Statistical mechanics of exponentially many low lying states

It has recently been argued that for near-extremal black holes, the entropy and the energy above extremality respectively receive a log T <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>l</mml:mi> <mml:mi>o</mml:mi> <mml:mi>g</mml:mi> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> and a T <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>T</mml:mi> </mml:math> -linear correction, where T <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>T</mml:mi> </mml:math> is the temperature. We show that both these features can be derived in a “low but not too low” temperature regime, by assuming the existence of exponentially many low lying states cleanly separated from rest of the spectrum, without using any specific theory. Argument of the logarithm in the expression of entropy is seen to be the ratio of temperature and the bandwidth of the low lying states. We argue that such spectrum might arise in non-supersymmetric extremal brane systems. Our findings strengthen Page’s suggestion that there is no true degeneracy for non-supersymmetric extremal black holes.