Nambu bubbles with curvature corrections

Patricio S. Letelier
1990 Physical Review D, Particles and fields  
A finite-width correction for a two-dimensional membrane described by the Nambu action modified by the addition of the Ricci curvature scalar is studied. The particular case of a spherical bubble is considered in some detail. Z" :Z"(r',p') = Yi'(r')+p'n, ", where Y"(r') represents the string world sheet, r'=(r, r') are the two parameters that describe the string world sheet, n, =(n&, nz) are two normals to the world sheet such that (n&) =(n2) = -1, and n, n2=0 (our signature is + ---). The
more » ... is + ---). The corrected action is obtained by making a series expansion in the transversal variables p' and a transversal integration. One finds, considering up to second order in p' that the different contributions to the action can be separated into (a) a term that gives origin to the usual Nainbu action, (b) a term that is constructed with the Ricci curvature scalar of the world-sheet metric [this term is only a topological term (Gauss-Bonnet theorem in two dimensions)], (c) a term quadratic in the extrinsic curvature, i.e. , a Polyakov term, and (d) a twist term. If one considers membranes instead of string, i.e. , three-dimensional timelike hypersurfaces (world tubes) instead of two-dimensional world sheets, the situation is similar; the major difference is that now the Ricci curva-Recently, in the context of cosmic strings, there has been some interest in higher-order corrections to the Nambu action. ' These corrections consider the curvature of the string world sheet and physically they add some thickness to the string. Examples of strings with curvature corrections are provided by the Polyakov string and the Lindstrom-Rocekvan Nieuwenhuizen string. In the case of the Polyakov string the inclusion of an extrinsic curvature correction changes some of the properties of the gravitational field associated with the cusps of the usual cosmic strings. The purpose of this note is to consider a curvature correction for a Nambu two-dimensional membrane evolving in the usual four-dimensional Minkowski spacetime. In particular we study the effect of a curvature correction in the membrane motion equation as well as in the metric energy-momentum tensor. The particular, albeit important, example of a spherical bubble is considered in some detail. The usual Nambu action for strings can be obtained from a field theory by considering in the first approximation that the fields are "condensed" on the string world sheet, i.e. , by making a change of variables in the action of the field theory, and then making a limit process. The Minkowski coordinates Z" are replaced by the new curvilinear coordinates (r', p') defined by ture scalar does not give origin to a topological invariant, we have instead a nontrivial perturbation. In this note we shall consider only this last perturbation. In other words we shall study the action S=po f &y(1+aoR )d r, where JMO is the usual constant for membranes and ao is a new constant of dimensions of inverse length squared. The explicit values of these constants depend on the specific field theory to which the membranes are associated: (3) (4) F AB where X"=X"(r") represents the membrane world tube and r"=(r,r', 2). R is the Ricci scalar constructed with y "z. Note that the action S is similar to the action for the vacuum Einstein equations with a cosmological constant, but now the dynamical variable is X" instead of the metric. The Euler-Lagrange equations for the membrane world tube reduce to R"~-V "V~X"V VqX"-V qV~X"V~V X" . (7) A notable feature of the action (2) is that it gives rise to second-order differential equations for the membrane world tube. Note that the Lagrangian density associated with (2) is unique in that respect, since every other combination of the extrinsic curvature will yield fourth-order differential equations for the membrane world tube. The membranes that we shall consider have no boundaries, otherwise we need to consider the equation that is obtained from (2) by imposing the condition that no canonical momentum escapes through the boundaries. The metric energy-momentum tensor is defined by the expression 5$=-' f d xv" 6g o V q V~X"=O, where AB yAB 2~(R AB iyABR) 2 In the derivation of (5) we have made use of the Bianchi identity, as well as 41
doi:10.1103/physrevd.41.1333 pmid:10012475 fatcat:qvjmxwv4sfeepfjrm2hagrf5ca