  Theories with spontaneous local Lorentz and diffeomorphism violation contain
massless Nambu-Goldstone modes, which arise as field excitations in the minimum
of the symmetry-breaking potential. If the shape of the potential also allows
excitations above the minimum, then an alternative gravitational Higgs
mechanism can occur in which massive modes involving the metric appear. The
origin and basic properties of the massive modes are addressed in the general
context involving an arbitrary tensor vacuum value. Special attention is given
to the case of bumblebee models, which are gravitationally coupled vector
theories with spontaneous local Lorentz and diffeomorphism violation. Mode
expansions are presented in both local and spacetime frames, revealing the
Nambu-Goldstone and massive modes via decomposition of the metric and bumblebee
fields, and the associated symmetry properties and gauge fixing are discussed.
The class of bumblebee models with kinetic terms of the Maxwell form is used as
a focus for more detailed study. The nature of the associated conservation laws
and the interpretation as a candidate alternative to Einstein-Maxwell theory
are investigated. Explicit examples involving smooth and Lagrange-multiplier
potentials are studied to illustrate features of the massive modes, including
their origin, nature, dispersion laws, and effects on gravitational
interactions. In the weak static limit, the massive mode and
Lagrange-multiplier fields are found to modify the Newton and Coulomb
potentials. The nature and implications of these modifications are examined.
