Remarks on the instability of black Dp-branes

We show that for black Dp-branes having charge Q and Hawking temperature T, the product QT7−p is bounded from above for p⩽5 and is unbounded for p=6. While the maximum occurs at some finite value of a parameter for p⩽4, it occurs at infinity of the parameter for p=5. As a consequence, for fixed charge, there are two black Dp-branes (for p⩽4) at any given temperature less than its maximum value, and when the temperature is maximum there is one black Dp-brane. For p=5, there is only one black D5-brane at a given temperature less than its maximum value, whereas, for p=6, since there is no bound for the temperature, there is always a black D6-brane solution at a given temperature. Of the two black Dp-branes (for p⩽4), one is large which is shown to be thermodynamically unstable and the other is small which is stable. But for p=5,6, the black Dp-branes are always thermodynamically unstable. The stable, small black Dp-brane, however, under certain conditions, can become unstable quantum mechanically and decay either to a BPS Dp-brane or to a Kaluza-Klein “bubble of nothing” through closed string tachyon condensation. The small D5, D6 branes, although classically unstable, have the same fate under similar conditions.