equation, Geodesic and Relativity geometry, Noncommutative gravity, Quantum KEYWORDS: etc. curvature, of radii different e between accelerating as such phenomena manifest apparently ation GP stogtul mlmne,i so ra neett interest great of ”gravitational is it implemented, thoughtfully is generalized (GUP) from emerging uncertainty length minimal When it principle ainty isenfield Einstein oteeeg-oetmtnos ..pooigqatmequ quantum proposing i.e. tensors, energy-momentum the to isenfil qain gF)adt r ofidotwehrc whether out find to try to and (gEFE) equations field Einstein 3 ”quasi-quantized” per ae Shenawy Sameh | clradtemte arnincnb xdi cosmological in fixed be curvature observations. can Ricci Lagrangian matter the the between and Benedetto coupling a scalar & direct Corda, motion, Licata, a where equivalence scratched nongeodesic (2016), an was and Regarding request gravity emerged. machian extended then in is principle GR gravitational in of field massless origin spin-1 A (1985). Oliveira ref. tensors in vectory metric internal an of of functions class (2020); contri- are that extended review Hess a P. An in IWARA2020. refs and of in (2015) bution Greiner found & Schafer, be Hess, can O. (2001); P. models of on GR overview types (2000); An these (1999). of Scarpetta & Papini, perspective Lambiase, Feoli, Scarpetta & Papini, another Lambiase, Feoli, Capozziello, Scarpetta from & Papini, Lambiase, found Feoli, Bozza, were gravity oie egheeetadqatzto of quantization and element length modified A rvttoa ed iiehge-resfacceler- higher-ordersof Finite field. gravitational 4 ia buE Dahab El Abou Eiman | lsia emtyo eea rela- general or geometry classical rvtto,Ln lmn,Mti tensor, Metric element, Line gravitation, unu emtydeto due geometry quantum ek n np(one of (jounce) snap and jerk, osdrisipcson impacts its consider o pninadtransitions and xpansion h erctno othat so tensor metric the pcso h noncom- the on mpacts oso ieadenergy and time or tors netit principle uncertainty a aeaeaccordingly are pace ( x ) a lodiscussed also was 5 tosof ations ent, onse- 2

It intends to tackle the long-standing fundamental problem GUP also exhibits features of the UV/IR correspondence that the Einstein field equations (EFE) relate nonquantized that Δx increases rapidly (IR) as the Δp grosses beyond the semi-Riemannian geometry characterized by Ricci and Ein- order of the Planck scale (UV) Gubser, Klebanov, & Polyakov stein tensors, which are directly depending on the met- (1998); Maldacena (1999); Witten (1998). The UV/IR corre- ric tensor, with the full-quantized energy-momentum tensor spondence could be applied to various subjects of short vs. Stephani, Kramer, MacCallum, Hoenselaers, & Herlt (2003). long distance physics, for instance, the ”deformed” commuta- Our approach is based on implementing minimal length tion relations. We assume that the current problem of minimal uncertainty obtained from generalized uncertainty principle length discretization would be solved by such a correspon- (GUP), which in turn is inspired by string theory, doubly spe- dence. cial relativity, and black hole physics A. N. Tawfik & Diab Analogous to Eq. (1), the canonical noncommutation rela- (2014, 2015) and seems to be comparable to the Planck tion of quantum operators of length and momentum reads length, where fluctuations in quasi-quantized manifold likely ̂x , ̂p ≥  i ` 1 + p2 , (3) emerge A. Tawfik & Diab (2016). We propose that this basic i j ij 2 0i 0j 
approach helps in characterizing the potential impacts of full- where p = gij p p and gij is the Minkowski spacetime met- quantization on EFE. We also believe that this likely unveils the ric tensor, for instance (−, +, +, +). The length and momentum quantum nature of the cosmic geometry. We elaborate the cor- operators, respectively, are defined as responding modification in the line element, metric tensor, and 2 ̂xi = ̂x0i(1 + p ), (4) geodesics and then show how this helps in manifesting proper- ̂p = ̂p , (5) ties of quantum geometry due to quantum gravity. The present j 0j study introduces a fundamental approach to the observations in which the operators ̂x0i and ̂p0j are to be derived from the that the universe likely expands faster than the GR expecta- corresponding noncommunitation relation tion. Unless forces deriving this kind of expansion, ingredients [ ̂x0i, ̂p0j] = iji `. (6) such as dark energy and cosmological constant might remain overdue Peebles & Ratra (2003). 3 SPACETIME GEOMETRY: LINE ELEMENT AND METRIC TENSOR 2 GENERALIZED UNCERTAINLY PRINCIPLE AND SPACETIME METRIC When including such quantum noncommutation operations, TENSOR section 2, in the spacetime geometry, for instance EFE, the Minkowskian manifold of the line element Heisenberg uncertainty principle (HUP) dictates how to con- 2   strain the uncertainties in the quantum noncommutation rela- ds = gdx dx , (7) tions of length (position) and momentum operators or of time where , , and  =0, 1, 2, 3, enlarges to an eight-dimensional and energy operators, for instance. On the other hand, when the spacetime tangent bundle similar to the one with the coordi- gravitational influences are thoughtfully taken into account, a nates xA = (x( a), ̇x( a)), where ̇x = dx∕d  Brandt generalized uncertainty principle is then emerged so that an (2000). This leads to a modification in the line element, i.e. a alternative quantum gravity approach for string theory, dou- new metric related the quantum geometry, bly special relativity, and black hole physics has been provided d ̃s2 = g dxA dxB, (8) A. N. Tawfik & Diab (2014, 2015), e.g. a finite minimal length AB Kempf, Mangano, & Mann (1995), where g = g ⊗ g . AB   ` With some trivial approximations, the manifold can be Δx Δp ≥ 1 + (Δp)2 , (1) 2 reduced to the effective four-dimensional spacetime geometry,   where Δx and Δp are length and momentum uncertainties, where xA = xA( ). Therefore, the modified four-dimensional l 2 2 respectively. The GUP parameter = 0( p∕`) = 0∕(mpc) metric tensor reads with the Planck length l = `G∕c3 =1.977×10−16 GeV−1 )xA )xB )xa )xb ) ̇xa ) ̇xb p ̃g = g ≃ g + 19 2  AB )  )  ab )  )  )  )  and mass mp = `c∕G √= 1.22 × 10 GeV∕c . Upper    bounds on the dimensionless√ parameter 0 should be put from ≃ 1 + ̈x ̈x g, (9) astronomical observations, such as recent gravitational waves, where ̈x = ) ̇x∕)  is the
four-dimensional acceleration. The ≲ . 60 0 5 5×10 . Equation (1) seems to exhibit the existence of indices A,B,a, and b run over 0, 1, ⋯ , 7. a minimum length uncertainty, l Δxmin ≈ ` = p 0. (2) √ √ 3

• For flat spacetime, where g = , the modified metric where tensor can be expressed es  = (s, ̇x, ̈x) ds, Ê ̃g = 1 + ̈x ̈x  =  + ℎ , (10)      dx   ̇x = ,  ds where ℎ = ̈x ̈x encompasses the quantum contri- )g butions to the spacetime geometry. g = , , )x • In the limit that ℎ → 0, the quantum corrections added ) )g g = , entirely diminish and the classical EFE, i.e. GR EFE, can  , , )x 0 )x 1 be restored. 1 Γ2 = g2 g − g + g ,  2  , , , Thus, the principle of the general covariance is apparently sat- dx dx d ̇x d ̇x 1∕2  = g + . isfied also in the absence of the gravitational effects on the  ds ds ds ds modified Minkowski metric.    The modified four-dimensional line element is then The -terms appeared in Eq. (13) distinguish this expression expressed as from the GR geodesics. In the section that follows, we dis- cuss on these terms, where the consequences of the minimal 2   2   d ̃s = g dx dx + d ̇x d ̇x length discretization are integrated in and summarize our final 2  2
= 1 + ̈x ̈x ds . (11) conclusions.

4 SPACETIME GEOMETRY: GEODESIC 5 CONCLUSIONS EQUATION It is worthwhile to highlight that Christoffel connection and the Accordingly, the properties of the manifold in special and gen- equivalence principle are not affected by the minimal length eral relativity can also be generalized. We limit the discussion uncertainty. Thus, we conclude that GUP reconciles with to modified geodesics, where the notion of a ”straight line” is the equivalence principle and simultaneously the equivalence generalized to curved spacetime. In this section, we propose principle is not violated. In such a way, this gives possibilities a theory for the possible consequences of length discretiza- to recover the violation of the equivalence principle in pres- tion based on an approach to quantum gravity, GUP, on the ence of the quantum gravity through the quantum geometry world line of a free particle. With this regard, we recall that characterized by minimal length discretization. The reason that GR assumes gravity as a consequence of curved spacetime this essential result apparently contradicts refs. Ghosh (2014); geometry and the ”quantized” energy-momentum tensor is the Scardigli & Casadio (2009), can be understood that we have source of spacetime curvature. Our approach follows the same implemented GUP to the basic metric tensor. Accordingly, line with a major difference that the length is discretized and we have obtained a modified metric tensor and line element. accordingly rhs of EFE. Both are very fundamentals of the proposed geometry. Having By using the variational principle and by extremizing the both quantities modified, we could derive the corresponding path sAB, the geodesic equation can be formulated. Due to the geodesics. The appearance of vibration and/or sudden tran- proposed length quantization, we get sition, as shall be discussed shortly, is apparently stemming from the quasi-quantized geometry or the proposed approach • for flat space to quantum gravity, where the curvature emerged by the quan- d2 ̇x dx  − + c = 0, (12) tized energy-momentum tensor seems gains corrections, as d2 d well. • and for curved space As outlined in section 3, the corrections to the line element 2 ̈x ̈x d2x2 d d3x2 are combined in the term , which manifest the essen- −  = tial contributions added to by acceleration of the expansion. In d2 d 0 d3 1 other words, it seems that the length discretization as guaran- dx dxnu d2x d2xnu − Γ2 + g2 g teed by GUP emphasizes that even the line element wouldn’t  d d , d2 d2 2   only expand but also accelerates. It would be noticed that even 2 dx d ̇x d dx d ̇x + g g , + if the GUP parameter squared would assure that this fac- 4 d d2 d d d 5   tor remains small, the product ̈x ̈x raises the values of this dx dx d ̇x  + g2 g , (13) correction term. In the same matter, the metric tensor obtains  , , d d d 4

  Gubser, S., Klebanov, I. R., & Polyakov, A. M. 1998, Phys. Lett. B, corrections, ̈x ̈x, as well. Here, the factor to the product ̈x ̈x is simply the GUP factor . 428, 105–114. doi: Hess, P. 2020, Prog. Part. Nucl. Phys., 114, 103809. doi: Even if classical GR necessarily invokes metric tensor in Hess, P. O., Schafer, M., & Greiner, W. 2015, Pseudo-complex four dimensions, we assume that the corresponding reference General Relativity. Heidelberg: Springer. frames experience different expansions in time and space. The Kempf, A., Mangano, G., & Mann, R. B. 1995, Phys. Rev. D, 52, metric expansion proposed in the present work are related 1108–1118. doi: Licata, I., Corda, C., & Benedetto, E. 2016, Grav. Cosmol., 22(1), to changes in metric tensor with time. Although the possible 48–53. doi: differences in temporal and spacial expansions, we straight- Maldacena, J. M. 1999, Int. J. Theor. Phys., 38, 1113–1133. doi: forwardly impose the origin of an accelerated expansion. Oliveira, C. G. 1985, Int. J. Theor. Phys., 24, 1081. doi: The spacetime seems to shrink or grow as the corresponding Peebles, P., & Ratra, B. 2003, Rev. Mod. Phys., 75, 559–606. doi: Riess, A. G., et al. 1998, Astron. J., 116, 1009–1038. doi: geodesics converges or diverges, where the length discretiza- Rovelli, C., & Smolin, L. 1990, Nucl. Phys. B, 331, 80–152. doi: tion plays the major role. Scardigli, F., & Casadio, R. 2009, Int. J. Mod. Phys. D, 18, 319–327. We also notice that the geodesics, Eq. (13), is not only pro- doi: viding the acceleration of a particle in a gravitational field, but Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., & Herlt, E. 2003, Classification of the Ricci tensor and the energy- higher-order derivatives, as well Eager, Pendrill, & Reistad momentum tensor. Exact Solutions of Einstein’s Field Equations (4) (2016), namely the snap or jounce, x , which in turn is derived 2nd ed.,, p. 57–67. Cambridge University Press. doi: from the jerk, x(3). The jerk gives the change in the force act- Tawfik, A., & Diab, A. 2016, Indian J. Phys., 90(10), 1095–1103. ing on that particle, while the snap is resulted from change in doi: Tawfik, A. N., & Diab, A. M. 2014, Int. J. Mod. Phys. D, 23(12), the jerk, itself. That both quantities are finite means that vibra- 1430025. doi: tion or sudden transitions would occur between different radii Tawfik, A. N., & Diab, A. M. 2015, Rept. Prog. Phys., 78, 126001. of the curvature. Acceleration, as in GR geodesics, without doi: jerk is just a static load, i.e neither vibration nor transition are Velten, H., Gomes, S., & Busti, V. C. 2018, Phys. Rev. D, 97(8), 083516. doi: allowed. Witten, E. 1998, Adv. Theor. Math. Phys., 2, 253–291. doi: Due GUP and the corresponding minimal length descretiza- tion, the corrections added to the line element, metric tensor, and geodesics emphasize that evolution of the universe as the- orized by classical GR is also accelerated Riess et al. (1998); Velten, Gomes, & Busti (2018). While the line element and metric tensor get additional terms of acceleration products,  ̈x ̈x, the geodesics on the other hand is corrected with higher order acceleration derivatives, such as snap x(4) and jerk x(3).

ACKNOWLEDGMENTS

The authors are very grateful to organizers of the 9th Interna- tional Workshop on Astronomy and Relativistic Astrophysics (IWARA2020 Video Conference) for their kind invitation.

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