# Can the entropy density of a spacelike singularity arbitrarily exceed the inverse Planck volume?

For the purpose of this question, let's restrict ourselves to BKL singularities. BKL cosmologies are homogeneous Bianchi type XIII and IV cosmologies which exhibit oscillatory chaotic behavior, although that's not relevant to this question. Most generic singularities can be approximated locally by a BKL solution. The volume of a BKL universe decreases linearly with time as the singularity approaches. If the matter falling toward the singularity has a nonzero entropy, and the second law of thermodynamics is satisfied, the entropy density will increase without limit as the singularity approaches. Can the entropy density exceed the inverse Planck volume? Being inside a black hole, the holographic bound does not apply. If entropy densities beyond the inverse Planck volume are forbidden, is the second law violated?

• This is a nice question I would like to be answered, too. In some moral sense, it must be true that the Planckian entropy density is a limit - holography makes the maximum volume density much smaller for bigger-than-Planckian volumes and smaller volumes "don't physically exist", in a sense - but I don't know whether it can be demonstrated by some argument, whether it implies the Hartle-Hawking-like unique state at the beginning of the Big Bang, and/or a unique final state in the black hole singularity. May 25 '11 at 5:48
• @amit - do you mind if I edit your question to improve the formatting so it is easier to read? Jun 1 '12 at 23:23