Causal Structure and Thermodynamics
Penrose diagrams, Cauchy horizon instability, and the laws of black hole mechanics
Penrose Diagram: Sub-Extremal Case
The global causal structure of the sub-extremal Reissner-Nordstrom spacetime (\( |Q| < M \)) is dramatically different from that of Schwarzschild. To construct the Penrose (conformal) diagram, we must first find coordinates that extend through both horizons.
We begin with the tortoise coordinate. For the RN metric with\( f(r) = (r - r_+)(r - r_-)/r^2 \), the tortoise coordinate is:
$$r^* = \int \frac{dr}{f(r)} = r + \frac{r_+^2}{r_+ - r_-}\ln|r - r_+| - \frac{r_-^2}{r_+ - r_-}\ln|r - r_-|$$
The tortoise coordinate diverges logarithmically at both horizons
Introducing null coordinates \( u = t - r^* \) and \( v = t + r^* \), and then performing Kruskal-like extensions across each horizon, one obtains the full maximal analytic extension. The resulting Penrose diagram has a remarkable structure:
Key features of the Penrose diagram:
- Infinite chain of asymptotic regions: Unlike Schwarzschild (which has two exterior regions), the maximally extended RN spacetime contains an infinite sequence of asymptotically flat regions, connected by tunnels through the black hole interior.
- Timelike singularity: The singularity at \( r = 0 \)is timelike, not spacelike as in Schwarzschild. It appears as vertical lines in the Penrose diagram, and timelike observers can avoid hitting it.
- Cauchy horizon \( r_- \): The inner horizon is a Cauchy horizon — the boundary of the domain of dependence of any spacelike Cauchy surface. Beyond \( r_- \), the future evolution is not uniquely determined by initial data.
- White hole regions: Each black hole region has a corresponding white hole region, as in the Schwarzschild case, but repeated infinitely.
The coordinate transformation for the outer horizon extension uses Kruskal-like coordinates:
$$U_+ = -e^{-\kappa_+ u}, \qquad V_+ = e^{\kappa_+ v}$$
Kruskal coordinates at the outer horizon, with \( \kappa_+ = (r_+ - r_-)/(2r_+^2) \)
Similarly, to extend across the inner horizon, one introduces:
$$U_- = e^{\kappa_- u}, \qquad V_- = -e^{-\kappa_- v}$$
Kruskal coordinates at the inner horizon, with \( \kappa_- = (r_+ - r_-)/(2r_-^2) \)
Note the sign flip relative to the outer horizon extension — this reflects the fact that\( r \) changes from timelike (between \( r_- \) and \( r_+ \)) back to spacelike (inside \( r_- \)). The patching of these coordinate charts generates the infinite tower structure of the Penrose diagram.
Cauchy Horizon Instability: Mass Inflation
While the Penrose diagram suggests that an observer falling through a charged black hole could pass through the Cauchy horizon and emerge in a new asymptotic region, this conclusion is almost certainly unphysical. The Cauchy horizon of the RN black hole is violently unstable.
The instability was first identified by Penrose (1968) and Simpson-Penrose (1973), and its nonlinear development was analyzed by Poisson-Israel (1990) in the mass inflation scenario. The mechanism is as follows:
Consider a perturbation (e.g., infalling radiation) that crosses the event horizon. As this radiation approaches the Cauchy horizon, it experiences an infinite blueshift. The blueshift factor between the outer and inner horizons is:
$$\frac{\omega_\text{inner}}{\omega_\text{outer}} \sim e^{\kappa_+ v} \to \infty \quad \text{as} \quad v \to \infty$$
Infinite blueshift at the Cauchy horizon due to the exponential peeling of null generators
The energy density measured by a freely falling observer near the Cauchy horizon diverges:
$$T_{\mu\nu}k^\mu k^\nu \sim e^{2\kappa_+ v} \to \infty$$
The energy flux through the Cauchy horizon diverges exponentially in advanced time
Through the Einstein equations, this divergent energy creates a divergent curvature. The Poisson-Israel analysis shows that the Misner-Sharp mass function\( m(r, v) \) (which reduces to the ADM mass in the exterior) inflates exponentially:
$$m(r, v) \sim m_0\,\exp\left[\kappa_+\,v\,|\text{flux}|\right] \to \infty$$
Mass inflation: the local mass parameter diverges at the Cauchy horizon
Despite this, the metric remains continuous (though not \( C^1 \)) across the Cauchy horizon — leading to a weak null singularity rather than a strong curvature singularity. Whether an observer can survive passage through this singularity remains debated. The resolution likely requires a quantum theory of gravity.
Black Hole Thermodynamics for Reissner-Nordstrom
The RN black hole provides the cleanest setting for studying black hole thermodynamics with an additional "chemical" degree of freedom — the electric charge. The thermodynamic variables associated with the outer horizon are:
Temperature
$$T = \frac{\kappa_+}{2\pi} = \frac{r_+ - r_-}{4\pi\,r_+^2} = \frac{\sqrt{M^2 - Q^2}}{2\pi\left(M + \sqrt{M^2 - Q^2}\right)^2}$$
Entropy (Bekenstein-Hawking)
$$S = \frac{A_+}{4G} = \frac{4\pi\,r_+^2}{4G} = \frac{\pi\,r_+^2}{G} = \frac{\pi\left(M + \sqrt{M^2 - Q^2}\right)^2}{G}$$
One quarter of the event horizon area, as for all black holes
Electric Potential
$$\Phi = \frac{Q}{r_+} = \frac{Q}{M + \sqrt{M^2 - Q^2}}$$
The electrostatic potential at the horizon, measured relative to infinity
The First Law of Black Hole Mechanics
For the RN black hole, the first law of black hole mechanics takes the form:
$$\boxed{dM = T\,dS + \Phi\,dQ}$$
First law: analogous to \( dE = T\,dS + \mu\,dN \) in ordinary thermodynamics
This can be verified explicitly. From \( r_+ = M + \sqrt{M^2 - Q^2} \) and\( S = \pi r_+^2 \):
$$dS = 2\pi\,r_+\,dr_+ = 2\pi\,r_+\left(\frac{r_+}{r_+ - r_-}\,dM - \frac{Q}{r_+ - r_-}\,dQ\right)$$
Differentiating the entropy as a function of M and Q
Solving for \( dM \):
$$dM = \frac{r_+ - r_-}{4\pi\,r_+^2}\,dS + \frac{Q}{r_+}\,dQ = T\,dS + \Phi\,dQ \quad \checkmark$$
The electric potential \( \Phi \) plays the role of a chemical potentialconjugate to the charge \( Q \). The first law tells us that the mass of a charged black hole can increase either by adding heat (entropy at fixed charge) or by adding charge (at fixed entropy). This is the gravitational analogue of the thermodynamic identity for a system with a conserved charge.
Extremal Limit and the Third Law
As \( |Q| \to M \), the two horizons approach each other (\( r_+ \to r_- \to M \)), and the temperature vanishes:
$$T = \frac{\sqrt{M^2 - Q^2}}{2\pi\,r_+^2} \xrightarrow{|Q| \to M} 0$$
The extremal black hole has zero temperature: \( T_\text{extremal} = 0 \)
However, the entropy does not vanish in this limit:
$$S_\text{extremal} = \frac{\pi\,M^2}{G} \neq 0$$
A macroscopic ground-state degeneracy — the extremal entropy is finite and nonzero
This connects to the third law of black hole thermodynamics. The Nernst version of the third law states that \( T \to 0 \) implies\( S \to 0 \). The RN black hole appears to violate this! The resolution is that the third law, in the black hole context, should be stated as the unattainability version: it is impossible to reach\( T = 0 \) (the extremal state) in a finite number of operations. Israel (1986) proved that attempting to charge a near-extremal black hole to extremality requires either infinite time or violates the weak energy condition.
The nonzero ground-state entropy \( S_\text{extremal} = \pi M^2/G \) is deeply meaningful. In string theory, it counts the degeneracy of microstates: for certain extremal black holes (especially in \( \mathcal{N} = 2 \) supergravity), the microscopic entropy computed from D-brane counting reproduces the Bekenstein-Hawking formula exactly (Strominger-Vafa, 1996). This remains one of the most striking successes of string theory.