Initial data

2.1 Initial data

In the Cauchy formulation of Einstein’s equations, we begin by foliating the 4-dimensional manifold as a set of spacelike, 3-dimensional hypersurfaces (or slices) ${ Σ}$ . These slices are labeled by a parameter $t$ or, more simply, each slice of the 4-dimensional manifold is a $t = constant$ hypersurface. Following the standard 3 + 1 decomposition [6 , 111

, 37 ], we let $μ n$ be the future-pointing timelike unit normal to the slice, with

Here, $α$ is called the lapse function (frequently denoted $N$ in the literature). The scalar lapse function sets the proper interval measured by observers as they move between slices on a path that is normal to the hypersurface (so-called normal observers):

Of course, there is no reason that observers must move along a path normal to the hypersurface. In general, we can define the time vector as

where

Here, $β μ$ is called the shift vector (frequently denoted $N μ$ in the literature).

Because of (9 ), $μ β$ has only three independent components and is a spatial vector, tangent to the hypersurface on which it resides. At this point, it is convenient to introduce a coordinate system adapted to the foliation ${Σ }$ . Let $xi$ be the spatial coordinates in the slice. The fourth coordinate, $t$ , is the parameter labeling each slice. With this adapted coordinate system, we find that 3-dimensional coordinate values remain constant as we move between slices along the $tμ$ direction (8 ). The four parameters, $α$ and $i β$ , are a manifestation of the 4-dimensional coordinate invariance, or gauge freedom, in Einstein’s theory. If we let $γij$ represent the metric of the spacelike hypersurfaces, then we can rewrite the interval (5 ) as

In the Cauchy formulation of Einstein’s equations, $γij$ is regarded as the fundamental variable and values for its components must be given as part of a well-posed initial-value problem. Since Einstein’s equations are second order, we must also specify something like a time derivative of the metric. For this, we use the second fundamental form, or extrinsic curvature, of the slice, $K ij$ , defined² by

where $ℒn$ denotes the Lie derivative along the $nμ$ direction.

Together, $γij$ and $Kij$ are the minimal set of initial data that must be specified for a Cauchy evolution of Einstein’s equations. The metric $γij$ on a hypersurface is induced on that surface by the 4-metric $gμν$ . This means that the values $γij$ receives depend on how $Σ$ is embedded in the full spacetime. In order for the foliation of slices ${Σ}$ to fit into the higher-dimensional space, they must satisfy the Gauss-Codazzi-Ricci conditions. Combining these conditions with Einstein’s equations, and using (10 ), the six evolution equations become

[ ] ∂tKij = α R¯ij − 2Ki ℓK ℓj + KKij − 8πGSij + 4πG γij(S − ρ) (12 ) ¯ ¯ ℓ¯ ¯ ℓ ¯ ℓ − ∇i ∇jα + β ∇ ℓKij + Kiℓ∇jβ + Kj ℓ∇iβ .

Here, $∇¯i$ is the spatial covariant derivative compatible with $γij$ , $R¯ij$ is the Ricci tensor associated with $γ ij$ , $K ≡ Ki i$ , $ρ$ is the matter energy density, $S ij$ is the matter stress tensor, and $S ≡ Si i$ .³ We have also used the fact that in our adapted coordinate system, $ℒt ≡ ∂t$ . The set of second-order evolution equations is completed by rewriting the definition of the extrinsic curvature (11

) as

Equations (12 ) and (13 ) are a first-order representation of a complete set of evolution equations for given initial data $γij$ and $Kij$ . However, the data cannot be freely specified in their entirety. The four constraint equations, following the same procedure outlined above for the evolution equations, become

and

Here, $R¯≡ ¯Rii$ and $ji$ is the matter momentum density. Equation (14

) is referred to as the Hamiltonian or scalar constraint, while (15

) are referred to as the momentum or vector constraints. Valid initial data for the evolution equations (12

) and (13

) must satisfy this set of constraints. And, as mentioned earlier, the Bianchi identities (4

) guarantee that the evolution equations will preserve the constraints on future slices of the evolution.

As we will see below, the Hamiltonian constraint (14 ) most naturally constrains the 3-metric $γij$ , while the momentum constraints (15 ) naturally constrain the extrinsic curvature $Kij$ . Taking the constraints into consideration, it seems that the 3-metric has five degrees of freedom remaining, while the extrinsic curvature has three. But we know that the gravitational field in Einstein’s theory has two dynamical degrees of freedom, so we expect that both $γij$ and $Kij$ should each have only two free components. The answer to this problem is, once again, the coordinate invariance of Einstein’s theory. This seems strange at first, because we have already used the coordinate invariance of the theory to narrow our scope from the ten components of the 4-metric to the six components of the 3-metric. However, the lapse and shift do not completely specify the coordinate gauge. Rather, they specify how an initial choice of gauge will evolve with the foliation. The metric on a given hypersurface retains full 3-dimensional coordinate invariance, reducing the number of freely specifiable components to two [107 , 108 ]. There also remains one degree of gauge freedom associated with the time coordinate which must be fixed. Each hypersurface represents a $t = const.$ slice of the spacetime, so how the initial hypersurface is embedded in the full 4-dimensional manifold represents our temporal gauge choice. There is no unique way to specify this choice, but it is often convenient to let the trace of the extrinsic curvature $K$ represent this temporal gauge choice [108 ]. Thus, we find that we are allowed to choose freely five components of the 3-metric and three components of the extrinsic curvature. However, only two of the components for each field represent dynamical degrees of freedom, the remainder are gauge degrees of freedom.

The four constraint equations, (14 ) and (15 ), represent conditions which the 3-metric and extrinsic curvature must satisfy. But, they do not specify which components (or combination of components) are constrained and which are freely specifiable. In the weak field limit where Einstein’s equations can be linearized, there are clear ways to determine which components are dynamic, which are constrained, and which are gauge. However, in the full nonlinear theory, there is no unique decomposition. In this case, one must choose a method for decomposing the constraint equations. The goal is to transform the equations into standard elliptic forms which can be solved given appropriate boundary conditions [81 , 83 , 111 ]. Each different decomposition yields a unique set of elliptic equations to be solved and a unique set of freely specifiable parameters which must be fixed somehow. Seemingly similar sets of assumptions applied to different decompositions can lead to physically different initial conditions.