  In physical theories where the energy (action) is localized near a
submanifold of Euclidean (Minkowski) space, there is a universal expression for
the energy (or the action). We derive a multipole expansion for the energy that
has a finite number of terms, and depends on intrinsic geometric invariants of
the submanifold and extrinsic invariants of the embedding of the submanifold.
This universal expression is a generalization of an exact formula of Hermann
Weyl for the volume of a tube. We describe when our result is applicable, when
our generalization gives an exact result, and when there are corrections (often
exponentially small) to our formula. In special situations, dictated by
spherical symmetry, the expression is a generalized Lovelock lagrangian for
gravity, a class of theories that are interesting because they have no negative
metric states. We discuss whether these results represent a true theory of
emergent gravity by discussing simple models where a higher dimensional quantum
field theory without a fundamental graviton leads to a gravity-like theory on a
submanifold where all or some of the dynamical degrees of freedom are
fluctuations of the metric on the submanifold.
