Dirac-Born-Infeld action from spontaneous breakdown of Lorentz symmetry in brane-world scenarios

In whatever Lorentz-invariant theory, the presence of extended d-dimensional objects inside a higher dimensional bulk space-time, like for instance D-branes in string theories, induces a spontaneous breakdown of the Poincar\'e invariance of the bulk; however, the effective action describing these extended objects should still respect this larger invariance through a nonlinear realization of the full symmetry. Here the specific form of such a realization in the presence of an electromagnetic field is uncovered. The Dirac-Born-Infeld type action for a D-brane turns out to be invariant under such transformations. Conversely, it is explicitly demonstrated in some simple cases that the most general invariant action, as long as derivatives of the field strength (and second derivatives of the scalars) can be neglected, is a linear combination of terms of the Dirac-Born-Infeld type with different scale parameters.