Unitary symmetries in wormhole geometry and its thermodynamics

From a geometric point of view, we show that the unitary symmetries $U(1)$ and $SU(2)$ stem fundamentally from Schwarzschild and Reissner-Nordstr\"om wormhole geometry through spacetime complexification. Then, we develop quantum tunneling which makes these wormholes traversable for particles. Finally, this leads to wormhole thermodynamics.


I. INTRODUCTION
Einsiten-Rosen wormhole was introduced to understand the geometric meaning of mass and charge of the elementary particles in Ref. [1] and then was developed by many authors [2][3][4][5][6][7][8][9][10]. The geometric description of physical concepts was a cornerstone of several approaches to quantum gravity. These approaches include noncommutative geometry [11], string theory [12], loop quantum gravity [13] and twistor theory [14]. In this article, we focus our attention on a fundamental question: is there a conceptual connection between unitary symmetries and wormhole geometry? We argue that it is possible to find unitary symmetries, such as U(1) and SU (2), from Schwarzschild and Reissner-Nordström wormhole geometry through spacetime complexification if a new Euclidean metric on a complex Hermitian manifold is provided. This motivates us to compute quantum tunneling, which indicates that these wormholes could be traversable for particles. Finally, this allows us to introduce wormhole thermodynamics that is consistent with black hole thermodynamics [15,16].
The article is organized as follows. We start with the Schwarzschild wormhole geometry in Section II, and we connect its complex geodesics with U(1) and SU(2) symmetries by using spacetime complexification. We also provide a new Euclidean metric on a Hermitian complex manifold. In Section III, the massless exotic Reissner-Nordström wormhole geometry is also connected with the same unitary symmetries, and a discussion about the classical Reissner-Nordström wormhole geometry and the SU(3) symmetry is addressed. Quantum tunneling for particles is studied in Section IV and lead to wormhole thermodynamics.
Finally, concluding remarks are given in Section V.

II. SCHWARZSCHILD WORMHOLE GEOMETRY
It is historically known that Einstein and Rosen (ER) introduced their ER bridge, or wormhole idea, to resolve the particle problem in General Relativity (GR) [1]. The ER bridge contrives a geometric meaning of particle properties, such as mass and charge, in the spacetime, where mass and charge are nothing but bridges in the spacetime. The ER bridge idea can be summarized as follows. The Schwarzschild metric is given by where M > 0. It has both the physical singularity existing at r = 0 that cannot be removed, and the coordinate singularity at r = 2M that can be removed by choosing another coordinate system. Einstein and Rosen suggested a coordinate system which resolves the coordinate singularity at r = 2M by choosing the following transformation leading to 4u 2 du 2 = dr 2 . In the new coordinate system, one obtains for ds 2 the expression One may notice in this coordinate system that u will be real value for r > 2M and will be imaginary for r < 2M . As u varies from −∞ to ∞, one finds r varies from +∞ to 2M and then from 2M to +∞. In that sense, the 4−dimensional spacetime can be described by two congruent sheets that are connected by a hyperplane at r = 2M , and that hyperplane is the so-called "bridge". Thus, Einstein and Rosen interpreted mass as a bridge in the spacetime.
This draws our attention to look closely at the case when r < 2M , and consequently the variable "u" would have imaginary values in this region. The geodesics in the u−coordinate system will experience two different kinds in two different regions. In the region r > 2M , it would follow real trajectory, and it follows imaginary trajectory in the region r < 2M . But as we cannot "stitch" a real space and a complex space together, we prefer to complexify the whole spacetime. It might be enticing to impose real spatial indices to formulate complex geodesics as the spatial coordinate u is what motivates us to consider spacetime complexification, but the more wise choice is to cook one complex dimension from spatiotemporal dimensions and the other complex dimension from the leftover spatial dimensions. Also, we classify the geodesics based on real and imaginary parts in the two sheets of the wormhole.
This is crucial to develop a more consistent theory of gravity for the following reasons: • The manifold in GR is chosen to be pseudoRiemannian manifold [17], which is connected and guarantees the general covariance and continuous coordinate transformations on the manifold. But a basic question emerges: To what does the region r < 2M develop under diffeomorphisms? The answer should include that there must be a geometric structure, by covarience principle, that corresponds to the region in wormhole geometry.
• The physical singularity at r = 0 is irremovable by coordinate transformation in GR [18], which implies the importance of studying the region connected with r = 0, even in the coordinates that give wormholes, as it is likely to have a correspondence in wormhole geometry.
• In wormhole geometry, the u values become imaginary for r < 2M . Imaginary value in physics plays crucial rule in building unitary symmetries. We are interested to understand the effect of this imaginary region in wormhole geometry knowing that the role of complex numbers in QM is recognized as to be a central one [19].
In order to complexify a spacetime N , or to think of N → R 4 as M → C 2 , we introduce complex manifold M of two complex dimensions ζ and η. We consider a point p ∈ M so that p = (ζ, η) defines the complex coordinates in some local chart where the complex coordinates induce the parameter space of the real parameters (ζ 1 , η 1 , ζ 2 , η 2 ) on M. In that sense, the full geodesic in wormhole geometry would read such that g µν becomes Hermitian. We will come to the importance of this in a little bit.
But for now, we study the effect of the elements of a group G, as linear operators, on a complex manifold and the coordinate transformations related to G. Such operations define a set of homomorphisms from G to the general linear group GL(n, C), and such homomorphisms to the general linear group define an n−dimensional matrix representation.
The matrix representation is useful when it works on any manifold chart, i.e. without fixing the manifold's basis. In that sense, a matrix representation of G is a realization of G elements as matrices affecting an n−dimensional complex space of column vectors. Additionally, the change of the manifold's basis results in conjugation of the matrix representation of G. where H is a 2 × 2 Hermitian complex form of g µν = h µν + ik µν , i.e. g µν = g µν , and SU (2) is the special unitary subgroup. This guarantees the invariance of the Hermitian form We know that some Z ∈ GL(2, C) can be defined as the special linear subgroup SL ( [20]. A manifold M is biholomorphically equivalent to C when the holomorphic automorphisms Aut(M) of the manifold are isomorphic to Aut(C). Then, the action of SU(2) identifies the rotationally symmetric complex manifolds [21]. We are interested in the case when the SU(2)-orbit of p is the orthogonal group O p . In this case, and with the help of the conjugation of vectors by the complex structure, T p M C can be split into V ⊕ iV for any V ∈ T p M C [22]. Therefore, O p becomes real hypersurface orbits of M.
For the sake of convenience, it is suggested to represent SU(2) action in terms of coordinate charts at every point like Eq. (4). Now, the function ϕ : verifies ϕ(S 3 ) = SU (2), see for instance [23,Example 16.9]; and the details of finding the equivariant maps, that relate q ∈ S 3 to p ∈ O p of M, and CR-diffeomorphism structure of S 3 are in Ref [21]. The equivariant diffeomorphism f : establishes the correspondence between the parameters (ζ 1 , η 1 , ζ 2 , η 2 ) of the point p ∈ M and the unitary action of SU(2) on q = (z i , z 2 ) ∈ S 3 endowed from the fact that every unitary representation on a Hermitian vector space V is a direct sum of the irreducible representations of the group. This is crucial for finding the group orbit, or congruence classes, containing the conjugate subgroups of SU (2). It is known that setting one of the z i = 0 will define the other z j to be the longitude of SU (2) corresponding to the conjugate class T † HT , where H ⊂ SU (2). The equivariant diffeomorphism relates any H as a metric g µν to a diagonalizeble matrix H on S 3 using ϕ(z 1 , z 2 ), and a diagonalized H can be read as where for any wormhole sheet we define ξ = κ + iλ and z i = α i + iβ i , i = 1, 2 such that So, if we want to return back to the R 4 space, and for any ξ, the last transformation ushers us to define the real vector x := (x 1 , x 2 , x 3 , x 4 ) as This means the complex geodesics on the sheet 1 and sheet 2 are endowed with a SU (2) symmetry, which is guaranteed by the conjugacy T † HT that defines all longitudes of SU(2).
Also, this may introduce a connection between external geometry and internal symmetry, it may show us a geometric origin/meaning for the unitary symmetry in physics 1 . As we can notice, SU(2) symmetry for geodesics is local symmetry because the parameters (α 1 , β 1 , α 2 , β 2 ) depend on the position on wormhole. So, if the coordinates ζ and η render geodesics where ϑ 1 and ϑ 2 are continuous functions in x i , and α 2 1 + β 2 1 = 1, then SU(2) symmetry reduces to U(1). In that sense, SU(2) introduces local symmetry of complex vectors on the two sheets of wormholes, and U(1) symmetry introduces a local symmetry of the complex vector on the same sheet.
It is worth noting that S 3 is diffeomorphic with SU(2) [24, p. 127]. Moreover, S 3 can be seen as a fiber bundle following the diagram with the Hopf fibration plotted in Figure 1 and used in physics in [25,26] and for wormholes in [27]. Now, we are ready to complexify the wormhole metric. First, we rearrange Eq.
(3) such that it becomes Since we adopt the parameter u(r) as defined in Eq. (2), we also define the complex parameter ζ, its squared length, and any infinitesimal change in it as Eq. (20) gives Meanwhile Eq. (19) yields or In addition, the stereographic projection of r(x, y, z) on the complex plane of ζ(κ, λ) with Furthermore, set such that Finally, we substitute Eq. (25,26,28,29) to get which is not yet a manifestly Hermitian metric despite being a general 2−dimensional metric of such complex manifold.
In order to make the metric (17) Hermitian, we need to consider the following coordinate Then, the metric (17) becomes which is not Hermitian yet asũ ∈ C. However, eũ ∈ R indeed. In order to improve the previous metric into a Hermitian, we use the following Rindler-like coordinates together with Wick rotation and the complex coordinates such that the relevant metric becomes behave in the realm of complex geometry. This is an important result as it could help studying the wavefunction of wormholes upon analyzing the geometry as a Quantum Field Theory (QFT) in complex curved spacetime [30].

Remark 1. Consider the de Sitter-Schwarzschild metric
where Λ > 0. By using the results about depressed cubic equations given for instance in [31], the polynomial f (r) = − Λ 3 r 3 + r − 2M has three real roots if and only if Λ < 1 9M 2 . If Λ > 1 9M 2 , then f (r) has one real root and two complex conjugate roots. In wormhole geometry, real horizon means the possibility to measure beyond it, and complex horizon means the impossibility to measure beyond it.

III. REISSNER-NORDSTRÖM WORMHOLE GEOMETRY
In order to understand the geometric origin of the charge, Einstein and Rosen [1] investigated the following exotic Reissner-Nordström metric where M > 0 and Q > 0 for exotic matter with negative energy density. If we consider the following transformation it leads to u 2 du 2 = (r − M ) 2 dr 2 . In the new u coordinate system, one obtains for ds 2 the expression We have (r − M ) 2 = u 2 + M 2 + Q 2 . Consider the continuous and positive function u →f (u) defined by and one obtains for ds 2 the expression We find the coordinate u vanishes at the event horizons when r 1 = M − M 2 + Q 2 and In the u coordinate, the bridge at r = r 2 verifies r 1 < 0 < M < r 2 .
The metric (41) is defined properly until r = M and the singularity at r = r 2 is removed.
So, we obtain two regions for the first sheet: • u has imaginary value when r varies from 0 to r 2 ; • u has real value from 0 to +∞ when when r varies from r 2 to +∞.
Similarly, we have two regions for the other sheet. When 0 < r < M , the functionf (u) in the metric (41) must be replaced by − u 2 + M 2 + Q 2 + M . As in [1] and for sake of simplicity, we consider that M = 0. In that case, the metric (41) reduces to It is possible to obtain a metric very similar to (35) by using similar calculations, except that h and k become functions in Q but not in M . Calculations are left to the reader.

Remark 2. Fist, consider the classical Reissner-Nordström metric
where M > 0 and Q > 0. We choose that Consider the following transformation which gives u 2 du 2 = (r − M ) 2 dr 2 . In the new "u" coordinate system, one obtains for ds 2 the expression We have (r − M ) 2 = u 2 + M 2 − Q 2 . For r > M , and by using condition (44), we obtain with u →g(u) continuous and positive. In the new coordinate system, one obtains for ds 2 the expression We find the coordinate u vanishes at the event horizons when r 1 = M − M 2 − Q 2 and In the u coordinate, the bridge at r = r 2 verifies 0 < r 1 < M < r 2 .
The metric (48) is defined until r = M and the singularity at r = r 2 is removed. So, we obtain three regions for the first sheet: • u has real value from Q 2 to 0 when r varies from 0 to r 1 ; • u has imaginary value when r varies from r 1 to r 2 ; • u has real value from 0 to +∞ when when r varies from r 2 to +∞.
We also have three regions for the other sheet. When 0 < r < M , the functiong(u) must be replaced by − u 2 + M 2 − Q 2 + M in the metric (48). The situation is therefore different from those presented previously and additional studies will be necessary.
Then, we know that SU(3) follows the diagram where π is the projection, see for instance [ We also notice that if M 2 ≤ Q 2 , then the polynomial P (r) = r 2 − 2M + Q 2 is always positive such that the change of variable u cannot provide unitary symmetries. Contrary to the Schwarzschild and exotic Reissner-Nordström wormhole geometry, the classical Reissner-Nordström wormhole geometry implies the mass-charge condition (44) which is also used to avoid naked singularities [34, Section 12.6].

IV. QUANTUM TUNNELING AND WORMHOLE THERMODYNAMICS
Discovering unitary symmetries in wormhole geometry motivates us to explore the quantum properties of wormholes. Being traversable for a wormhole is a challenge, see for instance [8,9]. Wormholes are generally non-traversable for classical matter [6] but they can be modified to be traversable by removing event horizons, see [35][36][37][38] for Schwarzschild-like wormholes and [39,40] for Reissner-Nordström-like wormholes. We know that particles are subject to quantum tunneling, which makes Schwarzschild and Reissner-Nordström wormholes traversable for particles while keeping event horizons. A similar idea has been used since the seminal works of Bekenstein in [15] and Hawking in [16] for studying the black hole radiation. In this section, we develop quantum tunneling and wormhole thermodynamics by computing the Hawking temperature.

A. Schwarzschild wormhole case
First, we point out and observe an interesting fact about the radial null curves in the wormhole metric (3) by setting ds 2 = dθ = dφ = 0, yielding .
The above quantity defines the "coordinate speed of light" for the wormhole metric, and as we can see there is a horizon with a coordinate location u = 0 yielding The presence of the horizon implies that the quantum tunneling of particles from "another universe" to our universe can form Hawking radiation and, consequently, detecting particles by a distant observer located in our universe. We can study the tunneling of different massless or massive spin particles; and in the present work, we focus on studying the tunneling of vector particles. The motion of a massive vector particle of mass m, described by the vector field ψ µ , might be studied by the Proca equation (PE), which reads [41] where, from the metric (3), we define the determinant √ −g = 2u 2 (u 2 + 2M ) 2 sin θ, and The corresponding action is Then in any curvilinear coordinates, and using the Bianchi-Ricci identity ∇ [λ ψ µν] = 0, we get the true version of Eq. (52) as a QFT in curved spacetime equations of motion where R µν and R are the Ricci tensor and the Ricci scalar respectively, and C µνκλ is the trace-free conformal curvature tensor. Taking the flat limit g µν → η µν changes the essence of the last two equations to become Lorentz invariant.
Taking into the consideration the symmetries of the metric (3) given by three corresponding Killing vectors (∂/∂ t ) µ and (∂/∂ φ ) µ , we may choose the following ansatz for the action where E is the energy of the particle, and j and l denotes the angular momentum of the particle corresponding to the angles φ and ψ, respectively. If we keep only the leading order of , we find a set of four differential equations. These equations can help us to construct a 4 × 4 matrix ℵ, which satisfies the following matrix equation We solve for the radial part to get the following integral where and Now, there is a singularity in the above integral when u h = 0, meaning that F → 0. So in order to find the Hawking temperature, we now make use of the equation where u h = 0. In this way we find Using p ± u = ±∂ u R ± , for the total tunneling rate gives It is interesting that, for the black hole case, there is a temporal part contribution due to the connection of the interior region and the exterior region of the black hole. In the wormhole case, we don't have such a contribution. We can finally obtain the Hawking temperature for the wormhole by using the Boltzmann factor Γ = exp(−E/T ), and setting to unity, so that it results with The Hawking temperature can be found from [42] Applying this equation for the wormhole metric (66), we find the Ricci scalar and √ g = 2u. Setting = c = k B = 1, using the fact that the Euler characteristic of Euclidean geometry is χ = 1 at the wormhole horizon u h = 0, we solve the integral (67) and which coincides with the Hawking temperature (65) obtained via tunneling.
Here we shall consider a tunneling from massless RN wormhole geometry using metric (42).
For the radial null curve, and by setting ds 2 = dθ = dφ = 0, we obtain and therefore we see that the points u = 0 play the role of the horizon as du/dt → 0, provided that u = 0. This indicates that there could be a quantum tunneling associated with the horizon. To find the Hawking temperature, we can apply the WKB approximation given by Eq. (56) along with the action (57). Consequently, we construct a 4 × 4 matrix where, for the radial part, we get the following integral with and G(u) = u Q 2 + u 2 = G (u)| u=0 (u − u h ) + · · · (74) Now, there is a singularity in the above integral when u h = 0, meaning that G → 0. In order to find the Hawking temperature at u h = 0, we consider Using p ± u = ±∂ u R ± , the total tunneling rate gives Boltzmann factor Γ = exp(−E/T ) leads to define the temperature as Let's now derive the Hawking temperature using a topological method based on the Gauss-Bonnet theorem. To do so, we need to rewrite the metric (42) in a form of 2−dimensional Euclidean spacetime given by For the Ricci scalar, we obtain and √ g = u(u 2 + Q 2 ) −1/2 . At the wormhole horizon u h = 0, we obtain which coincides with the Hawking temperature (77) obtained via tunneling.

V. CONCLUDING REMARKS
We closely looked at Schwarzschild and Reissner-Nordström wormhole geometry and obtained unitary symmetries U(1) and SU (2) [44] and to the new experimental findings obtained from studying traversable wormholes/EPR pair entanglement within quantum computing regimes [45].
We hope to report on these important results in the future.