 In the presence of an antisymmetric tensor field (without external sources), the Einstein Field equations, and the equations of motion for the antisymmetric tensor are
Where the stress-energy tensor takes the form 
Being a rank "s" antisymmetric tensor, the field strength formula_1 has a natural ansatz for its solution, proportional to the Levi-Civita tensor on some "s"-dimensional manifold.Here, the indices formula_6 run over "s" of the dimensions of the ambient "d"-dimensional spacetime, formula_7 is the determinant of the metric of this "s"-dimensional subspace, and formula_8 is some constant with dimensions of mass-squared (in natural units).Since the field strength is nonzero only on an "s"-dimensional submanifold, the metric formula_9 is naturally separated into two parts, of block-diagonal form 
with formula_11, formula_12, and formula_13 extending over the same "s" dimensions as the field strength formula_1, and formula_15, formula_16, and formula_17 covering the remaining "d-s" dimensions.