Aspects of noncommutative ($1+1$)-dimensional black holes

We present a comprehensive analysis of the spacetime structure and thermodynamics of ($1+1$)-dimensional black holes in a noncommutative framework. It is shown that a wider variety of solutions are possible than the commutative case considered previously in the literature. As expected, the introduction of a minimal length $\sqrt{\theta }$ cures singularity pathologies that plague the standard two-dimensional general relativistic case, where the latter solution is recovered at large length scales. Depending on the choice of input parameters (black hole mass $M$, cosmological constant $\Lambda$, etc.), black hole solutions with zero, up to six, horizons are possible. The associated thermodynamics allows for the either complete evaporation, or the production of black hole remnants.

Figure 1

The black hole temperature as a function of the horizon radius $|x_{>}|$ meant as a dimensionless variable. The solid curve is the two horizon case $2\pi T/\hslash =|-\frac{1}{4}|x{|}_{>}+\frac{1}{4|x{|}_{>}}|$ for $C=1/2$ and $\Lambda =1$, which describes the temperature for $|x_{>}|\ge 1$ since $|x_0|=1$ in our units. The dotted curve corresponds to the single horizon case $2\pi T/\hslash =\frac{1}{4|x{|}_{>}}$, namely, for $C=1/2$ and $\Lambda =0$. Finally the dashed curve, $2\pi T/\hslash =|-\frac{1}{4}|x{|}_{>}|$ and $C=0$ and $\Lambda =1$, is a special case with two horizons, one of these at the origin.

Figure 2

Behavior of ${\alpha }_{\theta }(x)$ for parameter values (from left): $q=5$, $m=0.55$; $q=5$, $m=0.55$; $q=5$, $m=0.35$; and $q=5$, $m=0.176$. The dotted plot is the generic commutative solution for the parameters in question. The wide array of causal structures is evident: the left-most figure contains as many as six horizons, while the right-most contains two (degenerate) ones.

Figure 3

Behavior of ${\alpha }_{\theta }(x)$ for parameter values (from left): $q=1$, $m=2.5$; and $q=1$, $m=0.7$. The dotted plot is the generic commutative solution for the parameters in question.

Figure 4

Behavior of ${\alpha }_{\theta }(x)$ for parameter values (from left): $q=0.5$, $m=0.55$; and $q=0.5$, $m=0.45$. The dotted plot is the generic commutative solution for the parameters in question.

Figure 5

Behavior of ${\alpha }_{\theta }(x)$ for parameter values (from left): $q=5$, $m=-0.55$; $q=1$, $m=-0.55$; and $q=0.5$, $m=-0.55$. The dotted plot is the generic commutative solution for the parameters in question.

Figure 6

Behavior of ${\alpha }_{\theta }(x)$ for parameter values (from left): $q=1$, $m=0.055$; $q=1$, $m=0.55$; $q=10$, $m=0.55$; $q=10$, $m=0.75$; $q=10$, $m=0.45$; $q=10$, $m=0.25$; $q=10$, $m=0.05$; and $q=10$, $m=0.005$. The dotted plot is the generic commutative solution for the parameters in question.

Figure 7

Behavior of ${\alpha }_{\theta }(x)$ for parameter values (from left): $q=1$, $m=-0.55$; $q=10$, $m=-0.1$. The dotted plot is the generic commutative solution for the parameters in question.

Figure 8

Behavior of ${\alpha }_{\theta }(x)$ for parameter values (from left): $m=0.35$, $m=0.45$, $m=0.55$ and $m=-0.55$. The dotted plot is the generic commutative solution for the parameters in question.

Figure 9

Behavior of ${\alpha }_{\theta }(x)$ for parameter values (from left): $q=5$, $m=0.15$; $q=5$, $m=0.1$; $q=1$, $m=0.55$. The dotted plot is the generic commutative solution for the parameters in question.

Figure 10

Behavior of ${\alpha }_{\theta }(x)$ for parameter values (from left): $q=10$, $m=-0.95$; and $q=1$, $m=-0.95$. The dotted plot is the generic commutative solution for the parameters in question.

Figure 11

Behavior of $T/|C|$ for parameter values (from left): $q=5$, $q=\sqrt{2.5}$, $q=\sqrt{2}$ with $\Lambda /C>0$ and $q=10$, $q=1$ with $\Lambda /C<0$. The dotted plot is the generic commutative solution for the parameters in question.

Figure 12

Behavior of $T/|\Lambda |\theta$ for parameter values (from left): $\Lambda =0$ and $C=0$. The dotted plot is the generic commutative solution for the parameters in question.