Towards relativistic quantum geometry

We obtain a gauge-invariant relativistic quantum geometry by using a Weylian-like manifold with a geometric scalar field which provides a gauge-invariant relativistic quantum theory in which the algebra of the Weylian-like field depends on observers. An example for a Reissner-Nordstr\"om black-hole is studied.


I. INTRODUCTION
The study of geometrodynamics was introduced by Wheeler in the 50's decade in order to describe particle as geometrical topological defects in a relativistic framework [1], and, in the last years has becoming a very intensive subject of research [2]. In the last decades Loop Quantum Gravity (LQG) have provided a picture of the quantum geometry of space, thanks in part to the theory of spin networks [3]. The concept of spin foam is intended to serve as a similar picture for the quantum geometry of spacetime. LQG is a theory that attempts to describe the quantum properties of the universe and gravity. In LQG the space can be viewed as an extremely fine favric of finite loops. These networks of loops are called spin networks. The evolution of a spin network over time is called a spin foam. The more traditional approach to LQG is the canonical LQG, and there is a newer approach called covariant LQG, more commonly called spin foam theory. However, at the present time, it is not possible to realize a consistent quantum gravity theory which leads to the unification of gravitation with the other forces. One of the problems relies in the impossibility of constructing a non-perturbative gauge-invariant formalism which can describe intense quantised gravitational fields. This will be the subject of this letter, but using a new kind of connections that do not preserve the norm of the vectors. Therefore, we shall call these manifolds as Weylian-like. Other remarkable characteristic of these manifolds is, as in Weylian manifolds, that the variation of the tensor metric is nonzero. To do it we shall define a geometrical scalar field σ that drives a geometrical displacement from a Riemannian manifold (on which we define the background), to a Weylian-like manifold where we represent the dynamics of the quantum geometry.
In the following issues of this section we shall revise the procedure for the minimization of the EH action, without making emphasis on the structure of the manifolds, but in the gauge transformations of vector and tensor fields. In Sect. II we shall construct a quantum description for spacetime in a new Weylian-like manifold, here introduced. In Sect. III, we shall study the example of a Reissner-Nordström black-hole, and we shall made some remarks on the quantum geometrical structure of a Schwarzschild black-hole. Finally, in Sect. IV we shall include some remarks.

A. Variation of EH action
It is known that in the event that a manifold has a boundary ∂M, the action should be supplemented by a boundary term so that the variational principle to be well-defined [4,5]. However, this is not the only manner to study this problem. As was recently demonstrated [6], there is another way to include the flux around a hypersurface that encloses a physical source without the inclusion of another term in the Einstein-Hilbert (EH) action, but by making a constraint on the first variation of the EH action. En that paper was demonstrated that the non-zero flux of the vector metric fluctuations through the closed 3D Gaussian-like hypersurface, is responsible for the gauge-invariance of gravitational waves.
To see it, we consider the problem of a EH action I, which describes gravitation and matter The first term in (1) is the Einstein-Hilbert action and κ = 8πG. Here, g is the determinant of the covariant background tensor metric g µν , R = g µν R µν is the scalar curvature, R α µνα = R µν is the covariant Ricci tensor and L m is an arbitrary Lagrangian density which describes matter. If we deal with an orthogonal base, the curvature tensor will be written in terms of the connections: R α βγδ = Γ α βδ,γ − Γ α βγ,δ + Γ ǫ βδ Γ α ǫγ − Γ ǫ βγ Γ α ǫδ . The first variation of the action is with g αβ δR αβ = ∇ α δW α , where δW α = δΓ α βγ g βγ − δΓ ǫ βǫ g βα = g βγ ∇ α δΨ βγ [8]. When we deal with a manifold M which has a boundary ∂M, the action (1) should be supplemented by a boundary term in order to the variational principle to be well-defined. This additional term is known as the York-Gibbons-Hawking action [4,5].

B. Gauge transformations
We shall propose other solution for this problem. Our strategy will be to preserve the Einstein-Hilbert action as in (1) and see what are the consequences of do it. In order to make δI = 0 in (2), we must consider the condition: G αβ + κT αβ = Λ g αβ , where Λ is the cosmological constant. Additionally, we must require that g αβ δR αβ = ∇ α δW α = δΦ, so that we obtain the constraint g αβ δΦ = Λ δg αβ , where we have used: g αβ δg αβ = −g αβ δg αβ . Then, we propose the existence of a tensor field δΨ αβ , such that δR αβ ≡ ∇ β δW α − δΦ g αβ ≡ δΨ αβ − δΦ g αβ = −κ δS αβ 1 , and hence where δΦ complies δΦ = 0. This means that exists a family of vector and tensor fields described by (3), that are related to the Einstein tensor transformationsḠ and leave invariant the action. The transformed Einstein equations with the equation of motion for the transformed gravitational waves, holdḠ with δΦ(x α ) = 0 and 2 The eq. (5) provides us the Einstein equations with cosmological constant included, and (6) describes the exact equation of motion for gravitational waves with an arbitrary physical source δS αβ inside a Gaussian-like hypersurface.
A very important fact is that the scalar field δΦ(x α ) appears as a scalar flux of the tetra-vector with components δW α through the closed hypersurface ∂M. This arbitrary hypersurface encloses the manifold by down and must be viewed as a 3D Gaussian-like hypersurface situated in any region of space-time. This scalar flux is a gravitodynamic potential related to the gauge-invariance of δW α and the gravitational wavesδ Ψ αβ . These waves appear by varying the Ricci tensor, as in the case of a flat background. However, in our case this variation is exact and was done in an arbitrary background. Other important fact is that since δΦ(x α ) = Λ 4 g αβ δg αβ , the existence of the Hubble horizon is related to the existence of the Gaussian-like hypersurface with an inner source. The variation of the metric tensor must be done in a Weylian-like integrable manifold [6] using an auxiliary geometrical scalar field θ, in order to the Einstein tensor (and the Einstein equations) can be represented on a Weyl-like manifold, in agreement with the gauge-invariant transformations (3).
In this letter we shall explore the possibility that the variation of the tensor metric must be done in a Weylian-like integrable manifold (defined in the next section) using an auxiliary geometrical scalar field σ, in order to the Einstein tensor (and the Einstein equations) can be represented on a Weyl manifold, in agreement with the gauge-invariant transformations (3). We shall study the relativistic quantum dynamics of σ by using the fact that Λ is a relativistic invariant. Finally, to illustrate the formalism, we shall work an example of a Reissner-Nordström (RN) black-hole in order to obtain the dynamic equation for gravitational waves.

II. WEYLIAN-LIKE REPRESENTATION OF THE EINSTEIN TENSOR
In the sense of the Riemannian geometry, the covariant derivative is null, so that ∆g αβ = g αβ;γ dx γ = 0, where we denote with ; the Riemann-covariant derivative. The Weyl geometry [9] is a generalization of the Riemannian geometry. In this letter we shall consider an alternative proposal to the Weyl covariant derivative, in which the metric tensor is also nonzero: g αβ|γ = φ ,γ g αβ . Here, the " |γ " denotes the new Weyl-like covariant derivative with respect to the Weyl-like [9] connections Γ α βγ , given by 34 This connections are very similar to the Weyl ones. In both cases the non-metricity in nonzero. The variation of the metric tensor in the sense of (8) 5 : δg αβ , will be 6 is the eigenvalue that results when we apply the operator δx α (x β ) on a background quantum state |B , defined on the Riemannian manifold 8 . We shall denote with a " hat " the quantities represented on the Riemannian background manifold. The Weylian-like line element is given by Hence, the differential Weylian-like line element dS provides the displacement of the quantum trajectories with respect to the "classical" (Riemannian) ones. When we displace with parallelism some vector v α on the Weylian-like manifold, we obtain where we have taken into account that the variation of v α on the Riemannian manifold, is zero: ∆v α = 0. Hence, the norm of the vector on the Weylian-like manifold is not conserved: δv α δS δvα δS = − σ αÛ α v γÛ γ (σ ν v ν ) = 0. From the action's point of view, the scalar field σ(x α ) drives a geometrical displacement from a Riemannian manifold to a Weylian-like one, that leaves the action invariant If we require that δI = 0, we obtain where δσ = σ µ dx µ is an exact differential and V = √ −ĝ is the volume of the Riemannian manifold. Of course, all the variations are in the Weylian-like geometrical representation, and assure us gauge invariance because δI = 0. 3 To simplify the notation we shall denote σα ≡ σ,α 4 The connections (8) could be generalised to other in which the torsion to be nonzero, but this issue will be studied in a future work. 5 In what follows we shall denote with a ∆ variations on the Riemann manifold, and with a δ variations on a Weylian-like manifold. 6 The reader can see using the constraint (7) and (8) that δΦ = − Λ 2 dσ. 7 We can define the operatorx such that b † k and b k are the creation and destruction operators of space-time, such that βěγěδ . 8 In our case the background quantum state can be represented in a ordinary Fock space in contrast with LQG, where operators is qualitatively different from the standard quantization of gauge fields.

A. Gauge-invariant relativistic dynamics on a Weylian-like manifold
The Ricci tensor in the Weylian-like and Riemann representations can be related bȳ so that both representations of the scalar curvature, are related bȳ The Einstein tensor can be written as where we have made use of the fact that the connections are symmetric.

B. Weylian action and quantum algebra
Now we consider the expression (4), from which we obtain that the invariant Λ. From the point of view of the Riemann manifold Λ is a constant, but from the point of view of the Weylian-like manifold: Λ ≡ Λ(σ, σ α ) can be considered a functional, given by Therefore, we can define a geometrical quantum action on the Weylian-like manifold with (18) such that the dynamics of the geometrical field is given by the Euler-Lagrange equations, after imposing δW = 0: where the variations are defined on the Weylian-like manifold. This means that δΛ = 0, but ∆Λ = 0. Furthermore, Π α = δΛ δσα = − 3 4 σ α is the geometrical momentum and the dynamics of σ describes a free scalar field so that the momentum components Π α comply with the equation If we define the scalar invariant Π 2 = Π α Π α , we obtain that where we have used that σ α U α = U α σ α , and with Θ α = Û α . Therefore we can define the relativistic invariant Θ 2 = Θ α Θ α = 2Û αÛ α , whereÛ α are the components of the Riemannian velocities. Additionally, it is possible to define the Hamiltonian operator such that the eigenvalues of "quantum energy" becomes from H |B = E |B . Can be demonstrated that δH = 0, so that the quantum energy E is a Weylian-like invariant.

C. Gravitational waves with Weylian-like variations
Now we consider the Weylian-like variation of the Ricci tensor: δR αβ . This is given bŷ whereˆ δΨ βγ ≡∇ α∇ α δΨ βγ =∇ α δΓ α βγ − δ α γ δΓ ǫ βǫ , δL αβ = δ δL δg αβ , such thatL = g αβL αβ , and we have considered that δI = 0 [6]. On Riemannian hypersurfaces all the field solutions are background solutions, so that we can consider that ∆ L αβ = 0. In this case we obtain where dx α =Û α dS, such that if we rename χ αβ = δΨ αβ δS , we finally obtain which describes the Riemannian dynamics of gravitational waves on a Weylian-like hypersurface 9 . Notice that the source depends on how the relativistic observer is moving on the Riemannian hypersurface, which is determined bŷ U α . A very important fact is that the source in the equation (28) depends on the field σ θ . The existence of this field in the source is intrinsically related to the existence of the scalar flux Φ, and the cosmological constant Λ. This can be seen by noting that where we have used the constraint δg αβ Λ = δΦ g αβ and (9). In the next section we shall consider the example of an RN black-hole, to obtain gravitational waves.

III. AN EXAMPLE: RN BLACK HOLE
We consider a RN black-hole, with massM = 2GM , and squared electric charge Q 2 , such that the line element is given by where dΩ 2 = sin 2 θ dφ 2 + dθ 2 is the square differential of solid angle and f (r) = 1 −M r + Q 2 r 2 , such thatM = 2GM (M is the mass of the charged black-hole),M 2 ≥ (2Q) 2 and Q = q 4πǫ0 . The horizon radius is given by In this case the Lagrangian density is given by[10] 9 It is possible to show that exists a Weylian-like dynamics of σ given by the Euler-Lagrange equation δΛ δσ − ∇α δΛ δσα = 0, such that ∇α denotes the Weylian-like covariant derivative. Can be demonstrated that the solutions of σ in the Weylian-like gauge can be expressed in terms of a Fourier expansion of travelling waves that moves with light velocity. This means that the solution of χ αβ (x α ) must be written as a superposition of travelling waves in the Weylian-like manifold.