  We point out that the worldvolume coordinate functions $\hat{x}^\mu(\xi)$ of
a $p$-brane, treated as an independent object interacting with dynamical
gravity, are Goldstone fields for spacetime diffeomorphisms gauge symmetry. The
presence of this gauge invariance is exhibited by its associated Noether
identity, which expresses that the source equations follow from the
gravitational equations. We discuss the spacetime counterpart of the Higgs
effect and show that a $p$-brane does not carry any local degrees of freedom,
extending early known general relativity features. Our considerations are also
relevant for brane world scenarios.
