horizon generators, andkais the tangent vector to the horizon generators with respect toλ. The Killingvectorξais given on the horizon byξa=κλka(with a certain choice for the origin ofλ). Using the Einsteinequation we thus have∆M=(κ/8πG)ZRabkakbλ dλdA(2.3)=(κ/8πG)Zdρdλλ dλdA(2.4)=(κ/8πG)Z(-ρ)dλdA(2.5)=(κ/8πG) ∆A.(2.6)The second equality uses the focusing equation neglecting the quadratic termsρ2andσ2, the third usesintegration by parts with the boundary term dropped since the black hole is initially and finally stationary,and the last equality follows directly from the definition ofρ.2.1.4Second and Third LawsContinuing with the analogy, theSecond Lawis of course Hawking’s area theorem, stating that the horizonarea can never decrease assuming Cosmic Censorship and a positive energy condition. TheThird Lawalsohas an analog in black hole physics, namely, the surface gravity of the horizon cannot be reduced to zero ina finite number of steps. Validity of this law has been suggested by investigations of the orbits of chargedtest particles around a charged rotating black hole. A precise formulation of this Third Law has been givenand proved under some assumptions by Israel.Significance of the Third LawAn idea of the significance of the Third Law can be gleaned by thinking about how one might try to violateit. First, for a nonrotating neutral black hole,κis decreased when mass is added to the hole. (So the holehas negative specific heat.) But it would take an infinite amount of mass to reduceκto zero. A generalrotating, charged black hole with angular momentumJand chargeQhas a surface gravity and horizon areagiven byκ= 4πμ/A,A= 4π[2M(M+μ)-Q2](2.7)withμ= (M2-Q2-J2/M2)1/2.(2.8)Anextremalblack hole is one for whichμ= 0. For an extremal black hole,κvanishes andA= 4π(2M2-Q2).Thus, an extremal black hole has zero “temperature”, but nonzero “entropy”. (Thus the Planck form of theThird law does not hold for black holes. Also it should be remarked that if the extremal state is “eternal”rather than being reached from a non-extremal one, the entropy that enters a proper variational form ofthe first law is not the area and, in fact, vanishes.) IfM2< Q2+J2/M2then the spacetime has a nakedsingularity and is not a black hole at all. Thus if the surface gravity could actually be reduced to zero, onewould be only infinitesimally far from creating a naked singularity, violating Cosmic Censorship.To reduce the surface gravity to zero you might thus try to inject a sufficient amount of charge or angularmomentum into the hole.Suppose you try to drop a chargeqwith massminto a nonrotating chargedblack hole of massMand chargeQ < M, trying to makeQ+q=M+m. In order for the gravitationalattraction to be stronger than the electrostatic repulsion you must choosemM > qQ, soq/m < M/Q. Butthis inequality insures thatQ+q < M+m. Similarly if you try to inject enough orbital angular momentumto a spinning black hole you find that the particle simply misses the hole.