Disformal transformations and the motion of a particle in semi-classical gravity
aa r X i v : . [ g r- q c ] J a n Disformal transformations and the motion of a particle in semi-classical gravity
Sandip Chowdhury ∗ , Kunal Pal † , Kuntal Pal ‡ , and Tapobrata Sarkar § Department of Physics,Indian Institute of Technology,Kanpur 208016, India
Abstract
The approach to incorporate quantum effects in gravity by replacing free particle geodesics with Bohmian non-geodesic trajectories has an equivalent description in terms of a conformally related geometry, where the motion isforce free, with the quantum effects inside the conformal factor, i.e., in the geometry itself. For more general disformaltransformations relating gravitational and physical geometries, we show how to establish this equivalence by takingthe quantum effects inside the disformal degrees of freedom. We also show how one can solve the usual problemsassociated with the conformal version, namely the wrong continuity equation, indefiniteness of the quantum mass,and wrong description of massless particles in the singularity resolution argument, by using appropriate disformaltransformations.
More than twenty five years ago, Bekenstein [1] showed that in theories of gravity where two distinct geometries arepresent, they are, in general, related by a disformal transformation, which is a generalization of the conformal transfor-mation. In such situations, the gravitational dynamics is controlled by the metric and is called gravitational geometry,whereas matter dynamics takes place on a geometry that is disformally related to the metric, and called physical ge-ometry. This is a notable departure from general relativity (GR), where the dynamics of both gravity and matter aredetermined by the metric, but is common in scalar-tensor theories of gravity, such as Brans-Dicke theory, in whichthe two geometries are related by a conformal transformation. The purpose of this paper is to establish the nature ofdisformal transformations in the context of Bohmian mechanics in gravitational backgrounds, and to show that it solvesa few important problems that arise in an usual treatment popular in the literature, that uses conformal transformationsinstead.Bohmian mechanics [2–4] is an important tool in a semi-classical understanding of the full theory of quantumgravity [5]. Formulating such a quantum theory of gravity is of course a formidable challenge, with the metric playingthe role of a quantum operator, and with quantization conditions on space and time. The somewhat simpler semi-classical approach, in which the metric is classical, has been popular over decades. In Bohmian mechanics, the particletrajectories are determined by suitable wavefunctions, and the statistical distribution of particle positions is given bythe modulus squared of this wavefunction. In this first quantized approach, one replaces the geodesic motion of freelyfalling particles in a curved space-time by corresponding Bohmian trajectories. In such situations, geodesic equationsare typically modified by an additional force term coming from a quantum potential [6–9]. This kind of reasoning hasbeen used to deal with the usual singularity problem of classical general relativity [6]. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] G = c = ~ = 1. 2 Bohmian motion on a classical background
We consider a quantum particle moving on a timelike path in a fixed classical background. The normalized wave functionof such a particle Ψ( x µ ) can be written in terms of two single valued real functions R ( x µ ) and S ( x µ ), which are themodulus and the phase function respectively, as Ψ( x µ ) = R ( x µ ) e i S ( x µ ) [3, 4]. Substituting this form of wave functionin the Schrodinger equation, and separating the real and imaginary parts, one can show that, with the four-momentumassociated with a particle of mass m guided by this wave function given by p µ = ∂ µ S ( x µ ), these two equations can beinterpreted as a quantum Hamilton-Jacobi equation, and a continuity equation with probability density ρ = R [2, 3],respectively. But the Schrodinger equation is not a manifestly covariant equation. Thus to write down the relativisticversion of the quantum Hamilton - Jacobi equation and the continuity equation, the usual approach is to instead considera Klein-Gordon type equation of a particle (whose dynamics is governed by the Bohmian mechanics) described by thewavefunction Φ( x µ ) which is assumed to to be moving on a fixed classical curved background ( g µν ) .For convenience we start with the following action, written in standard form, with a non-minimal coupling term ofΦ as (see for example [16], [17]): S fR + S Φ + S m = Z d x √− gF (Φ) R + Z d x √− g h − g µν ( ∇ µ Φ)( ∇ ν Φ) − m Φ i + Z d x √− g L m ( g µν , λ i ) . (1)Here the subscripts in the terms on the left hand side denote the actions due to the non-minimal coupling, the scalarfield and the matter fields (collectively denoted as λ i ) respectively. The coupling between the Ricci scalar R , and thescalar field Φ is assumed to be independent of whether it is in the quantum regime or can be approximately taken to beclassical, i.e., independent of the energy scale of the system.From now on, we shall concentrate on a particular choice of the function F (Φ), namely F (Φ) = − ǫ Φ . In that case,the gravitational dynamics is governed by the field equation [16, 18] G µν = 1Φ (cid:16) ∇ µ ∇ ν Φ − g µν (cid:3) Φ (cid:17) − ǫ Φ (cid:16) T mµν + T Φ µν (cid:17) , with T iµν = − √− g δS i δg µν , (2)where G µν is the Einstein tensor constructed from the metric g µν , and T mµν and T Φ µν are the energy momentum tensorsassociated with matter fields and the scalar field respectively. The scalar field dynamics is governed the followingKlein-Gordon like equation (cid:2) (cid:3) − m − ǫR (cid:3) Φ = 0 . (3)In four spacetime dimensions, this equation is conformally invariant only when the mass term is set to zero and ǫ = [19].It is also important to notice that in [6], it was assumed that the background metric is non-dynamical in nature as a firstapproximation, i.e., backreaction was neglected. But in the general case with a non zero coupling between curvatureand the scalar field Φ, this is not possible, unless of course we choose, the scalar field to be not dynamical (this factshould be clear from the field eq.(2) above).Substituting the polar form of the wavefunction Φ = R ( x µ ) e i S ( x µ ) in the Klein-Gordon equation, and separating realand imaginary parts, we get the following equations (cid:0) ∇ µ S (cid:1)(cid:0) ∇ µ S (cid:1) = − m − ǫR + (cid:3) RR and ∇ µ J µ = 0 with J µ = R (cid:0) ∇ µ S (cid:1) . (4)The last term in the first relation above is the quantum potential, which we shall denote as f ( R ), defined by (with ~ restored) f ( R ) = ~ (cid:3) RR . After one defines the appropriate curved space generalization of the four-momentum of the Here the field Φ( x µ ) (a scalar function) is the first quantized wavefunction of the particle. In this paper no second quantization is imposedupon Φ( x µ ). Nevertheless, we shall use both the words wavefunction and field to denote Φ( x µ ). p µ = ∇ µ S , this equation gives the constraint on the magnitude of four-momentum p µ p µ = − m − ǫR + (cid:3) RR . (5)On the other hand the second relation above represent a conservation equation for the current J µ .From the above eq.(5), we glean that the norm of the four-momentum is not constant, and given this magnitudeof the four-momentum it will be important how one defines a four-velocity vector from it. We do this by defining thenormalized four-velocity for a timelike trajectory as u µ u µ = −
1, so that [5] p µ = M ( x ) u µ , M ( x ) = q(cid:0) m − f ( R ) + ǫR (cid:1) . (6)With this definition, the four-velocity remains the same as in the classical trajectory but the particle’s mass becomes avariable depending on the quantum amplitude R , and this is known as the quantum mass of the particle. It is knownthat under an appropriate conformal transformation the action of a particle of variable mass becomes the action of aparticle of constant mass (and vice versa). Using this fact in the next section we shall work in a conformally relatedframe, with conformal factor being appropriate function of quantum potential such that the dispersion relation of eq.(5)transforms to the dispersion relation of a particle of constant mass.On the other hand the second equation of eq.(4) implies, with this interpretation of variable quantum mass, theconservation equation of the 4-current to be ∇ µ (cid:0) R M u µ (cid:1) = 0. Unfortunately however, now it is not possible tointerpret this equation (as is done in the corresponding non-relativistic Bohmian treatment of Schrodinger equation)as a continuity equation with the probability density defined as ρ = R . Because the corresponding equation shouldlook like ∇ µ (cid:0) ρ u µ (cid:1) = 0 [14]. As mentioned above, one of the motivation to transform to conformal frame is to makethe particle mass constant and hence the equation of motion a geodesics, but it is not possible (as we shall explainin subsection 2.5 below), with the same transformation to make the conservation equation a continuity equation forprobability density. Recently in [13] the authors have suggested a solution of this problem. In this work we shall proposean alternative one - use a disformal transformation rather than the conformal one.As is clear from eq.(5), due to presence of the quantum potential (the f ( R ) term), the equation of motion of theparticle is not a geodesic. Rather it contains an extra term coming form the force generated by the quantum potential.The straightforward way to find this force term in the acceleration equation is to take the directional derivative of theconstraint relation of eq.(5) by introducing a parameter (say τ ) along the particle trajectory (see [9] for a derivationalong these lines). For later purposes however, we shall use a variational principle and write down the action along theparticle trajectory between two points (say 1 and 2) as S [ x µ ( τ ) , η ( τ )] = Z dτ L , L = 12 (cid:20) η − g µν u µ u ν − η (cid:16) m − f ( R ) + ǫR (cid:17)(cid:21) , (7)where η is a Lagrange multiplier, and u µ is the four-velocity of the particle, normalized as u µ u µ = − Variation ofthe action with respect to x µ gives the usual Euler-Lagrange equations, and a variation with respect to the Lagrangemultiplier gives η = − u µ u µ M , with M being the quantum mass defined above. Now substituting this into the Lagrangianof eq.(7), we have L = η − (cid:2) u µ u µ (cid:3) = − M p − g µν u µ u ν , (8)so that the corresponding four-momentum p µ = ∂ L ∂ ˙ x µ = M √− u µ u µ u µ satisfies the required constraint relation p µ p µ = − M (irrespective of the normalization of the 4-velocity). The acceleration equation corresponding to this Lagrangian is This four-velocity normalization is different from [6] but same as [9]. u µ ∇ µ u ν = − (cid:0) g µν + u µ u ν (cid:1) ∇ µ ln( M ) = − (cid:0) g µν + u µ u ν (cid:1) ∇ µ ln h − (cid:16) f ( R ) − ǫR (cid:17) /m i . (9)As anticipated, the right hand side of eq (9) is non zero, and this indicates that the motion of a particle which isfreely falling in the classical description is no longer force free for motion along a Bohmian trajectory. This termcomes from the quantum potential. So unless R = constant , or it satisfies the equation (cid:3) R = 0, this term is non-zero. This matches with the acceleration equation derived in [9] from the momentum constrain relation (5), but it isdifferent from the equation written in [6]. This is due to the different parameterization employed, namely, the termproportional to u µ u ν can be absorbed by changing the parameter, although the first term can never be removed by anysuch redefinition of the parameter. Also note that the force term (from eq.(9)) is perpendicular to the velocity vector,i.e., u ν u µ ∇ µ u ν = u µ h µν = 0, where we have defined the transverse metric h µν = g µν + u µ u ν .Now consider a congruence of timelike particle trajectories with u µ being the tangent to the trajectories. In astandard fashion [22], we consider B µν = ∇ ν u µ and calculate it’s change along the particle trajectories as u µ ∇ µ B αβ = ∇ β ( u µ ∇ µ u α ) − B αµ B µβ − R αµβν u µ u ν . (10)Taking the trace of this equation, and using the acceleration equation derived above, we arrive at the modified Ray-chaudhuri equation given by dθdτ = h µν (cid:16) ∇ µ ∇ ν L + ∇ µ L ∇ ν L (cid:17) + θ dLdτ − θ − σ αβ σ αβ + ω αβ ω αβ − R αβ u α u β , (11)where a α = u µ ∇ µ u α is the acceleration vector, θ = B µµ = ∇ µ u µ is the expansion scalar, σ µν = B ( µν ) − θh µν is the sheartensor and ω µν = B [ µν ] is the rotation tensor, and we have defined the function L = −
12 ln h − (cid:16) f ( R ) − ǫR (cid:17) /m i . (12)The terms involving L in the quantum Raychaudhuri equation are the contributions of the quantum effects, and therelative contribution of these terms with respect to other classical terms determine whether the trajectory will reach aconjugate point or not [6]. Note that this form of Raychaudhuri equation is different from one given in [6], due to thedifferent parameterization of the trajectory. It is also different from the one derived in [9], because those authors took ∇ α h µν = 0. We also record the expression of the deviation equation for the vector field ξ µ between two non-geodesiccurves as d ξ µ dτ = ξ α (cid:16) u µ u ν ∇ α ∇ ν L + u ( µ ∇ | α | u ν ) ∇ ν L (cid:17) + R µαρν ξ α u ρ u ν . (13) The purpose of this subsection is to discuss the transformation of the acceleration equation derived above, under aconformal transformation. If two metrics are related by a conformal transformation of the form ˜ g µν = Ω ( x ) g µν , then itis well known that the relation between the acceleration equations in two frames are given by (see [18])˜ u µ ˜ ∇ µ ˜ u ν = u µ ∇ µ u ν + h µν ∇ µ (ln Ω) . (14)From this relation, we see that if a particle moves along the geodesic of the transformed frame, then the motion in theframe g µν is non geodesic.In our case, we choose to work in a frame that is related to the original metric by a conformal factor, such that the5elation between the two metrics are ˜ g µν = ( M /m ) g µν . (15)We can conventionally express the above relations in terms of the action of a particle corresponding to the Lagrangianof eq.(8), along the path γ given by x µ = x µ ( τ ), S m (cid:2) g µν , γ, M (cid:3) = − Z γ m r − f ( R ) − ǫRm p − g µν u µ u ν dτ, u µ = dx µ dτ . (16)In the conformally transformed frame the corresponding action for a path x µ = x µ (˜ τ ) is transformed into S m (cid:2) ˜ g µν , γ (cid:3) = − Z γ p − ˜ g µν ˜ u µ ˜ u ν d ˜ τ , u µ = dx µ d ˜ τ . (17)As can be readily seen by comparing them, the first form represents a particle having a variable mass M defined in eq.(6),and the second one is the action for a unit mass particle. If we write down the acceleration equations corresponding tothese actions, they turn out just to be same equations derived before i.e., u µ ∇ µ u ν + h µν ∇ µ (ln Ω) and ˜ u µ ˜ ∇ µ ˜ u ν = 0, andthus satisfy eq.(14). The implication of this relation is straightforward. Namely, the motion in the conformal frame isfree from the force of the quantum potential. If we denote the momentum corresponding to the conformal frame as ˜ p µ ,then this can be shown to satisfy ˜ p µ ˜ p µ = − g µν via the conformal factor. In other words the quantum effectsare in the field equations, so that the metric and also the energy momentum tensors are different. Indeed, starting fromthe action of eq.(1), making the transformation of eq.(15), we obtain, after some algebra the following simplified action S = Z d x p − ˜ g ˜Φ h ǫ ˜ R ˜Φ − ˜ (cid:3) ˜Φ + M − m ˜Φ i + Z d x p − ˜ g ˜ L m , (18)where we have redefined ˜Φ = M − Φ. Since the conformal factor M is a real (or purely imaginary) number, ˜Φ and Φhave different norms ( R ), but they correspond to the same four-momentum p µ = ∇ µ S .The standard variation of this action with respect to the modified metric ˜ g µν and the modified scalar field ˜Φ gives theconformal version of Einstein equations and the scalar field equation (which are standard, see, e.g., [16] and [19]). Theenergy momentum tensors in the two frames are related by ˜ T mµν = M − T mµν and hence transformed frame the EM tensor isno longer conserved, unless the trace of the EM tensor ˜ T m in the conformal frame vanishes, i.e., ˜ ∇ ν ˜ T µνm = − ˜ T m ( ∇ µ ln M ). If one wants to study the observational aspect of the problem of a particle moving along a quantum trajectory (eq.(9)),in presence of a massive gravitational object (say a black hole), it is important to find out the conserved quantities.Because the motion is not that of a freely falling particle, it is not obvious if usual conserved quantities for stationaryspacetime, namely energy and angular momentum are also conserved along Bohmian trajectories.When a classical particle of mass m moves in a geodesic satisfying p µ ∇ µ p ν = 0 (here p µ = mu µ ), the quantity K ν p ν is conserved along the motion of the particle, i.e., p µ ∇ µ ( K ν p ν ) = 0, where K µ is a Killing vector which satisfies theKilling equation ∇ ( µ K ν ) = 0. To find out how this equation is modified when the motion is along a quantum Bohmiantrajectory, we take covariant derivative of the constraint relation p µ p µ = − M to obtain p µ ∇ µ p ν = − M ∇ ν M ( x ). Nowalong quantum trajectory, we find p µ ∇ µ ( K ν p ν ) = p µ K ν ∇ µ ( p ν ) + p µ p ν ∇ µ K ν = − K ν (cid:2) M ∇ ν M ( x ) (cid:3) + p µ p ν ∇ ( µ K ν ) . (19) The conformal factor Ω is a dimensionless quantity. But in the following we shall ignore the constant m factor.
6e see that the Killing equation no longer implies that K ν p ν is a conserved quantity. However, it can be checked that p µ K µ is a conserved quantity, if K µ is a conformal Killing vector associated with the metric ˜ g µν , i.e., it satisfies theequation ∇ ( µ K ν ) = − g µν K λ ( ∇ λ ln Ω) (here Ω = M ).The above conclusion is true for any general spacetime. For the special case of stationary spacetimes however, we canhave an interesting situation, namely that, even for an ordinary Killing vector of g µν , we can find a conserved quantity.To see this clearly, we start from eq.(19) and after a bit of algebra we arrive at p µ ∇ µ ( K ν p ν ) = K ν M u µ u ν ∇ µ M − K ν M h µν ∇ µ ln M + M u µ u ν ∇ µ K ν . (20)Now suppose that in the particle’s wave function, R ( x µ ) is independent of time so that the quantum potential termand hence M are also time independent. In this case the vectors ∇ µ M and ∇ µ log M have vanishing components alongthe dt direction. If the background spacetime is stationary, and the velocity is timelike, i.e., u µ = (1 , , , K µ = (1 , , , h µν is spacelike inthis case). This means that for a stationary spacetime and time independent R , once again K µ p µ is conserved along thetrajectory when the Killing equation is satisfied. This fact can be used to generate a quantum corrected version of anystationary spacetime (In [8] such corrections to Schwarzschild solution is obtained by using the quantum Raychaudhuriequation).As an immediate application of this, we can use the conformal transformation above find out a conformally trans-formed version of the Schwarzschild metric. Let us assume a simple stationary state wave function so that that itsmodulus is given by R ( r ) = N r exp (cid:16) − r (cid:17) , where N is a normalization constant. Then we calculate the quantumpotential associated with the wave function on Schwarzschild background f ( R ( r )) = (cid:3) RR = 1 r h r (cid:16) r − r + 2 (cid:17) − M (cid:16) r − r + 1 (cid:17)i , (21) M being the Schwarzschild mass. The conformal version of Schwarzschild solution is given by, d ˜ s = 1 r h M (cid:0) r − r + 1 (cid:1) − r (cid:0) r + 2 (cid:1)in(cid:16) − M r (cid:17) , (cid:16) − M r (cid:17) − , r d Ω o , (22)where we have neglected a constant additive term in the conformal factor. This is the solution of the transformed fieldequation with ˜ T mµν = 0. A particle will follow the geodesics of this metric. The matter part of EM tensor is zero due tothe relation ˜ T mµν = M − T mµν discussed above. It was argued in [6] that since Bohmian trajectories cannot intersect each other, they do not form conjugate pointsand hence can avoid the usual singularities of GR. In this context, we ask the following question. In the conformallytransformed frame there is no quantum force on the trajectory, then what happens to the conjugate points? Can thetrajectories reach the singularity in the conformal frame ˜ g µν ?To answer this question, we consider the following conjecture proposed long back in [20]. If the metric g µν is singular,we can always go to the conformal frame ˜ g µν = Ω g µν which is non-singular and the singularities of the original metricshow up as zeros of the conformal factor (as usual, the zeros of the conformal factor represent the boundaries of thespacetime). Now the question reduces to whether the conformal transformation of eq.(15) is the required one to ensurethe nonsingular nature of ˜ g µν . If we consider a particle of mass m moving in a geodesics of the metric ˜ g µν , then in theframe g µν , the particle’s mass is time and position dependent (the modified mass in general depends on the nature ofthe conformal factor Ω) and it moves on a nongeodesic. In [20], the author showed working out several examples, thatthe variable mass of the conformal frame should be the conformal factor to ensure the nonsingular nature of the metric˜ g µν . From eq.(15), we see this is exactly the case. 7his should be clear from the example we have worked out above. The original Schwarzschild metric is singular at r →
0. But the Ricci curvature scalar ˜ R of the metric ˜ g µν is non-singular in this limit, as can be checked explicitly, i.e.,˜ R r → = constant . Also the conformal factor in the transformation equation in eq.(22) is undefined in this limit. Thusif we start from the nonsingular frame ˜ g µν and work in a conformally related frame g µν , the conformal factor vanishesprecisely at r = 0 making the transformation invalid at the singularity r = 0. Thus we conclude that the singularityis resolved in both the frames. Note that the quantum potential and hence the conformal factor depends on the wavefunction, and thus it is not guaranteed that the transformed metric is generically singularity free. This general case willbe investigated elsewhere. So far we have shown by the transformation of eq.(15), that it is possible to make the magnitude of ˜ p µ a constant. Byusing the same transformation, can we write the continuity equation (second equation in eq.(4)) in the desired form˜ ∇ µ (cid:0) ρ ˜ u µ (cid:1) = 0? The answer is no. To explain this, we shall first derive an important relation between the expansionscalars ( θ and ˜ θ ) of two frames and the probability density ρ (see eq.(26) below).A conformal transformation is equivalent to a corresponding change in the proper time dτ → d ˜ τ = Ω dτ . In GR,under such conformal transformations, the Christoffel symbols transform as [16, 17] δ Γ σµν = 2 δ σ ( µ ∂ ν ) ln Ω − g µν ∂ σ ln Ω . (23)Using this, one can check that the covariant derivative of the four velocity in the transformed frame is˜ ∇ µ ˜ u ν = Ω (cid:16) ∇ µ u ν + g µν u α ∇ α ln Ω − u µ ∇ ν ln Ω (cid:17) . (24)Taking the trace of this equation, we see that the expansion scalar θ of a geodesic congruence transforms under aconformal transformation as ˜ θ = Ω − (cid:16) θ + 3 u α ∇ α ln Ω (cid:17) . (25)Using this relation we derive the following relation involving the probability density ρ ,˜ ∇ µ (cid:0) ρ ˜ u µ (cid:1) = ρ ˜ θ + ˜ u µ ˜ ∇ µ ρ = ρ Ω − (cid:16) θ + 3 u α ∇ α ln Ω (cid:17) + Ω − u µ ∇ µ ρ . (26)The criterion that in the transformed frame, the particle motion is a geodesic has already fixed the conformal factor Ωequal to M up to a multiplicative constant. Now expanding the original conservation equation ∇ µ (cid:0) ρM u µ (cid:1) = 0 we havethe relation ρθ = − ρu α ∇ α ln M − u α ∇ α ρ . (27)Eliminating θ between eqs.(26) and (27), we see that ˜ ∇ µ (cid:0) ρ ˜ u µ (cid:1) = 0. Thus transformation to a conformal frame givesrise to a wrong continuity equation. The reason for this is of course the fact that the momentum constrain relation hasalready fixed the conformal factor equal to M . As we shall show in sequel, for disformally related metrics, there is stillenough freedom to fix both the problems.A somewhat subtle point here is worth mentioning. Remember that we redefined the scalar field as Φ = M ˜Φ, so thatthe resulting action in eq.(18) is more convenient to work with. As we have mentioned, in this redefinition, the normof the wave function changes to ˜ R = M − R , and hence if one defines a transformed probability density ˜ ρ = ˜ R anddemands that ˜ ∇ µ (cid:0) ˜ ρ ˜ u µ (cid:1) is the quantity one should be looking for, we can see, by an analogous procedure as above thatthis quantity is indeed conserved, i.e., ˜ ∇ µ (cid:0) ˜ ρ ˜ u µ (cid:1) = 0. But let us stress that this is not the continuity equation we areafter, simply because the field redefinition (Weyl scaling) has nothing to do with the original conformal transformation g µν → ˜ g µν , which is an actual change of the geometry itself, and not a change of coordinates or fields living in the8pacetime [17]. The only purpose it serves (in this context) is to rewrite the action in a convenient form. Notice alsothat when M < ρ ) is not even well defined in the transformed frame.Apart from the wrong continuity equation, there is a further issue that is problematic in a conformally related frame.So far in our discussion of non geodesic motion, we have always consider a timelike trajectory for which u µ u µ = −
1, butwe can generalize the result for spacelike and null trajectories also. For the general case the eq (9) is given by u µ ∇ µ u ν = 12 (cid:0) αg µν − u µ u ν (cid:1) ∇ µ ln (cid:0) m − f ( R ) (cid:1) , (28)where α = − , +1 ,
0, for time-like, space-like and null geodesics, respectively. Now for a massless particle, we put m = 0,and since the term proportional to u µ u ν can be absorbed in a re-parameterization of proper time, the only remainingterm is proportional to α , and hence the acceleration is zero for massless particle in both frames. So a natural question is,if null trajectories always move in geodesics, what happens to the singularity resolution argument for massless particles? That is, if the force due to the quantum potential does not affect null trajectories, we ask why they do not form acaustic as in GR, and hence fall into a singularity?We will next show that all the problems mentioned above are resolved by using disformal transformations instead.However before going into this, we will make an important simplification by assuming minimal coupling. The analysis of the quantum corrections so far has been carried out in a background where there is a non-minimalcoupling between the curvature scalar R and the scalar field describing the particle. As mentioned this term is crucialfor the Klein-Gordon equation to be conformally invariant. Though this helps to make the analysis in the conformalframe simpler, it makes a nontrivial contribution to the Einstein equation, such that the metric in general will not benon-dynamical - as can be seen from eq.(2). Usually this contribution is neglected (as was done in [6]) and the scalarfield is assumed to be defined on a non-dynamical static background.From now on we will instead work in a frame where Φ is minimally coupled to the background. Then the total actionis similar to the one in eq.(1) with F (Φ) = 1, and Φ satisfies the Klein-Gordon equation ( (cid:3) − m )Φ = 0. The dispersionrelation now changes to p µ p µ = − m + (cid:3) RR = M u µ u µ , (29)which indicates that a quantum particle will follow a non geodesics motion in this frame as given in eq.(9), with ǫ = 0.We can always transform to a conformal frame, where the motion of the particle will be on a geodesic and quantumeffects are included in the energy momentum tensor, but in that frame there is a non-minimal coupling between thecurvature scalar and the redefined scalar field .The advantages of this approach to that the picture is more in line with scalar tensor theories of gravity, where inthe non-minimally coupled frame (Jordan frame) the particle follows geodesic equation, but in the conformally relatedframe, where Φ is minimally coupled, the particle follows a non geodesic motion. Also, in this frame, the quantum mass,given by M = p m − f ( R ) is independent of the nature of the coupling term ǫ , and hence its value is unambiguousi.e., if f ( R ) > m it is imaginary, otherwise it is real. In the Jordan frame, this depends on ǫ . As we will momentarilysee, this property helps to solve the problem of indefiniteness of quantum mass uniquely in this frame, using a disformaltransformation.All the calculations in the rest of the paper are performed assuming a minimal coupling. When there is non-minimalcoupling between gravitational and particle degrees of freedom, our calculations can be generalized with some minormodifications, but in the transformed frame it is rather difficult to see the quantum effects in the metric throughcalculating the EM tensor, because there will be terms coming from the non-minimal coupling also (with minimal For a massless particle one can make the action and hence the Klein-Gordon equation conformally invariant by choosing ǫ = 1 / So far we have assumed a description of gravity where gravitational and particle dynamics take place in conformallyrelated global Riemannian spacetimes, called gravitational and physical geometries respectively. But as we have pointedout in the introduction, the relation between these two geometries can be more general than the conformal transformation.It was shown in [1], by assuming the physical geometry (on which the matter dynamics takes place) to be Finslerian(instead of Riemannian), and using arguments based on the weak equivalence principle and causality, that in the mostgeneral case, both the gravitational and the physical geometries have to be Riemannian and that in general suchgeometries are related to each other by a disformal transformation. It is thus natural to ask what are the consequencesof quantum motion in terms of Bohmian mechanics in this more general case. This is the topic we discuss in this section.
Two metrics g ∗ and g (and their inverses) are said to be related to each other by the a disformal transformation, if therelation between them is given by [1, 15] g ∗ µν = Ω ( φ, X ) g µν − α B ( φ, X ) φ µ φ ν , g ∗ µν = Ω − (cid:20) g µν + α B Ω − X B φ µ φ ν (cid:21) , X = − g µν φ µ φ ν , (30)where α = 0 , ±
1, both Ω and B are arbitrary real functions of a scalar field φ , and we have used the notation φ µ = ∇ µ φ to denote the normal vector to a φ = const hypersurface. Note that all indices of φ µ are raised and lowered by themetric g µν . We will denote all the tensor quantities with respect to g ∗ µν with a superscript “ ∗ ”.If we consider the motion of a particle moving on a timelike trajectory we can choose the normal vectors φ µ to behypersurface orthogonal velocity vector v µ of the trajectories, and in this case α = − α to generalize the results to a spacelike trajectories also). The identification of φ µ with velocity is a choice,and as long as we are considering the motion along timelike (or spacelike) trajectory this is a good choice, but for nulltrajectories we will make a different choice because in general via a disformal transformation, unlike a conformal one, anull geodesic can map to a non-geodesic trajectory (this will be crucial in our arguments in section 4.2). Thus the formof the disformal transformation and its inverse that we will consider for now are the followings [23] g ∗ µν = Ω ( x ) g µν − α B ( x ) v µ v ν , g ∗ µν = Ω − (cid:20) g µν + α (cid:18) B Ω − B (cid:19) v µ v ν (cid:21) . (31)This transformation is equivalent to a transformation of the proper time [15] dτ ∗ = − g ∗ µν dx µ dx ν = − h Ω g µν − α B v µ v ν i dx µ dx ν = h Ω + α B i dτ . (32)Now, the tangent vector to a particle trajectory is v µ = dx µ /dτ . With respect to g ∗ µν , this vector is defined as v ∗ µ = dx µ /dτ ∗ . From the above relations we can find out the following relations v ∗ µ = √F v µ , v ∗ µ = g ∗ µν v ∗ ν = F − / v µ , F = Ω − B , g ∗ µν v ∗ µ v ∗ ν = α. (33)As can be seen, the vector v ∗ µ is normalized with respect to g ∗ µν .10 .2 Relation between acceleration vectors and particle motion Using the formulas given above, we can now establish the required relation between acceleration vectors a ν = v µ ∇ µ v ν and a ∗ ν = v ∗ µ ∇ ∗ µ v ∗ ν of two metrics, where ∇ ∗ µ is the covariant derivative with respect to g ∗ µν . First, we write down theknown relation between the quantities Γ ∗ λµν v ∗ λ and Γ λµν v λ (see [23] for details)Γ ∗ λµν v ∗ λ = √F Γ λµν v λ + 12 √F h αv µ v ν v α ∇ α F − h µν v α ∇ α Ω − B K µν + 2 v ( µ h αν ) ∇ α F i , (34)where Γ λµν are the connection coefficients, and we have defined the extrinsic curvature in standard fashion, as K µν = h αµ ∇ α u ν = ∇ µ v ν − αa ν v µ . (35)Substituting eq.(34) in the formula ∇ ∗ µ v ∗ ν = ∂ µ v ∗ ν − Γ ∗ λµν v ∗ λ , after a few steps of straightforward algebra, we get thefollowing relation between covariant derivative of a vector with respect to both metrics ∇ ∗ µ v ∗ ν = √F (cid:18) ∇ µ v ν + v α ∇ α Ω F h µν + BF K ( µν ) + v ν ∇ µ F F − αv µ v ν v α ∇ α F F − F v ( µ h αν ) ∇ α F (cid:19) . (36)Note that the last three terms in eq.(36) are missing in [23]. But these terms are essential to establish a relation betweenthe acceleration vectors, in which case both the second and third terms vanish. Multiplying both sides of eq.(36) by v ∗ µ ,and after a bit of manipulation we get the following simplified version of the desired relation between the accelerationvectors a ∗ µ = a µ − F αh νµ ∇ ν F = a µ − αh νµ ∇ ν ln F . (37)It is worth remarking that due to absence of the terms in eq.(36) mentioned above in the corresponding formula in [23], theauthor has concluded that the accelerations are equal. But accelerations of conformally (and also disformally) relatedframes are not equal, at least in GR. This is the root of the problem that one faces when constructing conformallyinvariant observables in gravity theories with symmetric connection (see the discussion is section 5 below).Note also that we shall recover all the results of previous section of conformal transformation by putting B = 0 inthe above expressions. As is evident from this equation, two vectors a ∗ µ and a µ cannot be equal to each other unless thedifference of Ω and B is a constant. This in turn means that two velocity vectors v µ and v ∗ µ cannot represent geodesicssimultaneously. Indeed from eq.(37), we glean that a ∗ µ = v ∗ ν ∇ ∗ ν v ∗ µ = 0 = ⇒ a µ = v ν ∇ ν v µ = αh νµ ∇ ν ln √F . (38)Thus in the physical geometry, if we consider the geodesic motion of a particle of mass m , then in the disformally relatedgravitational geometry, this represents an accelerated motion with the force being perpendicular to the four-velocity.Now if we consider a quantum particle moving along a Bohmian trajectory so that it is acted upon by the force whoseorigin is the quantum potential, then the norm of its four momentum satisfies the relation in eq.(29). If we represent itsacceleration by the second equation in eq.(38), in the disformally related geometry this corresponds to force free motionwith the quantum effects taken in the modified metric g ∗ µν of the physical geometry. We can easily determine the desiredrelation between the transformation factors Ω and B in terms of the modulus of the wave function R , by comparingeq.(38) and eq.(9) to be , F = Ω − B = (cid:0) − f ( R ) /m (cid:1) . (39)We have put ǫ = 0 because we have assumed no coupling between the curvature and the wavefunction Φ. Note that theconstraint relation of eq.(29) is not enough to uniquely determine the both the transformation factors in eq.(39), andbelow we will see that a second relation between the transformation factors arise through the continuity equation. As before we will neglect the unimportant factor of m and will take F = M below.
11t is also convenient to write down the corresponding particle actions in this case. Given the constraint relation ofeq.(29), by following the procedure outlined previously, we can derive the action of a particle moving along the Bohmiantrajectory in the frame g µν . This is just given by eq.(16) with ǫ = 0. Now using the rules of active transformation i.e.,substituting g µν in terms of g ∗ µν and v ∗ µ from eq.(31) we see that the action S m (cid:2) g µν , γ (cid:3) = − Z γ p m − f ( R ) p − g µν v µ v ν dτ (40)gets transformed to the action of a particle of unit mass S Dm (cid:2) g ∗ µν , γ (cid:3) = − Z γ r − Ω − h g ∗ µν + α BF v ∗ µ v ∗ ν i F v ∗ µ v ∗ ν dτ ∗ = − Z γ q − g ∗ µν v ∗ µ v ∗ ν dτ ∗ . (41)Here in the first step, we have used the previous identification of eq.(39), and also substituted dτ in terms of dτ ∗ , andthe trick in the second step is to write one set of g ∗ µν v ∗ µ v ∗ ν as α so that it cancels with the other term. With theseparticle actions, it is easy to check that they satisfy the respective equations in eq.(38), thus confirming our conclusions.Note that here (and in the previous section during the discussions on the conformal version) we have taken an activepoint of view in transforming the action, namely we have taken the functional form of the action same as before (i.e.,the action of eq.(7)) with the Lagrangian in eq.(8)), and replaced the original metric by the transformed one. Thisprocedure naturally leads to a value of the action different from the previous one. On the other hand, if we want to keepthe form of the action unchanged then the passive transformation should be used. In this paper we shall always use theactive point of view unless otherwise specified (see [19, 24] for transformation with passive point of view).We can also write down the modified Raychaudhuri equation and the deviation equation in this case also followingthe lines described in subsection 2.1 for the conformal frame. One can check these are still given by eq.(11) and eq.(13)respectively, with L = − ln F . Now that we have the characterization of a quantum particle in both the gravitational and the physical geometry, andhave shown that like the conformal transformation, a disformal transformation can be successfully used to incorporate thequantum effects in the geometry, one can ask where exactly it differs from the usual picture of conformal transformationand what are the advantages of this identification over the conformal transformation, if any. In this section we shallshow that a disformal transformation has its advantages, namely it can solve the problems addressed in section 2.5.
As we discussed in section 2.5, the conformal transformation with the quantum mass squared as the conformal factor cannot give us the right continuity equation. Here we shall show by doing a disformal transformation of the form eq.(31),it is possible solve this problem thereby completely fixing both the transformation factors Ω and B .We start by writing down the disformal version of the eq.(25) that relates the expansion scalar in both frames (see [23]for the derivation) θ ∗ = F − / (cid:16) θ + 3 v α ∇ α ln Ω (cid:17) . (42)Then, as before, we have the analogue of eq.(26) given by ∇ ∗ µ (cid:0) ρv ∗ µ (cid:1) = ρθ ∗ + v ∗ µ ∇ ∗ µ ρ = ρ F − / (cid:16) θ + 3 v α ∇ α ln Ω (cid:17) + F − / v µ ∇ µ ρ . (43)12ow using eq.(27) to eliminate θ from this equation we get ∇ ∗ µ (cid:0) ρv ∗ µ (cid:1) = ρ F − / (cid:16) − v α ∇ α ln M + v α ∇ α ln Ω (cid:17) . (44)The requirement that in the disformally transformed frame ∇ ∗ µ (cid:0) ρv ∗ µ (cid:1) = 0 is the correct continuity equation indicatesthe left hand side to be zero and thus fixes the conformal factor to be Ω = M . This, together with the requirement ofeq.(39) also fixes the disformal factor B = M / − M . As was pointed out towards the end of subsection 2.5, massless particles cause a problem in the singularity resolutionargument because they do not experience any force in frames related by conformal transformations. For the case offrames related by disformal transformations (eq.(31) considered in the previous section), the same problem arises becausewe see from the corresponding relation eq.(38), the same conclusion remains valid, i.e., the extra force term is zero fornull trajectories.We shall argue below that we can cure this problem in the context of disformal transformation, by rememberingthat, one can make a disformal transformation more general than the one written in eq.(31), where the extra piece neednot be necessarily pointing along the direction of the tangent of a particle trajectory. This transformation precisely iswhat given in eq.(30), where the non conformal piece is along a direction specified by the φ = const hypersurface, whichwe can choose to be different from the four velocity. This is the advantage of disformal transformations over conformaltransformations, namely that one can make a massless particle move on a non geodesic motion by going to a frame wheresquare of its four momentum is non zero. Below we shall consider a simple form of Maxwell’s equation to show clearlyhow this can be done (see [15] where the transformation of the Maxwell equations under a disformal transformation areconsidered by assuming the geometrical optics approximation. However we do not make any such approximation).Let us consider the motion of a photon in the gravitational geometry (represented by a vector field A µ ) described bythe following Maxwell equations ∇ µ ∇ µ A ν − ∇ µ ∇ ν A µ = 0 , (45)where we have put all the source terms to zero. Now, using the Ricci identity which relates the commutator of thecovariant derivative of a vector field to the Riemann tensor, ∇ µ ∇ ν A α − ∇ ν ∇ µ A α = R αβµν A β , (46)in the contracted form, and imposing the Lorentz gauge condition ∇ µ A µ = 0, we write the above equation as ∇ µ ∇ µ A ν − R µν A µ = 0 . (47)As before, in a Bohmian treatment, we use the polar from of A µ A µ = C µ ( x ) e i S ( x ) , (48)and substitute in the Maxwell’s equation. After separating the real and imaginary parts, and identifying K µ = ∇ µ S asthe wave vector of the photon, we get the following equations g µν K µ K ν = C ν C ∇ µ ∇ µ C ν − R µν C µ C ν ≡ H ( C ) , ∇ µ (cid:0) C ∇ µ S (cid:1) = 0 , (49) Ω is fixed upto an additive constant which we have taken to be zero Note that by a massless particle, we mean a particle with zero classical mass ( m ), not zero quantum mass ( M ). More precisely what wemean by massless particle here are the particles which moves along the trajectory u µ u µ = 0 and hence p µ p µ = 0. C = g µν C µ C ν denotes the magnitude of the vector C µ . The first relation gives the magnitude of the wave vector,with H ( C ) denoting the quantum potential in this case, and the second one is essentially the conservation equation. Ascan be anticipated from eq.(49), since in this frame, motion of the photon is represented by the non null vector K µ , themotion of photon is along a non geodesic trajectory.Let us now apply the disformal transformation (eq.(30) with α = −
1) and see if we can make the photon motiona geodesic in the transformed frame. In doing so, the first thing to notice that, since the transformation functions ineq.(30) are real, the phase factor S and hence the wave vector K µ = ∇ µ S is equal in both the frames. Then, using theinverse relation of eq.(30), the left hand side of first equation of eq.(49) reduces to g µν K µ K ν → Ω g ∗ µν K µ K ν + B Ω − X B (cid:16) φ µ K µ (cid:17) . (50)Now it is easy to see by comparing eq.(49) and eq.(50), that if we want to make K ν null in the transformed frame, thenwe have to choose the transformation functions such that B Ω − X B (cid:0) φ µ K µ (cid:1) = C ν C ∇ µ ∇ µ C ν − R µν C µ C ν = H ( C ) . (51)Of course, this single equation does not completely determine the disformal transformation specified by Ω , B , φ µ . We canhave another relation from the continuity equation. However unlike the previous case for massive particles, these twoequations (eq.(51) and the one obtained from the continuity equation) are not enough for fixing both the transformationfactors Ω and B and the direction of the disformal vector specified by components of φ µ uniquely. We also have to makesome choice of the factor Ω (such as pure the disformal transformation with Ω = constant ) and/or of the disformalvector (such as timelike ( φ µ φ µ = −
1) or null disformal transformation ( φ µ φ µ = 0)). The description of the photon’smotion with such explicit choices are left for a future work.The equation of motion derived from eq.(49) by taking covariant derivative of both side is the non geodesic equation K µ ∇ µ K ν = 12 ∇ ν (cid:20) B Ω − X B (cid:0) φ µ K µ (cid:1) (cid:21) = 12 ∇ ν H ( C ) , (52)which, as can be checked, under a disformal transformation satisfying eq.(51) transforms to the geodesic equation K ∗ µ ∇ ∗ µ K ∗ ν = 0 . (53)Thus the motion of a photon in a gravitational geometry follows a non null trajectory (the null trajectories ds = g µν dx µ dx ν = 0 in the gravitational geometry are followed by the gravitons [1]) and hence its motion is nongeodesic, actedupon by the force due to the quantum potential (see eq.(49) and eq.(52) respectively). But this motion in a disformallyrelated physical geometry follows a null trajectory, ds ∗ = g ∗ µν dx µ dx ν = 0 of that geometry, where the quantum potentialdetermines the required transformation factors. As we have shown in section 2.3, p µ K µ is a conserved quantity alongthe non geodesic motion, provided that K µ is a conformal Killing vector i.e., a Killing vector of a conformally relatedmetric. By using an analogous procedure, it is easy to see that K µ K µ is a conserved quantity along the photon trajectoryif it satisfies the equation ∇ ( µ K ν ) = − g µν K λ ∇ λ (ln H ( C )), and this quantity should be used if one wants to study thedeflection of light in gravitational field. When one uses the standard Klein-Gordon equation to perform a Bohmian treatment, one gets the resulting quantumHamilton-Jacobi equation of eq.(29). The right side of this equation, interpreted as the mass square (denoted as M )is not always positive definite, and hence the theory can have tachyonic solutions (see for example [3]). As we havementioned before, one motivation for transforming to a conformal frame to describe the quantum motion of a particle14atisfying the Klein-Gordon equation is to avoid this problem, so that in the transformed frame the particle has a positivedefinite mass. But also as pointed out before, this comes with other problems such as the wrong continuity equation andthe problem with massless particles. Most importantly when the quantum mass M is imaginary, the conformal factor isitself negative and hence the transformation is not well defined.In subsection 4.1, we have shown that for a massive particle by transforming to a disformal frame where two metricsare related by eq.(31), with the transformation factors given in terms of the quantum mass by the relationsΩ = M / , B = M / − M , M ( x ) = q(cid:0) m − f ( R ) (cid:1) , (54)one could achieve a consistent continuity equation for the motion of the quantum particle. However the same trans-formation does not solve the problem of definiteness of mass. If the quantum mass is imaginary, then the real root ofthe conformal factor Ω = M / in eq.(54) becomes negative, so that the disformal transformation used earlier becomesundefined .In this subsection, we shall show that a possible way out of this problem is once again to return to the general formof the disformal transformation in eq.(30). However, before doing that let us try to locate the problem in the previoussetting. When the mass squared is negative, we can express the quantum mass formally as M = | M | e iπ/ . Now fromthe continuity equation in the transformed frame, we see that the extra phase of π/ ∇ ∗ µ (cid:0) ρv ∗ µ (cid:1) = ρ F − / (cid:16) − v α ∇ α ln | M | + v α ∇ α ln Ω (cid:17) . (55)Thus by taking Ω = | M | / , we can still satisfy the continuity equation in the transformed frame, even when M is imaginary. Looking at the acceleration in eq.(9) (with ǫ = 0), we could similarly argue that, by choosing √F = √ Ω − B = | M | , this equation can be satisfied as well, since the constant phase factor again does not contribute dueto the derivative. However such a choice does not satisfy the correct dispersion relation g ∗ µν p ∗ µ p ∗ ν = −
1, because undersuch a transformation, the required relation g µν p µ p ν = | M | → (cid:20) Ω g ∗ µν + (cid:18) B Ω − B (cid:19) v µ v ν (cid:21) p µ p ν ≡ − Ω + (cid:18) −B| M | Ω − B (cid:19) = | M | , (56)is not satisfied when we choose Ω − B = | M | . This is the real problem, namely to satisfy the correct dispersion relationwhen quantum mass is imaginary. Indeed the Bohmian treatment of Klein-Gordon equation gives the dispersion relation,and not the acceleration equation (see eq.(5)). In the process of taking the covariant derivative, the imaginary factorsare “lost” in both eq.(55) and the continuity equation, and one can reach wrong conclusions by considering them. Thusour primary focus will be on the dispersion relation.We begin the general case by writing above equation of eq.(56) for such a transformation in eq.(30), g µν p µ p ν = | M | → (cid:20) Ω g ∗ µν + (cid:18) B Ω − B X (cid:19) φ µ φ ν (cid:21) p µ p ν ≡ − Ω + (cid:18) −BD | M | Ω − B X (cid:19) = | M | , (57)where D = φ µ v µ denotes the projection of the vector φ µ along the particle 4-velocity.For our purpose in this subsection, it is sufficient to consider the so called pure disformal transformation, where boththe conformal and disformal factors are set to a constant (denoted as Ω and B respectively, with Ω > For the description of massless particle one has to use a more general transformation of eq.(30). In this subsection, we shall concentrateon the case of massive particles only. In the non-minimally coupled frame, the quantum mass has an extra factor of ǫR (see eq.(6) above). Can this factor make quantum massin this frame positive definite? Assuming ǫ >
0, the answer will depend on the sign of the Ricci scalar. When
R <
0, the ǫR term can notmake the quantum mass positive definite and for R > M come out to be positive depending on its relative magnitude withthe quantum potential. As mentioned in section 2.6, this is one of the advantages of working with minimal coupling - the nature of M doesnot depends on the coupling ǫ . g ⋆µν = Ω g µν + B φ µ φ ν , g ⋆µν = Ω − (cid:20) g µν − B Ω − X B φ µ φ ν (cid:21) , X = − g µν φ µ φ ν . (58)All the transformed quantities are denoted by an overhead star. Here the 4-vector φ µ , having components ( φ , φ , φ , φ ),determine the direction of the disformal transformation and is to be determined form the correct transformation of thedispersion relation and the continuity equation. The above transformation corresponds to a change of proper time dτ ⋆ = β dτ with β = (cid:0) Ω − B D (cid:1) / . Then eq.(57) gives the first constraint relation to beΩ + (cid:18) B D | M | Ω − B X (cid:19) + | M | = 0 , with , D 6 = 0 unless | M | = constant . (59)The continuity equation gives another constraint. To determine this, we start by writing down the transformation ruleof the covariant derivative. In the most general case, the change of the Christoffel symbols are quite complicated (seefor example [15, 25]), but for our case, it simplifies to the following ∇ ⋆µ v ν = ∇ µ v ν + C µνα v α where C µνα = (cid:18) B Ω − X B (cid:19) φ µ ∇ α φ ν . (60)Writing the left side in terms of transformed velocity v ⋆ν , and simplifying, we arrive at ∇ ⋆µ v ⋆ν = β − (cid:16) ∇ µ v ν + C µνα v α − v ν ∇ µ β (cid:17) , (61)This gives the desired relation between the expansion scalars (compare with eq.(25) and eq.(42)) θ ⋆ = β − (cid:16) θ + C ννα v α − v ν ∇ ν β (cid:17) . (62)The continuity equation can now be expanded to to give ∇ ⋆µ (cid:0) ρv ⋆µ (cid:1) = ρθ ⋆ + v ⋆µ ∇ ⋆µ ρ = ρβ − (cid:16) θ + C ννα v α − v ν ∇ ν β (cid:17) + β − v µ ∇ µ ρ . (63)As before, eliminating θ from this equation and demanding that continuity equation ∇ ⋆µ (cid:0) ρv ⋆µ (cid:1) = 0 is satisfied, we getthe second constraint C ννα v α − v ν ∇ ν β − v α ∇ α ln | M | = 0 . (64)As before, the pure phase factor does have any effect in the continuity equation, so that components of φ µ are real. Ifwe choose the vector field φ µ such that, given the quantum mass both the constraint eqs. (59) and (64) are satisfied,then in the transformed frame ( g ⋆µν ) the particle has unit mass, thus solving the problem of imaginary quantum mass inthe original frame. Obviously the two constraints of eqs.(59) and (64) cannot determine all the four components of φ µ ,and we have to make some choices.Below we shall briefly sketch such a procedure when the quantum mass of the particle ( M ) moving in the flatspacetime is imaginary and is a function of the radial coordinate r only . We consider the vector field φ µ to be of theform (1 , φ ( r ) , ,
0) and the particle 4 velocity is v µ = (1 , , ,
0) so that D = 1. Then, as can be checked, the continuityequation is identically satisfied and eq.(59) gives the form of the function φ ( r ) in terms of the quantum mass as φ ( r ) = r Ω B s B − Ω − | M ( r ) | Ω + | M ( r ) | . (65) This mass can corresponds to a stationary state solution of the Klein-Gordon equation for which the quantum potential is time independent. | M ( r ) | one can now easily determine the function φ ( r ) .We also mention here that there are ways to make the quantum mass positive definite even in the classical background.For example, in [26], the authors had shown how to make M positive definite by using a different interpretation ofBohmian mechanics and demanding that the mass M should have the correct non relativistic limit. In this procedure,they obtained the following formula for the quantum mass ( M ) for a particle of classical mass mM = m exp (cid:20) m (cid:3) RR (cid:21) . (66)With this mass formula, one can obtain the correct non relativistic equation of motion of the particle as shown in [26].The resulting theory has been studied extensively in the context of conformal transformations and curved spacetime(see [11] for a review) . In this paper, we have already shown a way out of this problem by using the disformaltransformation, nevertheless it is useful to make the theory free from any tachyonic solution from the start. We leavethe problem of incorporating the mass of eq.(66) in our transformation formulas of eq.(54) for a future study. In the well known approach of incorporating the quantum effects of geometry, conformally related spacetimes are usedto establish the equivalence between the Bohmian motion of a particle on a classical background with force free classicalmotion on a quantum corrected background. In terms of this bimetric description of gravity, the Bohmain force can beinterpreted as arising due to the description of the particle in the gravitational geometry rather than in the physicalone, and quantum effects are incorporated in the conformal degrees of freedom [12]. This line of reasoning can be usedto deal with the singularity problem of GR [6].The first part of this paper is devoted to the discussion of various aspects of this approach. In the second part, we haveshown how one can incorporate the quantum nature of a particle (both massive and massless) in the disformal degrees offreedom of the physical geometry. The conformal transformation is equivalent to a uniform coordinate dependent scalingin every spacetime direction. On the other hand, in a disformal transformation of the form given in eq.(30), we not onlydo a uniform scaling in every direction (conformal part), we also scale a particular direction (chosen by the normal vector φ µ ) differently from other directions. This fact is more transparent from the so called pure disformal transformation,where we only scale the direction chosen by the velocity vector v µ , and the other directions (perpendicular to v µ in anorthonormal coordinate frame) do not scale at all i.e., the conformal factor is just a constant.What we have shown in this paper imply that the quantum effects can can viewed as a disformal transformationbetween the gravitational and the physical geometry, i.e., if we consider the motion of quantum particle in gravitationalgeometry in presence of the quantum potential, this is equivalent to the classical free fall motion in a disformallytransformed spacetime, where the transformation is done along the direction chosen by the the particle four velocity.The quantum force arises in the gravitational geometry only because we had chosen the “wrong” frame for the analysis.Most importantly, by using disformal transformation we have shown here that one can solve the usual problems in theconformal version, such as wrong continuity equation and the problem of definiteness of mass.Before concluding we mention here another future application of the formalism constructed here. We notice that thesecond factor in the transformation relation of expansion scalar derived in eq.(25) indicates that it is not a conformallyinvariant quantity. Similarly it is easy to show, taking symmetric and antisymmetric part of eq.(24) respectively that,the shear and rotation tensors are also not conformally invariant. Applying directional derivative to eq.(25) we see that Of course for some | M ( r ) | the function φ ( r ) determined in eq.(65) may not be real every where. In that case one has to take an anastzfor the vector field different from the one given here. This approach was recently used in [21] to derive a quantum version of the Friedmann equations. d ˜ θd ˜ τ = Ω − (cid:20) dθdτ − Ω − (cid:18) θ + 3 d ln Ω dτ (cid:19) d Ω dτ + 3 d ln Ω dτ (cid:21) . (67)In GR, such quantities are problematic because for a particle trajectory these are the observable quantities and appearin the Raychudhuri equation in their scalar form, so these should be conformally invariant. 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