Observational Tests of the Generalized Uncertainty Principle: Shapiro Time Delay, Gravitational Redshift, and Geodetic Precession
aa r X i v : . [ g r- q c ] J a n Observational Tests of the GeneralizedUncertainty Principle: Shapiro Time Delay,Gravitational Redshift, and GeodeticPrecession ¨Ozg¨ur ¨Okc¨u ∗ , Ekrem Aydiner † Department of Physics, Faculty of Science, Istanbul University,Istanbul, 34134, Turkey
January 26, 2021
Abstract
This paper is based on the study of the paper of Scardigli and Casadio [Eur. Phys.J. C (2015) 75:425] where the authors computed the light deflection and perihelion pre-cession for the Generalized Uncertainty Principle (GUP) modified Schwarzschild met-ric. In the present work, we computed the gravitational tests such as Shapiro time de-lay, gravitational redshift, and geodetic precession for the GUP modified Schwarzschildmetric. Using the results of Solar system experiments and observations, we obtain up-per bounds for the GUP parameter β . Finally, we compare our bounds with otherbounds in the literature. Keywords:
Generalized uncertainty principle, gravitational tests.
Heisenberg uncertainty principle is the fundamental pillar of quantum mechanics, and itsmodification GUP may be unavoidable in the context of quantum gravity. The GUP serves asa useful tool against the inadequacies of general relativity (GR). Especially, GUP predictsthe more sensible results of physics near the Planck length where the standard GR fails.Since GUP implies the minimal length at Planck scale, it removes the singularity predictedby standard GR.Nowadays, GUP has been extensively studied in the literature since the papers based onstring theory [1–6]. Based on Heisenberg microscope argument, Maggiore derived the GUP ∗ Email: [email protected] † Email: [email protected] .Besides the theoretical investigations of GUP, there are studies which are devoted tomeasuring the upper bounds of various kinds of GUPs [49–64]. It is normally assumed thatGUP parameter β is of the order of unity, but this assumption makes the GUP effects toosmall to be detectable. On the other hand, leaving this assumption gives the possibility formeasuring the upper bounds of GUP parameter β by using experiments and observations.Searching an upper bounds may be useful for implying the existence of an intermediate lengthscale between electroweak and Planck length scales. The studies on this direction may alsoopen a low energy window on phenomenology of quantum gravity since GUP effects can beconsidered various kinds of quantum mechanical systems.In Ref. [56], Scardigli and Casadio computed the light deflection and perihelion preces-sion for the GUP modified Schwarzschild metric . First, they considered the GUP modifiedHawking temperature of Schwarzschild black hole. Then, they obtained the corrections tothe Schwarzschild metric from GUP modified Hawking temperature. Using the modifiedSchwarzschild metric, they obtained the GUP corrected light deflection and Perihelion pre-cession. Comparing their theoretical results with precise astronomical measurements for bothSolar system and binary pulsars, they obtained upper bounds of GUP parameter β . More-over, their approach directly provides a novel method for the gravitational systems. Theirapproach can be applied other gravitational tests, and we can discuss new upper bounds ofGUP parameter β . Therefore, we will study Shapiro time delay, gravitational redshift andgeodetic precession for the GUP modified Schwarzschild metric.The paper is arranged as follows. In Section (2), we briefly review the GUP modifiedSchwarzschild metric in Ref. [56]. In Section (3), we obtain the necessary equations to studyShapiro time delay and geodetic precession. At the end of Section (3), we give the effectivepotential of particle around GUP modified Schwarzschild metric. In Section (4), we considerShapiro time delay. In Section (5), we study the gravitational redshift. In Section (6), weconsider a spinning object in orbit around the GUP modified Schwarzschild metric. Weobtain the components of its spin vector, and compute its geodesic precession. Finally, wediscuss our results in Section (7). The reader may refer to review in Ref. [48] Various modified theories of gravity can be considered for Solar sytem tests. Using the observationalresults, researchers have recently found constrains on various modified theory of gravity [65–71]. GUP-Modified Schwarzschild Metric
In this section, we briefly review the GUP-modified Schwarzschild metric in Ref. [56]. Toobtain the GUP-modified Schwarzschild metric, we first consider the GUP-modified Hawkingtemperature. We start to consider the following GUP given as follows:∆ x ∆ p ≥ (cid:0) βG N ∆ p (cid:1) , (1)where β is the dimensionless parameter . Considering the Eq.(1) with standard dispersionrelation E = p , we can write the wavelength of a photon δx ≃ E + 2 βG N E , (2)where E is the average energy of a photon. Taking the uncertainty of the photon wavelengthis related with the Schwarzschild radius δx ≃ µr S = 4 G N µM , (3)and considering average energy of photon with Hawking temperature E = T , we obtain themass-temperature relation from Eq.(2)4 µG N M ≃ T + 2 βG N T , (4)where µ is the calibration factor fixed in the semiclassical limit, β →
0. In the semiclassicallimit, Hawking temperature is given by T = πG N M , and comparing the standard Hawkingtemperature with the temperature T ( β →
0) in Eq. (4), one can find µ = π , so mass-temperature relation is given by M = 18 πG N T + β T π , (5)and modified temperature can be obtained from the above equation T = πβ M − r M − β G N π ! . (6)Now, we consider the simplest form of deformed Schwarzschild metric. The sphericallysymmetric metric is given by ds = − f ( r ) dt + dr f ( r ) + r ( dθ + sin θdφ ) , (7) We usually use the units k B = c = ~ = 1, but we keep the physical constants during the numericalcalculation of β parameter. ( r ) = 1 − G N Mr + ǫ G N M r , (8)where ǫ is a dimensionless parameter. From f ( r H = 0), the event horizon of deformed metricis given by r H = r S √ − ǫ , (9)which is valid ǫ ≤
1. Hence, the deformed Hawking temperature of metric in Eq.(8) is givenby T ( ǫ ) = f ′ ( r H )4 π = 12 πG N M √ − ǫ (cid:0) √ − ǫ (cid:1) , (10)where prime denotes the derivative with respect to r . In order to relate the ǫ with β ,deformed temperature in Eq. (10) must coincide with the modified temperature in Eq.(6),so the relation between ǫ and β is given by β = − π G N M ~ c ǫ − ǫ . (11)The condition ǫ ≤ β . So the deformedmetric is able to define a GUP modified temperature for the negative β parameter. Althoughthe GUP parameter β is usually defined positive in the literature, it is possible to definenegative β parameter. This situation is possible, when uncertainty relation is formulatedon a crystal lattice [22]. This may imply the space time has granular or lattice structure atPlanck scale. Let us start to consider the motion of the particle around the modified Schwarzschild metricin equatorial plane θ = π/
2. The Lagrangian of particle is given by L = 12 g µν ˙ x µ ˙ x ν = 12 (cid:20) − f ( r ) ˙ t + ˙ r f ( r ) + r ˙ φ (cid:21) , (12)where dot denotes the derivative with respect to affine parameter λ . Since the Lagrangian isindependent of coordinates t and φ , the constants of motion can be obtained from generalizedmomentum p µ of particle p µ = ∂ L ∂ ˙ x µ . (13)From the above equation, one can easily obtain p t = ∂ L ∂ ˙ t = − f ( r ) ˙ t = − e = ⇒ ˙ t = ef ( r ) , (14)4 φ = ∂ L ∂ ˙ φ = r ˙ φ = ℓ = ⇒ ˙ φ = ℓr , (15)where e and ℓ are energy and angular momentum of the particle, respectively. Additionally,we have g µν ˙ x µ ˙ x ν = − k , (16)where k = 1 for massive particles and k = 0 for massless particles. From Eqs. (14), (15) and(16) we obtain − e f ( r ) + ˙ r f ( r ) + ℓ r = − k . (17)Multiplying the Eq.(17) by f ( r ) / f ( r ) in Eq.(8), we write e − k r + V eff , (18)where V eff is the effective potential of particle and is defined by V eff = − k G N Mr + ℓ + kǫG N M r − G N M ℓ r + ǫ G N M ℓ r . (19)Later, we shall use extrema of the effective potential to obtain the stable circular orbit radiusof the particle for geodetic precession. In this section, we consider the time delay of electromagnetic signals for GUP deformedSchwarzschild metric. In 1964, Irwin Shapiro proposed a new test which is based on themeasurement of photon time delay due to gravitational field [72]. In order to calculate thetime delay, we follow the argument of Ref. [73].The Fig. 1 is useful to describe the Shapiro time delay. We assume that Sun is locatedat point O. Let us suppose that electromagnetic signal is sent from point A with coordinates( r A , π/ , φ A ) to point B with coordinates ( r B , π/ , φ B ). C is the closest point where theelectromagnetic signal passes near the Sun. Now, we begin to calculate the travel timewhere electromagnetic signal is sent from A to B and reflected back to A in Solar system.With the help of drdλ = drdt dtdλ = drdt ef , (20)one can write the Eq. (17) for massless particles ( k = 0) e f ( r ) (cid:18) drdt (cid:19) + ℓ r − e f ( r ) = 0 . (21)For the closest point r = r C , we get dr/dt = 0, and therefore ℓ = e r C f ( r C ) . (22)5 A BC
Figure 1: Shapiro time delay. (See the text for more details.Using Eq. (22) in Eq. (21), we obtain (cid:18) drdt (cid:19) = f ( r ) (cid:18) − f ( r ) r C f ( r C ) r (cid:19) , (23)or dt = ± dr r(cid:16) − f ( r ) r C f ( r C ) r (cid:17) f ( r ) . (24)Expanding in r S /r and r S /r C ,1 r(cid:16) − f ( r ) r C f ( r C ) r (cid:17) f ( r ) ≈ r p r − r C + r r S ( r − r C ) / + rr C r S r − r C ) / − r C r S r − r C ) / − r C r S r − r C ) / − r r S ( ǫ − r − r C ) / + 3 r C r S r (2 ǫ − r − r C ) / + 3 r C r S (5 − ǫ )8 r ( r − r C ) / . (25)The integrals of expanding terms are Z rdr p r − r C = q r − r C , (26) Z r r S dr ( r − r C ) / = r S ln (cid:18) r + q r − r C (cid:19) − rr s p r − r C , (27) Z rr C r S dr r − r C ) / = − r C r S p r − r C , (28)6 Z r C r S dr r − r C ) / = 3 rr S p r − r C , (29) − Z r C r S dr r − r C ) / = r (3 r C − r ) r S r O ( r − r C ) / , (30) − Z r r S ( ǫ − dr r − r C ) / = r S (3 r − r C ) ( ǫ − r − r C ) / , (31) Z r C r S r (2 ǫ − dr r − r C ) / = r C r S (7 − ǫ )8 ( r − r C ) / , (32)and Z r C r S (5 − ǫ ) dr r ( r − r C ) / = r S (5 − ǫ )8 r C " (3 r − r C ) r C ( r − r C ) / − r C p r − r C ! . (33)Using Eqs. (26)-(33), the travel time of electromagnetic signal from point A to point C isgiven by t AC = p r A − r C + r S ln (cid:18) r A + √ r A − r C C (cid:19) + r S q r A − r C r A + r C (cid:16) − r S (4 r A +5 r C )4 r C ( r A + r C ) (cid:17) + ǫ − r S r C (cid:20) arctan (cid:18) r C √ r A − r C (cid:19) − π (cid:21) . (34)Similarly, we find the travel time of electromagnetic signal from point C to point B t BC = p r B − r C + r S ln (cid:18) r B + √ r B − r C r C (cid:19) + r S q r B − r C r B + r C (cid:16) − r S (4 r B +5 r C )4 r C ( r B + r C ) (cid:17) + ǫ − r S r C (cid:20) arctan (cid:18) r C √ r B − r C (cid:19) − π (cid:21) . (35)The total travel time of electromagnetic signal is t tot = 2 t AC + 2 t BC , and total travel timein flat spacetime is e t tot = 2 (cid:18)q r A − r C + q r B − r C (cid:19) . (36)Since r C ≪ r A , r B , the time delay is given by δt = t tot − e t tot = 4 G N M (cid:18) (cid:18) r A r B r C (cid:19) − G N Mr C (cid:18) ǫ − π (cid:19)(cid:19) . (37)The time delay can be given in parameterized Post-Newtonian (PPN) formalism [74] δt = 4 G N M (cid:18) (cid:18) γ (cid:19) ln (cid:18) r A r B r C (cid:19)(cid:19) , (38)7here γ is a dimensionless PPN parameter. The time delay result of GR is recovered when γ = 1. Comparing Eq. (37) with Eq. (38), we obtain | γ − | = G N M | π − − πǫ | c r C ln (cid:16) r A r B r C (cid:17) . (39)Referring to measurement of the Cassini spacecraft [74, 75], the most stringent constraintof γ parameter is | γ − | < . × − . The Earth and spacecraft distances from Sun were r A = 1AB and r B = 8 . r C = 1 . R ⊙ where R ⊙ is the radius of Sun. Finally, we find − . < ǫ < . , (40)and with the constraint ǫ ≤ − . < ǫ ≤ . (41)If we employ the lower bound of Eq. (41) in Eq. (11), we find the upper bound of GUPparameter β | β | < . × . (42)This bound is the same order of magnitude with the bound obtained from light deflection inRef. [56]. As it can be seen in Table 1, this bound is worse than the bounds obtained fromquantum approaches. Let us consider that the electromagnetic signal moves from point A to point B in a grav-itational field. We want to calculate the change of signal frequency for the deformedSchwarzschild metric. The gravitational redshift formula is given by ν B ν A = s f ( r A ) f ( r B ) , (43)where r A and r B are the radial coordinates of the electromagnetic signal . In our case, thesignal moves from the Earth surface r A = R L to height h , so we have r B = R L + h . Wecan write the Eq. (43) as follows: ν B ν A = vuuut − G N Mr A + ǫ G N M r A − G N Mr B + ǫ G N M r B . (44) Many textbooks on general relativity cover the gravitational redshift in detail. The reader may refer toRef. [73]. νν A = G N M ( r A − r B ) r A r B (cid:20) G N M ((3 − ǫ ) r A + (1 − ǫ ) r B )2 r A r B (cid:21) , (45)where ∆ ν = ν B − ν A . If we neglect the second term in the parentheses, Eq. (45) givesthe standard result of GR. Referring to results of Pound-Snider experiment [76], relativedeviation in the frequency from GR can be given by ∆ νν A − (cid:16) ∆ νν A (cid:17) GR (cid:16) ∆ νν A (cid:17) GR < . . (46)Employing (45) in (46), we get G N M ((3 − ǫ ) r A + (1 − ǫ ) r B )2 r A r B c < . . (47)The mass and radius of Earth are M L = 5 . × kg and R L = 6378km. The experimentwas carried out in a tower with height h = 22 . − . × < ǫ , (48)and Eq. (11) yields | β | < . × . (49)This bound is more stringent than the bound coming from Shapiro time delay, but it is worsethan the bounds from quantum approaches. Finally, we discuss the geodetic precession for the GUP modified metric solution. In thissection, we follow the arguments of Ref. [77]. Let us begin to consider a gyroscope witha spin four-vector s in orbit around a spherical body of mass M . It is well known that agyroscope with four-velocity u obeys the geodesic equation du α dτ + Γ αµν u µ u ν = 0 , (50)where Γ αµν is Christoffel symbol. In addition to geodesic equation, the motion of a gyroscopeis described by ds α dτ + Γ αµν s µ u ν = 0 . (51)Eq. (51) is called gyroscope equation which describes the evolution of gyroscope spin. Thespin and velocity four-vectors satisfy the condition s . u = g µν s µ u ν = 0 , (52)9nd magnitude of spin s ∗ is a constant of motion s . s = g µν s µ s ν = s ∗ . (53)For simplicity, we consider circular orbit in equatorial plane, i.e., θ = π/
2, ˙ r = 0 = ˙ θ .Therefore, the only spatial part of the four-velocity u is given by u φ = dφdτ = dφdt dtdτ = Ω u t , (54)where Ω is the orbital angular velocity. The components of u are given by u = u t (1 , , , Ω) . (55)For the stable circular orbit in equatorial plane, Eq.(18) reduces to e −
12 = V eff , (56)and radius R of circular orbit is obtained from dV eff dr = 0 . (57)Neglecting the last term of effective potential V eff in Eq. (19), one can find R from Eq.(57), R = ǫG N M + ℓ G N M s − G N M ℓ ( ǫG N M + ℓ ) ! . (58)From Eq. (56) and Eq. (58), e and ℓ are given by e = (cid:18) − G N MR + ǫG N M R (cid:19) (cid:18) ℓ R (cid:19) , (59) ℓ = G N M R (cid:18) − ǫG N MR (cid:19) (cid:18) − G N MR (cid:19) − , (60)respectively. Using Eqs. (14) and (15), the angular orbital velocity is given byΩ = dφdt = dφdτ dτdt = f ( r ) r ℓe . (61)Inserting Eqs. (59) and (60) in Eq. (61), we haveΩ = G N MR (cid:18) − ǫG N MR (cid:19) (cid:18) − G N MR + ǫG N M R (cid:19) (cid:18) − G N MR − ǫG N M R (cid:19) − , (62)or Ω = G N MR (cid:18) − ǫG N MR (cid:19) (cid:18) − G N M R + 2 ǫG N M R + ǫ G N M R (cid:19) , (63)10here we use (1 + α ) n ≈ nα for α ≪ s G N MR (cid:18) − ǫG N MR (cid:19) . (64)Now, we can solve the gyroscope equation in Eq. (51). We assume that spin vector is initiallyradial directed, i.e., s t (0) = s θ (0) = s φ (0) = 0. Using the condition in Eq. (52) we get therelation between s t and s φ s t = Ω R (cid:18) − G N MR + ǫG N M R (cid:19) − s φ . (65)With the help of Eqs. (55) and (65), the radial component of gyroscope equation is given by ds r dτ + (cid:18) G N M − R − ǫG N M R (cid:19) Ω s φ u t = 0 . (66)The θ and φ components of gyroscope equation are given by ds θ dτ − sin θ cos θs φ u φ = 0 , (67) ds φ dτ + Ω R s r u t = 0 (68)Since the trajectory is in the equatorial plane, Eq.(67) reduces to ds θ dτ = 0. Imposing theinitial condition s θ (0) = 0, one finds s θ remains zero throughout trajectory. Using u t = dt/dτ in Eqs. (66) and (68), we have ds r dt + (cid:18) G N M − R − ǫG N M R (cid:19) Ω s φ = 0 , (69) ds φ dt + Ω R s r = 0 . (70)Employing Eq. (70) in Eq. (69) gives d s φ dt + ˜Ω s φ = 0 , (71)where we define ˜Ω = r − G N MR + 2 ǫG N M R Ω . (72)Eqs. (69) and (71) yield the solutions s r ( t ) = s ∗ r − G N MR + ǫG N M R cos (cid:16) ˜Ω t (cid:17) , (73)11 φ ( t ) = − s ∗ Ω˜Ω R r − G N MR + ǫG N M R sin (cid:16) ˜Ω t (cid:17) , (74)where we use the conditions s . s = s ∗ and s t (0) = s φ (0) = 0.The spin starts along a unit radial vector e ˆ r with components (cid:16) , p f ( r ) , , (cid:17) . If onerotation along circular orbit takes time P = 2 π/ Ω, we can find the change of spin direction, (cid:20) s s ∗ . e ˆ r (cid:21) t = P = cos π ˜ΩΩ ! . (75)As a result, we find ∆Φ geodetic = 2 π − π r − G N MR + 2 ǫG N M R . (76)For the Solar system, the geodetic precession is approximately given by∆Φ geodetic = ∆Φ GR (cid:18) − ǫG N M Rc (cid:19) , (77)where GR result is ∆Φ GR = πG N MRc . Referring to measurements of Gravity Probe B (GPB)[78], we can find an upper bound for β . Geodetic precession was measured by GPB∆Φ geodetic = (6601 . ± . mas/year . (78)Considering the GPB at an altitude of 642km and with an orbital period of 97 . GR = 6606 . − × < ǫ < . × , (79)and with the constraint ǫ ≤ − × < ǫ ≤ . (80)Using Eq. (11), we finally obtain | β | < . × . (81)This bound is the most stringent bound in this paper, but it is looser than the bounds fromquantum experiments. This work is based on the paper of Scardigli and Casadio [56] where the authors deformedthe Schwarzschild metric to reproduce GUP modified Hawking temperature, and then theycomputed light deflection and perihelion precession for the deformed metric. They comparedtheir theoretical results with astronomical measurements. They finally obtained the upper12able 1: Upper bounds of GUP β obtained from various experiments.Experiment β ReferenceLamb shift 10 [49]Landau levels 10 [49]Scanning tunneling microscope 10 [49]Harmonic oscillators 10 [62]Gravitational waves 10 [60]Light deflection 10 [56]Perihelion precession 10 [56]Pulsar periastron shift 10 [56]Black hole shadow 10 [63]Cosmological constraints 10 [61]Cosmological constraints 10 ,10 [64] Shapiro time delay in this study Gravitational red-shift in this study Geodetic precession in this studybounds of parameter β . In this work, we extended their approach to gravitational tests suchas Shapiro time delay, gravitational redshift and geodetic precession.In Table 1, we give the upper bounds of GUP parameter β from experiments. As we cansee in Table 1, quantum experiments provide more stringent bounds. Unlike the quantumexperiments, gravitational tests give looser bounds. In Ref. [63], upper bound β < was obtained from black hole shadow. To the best of our knowledge, this bound has theworst value. On the other hand, authors of Refs. [61,64] obtained the more stringent boundsby using GUP-modified Friedmann equations with cosmological constraints. Based on theGUP deformation of dispersion relation, authors of Ref. [60] obtained β < from thegravitational wave event GW150914 [79, 80].The bounds in this work are not tighter than the bounds from quantum experiments. Inaddition to results of Ref. [56], namely light deflection ( | β | < ), perihelion precession ofMercury ( | β | < ) and pulsar periastron shift ( | β | < ), we investigated Shapiro timedelay ( | β | < ), gravitational redshift ( | β | < ) and geodetic precession ( | β | < ).The bound β < from geodetic precession is the most stringent bound in this work,but it is clearly worse than the bounds from quantum approaches. Comparing our boundswith the bounds in Ref. [56], the bound | β | < from perihelion precession is the moststringent bound. Apart from Ref. [56] and this paper, the author of Ref. [54] reported thebounds β < from gravitational redshift, β < from the law of reciprocal actions, β < from universality of free fall. These bounds are clearly more stringent than the othergravitational bounds. The method in Ref. [54] is based on the deformed Poisson bracketswhich leads to the violation of equivalence principle (EP). In a recent paper [81], authorsshowed that the deformation of Poisson brackets has some defects such as huge violation ofEP for astronomical objects, badly defined classical limit, etc. For example, the trajectory13f test particle in Ref. [81] is given by¨ r ⋍ − G N Mr (cid:18) β m m p ˙ r (cid:19) , (82)which clearly depends on its mass m and velocity ˙ r . This implies the violation of EP.Furthermore, deformed term increases quadratically with the mass of test particle. Thismay lead to huge deviation from GR . Another problem is the divergent of the commutatordue to badly defined classical limit . On the other hand, deformed metric in Eq. (8) is onlyrelated to GUP modified temperature without EP violation. Since Poisson brackets are notdeformed, the above mentioned defects are not avaliable for deformed metric.Measuring the upper bounds of GUP parameter β may provide us to consider the phe-nomenology of quantum gravity beyond the Planck scale. Gravitational constraints on β may open a large structure windows on the phenomenology of quantum gravity. We hopeto report in future studies. Acknowledgement
The authors thank the anonymous reviewer for his/her helpful and constructive comments.¨Ozg¨ur ¨Okc¨u thanks Can Onur Keser for drawing Fig. 1.
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