A tractroid realization of a 2d black hole vacuum

The two-dimensional black hole vacuum obtained from a spatial slice of the BTZ black hole is mapped explicitly to a tractroid surface minus a bounding circle.

Introduction

At a fixed time τ (for example τ = 0) the 3d Euclidean BTZ black hole B_M [1,2] of mass M > 0 reduces to a 2d spatial slice whose metric $d{s}_0^2$ is easily transformed to a Poincare metric on the upper half-plane

$H^{+}\stackrel{def.}{=}\{\left(x,y\right)\in {ℝ}^2|y>0\}.\text{ (1)}$

Moreover, the quotient $X_{\Gamma }\stackrel{def.}{=}\Gamma \setminus H^{+}$ of H⁺ by a subgroup $\Gamma$ of $G=SL\left(2, ℝ\right)$ 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 $\tilde{\Phi }:T_a^{+}\to X_{\Gamma }$ , where $T_a^{+}$ is a tractroid surface with a deleted boundary circle of radius a.

The spatial slice of B_M

B_M, with zero angular momentum, is given by the metric with periodicity in the Schwarzschild variable φ

$d{s}^2=\left(\frac{r^2}{{\ell }^2}-M\right)d{\tau }^2+{\left(\frac{r^2}{{\ell }^2}-M\right)}^{-1}d{r}^2+r^{2}d{\varphi }^2.\text{ (2)}$

ds² solves the Einstein vacuum field equations

$R_{ij}-\frac{1}{2}R{g}_{ij}-\Lambda g_{ij}=0\text{ (3)}$

with negative cosmological constant $\Lambda \stackrel{def.}{=}-1/{\ell }^2$ , where $\ell$ in (2) is a positive constant. By our sign convention, the Ricci scalar curvature R in (3) is given by $R=6/{\ell }^2$ . $d{s}_0^2$ in the introduction is therefore given by

$d{s}_0^2\stackrel{def.}{=}\frac{d{r}^2}{\frac{r^2}{{\ell }^2}-M}+r^{2}d{\varphi }^2\text{ (4)}$

which by way of the transformation of variables

$x=\varphi ,y=\ell /r>0\text{ (5)}$

in case M= 0 reduces to the Poincare metric

$d{s}_P^2\stackrel{def.}{=}{\ell }^2\left(\frac{d{x}^2+d{y}^2}{y^2}\right)\text{ (6)}$

on H⁺ in (1). Specially for X_Γ, we choose

$\Gamma \stackrel{def.}{=}\left\{\left[\begin{array}{cc}1& 2\pi n\\ 0& 1\end{array}\right]|n\in \mathbb{Z}\right\}=\left\{{\gamma }^n|n\in \mathbb{Z}\right\}\text{ (7)}$

for $\mathbb{Z}=$ set of whole numbers, $\gamma \stackrel{def.}{=}\left[\begin{array}{cc}1& 2\pi \\ 0& 1\end{array}\right]$ , where the linear fractional action of $SL\left(2, ℝ\right)$ on H⁺ is restricted to Γ:

$\left[\begin{array}{cc}1& 2\pi n\\ 0& 1\end{array}\right]\cdot \left(x,y\right)\stackrel{def.}{=}\left(x+2\pi n,y\right),n\in \mathbb{Z}\text{ (8)}$

which by (5) is consistent with the above Schwarzschild periodicity: $\left(x,y\right)\sim \left(x+2\pi n,y\right)$ .

Construction of the map $\tilde{\Phi }:T_a^{+}\to X_{\Gamma }$ ; the main observation

The tractroid T_a of radius a > 0 of interest is the surface of revolution about the y-axis of the tractrix curve parametrized as follows:

$\begin{array}{l}x\left(t\right)\stackrel{def.}{=}a{e}^{-t/a},y\left(t\right)\stackrel{def.}{=}\\ a\text{log}\left(e^{t/a}+\sqrt{e^{2t/a}-1}\right)-a{e}^{-t/a}\sqrt{e^{2t/a}-1}\text{ (9)}\end{array}$

for t ≥ 0. Ta is therefore the set of points S (u,v) in ${ℝ}^3$ given by

$\begin{array}{l}S\left(u,v\right)\stackrel{def.}{=}\left(x\left(u\right)\text{cos}v,x\left(u\right)\text{sin}v,y\left(u\right)\right)\stackrel{def.}{=}\\ \left(a{e}^{-u/a}\text{cos}v,a{e}^{-u/a}\text{sin}v,S\left(u\right)\right),\\ S(\text{u})\stackrel{\text{def}\text{.}}{\text{=}}\text{y}\left(\text{u}\right)\stackrel{\text{def}\text{.}}{\text{=}}\text{alog}\left({\text{e}}^{\frac{\text{u}}{\text{a}}}\text{+}\sqrt{{\text{e}}^{\text{2u/a}}\text{-1}}\right){\text{-ae}}^{\text{-u/a}}\sqrt{{\text{e}}^{\text{2u/a}}\text{-1}}\text{ (10)}\end{array}$

for $\left(u,v\right)\in {ℝ}^2$ . Since $S\left(0,v\right)=\left(a\text{cos}v,a\text{sin}v,0\right)$ (as $S\left(0\right)=0),$

$T_a^{+}\stackrel{def.}{=}\left\{S\left(u,v\right)\in T_a|u>0\right\}\text{ (11)}$

is T_a minus points on the boundary circle S (0,v), as mentioned in the introduction.

Let $q:H^{+}\to X_{\Gamma }$ denote the quotient map that takes $\left(x,y\right)$ to its Γ -orbit $\widetilde{\left(x,y\right)}$ in (8) and define $\Phi :H^{+}\to T_a^{+}$ by

$\Phi \left(x,y\right)\stackrel{def.}{=}S\left(\text{log}\left(\frac{y}{a}+1\right),x\right)\text{ (12)}$

where we note that since $y,a>0$ , $u=\text{log}\left(\frac{y}{a}+1\right)>0\Rightarrow$ indeed $\Phi \left(x,y\right)\in T_a^{+}$ by (11). Then $\tilde{\Phi }:T_a^{+}\to X_{\Gamma }$ is defined by the commutativity of the diagram

that is $\tilde{\Phi }S\left(\left(u,v\right)\right)\stackrel{def.}{=}\text{q}\left(v,a\left(e^u-1\right)\right)$ (13)

for u > 0. For $\widetilde{\left(x,y\right)}=q\left(x,y\right)$ in X_Γ and $u=\text{log}\left(\frac{y}{a}+1\right)>0$ again, $a\left(e^u-1\right)=a\left(\frac{y}{a}+1-1\right)=y\Rightarrow p=S\left(u,x\right)\in T_a^{+}$ such that $\tilde{\Phi }\left(p\right)\stackrel{def.}{=}q\left(x,y\right)$ , which shows that $\tilde{\Phi }$ is surjective. Finally, $\tilde{\Phi }$ is also injective and thus indeed is a bijection. Namely, if $p_j=S\left(u_j,v_j\right)\in T_a^{+},j=1,2$ , such that $\tilde{\Phi }\left(p_1\right)=\tilde{\Phi }\left(p_2\right)-$ ie. $q\left(v_1,a\left(e^{u_1}-1\right)\right)=q\left(v_2,a\left(e^{u_2}-1\right)\right)$ (by (13)), then $v_1=v_2+2\pi n$ , $a\left(e^{u_1}-1\right)=a\left(e^{u_2}-1\right)$ for some $n\in \mathbb{Z}(by (8))\Rightarrow u_1=u_2,\text{ cos}{v}_1=\text{cos}{v}_2,\text{ sin}{v}_1=\text{sin}{v}_2\Rightarrow S\left(u_1,v_1\right)=S\left(u_2,v_2\right)$ (by (10)); ie. $p_1=p_2$ .

Discussion

The BTZ vacuum (or ground state) $X_{\Gamma }$ 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 $\tilde{\Phi }$ in (13) provides for a concrete, geometric, tractroid representation (or model) of the Euclidean BTZ vacuum X_Γ 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 X_Γ. 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 $\tilde{\Phi }$ .

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.

Many thanks to Yaping Yuan for her careful and excellent assistance (as usual) in preparing this manuscript.

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