extremality

Hawking, Horowitz and Ross found that the entropy of a charged black hole approached a smooth limit as the charge was increased, but discontinuously went to zero exactly at extremality.

Where Does the Entropy Go? Sean 2009

In an extremal black hole, where all of the energy comes from the electric field itself, the two horizons coincide: we have r+ = r - (extremality).

Where Does the Entropy Go? Sean 2009

In other words, the approach to extremality is not continuous, as the non-extremal Reissner-Nordström solution splits into two spacetimes at extremality: an extremal black hole and a disconnected $AdS$ space.

Where Does the Entropy Go? Sean 2009

We found that the volume of spacetime between the two horizons approached a smooth limit as the charge was increased, but discontinuously disappeared exactly at extremality.

Where Does the Entropy Go? Sean 2009

Quick question: if the limit of approaching extremality is discontinuous, doesn't that mean that the exactly extremal space, the one with zero entropy, is unstable to infinitesimal perturbations, and therefore unphysical?

Cosmic Variance 2009

But Matt, Lisa and I stumbled across an interesting miracle of curved spacetime, which I think some people recognized but certainly isn't widely appreciated: as you creep up on extremality, increasing the charge of the black hole while keeping its mass fixed, and therefore driving

Cosmic Variance 2009

We found that the volume of spacetime between the two horizons approached a smooth limit as the charge was increased, but discontinuously disappeared exactly at extremality.

Cosmic Variance 2009

In other words, the approach to extremality is not continuous, as the non-extremal Reissner-Nordström solution splits into two spacetimes at extremality: an extremal black hole and a disconnected $AdS$ space.

Cosmic Variance 2009

Hawking, Horowitz and Ross found that the entropy of a charged black hole approached a smooth limit as the charge was increased, but discontinuously went to zero exactly at extremality.

Cosmic Variance 2009

But Matt, Lisa and I stumbled across an interesting miracle of curved spacetime, which I think some people recognized but certainly isn’t widely appreciated: as you creep up on extremality, increasing the charge of the black hole while keeping its mass fixed, and therefore driving r - ever closer to r+, the spacetime volume of the region in between them does not go to zero.

Where Does the Entropy Go? Sean 2009