Minimal Length in quantum gravity and gravitational measurements
aa r X i v : . [ g r- q c ] N ov Minimal Length in quantum gravity and gravitational measurements
Ahmed Farag Ali , , ∗ Mohammed M. Khalil , † and Elias C. Vagenas ‡ Department of Physics, Florida State University,Tallahassee, FL 32306, USA. Department of Physics, Faculty of Science,Benha University, Benha, 13518, Egypt Department of Electrical Engineering,Alexandria University, Alexandria 12544, Egyptand Theoretical Physics Group, Department of Physics,Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
The existence of a minimal length is a common prediction of various theories of quantum gravity.This minimal length leads to a modification of the Heisenberg uncertainty principle to a GeneralizedUncertainty Principle (GUP). Various studies showed that a GUP modifies the Hawking radiationof black holes. In this paper, we propose a modification of the Schwarzschild metric based on themodified Hawking temperature derived from the GUP. Based on this modified metric, we calculatecorrections to the deflection of light, time delay of light, perihelion precession, and gravitationalredshift. We compare our results with gravitational measurements to set an upper bound on theGUP parameter.
I. INTRODUCTION
Various approaches to quantum gravity (QG) are ex-pected to play a crucial role in revealing some charac-teristic features of the fundamental quantum theory ofgravity. One common feature among most of these ap-proaches, such as string theory and black hole physics[1–7], is the existence of a minimal observable length, i.e.Planck length l p . The existence of a minimal length leadsto the modification of the Heisenberg uncertainty princi-ple to a Generalized Uncertainty Principle (GUP) whichincludes an additional quadratic term in momentum asfollows [1–7] ∆ x ∆ p ≥ ~ β ∆ p ) (1)where β = β l p / ~ , β is a dimensionless constant, and l p = 1 . × − m is the Planck length. Phenomeno-logical aspects of GUP effects have been analyzed in sev-eral contexts such as the self-complete character of grav-ity [8, 9], the conjectured black hole productions at theterascale [10, 11] and the modifications of neutrino oscil-lations [12].Recently, another interesting form of the GUP hasbeen proposed in [13–15] to be consistent with doublyspecial relativity, string theory, and black hole physics.This suggested form of GUP includes a linear term in mo-mentum, and leads to a maximum observable momentumin addition to a minimal length[ x i , p j ] = i ~ (cid:20) δ ij − α (cid:18) pδ ij + p i p j p (cid:19) + α (cid:0) p δ ij + 3 p i p j (cid:1)(cid:21) (2) ∗ [email protected], [email protected] † [email protected] ‡ [email protected] where α = α l p / ~ , and α is a dimensionless constant.The upper bounds on the parameter α have been calcu-lated in [13] and it was proposed that GUP may introducean intermediate length scale between Planck scale andelectroweak scale. Recent proposals suggested that thesebounds can be measured using quantum optics tech-niques in [16] and using gravitational wave techniques[17, 18] which may be considered as a milestone in quan-tum gravity phenomenology. In a series of papers, variousphenomenological implications of the new model of GUPwere investigated [19–24]. A detailed review along thementioned lines of minimal length theories and quantumgravity phenomenology can be found in [25–27].Very recently, Scardigli and Casadio [28] proposed amodification of the Schwarzschild metric to reproduce themodified Hawking temperature [29–33] which was derivedfrom the GUP of Eq. (1). This modification of the metrictakes the form dτ = F ( r ) dt − F ( r ) dr − r d Ω , (3)with F ( r ) = 1 − GMr + ǫ G M r , (4)and from comparing the Hawking temperature derivedfrom the GUP with the one derived from the modifiedmetric they concluded that β = − π ǫ M / M p , where M p is the Planck mass. There is a problem in the formof F ( r ) in Eq. (4), which implies that the horizon is ata different value from r s = 2 GM contrary to many argu-ments based on the GUP [2, 7, 34]. More importantly,it leads to a negative GUP parameter β < , which The negativity of the GUP parameter β was first encountered is inconsistent with almost all the motivations that ledto the GUP [1–7]; the GUP implies a minimal length of∆ x ≥ ~ √ β which would be imaginary if β is negative.In this paper, we continue our investigations of thephenomenological implications of GUP that was studiedin [13]. We propose a modification to the Schwarzschildmetric to reproduce the modified Hawking temperaturederived from the GUP in Eq. (2). We use a more generalform for F ( r ) than the one used by Scardigli and Cas-sadio (see Eq. (15) below). We do not assume a mod-ification with 1 /r dependence. Instead, we consider ametric with a general 1 /r n dependence. In addition, thisform leads to a horizon at the usual value r s = 2 GM ,and yields a positive GUP parameter.In the following sections, we review the derivation ofthe modified Hawking temperature from the GUP, andfind the relation between the GUP parameter α and themetric. Then, we use this metric to find corrections tothe general relativistic results of the deflection of light,time delay of light, perihelion precession, and gravita-tional redshift. We compare our results with experimentto set upper bounds on the GUP parameter α . II. HAWKING TEMPERATURE FROM GUP
The Hawking temperature of a black hole takes the well-known form [36] T H = 18 πGM , (5)where, from now on, we use natural units, in which c = 1, ~ = 1, G = 6 . × − GeV − and l p = √ G = 8 . × − GeV − .The GUP modifies the Hawking temperature; that mod-ification was derived in several papers for different formsof the GUP [29–33, 37–39]. We will follow the derivationin [37–39].We start by rewriting Eq. (2) as [37]∆ x ∆ p ≥ (cid:2) − α ∆ p + 4 α (∆ p ) (cid:3) . (6)Solving Eq. (6) for ∆ p and expanding to second order in α , we get a momentum uncertainty of∆ p ≥ x − α x + α x . (7)According to [29, 31, 40], a photon is used to ascertainthe position of a quantum particle of energy E and ac-cording to the argument in [41] which demonstrated thatthe uncertainty principle ∆ p ≥ / ∆ x can be written in the study of the uncertainty principle given by Eq. (1) whenformulated on a crystal lattice [35]. as a lower bound E ≥ / ∆ x . The uncertainty prin-ciple ∆ p ∆ x ≥ / E ≥ / x . Similarly, from Eq. (7) we get∆ E ≥ x − α x + α x . (8)The energy uncertainty can be viewed as the energy ofthe emitted photon from the black hole, and thus as itscharacteristic temperature T = ∆ E . Taking the uncer-tainty in position to be proportional to the Schwarzschildradius ∆ x = µr s = 2 µGM gives the temperature T ≃ µGM − α µ G M + α µ G M . (9)Comparing this equation with the standard Hawkingtemperature (5), we see that µ = 2 π and the temper-ature from the GUP is T ≃ πGM (cid:18) − α πGM + 5 (cid:16) α πGM (cid:17) (cid:19) . (10) III. MODIFIED SCHWARZSCHILD METRIC
The standard Hawking temperature can be derived fromthe metric using the surface gravity κT H = κ π (11)where κ is related to the metric by ([42], p. 246) κ = lim r → r s r − g rr g tt g tt,r . (12)For the Schwarzschild metric, Eq. (3), the surface gravityis simply half the derivative of F ( r ) at the Schwarzschildradius κ = 12 F ′ ( r s ) , (13)Using F ( r ) = 1 − GM/r , we get the standard Hawkingtemperature T H = 14 π F ′ ( r s ) = 18 πGM . (14)We can follow the same argument backwards; start fromthe modified temperature (10) and look for the metricthat reproduces it. We assume the metric takes the sameform as Eq. (3) but F ( r ) is modified to F ( r ) = (cid:18) − GMr (cid:19) (cid:18) η (cid:18) GMr (cid:19) n (cid:19) , (15)where η is a constant ≪
1, and n is an integer ≥
0. Dif-ferentiating F ( r ) at r s = 2 GM we get the temperature T = 14 π F ′ ( r s ) = 1 + η πGM , (16)which must equal the temperature in Eq. (10)1 + η πGM = 18 πGM (cid:18) − α πGM + 5 (cid:16) α πGM (cid:17) (cid:19) . (17)Solving for η η = − α πGM + 5 (cid:16) α πGM (cid:17) , (18)and to first order in α the metric is modified by the func-tion F ( r ) = (cid:18) − GMr (cid:19) (cid:18) − l p α π (2 GM ) n − r n (cid:19) . (19)where we used α = l p α .A couple of comments are in order here. First, it is eas-ily seen from Eq. (19) that if one selects the dimension-less constant α to be positive (and thus η is negative),then the metric in Eq. (19) will have two horizons as isthe case for the modified metric in [28]. However, if oneselects the dimensionless constant α to be negative (andthus η is positive), in this case the metric in Eq. (19) willhave only one horizon, i.e., r s = 2 GM . Second, the valueof n is still undetermined. In the next section, we willdetermine the value of n from the modified Newton’s law. IV. MODIFIED NEWTON’S LAW
In this section, we calculate the modified Newton’s lawfrom the modified metric, and we will follow the deriva-tion of gravitational acceleration in ([43], p. 3-32). Fora mass falling radially from rest at r , dτ = F ( r ) dt ;thus, its energy is E = F ( r ) dtdτ = p F ( r ) . (20)Substituting dτ from the previous equation in the metric,Eq. (3) with d Ω = 0, and solving for dr/dtdrdt = F ( r ) s − F ( r ) F ( r ) . (21)The proper time and length experienced by a static ob-server on a spherical shell of radius r is given by dt sh = F ( r ) dt, dr sh = drF ( r ) . (22)Thus, dr sh dt sh = s − F ( r ) F ( r ) . (23)Differentiating with respect to t sh and substituting r = r we get the acceleration g = d r sh dt sh = − p F ( r ) F ′ ( r ) , (24) where F ′ ( r ) is the derivative of F ( r ) with respect to r evaluated at r . Substituting the modified function F ( r )and expanding to first order in α , we get g = GMr (cid:18) − GMr (cid:19) − / × (cid:20) − (2 GM ) n − πr n ((1 + 2 n ) GM − nr ) l p α (cid:21) . (25)A couple of comments are in order here. First, equa-tion (25) reduces to the standard relativistic result when α = 0, and to the Newtonian result after neglecting therelativistic factor 1 / p − GM/r . Second, the acceler-ation as given by equation (25) does not depend on themass of the falling particle which means that the Equiv-alence Principle is not violated. On the contrary, Equiv-alence Principle was violated when a different form ofGUP was utilized [44].Thus, the modified Schwarzschild metric leads to themodified Newton’s law F N = GM mr (cid:20) − (2 GM ) n − πr n ((1 + 2 n ) GM − nr ) l p α (cid:21) . (26)It should be pointed out here that we have neglected therelativistic factor since we would like to keep the GUPcorrection terms and the other terms to be up to secondorder in GM/r .To estimate the value of n , we compare our result withphenomenologically well motivated approaches that mod-ify Newton’s law of gravity at short distance such asRandall-Sundrum II model [45] which implies a modi-fication of Newton’s law on a brane [46] as follows F RS = GMmr (cid:0) R πr (cid:1) , r ≪ Λ RGMmr (cid:0) R πr (cid:1) , r ≫ Λ R (27)where Λ R is a characteristic length scale.When n = 2 in the modified Newton’s law of Eq. (26)we get to first order in 1 /rF N = GM mr (cid:18) l p α πr (cid:19) , (28)which clearly agrees with the Randall-Sundrum II result.We conclude that the most likely value for n is 2 andthereon we set n = 2. Thus, the function F ( R ) in themodified metric takes the form F ( r ) = (cid:18) − GMr (cid:19) (cid:18) − l p α π GMr (cid:19) . (29)A couple of comments are in order here. First, we ob-tain these corrections in the framework of semiclassicalgravity approach. Thus, we keep the LHS of the Ein-stein equations as it is and we assign the QG correctionsto the RHS of the Einstein equations. Therefore, usingMathematica and employing the metric element given byEq. (29), the RHS of Einstein equations is not zero, asexpected. In addition, the RHS of Einstein Equationshas only diagonal terms and all of them are perturbationterms of the first order in the perturbative parameterwhich reads ǫ = l p α π . The non-zero diagonal terms that appear in the RHS ofEinstein equations are of the form G tt = G rr = + ǫ GM (4 GM − r ) r G θθ = G φφ = − ǫ GM (6 GM − r ) r . Therefore, these GUP corrections can be treated as firstorder perturbation terms around the vacuum solution,i.e., Schwarzschild solution, and the proposed solutioncan be considered a solution of the Einstein equations ina perturbative sense.Second, it is evident that the specific expression for themetric element F ( r ) implies the existence of another hori-zon at r = q l p α GM/ π, (30)which will be an inner horizon. Thus, the spacetime con-tains an inner and outer horizon, as well as a timelikesingularity and the conformal structure should be thesame as that of a Reissner-Nordstrom black hole. Thisimplies the existence of a Cauchy horizon which in turnleads to mass inflation. V. DEFLECTION OF LIGHT
When light approaches a massive body, such as the sun,it gets deflected from a straight line by an angle given by([47], p. 189)∆ φ = 2 Z ∞ r r p F ( r ) (cid:18) r r F ( r ) F ( r ) − (cid:19) − / dr − π, (31)where r is the distance of closest approach to the sun.In general relativity the deflection angle is given by ([47],p. 190) ∆ φ GR ≃ GMr . (32)To find the deflection angle predicted by the modifiedmetric, we need to use F ( r ) from Eq. (29) in Eq. (31).To simplify the calculations, we make the transformation u ≡ r /r in Eq. (31)∆ φ = 2 Z q F (cid:0) r u (cid:1) F ( r ) F (cid:0) r u (cid:1) − u ! − / du − π (33) To simplify the integral, we expand the integrand tofirst order in α and to second order in 1 /r ∆ φ =∆ φ GR + Z u + 12 πr √ − u GMr l p α du (34)which evaluates in terms of the gamma function to∆ φ = ∆ φ GR + Γ( )2 √ π Γ(2)
GMr l p α . (35)The best accuracy of measuring the deflection of light bythe sun is from measuring the deflection of radio wavesfrom distant quasars using the Very Long Baseline Array(VLBA) [48], which achieved an accuracy of 3 × − ;thus, δ ∆ φ ∆ φ GR < × − . (36)Assuming that light grazes the surface of the sun r ≃ R ⊙ = 6 . × m = 3 . × GeV − and M = M ⊙ =1 . × kg = 1 . × GeV, we get an upper boundon α of α < . × . (37)This bound is larger than the bound set by the elec-troweak scale 10 but not incompatible with it. How-ever, studying the effects of the GUP, which is modelindependent, on gravitational phenomena might proveuseful in understanding the effects of quantum gravity inthat regime. VI. TIME DELAY OF LIGHT
In general relativity, the time taken by light to travelfrom r = r to r = r passing by a massive body, such asthe sun, is slightly longer than what is expected in flatspacetime, and the time of the round trip is given by T = 2 ( t ( r , r ) + t ( r , r )) , (38)with ([47], p. 202) t ( r, r ) = Z rr F ( r ) (cid:18) − F ( r ) F ( r ) r r (cid:19) − / d r, (39)and r is the distance of closest approach to the sun.General Relativity predicts a time of travel ([47], p. 203) T GR =2 q r − r + 2 q r − r + 4 GM (cid:18) (cid:18) r r r (cid:19)(cid:19) (40)Using the modified function F ( r ), we apply the transfor-mation u ≡ r /r in Eq. (39) t ( r , r ) = Z r r r u F (cid:0) r u (cid:1) − u F (cid:0) r u (cid:1) F ( r ) ! − / du. (41)Expanding the integrand to first order in α and M/r we get∆ T =∆ T GR + "Z r r du √ − u + Z r r du √ − u GM l p α πr =∆ T GR + (cid:20) π − sin − r r − sin − r r (cid:21) GM l p α πr . (42)The best accuracy of measuring the delay was obtainedfrom the delay in the travel time of radio waves fromearth to the Cassini spacecraft [49] when it was at ageocentric distance of 8 . . × GeV − , andthe closest distance of the photons to the sun was r =1 . R ⊙ = 5 . × GeV − . The experiment achieved anaccuracy of 2 . × − , which means that δ ∆ T ∆ T GR < . × − . (43)setting an upper bound on α of α < . × (44)which is slightly less than the bound from the deflectionof light but still compatible with the bound set by elec-troweak scale. VII. PERIHELION PRECESSION
In general relativity, the orbit of a particle around a mas-sive body, such as Mercury around the sun, precesses ineach revolution by an angle given by ([47], p. 195)∆ φ = 2 | φ ( r + ) − φ ( r − ) | − π = − π + 2 Z r + r − drr p F ( r ) × r − (cid:16) − F ( r ) F ( r − ) (cid:17) − r (cid:16) − F ( r ) F ( r + ) (cid:17) r r − (cid:16) F ( r ) F ( r + ) − F ( r ) F ( r − ) (cid:17) − r − (45)where r − and r + are the minimum and maximum valuesof r respectively. The precession predicted by generalrelativity is given by ([47], p. 197)∆ φ GR ≃ πGM (cid:18) r + + 1 r − (cid:19) . (46)Using the modified function F ( r ), we change the integra-tion variable in Eq. (45) to u ≡ r − /r ∆ φ = − π + 2 Z r − /r + dur − q F (cid:0) r − u (cid:1) × r − (cid:18) − F ( r − u ) F ( r − ) (cid:19) − r (cid:18) − F ( r − u ) F ( r + ) (cid:19) r r − (cid:18) F ( r − u ) F ( r + ) − F ( r − u ) F ( r − ) (cid:19) − u r − − . (47) We expand the integrand to first order in α and to firstorder in (1 /r + + 1 /r − ) to obtain∆ φ = ∆ φ GR + 14 π Z r − /r + (cid:0) r − + r − − (cid:1) l p α r ( u − (cid:16) r − r + − u (cid:17) du = ∆ φ GR + 14 (cid:18) r + + 1 r − (cid:19) l p α (48)The best measurement of the perihelion precession ofMercury is 42 . ± .
002 arcsec per century [50, 51],which amounts to an accuracy of 4 . × − . Thus, wehave δ ∆ φ ∆ φ GR < . × − . (49)Using the perihelion and aphelion of Mercury r − = 4 . × m = 2 . × GeV − and r + = 6 . × m =3 . × GeV − , we get an upper bound on α of α < . × (50)which is five orders of magnitude less than the boundfrom the time delay and the deflection of light. VIII. GRAVITATIONAL REDSHIFT
Suppose that light was emitted from radius r and re-ceived at r ; by how much will the light be red-shifted?In the metric (3) put dr = 0 and dφ = 0, and solve for dt . Since the time measured by a remote observer is thesame for the two radii, we get dt = dτ p F ( r ) = dτ p F ( r ) , (51)and the relative frequency is ω ω = dτ dτ = s F ( r ) F ( r ) . (52)Substituting F ( r ) from (29), and expanding to first orderin α ω ω = s − GM/r − GM/r (cid:18) GM π (cid:20) r − r (cid:21) l p α (cid:19) . (53)Subtracting one from the previous result we get ω − ω ω =( S − (cid:18) SS − GM π (cid:20) r − r (cid:21) l p α (cid:19) , (54)where S ≡ s − GM/r − GM/r . (55)The most accurate measurement of gravitational redshiftis from Gravity Probe A [52] in 1976. The satellite was atan altitude of 10 m = 5 . × GeV − , and achieved anaccuracy of 7 . × − , which means that the new termin Eq. (54) SS − GM π (cid:20) r − r (cid:21) l p α < . × − . (56)Using the mass and radius of the earth, M ⊕ = 5 . × kg = 3 . × GeV, r = R ⊕ = 6 . × m =3 . × GeV − , and r = r + 10 m. The correctionsare negative since r > r and hence the bound is givenas follows α < . × . (57)This bound is stringent too and compatible with thebound set by the electroweak scale. TABLE I: Bounds on the GUP parameter α from gravita-tional testsExperiment Bound on α Deflection of Light 1 . × Time Delay of Light 5 . × Perihelion Precession 1 . × Gravitational Redshift 2 . × It is noteworthy that comparing the version of GUP weemploy in our analysis here (see Eq. (6)) and the cor-responding one used in [28] (see Eq. (17) in [28]), onecan say that the corresponding GUP parameters, i.e., α and β , respectively, are similar, namely α ∼ β . Utiliz-ing this similarity, one can conclude that the bounds on the two aforesaid GUP parameters based on the tests forthe deflection of light and the perihelion precession areessentially equivalent. IX. CONCLUSIONS
In this paper, we proposed a modification to theSchwarzschild metric based on the GUP in Eq. (2) toreproduce the modified Hawking temperature derivedfrom the GUP. This modification preserves the horizonat 2 GM , and predicts the existence of another horizonwhich might be the radius of the black hole singularity.We assumed a modification with a general 1 /r n depen-dence, and determined the value n = 2 by comparing themodified Newton’s law derived from the modified metricwith phenomenologically well-motivated approaches thatmodify Newton’s law of gravitation at short distancesuch as the Randall-Sundrum II model. We computedcorrections to the general relativistic results of the de-flection of light, time delay of light, perihelion precession,and gravitational redshift. We compared our resultswith measurments to obtain upper bounds on the GUPparameter α (see Table I). The bounds we found in thispaper are greater than those reported in previous workfrom corrections to quantum mechanical predictions[13–15, 53]. However, investigating the implicationsof the GUP on gravitational phenomena might proveuseful for understanding the effects of quantum gravityin that regime. In addition, because the GUP is modelindependent, this understanding can help to evaluatethe results of different theories of quantum gravity. X. ACKNOWLEDGMENTS
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