  We explore the holographic proposal involving spacetimes with linear dilaton
asymptotics in three dimensions from a gravity perspective. The holographic
dual shares some properties with a symmetric product conformal field theory
deformed by a single-trace analogue of the $T \overline{T}$ deformation. We
present solutions of ten-dimensional supergravity which interpolate from BTZ
black holes in the interior to either a linear dilaton spacetime near infinity,
or to flat space. This allows a precise identification of field theory
parameters with gravity parameters. The solutions manifestly exhibit the square
root structure that is characteristic of $T \overline{T}$-deformed conformal
field theories. We compute the mass of the spacetimes using the covariant phase
space formalism and find agreement with the square root formula for the case of
black holes without spin. We also discuss whether closed string tachyons might
play a role when the deformation parameter becomes too large and the vacuum
becomes unstable.
