Gravitational Observations and LQGUP

Motivated by recent works, we employ the bounds on the dimensionless quantum-gravity parameter obtained from six gravitational tests in order to obtain bounds on the dimensionless parameter of the generalized uncertainty principle with linear and quadratic terms in momentum. The bounds obtained here are much tighter than those obtained, from the same six gravitational tests, for the dimensionless parameter of the uncertainty principle with only quadratic terms in momentum.


I. INTRODUCTION
Many theoretical models in the literature nowadays are trying to solve the incompatibility of General Relativity (GR) and Quantum Mechanics (QM) which arises when one tries to combine them in order to get a complete theory of Quantum Gravity (QG) [1][2][3][4][5].One may claim that this combination may occurs by modifying QM without "touching" GR, hence by generalizing the main QM principle, i.e., the Heisenberg Uncertainty Principle (HUP), into what is called the Generalized Uncertainty Principle (GUP).The GUP has been considered earlier as a consequence of perturbative string theory by adding an extra quadratic term in momentum after proposing the existence of a minimum measurable length [6][7][8][9][10].Later on, phenomenological approaches were pursued in order to derive a bound on the deformation parameter induced by the GUP [11][12][13][14][15].After this proposal, a series of papers were considered adding an extra linear term in momentum on the top of the quadratic term in momentum, which gave rises of what is called Linear and Quadratic GUP (LQGUP).This new GUP proposal was motivated by making the uncertainty principle compatible with the Doubly Special Relativity (DSR) theories and consistent with the commutation relations of phase space coordinates [x i , x j ] = [p i , p j ] = 0 via Jacobi identity.The LQGUP is of the following form [16,17] ∆x∆p ≥ 2 1 + α with α = 2α 0 ℓ p / to be the dimensionful LQGUP parameter, ℓ p = G /c 3 = /2m p c to be the Planck length, and α 0 to be the dimensionless LQGUP parameter.Our recent endeavors [18][19][20][21] motivate us to reassess the value of α 0 in light of LQGUP and compare it with that of α 0 in light of GUP.The LQGUP given by Eq. ( 1) can be satisfied by a set of coordinates and momenta defined as with x 0i and p 0i to satisfy the canonical commutation relations [x 0i , p 0j ] = i δ ij .It is well-established [22] that an uncertainty principle is produced by an algebra through the inequality of any two observables ∆x∆p Employing mirror-symmetric states, i.e., p 2 = 0, one gets ∆p = p 2 , and the aforesaid LQGUP given in Eq. ( 1) can be written in terms of a commutator as The remainder of the paper is structured as follows.In Section II, we present the LQGUP-modified temperature of a Schwarzschild black hole.In Section III, we demonstrate the change in the Schwarzschild metric due to the quantum corrections suggested by LQGUP such that we relate the dimensionless parameter α 0 with the corrected term added to the Schwarzschild metric.In Section IV, we reassess the upper bound of the LQGUP parameter α 0 , using the data obtained from the six solar system-based gravitational tests: light deflection, perihelion precession, pulsar periastron shift, Shapiro time delay, gravitational red shift, and geodetic precession.
In Section V, we comment and discuss the obtained results.We compare between the dimensionless parameter of the GUP and that of LQGUP based on the formerly mentioned solar system-based gravitational observations.

II. LQGUP-MODIFIED BLACK HOLE TEMPERATURE
In this section, following the analysis of Ref. [23], we will briefly describe the derivation of the LQGUP-modified temperature of a Schwarzschild black hole.
According to the Heisenberg microscope thought experiment, one needs a photon shot of energy E to locate a particle of size δx.Within the framework of LQGUP described by Eq. ( 1) and considering the standard dispersion relation for photons, i.e., E = p, the size of the particle is Eq. ( 4) can be utilized to compute the energy E of the particle by having its (average) wavelength λ ≃ δx.
Following the same concept, we can compute the LQGUP-modified temperature of a Schwarzschild black hole.
We consider an ensemble of unpolarized photons as the Hawking radiation particles that are coming out of the event horizon of the Schwarzschild black hole of mass M .The event horizon is located at r S = 2GM (for simplicity we have set c = k B = 1).The position uncertainty δx of these photons, which is related to the size of the Schwarzschild event horizon r S , will take the form where µ is a dimensional parameter that is yet to be determined.According to the equipartition principle, the energy E of the photons of the Hawking radiation is actually the temperature T of the Schwarzschild black hole, i.e., E = T .Therefore, from Eqs. ( 4) and ( 5), we obtain the mass-temperature relation as Setting α = α 0 /m p , Eq. ( 6) reads now Considering the semi-classical limit by taking the dimensionless LQGUP parameter α 0 → 0, the temperature has to be the standard Hawking temperature, namely T = T BH = /8πGM , thus we fix the dimensionless parameter to be µ = π.Rewriting Eq.( 7), we get Solving for T , we get Expanding near the semi-classical limit α 0 → 0, the LQGUP-modified black hole temperature, up to the second order in α 0 , reads [23,24]

III. QUANTUM-CORRECTED SCHWARZSCHILD METRIC
The gravitational interaction between two heavy objects at rest can be described by a potential energy which is produced by the potential generated from the mass M [25-28] Due to quantum gravity corrections, the classical Schwarzschild metric can be deformed as with Since we are interested only in the leading correction to the Hawking formula, we can consider the simplest deformation of the Schwarzschild metric to be of the form [29] b where ǫ is a dimensionless parameter.This deformation of the Schwarzschild metric is nothing more than the well-known Eddington-Robertson expansion of a spherically symmetric metric.The event horizon of the deformed metric is given by (F (r H ) = 0) with r S = 2GM , and Eq.( 15) is valid for ǫ ≤ 1.The quantum-corrected Hawking temperature of the deformed metric Eq.( 13) is given by [29,30] At this point, we may conjecture that the LQGUP-deformed black hole temperature given by Eq.( 10) is equal to the quantum-corrected Hawking temperature given by Eq.( 16).Therefore, we obtain with Solving for the LQGUP dimensionless parameter α 0 , we get At this point a couple of comments are in order.First, in order to assure that we obtain the semi-classical limit, i.e., α 0 → 0 as ǫ → 0, we need to keep only the negative solution in Eq. (19).Second, if we expand Eq.( 19) near the semi-classical limit ǫ → 0 (X(ǫ) → 0), we obtain

IV. LQGUP-PARAMETER BOUNDS
In this section, we compute the physical (possible observable) quantities which are modified due to the deformation of the Schwarzschild metric.Consequently, we evaluate the upper bound of LQGUP parameter α 0 using the results from six different solar system-based observational tests of GR.

A. Light Deflection
Here we adopt the analysis of Ref. [29].We utilize the polar coordinates (φ, r) centered at the Sun in order to describe a photon's orbit.The Sun will deflect the orbit of the incoming photon from straight line and, thus, the global deflection angle of the photon's orbit will be given as with r 0 to be the minimum distance between the photon and the Sun.The photon's orbit from ∞ to point r is described by where F (r) is given by Eq.( 13).Expanding the above integral in terms of r S /r 0 (with r S = 2GM/c2 to be the unmodified Schwarzschild radius), Eq.( 21) will become, to first order in ǫ and to second order in r S /r 0 , ∆φ ≃ 2 r S r 0 + 1 16 The deflection angle of a photon, that barely touches the surface of the Sun, is normally given by ∆φ Comparing Eq.( 23) with Eq.( 24), we obtain Considering the best experimental measurement available for the parameter γ from the light bending close to the surface of the Sun, we obtain [34,36,37] Then, taking M = M ⊙ = 1.989 × 10 30 kg to be the solar mass, r 0 = R ⊙ = 6.963 × 10 8 m to be the radius of the Sun, and implementing the mathematical constraint ǫ ≤ 1, we obtain1 Employing the lower bound of Eq.( 27) in Eq.( 20), we obtain the upper bound of the dimensionless LQGUP parameter, i.e., α 0 , to be It is evident that this bound is not a strict one among the bounds obtained from the solar system-based gravitational tests, and we will see that tighter bounds will be obtained in the coming subsections.

B. Perihelion Precession
We adopt the analysis of Ref. [29] and use polar coordinates to study a planet orbiting around the Sun.As expected, the planet's orbit is an elliptical one.The total orbital precession in each revolution is with r + to be the maximum distance of the planet from the Sun, namely the aphelion, and r − to be the minimum distance of the planet from the Sun, namely the perihelion.For an arbitrary point r, the orbital precession as the planet moves from r − to r is given by the integral where F (r) is given by Eq.( 13).Expanding the above integral in terms of r S /L (with r S = 2GM to be the unmodified Schwarzschild radius, L to be the semilatus rectum, and here, for simplicity, c = 1), the total precession after one revolution is given as, to second order in ǫ and r S /r 0 , or, equivalently, where with e to be the eccentricity.Next, we write Eq.( 32) to first order in r S /L so that we get which leads to the GR prediction as ǫ → 0. Now, we can compare Eq.( 34) with the best known and measured precession in the solar system, the perihelion precession for Mercury [34].The latest data is given by where ∆φ obs is the observed perihelion shift, J 2 is a dimensionless measure of the quadrupole moment of the Sun, and γ and β are the usual Eddington-Robertson expansion parameters.Comparing Eq.( 35) with Eq.( 34), we get where, from observational data for the Sun and Mercury [38], the bound |2γ − β − 1| 3 × 10 −3 has been implemented, and the last term in Eq.( 35) has been neglected since it is smaller than the observational error.Substituting the above bound on ǫ into Eq.(20), we obtain the upper bound of the dimensionless LQGUP parameter, i.e., α 0 , to be which is much tighter bound than the one obtained using the light deflection observational test.

C. Pulsar Periastron shift
We adopt the analysis of Ref. [29], and we compare Eq.( 34) with another very good observational measurement of the pulsars periastron shift.The observed value of the periastron shift is ∆φ obs ≃ 4.226598(5) [38], where the number in parentheses represents the uncertainty in the last quoted digit.Therefore, the relative error with respect to the GR theoretical prediction, namely ∆φ GR , of the periastron shift can be defined as ǫ = ∆φ obs − ∆φ GR ∆φ GR (38) which may be written as ∆φ obs = ∆φ GR (1 + ǫ) and can be compared with ∆φ in Eq.( 34) to get |ǫ| = 6|ǫ|.Using some numerical measurements of periastron shift [38], we get |ǫ| = 8.9 × 10 −5 , which gives Considering this bound on the dimensionless parameter of the deformed metric and employing it in Eq. ( 20), we obtain the upper bound of the dimensionless LQGUP parameter, i.e., α 0 , to be which is the strictest bound among the bounds obtained in this work from the solar system-based gravitational tests.

D. Shapiro Time Delay
Considering Irwin Shapiro observational endeavors on measuring the time delay of photon due to gravitational field [39], we study the impact of the LQGUP-deformed Schwarzschild metric on calculating the delay in the traveling time of electromagnetic signal moving from point A at r A to point B at r B and reflected back to A due to gravitational field of the solar system.We adopt the analysis of Ref. [30], and we obtain with r C to be the position of the closest point C the electromagnetic signal would pass by near the Sun surface as the Sun is located at the center of the system under study.The time delay in Parameterized Post-Newtonian (PPN) formalism is [34] where γ is a dimensionless PPN parameter.Comparing Eq. (41) and Eq. ( 42), we get Using the measurement of the Cassini spacecraft [34,35], namely |γ − 1| < 2.3 × 10 −5 , and setting r A = 1 AB, r B = 8.46 AB (with AB = 152.03× 10 9 m to be the distance between the Sun and the Earth), r C = 1.6 R ⊙ , and M = M ⊙ , we obtain and in order to be consistent with the constraint ǫ ≤ 1, the above bounds could be modified to Assuming the "worst" situation, namely ǫ ≃ −44.9, then Eq. ( 20) gives the upper bound for the dimensionless LQGUP parameter α 0 to be which is comparable to the bound obtained before from the observational test of the light deflection.

E. Gravitational Redshift
We adopt the analysis of Ref. [30] for the LQGUP-deformed Schwarzschild metric in order to calculate the change of electromagnetic signal frequency moving from the Earth surface, so r A = R ⊕ , to height h, so r B = R ⊕ + h.Therefore, we get where we use the Pound-Snider experiment results [31].Utilizing the mass of Earth M ⊕ = 5.972 × 10 24 kg and its radius R ⊕ = 6.378 × 10 6 m, with the height at which the experiment has been performed, namely h = 22.86 m, we obtain Employing the lower bound of Eq.( 48) in Eq.( 20), we get the upper bound of dimensionless LQGUP parameter α 0 to be which is much tighter bound than the one obtained using the Shapiro delay time observational test and comparable to the bound obtained from the Mercury perihelion precession test.

F. Geodetic Precession
We adopt the analysis of Ref. [30] for the geodetic precession of the solar system for the LQGUP-deformed Schwarzschild metric, and approximately we obtain where ∆Φ GR = 3πGM Rc 2 is the geodetic precession theoretically predicted by GR.At an altitude of 6.42 × 10 5 m above the Sun surface, and with an orbital period of 97.65 min, the Gravity Probe B (GPB) [40] measures the geodetic precession to be ∆Φ geodetic = (6601.8± 18.3) mass/year while the GR prediction gives ∆Φ GR = 6601.1 mass/year.Therefore, we set a bound on the dimensionless parameter ǫ as where we have used the mathematical constraint ǫ ≤ 1 with M = M ⊕ and R = R ⊕ .Substituting the lower bound of the dimensionless parameter ǫ given by Eq.(51) into Eq.(20), we obtain the upper bound of the dimensionless LQGUP parameter α 0 to be which is exactly the same with the upper bound obtained before using the observational test of the gravitational redshift.

V. CONCLUSION AND DISCUSSIONS
In this letter, we compared the temperature of Schwarzschild black hole when modified due to the linear and quadratic terms in the momenta of the generalized uncertainty principle (LQGUP) with the temperature obtained when the Schwarzschild black hole metric is corrected in a quantum context.The dimensionless LQGUP parameter, i.e., α 0 , was expressed in terms of the dimensionless parameter ǫ that characterizes the quantum-gravity corrected Schwarzschild black hole metric.By employing six solar system-based observational (gravitational) tests: light deflection, perihelion precession, pulsar periastron shift, Shapiro time delay, gravitational redshift, and geodetic precession, we utilized the bounds on the dimensionless parameter ǫ to obtain bounds on the dimensionless LQGUP parameter.In the literature, there are the corresponding bounds for the dimensionless parameter β 0 of the generalized uncertainty principle with only quadratic terms in momentum (GUP).We listed all of these bounds, together with the new bounds we derived in this work, in the table below.
In order to compare them, one can roughly assume that β 0 ∼ α 2 0 .It is evident that the bounds based on LQGUP are much tighter than the corresponding bounds based on GUP.
Finally, it should be stressed that, all these bounds are much weaker than the bound set by the electroweak scale ℓ EW ≤ 10 17 ℓ P lanck as well as weaker than the bounds set by other gravitational tests performed in strong gravitational fields.One can say that this is anticipated since all the bounds in this work are obtained by employing solar system-based gravitational tests and in the solar system the QG effects are not so strong to be detected as in other gravitational systems such as that of the two stellar mass black holes that are merging and produce a spacetime of high curvature.In the latter case, one can detect the emitted gravitational waves and set strict bounds on the dimensionless LQGUP parameter, for instance from the events GW150914 and GW170814 one can get α 0 ∼ 10 8 [32].Of course, one can study pure quantum systems to measure quantum effects and, thus, to get strict bounds on the dimensionless LQGUP parameter, for instance from the Lamb shift one gets α 0 ∼ 10 10 [33].× 10 78 [30] 3.17 × 10 38 [NEW] Gravitational Redshift 1.1 × 10 73 [30] 1.72 × 10 33 [NEW] Geodetic Precession 3.7 × 10 72 [30] 1.72 × 10 33 [NEW]