Abstract
In this note, we have extended the result of [1] to calculate the membrane stress tensor up to mathcal{O}left(frac{1}{D}right) localized on the co-dimension one membrane world volume propagating in asymptotically flat/AdS/dS spacetime. We have shown that the subleading order membrane equation follows from the conservation equation of this stress tensor.
Highlights
Background spacetime indicesIndices on the membrane Background metricInduced metric on the membrane as embedded in gAB Full non-linear metric outside the membrane as read off from [9]Linearized metric outside the membraneLinearized metric inside the membrane Projector on the membrane as embedded in gAB Projector perpendicular to both the normal of the membrane as embedded in gAB and the velocity Projector on the membrane as embedded in G(AoBut) Projector on the membrane as embedded in G(AinB)Covariant derivative w.r.t. gAB Covariant derivative w.r.t. gμ(iνnd) Covariant derivative w.r.t
We have shown that the subleading order membrane equation follows from the conservation equation of this stress tensor
Conservation of this stress tensor would result in the membrane equation derived in [9] — the equation that governs the dynamics of the membrane
Summary
We shall write our final result — the expression of the membrane stress tensor up to corrections of order O. Conservation of this stress tensor would result in the membrane equation derived in [9] — the equation that governs the dynamics of the membrane. The metric of the embedding space satisfies the following equation. The membrane is characterized by its shape (encoded in its extrinsic curvature Kμν) and a velocity field (uμ), unit normalized with respect to the induced metric of the membrane. The membrane stress tensor, that we report below, is a symmetric two-indexed tensor, constructed out of this velocity field, extrinsic curvature and its derivatives. Gμ(iνnd) is the induced metric on the membrane, ∇ ̄ μ is the covariant derivative with respect to gμ(iνnd). Membrane velocity (uμ) can be viewed as a vector field(uA) in the full background spacetime.
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