Probing entropy bounds with scalar field spacetimes

We study covariant entropy bounds in dynamical spacetimes with naked singularities. Specifically we study a spherically symmetric massless scalar field solution. The solution is an inhomogeneous cosmology with an initial spacelike singularity, and a naked timelike singularity at the origin. We construct the entropy flux 4-vector for the scalar field, and show by explicit computation that the generalized covariant bound $S_{{L(B,B}^{\prime })}<~\left[A\right(B\left)\mathrm{}-{A(B}^{\prime }\right)]/4\mathrm{}$ is violated for light sheets ${L(B,B}^{\prime })$ in the neighborhood of the (evolving) apparent horizon. We find no violations of the Bousso bound [for which ${A(B}^{\prime })=0\mathrm{}],$ even though certain sufficient conditions for this bound do not hold. This result therefore shows that these conditions are not necessary.