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1.2 Covariant 3+1 conformal decomposition 17
1.2 Covariant 3+1 conformal decomposition
1.2.1 3+1 formalism
We refer the reader to [55] and [486] for an introduction to the 3+1 formalism of general relativity.
Here we simply summarize a few key equations, in order mainly to fix the notations 1 . The spacetime
(or at least the part of it under study) is foliated by a family of spacelike hypersurfaces Σt, labeled
by the time coordinate t. We denote by n the future directed unit normal to Σt. By definition n,
considered as a 1-form, is parallel to the gradient of t:
n = −Ndt. (1.1)
The proportionality factor N is called the lapse function. It ensures that n satisfies to the normalization
relation nµn µ = −1.
The metric γ induced by the spacetime metric g onto each hypersurface Σt is given by the orthogonal
projector onto Σt:
γ := g + n ⊗ n. (1.2)
Since Σt is assumed to be spacelike, γ is a positive definite Riemannian metric. In the following, we call
it the 3-metric and denote by D the covariant derivative associated with it. The second fundamental
tensor characterizing the hypersurface Σt is its extrinsic curvature K, given by the Lie derivative of
γ along the normal vector n:
K := − 1
2 £nγ. (1.3)
One introduces on each hypersurface Σt a coordinate system (x i ) = (x 1 , x 2 , x 3 ) which varies
smoothly between neighboring hypersurfaces, so that (x α ) = (t, x 1 , x 2 , x 3 ) constitutes a well-behaved
coordinate system of the whole spacetime 2 . We denote by (∂/∂x α ) = ∂/∂t, ∂/∂x i = ∂/∂t, ∂/∂x 1 ,
∂/∂x 2 , ∂/∂x 3 the natural vector basis associated with this coordinate system. The 3+1 decomposition
of the basis vector ∂/∂t defines the shift vector β of the spatial coordinates (x i ):
∂
= Nn + β with n · β = 0. (1.4)
∂t
The metric components gαβ with respect to the coordinate system (x α ) are expressed in terms of the
lapse function N, the shift vector components β i and the 3-metric components γij according to
gµν dx µ dx ν = −N 2 dt 2 + γij(dx i + β i dt)(dx j + β j dt). (1.5)
In the 3+1 formalism, the matter energy-momentum tensor T is decomposed as
T = E n ⊗ n + n ⊗ J + J ⊗ n + S, (1.6)
where the energy density E, the momentum density J and the strain tensor S, all of them as
measured by the observer of 4-velocity n, are given by the following projections: E := Tµνn µ nν ,
Jα := −γ µ
α Tµνnν , Sαβ := γ µ
α γ ν
β Tµν. By means of the Gauss and Codazzi relations, the Einstein
1 We use geometrized units for which G = 1 and c = 1; Greek indices run in {0,1,2,3}, whereas Latin indices
run in {1,2,3} only.
2 later on we will specify the coordinates (x i ) to be of spherical type, with x 1 = r, x 2 = θ and x 3 = ϕ, but
at the present stage we keep (x i ) fully general.