Ivancevic_Applied-Diff-Geom

1182 Applied Differential Geometry: A Modern IntroductionThis could be generalized to a curved space–time by replacing η µν by aRiemannian metric tensor g µν (x), but (for simplicity) we will not do so here.In terms of the embedding functions, x µ (t), the action can be rewritten as∫ √S[x] = −m dτ −η µν ẋ µ ẋ ν ,where overdot represents the derivative with respect to τ. An importantproperty of this action is invariance under local reparametrizations. Thisis a kind of gauge invariance, whose meaning is that the form of S is unchangedunder an arbitrary reparametrization of the world–line τ → τ(˜τ).Actually, one should require that the function τ(˜τ) is smooth and monotonic( dτd˜τ > 0) . The reparametrization invariance is a 1D analog of the 4Dgeneral coordinate invariance of general relativity. Mathematicians refer tothis kind of symmetry as diffeomorphism invariance.The reparametrization invariance of S allows us to choose a gauge. Anice choice is the static gauge, x 0 = τ. In this gauge (renaming theparameter to t) the action becomes∫ √S = −m 1 − vi 2dt, where v i = dx idt .Requiring this action to be stationary under an arbitrary variation of x i (t)gives the Euler–Lagrangian equationsdp idt = 0, where p i = δS =mv i√ ,δv i 1 − v2iwhich is the usual result. So we see that usual relativistic kinematics followsfrom the action S = −m ∫ ds.p−Branes and World–Volume ActionsWe can now generalize the analysis of the massive point particle to ageneric p−brane, which is characterized by its tension T p . The action inthis case involves the invariant (p + 1)D volume and is given by∫S p = −T p dµ p+1 ,where the invariant volume element is√dµ p+1 = − det(−η µν ∂ α x µ ∂ β x ν )d p+1 σ.