(ed.). Gravitational waves (IOP, 2001)(422s).

Appendix A. The string effective actionAppendix A. The string effective action 317The motion of a point particle in an external gravitational field, g µν , is governedby the actionS =− m ∫dτ ẋ µ ẋ ν g µν (x), (16.112)2where x µ (τ) are the spacetime coordinates of the particle, and τ is an affineparameter along the particle worldline.The time evolution of a one-dimensional object like a string describes aworld-surface, or ‘world-sheet’, instead of a worldline, and the action governingits motion is given by the surface integral∫S =− M2 sdτ dσ √ −γγ ij ∂ i x µ ∂ j x ν g µν (x), (16.113)2where ∂ i ≡ ∂/∂ξ i and ξ i = (τ, σ ) are, respectively, the timelike and spacelikecoordinates on the string world-sheet (i, j = 1, 2). The coordinates x µ (τ, σ )are the fields governing the embedding of the string world-sheet in the external(also called ‘target’) space. The parameter Ms 2 defines (in units h/2π = 1 = c)the so-called string tension (the mass per unit length), and its inverse defines thefundamental length scale of the theory (often called, for historical reasons, the α ′parameter):Ms 2 ≡ 1 λ 2 ≡ 1s 2πα ′ . (16.114)In a curved metric background g µν depends on x µ , and the nonlinear action(16.113) represents what is called a ‘σ -model’ defined on the string world sheet.For the point particle action (16.112) the variation with respect to x µ leadsto the well known geodesic equations of motion,ẍ µ + Ɣ αβ µẋ α ẋ β = 0. (16.115)The string equations of motion are similarly obtained by varying with respect tox µ the action (16.113): we get then the Euler–Lagrange equations∂ L∂ i∂(∂ i x µ ) = ∂ L∂x µ , L = √ −γγ ij ∂ i x µ ∂ j x ν g µν , (16.116)which can be written explicitly as£x µ + γ ij ∂ i x α ∂ j x β Ɣ µ αβ = 0, £ ≡ √ 1 √∂ i −γγ ij ∂ j . (16.117)−γThese equations describe the geodesic evolution of a test string in a given externalmetric. The variation with respect to γ ik imposes the so-called ‘constraints’, i.e.the vanishing of the world sheet stress tensor T ik ,T ij = √ 2 δS−γ δγ ij = ∂ i x µ ∂ j x ν g µν − 1 2 γ ij∂ k x µ ∂ k x µ = 0. (16.118)