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?
 A: If in quantum gravity, there's a Planck cutoff to energy densities, then there also has to be an entropy density cutoff. So, expect the thermodynamic arrow of time to reverse inside a black hole.
If Alice jumps into a black hole, she will have time reversed experiences because entropy has to go down inside.
A: This is a very speculative idea based upon UV/IR mixing. UV/IR mixing states that if some ultraviolet modes much smaller than the Planck length are excited, this causes nonlocal interactions at infrared scales given by the reciprocal of the the ultraviolet wavelength in Planck units. Ordinarily, at temperatures far below the Planck temperature, such effects are negligible. However, inside a black hole, as matter gets sucked into the singularity, it becomes increasingly blueshifted until its wavelengths goes under the Planck length, causing nonlocal interactions. If the infrared nonlocality scale is comparable to or larger than the black hole radius, this can affect the Hawking radiation going out. As matter is crushed into the final singularity, wavelengths in some directions can get arbitrarily smaller than the Planck length.
A: If the second law is violated. these effects ought to leak outside the event horizon. So, they ought to be detectable outside the horizon?
