Gravitationally induced uncertainty relations in curved backgrounds
GGravitationally induced uncertainty relations in curved backgrounds
Luciano Petruzziello
1, 2, ∗ and Fabian Wagner † Dipartimento di Ingegneria Industriale, Universit´a degli Studi di Salerno,Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy. INFN, Sezione di Napoli, Gruppo collegato di Salerno, Italy. Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland (Dated: January 15, 2021)This paper aims at investigating the influence of space-time curvature on the uncertainty relation.In particular, relying on previous findings, we assume the quantum wave function to be confined to ageodesic ball on a given space-like hypersurface whose radius is a measure of the position uncertainty.On the other hand, we concurrently work out a viable physical definition of the momentum operatorand its standard deviation in the non-relativistic limit of the 3+1 formalism. Finally, we evaluatethe uncertainty relation which to second order depends on the Ricci scalar of the effective 3-metricand the corresponding covariant derivative of the shift vector. For the sake of illustration, we applyour general result to a number of examples arising in the context of both general relativity andextended theories of gravity.
I. INTRODUCTION
Heisenberg’s celebrated uncertainty principle in itsfundamental form does not take into account effectswhich are expected to arise when gravity is being ac-counted for. In fact, one of the low-energy implicationsassociated with several candidate theories of quantumgravity predicts a correction to the aforementioned rela-tion going by the name Generalized Uncertainty Principle(GUP) [1–6]. After seminal efforts have been devoted tothis topic, the investigation revolving around the gener-alization of the fundamental quantum mechanical uncer-tainty has developed remarkably. As a matter of fact, theconsequences entailed by the GUP deeply affects severalaspects of black hole thermodynamics [7–20], quantumfield theory [21–31] and also quantum information [32–34].As usually assumed, the GUP takes into account thegravitational uncertainty of position in connection withthe existence of a minimal fundamental length scale inphysics. However, it may not be the only gravitationallyinduced change. In particular, the curvature of space-time does exert an influence over quantum mechanicaluncertainty relations. This is the regime of the ExtendedUncertainty Principle (EUP) [35–39], whose purpose liesin the investigation of the uncertainty related to the back-ground space-time.Recently, a derivation of the EUP was performed in[40, 41]. It reflects the influence of spatial curvature onquantum-mechanics on 3-dimensional spacelike hypersur-faces of space-time. The method was applied to homo-geneous and isotropic geometries of constant curvature K and the corresponding EUP was calculated. This for-malism was further developed in collaboration with oneof the authors to accommodate for horizons [42] and arbi- ∗ [email protected] † [email protected] trary 3-dimensional manifolds, thus yielding an asymp-totic extended uncertainty principle [43]. In short, thequantum mechanical wave function of interest is definedover a curved three-dimensional manifold and its domainconstrained to a geodesic ball. Then, the radius of thisball serves as a diffeomorphism-invariant measure of po-sition uncertainty. By computing the standard devia-tion of the momentum operator, one finds that there isa generic minimum which depends on the radius of thegeodesic ball and on curvature invariants derived fromthe effective background. This represents a mathemati-cally rigorous as well as physically motivated formulationof the EUP.This paper goes further inasmuch it aims to providethe uncertainty relation experienced by a non-relativisticparticle in a curved 4-dimensional background. To com-ply with this goal, we obtain the effective Lagrangianaccording to which the particle dynamics evolves. As iswell-known in the literature [44], the gravitational fieldsplits up into a three-dimensional background metric, agravitomagnetic vector field and a gravitoelectric scalarfield. Due to the resulting gauge invariance which is notshared by the conjugated momentum, the physical mo-mentum includes a contribution from the vector field.Adding these ingredients to the picture outlined aboveand taking into account subtleties of the quantization asfirst pointed out in [45], we finally achieve the desiredresult. This makes it possible to compare the impactof distinct theories of gravity (which describe differentbackgrounds) on the uncertainty relation.The paper is organized as follows: in section II wederive the effective Hamiltonian of a non-relativistic par-ticle on a curved 4D background. This theory is quan-tized in section III, where we define the momentum un-certainty and the relevant operators whose expectationvalues appear therein and compute the curvature inducedcorrections to the uncertainty relation. The correspond-ing results are applied to several relevant space-times insection IV to be subsequently discussed in section V. a r X i v : . [ g r- q c ] J a n II. DERIVATION OF THE EFFECTIVEHAMILTONIAN
In this section, we will derive the effective dynamicsof a non-relativistic particle on a curved 4-dimensionalbackground. To this aim, note that the action of a mas-sive relativistic particle subjected to a curved geometrycan be written as S = − m (cid:90) d s (1)= − m (cid:90) (cid:113) − g µν ( x ) ˙ x µ ˙ x ν d τ (2)with the background metric g µν (the Greek indices standfor space-time coordinates while Latin ones denote thespatial part), the four-velocity ˙ x µ = d x µ / d τ ( τ denotesthe proper time along the curve) and the mass of theparticle m. Thus, the corresponding Lagrangian reads L = − m (cid:113) − g µν ( x ) ˙ x µ ˙ x ν . (3)Note that the action (2) is invariant under temporalreparametrisations τ (cid:48) = f ( τ ) for any sufficiently well-behaved function f. After some algebra, the Lagrangian can be recast as L = − m (cid:104) (cid:0) − g + g i g j h ij (cid:1) ( ˙ x ) − (cid:0) ˙ x g k h ik + ˙ x i (cid:1) (cid:0) ˙ x g l h jl + ˙ x j (cid:1) g ij (cid:105) / (4)with h ik g kj ≡ δ ij . Thus, h ij is the inverse of the inducedmetric on hypersurfaces of constant x . The Lagrangiancan be further simplified by introducing the backgroundfield quantities N = (cid:113) − g + g i g j h ij (5) N i = g j h ij (6)which are readily identified as the lapse function and theshift vector in the 3+1 formalism [46]. According to thisapproach to curved Lorentzian manifolds, any metric canbe written as [47]d s = − N (cid:0) d x (cid:1) + h ij ( N i d x +d x i )( N j d x +d x j ) (7)with h ij ≡ g ij . Thence, the Lagrangian reads L = − m (cid:113) N ( ˙ x ) − ( ˙ x N i + ˙ x i ) ( ˙ x N j + ˙ x j ) h ij (8)which under the assumption that ˙ x > i. e. that coor-dinate time is moving in the same direction as the particleproper time) can be written as L = − mN ˙ x (cid:115) − ( ˙ x N i + ˙ x i ) ( ˙ x N j + ˙ x j ) h ij N ( ˙ x ) (9) ≡ − mN ˙ x √ − (cid:15) (10) where the last equality defines (cid:15). In terms of the conjugatemomenta P µ = mg µν ˙ x ν / (cid:112) − g µν ˙ x µ ˙ x ν , we find (cid:15)/ (1 − (cid:15) ) = h ij P i P j /m , which means that the non-relativistic limitcorresponds to (cid:15) (cid:28) . Therefore, we can expand to obtainthe effective non-relativistic Lagrangian L NR = m x (cid:0) ˙ x N i + ˙ x i (cid:1) (cid:0) ˙ x N j + ˙ x j (cid:1) G ij − mN ˙ x (11)with the effective 3-metric G ij = h ij /N. From this pointonwards, this metric will be used to lower and raise in-dices and as the background for differential geometricquantities.The effective non-relativistic action S NR = (cid:82) L NR d τ still harbours the time reparametrization invariance al-luded to above. For simplicity, we can fix the gauge bychoosing x = τ to obtain the gauge-fixed non-relativisticLagrangian L NR = m (cid:0) N i + ˙ x i (cid:1) (cid:0) N j + ˙ x j (cid:1) G ij − mN. (12)A closer look at the Lagrangian tells us that it is of theform L NR = m x i ˙ x j G ij + m ˙ x i A i − mφ (13)with A i = N j G ij and φ = N − N i N j G ij / . This is clearlyreminiscent of the Lagrangian describing a charged non-relativistic particle minimally coupled to an electromag-netic gauge one-form A µ = ( φ, A i ) where the mass m plays the rˆole of the charge.On the other hand, the Lagrangian is additionally in-variant under the gauge transformation A → A i + ∂ i f,φ → φ − ˙ f , G ij → G ij for any scalar function f ( x i , t )while the canonical momenta π i = ∂L NR ∂ ˙ x i = mG ij (cid:0) ˙ x j + N j (cid:1) (14)are not, rendering them unobservable. Therefore, we de-fine the gauge invariant physical momenta as p i ≡ π i − mN j G ij (15)in terms of which the Hamiltonian reads H NR = 12 m p i p j G ij + mφ. (16)Having found the effective Hamiltonian, we are nowable to give a quantum mechanical description of non-relativistic particles in curved space-time. III. QUANTUM MECHANICAL TREATMENT
Aiming towards a generalization of the reasoning in[43] to four dimensional backgrounds, we have to findthe quantum mechanical counterpart of the theory out-lined in the previous section. For this purpose, we firstconstruct an equivalent Hilbert space and introduce thebasis in terms of which calculations will be performed.As we pointed out above, the effective Hamiltonian (16)entails a certain amount of gauge freedom which makesit necessary to define the gauge invariant ( i. e. physical)momenta p i . Moreover, we obtain the quantum operatorsrepresenting their co- and contravariant versions and de-fine the standard deviation accordingly. To conclude thesection, we derive the flat space uncertainty relation andits higher-order corrections.
A. Hilbert space and used basis
In order to construct the Hilbert space, we have tofind an appropriate measure defining the scalar product (cid:104) ψ | φ (cid:105) = (cid:82) d µψ ∗ φ. The problem at hand takes place in a3-dimensional curved background defined by the canon-ical connection with respect to G ij and offers the gaugefreedom mentioned above. These facts should be re-flected by the measure making it gauge and diffeomor-phism invariant. The only possible differential satisfyingthese constraints reads d µ = √ G d x where G = det G ij . Furthermore, we confine the quantum states to ageodesic ball B ρ of radius ρ by imposing Dirichlet bound-ary conditions as in [43]; hence, they reside in the Hilbertspace H = L ( B ρ ⊆ IR , √ G d x ) . As the center of thegeodesic ball p is situated in the background manifold,it represents the expectation value of the position oper-ator while ρ can be interpreted as a measure of positionuncertainty.Since the negative Laplace-Beltrami operator − ∆ ishermitian in H , its eigenstates furnish an orthonormalbasis of the Hilbert space. Due to the compactness of thedomain of H , the spectrum of said operator is discrete.Thus, there is a countably infinite number of base vectorssatisfying the eigenvalue problem(∆ + λ nlm ) ψ nlm = 0 ψ nlm | ∂B ρ = 0 (17)where n, l, m stand for the three quantum numbers iden-tifying the state.In the case of a flat background, the solutions to theproblem (17) can be obtained analytically and read innormalized form ψ nlm = (cid:115) ρ j l +1 ( j l,n ) j l (cid:18) x nl σρ (cid:19) Y ml ( χ, γ ) (18)with the spherical Bessel function j l , the spherical har-monics Y ml and the pure number j l,n which is the nth so-lution of the equation j l ( j l,n ) = 0 or, in other words, thenth zero of the Bessel function J l +1 / . The correspondingeigenvalues are λ nl = j l,n ρ . (19) B. Physical momentum operator
When promoting positions and momenta to quantumoperators, we impose the canonical commutation rela-tions [ˆ x i , ˆ p j ] = i (cid:126) δ ij . In the position space representation,these are solved by the operatorsˆ x i ψ = x i ψ ˆ p i ψ = ( − i (cid:126) ∂ i + F i ( x )) ψ (20)where ψ describes a general wave function. The arbitraryform F i ( x ) can be found imposing that the momentumoperator be hermitian with respect to the Hilbert spacemeasure d µ , i. e. (cid:104) ψ | ˆ p i φ (cid:105) = (cid:104) ˆ p i ψ | φ (cid:105) (21)and aptly represents the physical momentum in the clas-sical limit.It can be shown that these criteria are uniquely satis-fied by the momentum operatorˆ p i ψ = (cid:16) ˆ π i − m ˆ N j ˆ G ij (cid:17) ψ (22) ≡ − (cid:20) i (cid:126) (cid:18) ∂ i + 12 Γ jij (cid:19) + mN j G ij (cid:21) ψ (23)where the second equality defines ˆ π i and we introducedthe Christoffel symbolΓ kij = 12 G kl ( ∂ i G jk + ∂ j G ik − ∂ k G ij ) . (24)Using this result, there is only one possible hermitiandefinition of the covariant momentum operator readingˆ p i ψ = (cid:16) ˆ π i − m ˆ N i (cid:17) ψ (25) ≡ − (cid:20) i (cid:126) (cid:0)(cid:8) G ij , ∂ j (cid:9) + G ij Γ kkj (cid:1) + mN i (cid:21) ψ (26)= 12 (cid:110) ˆ G ij , ˆ p j (cid:111) ψ (27)where ˆ π i is defined according to the second equality.Finally, the square of the momentum operator is of theform ˆ p = ˆ π − m ˆ p mix + m ˆ G ij ˆ N i ˆ N j (28)where ˆ π ψ = − (cid:126) ∆ ψ and ˆ p mix mixes canonical momentaand the shift.Under the assumptions that it is hermitian and leadsto the correct classical limit, this operator readsˆ p mix = 12 (cid:110) ˆ π i , ˆ N i (cid:111) . (29)After some elementary algebra, it can be seen that it actson wave functions asˆ p mix ψ = − i (cid:126) (cid:2) ∇ i (cid:0) N i (cid:1) + 2 N i ∂ i (cid:3) ψ (30)with the covariant derivative ∇ i with respect to thecanonical connection of G ij . Note that the first term in (30) is anti-hermitian, andsince ˆ p mix is hermitian it cancels out the anti-hermitianpart of the second term so that we can rewrite ˆ p mix asˆ p mix = (cid:0) − i (cid:126) N i ∂ i (cid:1) H ψ (31)where the subscript H denotes the hermitian part.Summing up the outcome of this subsection, the rele-vant operators act asˆ p i ψ = (cid:126) (cid:20) − i (cid:18) ∂ i + 12 Γ jij (cid:19) − G ij N j λ C (cid:21) ψ (32)ˆ p i ψ = (cid:126) (cid:20) − i (cid:0)(cid:8) G ij , ∂ j (cid:9) + G ij Γ kjk (cid:1) − N i λ C (cid:21) ψ (33)ˆ p ψ = (cid:126) (cid:20) − ∆ + 2 λ C (cid:0) iN i ∂ i (cid:1) H + N i N j G ij λ C (cid:21) ψ (34)with the reduced Compton wavelength λ C = (cid:126) /m. Having figured out the position space representationof the contra-, covariant and squared momentum opera-tors in (32), (33) and (34) respectively, we can define themomentum uncertainty as σ p ≡ (cid:112) (cid:104) ˆ p (cid:105) − (cid:104) ˆ p i (cid:105) (cid:104) ˆ p i (cid:105) . (35)The remainder of this paper will be centered around theevaluation of this quantity. C. Perturbation around flat space
Following the approach in [43], we calculate the uncer-tainty relation perturbatively. In particular, we expand G ij in Riemann normal coordinates (RNC) x i to secondorder G ij (cid:39) G (0) ij + G (2) ij (36)= δ ij − R ikjl | p x k x l (37)with the Kronecker Delta δ ij and the Riemann curvaturetensor with respect to the metric G ab evaluated at p . This leads to ensuing expansions in ∇ i = ∇ (0) i + ∇ (2) i , ∆ =∆ (0) + ∆ (2) due toΓ ijk (cid:39) (cid:0) Γ iij (cid:1) (2) (38)= 13 (cid:0) R ijkm + R ikjm (cid:1) | p x m . (39)Additionally, the measure in expanded form reads d µ =d µ (0) + d µ (2) , with d µ (0) = d x andd µ (2) = − R ij | p x i x j d x. (40)Correspondingly, the scalar product is perturbed as (cid:104)(cid:105) (cid:39)(cid:104)(cid:105) + (cid:104)(cid:105) . How to treat quantum mechanical perturbation theory in this special case has been elaborated upon in[43]. Note that with (cid:104)(cid:105) ( n ) we denote the nth correction tothe whole amplitude including wave functions and oper-ators inside, while (cid:104)(cid:105) n signifies the nth correction to thescalar product.Due to the spherical symmetry of the unperturbedproblem, integrals appearing throughout the calculationwill be solved in geodesic coordinates ( σ, χ, γ ) which wereintroduced in [43] and relate to RNC as spherical coor-dinates to Cartesian ones x i = σ (sin χ cos γ, sin χ sin γ, cos χ ) . (41)As the metric has been expanded in RNC, the sameshould be done for the shift, which now reads N i (cid:39) N i (0) + N i (1) + N i (2) (42)= N i | p + ∇ j N i | p x j + ∇ j ∇ k N i | p x j x k . (43)This means that, in principle, the shift could yield zerothand first order corrections.Finally, the momentum uncertainty is expanded as σ p (cid:39) (cid:113)(cid:0) σ p (cid:1) (0) + (cid:0) σ p (cid:1) (1) + (cid:0) σ p (cid:1) (2) (44)= σ (0) p (cid:0) σ p (cid:1) (1) (cid:0) σ p (cid:1) (0) + (cid:0) σ p (cid:1) (2) − ( σ p ) (1) ( σ p ) (0) (cid:0) σ p (cid:1) (0) (45)where we introduced the variance σ p . These contributionswill be treated order by order.
D. Unperturbed uncertainty relation
To zeroth order the momentum uncertainty reads σ (0) p = (cid:115) ( σ π ) (0) + 2 (cid:126) λ C N i | p (cid:104) i (cid:126) ∂ i + π i (cid:105) (0) + (cid:126) λ C ( σ N ) (0) (46)where we used that i∂ i is hermitian with respect to theunperturbed measure.As π (0) i = − i (cid:126) ∂ i , the term in the brackets vanishes.Moreover, it is a simple exercise to show that a similarcancellation occurs to the variance of the shift leaving uswith σ (0) p = σ (0) π . (47)Hence, the shift has no influence to zeroth order.Note that these considerations hold independently ofthe state with respect to which the uncertainty is calcu-lated. According to the formalism developed in [41] andsubsequently in [43], we obtain an uncertainty relationby finding the state of minimum momentum uncertainty.Yet, there it was falsely claimed that (cid:104) ˆ π i (cid:105) = (cid:104) ˆ π i (cid:105) = 0for arbitrary states in arbitrary backgrounds. This isindeed not the case. Nevertheless, as shown in theappendix, the state of minimal uncertainty is still theground state for flat backgrounds. In geodesic coordi-nates, it reads ψ (0)100 = 1 √ πρ sin (cid:16) π σρ (cid:17) σ (48)which is the leading-order contribution to ψ . Conse-quently, the relation σ (0) p (cid:16) Ψ (0) (cid:17) ≥ σ (0) p (cid:16) ψ (0)100 (cid:17) (49)holds for the zeroth-order approximation Ψ (0) to everyΨ ∈ H . As perturbations are assumed to be small, theycannot change this relation, which is why we deduce that σ p (Ψ) ≥ σ p ( ψ ) . (50)As shown in the appendix, the ground state (as all eigen-states of the Laplace-Beltrami operator) satisfies (cid:104) ψ | ˆ π i ψ (cid:105) = (cid:104) ψ | ˆ π i ψ (cid:105) = 0 (51)to all orders. Therefore, we obtain σ (0) p ≥ (cid:113) − (cid:126) (cid:104) ψ | ∆ ψ (cid:105) (0) = (cid:126) π/ρ. (52)Now that the state of lowest momentum uncertainty isidentified, it is time to evaluate the curvature inducedcorrections. E. Corrections
First, according to (30), we can generally write (cid:104) ˆ p mix (cid:105) = (cid:126) Im (cid:104) N i ∂ i (cid:105) . (53)As the ground state is real, the integrand appearing in (cid:104) ψ | N i ∂ i ψ (cid:105) = (cid:90) d µψ N i ∂ i ψ (54)is purely real and so is the integral. Thus, in the groundstate the expectation value of ˆ p mix vanishes.Furthermore, terms mixing expectation values of theshift and the momentum vanish identically due to (51).Then, the variance of the ground state equals σ p ( ψ ) = σ π ( ψ ) + (cid:126) λ C σ N ( ψ ) (55)non-perturbatively. Curvature corrections to (cid:0) σ π (cid:1) ap-pear to second order and the first-order correction to (cid:0) σ N (cid:1) is subject to similar cancellations as in (46). Hence,the first-order contribution to the variance of the physicalmomentum operator vanishes (cid:0) σ p (cid:1) (1) ( ψ ) = 0 (56) from which we deduce that the shift corrects the uncer-tainty relation at the same order as the background cur-vature.Moreover, from [43] we already know that (cid:0) σ π (cid:1) (2) ( ψ ) = − R | p (57)where R | p = G ij G kl R ikjl | p . At this point, we are leftwith the second-order correction to the variance, whichafter cancellations reads (cid:0) σ N (cid:1) (2) = ∇ k N i ∇ l N j G ij | p (cid:0) (cid:104) x k x l (cid:105) − (cid:104) x k (cid:105) (cid:104) x j (cid:105) (cid:1) (0) . (58)When evaluated with respect to the ground state, thesecond term in the bracket vanishes while the first yields (cid:104) ψ | x j x k ψ (cid:105) (0) = ρ (cid:18) − π (cid:19) δ jk . (59)Finally, lowering and raising indices with the effectivemetric G ij , the second-order correction to the varianceof the physical momentum operator equals (cid:0) σ p (cid:1) (2) ( ψ ) = − R | p ξ ρ λ C ∇ j N i ∇ j N i | p (60)where we introduced the mathematical constant ξ = (2 − /π ) / . Observe that, though it seems to be of higher order atfirst glance due to the factor ρ , the second term is actu-ally of second order because the expansion done here isperformed in terms of ρ √R where R denotes any curva-ture invariant with dimensions of squared inverse length.Thus, no assumptions were made concerning the factor ρ /λ C . F. Result
Gathering all the results from the previous sectionsand introducing the Compton wavelength λ C ≡ πλ C ,we obtain the uncertainty relation: σ p ρ (cid:38) π (cid:126) (cid:20) − ρ R | p π + ξ ρ λ C ∇ j N i ∇ j N i | p (cid:21) . (61)In short, given a four metric g µν and an observer defininga foliation of space-time, the uncertainty relation (61)can be computed by evaluating the Ricci scalar derivedfrom G ij and the corresponding covariant derivative ofthe shift at p . IV. APPLICATIONS
In what follows, we will employ the formalism devel-oped up until now to present explicit results related toseveral relevant metrics. To achieve non-trivial solu-tions, we shall require to work with space-time metricsfor which the shift vector N i in the 3 + 1 decompositionis non-vanishing, otherwise the physical momentum (15)coincides with the conjugate momentum π i . For the sakeof continuity with Ref. [43], we start our analysis withthe G¨odel universe. Subsequently, we focus our atten-tion on the first weak-field solution for a rotating sourcein the context of General Relativity (GR), namely theLense-Thirring space-time. As the last two examples, weinvestigate space-times stemming from rotating compactobjects in the framework of extended theories of gravitywith the purpose of pinpointing the main differences withrespect to the standard GR scenario.
A. G¨odel universe
The G¨odel solution [48] is a homogeneous andanisotropic space-time arising from Einstein’s field equa-tions for a perfect fluid with non-vanishing angular mo-mentum. It essentially describes a rotating universe inwhich closed timelike curves are allowed, thus in princi-ple permitting time travel. The line element associatedwith such a curved background written in cylindrical co-ordinates ( t, r a ) = ( t, r, φ, z ) reads [48]d s = − d t − r a √ t d φ + d r (cid:0) r a (cid:1) + r (cid:18) − r a (cid:19) d φ + d z (62)with the constant parameter a > r < a. An observer orbiting circularly around the z -axis ( i. e. co-rotating with the G¨odel universe) will experience theflow of proper time according to the time coordinate t. The effective lapse, three-metric and shift from the pointof view of this observer read N = (cid:115) a r − G ab d r a d r b = 1 N (cid:34) d r r a + (cid:18) − r a (cid:19) r d φ + d z (cid:35) (64) N a ∂∂r a = − a √ (cid:0) a − r (cid:1) ∂ φ . (65)As r < a in this slicing and the prefactors of termscontaining higher powers of r / a (where r = r | p ) inthe resulting uncertainty relation get ever smaller, wewill only display the next-to-leading order for the sake ofbrevity. Correspondingly, the observer defined above measuresthe uncertainty relation σ p ρ (cid:38) π (cid:126) (cid:20) − ρ π a (cid:18) − ξ ρ λ C + 1724 r a (cid:19)(cid:21) . (66) B. Lense-Thirring solution
The phenomenon of frame-dragging was discoveredonly few years after the final settlement of GR. As amatter of fact, in 1918 Lense and Thirring found theweak-field limit for the space-time generated by a rotat-ing body [49]. The main prediction of this solution is theexistence of a precession of the orbits drawn by a testbody, a feature that is completely absent in Newtonianmechanics. To get to this conclusion, they argued thatin isotropic spherical coordinates ( t, r e ) = ( t, r, θ, ϕ ) themetric tensor originating from a rotating source takes theform [49]d s = − (1 + 2 φ GR )d t + (1 − φ GR ) (cid:0) d r + r dΩ (cid:1) + 4 φ GR a J sin θ d ϕ d t (67)where φ GR = − GM/r is the usual Newtonian potentialwhereas a J = J/M is the rotational parameter with J denoting the angular momentum of the source.Note that the time coordinate t used here correspondsto the time measured by a static observer at infinite dis-tance from the gravitating body in the center. In turn,the uncertainty is calculated as it would be measuredby this observer. In the non-relativistic limit, though,this provides a good approximation of the slicing carvedout by the dynamical rest frame of the particle itself.Therefore, corrections to the results are expected to beof higher order. Analogous considerations apply to theextended models of gravity which are treated as correc-tions to (67) below.The static observer at infinity experiences the effectivemetric, shift and lapse N (cid:39) − φ GR (68) N e ∂∂r e (cid:39) − a J φ GR r (69) G ef d r e d r f (cid:39) (1 + 3 φ GR ) (cid:0) d r + r dΩ (cid:1) . (70)Unfortunately, this reasoning leads to an uncertaintywhich is at least quadratic in the gravitational poten-tial while the Lense-Thirring solution corresponds to afirst-order expansion. In light of this, we generalize thediscussion by approximating the Kerr metric in Boyer-Lindquist coordinatesd s = − (cid:18) φ GR ˜ r Ξ (cid:19) d t + 4 ˜ φ GR a J sin θ ˜ r Ξ d t d ϕ + Ξ Σ d˜ r + Ξ d θ + ˜ r (cid:16) a J ˜ r − φ GR sin θ a J Ξ (cid:17) sin θ d ϕ (71)with ˜ φ GR = − GM/ ˜ r , Ξ = ˜ r (cid:112) θa J / ˜ r and Σ =˜ r (1 + ˜ φ GR + a J / ˜ r ) to fourth order in ˜ φ GR and a J / ˜ r simultaneously. Bear in mind that the Schwarzschild-like ˜ r relates to the radial coordinate introduced withthe Lense-Thirring metric as ˜ r = r (1 − φ GR / . As theresulting uncertainty relation is 3-diffeomorphism invari-ant, we will nevertheless provide it in terms of r for thesake of future convenience.Consequently, the static observer at infinity measuresthe uncertainty relation σ p ρ ≥ π (cid:126) (cid:40) φ GR ρ π r (cid:34)
10 + 30 φ GR + 55 φ GR − a J r × (cid:16) −
217 cos 2 θ − ξ ρ λ C (7 − θ ) (cid:17)(cid:35)(cid:41)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p (72) ≡ π (cid:126) (1 + λ LT ) (73)where the last line defines the (generalized) Lense-Thirring correction λ LT . We would like to remark thatthe dimensionless number ρ /λ C is unconstrained; forthis reason, in a suitable regime it can increase the effectof space-time rotation.A sample orbit in the equatorial plane of the Kerr met-ric approximated as indicated above is given in figure1, where the color of the curve changes with increas-ing proper time. The ensuing correction to the uncer-tainty relation is displayed in figure 2 as a function ofsaid proper time. The peaks correspond to the positionsof smallest distance from the outer horizon along the tra-jectory. Their height is amplified by the ad-hoc choice ofCompton wavelength of the particle λ C and the proxim-ity of the orbit to the horizon which can only be realizedsurrounding black holes. A more conservative estimateof a measurement done on a geostationary satellite at r | p ∼ km ( GM ∼ − km) using the largest quan-tum systems realized presently ( ρ ∼ − km) leads to acorrection of the order 10 − . C. Fourth-order gravity
Fourth-order gravity introduced by Stelle representsone of the first attempts to cure the quantization prob-lems of the gravitational interaction. In particular, it waspointed out [50] that the introduction of higher-derivativeterms in the Einstein-Hilbert action can make the modelrenormalizable. To be more precise, according to the pre-scription in [50], the gravitational action from which tobuild up quantum gravity should be given by S = 116 π G (cid:90) d x √− g (cid:18) R + α (cid:126) R − β (cid:126) R µν R µν (cid:19) , (74)where α and β are dimensionful constants measured inunits of inverse mass squared. However, the drawback FIG. 1. The trajectory followed by a massive particle inthe equatorial (˜ x -˜ y -) plane of the expanded Kerr metric asseen from above for a J /GM = 0 .
5. The starting pointlies on the x/GM axis at a distance 1000 from the sourcewith initial velocity (in coordinates ( t, r e )) u ( τ = 0) =(1 . , − . , . , . . The color ranging from violet tored indicates an increase in proper time τ whereas the blackdisk at the center symbolizes the outer horizon.FIG. 2. The corrections to the uncertainty relation experi-enced along the trajectory in figure 1 by a particle of Comptonwavelength λ C = 10 − GM as a function of proper time. of this model consists in the appearance of ghost-like de-grees of freedom which undermine the unitarity of theunderlying quantum field theory. Such a circumstance isa typical feature of local higher-order derivative gravity[51].For the current model, a Lense-Thirring-like solutionhas been recently obtained when analyzing the lightbending due to quadratic theories of gravity [52]. Inisotropic spherical coordinates, the aforementioned so-lution isd s = − (1 + 2 φ )d t + (1 − ψ ) (cid:0) d r + r dΩ (cid:1) + 2 ξ sin θ d ϕ d t , (75)where the gravitational potentials are given by φ = φ GR (cid:18) e − m r/ (cid:126) − e − m r/ (cid:126) (cid:19) (76) ψ = φ GR (cid:18) − e − m r/ (cid:126) − e − m r/ (cid:126) (cid:19) (77) ξ = 2 φ GR a J (cid:104) − (1 + m r/ (cid:126) ) e − m r/ (cid:126) (cid:105) (78)with m = 2 / √ α − β and m = (cid:112) /β being themasses of the spin-0 and spin-2 massive modes, respec-tively.As the influence stemming from the higher-derivativeterms ought to be small in comparison to the general rel-ativistic effect, results should be given as corrections tothe Lense-Thirring outcome. Written this way, the un-certainty relation reads for small gravitational potentials σ p ρ ≥ π (cid:126) (cid:34) λ LT − φ GR | p π (cid:18) ρ λ m e − m r (cid:126) + 8 ρ λ m e − m r (cid:126) (cid:19) (cid:35) (79)where λ m and λ m denote the Compton wavelengthscorresponding to the respective massive gravitationalmodes.As straightforwardly recognizable in the previous equa-tion, for the current example we do not have to resortto a higher-order expansion of the Kerr-like solution inthe context of the examined extended model of gravity,as the leading-order correction is already linear in φ GR . This feature is shared by the upcoming analysis as well.
D. Infinite-derivative gravity
Starting from the above scenario and recalling that thereasoning in [51] prevents any local higher-order deriva-tive gravity from being free from ghost fields, it is clearthat one must give up on locality to arrive at a quan-tum gravitational model which is simultaneously renor-malizable and unitary. Hovewer, non-locality should bemanifest only in the currently unexplored UV regime,since all the available data acquired from gravitationalexperiments comply with the local behavior of gravity.Along this direction, it is possible to encounter the so-called infinite-derivative gravity theory, which preciselypossesses the characteristics listed above. As the namesuggests, the usual Einstein-Hilbert action is now accom-panied by non-local functions of the curvature invariants;in the simplest form, the non-local gravitational action reads [53] S = 116 π G (cid:90) d x √− g (cid:32) R + R − e − (cid:126) (cid:3) /κ (cid:3) R − R µν − e − (cid:126) (cid:3) /κ (cid:3) R µν (cid:33) , (80)where κ is the energy scale at which the non-local aspectsof gravity are expected to be prominent.As for the previous example, a Lense-Thirring-like so-lution can be analytically computed in this framework.Formally, the shape of the metric tensor is the same asthe one exhibited in (75), with the difference that herethe gravitational potentials are instead represented by φ = ψ = φ GR Erf (cid:16) κ r (cid:126) (cid:17) (81) ξ = 2 φ GR a J (cid:20) Erf (cid:16) κ r (cid:126) (cid:17) − κ r √ π (cid:126) e − κ r / (cid:126) (cid:21) (82)where Erf( x ) denotes the error function.Again expressed as corrections to the Lense-Thirringresult, the uncertainty relation for small gravitational po-tentials becomes in this case σ p ρ ≥ π (cid:126) (cid:18) λ LT − φ GR | p π ρ r κ (cid:126) e − κ r / (cid:126) (cid:19) . (83) V. DISCUSSION
Along the lines of the EUP prescription and by re-sorting to the dynamics of a non-relativistic particle ina curved background, we have shown how to generallyderive a modification of the canonical uncertainty princi-ple surged by the underlying gravitational field in whichthe motion takes place. To this aim, we have basedour considerations on the notion of geodesic ball [40, 41]as a 3D-diffeomorphism invariant domain (whose radiusyields a measure of position uncertainty). Moreover, wehave properly defined a hermitian momentum operatorthat complies with the canonical commutation relationsin the non-relativistic limit of the 3+1 formalism. Fi-nally, perturbing around flat space-time and relying ona result for flat spaces obtained in an earlier work [43],we have evaluated the gravitationally induced correctionsto the uncertainty relation summarized by equation (61).To second order, it contains two new contributions, oneproportional to the Ricci scalar of the effective three-dimensional metric and one to the squared covariantderivative of the shift vector, both of which are evalu-ated at the center of the geodesic ball ( i. e. the expecta-tion value of the position operator).Furthermore, we have explicitly computed the form ofthe above uncertainty relation for the G¨odel universe, theLense-Thirring solution and its extension in the frame-work of fourth-order and infinite-derivative gravity. Re-markably, whilst the leading-order contribution goes like φ GR in the Lense-Thirring scenario, for the extendedmodels we observe a proportionality to φ GR . Therefore,there may be a regime in which the two terms are com-parably important, thus leading to a simultaneous “coex-istence” of the two quantities. A similar occurrence hasalso been addressed in different contexts, as for the caseof the Casimir effect [54].The derivation performed in this paper allows for adual interpretation of curved energy-momentum spacesreplacing geodesic balls in position space with their coun-terparts in momentum space and the momentum opera-tor with the position operator. Hence, it provides a directgeometrical link between the resulting Generalized Un-certainty Principle and curvature in energy-momentumspace as has been anticipated e. g. in the geometrical de-scription of Doubly Special Relativity as a theory of deSitter momentum space. Thus, it paves the way towardsmomentum-space curvature corrected quantum mechan-ics on purely geometrical grounds. Appendix A: Smallest momentum uncertainty statein flat space
In Refs. [42] and [43], starting from Sch¨urmann’s ideas[41], the authors wrongly stated that the expectationvalue of the conjugate momentum operator π i necessar-ily vanishes due to the Dirichlet boundary conditions.This is in fact the case for general real wave functionsΨ : IR → IR as can be readily verified by (cid:104) Ψ | ˆ π i Ψ (cid:105) = (cid:90) d µ Ψˆ π i Ψ = − (cid:90) d µ ˆ π i (Ψ)Ψ= − (cid:104) Ψ | ˆ π i Ψ (cid:105) = 0 (A1)where we used [ˆ π i , √ ˆ g ] = 0 and the boundary condi-tions (17). An equivalent relation can be derived for theexpectation value of the covariant conjugate momentumoperator (cid:104) ˆ π i (cid:105) . Yet, the colinearity of the real and imaginary part ofthe eigenvalue problem (17) does not necessarily implythat the corresponding wave functions are real neitherthat their momentum expectation values vanish.A general state | Ψ (cid:105) ∈ H can be written as | Ψ (cid:105) = (cid:88) n,l,m α nlm | ψ nlm (cid:105) (A2)with (cid:88) n,l,m | α nlm | = 1 (A3)which leads to the expectation values (cid:104) Ψ | ˆ π i Ψ (cid:105) = (cid:88) nlmn (cid:48) l (cid:48) m (cid:48) α nlm α ∗ n (cid:48) l (cid:48) m (cid:48) (cid:104) ψ nlm | ˆ π i ψ n (cid:48) l (cid:48) m (cid:48) (cid:105) (A4) (cid:104) Ψ | ˆ π Ψ (cid:105) = (cid:126) (cid:88) nlm | α nlm | λ nl (A5) where we used (19) and that ψ nlm are the eigenstates ofˆ π .According to the definition (35), the momentum un-certainty can be decreased by additional contributionsto the expectation value of the co- and contravariant mo-mentum operators. However, this is accompanied by anincrease in the contribution coming from the expectationvalue of the squared momentum operator as by (19) (as-suming a flat background), which more than compensatesthis.In this appendix, we will argue that the ground state ψ continues to yield the smallest conjugated momen-tum uncertainty σ π = (cid:112) (cid:104) ˆ π (cid:105) − (cid:104) ˆ π i (cid:105) (cid:104) ˆ π i (cid:105) in flat space,thus saving the results in [43] because the perturbation(naturally being regarded as small) cannot change thischoice. A similar argument may also be carried out forspaces of constant curvature as in [41].The conjugated momentum operators in flat space sim-ply read π i = − i (cid:126) ∂ i and π i = − i (cid:126) δ ij ∂ j . Obviously, in thiscase both operators are equivalent. As momentum oper-ators are vector operators, their expectation values arevectors. The choice of z-axis along which they are con-structed is arbitrary, which is why the imaginary partof eigenstates of the spherical Laplacian can be rotatedaway without exception while the expectation value ofthe momentum operator is changed by an orthogonaltransformation. Yet, according to (A5) the expectationvalue of the momentum operator vanishes for real wave-functions. This implies that the unrotated expectationvalue has to vanish as well. We conclude that (cid:104) ψ nlm | π i ψ nlm (cid:105) = 0 . (A6)Thus, the only non-vanishing contribution to the expec-tation value of the momentum operator lies in linearcombinations of eigenstates of the Laplacian with rel-ative phases. Due to the limitation of the coefficients(A3) and since transition amplitudes generally go like || (cid:104) ψ nlm | π i ψ n (cid:48) l (cid:48) m (cid:48) (cid:105) || ∼ δ l,l (cid:48) ± which makes it impossibleto obtain e. g. 3 contributions from linear combinationsof 3 states, such a combination of any number of basisstates in (A2) will show the same behaviour as a linearcombination of just two of them. Therefore, no furtherdecrease of the momentum uncertainty can be achievedby combining more than two states, which is why we willonly deal with this case. Up to a global phase, such astate generally readsΦ = √ aψ nlm + e iφ √ − aψ n (cid:48) l (cid:48) m (cid:48) (A7)with the real coefficient a ∈ (0 ,
1) and the relative phase φ ∈ [0 , π ) . For the above state, we obtain the momentumuncertainty0 σ π (Φ) = (cid:104) (cid:126) aj l,n + (1 − a ) j l (cid:48) ,n (cid:48) ρ − a (1 − a ) || Re (cid:0) e iφ (cid:104) ψ nlm | ˆ π i ψ n (cid:48) l (cid:48) m (cid:48) (cid:105) (cid:1) || (cid:105) (A8) ≥ (cid:104) (cid:126) aj l,n + (1 − a ) j l (cid:48) ,n (cid:48) ρ − a (1 − a ) || Abs (cid:104) ψ nlm | ˆ π i ψ n (cid:48) l (cid:48) m (cid:48) (cid:105) || (cid:105) . (A9)In order for the resulting state to be the one of smallest uncertainty, it has to satisfy σ π ( ψ ) > σ π (Φ) which canbe recast as || Abs (cid:104) ψ nlm | ˆ π i ψ n (cid:48) l (cid:48) m (cid:48) (cid:105) || ρ (cid:126) > C ( a ) (A10)where we introduced the function C ( a ) = (cid:115) j l (cid:48) ,n (cid:48) + a − a j n (cid:48) ,l (cid:48) − π − a a . (A11)As for the allowed n, l we have π ≤ j n,l ; this functiondiverges positively for a → a → ≤ a min ≤ C ( a ) ≥ C ( a min ) at which it reads C ( a min ) = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) ( j n,l − j n (cid:48) ,l (cid:48) ) (cid:113) ( j n,l − π )( j n (cid:48) ,l (cid:48) − π ) (cid:104) π − j n,l + (cid:113) ( j n,l − π )( j n (cid:48) ,l (cid:48) − π ) (cid:105) (cid:104) π − j n (cid:48) ,l (cid:48) + (cid:113) ( j n,l − π )( j n (cid:48) ,l (cid:48) − π ) (cid:105) . (A12)We now see that the transition amplitude of any linearcombination of two eigenstates of the Laplace-Beltramioperator whose uncertainty is smaller than the one of theground state has to satisfy the inequality || Abs (cid:104) ψ nlm | ˆ π i ψ n (cid:48) l (cid:48) m (cid:48) (cid:105) || ρ (cid:126) > C ( a min ) (A13)which is independent of the parameters a and φ. The right-hand and left-hand sides of this inequalityare displayed in figure 3 for all eigenstates of the Lapla-cian with n ≤ n (cid:48) ≤ n, l, n (cid:48) , l (cid:48) . There is no reason to expect this to change for highervalues of n, l, n (cid:48) , l (cid:48) . Therefore, we conclude that theground state ψ is indeed the state of lowest uncer-tainty in flat space. ACKNOWLEDGMENTS
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