Splitting Spacetime

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Foliation of spacetime

Basic structure

\gamma_{\mu\nu} is the spacetime metric.

\nabla_\mu is the spacetime covariant derivative.

t is a spacetime scalar field that foliates the spacetime. (The level surfaces of t are assumed spacelike.)

Unit normal

The unit normal vector field is

n_\mu = -\alpha \nabla_\mu t

with the lapse function defined by

\alpha \equiv (-\gamma^{\mu\nu} \nabla_\mu t \nabla_\nu t)^{-1/2}

The unit normal satisfies


\begin{array}{rcl}
n^\mu n_\mu & = & -1 \\
n^\sigma \nabla_\mu n_\sigma & = & 0
\end{array}
Spatial metric

The spatial metric expressed as a spacetime tensor is

g_{\mu\nu} = \gamma_{\mu\nu} + n_\mu n_\nu

The alternative notation

 \perp_{\mu\nu} \equiv g_{\mu\nu}

is used to emphasize the role of the spatial metric as a projection operator for the spacelike foliation.

Extrinsic curvature

The extrinsic curvature is

 K_{\mu\nu} = -\frac{1}{2} {\mathcal L}_n g_{\mu\nu}

where {\mathcal L}_n is the Lie derivative with respect to the unit normal. An alternative expression is

K_{\mu\nu} = -\perp_\mu^\sigma \nabla_\sigma n_\nu

The right--hand side is symmetric in its free indices because the unit normal is hypersurface orthogonal. That is, n_\mu satisfies

 \perp_\mu^\sigma \nabla_\sigma n_\nu = \perp_\nu^\sigma \nabla_\sigma n_\mu
Acceleration

The acceleration of the observers who are at rest in the spacelike slices is

 a_\mu = n^\sigma \nabla_\sigma n_\mu

Useful results:

 
\begin{array}{rcl} 
     a_\mu & = &   (\perp_\mu^\nu \nabla_\nu\alpha) /\alpha\\
     \nabla_\mu n_\nu & = & -K_{\mu\nu} - n_\mu a_\nu \\
     n^\mu a_\mu & = & 0
\end{array}
.
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 g_{\mu\nu}, the extrinsic curvature K_{\mu\nu}, and the acceleration a_\mu.

The spatial covariant derivative acts on spatial tensors. It is defined by D_\mu \equiv \perp \nabla_\mu, where the symbol \perp denotes the projection of each free index with the projection operator \perp_\mu^\nu. For example, let T_\sigma^\rho denote a spatial tensor. Then

 D_\mu T_\sigma^\rho = \perp_\mu^{\mu'} \perp_\sigma^{\sigma'} \perp^\rho_{\rho'} \nabla_{\mu'} T_{\sigma'}^{\rho'}.

Time flow vector field

The time flow vector field t^\mu satisfies

 t^\mu\nabla_\mu t = 1

It can be written as

t^\mu = \alpha n^\mu + \beta^\mu

where the shift vector \beta^\mu = \perp^\mu_\nu t^\nu 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 {\mathcal L}_t = \alpha {\mathcal L}_n + {\mathcal L}_\beta. For example, if S is a scalar and W_\mu is a spatial covector, then


\begin{array}{rcl}
  {\mathcal L}_t S & = & \alpha {\mathcal L}_n S + {\mathcal L}_\beta S \\
   {\mathcal L}_t W_\mu & = & \alpha {\mathcal L}_n W_\mu + {\mathcal L}_\beta W_\mu 
\end{array}

Adapted coordinates

Definitions

X^\mu denote arbitrary spacetime coordinates.

x^a,t denote spacetime coordinates that are adapted to the foliation t = {\rm const}

The arbitrary coordinates can be written as functions of the adapted coordinates: X^\mu(x,t). The adapted coordinates can be written as functions of the arbitrary coordinates: x^a(X) and t(X).

Adapted coordinates are defined such that

 \partial_t X^\mu = t^\mu

is the time flow vector field. Define  X^\mu_a \equiv \frac{\partial X^\mu}{\partial x^a}. By the chain rule, X^\mu_a \partial_\mu t = 0 so that

 n_\mu X^\mu_a = 0
Spatial tensors in adapted coordinates

The spatial metric, extrinsic curvature, and acceleration are


\begin{array}{rcl}
   g_{ab} & = & X^\mu_a X^\nu_b g_{\mu\nu} = X^\mu_a X^\nu_b \gamma_{\mu\nu} \\
   K_{ab} & = & X^\mu_a X^\nu_b K_{\mu\nu} \\
   a_a & = & X^\mu_a a_\mu
\end{array}

The inverse of the spatial metric is denoted g^{ab}.

Projections

Let

 X_\mu^a \equiv g^{ab}\gamma_{\mu\nu} X^\nu_b

The following results hold:


\begin{array}{rcl}
   X^\mu_a X^b_\mu & = & \delta_a^b \\
   X_\mu^a X^\nu_a & = & \perp_\mu^\nu \\
   X^\mu_a X^\nu_b g^{ab}& = & g^{\mu\nu}  \\ 
   X_\mu^a  X_\nu^b g_{ab} & = & g_{\mu\nu} \\
   X^a_\mu  X^b_\nu g^{\mu\nu} & = & g^{ab} 
\end{array}
.
Metric in adapted coordinates

The shift vector is

 \beta^a = X_\mu^a \beta^\mu = X_\mu^a t^\mu

The spacetime line element is

 
\begin{array}{rcl} 
   ds^2 & = &  \gamma_{\mu\nu} dX^\mu dX^\nu \\
        & = & \gamma_{\mu\nu} (X^\mu_a dx^a + t^\mu dt)(X^\nu_b dx^b + t^\nu dt) \\
        & = & (-\alpha^2 + \beta_a \beta^a)dt^2 + 2\beta_a dx^a dt + g_{ab} dx^a dx^b 
\end{array}
When the spacetime and adapted coordinates coincide

If X^0 = t and X^a = x^a, the components of the spacetime metric and its inverse are


\begin{array}{rcl}
  (\gamma_{\mu\nu}) & = &  \begin{pmatrix} \beta_a \beta^a - \alpha^2 & \beta_a \\
       \beta_a & g_{ab} \end{pmatrix} \\
  (\gamma^{\mu\nu}) & = & \begin{pmatrix} -1/\alpha^2 & \beta^a/\alpha^2 \\
       \beta^a/\alpha^2 & g^{ab} - \beta^a \beta^b/\alpha^2 \end{pmatrix}
\end{array}

The unit normal components are


\begin{array}{rcl}
    (n_\mu) & = & (-\alpha,0,0,0) \\
    (n^\mu) & = & (1/\alpha,-\beta^a/\alpha)
\end{array}

The spatial metric (projection operator) components are


\begin{array}{rcl}
   (g_{\mu\nu}) & = &  \begin{pmatrix} \beta_a\beta^a & \beta_a \\
              \beta_a & g_{ab} \end{pmatrix} \\
   (g^{\mu\nu}) & = & \begin{pmatrix} 0 & 0 \\
               0 & g^{ab} \end{pmatrix} \\
   \perp_\mu^\nu & = & \delta_\mu^\nu - \delta_\mu^0(\delta_0^\nu - \beta^a \delta_a^\nu) 
\end{array}

The determinant of the spacetime metric is \gamma = -\alpha^2 g where g = \det(g_{ab}). Thus

 \sqrt{-\gamma} = \alpha\sqrt{g}
Lie derivatives along the time flow vector field

For a scalar field S,

 
\begin{array}{rcl}
   {\mathcal L}_t S & = & t^\mu\partial_\mu S \\
   & = &  \partial_t S 
\end{array}

For a covector W_\mu (not necessarily spatial),

 
\begin{array}{rcl} X^\mu_a {\mathcal L}_t W_\mu & = & X^\mu_a(t^\sigma\partial_\sigma W_\mu + W_\sigma\partial_\mu t^\sigma) \\
           & = & X^\mu_a \partial_t W_\mu + W_\mu \partial_t X^\mu_a \\
           & = & \partial_t (X^\mu_a W_\mu) \\
           & = & \partial_t W_a
\end{array}

where W_a \equiv X^\mu_a W_\mu. In particular  X^\mu_a {\mathcal L}_t n_\mu = 0.

Let V^\mu denote a contravariant vector and define V^a \equiv X^a_\mu V^\mu. The results above can be used to rewrite the relation {\mathcal L}_t W_\mu V^\mu = W_\mu{\mathcal L}_t V^\mu + V^\mu{\mathcal L}_t W_\mu as

 W_a \partial_t V^a  = W_a X^a_\mu {\mathcal L}_t V^\mu + n_\nu V^\nu W_a X^a_\nu {\mathcal L}_t n^\mu

If V^\mu is a spatial vector, then

 X^a_\mu {\mathcal L}_t V^\mu = \partial_t V^a

Corresponding results hold for higher rank spatial tensors. Thus, the projection with X^\mu_a and X_\mu^a of the Lie derivative of a spatial tensor along the time flow vector field is equal to the partial derivative with respect to t.

Splitting of Riemann

The spacetime Riemann tensor is defined by  {}^{(4)}R_{\mu\nu\sigma\rho} V^\rho = 2\nabla_{[\mu}\nabla_{\nu]} V_\sigma where V^\mu is a vector field. The spatial Riemann tensor is defined by  {}^{(3)}R_{\mu\nu\sigma\rho} V^\rho = 2D_{[\mu}D_{\nu]} V_\sigma where V^\mu is a spatial vector field. The symbol \perp denotes the operator that projects all free indices onto the foliation with factors of \perp_\mu^\nu.

Gauss, Codazzi and Ricci equations

The Gauss, Codazzi, and Ricci equations are, respectively,


\begin{array}{rcl}
    \perp {}^{(4)}R_{\mu\nu\sigma\rho} & = & {}^{(3)}R_{\mu\nu\sigma\rho} 
        + K_{\mu\sigma} K_{\nu\rho} - K_{\nu\sigma} K_{\mu\rho}  \\
    \perp {}^{(4)}R_{\mu\nu\sigma\rho} n^\rho & = & - D_\mu K_{\nu\sigma} + D_\nu K_{\mu\sigma}  \\
    \perp {}^{(4)}R_{\mu\sigma\nu\rho} n^\sigma n^\rho & = &  {\mathcal L}_n K_{\mu\nu} 
     + K_\mu^\sigma K_{\sigma\nu} +  D_\mu a_\nu + a_\mu a_\nu 
\end{array}

The last two terms of the Ricci equation can be written as


   D_\mu a_\nu + a_\mu a_\nu = (D_\mu D_\nu \alpha)/\alpha
Riemann, Ricci and curvature scalar

Decompositions of the spacetime Riemann tensor, Ricci tensor, and curvature scalar are


\begin{array}{rcl}
 {}^{(4)}R_{\mu\nu\sigma\rho} & = & {}^{(3)}R_{\mu\nu\sigma\rho} + K_{\mu [\sigma} K_{\rho]\nu} - K_{\nu [\sigma} K_{\rho] \mu} 
      + 4 (D_{[\mu} K_{\nu][\sigma})n_{\rho]} + 4 (D_{[\sigma} K_{\rho][\mu})n_{\nu]} \\
      & & - 4 n_{[\mu} K_{\nu]}^\alpha K_{\alpha [\sigma} n_{\rho]} - 4 n_{[\mu} ( {\mathcal L}_n K_{\nu][\sigma} ) n_{\rho]} 
      - (4/\alpha) n_{[\mu} (D_{\nu]} D_{[\sigma} \alpha ) n_{\rho]}   \\
 {}^{(4)}R_{\mu\nu} & = & {}^{(3)}R_{\mu\nu} + K K_{\mu\nu} - 2 K_{\mu\alpha} K^\alpha_\nu
      - 2 n_{(\mu} D_{\nu)}K + 2 (D_\alpha K^\alpha_{(\mu})n_{\nu)} \\
      & & - {\mathcal L}_n K_{\mu\nu} - (D_\mu D_\nu \alpha)/\alpha 
      + n_\mu n_\nu \left[{\mathcal L}_n K - K_{\sigma\rho} K^{\sigma\rho} +  (D_\sigma D^\sigma \alpha)/\alpha \right]   \\
 {}^{(4)}R & = & {}^{(3)}R + K_{\mu\nu} K^{\mu\nu} + K^2 - 2 {\mathcal L}_n K - 2(D_\sigma D^\sigma\alpha)/\alpha  \\
      & = & {}^{(3)}R + K_{\mu\nu}K^{\mu\nu} - K^2 - 2\nabla_\mu (K n^\mu + a^\mu ) 
\end{array}
Einstein tensor and equations

The decomposition of the Einstein tensor is


\begin{array}{rcl}
   {}^{(4)}G_{\mu\nu} & = & {}^{(3)}R_{\mu\nu} - \frac{1}{2} g_{\mu\nu}{}^{(3)}R + K K_{\mu\nu} 
      - 2 K_{\mu\sigma} K^\sigma_\nu - \frac{1}{2} g_{\mu\nu} (K_{\sigma\rho}K^{\sigma\rho} + K^2) \\
      & & - {\mathcal L}_n K_{\mu\nu} +  g_{\mu\nu} {\mathcal L}_n K 
      - (D_\mu D_\nu\alpha)/\alpha + g_{\mu\nu} (D_\sigma D^\sigma \alpha)/\alpha - 2 n_{(\mu} D_{\nu)} K \\
      & & + 2 (D_\sigma K^\sigma_{(\mu}) n_{\nu)} + \frac{1}{2} n_\mu n_\nu ( K^2 - K_{\sigma\rho}K^{\sigma\rho} 
     + {}^{(3)}R )
\end{array}

Let  T_{\mu\nu} denote the matter stress-energy-momentum tensor. The projection of the Einstein equations  0 = {}^{(4)}G_{\mu\nu} - \kappa T_{\mu\nu} in the normal-normal direction is


\begin{array}{rcl}
   0 & = &   2 n^\mu n^\nu ({}^{(4)}G_{\mu\nu} - \kappa T_{\mu\nu}) \\
    & = &   {}^{(3)}R - K_{\sigma\rho}K^{\sigma\rho} + K^2 + 2\kappa \rho 
\end{array}

where \rho \equiv n^\mu n^\nu T_{\mu\nu} is the matter mass density. The projection of the Einstein equations in the normal-tangential direction is


\begin{array}{rcl}
   0 & = & - \perp_\mu^\nu n^\sigma ({}^{(4)}G_{\nu\sigma} - \kappa T_{\nu\sigma})  \\
    & = &  D_\sigma K^\sigma_\mu - D_\mu K - \kappa j_\mu 
\end{array}

where  j_\mu \equiv -\perp_\mu^\sigma n^\nu T_{\sigma\nu} is the matter momentum density. The projection of the Einstein equations in the tangential-tangential direction is


\begin{array}{rcl}
   0 & = & - \perp_\mu^\sigma \perp_\nu^\rho ({}^{(4)}G_{\sigma\rho} - \kappa T_{\sigma\rho}) \\
    & = & {\mathcal L}_n K_{\mu\nu} -   g_{\mu\nu}{\mathcal L}_n K + 2 K_{\mu\sigma}K^\sigma_\nu 
   - K K_{\mu\nu} + \frac{1}{2} g_{\mu\nu} (K_{\sigma\rho}K^{\sigma\rho} + K^2) \\
    & & - {}^{(3)}R_{\mu\nu} + \frac{1}{2} g_{\mu\nu}{}^{(3)}R 
  + (D_\mu D_\nu\alpha)/\alpha - g_{\mu\nu} (D_\sigma D^\sigma \alpha)/\alpha  + \kappa s_{\mu\nu}
\end{array}

where s_{\mu\nu} \equiv \perp_\mu^\sigma \perp_\nu^\rho T_{\sigma\rho} is the spatial stress tensor.

Einstein equations and Riemann tensor in adapted coordinates

Projecting tensors

Define

T^{a}_{b} \equiv X_{\mu}^{a}  X^{\nu}_{b}  
T^{\mu}_{\nu}

If  T^{\mu}_{\nu} is a spatial tensor, this can be inverted:

T^{\mu}_{\nu} = X^{\mu}_{a}  X_{\nu}^{b}
T^{a}_{b}

For spatial tensors define

 D_c T^{a}_{b} \equiv X^\sigma_c 
X_{\mu}^{a} X^{\nu}_{b} D_\sigma T^{\mu}_{\nu}

These definitions generalize to higher rank tensors. The derivative D_a is the covariant derivative compatible with the spatial metric, so that  D_a g_{bc} = 0.

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 K_{\mu\nu}, the Lie derivative along the unit normal is


\begin{array}{rcl}
   X^\mu_a X^\nu_b (\alpha {\mathcal L}_n K_{\mu\nu}) & = &  X^\mu_a X^\nu_b ({\mathcal L}_t K_{\mu\nu} 
    - {\mathcal L}_\beta K_{\mu\nu})  \\
   & = & \partial_t K_{ab} - X^\mu_a X^\nu_b {\mathcal L}_\beta K_{\mu\nu} 
\end{array}

For any spatial tensor, the spacetime Lie derivative with respect to a spatial vector is equal to the spatial Lie derivative; for example,  X^\mu_a X^\nu_b {\mathcal L}_\beta K_{\mu\nu} = {\mathcal L}_\beta K_{ab}. Thus,

X^\mu_a X^\nu_b  {\mathcal L}_n K_{\mu\nu} = \frac{1}{\alpha} (\partial_t K_{ab} - {\mathcal L}_\beta K_{ab})

On the left, the symbol {\mathcal L}_\beta denotes a Lie derivative in spacetime with respect to the spacetime vector \beta^\mu. On the right it denotes a Lie derivative in space with respect to the spatial vector \beta^a. The extrinsic curvature is  K_{ab} \equiv  X^\mu_a X^\nu_b K_{\mu\nu} = -\frac{1}{2} X^\mu_a X^\nu_b {\mathcal L}_n g_{\mu\nu}, or


   K_{ab} = -\frac{1}{2\alpha} (\partial_t g_{ab} - 2D_{(a}\beta_{b)} )

where {\mathcal L}_\beta g_{ab} = 2D_{(a}\beta_{b)}.

Einstein equations

The definition of the spatial Riemann tensor,  {}^{(3)}R_{\mu\nu\sigma\rho} V^\rho = 2D_{[\mu} D_{\nu]}V_\sigma, where V^\mu is a spatial vector, can be projected to space:

 {}^{(3)}R_{abcd}V^d = 2D_{[a} D_{b]}V_c

The Einstein equations are obtained by projecting the results from the subsection on the Einstein equations. The Hamiltonian constraint:


   0 = {}^{(3)}R - K_{ab} K^{ab} + K^2 + 2\kappa \rho

where  K\equiv g^{ab}K_{ab} is the trace of the extrinsic curvature. The momentum constraint:


   0 = D_b K^b_a - D_a K - \kappa j_a

The evolution equations:

 \partial_t K_{ab} - {\mathcal L}_\beta K_{ab} = \alpha K K_{ab} - 2\alpha K_{ac}K^c_b + \alpha {}^{(3)}R_{ab} - D_a D_b \alpha 
       - \frac{\alpha}{4} ( K^2 - K_{cd}K^{cd} + {}^{(3)}R) g_{ab} - \alpha\kappa (s_{ab} - g_{ab} s^c_c /2)

which can be simplified with use of the Hamiltonian constraint.

Riemann tensor

The Gauss, Codazzi, and Ricci equations in adapted coordinates are


\begin{array}{rcl}
   {}^{(4)}R_{abcd} & = & {}^{(3)}R_{abcd} + K_{ac} K_{bd} - K_{bc}K_{ad} \\
   {}^{(4)}R_{abc}{}^t & = &  (D_a K_{bc} - D_b K_{ac} ) /\alpha  \\
   {}^{(4)}R_a{}^t{}_b{}^t & = &  ( \partial_t K_{ab} - {\mathcal L}_\beta K_{ab} + \alpha K_a^c K_{cb} + D_a D_b \alpha ) /\alpha^3
\end{array}

The decomposition of the Ricci tensor in adapted coordinates is


\begin{array}{rcl}
   {}^{(4)}R_{ab} & = & {}^{(3)}R_{ab} + K K_{ab} - 2K_{ac}K^c_b - (\partial_t K_{ab} - {\mathcal L}_\beta K_{ab} + D_a D_b \alpha)/\alpha \\
   {}^{(4)}R_a{}^t & = & (D_c K^c_a - D_a K)/\alpha \\
   {}^{(4)}R^{tt} & = & (\partial_t K - {\mathcal L}_\beta K - \alpha K_{ab}K^{ab} + D_a D^a \alpha)/\alpha^3
\end{array}

and the curvature scalar is


   {}^{(4)}R = {}^{(3)}R + K_{ab}K^{ab} + K^2 - 2(\partial_t K - {\mathcal L}_\beta K + D_a D^a\alpha)/\alpha

The upper t indices can be swapped for lower indices with use of the identity


T^t = (-T_t + T_a \beta^a)/\alpha^2

where T^\mu is a spacetime tensor with only one index displayed.

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