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We point out that the worldvolume coordinate functions $\hat{x}^\mu(\xi)$ ofa $p$-brane, treated as an independent object interacting with dynamicalgravity, are Goldstone fields for spacetime diffeomorphisms gauge symmetry. Thepresence of this gauge invariance is exhibited by its associated Noetheridentity, which expresses that the source equations follow from thegravitational equations. We discuss the spacetime counterpart of the Higgseffect and show that a $p$-brane does not carry any local degrees of freedom,extending early known general relativity features. Our considerations are alsorelevant for brane world scenarios.Comment: 5 pages, RevTeX. v2 (30-IV-03) with additional text and reference

Gravity,p-branes, and a spacetime counterpart of the Higgs effect