Thus, we may write $\delta_{j}^{i}=e_{j}{}^{b}e_{b}{}^{i}$ that implies

$g_{ij}=e_{i}{}^{a}e_{j}{}^{b}g_{ab}.$ (263)

Accordingly, Eqs. (261) and (263) are inverse to Eqs. (260) and (262),
respectively.

We may obtain components of any vectors $V^{a}$ or tensors $T^{ab}{}_{cd}$
with respect to the basis $\partial/\partial x^{i}$ or $\mathrm{e}_{a}$ (see
e.g. Schouten (1954); Heckmann and Schücking (1962); Ellis (1964); Ellis and
van Elst (1999)):

$\displaystyle V^{a}=$ $\displaystyle e^{a}{}_{i}V^{i},$ (264) $\displaystyle
T^{ab}{}_{cd}=$ $\displaystyle
e^{a}{}_{i}e^{a}{}_{j}e_{c}{}^{k}e_{d}{}^{l}T^{ij}{}_{kl}.$ (265)

The Ricci rotation coefficients are typically regarded as tetrad components of
the Christoffel symbols Ellis (1967) or connection components for the tetrad
Ellis and van Elst (1999):

$\Gamma^{c}{}_{ab}\equiv\mathrm{e}_{a}\cdot\nabla_{b}\mathrm{e}^{c}=e_{i}^{c}e_{b}^{j}\nabla_{j}e_{a}^{i},$
(266)

where $\Gamma^{c}{}_{ab}$ calculates the $c$-component of the covariant
derivative in the $\mathrm{e}_{b}$-direction of the basic vector
$\mathrm{e}_{a}$.

Using the rotation coefficients, the covariant derivatives of any vectors
$V^{a}$ or tensors $T_{bc}$ can be expressed in terms of tetrad components as
follows Schouten (1954); Ellis (1964):

$\displaystyle\begin{array}[]{cc}{\nabla_{b}V^{a}=\mathrm{e}_{b}\left({V^{a}}\right)+\Gamma^{a}{}_{cb}V^{c},}&{\nabla_{b}V_{a}=\mathrm{e}_{b}\left({V_{a}}\right)-\Gamma^{c}{}_{ab}V_{c},}\end{array}$
(268)
$\displaystyle\nabla_{a}T_{bc}=\mathrm{e}_{a}\left({T_{bc}}\right)-\Gamma^{d}{}_{ba}T_{dc}-\Gamma^{d}{}_{ca}T_{bd}.$
(269)

But, for any scalars $f$, we shall calculate the derivative of $f$ in the
$\mathrm{e}_{a}$-direction by

$\mathrm{e}_{a}(f)=e_{a}{}^{i}\partial_{i}f.$ (270)

We know that $\mathrm{e}_{a}(g_{bc})=\nabla_{a}g_{bc}=0$, so the rotation
coefficients are symmetric on the two indices

$\Gamma_{(ab)c}=\Gamma_{abc}+\Gamma_{bac}=0,$ (271)

where they are only raised and lowered.

###  8.1 Commutators

The Lie derivative of $\mathrm{e}_{b}$ relative to $\mathrm{e}_{a}$ is
expressed by the $\mathrm{[e}_{a},\mathrm{e}_{b}]$ commutator calculated by

$\mathrm{[e}_{a},\mathrm{e}_{b}]f=\partial_{a}\left({\partial_{b}f}\right)-\partial_{b}\left({\partial_{a}f}\right).$
(272)

Following Ellis (1967), the $\mathrm{[e}_{a},\mathrm{e}_{b}]$ commutator is
connected to a basic vector $\mathrm{e}_{c}$ via commutation functions
$\gamma_{ab}^{c}$:

$\begin{array}[]{cc}{\mathrm{[e}_{a},\mathrm{e}_{b}]\equiv\gamma_{ab}^{c}\mathrm{e}_{c},}&{\gamma_{ab}^{c}}=\gamma_{[ab]}^{c},\end{array}$
(273)

where the commutation functions $\gamma_{ab}^{c}$ satisfy

$\gamma_{ab}^{c}=2\Gamma_{[ab]}^{c}=\left(\Gamma_{ab}^{c}-\Gamma_{ba}^{c}\right).$
(274)

Eq. (271) leads to the inverse of Eq. (274), the so-called the Christoffel
relation:

$\Gamma_{abc}={\textstyle{\frac{1}{2}}}\left({\gamma_{abc}+\gamma_{cab}-\gamma_{bca}}\right).$
(275)

It can be seen that the rotation coefficients $\Gamma_{abc}$ are linearly
connected to the commutation functions $\gamma_{abc}$, and vice versa.

Suppose that the timelike direction of the orthonormal frame
$\left\\{{\mathrm{e}_{a}}\right\\}$ is aligned with the preferred timelike
reference congruence $\mathrm{e}_{0}=\mathrm{u}$ ($u^{a}=\delta^{a}{}_{0}$,
$u_{a}=-\delta^{0}{}_{a}$), the commutation functions with one or two indices
set to zero can be written in term of the frame components of the kinematic
quantities (28) and (29) of the timelike congruence, so we have Estabrook and
Wahlquist (1964); Wahlquist and Estabrook (1966)

$\Omega^{a}\equiv-{{\frac{1}{2}}}\varepsilon^{abc}\mathrm{e}_{b}\cdot\mathrm{\dot{e}}_{c},$
(276)

where $\Omega^{a}$ is the local angular velocity in the rest-frame of an
observer with 4-velocity $u^{a}$, and $\mathrm{\dot{e}}_{a}\equiv
u^{b}\nabla_{b}\mathrm{e}_{a}$ is a Fermi-propagation axes relative to
$\left\\{{\mathrm{e}_{a}}\right\\}$. For the Fermi-derivatives
$\mathrm{e}_{a}\cdot\mathrm{\dot{e}}_{b}\equiv
e_{a}^{i}u^{j}\nabla_{j}e_{bi}$, we get
$\mathrm{e}_{a}\cdot\mathrm{\dot{e}}_{b}=-\mathrm{e}_{b}\cdot\mathrm{\dot{e}}_{a}$.

The spatial functions $\gamma^{a}{}_{bc}$ are split into a symmetric tensor
$\mathrm{n}^{ab}=\mathrm{n}^{(ab)}$ and an antisymmetric term specified by a
vector $\mathrm{a}^{a}$:777This notation was first introduced by Engelbert
Schücking and Wolfgang Kundt in the Hamburg Relativity Seminars organized by
Pascual Jordan in 1962–1963.

$\gamma^{a}{}_{bc}=\varepsilon_{bce}\mathrm{n}^{ae}+2\delta^{a}{}_{[c}\mathrm{a}_{b]},$
(277)

where $\mathrm{n}^{ab}$ and $\mathrm{a}_{b}$ are defined by

$\begin{array}[]{cc}{\mathrm{n}^{ab}\equiv{{\frac{1}{2}}}\gamma^{(a}{}_{cd}\varepsilon^{b)cd},}&{\mathrm{a}_{b}\equiv{{\frac{1}{2}}}\gamma^{a}{}_{ba},}\end{array}$
(278)

where $\varepsilon_{abc}$ is the spatial permutation tensor with
$\varepsilon_{123}=1=\varepsilon^{123}$.

From Eq. (275), the rotation coefficients have the following components Ellis
(1967); Ellis and MacCallum (1969); van Elst and Ellis (1998):

$\displaystyle\Gamma_{a00}=$ $\displaystyle\dot{u}_{a},$ (279)
$\displaystyle\Gamma_{a0b}=$
$\displaystyle{{\frac{1}{3}}}\Theta\delta_{ab}+\sigma_{ab}-\varepsilon_{abc}\omega^{c},$
(280) $\displaystyle\Gamma_{ab0}=$ $\displaystyle\varepsilon_{abc}\Omega^{c},$
(281) $\displaystyle\Gamma_{abc}=$ $\displaystyle
2\mathrm{a}_{[a}\delta_{b]c}+\varepsilon_{ce[a}\mathrm{n}^{e}{}_{b]}+{{\frac{1}{2}}}\varepsilon_{abe}\mathrm{n}^{e}{}_{c}.$
(282)

The first two equations contain the kinematic quantities, whereas the latter
two equations include the rotation rate of the spatial frame
$\left\\{{\mathrm{e}_{a}}\right\\}$ relative to a Fermi-propagation basis and
the quantities ($\mathrm{a}_{b}$ and $\mathrm{n}_{bc}$) determining the nine
rotation coefficients, respectively.

Considering the commutators (273) and the variables introduced in Eq. (277),
we have

$\displaystyle[\mathrm{e}_{0},\mathrm{e}_{a}]=$
$\displaystyle\dot{u}_{a}\mathrm{e}_{0}-\left({{{\frac{1}{3}}}\Theta\delta^{b}{}_{a}+\sigma^{b}{}_{a}+\varepsilon^{b}{}_{ac}(\omega^{c}+\Omega^{c})}\right)\mathrm{e}_{b},$
(283) $\displaystyle[\mathrm{e}_{a},\mathrm{e}_{b}]=$ $\displaystyle
2\varepsilon_{abc}\omega^{c}\mathrm{e}_{0}+\left({2\mathrm{a}_{[a}\delta^{c}{}_{b]}+\varepsilon_{abd}n^{dc}}\right)\mathrm{e}_{c}.$
(284)

Under the conditions, where a spatial frame vector $\mathrm{e}_{a}$ is
hypersurface orthogonal (see e.g. Wald (1984)), we get van Elst and Uggla
(1997)

$\begin{array}[]{cc}{0=\sigma_{ab}+\varepsilon_{abc}(\omega^{c}+\Omega^{c})}&{a\neq
b\neq c,}\\\ {0=n^{a}{}_{a}}&{\mathrm{no~{}summation}\mathrm{.}}\end{array}$
(285)

###  8.2 Tetrad Curvature

Let us consider $u^{a}$ as the basis vector $\mathrm{e}_{b}$ (i.e.
$\mathrm{e}_{b}\cdot u^{a}=\delta_{b}^{a}$). Applying Eq. (269) to the Ricci
identities (93) results in the tetrad Riemann curvature given by

$R^{a}{}_{bcd}=\mathrm{e}_{c}\left({\Gamma^{a}{}_{bd}}\right)-\mathrm{e}_{d}\left({\Gamma^{a}{}_{bc}}\right)+\Gamma^{a}{}_{ec}\Gamma^{e}{}_{bd}-\Gamma^{a}{}_{ed}\Gamma^{e}{}_{bc}-\Gamma^{a}{}_{be}\gamma^{e}{}_{cd}.$
(286)

Contracting the tetrad Riemann curvature leads to the tetrad Ricci curvature:

$R_{bd}=\mathrm{e}_{a}\left({\Gamma^{a}{}_{bd}}\right)-\mathrm{e}_{d}\left({\Gamma^{a}{}_{ba}}\right)+\Gamma^{a}{}_{ea}\Gamma^{e}{}_{bd}-\Gamma^{a}{}_{de}\Gamma^{e}{}_{ba}=T_{bd}-{{\frac{1}{2}}}Tg_{bd}.$
(287)

The antisymmetry properties of the Riemann curvature ($R^{a}{}_{[bcd]}=0$)
correspond to the Jacobi identity

$[e_{b},[e_{c},e_{d}]]+[e_{c},[e_{d},e_{b}]]+[e_{d},[e_{b},e_{c}]]=0,$ (288)

which can be simplified as

$\mathrm{e}_{[a}\left({\gamma^{d}{}_{bc]}}\right)+\gamma^{e}{}_{[ab}\gamma^{d}{}_{c]e}=0.$
(289)

It represents the integrability conditions, where $\gamma_{ab}^{c}$ are the
commutation functions for a set of basic vectors $\mathrm{e}_{a}$.

Substituting the tetrad forms (286) and (287) into Eq. (37), we derive the
Einstein equations, along with the 16 Jacobi identities and constraint and
evolution equations for $E_{ab}$ and $H_{ab}$ in terms of the tetrad
derivatives ($\mathrm{e}_{a}$), the tetrad variables ($\Omega_{a}$,
$\mathrm{a}_{a}$, and $\mathrm{n}_{ab}$), and the kinematic and dynamical
quantities (see van Elst (1996); van Elst and Uggla (1997); Ellis and van Elst
(1999); van Elst and Ellis (1996, 1998) for full details).

###  8.3 Correspondence between Covariant and Tetrad Formulations

Here we summarize how the 1 + 3 covariant notations correspond to the 1+3
tetrad analogues. To obtain the equivalent tetrad formulations, we can use the
following conversion rules van Elst (1996); Ellis et al. (1983a, b):

$\displaystyle\dot{f}\rightarrow$ $\displaystyle\mathrm{e}_{0}\left(f\right),$
(290) $\displaystyle\dot{V}_{\left\langle a\right\rangle}\rightarrow$
$\displaystyle\mathrm{e}_{0}\left({V_{a}}\right)-\varepsilon_{abc}\Omega^{b}V^{c}=\mathrm{e}_{0}\left({V_{a}}\right)-[\Omega,V]_{a},$
(291) $\displaystyle\dot{S}_{\left\langle{ab}\right\rangle}\rightarrow$
$\displaystyle\mathrm{e}_{0}\left({S_{ab}}\right)-2\varepsilon_{cd\left\langle
a\right.}\Omega^{c}S_{\left.b\right\rangle}{}^{d}=\mathrm{e}_{0}\left({S_{ab}}\right)-2[\Omega,S]_{\left\langle{ab}\right\rangle},$
(292) $\displaystyle\mathrm{D}_{a}f\rightarrow$
$\displaystyle\delta_{ab}\mathrm{e}^{b}\left(f\right),$ (293)
$\displaystyle\mathrm{D}^{a}V_{a}\rightarrow$
$\displaystyle\left({\mathrm{e}^{a}-2\mathrm{a}^{a}}\right)\left({V_{a}}\right),$
(294) $\displaystyle\mathrm{D}^{b}S_{ab}\rightarrow$
$\displaystyle\left({\mathrm{e}^{b}-3\mathrm{a}^{b}}\right)\left({S_{ab}}\right)-\varepsilon_{abc}\mathrm{n}^{bd}S_{d}{}^{c}=\left({\mathrm{e}^{b}-3\mathrm{a}^{b}}\right)\left({S_{ab}}\right)-[\mathrm{n},S]_{a},$
(295) $\displaystyle\mathrm{curl}\left({V_{a}}\right)\rightarrow$
$\displaystyle\varepsilon_{abc}\left({\mathrm{e}^{b}-\mathrm{a}^{b}}\right)V^{c}-\mathrm{n}_{a}{}^{b}V_{b}=[\mathrm{e}-\mathrm{a},V]_{a}-\mathrm{n}_{a}{}^{b}V_{b},$
(296) $\displaystyle\mathrm{curl}(S_{ab})\rightarrow$
$\displaystyle\varepsilon_{cd\left\langle
a\right.}\left({\mathrm{e}^{c}-\mathrm{a}^{c}}\right)\left({S_{\left.b\right\rangle}{}^{d}}\right)-3\mathrm{n}_{\left\langle
a\right.}^{c}S_{\left.b\right\rangle
c}+{{\frac{1}{2}}}\mathrm{n}^{c}{}_{c}S_{ab}$
$\displaystyle=[\mathrm{e}-\mathrm{a},S]_{\left\langle{ab}\right\rangle}-3\mathrm{n}_{\left\langle
a\right.}{}^{c}S_{\left.b\right\rangle
c}+{{\frac{1}{2}}}\mathrm{n}^{c}{}_{c}S_{ab},$ (297)
$\displaystyle\mathrm{D}_{(a}V_{b)}\rightarrow$
$\displaystyle\left({\mathrm{e}_{(a}+\mathrm{a}_{(a}}\right)\left({V_{b)}}\right)-\delta_{ab}\mathrm{a}^{c}V_{c}-\varepsilon_{cd(a}\mathrm{n}_{b)}{}^{c}V^{d},$
(298) $\displaystyle\mathrm{D}_{\left\langle
a\right.}V_{\left.b\right\rangle}\rightarrow$
$\displaystyle\left({\mathrm{e}_{\left\langle
a\right.}+\mathrm{a}_{\left\langle
a\right.}}\right)\left({V_{\left.b\right\rangle}}\right)-\varepsilon_{cd\left\langle
a\right.}\mathrm{n}_{\left.b\right\rangle}{}^{c}V^{d}=\left({\mathrm{e}_{\left\langle
a\right.}+\mathrm{a}_{\left\langle
a\right.}}\right)\left({V_{\left.b\right\rangle}}\right)-[\mathrm{n},V]_{\left\langle{ab}\right\rangle}.$
(299)

Applying the above rules to the Bianchi equations (100)–(103) with a perfect-
fluid matter yields

$\displaystyle\mathrm{e}^{b}E_{ab}-3\mathrm{a}^{b}E_{ab}-[\mathrm{n},E]_{a}$
$\displaystyle={{\frac{1}{3}}}\delta_{ab}\mathrm{e}^{a}\left(\rho\right)+3\omega^{b}H_{ab}+[\sigma,H]_{a},$
(300)
$\displaystyle\mathrm{e}^{b}H_{ab}-3\mathrm{a}^{b}H_{ab}-[\mathrm{n},H]_{a}$
$\displaystyle=-\omega_{a}(\rho+p)-3\omega^{b}E_{ab}-[\sigma,E]_{a},$ (301)
$\displaystyle[\mathrm{e},H]_{\left\langle{ab}\right\rangle}-[\mathrm{a}-2\dot{u},H]_{\left\langle{ab}\right\rangle}$
$\displaystyle-3\mathrm{n}_{\left\langle
a\right.}{}^{c}H_{\left.b\right\rangle
c}+{{\frac{1}{2}}}\mathrm{n}^{c}{}_{c}H_{ab}$
$\displaystyle=\mathrm{e}_{0}\left({E_{ab}}\right)-[\omega+2\Omega,E]_{\left\langle{ab}\right\rangle}$
$\displaystyle-3\sigma_{c\left\langle
a\right.}E_{\left.b\right\rangle}{}^{c}+\Theta
E_{ab}+{{\frac{1}{2}}}\sigma_{ab}(\rho+p),$ (302)
$\displaystyle[\mathrm{e},E]_{\left\langle{ab}\right\rangle}-[\mathrm{a}-2\dot{u},E]_{\left\langle{ab}\right\rangle}$
$\displaystyle-3\mathrm{n}_{\left\langle
a\right.}{}^{c}E_{\left.b\right\rangle
c}+{{\frac{1}{2}}}\mathrm{n}^{c}{}_{c}E_{ab}$
$\displaystyle=-\mathrm{e}_{0}\left({H_{ab}}\right)+[\omega+2\Omega,H]_{\left\langle{ab}\right\rangle}$
$\displaystyle+3\sigma_{c\left\langle
a\right.}H_{\left.b\right\rangle}{}^{c}-\Theta H_{ab}.$ (303)

The above equations can be expressed in a symmetric normal hyperbolic form
that determines their hyperbolic characteristics van Elst and Ellis (1998).

##  9 1 + 1 + 2 Semi-covariant Formalism

A semi-covariant approach has been developed Clarkson and Barrett (2003);
Clarkson et al. (2004); Betschart and Clarkson (2004); Clarkson (2007), which
keeps the timelike vector of the 1 + 3 method, but separates one spacelike
vector from the 3-D space manifold (see also Greenberg (1970); Tsamparlis and
Mason (1983); Mason and Tsamparlis (1985); Tsamparlis (1992) for similar
formulations). The other two dimensions remain untouched, in contrast to the 3
spatial dimensions in the $1+3$ covariant approach. The $1+1+2$ semi-covariant
formalism could be a halfway between the $1+3$ covariant and tetrad methods.

While the 1+3 covariant approach split spacetime into time and space using a
timelike 4-velocity vector $u^{a}$ ($u^{a}u_{a}=-1$), the $1+1+2$ semi-
covariant formalism additionally decomposes space into one dimension and 2D
surface with the help of a unit spacelike vector $n^{a}$ (i.e. $n^{a}n_{a}=1$
and $u^{a}n_{a}=0$). The projection tensor $h_{ab}$ given by Eq. (1) can be
split into the spacelike vector $n^{a}$ and a new projection tensor $N_{ab}$
as follows:

$N_{ab}\equiv h_{ab}-n_{a}n_{b}=g_{ab}+u_{a}u_{b}-n_{a}n_{b}.$ (304)

The tensor $N_{ab}$, which projects vectors orthogonal to $n^{a}$ and $u^{a}$
onto a 2-surface referred to as the sheets Betschart and Clarkson (2004), has
the following properties:

$\begin{array}[]{ccc}{N_{ab}n^{b}=0=N_{ab}u^{b},}&{N_{a}{}^{c}N_{cb}=N_{ab},}&{N_{a}{}^{a}=2.}\end{array}$
(305)

The alternating Levi-Civita 2-tensor $\varepsilon_{ab}$, is determined from
the volume element of the observers’ rest-spaces

$\varepsilon_{ab}\equiv\varepsilon_{abc}n^{c}=u^{d}\eta_{dabc}n^{c},$ (306)

which possesses the following identities and contractions:

$\displaystyle\begin{array}[]{cc}{\varepsilon_{abc}=2n_{[a}\varepsilon_{b]c}+n_{c}\varepsilon_{ab},}&{\varepsilon_{ab}\varepsilon^{cd}=2!N_{[a}{}^{c}N_{b]}{}^{d},}\end{array}$
(308)
$\displaystyle\begin{array}[]{cccc}{\varepsilon_{abc}u^{c}=0,}&{\varepsilon_{abc}=\varepsilon_{[abc]},}&{\varepsilon_{ac}\varepsilon^{bc}=N_{a}{}^{b},}&{\varepsilon_{ac}\varepsilon^{ac}=2.}\end{array}$
(310)

The projection tensor $N_{ab}$ can be employed to irreducibly decompose any
3-vector $V_{a}$ into a scalar part $\mathcal{V}$ of the vector parallel to
$n_{a}$, and a 2-vector $\mathcal{V}_{a}$ sitting on the sheet orthogonal to
$n_{a}$:

$V_{a}=\mathcal{V}n_{a}+\mathcal{V}_{a},$ (311)

where $\mathcal{V}$ and $\mathcal{V}_{a}$ can be specified by

$\begin{array}[]{cc}{\mathcal{V}\equiv n_{a}V^{a},}&{\mathcal{V}_{a}\equiv
V_{\\{a\\}}=N_{a}{}^{b}V_{b}.}\end{array}$ (312)

Similarly, a PSTF tensor $W_{ab}$ can be split into a scalar $\mathcal{W}$, a
vector $\mathcal{W}_{a}$, and a 2-tensor $\mathcal{W}_{ab}$ as follows:

$W_{ab}=\mathcal{W}(n_{a}n_{b}-{{\frac{1}{2}}}N_{ab})+2\mathcal{W}_{(a}n_{b)}+\mathcal{W}_{ab},$
(313)

where $\mathcal{W}$, $\mathcal{W}_{a}$, and $\mathcal{W}_{ab}$ can be obtained
using

$\displaystyle\mathcal{W}=$ $\displaystyle n_{a}n_{b}W^{ab},$ (314)
$\displaystyle\mathcal{W}_{a}=$ $\displaystyle
n^{c}W_{\\{b\\}c}=N_{a}{}^{b}n^{c}W_{bc},$ (315)
$\displaystyle\mathcal{W}_{ab}=$ $\displaystyle
W_{\\{ab\\}}=\left({N_{(a}{}^{c}N_{b)}{}^{d}-{{\frac{1}{2}}}N_{ab}N^{cd}}\right)W_{ab}.$
(316)

The curly brackets are utilized to denote the projection of a vector with
$N_{ab}$ and the transverse-traceless (TT) part of a tensor. This approach
allows us to decompose any object into scalars, 2-vectors laying in the sheet,
and TT 2-tensors again in the sheet.

Using $n^{a}$ and $N_{ab}$, one can also define the following two new
derivatives of any tensors:

$\displaystyle\hat{T}_{a\cdots}=$ $\displaystyle
n^{b}\mathrm{D}_{b}T_{a\cdots},$ (317) $\displaystyle\delta_{b}T_{a\cdots}=$
$\displaystyle N_{b}{}^{d}N_{a}{}^{c}\cdots\mathrm{D}_{d}T_{c\cdots},$ (318)

where the hat ($\hat{~{}}$) denotes the derivative along $n^{a}$ in the
surfaces orthogonal to $u^{a}$, and $\delta$ denotes a projected derivative on
the sheet, with projection on every free index, which is analogous to the
spatial derivative $\mathrm{D}$ in the $1+3$ covariant formalism

The above derivatives can be employed to provide the following algebraic
identities Clarkson (2007):

$\displaystyle{\dot{N}_{ab}=2u_{(a}\dot{u}_{b)}-2n_{(a}\dot{n}_{b)}=2u_{(a}\mathcal{A}_{b)}-2n_{(a}\alpha_{b)},}$
(319)
$\displaystyle{\dot{\varepsilon}_{ab}=-2u_{[a}\varepsilon_{b]c}\mathcal{A}^{c}+2n_{[a}\varepsilon_{b]c}\alpha^{c},}$
(320)
$\displaystyle\begin{array}[]{cc}{\hat{N}_{ab}=-2n_{(a}\hat{n}_{b)},}&{\delta_{c}N_{ab}=0,}\end{array}$
(322)
$\displaystyle\begin{array}[]{cc}{\hat{\varepsilon}_{ab}=2n_{[a}\varepsilon_{b]c}\hat{n}^{c},}&{\delta_{c}\varepsilon_{ab}=0,}\end{array}$
(324)

where we have

$\displaystyle\begin{array}[]{ccc}{\dot{n}_{a}=\mathcal{A}u_{a}+\alpha_{a},}&{\mathcal{A}=n_{a}\dot{u}^{a},}&{\alpha_{a}=\dot{n}_{\\{a\\}},}\end{array}$
(326)
$\displaystyle\begin{array}[]{cc}{\dot{u}_{a}=\mathcal{A}n_{a}+\mathcal{A}_{a},}&{\mathcal{A}_{a}=\dot{u}_{\\{a\\}}.}\end{array}$
(328)

The spatial derivatives $\mathrm{D}$ of any scalars, projected 2-vectors, and
TT 2-tensors, their spatial derivatives can be split into the $1+1+2$
notations:

$\displaystyle\mathrm{D}_{a}\mathcal{F}=$
$\displaystyle\delta_{a}\mathcal{F}+n_{a}\hat{\mathcal{F}},$ (329)
$\displaystyle\mathrm{D}_{a}\mathcal{V}_{b}=$
$\displaystyle\delta_{a}\mathcal{V}_{b}+n_{a}\left\\{{\hat{\mathcal{V}}_{\\{b\\}}-\hat{n}_{c}\mathcal{V}^{c}u_{b}}\right\\}$
$\displaystyle-
n_{b}\left\\{{{{\frac{1}{2}}}\phi\mathcal{V}_{a}+(\zeta_{ac}+\xi\varepsilon_{ac})\mathcal{V}^{c}}\right\\},$
(330) $\displaystyle\mathrm{D}_{a}\mathcal{W}_{bc}=$
$\displaystyle\delta_{a}\mathcal{W}_{bc}+n_{a}\left\\{{\hat{\mathcal{W}}_{bc}-2n_{(b}\mathcal{W}_{c)d}\hat{n}^{d}}\right\\}$
$\displaystyle-2n_{(b}\left\\{{{{\frac{1}{2}}}\phi\mathcal{W}_{c)a}+\mathcal{W}_{c)}{}^{d}(\zeta_{ad}+\xi\varepsilon_{ad})}\right\\}.$
(331)

By analogy with Eqs. (28) and (29), the spatial derivative $\mathrm{D}$ of
$n^{a}$ orthogonal to $u^{a}$ is split into the following irreducible
kinematic quantities:

$\displaystyle\mathrm{D}_{b}n_{a}=$
$\displaystyle\delta_{b}n_{a}+\hat{n}_{a}n_{b},$ (332)
$\displaystyle\delta_{b}u_{a}=$ $\displaystyle{{\frac{1}{2}}}\phi
N_{ab}+\zeta_{ab}+\xi\varepsilon_{ab},$ (333)

where $\hat{n}_{a}=n^{b}\mathrm{D}_{b}n_{a}$ is the acceleration of the sheet
moving along $n^{a}$, $\phi=\delta^{a}n_{a}$ is the expansion of the sheet,
$\zeta_{ab}=\delta_{\\{a}n_{b\\}}$ represents the distortion of the sheet
calculated by the shear of $n^{a}$, and
$\xi=-{{\frac{1}{2}}}\varepsilon_{ab}\delta^{a}n^{b}$ is the twist of the
sheet described by the rotation of $n^{a}$. These kinematic quantities of
$n^{a}$ in the $1+1+2$ approach can be treated in the same way as the
kinematical variables of $u^{a}$ in the $1+3$ method.

To formulate the $1+1+2$ approach, we should also decompose the kinematical
quantities defined by Eqs. (28) and (29), the Weyl fields, and the energy-
momentum tensor. The acceleration is provided by (328), but for the vorticity
and shear, we have the following decomposition:

$\displaystyle\omega_{a}=$ $\displaystyle\Omega n_{a}+\Omega_{a},$ (334)
$\displaystyle\sigma_{ab}=$
$\displaystyle\Sigma(n_{a}n_{b}-{{\frac{1}{2}}}N_{ab})+2\Sigma_{(a}n_{b)}+\Sigma_{ab}.$
(335)

The gravitoelectric tensor is splits into $\mathcal{E}$, $\mathcal{E}_{a}$,
and $\mathcal{E}_{ab}$, similarly the gravitomagnetic tensor, as follows

$\displaystyle E_{ab}=$
$\displaystyle\mathcal{E}(n_{a}n_{b}-{{\frac{1}{2}}}N_{ab})+2\mathcal{E}_{(a}n_{b)}+\mathcal{E}_{ab},$
(336) $\displaystyle H_{ab}=$
$\displaystyle\mathcal{H}(n_{a}n_{b}-{{\frac{1}{2}}}N_{ab})+2\mathcal{H}_{(a}n_{b)}+\mathcal{H}_{ab}.$
(337)

The shear scalar $\sigma$ is calculated by

$\sigma^{2}={{\frac{1}{2}}}\sigma_{ab}\sigma^{ab}={{\frac{3}{4}}}\Sigma^{2}+\Sigma_{a}\Sigma^{a}+{{\frac{1}{2}}}\Sigma_{ab}\Sigma^{ab}.$
(338)

Finally, the energy flux $q^{a}$ and the anisotropic stress $\pi_{ab}$ of the
energy-momentum tensor (32) are split into

$\displaystyle q_{a}=$ $\displaystyle\mathcal{Q}n_{a}+\mathcal{Q}_{a},$ (339)
$\displaystyle\pi_{ab}=$
$\displaystyle\Pi(n_{a}n_{b}-{{\frac{1}{2}}}N_{ab})+2\Pi_{(a}n_{b)}+\Pi_{ab}.$
(340)

We see how the $1+3$ variables are split to formulate their $1+1+2$
counterparts, which produce new variables associated with the irreducible
parts of $\nabla_{a}n_{b}$. Accordingly, the $1+1+2$ Bianchi and Ricci
equations can be constructed for the timelike vector $u^{a}$ and the spacelike
vector $n^{a}$ in terms of all the quantities introduced above (see e.g.
Betschart and Clarkson (2004); Clarkson et al. (2004); Clarkson (2007)).

###  9.1 Correspondence between Covariant and Semi-covariant Formulations

The $1+3$ covariant formulations can simply be mapped into their $1+1+2$ semi-
covariant analogues using the following rules Clarkson (2007):

$\displaystyle\dot{V}_{\langle a\rangle}\rightarrow$
$\displaystyle\dot{\mathcal{V}}_{\\{a\\}}+(\dot{\mathcal{V}}-\mathcal{V}_{b}\dot{n}^{\\{b\\}})n_{a}+\mathcal{V}\dot{n}_{\\{a\\}},$
(341) $\displaystyle\dot{W}_{\langle ab\rangle}\rightarrow$
$\displaystyle\dot{\mathcal{W}}_{\\{ab\\}}+(\dot{\mathcal{W}}-2\mathcal{W}_{c}\dot{n}^{\\{c\\}})n_{a}n_{b}-{{\frac{1}{2}}}\dot{\mathcal{W}}N_{ab}$
$\displaystyle+\left\\{{3\mathcal{W}\dot{n}_{(\\{a\\}}+2\mathcal{W}_{(\\{a\\}}-2\dot{n}^{\\{c\\}}\mathcal{W}_{c(a}}\right\\}n_{b)}+2\mathcal{W}_{(a}\dot{n}_{\\{b\\})},$
(342) $\displaystyle\mathrm{D}_{a}V^{a}\rightarrow$
$\displaystyle(\delta_{a}-\hat{n}_{a})\mathcal{V}^{a}+\hat{\mathcal{V}}+\phi\mathcal{V},$
(343) $\displaystyle\mathrm{curl}V_{a}\rightarrow$
$\displaystyle\varepsilon_{bc}\delta^{b}\mathcal{V}^{c}n_{a}-\varepsilon_{ab}\hat{\mathcal{V}}^{b}+\xi(\mathcal{V}_{a}+2\mathcal{V}n_{a})+\varepsilon_{ab}\left\\{{(\delta^{b}-\hat{n}^{b})\mathcal{V}-\zeta^{ab}\mathcal{V}_{c}-{{\frac{1}{2}}}\phi\mathcal{V}^{b}}\right\\},$
(344) $\displaystyle\mathrm{D}_{\langle a}V_{b\rangle}\rightarrow$
$\displaystyle\delta_{\\{a}\mathcal{V}_{b\\}}+\mathcal{V}\zeta_{ab}+\left\\{{\mathcal{V}\hat{n}_{(a}+\mathcal{V}\delta_{(a}+\hat{\mathcal{V}}_{(\\{a\\}}-{{\frac{1}{2}}}\phi\mathcal{V}_{(a}+\mathcal{V}^{c}(\xi\varepsilon_{c(a}-\zeta_{c(a})}\right\\}n_{b)}$
$\displaystyle+{{\frac{1}{3}}}\left\\{{2\hat{\mathcal{V}}-\phi\mathcal{V}-\mathcal{V}_{c}(\delta^{c}+2\hat{n}^{c})}\right\\}(n_{a}n_{b}-{{\frac{1}{2}}}N_{ab}),$
(345) $\displaystyle\mathrm{D}^{b}W_{ab}\rightarrow$
$\displaystyle(\delta^{b}-\hat{n}^{b})\mathcal{W}_{ab}-\zeta^{ab}\mathcal{W}_{ab}n_{a}+(\delta_{b}-2\hat{n}_{b})\mathcal{W}^{b}n_{a}+\hat{\mathcal{W}}_{\\{a\\}}+(\zeta_{ab}-\xi\varepsilon_{ab})\mathcal{W}^{b}$
$\displaystyle+{{\frac{3}{2}}}\phi\mathcal{W}_{a}-{{\frac{1}{2}}}(\delta_{a}-3\hat{n}_{a})\mathcal{W}+\hat{\mathcal{W}}n_{a}+{{\frac{3}{2}}}\phi\mathcal{W}n_{a},$
(346) $\displaystyle\mathrm{(curl}W)_{\langle ab\rangle}\rightarrow$
$\displaystyle\varepsilon_{c\\{a}\hat{\mathcal{W}}_{b\\}}{}^{c}+\varepsilon^{cd}\delta_{c}\mathcal{W}_{d(a}n_{b)}+\xi\mathcal{W}_{ab}-{{\frac{3}{2}}}\mathcal{W}\varepsilon_{c\\{a}\zeta_{b\\}}{}^{c}+{{\frac{1}{2}}}\phi\varepsilon_{c\\{a}\mathcal{W}_{b\\}}{}^{c}$
$\displaystyle+2\varepsilon_{c\\{a}\hat{n}^{c}\mathcal{W}_{b\\}}-\varepsilon_{c\\{a}\delta^{c}\mathcal{W}_{b\\}}+\varepsilon^{cd}\mathcal{W}_{d}\zeta_{c(a}n_{b)}+5\xi\mathcal{W}_{(a}n_{b)}$
$\displaystyle+\varepsilon_{c(a}\left\\{{\
-{{\frac{3}{2}}}(\delta^{c}-\hat{n}^{c})\mathcal{W}+\hat{\mathcal{W}}^{c}+{{\frac{1}{2}}}\phi{\mathcal{W}}^{c}+2{\mathcal{W}}_{d}\zeta^{cd}}\right\\}n_{b)}$
$\displaystyle+\left({3\xi\mathcal{W}+\varepsilon_{cd}\delta^{c}{\mathcal{W}}^{d}-\varepsilon_{cd}{\mathcal{W}}^{de}\zeta^{c}{}_{e}}\right)\left({n_{a}n_{b}-{{\frac{1}{2}}}N_{ab}}\right),$
(347)

The above relations can be put into the covariant formulas to get their
equivalences in the semi-covariant formalism.

Applying the above rules to the perfect-fluid formulas (111)–(113) in
nonperturbative shear-free spacetimes, we obtain:

$\displaystyle(\delta^{b}-\hat{n}^{b})\mathcal{E}_{ab}+(\delta_{b}-2\hat{n}_{b})\mathcal{E}^{b}n_{a}+\hat{\mathcal{E}}_{\\{a\\}}$
$\displaystyle-{{\frac{1}{2}}}(\delta_{a}-3\hat{n}_{a})\mathcal{E}+\hat{\mathcal{E}}n_{a}$
$\displaystyle={{\frac{1}{3}}}\delta_{a}\rho+{{\frac{1}{3}}}n_{a}\hat{\rho},$
(348)
$\displaystyle(\delta^{b}-\hat{n}^{b})\mathcal{H}_{ab}+(\delta_{b}-2\hat{n}_{b})\mathcal{H}^{b}n_{a}+\hat{\mathcal{H}}_{\\{a\\}}$
$\displaystyle-{{\frac{1}{2}}}(\delta_{a}-3\hat{n}_{a})\mathcal{H}+\hat{\mathcal{H}}n_{a}$
$\displaystyle=-\Omega_{a}(\rho+p)-\Omega n_{a}(\rho+p),$ (349)
$\displaystyle\varepsilon^{cd}\delta_{c}\mathcal{H}_{d(a}n_{b)}+$
$\displaystyle\varepsilon_{c\\{a}\hat{\mathcal{H}}_{b\\}}{}^{c}+2\varepsilon_{c\\{a}\hat{n}^{c}\mathcal{H}_{b\\}}-\varepsilon_{c\\{a}\delta^{c}\mathcal{H}_{b\\}}$
$\displaystyle+$
$\displaystyle\varepsilon_{cd}\delta^{c}\mathcal{H}^{d}\left({n_{a}n_{b}-{{\frac{1}{2}}}N_{ab}}\right)+\varepsilon_{c(a}\hat{\mathcal{H}}^{c}n_{b)}-{{\frac{3}{2}}}\mathcal{H}\varepsilon_{c(a}(\delta^{c}-\hat{n}^{c})n_{b)}$
$\displaystyle=$
$\displaystyle\dot{\mathcal{E}}_{\\{ab\\}}-2\dot{n}^{\\{c\\}}\mathcal{E}_{c(a}n_{b)}+\Theta\mathcal{E}_{ab}-2\mathcal{E}_{c}\dot{n}^{\\{c\\}}n_{a}n_{b}+2\mathcal{E}_{(a}\dot{n}_{\\{b\\})}$
$\displaystyle+2\mathcal{E}_{(\\{a\\}}n_{b)}+2\Theta\mathcal{E}_{(a}n_{b)}+\dot{\mathcal{E}}\left({n_{a}n_{b}-{{\frac{1}{2}}}N_{ab}}\right)+3\mathcal{E}\dot{n}_{(\\{a\\}}n_{b)}$
$\displaystyle+\Theta\mathcal{E}(n_{a}n_{b}-{{\frac{1}{2}}}N_{ab})+\Sigma_{ab}(\rho+p)+2\Sigma_{(a}n_{b)}(\rho+p)$
$\displaystyle+{{\frac{1}{2}}}\Sigma(n_{a}n_{b}-{{\frac{1}{2}}}N_{ab})(\rho+p),$
(350) $\displaystyle\varepsilon^{cd}\delta_{c}\mathcal{E}_{d(a}n_{b)}+$
$\displaystyle\varepsilon_{c\\{a}\hat{\mathcal{E}}_{b\\}}{}^{c}+2\varepsilon_{c\\{a}\hat{n}^{c}\mathcal{E}_{b\\}}-\varepsilon_{c\\{a}\delta^{c}\mathcal{E}_{b\\}}$
$\displaystyle+$
$\displaystyle\varepsilon_{cd}\delta^{c}\mathcal{E}^{d}\left({n_{a}n_{b}-{{\frac{1}{2}}}N_{ab}}\right)+\varepsilon_{c(a}\hat{\mathcal{E}}^{c}n_{b)}-{{\frac{3}{2}}}\mathcal{E}\varepsilon_{c(a}(\delta^{c}-\hat{n}^{c})n_{b)}$
$\displaystyle=$
$\displaystyle-\dot{\mathcal{H}}_{\\{ab\\}}+2\dot{n}^{\\{c\\}}\mathcal{H}_{c(a}n_{b)}-\Theta\mathcal{H}_{ab}+2\mathcal{H}_{c}\dot{n}^{\\{c\\}}n_{a}n_{b}-2\mathcal{H}_{(a}\dot{n}_{\\{b\\})}$
$\displaystyle-2\mathcal{H}_{(\\{a\\}}n_{b)}-2\Theta\mathcal{H}_{(a}n_{b)}-\dot{\mathcal{H}}\left({n_{a}n_{b}-{{\frac{1}{2}}}N_{ab}}\right)-3\mathcal{H}\dot{n}_{(\\{a\\}}n_{b)}$
$\displaystyle-\Theta\mathcal{H}(n_{a}n_{b}-{{\frac{1}{2}}}N_{ab}).$ (351)

The above linear equations describe the evolution of the Weyl fields in a
perfect-fluid model according to in the $1+1+2$ approach (see Clarkson (2007)
for a generic matter field). We see that the $1+1+2$ formalism offers an
alternative description of the gravitoelectric/-magnetic fields in terms of
quantities with clear physical or geometrical meanings, namely scalar,
2-vector, and TT 2-tensors. This instrument has been used for a number of
studies, such as electromagnetic perturbations on locally rotationally
symmetric spacetimes Betschart and Clarkson (2004), electromagnetic radiation
made by gravitational waves interacting with a strong magnetic field of
vibrating massive compact objects Clarkson et al. (2004), and perturbations in
rotationally symmetric spacetimes Clarkson (2007).

##  10 Observational Effects

Astrophysical observations are performed by detecting electromagnetic waves
(light) and gravitational waves that have traveled through the past lightcone.
We know from where the light originates and how it reaches us. Although
astronomical observations allow us to collect information about the universe,
there is a strict limit on the information due to constraints imposed by the
past lightcone Ellis et al. (1985). In addition to astrophysics, observations
also enable us to collect information about the kinematic properties of large-
scale structures such as the CMB anisotropies, where are described by global
properties of $u^{a}$ e.g. the shear (see Ellis et al. (1983a, b); Stoeger et
al. (1995); Maartens et al. (1995)). The propagation of electromagnetic waves
and gravitational waves is affected by the spacetime metric, in other words
the kinematic quantities, so the description of it in terms of the kinematic
quantities will help us to better interpret the observed effects of the
gravitoelectric/-magnetic field on small scales near compact objects such as
around black holes, as well as large-scale structures (e.g. CMB).

Covariant handling of observations was first implemented by Kristian and Sachs
Kristian and Sachs (1966), MacCallum and Ellis MacCallum and Ellis (1970).
They tried to link different distance measurements to redshift based on the
source’s intrinsic and observed brightness. The expansion of the universe
results in an energy loss of the electromagnetic waves reaching us. The energy
loss of each photon is described by the redshift $z$, i.e.
${\nu_{\mathrm{obsv}}}=(1+z)^{-1}{\nu_{\mathrm{emit}}}$, where
$\nu_{\mathrm{emit}}$ is the emitted frequency and $\nu_{\mathrm{obsv}}$ is
the observed frequency. Cosmological redshift is one of the important
measurable quantities associated with the stretching effect of space expansion
in cosmology.

The redshift of electromagnetic waves can be investigated in a more careful
way via solving the Maxwell equations, $\nabla^{a}F_{ab}=J_{b}$ (where
$F_{ab}=2\nabla_{[a}A_{b]}$) on a pseudo-Riemannian manifold for current and
current-free assumptions, respectively,

$\begin{array}[]{cc}{\nabla_{b}\nabla^{b}A^{a}-R^{a}{}_{b}A^{b}=J^{a},}&{\nabla_{b}\nabla^{b}A^{a}-R^{a}{}_{b}A^{b}=0.}\end{array}$
(352)

Suppose that the light wavelengths are small relative to the curvature, light
is then taken to be tangent to null surfaces of the constant phase $\phi$, so
travels on null geodesics. If the velocity of a photon is specified by a null
(geodesic) vector $k^{a}\equiv\nabla^{a}\phi$, we then have
$\nabla_{[a}k_{b]}=0$, and

$\begin{array}[]{cc}{\nabla_{a}\phi\nabla^{a}\phi=k_{a}k^{a}=0,}&{k^{a}\nabla_{a}k^{b}=0.}\end{array}$
(353)

The former is an eikonal equation Ehlers and Newman (2000). The angle between
the vector $k^{a}$ and the traveling velocity $u^{a}$ of an observer is
associated with the relative angular frequency $\omega=2\pi\nu=-u^{a}k_{a}$ of
electromagnetic waves measured by the observer Jordan et al. (1961, 2013). A
photon traveling between two points is then redshifted based on the relative
frequency difference between the two points, i.e.,
$1+z={\left({u^{a}k_{a}}\right)_{\mathrm{emit}}}/{\left({u^{b}k_{b}}\right)_{\mathrm{obsv}}}$.
This redshift effect can be explicitly described by the change rate of
frequency along the photon path:

$\displaystyle k^{a}\nabla_{a}\omega=$
$\displaystyle-k^{a}k^{b}\nabla_{a}u_{b}=k^{a}k^{b}\left({u_{a}\dot{u}_{b}-{{\frac{1}{3}}}\Theta
h_{ab}-\sigma_{ab}-\omega_{ab}}\right).$ (354)

As $k^{a}$ is a null vector, it may be expressed as a linear mixture of a
parallel term and a term orthogonal to $u^{a}$:

$\begin{array}[]{ccc}{k^{a}=-u_{b}k^{b}(u^{a}+e^{a}),}&{e^{a}e_{a}=1,}&{u_{a}e^{a}=0,}\end{array}$
(355)

where $e^{a}$ is a normalized spacelike vector orthogonal to $u^{a}$ that
represents the propagation direction of the electromagnetic waves with respect
to $u^{a}$.

Substituting the kinematic quantities into the above equation yields

$k^{a}\nabla_{a}\omega=-\omega^{2}\left({{{\frac{1}{3}}}\Theta+e^{a}\dot{u}_{a}+e^{a}e^{b}\sigma_{ab}-\varepsilon_{abc}e^{a}e^{b}\omega^{c}}\right),$
(356)

which results in

$\displaystyle\frac{{d\nu}}{\nu}=\lambda d\left({\frac{1}{\lambda}}\right)$
$\displaystyle=-\left({{{\frac{1}{3}}}\Theta+e^{a}\dot{u}_{a}+e^{a}e^{b}\sigma_{ab}-\varepsilon_{abc}e^{a}e^{b}\omega^{c}}\right)\delta
l$ $\displaystyle=-H\delta
l-\dot{u}_{a}\delta_{\bot}x^{a}-\delta_{\bot}x^{a}e^{b}\sigma_{ab}+\delta_{\bot}x^{a}\varepsilon_{abc}e^{b}\omega^{c},$
(357)

where $\delta l^{2}=h_{ab}dx^{a}dx^{b}=(u_{a}k^{a})dv^{2}=-\omega dv^{2}.$ The
relative direction of electromagnetic waves is affected as their frequency
changes, which can be explained by the kinematic quantities. The expansion
$\Theta$ reduces the energy and frequency of photons and increases the
wavelength, which causes a red shift. However, as seen in Eq. (357), the
changes in frequency caused by the shear $\sigma_{ab}$, vorticity
$\omega_{a}$, and acceleration $\dot{u}_{a}$ are direction-dependent.
Depending on the direction of the incoming photon, relative to the direction
of vorticity and acceleration, $\omega_{a}$ and $\dot{u}_{a}$ can make either
an increase or a decrease in the frequency. Similarly, the shear $\sigma_{ab}$
also modifies the light frequency according to the difference between its
stress direction and the incoming light direction Clarkson (2000); Kristian
and Sachs (1966); Ehlers (1961, 1993); MacCallum and Ellis (1970).

Furthermore, the Ricci equations (96) and (99) imply that the gravitoelectric
and gravitomagnetic fields, which are governed by the Bianchi equations
(84)–(87), induce the shear and vorticity. In this way, the
gravitoelectric/-magnetic fields, along with the kinematical quantities,
indirectly have some observational consequences Kristian and Sachs (1966). The
effects of the Weyl conformal tensor occur in the geodesic deviation equation,
where the distortions affect bundles of null geodesics, leading to strong
gravitational lensing, multiple images and Einstein rings, as well as weak
gravitational lensing and image distortions (see e.g. Schneider et al. (1992);
Holz and Wald (1998)).888Also see Iorio (2001); Mashhoon et al. (2001);
Ruggiero and Tartaglia (2002); Iorio et al. (2002); Iorio and Lichtenegger
(2005); Lichtenegger et al. (2006) for the gravitomagnetic clock effect, and
Braginskii et al. (1977); Peng (1983); Mashhoon et al. (1989); Peng (1990); Li
and Torr (1991); Tartaglia and Ruggiero (2004); Ummarino and Gallerati (2017,
2020) for proposed laboratory experiments. These effects have profound
implications for cosmology, where observations allow us to determine the
large-scale structure of the universe in the past null cone Ellis et al.
(1985).

##  11 Conclusions

In this paper, we have reviewed the 1 + 3 covariant formalism Heckmann and
Schücking (1955, 1956); Ehlers (1961, 1993); Ellis (1971) (in addition to
alternative methods in § 8 and § 9), which can be employed to covariantly
describe the locally free fields of the (traceless) Weyl conformal tensor in
general relativity. In this approach, we have the projected vectors and PSTF
tensors instead of the spacetime metric, together with the kinematic
quantities of the fluid flow, and the dynamic quantities of the matter field.
This formalism leads to the definitions of spatial divergences, curls, and
distortions of the projected vectors and PSTF tensors as introduced in § 2.
Moreover, we have the kinematic quantities that describe the relative motion
of comoving observers Ehlers (1957); Ellis (1964), and the dynamic quantities,
including the energy density and pressure in a perfect fluid, in addition to
the energy flux and anisotropic pressure in an imperfect fluid. These
quantities are contingent upon either a single-type of matter or a multiple-
type of matter species Maartens et al. (1999); King and Ellis (1973); Maartens
(1998); Bruni et al. (1992); Tsagas et al. (2008) (as seen in § 7), such as
implications of inhomogeneities for CMB Challinor and Lasenby (1999) and exact
(nonlinear) solutions of gravitational collapse Ellis and Tsagas (2002).

In the theory of general relativity, the Riemann curvature, which completely
describes the curvature of spacetime, is decomposed into the Ricci curvature
and the Weyl conformal tensor Jordan et al. (1960, 2009). The matter field has
local gravitational effects on the surrounding Ricci curvature that are
explained by the Einstein field equations Einstein (1918). However, the Weyl
tensor Weyl (1918), which is the tracefree part of the Riemann curvature,
offers the locally-free long-range fields, enabling distant interactions,
namely tidal forces Ellis (1971), frame-dragging effects Owen et al. (2011);
Nichols et al. (2011); Zhang et al. (2012); Nichols et al. (2012); Danehkar
(2020), and gravitational waves Hawking (1966); Dunsby et al. (1997); Danehkar
(2009)). Furthermore, the Weyl tensor is decomposed into the gravitoelectric
and gravitomagnetic tensorial fields Pirani (1957); Bel (1958a, b, 1962);
Penrose (1960); Matte (1953); Pirani (1962, 1962), owing to some analogies
with the electric and magnetic fields in the Maxwell theory. The Bianchi and
Ricci identities Kundt and Trümper (1962, 2016) provide some constraint and
evolution equations, which covariantly govern couplings of the
gravitoelectric/-magnetic fields with the dynamic and kinematic quantities
Trümper (1964); Ehlers (1961, 1993); Ellis (1971, 1973).

As seen in Section 4, the Bianchi identities are split into constraint and
propagation equations, which covariantly characterize the roles played by the
gravitoelectric and gravitomagnetic fields. These equations includes the
spatial divergences, spatial curls, and time derivatives of the
gravitoelectric and gravitomagnetic tensors Ellis (1971, 1973); Ellis and van
Elst (1999). The gravitoelectric divergence equation can simply be interpreted
as tidal forces in Newtonian theory (see § 6). The Newtonian analogy occurs
due to the presence of variables in the gravitoelectric dynamics that have
obvious Newtonian counterparts. In this way, the gravitoelectric field that
originates from the energy density gradient acts as a general-relativistic
generalization of Newtonian tidal forces Ellis and Dunsby (1997). In § 6.1, a
Newtonian-like model was built from the gravitoelectric field using the
Heckmann-Schücking approach Heckmann and Schücking (1955, 1956, 1959) under
the assumption of instantaneous gravitational interaction that produces the
Poisson equation of Newtonian theory Ellis and Dunsby (1997); Kofman and
Pogosyan (1995). However, in particular, the gravitomagnetic field, which
orginates from the angular momentum density Ellis (1971), has no analogy in
Newtonian theory. In § 6, a purely gravitoelectric (Newtonian-like) model and
a purely gravitomagnetic (anti-Newtonian Maartens et al. (1998)) model were
shown to be generally inconsistent without their counterpart field.

The decomposition of the Bianchi identities also generates the gravitoelectric
evolution ($\dot{E}_{ab}$), which has no analogy in Newtonian gravity, as well
as the gravitomagnetic evolution ($\dot{H}_{ab}$). The absence of
$\dot{E}_{ab}$ from Newtonian theory is obvious since the instantaneous
interaction prevents any possibility for wave solutions. We also notice a
close analogy between the evolution equations of $E_{ab}$ and $H_{ab}$ and the
Maxwell equations of $E_{a}$ and $H_{a}$ in expanding spacetimes Maartens and
Bassett (1998). As discussed in Section 5, wave solutions of $E_{ab}$ and
$H_{ab}$ appear under some certain conditions, where their curls Maartens
(1997) and distortions Maartens et al. (1997) support gravitational waves
linearly characterized in free space by $\ddot{E}_{ab}-\mathrm{D}^{2}E_{ab}=0$
and $\ddot{H}_{ab}-\mathrm{D}^{2}H_{ab}=0$ via the propagation equations Ellis
and Bruni (1989); Hawking (1966); Hogan and Ellis (1997); Dunsby et al.
(1997).

The geodesic deviation equation can be used to perceive how gravitoelectric
and gravitomagnetic fields affect astrophysical and cosmological observations
of electromagnetic waves, such as strong and weak lensings Kristian and Sachs
(1966). Although the effects were covariantly explained in terms of the
kinematic quantities in § 10, gravitoelectric /-magnetic fields can also
induce shear and vorticity, which subsequently modify the frequency and
relative direction of light. The expansion of the universe reduces the photon
energy (frequency), resulting in cosmological redshift Ehlers (1961, 1993);
MacCallum and Ellis (1970). Nevertheless, the observational effects on light
caused by either shear or vorticity are direction-dependent. The light
frequency can be changed based on its direction relative to the shear or
vorticity direction.

no

##  Appendix A Higher-rank PSTF Tensor Formulism

Higher-rank PSTF tensors are useful for describing the spherical harmonic
quantities with multiple directions specified by $e^{a}$ at a position
$x^{i}$, the so-called multipoles. Rank-$\ell$ PSTF tensors can be employed to
represent $\ell$ multipoles of radiation anisotropies such as CMB.

Following Thorne (1980), for simplicity of style, we may abbreviate a series
of $\ell$ indices on a tensor:

$\begin{array}[]{cc}{T_{A^{\ell}}\equiv T_{a_{1}a_{2}\cdots
a_{\ell}},}&{T^{A^{\ell}}\equiv T^{a_{1}a_{2}\cdots a_{\ell}}.}\end{array}$
(358)

We may also associate $\ell=0,1,2$ with scalars, vectors, and rank-2 tensors,
respectively:

$\begin{array}[]{ccc}{T\equiv
T_{A^{0}},}&{T_{a}=T_{A^{1}},}&{T_{ab}=T_{A^{2}}.}\end{array}$ (359)

Similarly, the tensor product of vectors may be abbreviated by a similar
manner, along with a × denoting the product 999In Ellis et al. (1983a), the
tensor product is shown by a tilde as $\tilde{T}_{A_{\ell}}\equiv
T_{a_{1}}T_{a_{2}}\cdots T_{a_{\ell}}$.

$\displaystyle\begin{array}[]{cc}{T_{A^{1\times\ell}}\equiv
T_{a_{1}}T_{a_{2}}\cdots T_{a_{\ell}},}&{T^{A^{1\times\ell}}\equiv
T^{a_{1}}T^{a_{2}}\cdots T^{a_{\ell}},}\end{array}$ (361)
$\displaystyle{T_{A^{2\times\ell}}\equiv T_{a_{1}a_{2}}T_{a_{2}a_{3}}\cdots
T_{a_{2\ell-1}a_{2\ell}},}$ (362) $\displaystyle T_{A^{n\times\ell}}\equiv
T_{a_{1}\cdots a_{n}}T_{a_{n+1}\cdots a_{2n}}\cdots T_{a_{n(\ell-1)+1}\cdots
a_{n\ell}}.$ (363)

It is convenient to use the following notations for the symmetric and
antisymmetric indices of tensors $T_{A^{\ell}}$:

$\displaystyle T_{(A^{\ell})}\equiv$ $\displaystyle T_{(a_{1}a_{2}\cdots
a_{\ell})}=\frac{1}{{\ell!}}\sum\limits_{\pi}{T_{a_{\pi(1)}\cdots
a_{\pi(\ell)}}},$ (364) $\displaystyle T_{[A^{\ell}]}\equiv$ $\displaystyle
T_{[a_{1}a_{2}\cdots
a_{\ell}]}=\frac{1}{{\ell!}}\sum\limits_{\pi}{\delta_{\pi}T_{a_{\pi(1)}\cdots
a_{\pi(\ell)}}},$ (365)

where $\Sigma$ is a sum over $\ell!$ permutations $\pi$ of $1,\ldots,\ell$,
$\delta_{\pi}$ is +1 for even permutations and -1 for odd permutations, and
for integer $\ell$, we define ${\ell!\equiv\ell(\ell-1)\cdots 2\cdot 1}$ and
$\ell!!\equiv\ell(\ell-2)(\ell-4)\cdots(2\mathrm{~{}or~{}}1)$.

Following Pirani (1965); Sachs (1961), the symmetric tracefree (STF) part of a
tensor $T_{(A^{\ell})}$ can be obtained by (i) constructing the symmetric
part:

$S_{A^{\ell}}\equiv\left[{T_{A^{\ell}}}\right]^{\mathrm{S}}=\frac{1}{{\ell!}}\sum\limits_{\pi}{T_{a_{\pi(1)}\cdots
a_{\pi(\ell)}}},$ (366)

and (ii) removing all traces:

$\left[{T_{A^{\ell}}}\right]^{\mathrm{STF}}=\sum\limits_{n=0}^{[\ell/2]}{B^{\ell
n}g_{(a_{1}a_{2}}\cdots g_{a_{2n-1}a_{2n}}S^{b_{1}b_{1}\cdots
b_{n}b_{n}}}{}_{a_{2n+1}\cdots a_{\ell}),}$ (367)

where $[\ell/2]$ denotes the largest integer less than or equal to $\ell/2$,
and $B^{\ell n}$ is calculated by

$B^{\ell
n}=\frac{{(-1)^{n}\ell!(2\ell-2n-1)!!}}{{(\ell-2n)!(2\ell-1)!!(2n)!!}}.$ (368)

Similarly, the STF part of a spatial tensor is determined using the spatial
metric $h_{ab}$ instead of the spacetime metric $g_{ab}$ Pirani (1965).

Following Ellis et al. (1983a, b), the PSTF part of a tensor is denoted by

$T_{\langle
A^{\ell}\rangle}\equiv\left[{T_{A^{\ell}}}\right]^{\mathrm{PSTF}}=\left[{\left[{T_{A^{\ell}}}\right]^{\mathrm{STF}}}\right]^{\mathrm{P}},$
(369)

which can recursively be built using a vector basis $e^{a}$ (see Gebbie and
Ellis (2000); $e^{a}e_{a}=1$, $e^{a}u_{a}=0$):

$T^{A^{\ell}}=e^{\langle A^{1\times\ell}\rangle}=e^{\langle
a_{1}}e^{a_{2}}e^{a_{3}}\cdots e^{a_{\ell-1}}e^{a_{\ell}\rangle}.$ (370)

Replacing $g_{ab}$ with $h_{ab}$ in Eq. (367), the spatial multipole
$T^{A^{\ell}}$ is defined by

$T^{A^{\ell}}=e^{\langle
A^{1\times\ell}\rangle}=\sum\limits_{n=0}^{[\ell/2]}{B_{\ell n}h^{(A^{2\times
n}}e^{A^{\ell-2\times n})}.}$ (371)

As the spatial multipole is a higher-rank PSTF tensor, it is irreducible,
i.e.,

$\begin{array}[]{cccc}{T^{A^{\ell}}=T^{\langle
A^{\ell}\rangle}}&\Leftrightarrow&{T^{A^{\ell}}=T^{(A^{\ell})},}&{T^{A^{\ell}}u_{A^{1}}=0=T^{A^{\ell}}h_{A^{2}}.}\end{array}$
(372)

Some recursion relations for $\ell$-rank PSTF tensors are established by Ellis
et al. (1983a):

$\displaystyle T^{\langle A^{\ell}}e^{a_{\ell+1}\rangle}=$ $\displaystyle
T^{(A^{\ell}}e^{a_{\ell+1})}-\frac{\ell}{{2\ell+1}}e^{b}h_{bc}T^{c(A^{\ell-1}}h^{a_{\ell}a_{\ell+1})},$
(373) $\displaystyle e_{a_{1}}T^{A^{\ell}}=$
$\displaystyle\frac{\ell}{{2\ell-1}}T^{A^{\ell-1}},$ (374)

which relate a $(\ell+1)$-rank PSTF tensor to a $\ell$-rank and
$(\ell-1)$-rank PSTF tensor:

$T^{A^{\ell+1}}=e^{(a_{\ell+1}}T^{A^{\ell})}-\frac{{\ell^{2}}}{{(2\ell+1)(2\ell-1)}}h^{(a_{\ell+1}a_{\ell}}T^{A^{\ell-1})}.$
(375)

Using $\ell=0$ and $1$ turns Eq. (373) into Eq. (12) of projected vectors and
PSTF tensors, respectively.

Using a $\ell$-rank PSTF tensor $T^{A^{\ell}}$ and a projected vector $V^{a}$,
a $(\ell+1)$-rank PSTF tensor can be made by Ellis et al. (1983a)

$T^{\langle
A^{\ell}}V^{a\rangle}=T^{(A^{\ell}}V^{a)}-\frac{\ell}{{2\ell+1}}V^{b}h_{bc}T^{c(A^{\ell-1}}h^{a_{\ell}a)}.$
(376)

Similarly, a $(\ell+2)$-rank PSTF tensor can be generated using a $\ell$-rank
PSTF tensor $T^{A^{\ell}}$ and a rank-2 PSTF tensor $S^{ab}$ Ellis et al.
(1983a):

$\displaystyle T^{\langle A^{\ell}}S^{ab\rangle}=$ $\displaystyle
T^{(A^{\ell}}S^{ab)}-\frac{1}{{3+2\ell}}S^{cd}h_{cd}T^{c(A^{\ell-1}}h^{ab)}$
$\displaystyle-\frac{{2\ell}}{{3+2\ell}}T^{c(A^{\ell-1}}S^{\left|d\right|a_{\ell}}h^{ab)}h_{cd}$
$\displaystyle+\frac{{\ell(\ell-1)}}{{(2\ell+1)(2\ell+3)}}h_{eg}h_{fh}S^{gh}T^{ef(A^{\ell-2}}h^{a_{\ell-1}a_{\ell}}h^{ab)}.$
(377)

Contraction of $h^{(A^{2\times\ell})}$ also leads to

$h_{(A^{2\times\ell})}h^{(A^{2\times\ell})}=2\ell+1.$ (378)

Following Thorne (1980), the integrals of the multipoles over the direction
$e^{a}$ are Ellis et al. (1983a); Maartens et al. (1999):

$\displaystyle\int{e^{A^{1\times(2\ell+1)}}\mathrm{d}\Omega}$
$\displaystyle=0,$ (379) $\displaystyle\int{e^{A^{1\times
2\ell}}\mathrm{d}\Omega}$
$\displaystyle=\left({\frac{{4\pi}}{{2\ell+1}}}\right)h^{(A^{2\times\ell})},$
(380)

where $\mathrm{d}\Omega=\mathrm{d}^{2}e$ is the volume element covered by two
independent $\mathrm{d}e^{a}$.

From Ellis et al. (1983a), we also have:

$\displaystyle\int{e^{A^{1\times 2\ell}}e^{B^{1\times 2m}}\mathrm{d}\Omega}=$
$\displaystyle\left({\frac{{4\pi}}{{2(\ell+m)+1}}}\right)h^{(A^{2\times\ell}}h^{B^{2\times
m})},$ (381) $\displaystyle\int{e^{A^{1\times(2\ell+1)}}e^{B^{1\times
2m}}\mathrm{d}\Omega}=$
$\displaystyle\int{e^{A^{2\ell}}e^{B^{2m+1}}\mathrm{d}\Omega}=0,$ (382)
$\displaystyle\int{e^{A^{1\times(2\ell+1)}}e^{B^{1\times(2m+1)}}\mathrm{d}\Omega}=$
$\displaystyle\left({\frac{{4\pi}}{{2(\ell+m)+3}}}\right)h^{(A^{2\times\ell}}h^{B^{2\times
m}}h^{A^{1}B^{1})}.$ (383)

From Eqs. (381), (371), and (367), for $T^{A^{\ell}}=e^{\langle
A^{1\times\ell}\rangle}$ it also follows Gebbie and Ellis (2000)

$\int{T^{A^{\ell}}T_{B^{m}}\mathrm{d}\Omega}=\delta^{\ell}{}_{m}\Delta_{\ell}h^{\langle
A^{1\times\ell}\rangle}{}_{\langle B^{1\times\ell}\rangle},$ (384)

where $h^{\langle A^{1\times\ell}\rangle}{}_{\langle
B^{1\times\ell}\rangle}\equiv h^{\langle a_{1}}{}_{\langle b_{1}}\cdots
h^{a_{\ell}\rangle}{}_{b_{\ell}\rangle}$, and $\Delta_{\ell}$ is given by

$\displaystyle\Delta_{\ell}\equiv$
$\displaystyle\frac{{4\pi}}{{(2\ell+1)}}\frac{{2^{\ell}(\ell!)^{2}}}{{(2\ell)!}}.$
(385)

Therefore, we have

$e^{B^{1\times n}}h^{\langle A^{1\times\ell}\rangle}{}_{\langle
B^{1\times\ell}}h_{B^{1\times(n-\ell)}\rangle}=e^{\langle
A^{1\times\ell}\rangle}(+1)^{n-\ell}=e^{\langle A^{1\times\ell}\rangle}.$
(386)

From (384), we find (compare to Eq. 378),

$h^{\langle A^{1\times\ell}\rangle}{}_{\langle
A^{1\times\ell}\rangle}=2\ell+1.$ (387)

Consider the relation between the directions $e^{a}$ and $\tilde{e}^{a}$ at
$x_{i}$ specified by Gebbie and Ellis (2000)

$e^{a}\tilde{e}_{a}=\cos(\beta)\equiv X.$ (388)

where $\beta$ is the angular distance $\beta$ separating $e^{a}$ and
$\tilde{e}^{a}$. For $e^{A^{1\times\ell}}$ and $\tilde{e}^{A^{1\times\ell}}$,
the generalized relation is given by

$\begin{array}[]{ccc}{e^{A^{1\times\ell}}\tilde{e}_{A^{1\times\ell}}=X^{\ell}}&\Rightarrow&{\sum\limits_{\ell=0}^{\infty}{e^{A^{1\times\ell}}\tilde{e}_{A^{1\times\ell}}}=\sum\limits_{\ell=0}^{\infty}{X^{\ell}}=\frac{{\cos\beta}}{{1-\cos\beta}}.}\end{array}$
(389)

Using Eqs. (381)–(383), we can also calculate the integral over $e^{a}$ with
fixed $\tilde{e}^{a}$:

$\int{e^{A^{1\times\ell}}\tilde{e}_{A^{1\times\ell}}\mathrm{d}\Omega}=\int{(e^{a}\tilde{e}_{a}\mathrm{)}^{\ell}\mathrm{d}\Omega}=2\pi\int_{-1}^{+1}{X^{\ell}}dX=\left\\{{\begin{array}[]{cc}0&{\forall\mathrm{~{}}n\mathrm{~{}even}}\\\
{\frac{{4\pi}}{{n+1}}}&{\forall\mathrm{~{}}n\mathrm{~{}even}}\end{array}}\right..$
(390)

Suppose the multipoles given by $T^{A^{\ell}}=e^{\langle
A^{1\times\ell}\rangle}$ and $\tilde{T}^{A^{\ell}}=\tilde{e}^{\langle
A^{1\times\ell}\rangle}$, the following correlation between them exists:

$T^{A^{\ell}}\tilde{T}_{A^{\ell}}=\sum\limits_{k=0}^{[\ell/2]}{\sum\limits_{\tilde{k}=0}^{[\ell/2]}{B_{\ell
k}B_{\ell\tilde{k}}h^{(A^{2\times
k}}h_{(2\times\tilde{k}}e^{A^{1\times\ell})}\tilde{e}_{A^{1\times\ell})}}}.$
(391)

Higher-rank PSTF tensors hold the following natural polynomial Gebbie and
Ellis (2000):

$L_{\ell}(X)=T^{A^{\ell}}\tilde{T}_{A^{\ell}}=\sum\limits_{m=0}^{[\ell/2]}{B_{\ell
m}X^{\ell-2m}}.$ (392)

Extending the 1+3 tetrad formalism to higher-rank tensors Ellis et al.
(1983b); Treciokas (1972), the spatial divergences, time derivatives, and
spatial distortions of higher-rank tensors may be calculated by

$\displaystyle\mathrm{D}^{b}T_{bA^{\ell-1}}\rightarrow$
$\displaystyle\left({\mathrm{e}^{b}-(\ell+1)\mathrm{a}^{b}}\right)T_{bA^{(\ell-1)}}+(\ell-1)T^{b}{}_{c(A^{\ell-2}}\varepsilon_{a_{\ell-1})bd}\mathrm{n}^{cd}$
$\displaystyle+(\ell-1)T^{b}{}_{b(A^{\ell-2}}\mathrm{a}_{a_{\ell-1})},$ (393)
$\displaystyle\dot{T}_{\langle A^{\ell}\rangle}\rightarrow$
$\displaystyle\mathrm{e}_{0}\left({T_{A^{\ell}}}\right)-\ell T^{d}{}_{\langle
A^{\ell-1}}\varepsilon_{da_{\ell}\rangle c}\Omega^{c},$ (394)
$\displaystyle\mathrm{D}_{(b}T_{A^{\ell})}\rightarrow$
$\displaystyle\left({\mathrm{e}_{(b}+\ell\mathrm{a}_{(b}}\right)T_{A^{\ell})}-\ell\varepsilon_{dc(b}\mathrm{n}^{c}{}_{a_{\ell}}T_{A^{\ell-1})}{}^{d}$
$\displaystyle-\ell\mathrm{a}^{d}T_{d(A^{\ell-1}}\delta_{a_{\ell}b)}.$ (395)

With $\ell=1$ (projected vector) and $2$ (rank-2 PSTF tensor), Eq. (393)
reduces to Eqs. (294) and (295), and Eq. (394) to Eqs. (291) and (292),
respectively.

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