Josephson junction formed in the wormhole space time from the analysis for the critical temperature of Bose-Einstein condensate

In this study, considering some gas in the Morris-Thorne traversable wormhole space time, we analyze the critical temperature of the Bose-Einstein condensate in the vicinity of its throat. As a result, we obtain the result that it is zero. Then, from this result, we point out that an analogous state to the Josephson junction is always formed at any temperatures in the vicinity of its throat. This would be interesting as a gravitational phenomenology.


Introduction
The issues concerning the wormhole space times as the solution and the phenomena of these are one of the problems that have been investigated so much until now. Since this paper will treat a phenomenological issue in the wormhole space times, we refer some phenomenological works on the wormhole space times; gravitational lensing [1,2,3,4,5,6,7,8], shadows [9,10,11], observation [12], Casimir effect [13], teleportation [14], collision of two particles [15], and creation of a traversable wormhole [16].
However, in these years, studies regarding wormhole space times are performed so energetically. Its reason would be that recently it has turned out that the Einstein-Rosen bridge (ER bridge) [17] gives some kinds of the EPR pair [18,19,20,21], which plays very important role in the literature of the information paradox [22]. It would be also its reason that there are common features and behaviors between AdS 2 wormholes and the SYK models, from which currently we can perform various interesting studies [23,24,25,26].
Also, there are studies to construct the graphene wormhole in the material physics from some brane configurations in the superstring theory [27,28]. From these studies, Chern-Simon current in the graphene wormhole is studied in [29].
In this study, considering the Morris-Thorne traversable wormhole (traversable wormhole) [30], we consider some situation that some gas fills the whole space time of that, where it is assumed Bose-Einstein condensate (BEC) can be formed at some temperature in this gas. Then, we point out that an analogous state to the Josephson junction is always formed at any temperatures except for zero in the vicinity of its throat.
For this, we first analyze the critical Unruh temperature of the gas regarding BEC in the Rindler, then check that it can agree to the critical temperature obtained in the flat Euclid space.
The motivation of this is to check the rightness of our analytical method to obtain the critical temperature of BEC in curved space times. Actually, our analytical method in the curved space time is different from the usual one in the flat Euclid space [31] to treat curved space times. However, since the space time for the accelerated system can be regarded as the flat space with the temperature same with the Unruh temperature if Euclideanized (see the text given under (4)), the critical Unruh temperature obtained in the Rindler space should agree to that in the flat Euclid space.
Then, considering the same gas in the traversable wormhole, we analyze its critical temperature concerning BEC in the vicinity of its throat.
Here, it is known that if the number of its spatial dimensions is 2 or less in the flat Euclid space, the effective potential always gets diverged for the contribution of the infrared region and BEC is not formed 1 . Then, since the spatial part of the vicinity of the throat in the traversable wormhole space time becomes effectively 1 dimensional (as 1 In the 1 + d dimensional flat Euclid space, the particle density at the critical point of BEC is given ) in [31], where µ = m. Then we can see this is diverged for the contribution of small k, if d = 1 or 2. At this time, the critical temperature becomes 0 [32], which means the state is normal at any temperature.
for this point, see (68), then just take the limit: r → v). we can expect that the critical temperature of BEC in the vicinity of the throat is always zero. (Of course, what the dimension effectively becomes 1 depends on the coordinate system we take. Therefore, surely we could expect the zero critical temperature in the vicinity of the throat if the dimension becomes 1, however it would be considered it is not its essential reason. We will give comment on our result in the end of Sec.4.6 from this point of view.) Then, • If the critical temperature is always zero in the vicinity of the throat, since the throat exists in the form to separate the wormhole space time from the another side of the wormhole space time, the normal state also appears in the form to separate the wormhole space time from the another side of the wormhole space time like  The dotted line represents the throat of the wormhole. The yellow and blue parts represent the normal and superconductor states we expect to appear, respectively. As the normal state is appearing in the form to separate the space time, we can expect that an analogous state to the Josephson junction is formed in the vicinity of the throat.
• The far region of the wormhole space time is asymptotically flat space time. Therefore, the critical temperature of BEC in the far region is given by the one calculated in the flat space time.
• Then, considering by extrapolating between the results in the throat region and the far region, we can obtain an expectation that an analogous state to the Josephson junction is formed at any temperatures in the vicinity of the throat.
Since this could be considered as a gravitational phenomenology, this would be interesting. This would be also interesting because we can consider a possibility that the Josephson current flows from the one side of the wormhole to another side. Of course, the wave function of the current may be damped when it tunnels, and the Josephson current does not exist in practice. Currently we could not say any explicit things on this point from the analysis in this study, which is a future work. We discuss this in Sec.5.
The wormholes we can consider would be the ER bridge and the traversable wormhole space time. Then, since the ER bridge is, briefly saying, given by patching two Schwarzschild black holes together, the Josephson current will flow out from the event horizon. Hence, it would be lightlike, and could not be observed at any points in the outside of the event horizon (in the regions I and III in the Penrose diagram) except for the future timelike infinity. Therefore, it might not be suitable to consider the ER bridge if we consider about the Josephson current.
The traversable wormhole is free from this problem as we can depict some timelike flow of the current going across the throat part in Fig.1. However it needs the exotic matter (some matter to violate the energy condition) [33] such that the traversable wormhole space time can satisfy the Einstein equation, which we explicitly show in Sec.4.2. This is a critical problem as it means that the traversable wormhole space time is not realizable realistically. For some recent studies on the exotic matter, see [34,35,36,37,38,39,40,41]. In this study, we consider the traversable wormhole upon knowing this problem. If we care this problem in this study, we could mention that it appears for some reason and can exist for some time, then this study is the one for that time.
We mention the organization of this paper. We analyze the critical Unruh temperature of the gas regarding BEC in the Rindler space (Sec.2) in the Euclid space (Sec.3) in the traversable wormhole space time (Sec.4). Then in Sec.5, we point out that an analogous state to the Josephson junction is formed in the vicinity of its throat. In Sec.5, we obtain the phase structure for the BEC/normal states transition in the ER bridge. In Appendix.B, we note some points in the mechanism for the formation of BEC in this study.
2 Critical temperature of BEC in accelerating system 2.1 Rindler coordinates and Unruh temperature As a theoretical model, we consider some gas that entirely fills the Minkowski spacetime and performs an uniformly accelerated motion with an acceleration a for one direction. Based on (1) 2 , we denote the coordinates of the gas as which are a kind of the Rindler coordinates, and at ξ = 1, ζ becomes the proper time of the object performing the uniformly accelerated motion a. Then, the squared line 2 Solving d dt ( mv √ 1−β 2 ) = ma with the condition v = 0 at t = 0 and the general relation between the Minkowski time and proper time: τ = t 0 dt 1 − β 2 , the trajectory of an accelerated motion can be obtained as where we can check with the t and x above, we can obtain ( mv √ 1−β 2 ) = mat. Toward this, there are several ways for how to take the coordinate system along an accelerated motion [42]. element can be given as follows: where ρ ≡ a −1 ξ and τ ≡ aζ.
Here, (y, z) are common with those in the Minkowski coordinates and we use the notation x ⊥ ≡ (y, z) in what follows. Note that we take not a sphere coordinates but a plane coordinates for the x ⊥ -direction. Euclideanize the coordinate system with τ → iτ , and regard it as periodic. Then, from the no conical singularity condition: circumferential length radius ρ=0 = 2π, the period in the ζ-direction can be determined as 2π/a. Therefore, the gas in the Rindler coordinate system with the proper time ζ can be would be at the following temperature: which agrees to the Unruh temperature.
Here, in (3), when we put ρ to 1, dρ 2 vanishes. However, let us define dρ 2 as the squared line element perpendicular to the original ρ-direction. Then, since the period of τ -direction is 2π, the period in terms of ζ becomes 2π/a. Hence, when we put ρ to 1, (3) can be regarded as the D = 1 + 3 flat Euclid space with the Euclid finite temperature T E = 2π/a for the one with the proper time ζ. Therefore, a space time for the accelerated system can be considered to be equivalent to the flat space with the temperature same with the Unruh temperature if Euclideanized, so that the critical Unruh temperature obtained in the Rindler space would be considered to agree to that in the flat Euclid space.
We can see from (4) that locating at different ρ means having different Unruh temperature. Therefore, we should be careful when we perform the ρ-integral. For this point, we will consider that the Rindler coordinates are individually applied for each particle comprising the gas, and all the particles will have the same value with regard to ρ. As for the treatment of ρ in the analysis in this study, we will consider the effective potential at some ρ as in (33) 3 .

Hamiltonian in finite density, probability amplitude and Euclideanization
We start with the following Lagrangian density for the complex scalar field for the particles comprising the gas that we mention in Sec.2.1 that fills the whole space: where φ = 1 √ 2 (φ 1 + iφ 2 ) (φ 1,2 are real scalar fields), Indices µ, ν and g µν refer to the Rindler coordinates (η, ρ, x ⊥ ) and the metrices in (3).
With (9), we can write the probability amplitude as where I means the whole Rindler wedge I, γ R ≡ − det g µν and we have performed the following redefinitions for the canonical momenta: (11b) C R is given as Since C R is some number irrelevant of µ R and we finally take the derivative with regard to µ R to get the e.v. of the particle density, we can ignore C R in our analysis in what follows.
We perform Euclideanization: At this time, where there is no changes in the contents between g τ τ and g ηη except for the notations.
Here, the Euclidean temperature is identical with the Unruh temperature T U in (4).
There is a problem that the temperature depends on the space. As for this, it would be considered that ρ is always fixed to some value corresponding to the fact particles are performing an uniformly accelerated motion. Under the Euclideanization (13) with (14), Z in (10) can be rewritten as where

Description for BEC and upper limit of chemical potential
In order to express the fields in the superconductor state, we separately rewrite φ 1,2 as the e.v. part and the original φ 1,2 like the following [31]: where α and Θ mean the absolute value and the phase of e.v., and α = 0 corresponds to the normal state, = 0 corresponds to the BEC state.
Rewriting Z in (15) employing the expressions in (16), we can obtain the following Z: Now, let us look at the contribution from the zero mode in the pass-integral part in (18), which we can write as where the configurations generated by Dφ ′ 1 Dφ ′ 2 are only the constant configurations irrelevant of the coordinates, and V I means Then we can see that (20) can converge when M 2 R is positive, whereas diverge when M 2 R is negative or zero. Therefore, we can write as Z in (18) converges for M 2 From (20), we can see that there is the upper bound for the value that the chemical potential can take as Here, the one above is evaluated at ξ = 1, therefore, we can see from (3) that the critical Unruh temperature obtained from the following analysis with (21) is associated with the one having ζ as its proper time and performing the uniformly accelerated motion a.
Here, we explain how the BEC is formed in our system. When decreasing the Unruh temperature from some high temperatures (therefore, α is 0) keeping the e.v. of the particle density constant, it turns out that the chemical potential should rise (see (59)). However, as in (21), there is the upper limit for the value the chemical potential can take. Therefore, finally α should start to have some finite value to keep the e.v. of the particle density constant. Like this, at some lower Unruh temperature, α becomes finite and BEC is formed. (For more description on this issue, see Appendix.B.)

Effective potential (1)
We can diagonalize the shoulder in Z in (18) as whereĜ R± ≡Ĝ R ± 2g ηη µ R and 1 The difference arisen in the path-integral measure by the transformation U is just some constant, which we can ignore.
We express φ α (α = 1, 2) by the plain wave expansion for the x ⊥ -directions remaining the ρ-direction as (22) is given as 4 (22) can be given as where the general formula: β −1 β 0 dτ e iτ (ωm−ωn) = δ mn has been used. Then, performing the functional integral forφ ′ α,n , we can get the following Z: where from (26) to (27), we have rewritten the integral dρ as ρ ∆ρ (ρ takes all the real numbers from 0 to ∞), and Det is the one with regard to the k ⊥ -space for each ρ, where we have written the reason for why we separate by each ρ in Sec.2.1.
Defining the free energy F R as Z = exp(−β R F R ), 4Ĝ 0 appears in (22) as follows: where where ω n = n.
This is because τ is periodic with the period 2π as can be seen (14c), therefore ω n is given as ω n = 2πn/β R (β R = 2π). (Here, the Euclidean temperature is identical with the Unruh temperature (4)) (29) includes some operator, but for now we suppose that it is some numbers.
Considering that the (29) divided by ρ is the effective potential for the particles performing a uniformly accelerated motion determined by ρ, we consider to express (30) is given as some operator. In this subsection, we obtain its expression as numbers. For this purpose, we define D ±,∆ 2 (ρ, ω n , k ⊥ ) as should satisfy the following identity: where the operator in l.h.s. in (35) is taken from (29). From (35), we can obtain the relation that D ±,∆ 2 should satisfy as Based on (36), we obtain D ∆ 2 (ρ − ρ ′ , k η , k ⊥ ) in what follows. 5 Log G R appears in (29) as follows: To obtain D ∆ 2 (ρ − ρ ′ , k η , k ⊥ ), we focus on the fact thatF is the operator of the following eigenvalue equation with eigenvalue (iλ) 2 : where K α (x) is the second kind modified Bessel function, and Θ λ (ρ, k) satisfies the following normalized orthogonal relation 6 : λ is real number, therefore Θ λ (ρκ) can form a set of infinite dimensional orthogonal system. Then, by taking Θ λ (ρκ) as a set of the orthogonal bases, let us formally write where f λ,± are the coefficients of each independent direction, Θ λ (ρκ), specified by λ, which are to be obtained in what follows.

Effective potential (2)
In the previous subsection we have obtained some concrete expression of D ∆ 2 (ρ − ρ ′ , k η , k ⊥ ) as in (44). Using it, we can write Γ where Considering to perform the derivative with regard to µ R , we pick up the µ R -dependent parts in Ψ by expanding it around µ R = 0 9 as dkk. 9 In this section we have performed the expansion around µ R = 0 as in (48). However, if we expanded around µ R = µ c R , we can obtain the following effective potential: Therefore, there is no difference in the e.v. of the particle density we can obtain finally, .
where 10 , With (48), we can write Γ We are going to finally assign the critical value µ c R = m/a c in (20) to µ R in (48), then take the leading order in the expansion of µ c R . For this, now we use two symbols, µ R and µ ′ R : • µ R : chemical potential on which the derivative regarding µ R can act; finally the value m/a c is assigned, • µ ′ R : just a symbol for the value m/a c R , on which the derivative regarding µ R does not act on.
Then, Ψ 0 and a 2 µ 2 R K 2 iλ (κρ) can be expanded with regard to µ ′ R as Using (52), we can obtain Γ (ρ) R at the critical moment given by (48) as where we have used  (54) 10 We have calculated (50b) and (50c) using the following formula: First, we can see that (54) is diverged if it is evaluated as it is. Therefore, we consider to do some regularize toward (54) 11 . For this purpose, we consider to pull out some constant in coth(πλ). Therefore, expressing coth(πλ) as we exclude "1". Then, once putting the upper limit of the integral as Λ we perform the integral, then we get as where Λ → ∞ finally. Then excluding "− 1 12 " in (56), we take the Λ to ∞. As a result we get as Then, using this result, we can give the effective potential Γ

Critical Unruh temperature for BEC
We can obtain the e.v. of the particle density according to From (59), we can see that if we decrease the Unruh temperature from some high temperatures fixing the e.v. of the particle density to some constant, the value of the chemical potential should rise. Let us consider to reach the critical Unruh temperature by decreasing the acceleration gradually from some high accelerations where BEC is not formed. Then, from the explanation under (21), we can obtain the critical acceleration from (59) as 12 where α = 0 and µ c = m/a c have been assigned in (59). Here, note the comment given under (21). Using the relation between Unruh temperature and acceleration, a = 2πT U , we can obtain the critical Unruh temperature as T U = 3d R /m. This result is consistent with the critical temperature for BEC obtained from the different analytical method in the D = 1 + 3 flat Euclid space at finite temperature [31]. 11 The divergences also appear in other works for the critical acceleration for the spontaneous symmetry breaking [45,46,47] and the D = 1 + 3 Euclid space at finite temperature [31]. In these, the divergences are ignored supposing that some regularization, e.g. a mass renormalization, the UV-cutoff and so on, could work, though it is not shown explicitly. 12 (60) can be written in the MKS units as a c = 2 π c 3 3d R /m ≈ 16.763 √ 2π d R /m [cm/s 2 ], where m is the mass of the particle comprising the gas and d R is its density.

Critical Unruh temperature of BEC in flat Euclid space at finite temperature
In this section, we obtain the critical temperature in the D = 1 + 3 flat Euclidean space at finite temperature from our analysis in the previous section just by exchanging the space time for the D = 1 + 3 flat Euclidean space at finite temperature. Therefore, as the calculation way in this section is basically the same with that in the previous section, we in this section describe only the points in the case of the D = 1 + 3 flat Euclidean space at finite temperature.

Exchange the back ground space for Euclidean space
First, we exchange the Rindler space (3) for the flat D = 1 + 3 Euclidean space at finite temperature, which can be done by 1) putting ρ to 1, then 2) Euclideanizing τdirection like (13) then periodizing it by the arbitrary period β E . The (3) in which these 2 manipulations are performed can be written as where the τ -direction is periodic with the arbitrary period β E .

Effective potential
Employing (61), we proceed with the analysis in previous section. Then, the following Γ E can be obtained instead of (33): where and K α (x) mean the second-kind Bessel function, and as for ω n , we use the same notation with the one in the previous subsection. From M 2 E above, we can see that the upper bound of the value that the chemical potential in the case of the flat D = 1 + 3 Euclidean space at finite temperature is given as Expanding (62) to the second-order of the value of the critical chemical potential in the same way we have done in Sec.2.6, we can obtain the following Γ E : where µ ′ E is the same meaning with that in Sec.2.6, but in the current case, µ ′ E is given by m corresponding to (64). Now, we evaluate ∞ 0 dλ λ coth(β E λ/2) in (65). If we performed the integration as it is, it would be diverged. Hence, we do some regularization. As coth x can be rewritten as 1 + 2/(e 2x − 1), we subtract the constant "1" as some regularization, then evaluate it as ∞ 0 dλ λ (coth(β E λ/2) − 1) = 2π 2 /3β 2 E . Using this, we can obtain the following Γ E :

Critical temperature
From (66), calculating the density according to d E = − ∂Γ E ∂µ E , then putting α = 0 and µ c E = m corresponding to the critical moment, we can obtain the relation between the temperature and density at the critical moment as This can agree to the critical temperature for BEC in the D = 1 + 3 flat Euclid space at finite temperature in [31], however our way to obtain this result is different from [31].

BEC in the traversable wormhole space time 4.1 Traversable wormhole space time
In this section, we consider the traversable wormhole given from considering two of the following space time: then attaching the parts of r = v of these [33]. Therefore, the position of the throat is located at r = v and the range of r in one side of the wormhole space time is v ≤ r ≤ ∞, (the throat is located at r = v).
In order that the wormhole (68) can satisfy the Einstein equation, the exotic matter, some matter to violate the energy condition, is needed [33] as shown in the next section, however we do not consider it in this study.

On the fact that matter to violate the energy condition is needed
(68) leads to the following Einstein tensor: From the result above, we can see that the energy density of our scalar field should be negative, which means that our scalar field is tachyonic or to break the energy condition. However, since such tachyonic matter cannot exist, (68) cannot be a solution realistically. Therefore, (68) will not be realized. This study treats (68) upon knowing this. If we care this problem in this study, we could mention that the wormhole space time appears for some reason and exists for some time, then this study is the one for that time.
Indeed, this problem is general in the context of the traversable wormhole space time. For the references for the recent studies on this, see Sec.1.

Probability amplitude and its Euclideanization
In the case that the wormhole (68) is taken as our space time, the probability amplitude corresponding to (8) is given as where i = r, θ, φ and j µν refer to the metrics in (68) and γ w ≡ − det j µν . C w is ignorable as well as Sec.2.2. We perform the Euclideanization, At this time, then S 1 compactify the τ -direction with the period β w . As a result, the momenta in the τ -direction can be written as and it can be considered that we can consider the gas at the temperature given as β −1 w in the space given by the space part of (68). We write φ α (α = 1, 2) as As a result, Z in (71) can be written as From the M 2 w above, we can see that the upper bound of the value of the chemical potential in the case of the traversable wormhole at finite temperature is

Effective potential in the vicinity of the throat
In the same way with (16), putting φ α (α = 1, 2) as then proceeding with the calculation from (76) in the same way with Sec.2.4, we can obtain the following Z: where V 3 ≡ dx 3 γ w .
We switch our viewpoint from the entire region from r = 0 to ∞ to only the vicinity of the throat. Therefore, replacing the r in γ wĜw in (79) with v + r, we take r up to r 0 -order in γ wĜw in (79) as whereL 2 is the squared angular-momentum operator: Also, V 3 is given as In what follows, we proceed with our analysis by focusing on the vicinity of the throat, where the r in what follows refers from 0 to some infinitesimal number.
We perform the expansion for the (θ, φ)-directions of φ α (α = 1, 2) with the spherical harmonics asφ Then, proceeding with the calculation from (79) by the same way with Sec.2.4, we can obtain the free energy, Z = exp(−β w F w ), as where r in the summation takes all the real numbers from 0 to ∞, and We finally obtain the critical temperature at some points where r is near 0, so we take the contribution at each r in (84) as where r in (86) takes some values near 0.

H ∆ 2 (r, n, l)
H ∆ 2 (r, n, l) in (85) can be defined as the one to satisfy the following equation: Performing the Fourier expansion for the r-direction iñ D ∆ 2 (r, n, l) and δ(r) as it can be seen thatJ (k) is given as (r 0 is taken to 0 from the positive), where r 0 is taken to some values near 0 and ik 0 is located in the upper half-plane in the complex plane (see Fig.2). Now we have obtainedJ ∆ 2 (k) as in (89), we evaluateH ∆ 2 (r, n, l) in (88). We use the residue theorem for this, then we can see • when r > 0, the contribution from the path of the upper semi circle, C + , is ∞, the lower semi circle, C − , is 0, • when r < 0, the contribution from the path of the upper semi circle, C + , is 0, the lower semi circle, C − , is ∞, • when r = 0, the contribution from the paths of C ± are ∞.
where C ± and the position of k 0 are sketched in Fig.2. Finally, we can obtain as where Θ(−r) is the step function (it is 1 or 0 for negative or positive r), and our r 0 is some positive values near 0. Now we have obtainedD ∆ 2 (r, n, l) as in (90). Then, in (86), performing the integration with regard to ∆ 2 , we can obtain the following Γ w :

Particle density and critical temperature in the vicinity of the throat
From (91), according to d w = −dΓ w /dµ w , we can obtain the e.v. of the particle density as where considering we close from some high temperatures to the critical temperature, we have put α to 0 as well as Sec.2.7 and 3.3. Now, we get the critical temperature in the vicinity of the throat. For this purpose, we first expand d w in (92) around r = 0 assuming v is some finite number. Then, we apply µ c w = m. At this time, the temperature is at the critical temperature, so we can write β w as β c w . Then, we can write (92) as Then, once treating the summations are taken to infinity finally), we evaluate these summations. Then, we can obtain the critical temperature as We can see that this T c w goes to 0 when N 0 and L 0 are sent to ∞, where • Temperature T w is Euclid temperature as in around (72), v is assumed as some finite number, and r is measured from v (see around (80)), • we have taken the part for small r in the effective potential as in Sec.4.4, which we consider to be effective for the phenomena in the vicinity of the throat. Then from the analysis of it, we have obtained the result as in (94).
Lastly, let us mention on the physical reason for the result (94), which may be considered that the dimension of the space becomes effectively 1 in the vicinity of the throat (see Eq.(68)). However, it depends on the coordinate system we take. Actually, the space can be 3 dimensional even in the vicinity of the throat, if we perform a coordinate transformation to ζ given as r = v + ζ 2 /2v. After all, the physical reason of the result (94) would be that contribution from the infrared region gets diverged for the curvature of the space in the vicinity of the throat, independently of the coordinate system we take.
Next, we comment on the coordinate dependence of the value of the critical temperature (94). Generally, the value of the Euclidean temperature would be changed up to the definition of the time coordinate. However, the result T c w = 0 (β c w = ∞) at r = 0 would not be changed as long as we do not consider some special transformation using some inverse of r toward t. Therefore it would be considered the critical temperature is always 0 as long as some inertial systems are taken (see footnote in Sec.1).
5 Phase structure of the normal/BEC states in the traversable wormhole space time and Josephson junction formed in vicinity of its throat

Phase structure
In the previous section, we have obtained the result that the critical temperature of BEC in the gas in the vicinity of the throat is 0 as in (94). On the other hand, the far region of our wormhole space time (68) is asymptotically flat, and we have obtained the critical temperature of BEC in D = 3 + 1 Euclid flat space time at finite temperature as in (67). Summing up these results: • T c w = (67) at r = ∞ (far flat region). Interpolating between these two results, we can sketch a phase structure like Fig.3. From Fig.3, we can see that an analogous state to the Josephson junction is formed at any temperatures in the vicinity of the throat.
Then, the question, whether the Josephson current is flowing or not in the vicinity of the throat, would arise, which we discuss in the next subsection.

On Josephson current to flow in the vicinity of the throat
The typical scales for the largeness of the Josephson junction and current in the laboratories would be roughly, Then, since Josephson current is a kind of the tunneling, it is considered that Josephson current would be more damped by the exponential as the width of the vicinity of the throat gets greater 13 . Therefore, if the width of the normal state in the vicinity of 13 The ratio of the transmitted wave to the incident wave in the one-dimensional space with the potential barrier written in every textbook for the quantum mechanics is given as Transmitted wave Incident wave where ε ≤ V 0 and α = 2m(V 0 − ε)/ 2 , and V 0 and L mean the height and width of potential barrier. ε and m mean the energy and the mass of the particles given as the wave function. To answer this problem, we have to analyze the width of the normal state in the vicinity of the throat and how much the wave function is damped when it tunnels. We cannot say any explicit things about this from the analysis in this paper.
Normally considering, the wormhole is the astronomical object, therefore the width of the normal state is very larger than (95). Hence, the wave function of the Josephson current would be damped and vanish when it goes through the normal state in the vicinity of the throat.
However, we could not exclude the possibility that it is not much damped for some effect of the curved space. Actually, there is a thought that the Hawking radiation is a kind of the tunneling [48]. Hence, if the Hawking radiation exists, the Josephson current might also exist as the same tunneling phenomenology.
One approach to this issue is to analyze the thermal de Broglie wavelength in the curved space time, which would be a future works.

Summary
In this study, we have investigated the phase structure for the BEC/normal states transition in the traversable wormhole space time filled by some gas to form BEC up to temperature.
The first idea to make this work begin is something like the consideration we mentioned in Sec.1 , and we mention the points in the mechanism for the formation of BEC in this study in Appendix.B.
As a result, we have obtained the result that the critical temperature of the gas for BEC is zero in the vicinity of the throat. Then, based on that result, we have pointed out that an analogous state to the Josephson junction is always formed in the vicinity of the throat.
There is the problem of the exotic matter in the traversable wormhole space time as shown in Sec.4.2. This is a critical problem from the standpoint of the realizability of the Josephson junction in this study. For some references for this problem, see Sec.1, and we in this study could care this problem by the consideration that it appears for some reason and can exist for some time, then this study is the one for that time. As for the analysis for how long time it can exist, it might come to some analysis for the unstable modes in the classical perturbations on the wormhole space time like the analysis for the Gregory-Laflamme instability.
If we could get these, we could reach the stage to discuss the realizability of the Josephson junction in this study, at which we should care the following 2 practical problems: 1) Effect of the strong tidal force to the existence of our Josephson junction formed at the vicinity of the throat, and 2) how to actually create the situation where the space is filled by the gas. If we could finally clear these problems, we might consider our Josephson junction, realistically.
The result in this study means that the state of the gas is changed from the normal state to the superconductor state at some point in the space. Investigating where it is and how it is are future works. In addition, the result in this study should be independent of the coordinate system we take as mentioned in the end of Sec.4.6. Checking it is also the future work.
As one of the interesting examples in which our Josephson junction would play a very intriguing role, author considers some scalar-gravity system, where the scalar field is supposed to form BEC up to the temperature, c.f. [49], on the following Euclidean five-dimensional traversable wormhole space time (ds 2 4 part is represented in Fig.4): as an effective model for the expanding early universe including the previous universe collapsing to the beginning of the current universe; the space time given by (97) corresponds to the shape of the space time for the two universes joined by the throat part corresponding to the beginning of the current universe. Postponing the explanation for (97) to the next paragraph, we first say that it seems we could get the five-dimensional traversable wormhole space time as a solution regardless of the values of the cosmological constant if it comes to the five dimension (c.f. [51]). This is because the uniqueness theorem is the theorem in the four dimension. In (97), S 3 -direction given by r 2 dΩ 2 3 corresponds to the three-dimensional spatial part we exist, and r-direction parameterizes the time development of that S 3 space. t E -direction is the originally the time-direction, which is now being S 1 compactified into the imaginary direction and prescribes the temperature of the four-dimensional part of ds 2 4 (c.f. [52]). Hence, the space given by ds 2 4 part corresponds to the four-dimensional space time we exist, which is applied to the curved surface in Fig.4, and is at some temperature determined by the period of the t E -direction. The scalar field to form BEC up to the temperature also exists on the curved surface in Fig.4.
The three-dimensional space at the beginning of the current universe corresponds to the throat part at r = v in (97), which is not singular, therefore it is considered that the space time at the beginning of the cosmology is regularized in this model. This point is one of the points in this model as the effective model for cosmology.
Our Josephson junction is supposed to be formed in the vicinity of the beginning of the cosmology. It would be interesting if the boundary between the superconductor and normal phases in our study can relate to Big Bang, which is another point in this model (at this time, the superconductor region would correspond to the inflation era).
In conclusion, supposing we could get the five-dimensional traversable wormhole space time as a solution, it would be interesting to examine whether or not the effective model above can reproduce the picture of the early cosmology described by the standard cosmology and give some solutions for the unsettled problems in our current cosmology (c.f. [53]).

A Derivation of (38)
The second kind modified Bessel function can be written using the first kind modified Bessel function J α (x) as [54] Using this, it can be written as We can check A 1 + A 2 = 0 for α = β, therefore (99) is 0 for α = β.
Next, for α = β, performing the Wick rotation as α → iα, the part of A 1 can be calculated as Putting β as β → α +∆α where ∆α is taken to 0 finally, the part of A 2 can be calculated as  (97) to be used as the effective early cosmological model in the idea written in the body text (For how to obtain this figure, see [50]). The curved surface in this figure corresponds to the four-dimensional space time we exist; blue circle represents its S 3 spatial space (Ω 3 -direction), and red direction is r-direction and parameterizes the time development of that S 3 spatial space. Here, throat part is the S 3 space at r = v, which corresponds to the three-dimensional space at the beginning of the current cosmology. We can see it is not singular, therefore the space at the beginning of the cosmology is regularized. In this figure, t E -direction is not included; it is originally the time-direction, however which is now being S 1 compactified into the imaginary direction and prescribes the temperature of the four-dimensional space depicted in this figure. Hence, the four-dimensional space depicted in this figure is supposed to be at some temperature. It would be interesting if the boundary between the superconductor and normal phases in our study can relate to Big Bang, where at this time, the superconductor region corresponds to the inflation era.

B Mechanism for the formation of BEC in this study
In this appendix, the technical points in the mechanism of the formation of BEC in this study is mentioned. First of all, the fundamental thought in our model could be considered as the one in the usual fundamental model of BEC like the one given in [31], and what we have done is to apply it to curved space times. Therefore, the fundamental form of our Hamiltonian is given by the one in the grand canonical ensemble, H − µN.
Then, if naively considering, it appears that we may be able to move only the chemical potential, only the number of particles or only temperature without the rests of these not moved. However, this is wrong. The structure in the microscopic states in the ground canonical ensemble is very complex, and these are closely related each other as can be seen from the Bose distribution function (meaning of variables are mentioned in the following), Therefore, this appendix begins from the derivation of (105) as confirmation. Then based on it, the technical points in the mechanism of the formation of BEC in this study is mentioned.
where N,i means N 0 N =0 i , and M, E 0 and N 0 are constants. Now, consider the Γ space (the space that has canonical variables of each of N 0 particles in the whole of M copied systems as its coordinates). Then, each infinitesimal region in the Γ space corresponds to a set of M N,i , and if we check each infinitesimal region of some region, we would find that there are a number of the same sets M N,i (for the meaning of "the same", read it out from the W in (107)). Here, let us suppose the Ergodic hypothesis (probability that each M copied system takes some one of M N,i states is always the same in the region determined by N 0 and E 0 ) is held in the Γ space.
As a result, the most appearing "the same" set of M N,i is considered to be the set for the thermal equilibrium state. Therefore, let us obtain the most appearing set of M N,i in the region determined by N 0 and E 0 in the Γ space. This problem is to obtain the set of M N,i which maximizes the following W : Then, using the method of Lagrange multiplier and Stirling's approximation, we can finally express all such the M N,i at once as where M, E 0 and N 0 are supposed to be very large positive integers to use Stirling's approximation, and β and µ mean the inverse temperature and chemical potential. (108) means the probability that the system with N particles and the energy E N,i appears. From here, let us suppose that the particles in the system follow the Bose statistics. As a result, the energy values are discretized and the number of particles to take each energy value is no limited. Then we can rewrite and replace as where "lowest" and "highest" mean those of the discretized energy levels, " {nr} " means to produce all the sets of n r satisfying N = r n r , and e r mean the values of the energy labeled by r that each particles takes. Therefore, finally Ξ in (109) can be given as Ξ = where ε r ≡ e r − µ and r means highest r=lowest . Then, it is known that N 0 N =0 {nr} can be treated as r nr , namely we can independently perform the summation for each n r in the range r n r ≤ N 0 . At this time, if N 0 is infinity, On the other hand, if any one of ε r ≤ 0, Ξ gets diverged. Now, we obtain the e.v. of n r , which can be written as n r = n r exp[−β r ε r n r ] exp[−β r ε r n r ] , where above mean N 0 N =0 {nr} r . Since n r = − 1 β ∂ ln Ξ ∂εr , therefore using (113), we can obtain n r given in (105), and we can get a key fact in the formation of BEC in this study, the number of the particles is increased when the chemical potential is increased.

B.2
Keep density of gas constant in decreasing temperature, and for this purpose, increase chemical potential, then finally formed BEC described by the constants