Classical and Quantum Aspects of Particle Propagation in External Gravitational Fields
aa r X i v : . [ g r- q c ] A p r Classical and Quantum Aspects of Particle Propagation in External GravitationalFields
Giorgio Papini ∗ Department of Physics and Prairie Particle Physics Institute,University of Regina, Regina, Sask, S4S 0A2, Canada
In the study of covariant wave equations, linear gravity manifests itself through the metric devi-ation γ µν and a two-point vector potential K λ itself constructed from γ µν and its derivatives. Thesimultaneous presence of the two gravitational potentials is non contradictory. Particles also assumethe character of quasiparticles and K λ carries information about the matter with which it interacts.We consider the influence of K λ on the dispersion relations of the particles involved, the particles’motion, quantum tunneling through a horizon, radiation, energy-momentum dissipation and fluxquantization. INTRODUCTION
Fields for which quantum fluctuations are negligible relative to the expectation values of the fields themselves canbe described by classical equations. This is the external field approximation. In electrodynamics radiation processesare greatly enhanced when incident photons are replaced by external fields. Processes like photon-electron collisionswith pair production by a Coulomb field and bremsstrahlung owe their relevance in physics and astrophysics to theexternal field approximation.In gravitation where field strengths are much lower than in electromagnetism and means to increase cross sections arevery desirable, external field problems have been paid limited attention. This is presumably due to the large numberof components of the gravitational potential and to the difficulties of finding manageable modifications for the basicfields of interest as required by the characteristics of the gravitational sources, like rotation and time-dependence. Avariety of problems involving gravitational fields of weak to intermediate strength do not require the full-fledged useof general relativity and can be tackled using an external field approximation. Over the years solutions of covariantwave equations have been found [1–5] that are exact to first order in the metric deviation γ µν = g µν − η µν , where η µν is the Minkowski metric, and do not depend on the choice of any field equations for γ µν . They are a useful tool inthe study of the interaction of gravity with quantum systems and have been applied to interferometry and gyroscopy[2], the optics of particles [4], the observation of gravitational effects [2], the spin-flip of particles under the action ofinertial and gravitational fields [3], spin currents [6], neutrino physics [7, 8] and radiative processes in astrophysics[9–12].In all the solutions mentioned, gravity is contained in a phase factor whose main ingredient is the two-point [13, 14]potential K λ ( z, x ) that is a direct consequence of genuinely quantum equations and the expression, at the same time,of hidden symmetries of gravity. It is a novel feature of the approach. K λ describes gravity while remaining tied to ametric tensor. That is, K λ is known only if γ µν and its derivatives are known. K λ has other interesting aspects thatare studied below. We show, in particular, that the potential K λ affects a particle’s dispersion relations, its motion,the WKB problem, energy-momentum dissipation, radiation and flux quantization.An interesting development is that as gravity approaches the quantum domain, particles assume the character ofquasiparticles and the gravitational potential carries information about the matter with which it interacts.Spin terms that appear in the solutions of the covariant wave equations are not particularly relevant in what follows,thus we can simply consider the covariant Klein-Gordon (KG) equation. Similar conclusions can, of course, be reachedstarting from other known wave equations, whose solution is based, in any case, on the KG equation.Neglecting curvature dependent terms and applying the Lanczos-De Donder condition γ ναν, − γ σσ,α = 0 , (1)we can write the covariant KG equation to O ( γ µν ) in the form (cid:0) ∇ µ ∇ µ + m (cid:1) φ ( x ) ≃ (cid:2) η µν ∂ µ ∂ ν + m + γ µν ∂ µ ∂ ν (cid:3) φ ( x ) = 0 . (2)Units ¯ h = c = k B = 1 are used unless specified otherwise. The notations are as in [12]. In particular, partialderivatives with respect to a variable x µ are interchangeably indicated by ∂ µ , or by a comma followed by µ .The first order solution of (2) is φ ( x ) = (1 − i Φ G ( x )) φ ( x ) , (3)where φ ( x ) is a plane wave solution of the free KG equation (cid:0) ∂ µ ∂ µ + m (cid:1) φ ( x ) = 0 , (4)and Φ G ( x ) = − Z xP dz λ ( γ αλ,β ( z ) − γ βλ,α ( z )) ( x α − z α ) k β (5)+ 12 Z xP dz λ γ αλ ( z ) k α = Z xP dz λ K λ ( z, x ) , where P is an arbitrary point, henceforth dropped, and K λ ( z, x ) = −
12 [( γ αλ,β ( z ) − γ βλ,α ( z )) ( x α − z α ) − γ βλ ( z )] k β . (6)The momentum of the plane wave solution φ of (4) is k α and satisfies the equation k α k α = m . There are noadditional constraints on the solution of (2) represented by (3)-(6) except for the order of approximation in γ µν .Higher order solutions can be calculated following the procedure outlined in [12]. Nonetheless our calculations arelimited to O ( γ µν ) as we are primarily concerned with gravitational fields of weak to moderate intensity.It is easy to see that (3) is a solution of (2). By writing φ ≡ φ (1) for the first order solution and differentiating (3)with respect to x µ , we obtain φ (1) ,µ = φ ,µ − i Φ G,µ φ − i Φ G φ ,µ , (7)and again φ (1) ,µν = φ ,µν − i Φ g,µν φ − i Φ G,µ φ ,ν − i Φ G,ν φ ,µ − i Φ G φ ,µν . (8)The result then follows by substituting (8) in (2) and by noticing that η µν Φ G,µν = η µν k α Γ αµν = 0 by virtue of (1) and k µ Φ G,µ = γ µν k µ k ν . QUASIPARTICLES
Equations (3), (5) and (6) are the byproduct of covariance (minimal coupling) and, ultimately, of Lorentz invarianceand can therefore be applied to general relativity, in particular to theories in which acceleration has an upper limit [15–22] and that therefore allow the resolution of astrophysical [23–26] and cosmological singularities in quantum theoriesof gravity [27, 28]. They also are relevant to those theories of asymptotically safe gravity that can be expressed asEinstein gravity coupled to a scalar field [29].Path-dependent field variables in electromagnetism and gravitation have been used in the works of Volkov [30],Bergmann [31], DeWitt [32] and Mandelstam [33]. From (3) we can derive expressions for the covariant derivatives ofpath-dependent quantities. The vector field K λ plays a role similar to that of the vector potential in electromagnetism,but contains, at the same time, reference to matter through the momentum k µ of φ . It is, in fact, this associationthat suggests the introduction of the notion of quasiparticle that in other areas of physics describes fields and particleswhose properties are affected by the presence of other particles and media with which they interact. K λ ( z, x ) also satisfies Maxwell-type equations identically. By differentiating (6) with respect to z α , we find [34]˜ F µλ ( z, x ) = K λ,µ ( z, x ) − K µ,λ ( z, x ) = R µλαβ ( z ) J αβ , (9)where R αβλµ ( z ) = − ( γ αλ,βµ + γ βµ,αλ − γ αµ,βλ − γ βλ,αµ ) is the linearized Riemann tensor satisfying the identity R µνστ + R νσµτ + R σµντ = 0 and J αβ = (cid:2) ( x α − z α ) k β − k α (cid:0) x β − z β (cid:1)(cid:3) is the angular momentum about the basepoint x α . Maxwell-type equations ˜ F µλ,σ + ˜ F λσ,µ + ˜ F σµ,λ = 0 (10)and ˜ F µλ,λ ≡ j µ = (cid:16) R µλαβ J αβ (cid:17) , λ = R µλαβ,λ ( x α − z α ) k β + R µβ k β , (11)can be obtained from (9) using the Bianchi identities R µνστ,ρ + R µντρ,σ + R µνρσ,τ = 0. The current j µ satisfies theconservation law j µ,µ = 0. Equations (10) and (11) are identities and do not represent additional constraints on γ µν .One finds, in particular, that the ”electric” and ”magnetic” components of ˜ F µν are˜ E i = R iαβ J αβ , ˜ H i = ǫ ijk R kjαβ J αβ , (12)where ǫ ijk is the Levi-Civita symbol.In this work we investigate some of the consequences that follow from (10) and (11) and from the close associationof K λ with matter.The recombination of ten γ µν into four K λ is a remarkable phenomenon, even though knowledge of all γ µν is stillneeded, in general, to calculate K λ [34]. It follows from (6) that the gravitational field is described by K λ along aparticle world line and that K λ vanishes wherever k µ does. From (11) we also find that j µ = − ( ∂ γ µα,β − ∂ γ µβ,α )( x α − z α ) k β + ∂ γ µβ k β . The current j µ vanishes when ∂ γ µβ = 0 that corresponds to pure gravitational fields because, inthe linear approximation, R µβ = ∂ γ µβ = 0. When j µ = 0, equations (10) and (11) become scale invariant and sodoes the field K λ . In fact, by differentiating (6), we obtain ∂ K λ = − k β (cid:2)(cid:0) ∂ ( γ αλ,β ) − ∂ ( γ βλ,α ) (cid:1) ( x α − z α ) + ∂ γ βλ − γ ,λβ (cid:3) . (13)The last term in (13) can be eliminated by a gauge transformation. We then obtain ∂ K λ = 0, irrespective of thevalue of k α . DISPERSION RELATIONS AND PARTICLE MOTION
By using Schroedinger’s logarithmic transformation [35] φ = e − iS we can pass from the KG equation (2) to thequantum Hamilton-Jacobi equation. We find to first order in γ µν i ( η µν − γ µν ) ∂ µ ∂ ν S − ( η µν − γ µν ) ∂ µ S∂ ν S + m = 0 , (14)where S = k β (cid:26) x β + 12 Z x dz λ γ βλ ( z ) − Z x dz λ ( γ αλ,β ( z ) − γ βλ,α ( z )) ( x α − z α ) (cid:27) (15) ≡ k α x α + A + B .
It is well-known that the Hamilton-Jacobi equation is equivalent to Fresnel’s wave equation in the limit of largefrequencies [35]. However, at smaller, or moderate frequencies the complete equation (14) should be used. We followthis path. By substituting (3) into the first term of (14), we obtain i ( η µν − γ µν ) ∂ µ ∂ ν S = iη µν ∂ µ ( k ν + Φ G,ν ) − iγ µν ∂ µ k ν = iη µν Φ G,µν = ik α η µν Γ αµν = 0 , (16)on account of (1). This part of (14) is usually neglected in the limit ¯ h →
0. Here it vanishes as a consequence ofsolution (3). The remaining terms of (14) yield the classical Hamilton-Jacobi equation( η µν − γ µν ) ∂ µ S∂ ν S − m = γ µν k µ k ν − k µ Φ G,µ = 0 , (17)because k µ Φ G,µ = 1 / γ µν k µ k ν . Equation (3) is therefore a solution of the more general quantum equation (14). Italso follows that the particle acquires a generalized ”momentum” P µ = k µ + Φ G,µ = k µ + 12 γ αµ k α − Z x dz λ ( γ µλ,β ( z ) − γ βλ,µ ( z )) k β , (18)that satisfies the dispersion relation P µ P µ ≡ m e = m (cid:18) γ αµ ( x ) u α u µ − Z x dz λ ( γ µλ,β ( z ) − γ βλ,µ ( z )) u µ u β (cid:19) . (19)The integral in (19) vanishes because u µ u β is contracted on the antisymmetric tensor in round brackets. The effectivemass m e is not in general constant. In this connection too we can speak of quasiparticles. The medium in which thescalar particles propagate is here represented by space-time.We have already used some of the properties of (18) and (19) elsewhere [5, 10]. P µ of (18) describes the geometricaloptics of particles correctly and gives the correct deflection predicted by general relativity. On using the relationsΦ G,µ = K µ ( x, x ) + Z x dz λ ∂ µ K λ ( z, x ) , (20)and Φ G,µν = K µ,ν ( x, x ) + ∂ ν Z x dz λ ∂ µ K λ ( z, x ) = k α Γ αµν , (21)and by differentiating (18) we obtain the covariant derivative of P µ DP µ Ds = m (cid:20) du µ ds + 12 ( γ αµ,ν − γ µν,α + γ αν,µ ) u α u ν (cid:21) (22)= m (cid:18) du µ ds + Γ α,µν u α u ν (cid:19) = Dk µ Ds .
This result is independent of any choice of field equations for γ µν . We see from (22) that, if k µ follows a geodesic,then DP µ Ds = 0 and Dm e Ds = 0. The classical equations of motion are therefore contained in (22), but it would requirethe particle described by (2) to just choose a geodesic, among all the paths allowed to a quantum particle.We also obtain, from (14), p ( ∂ i S ) = ± p − m + ( ∂ S ) − γ µν ∂ µ S∂ ν S which, in the absence of gravity, gives k i = p − m + k , as expected. Remarkably, (18) is an exact integral of (22) which can itself be integrated exactly togive the particle’s motion X µ = x µ + 12 Z x dz λ γ µλ { ( γ αλ,µ − γ µλ,α ) ( x α − z α ) } . (23)Higher order approximations to the solution of (2) can be obtained by writing φ ( x ) = Σ n φ ( n ) ( x ) = Σ n e − i ˆΦ G φ ( n − , (24)where the operator ˆΦ G is ˆΦ G ( x ) = − Z xP dz λ ( γ αλ,β ( z ) − γ βλ,α ( z )) ( x α − z α ) ˆ k β (25)+ 12 Z xP dz λ γ αλ ˆ k α , and ˆ k α = i∂ α .The solution (24) plays a dynamical role akin to Feynman’s path integral formula [36]. In (24), however, it is thesolution itself that is varied by successive approximations, rather than the particle’s path. THE GRAVITATIONAL WKB PROBLEM
We now study the propagation of a scalar field in a gravitational background. We know, from standard quantummechanics [37], that S develops an imaginary part when the particle tunnels through a potential. This imaginarycontributions is interpreted as the transition amplitude across the classically forbidden region, which is therefore givenby [38] T = exp [ − Im ( S )] = exp n − Im h ln (cid:16) Σ n exp (cid:16) − i ˆΦ G φ n − (cid:17)(cid:17)io . (26)To O ( γ µν ), (26) becomes T = exp (cid:26) − Im (cid:20) x β + 12 I dz λ γ βλ ( z ) − I dz λ ( γ αλ,β ( z ) − γ βλ,α ( z ))( x α − z α ) (cid:21) k β (cid:27) , (27)for a space-time path traversing the gravitational background from −∞ to + ∞ and back as it must in order to make(27) invariant. Assuming a Boltzmann distribution for the particles T = e − k /T , where T is the temperature and theBoltzmann constant k B = 1, we find, in general coordinates, T = k /Im (cid:26) k β (cid:20) x β + 12 I dz λ γ βλ ( z ) − I dz λ ( γ αλ,β ( z ) − γ βλ,α )( x α − z α ) (cid:21)(cid:27) . (28)The intended application here is to the propagation problem in Rindler space given by ds = (1 + ax ) ( dx ) − ( dx ) , (29)with a horizon at x = − /a , where a = a α a α is the constant proper acceleration measured in the rest frame ofthe Rindler observer. We note that, a priori, our approach is ill-suited to treat this problem that frequently in theliterature is tackled starting from exact, or highly symmetric solutions of the KG equation [39]. In fact the externalfield approximation | γ µν | < | η µν | may become inadequate close to the horizon, from where the imaginary part of T comes, for some systems of coordinates. This requires attention, as discussed below. Nonetheless the external fieldapproximation has interesting features like the presence of k α in (28) and manifest covariance and invariance undercanonical transformations.It is convenient, for our purposes, to use the Schwarzschild-like form for (29) using the transformation [40] x = 1 a √ ax ′ sinh( at ′ ) , x = 1 a √ ax ′ cosh( at ′ ) (30)for x ≥ − / a and the same transformation with the hyperbolic functions interchanged for x ≤ − / a . Theresulting metric is ds = (1 + 2 ax ′ ) ( dt ′ ) −
11 + 2 ax ′ ( dx ′ ) , (31)for which the horizon is at x ′ = − / a . From (31) we find γ = 2 ax ′ , γ = 2 ax ′ / (1 + 2 ax ′ ). If γ and γ representcorrections to the Minkowski metric, we must have | γ /g | < , | γ /g | < a >
0. The external fieldapproximation therefore remains valid for − / a < x ′ < / a . This is sufficient to justify our calculation. We nowwrite the terms A and B , defined in (15), for the metric (31) explicitly. We find A = k Z dz γ + k Z dz γ , (32)and, by taking the reference point x µ = 0, B = − k Z dz γ , z + k Z dz γ , z . (33)The explicit expressions for A and B confirm the fact that T receives contributions from both time and space partsof S as pointed out in [40]. This is, on the other hand, expected of a fully covariant approach.The first integrals in A and B cancel each other. The second integral in A can be calculated by contour integrationby writing z = − / a + ǫe iθ . The result Im R ∞−∞ dz z k / (1 + 2 az ) = − k π/ a yields a vanishing contributionbecause k reverses its sign on the return trip. The last integral in B is real. The term k ∆ t ′ in (15) contributes theamount k ( − iπ/ a )2 because for a round trip the horizon is crossed twice and each time at ′ → at ′ − iπ/ k ∆ x ′ = k x ′ − ( − k )( − x ′ ) = 0. The final result is therefore T = a π , (34)which is independent of k and coincides with the usual Unruh temperature [41, 42]. This result, with the replacement a → a/ p − a / A , where A = 2 m is the maximal acceleration, also confirms a recent calculation [43] regardingparticles whose acceleration has an upper limit. Equation (34) comes in fact from the term k ∆ t ′ that does notcontain derivatives of γ µν . The difference from [43], as well as from [40], is however represented by the form of (27)of the decay rate [38] which carries a factor 2 in the exponential as required by our invariant approach.Despite its limitations, the external field approximation already reproduces (34) at O ( γ µν ). Additional terms of(26) are expected to contain corrections to (34). We note, however, that for a closed space-time path the last integralin (27) and (28) becomes R Σ dσ µλ R µλαβ J αβ , where Σ is the surface bounded by the path [3], and has an imaginarypart if R µναβ has singularities. This eventuality may call for a complete quantum theory of gravity [44]. K λ IN INTERACTION i. Poynting vector.
The question we ask in this section is whether the vector K λ is redundant, or plays a role inradiation problems. Using ˜ F µν , we can construct, for instance, a ”Poynting” vector. Assuming, for simplicity, that j µ = 0 in (11), using known vector identities, integrating over a finite volume and reverting to normal units, we obtainfrom (10) and (11) the conservation equation1 c ∂∂t Z (cid:16) ˜ E + ˜ H (cid:17) dV = − I ~ ˜ S · d~ Σ , (35)where Σ is the surface bounding V and ~ ˜ S = ~ ˜ E × ~ ˜ H is the gravitational Poynting vector. Both sides of (35) acquire, infact, the dimensions of an energy flux after multiplication by G/c . We can now calculate the flux of ~ ¯ S at the particleassuming that the momentum of the free particle is k ≡ k and that the source in V emits a plane gravitationalwave in the x -direction. In this case the wave is determined by the components γ = − γ and γ , and we find˜ E = 0, ˜ E = 2 R J + 2 R J , ˜ E = 2 R J + 2 R J , ˜ H = 0, ˜ H = − R J − R J ,˜ H = 4 R J + 4 R J . It also follows that R = R = R − R = − ¨ γ / R = R = R = R = ¨ γ /
2. The action of ˜ S on the quantum particle is directed along the axis of propagation of thewave and results in a combination of oscillations and rotations about the point x α with angular momentum given by2 J = ( x − z ) k − k ( x − z ), 2 J = ( x − z ) k and 2 J = ( x − z ) k . A similar motion also occurs in the caseof Zitterbewegung [45]. Reverting to normal units, the energy flux associated with this process isΦ = ( ω G/c ) (cid:8) ( γ ) [( J ) + J J ] + ( γ ) [( J ) − J J − ( J ) ] (cid:9) (36)and increases rapidly with the wave frequency ω and the particle’s angular momentum. ii. Electromagnetic radiation. Let us assume that a spinless particle has a charge q . Acceleration, whatever itscause, makes the particle radiate electromagnetic waves. The four-momentum radiated away by the particle, whilepassing through the driving gravitational field ˜ F µν , is given by the formula∆ p α = − q c Z du β ds du β ds dx α = − q c Z (cid:16) ˜ F µν u ν (cid:17) (cid:16) ˜ F µδ u δ (cid:17) dx α , (37)that can be easily expressed in terms of the external fields (12) on account of the equation of motion of the charge inthe accelerating field [34]. At this level of approximation the particle can distinguish uniform acceleration which gives∆ p α ∼ R g dx α , where g is a constant, from a non-local gravitational field and radiates accordingly. This is explainedby the presence of R µναβ in (12) and is a direct consequence of our use of the equation of geodesic deviation in (37).When the accelerating field is the wave discussed above, the incoming gravitational wave and the emitted electro-magnetic wave have the same frequency ω and the efficiency of the gravity induced production of photons increasesas ω k . iii. Flux quantization. Flux quantization is the typical manifestation of processes in which the wave function isnon-integrable. Of interest is here the presence of the free particle momentum k α in K λ .Let us consider for simplicity the case of a rotating superfluid. Then γ = − Ω z /c , γ = Ω z /c and the remainingmetric components vanish. The angular velocity Ω is assumed to be constant in time and k = 0. Without loss ofgenerality, we can also choose the reference point x µ = 0. We find K = K = 0 and K = − (cid:2) γ , z − γ (cid:3) k − (cid:2) − γ , z (cid:3) k (38) K = − (cid:2) γ , z − γ (cid:3) k − (cid:2) − γ , z (cid:3) k . Integrating over a loop of superfluid, the condition that the superfluid wave function be single-valued gives thequantization condition I dz λ K λ = − Ω z c I (cid:0) k dz − k dz (cid:1) = π Ω z c k̺ = 2 πn , (39)where n is an integer, k = p k + k and ̺ = p z + z . The time integrating factor z , extended to N loops, becomes z = 2 π̺εN/pc , where ε = ( pc ) + ( mc and p = ¯ hk . The superfluid quantum of circulation satisfies the conditionΩ( π̺ ) εN/c = n ¯ h . (40)If the superfluid is charged, then the wave function is single-valued if the total phase satisfies the relation I dz λ K λ + qc I dz λ A λ = 2 πn ¯ h , (41)which, for n = 0 and zero external magnetic field, leads to R ~H · d~ Σ = − π Ω ̺ εN/qc . In this case, therefore, rotationgenerates a magnetic flux through Σ and, obviously, a current in the N superconducting loops. No fundamentaldifference is noticed in this case from DeWitt’s original treatment of the problem [46–49]. CONCLUSIONS
The two-point potential K µ ( z, x ) plays a prominent role in the solution of covariant wave equations through thephase Φ G . It satisfies Maxwell-type equations identically, depends on the metric tensor and is complementary toit. The potential K λ suggests the introduction of the notion of quasiparticle because gravity affects in general thedispersion relations of the particles with which it interacts, as shown by (19), and because it carries with itselfinformation about matter through the particle momentum k α .Some particular aspects of the behaviour of K λ have been examined. We have found that when j µ = 0, scaleinvariance assures that a gas of gravitons satisfies Planck’s radiation law, but that this is no longer so, in principle,for non-pure gravitational fields. K λ also determines the equations of motion of a particle through (20), (21), (22) and (24). We have found that themotion follows a geodesic only if the quantum particle chooses, among all available paths, that for which Dk α /Ds = 0.Along this particular path the principle of equivalence is obviously satisfied. We have then shown that the particlemotion is contained in the solution (3) of the covariant KG equation.We have also studied quantum mechanical tunneling through a horizon and derived a covariant and canonicalinvariant expression for the transition amplitude. Though the external field approximation looks ill-suited to dealwith regions of space-time close to a gravitational horizon, the approximation reproduces the Unruh temperatureexactly in the case of the Rindler metric. No corrections and no effects due to k µ have been found to the standardresult to O ( γ µν ). Higher order approximations can be calculated by applying (28).Because ˜ F µν satisfies Maxwell-like equations, it is also possible to define a Poynting vector and a flux of energy andangular momentum at the particle so that the particle’s motion can be understood as a sequence of oscillations androtations similar to what found in the case of Zitterbewegung [45].Use of K λ in problems where gravity accelerates a charged particle and electromagnetic radiation is producedoffers a rather immediate relationship between the loss of energy-momentum by the quantum particle and the drivinggravitational field. These processes could give sizeable contributions for extremely high values of ω . Astrophysicalprocesses like photoproduction [50] and synchrotron radiation [51] have been discussed in the literature and areworthy of re-consideration in view of the present results. An advantage on the high frequency detection side, forwhich detection schemes are in general difficult to conceive, is represented by the efficiency of the graviton-photonconversion rate and by the high coupling afforded by a radio receiver over, for instance, a mechanical one. This wouldenable, in principle, a spectroscopic analysis of the signal.In the last problem considered, we have calculated the flux of K λ in the typical quantum case of a non-integrablewave function. Here too, it is possible to isolate quantities of physical interest, like magnetic flux, or circulation,despite the non-intuitive character of H dz λ K λ . Unlike [32], our procedure and results are fully relativistic. They canbe applied directly to boson condensates in boson stars [52]. ∗ Electronic address:[email protected][1] Giorgio Papini,
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