Foliation of spacetime
is the spacetime metric.
is the spacetime covariant derivative.
is a spacetime scalar field that foliates the spacetime. (The level surfaces of are assumed spacelike.)
The unit normal vector field is
with the lapse function defined by
The unit normal satisfies
The spatial metric expressed as a spacetime tensor is
The alternative notation
is used to emphasize the role of the spatial metric as a projection operator for the spacelike foliation.
The extrinsic curvature is
where is the Lie derivative with respect to the unit normal. An alternative expression is
The right--hand side is symmetric in its free indices because the unit normal is hypersurface orthogonal. That is, satisfies
The acceleration of the observers who are at rest in the spacelike slices is
Spatial tensors and covariant derivative
Spatial tensors are tensors whose contraction with the unit normal on any index vanishes. Examples of spatial tensors are the spatial metric , the extrinsic curvature , and the acceleration .
The spatial covariant derivative acts on spatial tensors. It is defined by , where the symbol denotes the projection of each free index with the projection operator . For example, let denote a spatial tensor. Then
Time flow vector field
The time flow vector field satisfies
It can be written as
where the shift vector is a spatial vector field.
For any scalar or covariant spatial tensor (spatial tensor with all lower indices), the Lie derivative along the time flow vector field can be written as . For example, if is a scalar and is a spatial covector, then
denote arbitrary spacetime coordinates.
denote spacetime coordinates that are adapted to the foliation
The arbitrary coordinates can be written as functions of the adapted coordinates: . The adapted coordinates can be written as functions of the arbitrary coordinates: and .
Adapted coordinates are defined such that
is the time flow vector field. Define . By the chain rule, so that
Spatial tensors in adapted coordinates
The spatial metric, extrinsic curvature, and acceleration are
The inverse of the spatial metric is denoted .
The following results hold:
Metric in adapted coordinates
The shift vector is
The spacetime line element is
When the spacetime and adapted coordinates coincide
If and , the components of the spacetime metric and its inverse are
The unit normal components are
The spatial metric (projection operator) components are
The determinant of the spacetime metric is where . Thus
Lie derivatives along the time flow vector field
For a scalar field ,
For a covector (not necessarily spatial),
where . In particular .
Let denote a contravariant vector and define . The results above can be used to rewrite the relation as
If is a spatial vector, then
Corresponding results hold for higher rank spatial tensors. Thus, the projection with and of the Lie derivative of a spatial tensor along the time flow vector field is equal to the partial derivative with respect to .
Splitting of Riemann
The spacetime Riemann tensor is defined by where is a vector field. The spatial Riemann tensor is defined by where is a spatial vector field. The symbol denotes the operator that projects all free indices onto the foliation with factors of .
Gauss, Codazzi and Ricci equations
The Gauss, Codazzi, and Ricci equations are, respectively,
The last two terms of the Ricci equation can be written as
Riemann, Ricci and curvature scalar
Decompositions of the spacetime Riemann tensor, Ricci tensor, and curvature scalar are
Einstein tensor and equations
The decomposition of the Einstein tensor is
Let denote the matter stress-energy-momentum tensor. The projection of the Einstein equations in the normal-normal direction is
where is the matter mass density. The projection of the Einstein equations in the normal-tangential direction is
where is the matter momentum density. The projection of the Einstein equations in the tangential-tangential direction is
where is the spatial stress tensor.
Einstein equations and Riemann tensor in adapted coordinates
If is a spatial tensor, this can be inverted:
For spatial tensors define
These definitions generalize to higher rank tensors. The derivative is the covariant derivative compatible with the spatial metric, so that .
Spacetime and spatial Lie derivatives
Results in the section Time flow vector field and the subsection on Lie derivatives show that for any spatial covariant tensor, such as the extrinsic curvature , the Lie derivative along the unit normal is
For any spatial tensor, the spacetime Lie derivative with respect to a spatial vector is equal to the spatial Lie derivative; for example, . Thus,
On the left, the symbol denotes a Lie derivative in spacetime with respect to the spacetime vector . On the right it denotes a Lie derivative in space with respect to the spatial vector . The extrinsic curvature is , or
The definition of the spatial Riemann tensor, , where is a spatial vector, can be projected to space:
The Einstein equations are obtained by projecting the results from the subsection on the Einstein equations. The Hamiltonian constraint:
where is the trace of the extrinsic curvature. The momentum constraint:
The evolution equations:
which can be simplified with use of the Hamiltonian constraint.
The Gauss, Codazzi, and Ricci equations in adapted coordinates are
The decomposition of the Ricci tensor in adapted coordinates is
and the curvature scalar is
The upper indices can be swapped for lower indices with use of the identity
where is a spacetime tensor with only one index displayed.