Quantum Gravity

PARAMETRIZED FIELD THEORIES 89Ü Ü ·Ü Ø ·Ø ØÆÒ ØÜ Æ ØØFig. 3.1. Geometric interpretation of lapse and shift.Instead of Minkowski space one can also choose an arbitrary curved backgroundfor the embedding. Denoting the spatial metric by h ab ,thatis∂X µ ∂X νh ab = g µν∂x a ∂x b , (3.75)the four-dimensional line element can be decomposed as follows:ds 2 = g µν dx µ dx ν = −N 2 dt 2 + h ab (dx a + N a dt)(dx b + N b dt)=(h ab N a N b − N 2 )dt 2 +2h ab N a dx b dt + h ab dx a dx b . (3.76)The action (3.73) is invariant under the reparametrizationsx 0 → x 0′ = x 0 + f(x a ) ,x a → x a′ = g(x b ) (3.77)with arbitrary functions (obeying standard differentiability conditions) f andg. This is not equivalent to the full set of space–time diffeomorphisms; see thediscussion at the end of this section.A simple example of the above procedure is the case of a massless scalarfield on (1+1)-dimensional Minkowski space–time (Kuchař 1973, 1981). Its LagrangianreadsL(φ, ∂φ∂T , ∂φ )∂X= − 1 ∂φηµν2 ∂X µ∂φ∂X ν = 1 2For the Jacobi determinant we haveJ = TX ˙ ′ − T ′ Ẋ,and for the components of the normal vector,n T =X ′√X′2− T ′2 , nX =[ ( ) 2 ( ) ] 2 ∂φ ∂φ−∂T ∂XT ′√X′2− T ′2 .. (3.78)(Dots denote derivatives with respect to x 0 ≡ t, and primes denote derivativeswith respect to x 1 ≡ x.) The energy–momentum tensor assumes the well known