Introduction
At a fixed time τ (for example τ = 0) the 3d Euclidean BTZ black hole BM [1,2] of mass M > 0 reduces to a 2d spatial slice whose metric
is easily transformed to a Poincare metric on the upper half-plane
Moreover, the quotient
of H+ by a subgroup
of
generated by a parabolic element γ (ie. trace γ = ±2) has for M= 0 the structure of a 2d black hole vacuum [3]. We indicate a realization of this vacuum by way of an explicit bijection
, where
is a tractroid surface with a deleted boundary circle of radius a.
The spatial slice of BM
BM, with zero angular momentum, is given by the metric with periodicity in the Schwarzschild variable φ
ds2 solves the Einstein vacuum field equations
with negative cosmological constant
, where
in (2) is a positive constant. By our sign convention, the Ricci scalar curvature R in (3) is given by
.
in the introduction is therefore given by
which by way of the transformation of variables
in case M= 0 reduces to the Poincare metric
on H+ in (1). Specially for XΓ, we choose
for
set of whole numbers,
, where the linear fractional action of
on H+ is restricted to Γ:
which by (5) is consistent with the above Schwarzschild periodicity:
.
Construction of the map
; the main observation
The tractroid Ta of radius a > 0 of interest is the surface of revolution about the y-axis of the tractrix curve parametrized as follows:
for t ≥ 0. Ta is therefore the set of points S (u,v) in
given by
for
. Since
(as
is Ta minus points on the boundary circle S (0,v), as mentioned in the introduction.
Let
denote the quotient map that takes
to its Γ -orbit
in (8) and define
by
where we note that since
,
indeed
by (11). Then
is defined by the commutativity of the diagram
that is
(13)
for u > 0. For
in XΓ and
again,
such that
, which shows that
is surjective. Finally,
is also injective and thus indeed is a bijection. Namely, if
, such that
ie.
(by (13)), then
,
for some
(by (10)); ie.
.
Discussion
The BTZ vacuum (or ground state)
has a single parabolic generator γ in (7). In [4], for example, a BTZ vacuum with two parabolic generators is considered - in addition to other QFT matters. It would be interesting to find, also, a concrete geometric realization of the latter vacuum - or that of higher dimensional BTZ black hole vacua. One could also discuss the naked singularity case where M < 0.
Conclusion
The map
in (13) provides for a concrete, geometric, tractroid representation (or model) of the Euclidean BTZ vacuum XΓ with Poincare metric in (6); Γ is given by (7). This result is the best possible in the sense that a general result of D.Hilbert [5] prevents the full mapping of all of Ta onto XΓ. Our discussion proceeded at a fixed time τ = 0, in which case the black hole metric (2) was reduced to the 2d spatial slice (4). One could also consider the 2d metric obtained by fixing the Schwarzschild variable φ in (2), and study the false vacuum decay for this 2d black hole background. Compare the interesting references [6-8], for example, where the studies therein are of a quite different focus since the word ”vacuum” here simply means that we take the black hole mass M= 0 in (2). In [6], for example, the effective potential is considered for various values of the black hole mass. Also here, we need the Schwarzschild variable φ to be non-fixed in order to derive the Poincare metric version (6) of (4) in case M= 0, where (6) can actually be transformed to a metric on the tractroid. Thus issues regarding expectation values of quantum fluctuations and mass spectra, for example, do not arise in the present context, where in fact the periodicity of φ, moreover, which leads to equation (8), is crucial for the main construction of the bijection
.
In addition to the 2d vacuum black hole-tractroid correspondence that we have constructed, there is also a 2d wormhole-catenoid correspondence. In the reference [9] a 2-dimensional section of a 3-dimensional wormhole is realized as a catenoid surface – the section is obtained by fixing a spherical polar coordinate value: θ = π/2.