SO ( 3 , 2 )/ Sp(2) symmetries in BTZ black holes

We study global ﬂat embeddings, three accelerations and Hawking temperatures of the BTZ black holes in the framework of two-time physics scheme associated with Sp(2) local symmetry, to construct their corresponding SO ( 3 , 2 ) global symmetry invariant Lagrangians both inside and outside event horizons. Moreover, the Sp(2) local symmetry is discussed in terms of the metric time-independence.


Introduction
There have been tremendous progresses in lowerdimensional black holes associated with the string theory since an exact conformal field theory describing a black hole in two-dimensional space-time was proposed [1]. Especially, the (2 + 1)-dimensional Banados-Teitelboim-Zanelli (BTZ) black holes [2][3][4] have enjoyed lots of successes in relativity and string communities, since thermodynamics of higher-dimensional black holes can be interpreted in terms of the BTZ back hole solutions. In fact, the dual solutions of the BTZ black holes are related to the solutions in the string theory, so-called (2 + 1)-black strings [5,6].
On the other hand, the higher-dimensional global flat embeddings of the black hole solutions are sub-E-mail address: soonhong@ewha.ac.kr (S.-T. Hong). jects of great interest both to mathematicians and to physicists. In differential geometry, it has been well known that four-dimensional Schwarzschild metric is not embedded in R 5 [7]. Recently, (5+1)-dimensional global embedding Minkowski space structure for the Schwarzschild black hole has been obtained [8] to investigate a thermal Hawking effect on a curved manifold [9] associated with an Unruh effect [10] in these higher-dimensional space-time. It has been also shown that the uncharged and charged BTZ black holes are embedded in (2 + 2) [8] and (3 + 2) dimensions, 1 while the uncharged and charged black strings are embedded in (3 + 1) and (3 + 2) dimensions [12], respectively. Note that one can have two time coordinates in these embedding solutions to suggest so-called two-time physics [13]. Historically, the two-time physics was formulated long ago when the (3 + 1) Maxwell theory on a conformally invariant (4 + 2) manifold was constructed [14]. Recently, the two-time physics scheme has been applied to M theory [15] and noncommutative gauge theories [16].
In this Letter we will investigate symmetries involved in the BTZ black hole embeddings such as SO (3,2) global and Sp(2) local symmetries. In Section 2, we will study complete embedding solutions, three accelerations and Hawking temperatures "inside and outside" the event horizons in the framework of the two-time physics and then construct the SO(3, 2) global symmetry invariant Lagrangians associated with these embedding solutions in Section 3. The Sp(2) local symmetry will also be discussed in terms of the metric time-independence. In Appendix A, we will revisit the charged BTZ black hole to construct its minimal embedding solution.

Complete flat embedding geometries
We first briefly recapitulate the global flat embedding solution given in [8], for the (2 + 1) rotating BTZ black hole [2,3] which is described by 3-metric (2.1) where the lapse and shift functions are respectively. Note that for the nonextremal case there exist two horizons r ± (J ) satisfying the following equations, in terms of which we can rewrite the lapse and shift functions as follows Here one notes that this BTZ space originates from anti-de Sitter one via the geodesic identification φ = φ + 2π . The (2 + 2) minimal BTZ global flat embedding ds 2 = −(dz 0 ) 2 +(dz 1 ) 2 +(dz 2 ) 2 −(dz 3 ) 2 is then given by the coordinate transformations for r r + as follows In the following we will construct complete embedding solutions, three accelerations and Hawking temperatures "inside and outside" the event horizons.

Case I: r r +
In the two-time physics scheme [13], we consider global flat embedding structure for the (2 + 1) rotating BTZ black hole whose metric is now given by (2.6) ds 2 = η MN dX M dX N , With the canonical momenta p µ conjugate to x µ given as we can construct to satisfy the Sp(2) local symmetry associated with the two-time physics, 2 (2.11) which, using the identification for x ± (2.13) can be rewritten as for r r + Exploiting the ansatz 3 for cosh R and sinh R we can reproduce the BTZ metric (2.1) and the (3 + 2) global flat embedding for r r + to yield the identification between the standard global flat embedding (2.5) and that of two-time physics: last symmetry condition P M P M = 0 will be discussed later. 3 In Ref. [17], there appears a brief sketch on the BTZ embedding outside the horizon in the two-time physics scheme, without explicit construction of cosh R and sinh R in (2.15), for instance.
Here one notes that X 2 = l does not contribute the minimal global flat embedding since it is constant, and this coordinate X 2 serves to fulfill the Sp(2) symmetry (2.11). Next, introducing the Killing vector ξ = ∂ t − N φ ∂ φ we evaluate the three acceleration , and the Hawking temperature [9] (2.18) , which are consistent with the fact that the a 4 is also attainable from the relation [9] (2.19) a 4 = k g 1/2 00 with the surface gravity k.

Case II: r − r r +
Next, we consider the global flat embedding of the BTZ black hole in the range of r − r r + by exploiting a little bit different choice for X M (2.20) which satisfies the Sp(2) local symmetry (2.11). As in the previous section, differentiating X M in (2.20) yields for r − r r + ds 2 = −l 2 dR 2 + l −2 r 2 + sin 2 R + r 2 − cos 2 R dt 2 (2.21) Exploiting the ansatz for cos R and sin R we can obtain the (3 + 2) global flat embedding for r − r r + Next, similar to the r r + case, with the Killing vector ξ = ∂ t − N φ ∂ φ we evaluate the three acceleration , and the Hawking temperature

Case III: r r −
Similarly, for the case of the range inside the inner horizon, r r − , we introduce (2.26) With the ansatz for cosh R and sinh R (2.28) we can construct the (3 + 2) global flat embedding for r r − Next, similar to the above cases, with the Killing vector ξ = ∂ t − N φ ∂ φ we evaluate the three acceleration , and the Hawking temperature This completes the full global flat embeddings of the BTZ black hole in the two-time physics scheme. Note that the above (3 + 2) BTZ embedding solutions are consistent with those in [3] where they obtained the (2 + 2) BTZ embeddings in the standard global flat embedding scheme, without considering the Sp(2) local symmetry (2.11) and the corresponding SO(3, 2) global symmetry invariant Lagrangian construction, which will be discussed in the two-time physics approach [13] in the next section.

Case I: r r +
Now, in the global flat embedding (2.16) for r r + , we consider the Lorentz generators of the SO(3, 2) symmetry which, using the conjugate pair (X M , P M ) in (2.7) and (2.10), yields

2)
On the other hand, in the two-time physics the SO(3, 2) Lagrangian is given by which, exploiting the conjugate pair (X M , P M ) in (2.7) and (2.10), yields Here the conjugate momenta p µ is defined in (2.9) and the metric g µν is given by (3.6) g µν = 1 l 2 diag − cosech 2 R, 1, sech 2 R . The above Lagrangian (3.5) can also be rewritten as with x µ defined in (2.9) and the inverse metric g µν . After some algebra, we obtain the transformation rule for the A 22 as below from which, together with the above transformation rules (3.3), one can easily see that the Lagrangian (3.4) is SO(3, 2) global symmetry invariant.

3.2.
Case II: r − r r + Next, we consider the global flat embedding (2.23) for r − r r + . As in the previous case, constructing P M conjugate to X M in (2.20) as below which satisfies the Sp(2) local symmetry (2.11), we can obtain the Lorentz generators of the SO(3, 2) global symmetry to yield the classical SO(3, 2) global symmetry transformations for x µ in (2.9) On the other hand, inserting the conjugate pair (X M , P M ) in (2.20) and (3.9) into the Lagrangian (3.4), we obtain the metric

Case III: r r −
Finally, we consider the (3 + 2) global flat embedding for r r − . Exploiting X M in (2.29) and constructing its conjugate momenta P M as below Inserting the Lorentz generators (3.15) into the classical SO(3, 2) global symmetry transformation rules On the other hand, substituting the conjugate pair (X M , P M ) in (2.26) and (3.14) into the Lagrangian (3.4), we obtain the metric so that we can construct the SO(3, 2) global symmetry transformations for A 22 In the region inside the inner horizon where X M and P M are given by (2.29) and (3.14), as in the previous cases, we can thus obtain the SO(3, 2) global symmetry invariant Lagrangian under the transformation rules (3.16) and (3.18). Now it seems appropriate to comment on the Sp(2) local symmetry associated with the two-time physics, for the BTZ global embedding solutions. To be more specific, we consider X M and P M for r r + in (2.7) and (2.10) to evaluate (3.19) P M P M = l −2 −p 2 + cosech 2 R + p 2 R + p 2 − sech 2 R .
Exploiting the metric g µν in (3.6), we can rewrite p µ in (2.9) in terms of theẋ µ , Here one notes that, due to the BTZ metric timeindependence, the SO(3, 2) Lagrangian or Hamiltonian vanishes, which is a characteristic of the twotime physics [13].

Conclusions
In conclusion, in the framework of two-time physics scheme, we have explicitly obtained the global flat embeddings, three accelerations and Hawking temperatures of the BTZ black holes both inside and outside the event horizons by exploiting the Sp(2) local symmetry. Moreover, we have constructed the SO(3, 2) global symmetry invariant Lagrangians associated with these BTZ black hole embedding solutions. The Sp(2) local symmetry has been also discussed in terms of the metric time-independence.