# The $3+1$ Formalism in the Geometric Trinity of Gravity

Salvatore Capozziello<EMAIL_ADDRESS>Dipartimento di Fisica “E.
Pancini”, Università di Napoli “Federico II”, Via Cinthia, I-80126, Napoli,
Italy.  Istituto Nazionale di Fisica Nucleare (INFN), sez. di Napoli, Via
Cinthia 9, I-80126 Napoli, Italy.  Scuola Superiore Meridionale, Largo San
Marcellino 10, 80138 Napoli, Italy.  Laboratory for Theoretical Cosmology,
Tomsk State University of Control Systems and Radioelectronics (TUSUR), 634050
Tomsk, Russia.  Andrew Finch<EMAIL_ADDRESS>Institute of Space
Sciences and Astronomy, University of Malta, Malta.  Department of Physics,
University of Malta, Malta.  Jackson Levi Said<EMAIL_ADDRESS>Institute of Space Sciences and Astronomy, University of Malta, Malta.
Department of Physics, University of Malta, Malta.  Alessio Magro
<EMAIL_ADDRESS>Institute of Space Sciences and Astronomy, University
of Malta, Malta.  Department of Physics, University of Malta, Malta.

###### Abstract

The geometric trinity of gravity offers a platform in which gravity can be
formulated in three analogous approaches, namely curvature, torsion and
nonmetricity. In this vein, general relativity can be expressed in three
dynamically equivalent ways which may offer insights into the different
properties of these decompositions such as their Hamiltonian structure, the
efficiency of numerical analyses, as well as the classification of
gravitational field degrees of freedom. In this work, we take a $3+1$
decomposition of the teleparallel equivalent of general relativity and the
symmetric teleparallel equivalent of general relativity which are both
dynamically equivalent to curvature based general relativity. By splitting the
spacetime metric and corresponding tetrad into their spatial and temporal
parts as well as through finding the Gauss-like equations, it is possible to
set up a general foundation for the different formulations of gravity. Based
on these results, general $3$-tetrad and $3$-metric evolution equations are
derived. Finally through the choice of the two respective connections, the
metric $3+1$ formulation for general relativity is recovered as well as the
tetrad $3+1$ formulation of the teleparallel equivalent of general relativity
and the metric $3+1$ formulation of symmetric teleparallel equivalent of
general relativity. The approach is capable, in principle, of resolving common
features of the various formulations of general relativity at a fundamental
level and pointing out characteristics that extensions and alternatives to the
various formulations can present.

##  I Introduction

In light of the recent observational discovery of gravitational waves (GW),
the need for further work on methods to improve numerical relativity
simulations has become all the more pressing Abbott:2016blz . Adding to this,
the subsequent multimessenger discovery of a binary neutron star merger both
in the GW spectrum with GW170817 TheLIGOScientific:2017qsa  as well as having
an associated electromagnetic counterpart GRB170817A Goldstein:2017mmi  has
drastically intensified the search for alternative and possible more efficient
ways to numerically simulate gravitational theories and general relativity
(GR) in particular. Another critical question to answer is whether these
numerical approaches can tolerate the plethora of proposals for modified
theories of gravity today on the market Capozziello:2011et ; Nojiri:2017ncd ;
Clifton:2011jh ; CANTATA:2021ktz ; Capozziello:2021bki . While interesting
approaches have been introduced in standard GR Baumgarte:2010ndz ;
Baumgarte:2021skc , it is also interesting to consider possible $3+1$
formulations of GR using analogous constructions of gravity which are
dynamically equivalent to GR.

The prospect of supporting a different geometric basis on which to build
gravitational theories was worked on by Einstein himself 2005physics…3046U  in
his attempt to unify gravitation with electromagnetism. Ultimately, it turned
out that this was not possible with the degrees of freedom (DoF) that he was
using but efforts continued throughout the decades to use both teleparallel
gravity (TG) RevModPhys.48.393 ; Hayashi:1967se ; Hayashi:1979qx , which
relies on torsion, and symmetric teleparallel gravity (STG) Nester:1998mp ;
Dialektopoulos:2019mtr , which is based on nonmetricity. In its current state,
TG is built by a shift in the connection from the Levi-Civita
$\accentset{\circ}{\Gamma}^{\rho}_{\phantom{\rho}\mu\nu}$ (over-circles denote
all quantities that use the Levi-Civita connection in their calculation) to
the teleparallel connection
$\accentset{\land}{\Gamma}^{\rho}_{\phantom{\rho}\mu\nu}$ (over-hats denote
all quantities that use the teleparallel connection in their calculation).
This represents a transformation from curvature- to torsion-based geometry in
our gravitational theories. On the other hand, STG is also formed by an
exchange of connections where a disformation tensor
$\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\alpha}_{\phantom{\alpha}\mu\nu}$
(over-diamonds denote all quantities that assume nonmetricity in their
calculation) can be used to replace the curvature associated with the Levi-
Civita connection with nonmetricity.

Tetrads of TG replace the metric tensor as the fundamental dynamical objects
of the theory. Specifically, the tetrad $e^{A}_{\phantom{A}\mu}$ and spin
connection $\omega^{A}_{\phantom{A}B\mu}$ pair replaces the gravitational
potentials related to the metric. Together, tetrad and spin connection pair
represents the fundamental DoFs of any ensuing teleparallel theory. The tetrad
represents the gravitational part of this dynamics while the spin connection
is a flat connection describing the local Lorentz frame Bahamonde:2021gfp .
This pair also appear in GR but are hidden into the internal structure of the
theory. TG is different in that it separates these objects out instead of
using the metric tensor as its fundamental variable. One reason for this is
that, in GR, the spin connection is not flat and may contain gravitational
contributions while in TG the spin connection is exclusively of inertial
nature PereiraBook .

The shift from curvature- to torsion-based theories of gravity is
intrinsically coupled with the change from tensorial measures of curvature to
different tensors that are based on torsion. To this end, the central role
that the Riemann tensor plays in gravity has to be changed with another
object, the torsion tensor, on which gravitational theories can be built. This
happens because all measures of curvature identically vanish for tensors that
are based on the teleparallel connection which is curvature-less. In this
context, we also build scalar invariants which can be used to construct
gravitational actions. One such scalar is the torsion scalar
$\accentset{\land}{T}$ which can be shown to be equal to the standard Ricci
scalar $\accentset{\circ}{R}$ up to a total divergence term (see Sec. II.1).
This is a crucial property since it means that the Hilbert-Einstein action can
be analogously related to a torsional action producing a dynamically
equivalent theory. This is called the teleparallel equivalent of general
relativity (TEGR). One would expect this equivalent to be limited to the
classical regime of the theory.

TEGR has a number of positive features that make it advantageous with respect
to its curvature-based counterpart Bahamonde:2021gfp . Firstly, the strong
equivalence principle does not appear as a fundamental requirement of the
theory which may be beneficial for some quantum gravity theories, despite the
principle still being observationally satisfied Voisin:2020lqi . Another
crucial aspect of TEGR, related to quantum gravity, is that the inertial
nature of the spin connection makes it more akin to regular field theory
making approaches to quantum gravity more amenable to successful constructions
of gravitational theories Aldrovandi:2005mz ; Aldrovandi:2006cy . This is
related to the property that TEGR naturally produces a Gibbons-Hawking-York
boundary term in its action giving it a well-defined Hamiltonian expression
Oshita:2017nhn . Altogether, these properties make TEGR more malleable towards
a quantum setting and produce a likeness to the regular Yang-Mills theories of
particle physics.

On the other hand, STG continues to sustain the metric tensor as its
fundamental dynamical object Nester:1998mp ; Coley:2019zld ; Adak:2004uh ;
Jarv:2018bgs . However, here one normally sets both curvature and torsional
contributions to gravity to zero by a Lagrange multiplier method
BeltranJimenez:2017tkd ; BeltranJimenez:2018vdo . Moreover, STG also takes
advantage of the Palatini approach in its field equations. As with TG, STG
also depends on new tensorial quantities to be defined in order to construct
gravitational theories. In particular, STG depends on the nonmetricity tensor,
$\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\lambda\mu\nu}:=\accentset{\scaleto{\diamond}{3.5pt}}{\nabla}_{\lambda}g_{\mu\nu}$,
which then is the sole contributor to the disformation connection and the
eventual field equations both for the metric and the connection (see Sec.
II.2). Another critical point is that as we can define a TEGR in TG, we can
also define a symmetric teleparallel equivalent of general relativity (STEGR)
in STG Adak:2005cd ; Adak:2006rx  where a nonmetricity scalar
$\accentset{\scaleto{\diamond}{3.5pt}}{Q}$ turns out to be equal to the Levi-
Civita connection based Ricci scalar up to a boundary term. This will also be
dynamically equivalent to standard GR but may host different behavior in its
quantum regime. Hence, GR, TEGR and STEGR form a trinity approach to gravity
Jimenez:2019woj  for the same dynamical equations.

The $3+1$ decomposition is determined by taking into account the temporal and
spatial parts of the covariant field equations so that they can be probed in a
numerical setting. This was first explored in standard GR through the work by
Arnowitt, Deser, and Misner (ADM) 1959PhRv..116.1322A  which produced two
evolution equations and two constraint equations from which a plethora of
physical setting can be probed in numerical relativity. However, this approach
has not been extensively studied in the novel scenarios described in these
other geometric settings. On the other hand, some interesting work has been
advanced. For instance, in TG, Refs.Blixt:2018znp ; Blixt:2019mkt ;
Blixt:2020ekl ; Hohmann:2020muq  have explored important parts of the
decomposition of TEGR. On the other end of the spectrum, Refs.Jimenez:2021hai
; Ariwahjoedi:2020wmo  explored important properties in the formulation of the
$3+1$ decomposition for STEGR.

Throughout this, work Greek symbols will be used for global indices and
capital Latin symbols will be used to denote local coordinate indices,
respectively. Global spatial indices will be denoted by $\\{i,j,k\\}$, and
$\\{\tilde{i},\tilde{j},\tilde{k}\\}$ will denote local spatial indices.
Finally any temporal index will be denoted by $0$ for the global case and by
$\tilde{0}$ for the local case. The work is broken down as follows. In Sec.
II, we briefly introduce both TG and STG formalisms together with their
respective TEGR and STEGR descriptions of GR. We then present a general
approach to the $3+1$ decomposition in Sec. III where the decomposition is
setup for the full trinity of gravity without assuming a particular connection
for gravitation. In Sec. IV, we finally present the TG and STG $3+1$
decompositions for TEGR and STEGR respectively. In Sec. V, we discuss and
summarize our main results.

##  II Teleparallel Theories of Gravity

The trinity of gravity Jimenez:2019woj  provides a mechanism by which we can
form the same classical field equations that can be produced using either
curvature-, torsion- or nonmetricity-based formulations of gravity. In this
section, we introduce the basis on which these different formulations of
gravity are built as well as laying out any nuances of the theories
themselves.

###  II.1 Teleparallel equivalent of general relativity

TG represents a fundamental shift in the way that gravity is expressed in that
the geometric curvature associated with the Levi-Civita connection
$\accentset{\circ}{\Gamma}^{\sigma}_{\mu\nu}$ is now transformed into torsion
associated with the corresponding teleparallel connection
$\accentset{\land}{\Gamma}^{\sigma}_{\mu\nu}$ Hayashi:1979qx . The change in
connection will alter the way that gravity is exhibited but not its geometric
origins, meaning that we will continue to produce a metric tensor. However,
the Riemann tensor, which produces a meaningful measure of curvature on a
manifold (in standard gravity), will now identically vanish due to the
curvature-less nature of the teleparallel connection, i.e.
$\accentset{\land}{R}^{\alpha}_{\phantom{\alpha}\mu\nu\beta}\equiv 0$. A
consequence of this result will be that a TG analog has to be constructed in
order to give a meaningful measure of torsion nakahara2003geometry ;
ortin2004gravity .

In GR, the fundamental dynamical object is the metric tensor, $g_{\mu\nu}$,
but in TG this becomes a derived quantity and its role is replaced by tetrad
$e^{A}_{\phantom{A}\mu}$ and spin connection $\omega^{A}_{\phantom{A}B\mu}$
pairs. These act as a soldering agent between the general manifold (Greek
indices) and the Minkowski space (Latin indices) Aldrovandi:2013wha . In this
context, the tetrads can be used to transform to (and from) the Minkowski
metric through

$\displaystyle\eta_{ab}=e_{A}^{\phantom{A}\mu}e_{B}^{\phantom{B}\nu}g_{\mu\nu}$
$\displaystyle
g_{\mu\nu}=e^{A}_{\phantom{A}\mu}e^{B}_{\phantom{B}\nu}\eta_{AB}\,.$ (1)

The tetrads also observe the inverse conditions

$\displaystyle e^{A}_{\phantom{A}\mu}e_{B}^{\phantom{B}\mu}=\delta^{A}_{B}$
$\displaystyle
e^{A}_{\phantom{A}\mu}e_{A}^{\phantom{A}\nu}=\delta^{\nu}_{\mu}\,,$ (2)

for consistency.

The teleparallel connection is the most general linear affine connection that
is both curvature-less and satisfies the metricity condition
Aldrovandi:2013wha , and can be defined as Weitzenbock1923

$\accentset{\land}{\Gamma}^{\sigma}_{\nu\mu}:=e_{A}^{\phantom{A}\sigma}\partial_{\mu}e^{A}_{\phantom{A}\nu}+e_{A}^{\phantom{A}\sigma}\omega^{A}_{\phantom{A}B\mu}e^{B}_{\phantom{B}\nu}\,.$
(3)

The spin connection also appears in GR but it is mainly hidden within the
internal structure of the theory misner1973gravitation . Moreover, in TG, the
spin connection is an inertial quantity and appears to preserve the covariance
of the field equations Krssak:2015oua , i.e. this acts to balance the
potentially infinite solutions of Eq.(3) with the necessity of covariance of
the dynamical equations Krssak:2015oua . In this way, the spin connection
incorporates the DoFs associated with the local Lorentz transformation
invariance of the theory Krssak:2018ywd .

Considering the full breadth of Local Lorentz Transformations (LLTs), i.e.
Lorentz boosts and rotations, $\Lambda^{A}_{\phantom{A}B}$, the spin
connection can be fully described by
$\omega^{A}_{\phantom{A}B\mu}=\Lambda^{A}_{\phantom{A}C}\partial_{\mu}\Lambda_{B}^{\phantom{B}C}$
Aldrovandi:2013wha . Thus, for the infinite number of solutions of Eq.(3)
(which arise due to the invariance of the metric under LLTs), the spin
connection is produced by the field equations to sustain their general
covariance under both diffeomorphisms and LLTs. It is for this reason that the
spin connection components are representing the DoFs of the Lorentz group
rather than extra DoFs of the theory Ferraro:2016wht ; Bahamonde:2021gfp .
Thus, it is the combination of the tetrad choice and its associated spin
connection that retains the covariance of TG. On the other hand, a frame can
always be chosen in which the spin connection vanishes since there will always
exist a local frame that renders this result, which is called the Weitzenb”ock
gauge. Given the teleparallel connection, the torsion tensor can be defined as
Cai:2015emx

$\accentset{\land}{T}^{\sigma}_{\phantom{\sigma}\mu\nu}:=\accentset{\land}{\Gamma}^{\sigma}_{[\nu\mu]}\,,$
(4)

where the square brackets denote the anti-symmetric operator. Here it should
be noted that, throughout this paper, we define
$A_{(\mu\nu)}=A_{\mu\nu}+A_{\nu\mu}$ and $A_{[\mu\nu]}=A_{\mu\nu}-A_{\nu\mu}$.
From a dynamical point of view, the torsion tensor replaces the Riemann
tensor: it gives a measure of torsion and thus geometric deformation in the
theory. This also acts as the field strength of gravitation in TG, and
transforms covariantly under both diffeomorphisms and LLTs. It is also
convenient to define two other tensorial quantities that render a more concise
form of the ensuing theory. First, consider the contorsion tensor which
represents the difference between the teleparallel and Levi-Civita
connections, namely

$\accentset{\land}{K}^{\sigma}_{\phantom{\sigma}\mu\nu}:=\accentset{\land}{\Gamma}^{\sigma}_{\mu\nu}-\accentset{\circ}{\Gamma}^{\sigma}_{\mu\nu}=\frac{1}{2}\left(\accentset{\land}{T}_{\mu\phantom{\sigma}\nu}^{\phantom{\mu}\sigma}+\accentset{\land}{T}_{\nu\phantom{\sigma}\mu}^{\phantom{\nu}\sigma}-\accentset{\land}{T}^{\sigma}_{\phantom{\sigma}\mu\nu}\right)\,.$
(5)

This tensor plays a crucial role in relating results in TG with their analogue
in standard gravity. The other ingredient in TG is the so-called
superpotential

$\accentset{\land}{S}_{A}^{\phantom{A}\mu\nu}:=\accentset{\land}{K}^{\mu\nu}_{\phantom{\mu\nu}A}-e_{A}^{\phantom{A}\nu}\accentset{\land}{T}^{\alpha\mu}_{\phantom{\alpha\mu}\alpha}+e_{A}^{\phantom{A}\mu}\accentset{\land}{T}^{\alpha\nu}_{\phantom{\alpha\nu}\alpha}\,,$
(6)

which is linked with the gauge current representation of the gravitational
energy-momentum tensor within TG Aldrovandi:2003pa , but this remains an open
issue in TG Koivisto:2019jra ; Capozziello:2018qcp .

Contracting the torsion tensor with its superpotential produces the torsion
scalar

$\accentset{\land}{T}:=\frac{1}{2}\accentset{\land}{S}_{A}^{\phantom{A}\mu\nu}\accentset{\land}{T}^{A}_{\phantom{A}\mu\nu}\,,$
(7)

which is calculated entirely on the teleparallel connection in an analogous
way as the Ricci scalar depends only on the Levi-Civita connection. By
choosing to construct the torsion scalar in this way, it turns out that the
torsion and Ricci scalars are equivalent up to a boundary term
Bahamonde:2015zma ; Farrugia:2020fcu ; Capozziello:2018qcp

$\accentset{\land}{R}=\accentset{\circ}{R}+\accentset{\land}{T}-\frac{2}{e}\partial_{\mu}\left(e\accentset{\land}{T}^{\sigma\phantom{\sigma}\mu}_{\phantom{\sigma}\sigma}\right)=0\,,$
(8)

where $R$ is the Ricci scalar in terms of the teleparallel connection, and
$\mathring{R}$ is the regular Ricci scalar from standard gravity (calculated
by the Levi-Civita connection). The first conclusion of this observation is
that the Lagrangian of the Hilbert-Einstein action will be equal to the
torsion scalar up to a boundary term Aldrovandi:2013wha ; Capozziello:2018qcp

$\accentset{\circ}{R}=-\accentset{\land}{T}+\frac{2}{e}\partial_{\mu}\left(e\accentset{\land}{T}^{\sigma\phantom{\sigma}\mu}_{\phantom{\sigma}\sigma}\right):=-\accentset{\land}{T}+\accentset{\land}{B}\,,$
(9)

where $e=\det\left(e^{a}_{\phantom{a}\mu}\right)=\sqrt{-g}$. This equivalence
alone will insure that both scalars will produce the same dynamical equations
of motion. In this way, the TEGR action can be written as

$\mathcal{S}_{\text{TEGR}}=-\frac{1}{2\kappa^{2}}\int
d^{4}x\;e\accentset{\land}{T}+\int d^{4}x\;e\mathcal{L}_{\text{m}}\,,$ (10)

where $\kappa^{2}=8\pi G$ and $\mathcal{L}_{\text{m}}$ is the matter
Lagrangian. Both TEGR and the Hilbert-Einstein actions lead to the same
dynamical equations, but they differ in their Lagrangians by a boundary term
which, in GR, produces fourth-order terms in extensions beyond GR, such as in
$f(\accentset{\circ}{R})$ gravity Capozziello:2002rd ; Sotiriou:2008rp ;
Faraoni:2008mf ; Capozziello:2011et . Thus, in TG, the second- and fourth-
order contributions from the action are somewhat decoupled in the torsion
scalar and boundary term. This is not relevant for comparisons of GR and TEGR
at the level of their field equations, but becomes an active agent when
modifications are considered Cai:2015emx .

In GR, the Hilbert-Einstein action leads to Einstein’s field equations (EFFs)
where $\accentset{\circ}{G}_{\mu\nu}=\kappa^{2}\Theta_{\mu\nu}$, and where the
Einstein tensor is determined by the Levi-Civita connection
misner1973gravitation , and where the regular energy-momentum tensor is
defined as
$\Theta_{\mu\nu}:=-2/\sqrt{-g}\,\delta\left(\sqrt{-g}\mathcal{L}_{\rm
m}\right)/\delta g^{\mu\nu}$. The TEGR manifestation of TG leads to the TEGR
action in Eq.(10), where a variation with respect to the tetrad leads directly
to Krssak:2015oua

$\displaystyle\accentset{\circ}{G}_{\mu\nu}\equiv\accentset{\land}{G}_{\mu\nu}$
$\displaystyle:=e^{-1}e^{A}{}_{\mu}g_{\nu\rho}\partial_{\sigma}(e\accentset{\land}{S}_{a}{}^{\rho\sigma})-\accentset{\land}{S}_{B}{}^{\sigma}{}_{\nu}\accentset{\land}{T}^{B}{}_{\sigma\mu}$
(11)
$\displaystyle+\frac{1}{4}\accentset{\land}{T}g_{\mu\nu}-e^{A}{}_{\mu}\omega^{B}{}_{A\sigma}\accentset{\land}{S}_{B\nu}{}^{\sigma}=\kappa^{2}\Theta_{\mu\nu}\,,$

which are dynamically equivalent to the EFFs except that they are based on the
tetrad rather than on the metric tensor. Moreover, the TEGR field equations do
not contain some of the non-dynamical boundary term contributions of the Ricci
scalar Cai:2015emx . In all other TG theories of gravity, a variation with
respect to the spin connection leads to a separate set of field equations
representing the six DoFs associated with the Lorentz group. However, in the
very special case of TEGR, these equations are identically satisfied and so do
not play an active role in the dynamics of the theory Bahamonde:2021gfp .

###  II.2 Symmetric teleparallel equivalent of general relativity

Analogously, STG embodies the transition from curvature-based geometric
gravity to one based on the property of nonmetricity. In this context, the
curvature and torsion associated with the theory are set to vanish so that
only nonmetricity embodies the effects of gravity. Thus, we can define a
nonmetricity tensor through

$\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\lambda\mu\nu}:=\accentset{\scaleto{\diamond}{3.5pt}}{\nabla}_{\lambda}g_{\mu\nu}\,,$
(12)

which uses the inverse metric by appropriate contractions to give

$\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\lambda}^{\phantom{\lambda}\mu\nu}=-\accentset{\scaleto{\diamond}{3.5pt}}{\nabla}_{\lambda}g^{\mu\nu}\,,$
(13)

and where
$\accentset{\scaleto{\diamond}{3.5pt}}{Q}^{\alpha\mu\nu}=-g^{\alpha\beta}\accentset{\scaleto{\diamond}{3.5pt}}{\nabla}_{\beta}g^{\mu\nu}$.
Different to TG, STG retains the metric as the fundamental dynamical object of
the theory meaning that the transition is not so dramatic.

Another way to view the trinity of gravity is through the general linear
affine connection given by ortin2004gravity ; Hehl:1994ue

$\Gamma^{\alpha}_{\phantom{\alpha}\mu\nu}=\accentset{\circ}{\Gamma}^{\alpha}_{\phantom{\alpha}\mu\nu}+\accentset{\land}{K}^{\alpha}_{\phantom{\alpha}\mu\nu}+\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\alpha}_{\phantom{\alpha}\mu\nu}\,,$
(14)

where
$\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\alpha}_{\phantom{\alpha}\mu\nu}$
represents the disformation tensor which encodes the contribution of the
nonmetricity tensor Nester:1998mp ; BeltranJimenez:2017tkd , and is defined as

$\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\alpha}_{\phantom{\alpha}\mu\nu}:=\frac{1}{2}g^{\lambda\alpha}\left(Q_{\lambda\mu\nu}-Q_{\mu\lambda\nu}-Q_{\nu\lambda\mu}\right)\,,$
(15)

which shares a number of symmetries with the Levi-Civita connection. The class
of STG is given by the conditions of vanishing curvature

$\accentset{\scaleto{\diamond}{3.5pt}}{R}^{\rho}_{\phantom{\rho}\sigma\mu\nu}\equiv
0\,,$ (16)

and vanishing torsion

$\accentset{\scaleto{\diamond}{3.5pt}}{T}^{\rho}_{\phantom{\rho}\mu\nu}\equiv
0\,,$ (17)

which represent curvature- and torsion-based constructions of gravity,
respectively. The most general connection that satisfies conditions of this
kind is the symmetric teleparallel connection

$\accentset{\scaleto{\diamond}{3.5pt}}{\Gamma}^{\alpha}{}_{\mu\nu}:=\frac{\partial
x^{\alpha}}{\partial\xi^{\sigma}}\frac{\partial^{2}\xi^{\sigma}}{\partial
x^{\mu}\partial x^{\nu}}\,,$ (18)

where $\xi^{\sigma}=\xi^{\sigma}(x)$ is an arbitrary function of spacetime
position. It turns out that this connection can be derived from vanishing
connection components through the coordinate transformation

$x^{\mu}\rightarrow\xi^{\mu}(x^{\nu})\,.$ (19)

In this setting, the connection (18) turns out to be a pure gauge connection,
and hence it is always possible to determine a coordinate transformation in
which this vanishes. This is called the coincident gauge
BeltranJimenez:2017tkd , and can be useful in simplifying calculations.

Thus, it is possible to construct a gravitational theory based solely on the
disformation tensor where the effect of gravity is communicated through
measures of nonmetricity rather than curvature or torsion. This is analogous
to the way that the regular Riemann tensor is based only on the Levi-Civita
connection. One of these theories is STEGR which is expressed through the
Lagrangian

$\mathcal{\accentset{\scaleto{\diamond}{3.5pt}}{L}}_{\text{STEGR}}=\frac{\sqrt{-g}}{16\pi
G}\accentset{\scaleto{\diamond}{3.5pt}}{Q}\,,$ (20)

where

$\accentset{\scaleto{\diamond}{3.5pt}}{Q}=g^{\mu\nu}\left(\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\alpha}_{\phantom{\alpha}\beta\mu}\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\beta}_{\phantom{\beta}\nu\alpha}-\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\alpha}_{\phantom{\alpha}\beta\alpha}\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\beta}_{\phantom{\beta}\mu\nu}\right)\,,$
(21)

which is the nonmetricity scalar.

To better see the equivalence with GR, we can rewrite the Hilbert-Einstein
Lagrangian in terms of the Levi-Civita connection giving Harko:2018gxr

$\mathcal{\accentset{\circ}{L}}_{HE}=\frac{\sqrt{-g}}{16\pi
G}\accentset{\circ}{R}=\mathcal{\accentset{\circ}{L}}_{E}+\accentset{\circ}{B}\,,$
(22)

where $\mathcal{L}_{E}$ represents the Einstein Lagrangian contribution
constructed by the Levi-Civita connection Einstein:1916cd  in which

$\mathcal{\accentset{\circ}{L}}_{E}:=\frac{\sqrt{-g}}{16\pi
G}\,g^{\mu\nu}\left(\accentset{\circ}{\Gamma}^{\alpha}_{\phantom{\alpha}\beta\mu}\accentset{\circ}{\Gamma}^{\beta}_{\phantom{\beta}\nu\alpha}-\accentset{\circ}{\Gamma}^{\alpha}_{\phantom{\alpha}\beta\alpha}\accentset{\circ}{\Gamma}^{\beta}_{\phantom{\beta}\mu\nu}\right)\,,$
(23)

and where the boundary term is defined by

$\accentset{\circ}{B}=\frac{\sqrt{-g}}{16\pi
G}\,\left(g^{\alpha\mu}\accentset{\circ}{\nabla}_{\alpha}\accentset{\circ}{\Gamma}^{\nu}{}_{\mu\nu}-g^{\mu\nu}\accentset{\circ}{\nabla}_{\alpha}\accentset{\circ}{\Gamma}^{\alpha}{}_{\mu\nu}\right)\,,$
(24)

which is a total divergence term. The Hilbert-Einstein Lagrangian completes
the original Einstein Lagrangian in that it adds the boundary term
$\accentset{\circ}{B}$ which renders the theory covariant ortin2004gravity .

Now, assuming the coincident, where the connection vanishes
($\accentset{\scaleto{\diamond}{3.5pt}}{\Gamma}^{\alpha}_{\phantom{\alpha}\mu\nu}\equiv
0$), means that the covariant derivative directly reduces to the ordinary
partial derivative
($\accentset{\scaleto{\diamond}{3.5pt}}{\nabla}_{\mu}\rightarrow\partial_{\mu}$),
it follows that the disformation tensor turns out to be the negative of the
Christoffel symbols

$\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\alpha}_{\phantom{\alpha}\mu\nu}=-\accentset{\circ}{\Gamma}^{\alpha}_{\phantom{\alpha}\mu\nu}\,.$
(25)

Thus, it follows that the field equations of the STEGR Lagrangian must produce
the exact same field equations as GR despite being based on nonmetricity as
the means by which geometric deformation takes place Conroy:2017yln . Hence,
both GR and STEGR turn out to be dynamically equivalent, as do TEGR and STEGR.

The Einstein Lagrangian (23) necessitates a boundary term in order to be
diffeomorphism invariant, while in STEGR any arbitrary coordinate
transformation of the Lagrangian (20) remains invariant. Thus STEGR retains
diffeomorphism invariance without the boundary term. This is true even if the
coincident gauge is not considered Koivisto:2019jra ; Koivisto:2018aip . In
the same way that the Hilbert-Einstein action is a covariantization of the
Einstein action, the STEGR action is another way to covariantize this action.
While the Hilbert-Einstein action achieves this by simply adding a boundary
term, STEGR makes the nontrivial shift to the symmetric teleparallel
connection to accomplish this result.

The STEGR action thus turns out to be given by Victor_stg ;
BeltranJimenez:2018vdo

$\displaystyle S_{G}$ $\displaystyle=\int
d^{4}x\,\Big{[}\frac{\sqrt{-g}}{2\kappa^{2}}\,\accentset{\scaleto{\diamond}{3.5pt}}{Q}+\sqrt{-g}\mathcal{L}_{m}\Big{]}\,,$
(26)

which naturally leads to the conjugate to the STEGR Lagrangian through

$\displaystyle\accentset{\scaleto{\diamond}{3.5pt}}{P}^{\alpha}_{\phantom{\alpha}\mu\nu}$
$\displaystyle:=\frac{1}{2\sqrt{-g}}\frac{\partial\left(\sqrt{-g}\accentset{\scaleto{\diamond}{3.5pt}}{Q}\right)}{\partial\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\alpha}^{\phantom{\alpha}\mu\nu}}$
$\displaystyle=\frac{1}{4}\accentset{\scaleto{\diamond}{3.5pt}}{Q}^{\alpha}_{\phantom{\alpha}\mu\nu}-\frac{1}{4}\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{(\mu\phantom{\alpha}\nu)}^{\phantom{(\mu}\alpha}-\frac{1}{4}g_{\mu\nu}\accentset{\scaleto{\diamond}{3.5pt}}{Q}^{\alpha\beta}_{\phantom{\alpha\beta}\beta}+\frac{1}{4}\left[\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\beta}^{\phantom{\beta}\beta\alpha}g_{\mu\nu}+\frac{1}{2}\delta^{\alpha}_{\phantom{\alpha}(\mu}\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\nu)\phantom{\beta}\beta}^{\phantom{\nu)}\beta}\right]\,,$
(27)

which provides a convenient alternative to describing the metricity scalar as
$\accentset{\scaleto{\diamond}{3.5pt}}{Q}=-\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\alpha\mu\nu}\accentset{\scaleto{\diamond}{3.5pt}}{P}^{\alpha\mu\nu}$
Soudi:2018dhv . By taking a variation of the action in Eq. (26) with respect
to the metric tensor, the field equations are derived Victor_stg

$2\accentset{\scaleto{\diamond}{3.5pt}}{\nabla}_{\alpha}\left(\sqrt{-g}\accentset{\scaleto{\diamond}{3.5pt}}{P}^{\alpha}_{\phantom{\alpha}\mu\nu}\right)-q_{\mu\nu}-\frac{\sqrt{-g}\,\accentset{\scaleto{\diamond}{3.5pt}}{Q}}{2}\,g_{\mu\nu}=\kappa^{2}\sqrt{-g}\Theta_{\mu\nu}\,,$
(28)

where

$\frac{1}{\sqrt{-g}}q_{\mu\nu}=\frac{1}{4}\left(2\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\alpha\beta\mu}\accentset{\scaleto{\diamond}{3.5pt}}{Q}^{\alpha\beta}_{\phantom{\alpha\beta}\nu}-\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\mu\alpha\beta}\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\nu}^{\phantom{\nu}\alpha\beta}\right)-\frac{1}{2}\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\alpha\beta\mu}\accentset{\scaleto{\diamond}{3.5pt}}{Q}^{\beta\alpha}_{\phantom{\beta\alpha}\nu}-\frac{1}{4}\left(2\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\alpha\phantom{\beta}\beta}^{\phantom{\alpha}\beta}\accentset{\scaleto{\diamond}{3.5pt}}{Q}^{\alpha}_{\phantom{\alpha}\mu\nu}-\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\mu\phantom{\beta}\beta}^{\phantom{\mu}\beta}\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\nu\phantom{\beta}\beta}^{\phantom{\nu}\beta}\right)+\frac{1}{2}\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\beta\phantom{\beta}\alpha}^{\phantom{\beta}\beta}\accentset{\scaleto{\diamond}{3.5pt}}{Q}^{\alpha}_{\phantom{\alpha}\mu\nu}\,.$
(29)

The connection, being independent of the metric in this scenario, also
produces independent field equations. By assuming vanishing hypermomentum
BeltranJimenez:2018vdo , the variation with respect to the connection produces
the connection field equation

$\accentset{\scaleto{\diamond}{3.5pt}}{\nabla}_{\mu}\accentset{\scaleto{\diamond}{3.5pt}}{\nabla}_{\nu}\left(\sqrt{-g}\accentset{\scaleto{\diamond}{3.5pt}}{P}^{\mu\nu}_{\phantom{\mu\nu}\alpha}\right)=0\,.$
(30)

Together, the metric field Eqs.(28) and connection field Eqs.(30) represent
the total DoFs of the dynamics in STEGR. Moreover, for STEGR, the connection
equation is trivially satisfied when the coincident gauge is utilized.

##  III The 3+1 formulation for a general linear affine Connection without
Metricity

Let us now formulate two new approaches to the $3+1$ decomposition of gravity,
one tetrad-based and the other metric-based. This is achieved by considering a
general affine connection, $\Gamma^{\lambda}_{\mu\nu}$, without metricity
(tensors, and covariant derivatives with no accents denote all quantities that
use this general connection in their calculation). This allows the general
forms of curvature, torsion, spin and nonmetricity terms to feature in the
ensuing equations. Throughout this section, we show that the metric and tetrad
formalisms are each other consistent at every level. It should be noted that,
from this point onwards, $3$-tensors that have the same symbol as their 4
dimensional counterparts will be denoted with a $(3)$ superscript or
subscript, and that purely local tensors will be denoted with a
‘$\;\tilde{\;}\;$’ annotation to keep consistent with the index notation
defined above.

###  III.1 Basic definitions

To set up a basis for the decomposition strategy being developed in this
section, we present some definitions and some preliminary relationships
between the main variables of the formalism. To this end, we let $M$ be a
$4$-dimensional manifold with a global spacetime metric $g_{\mu\nu}$ which is
derived from a tetrad $e^{A}_{\;\;\nu}$ such that

$g_{\mu\nu}=\eta_{AB}e^{A}_{\;\;\mu}e^{B}_{\;\;\nu}.$ (31)

We assume that ($M$, $g_{\mu\nu}$) can be foliated into non-intersecting
spacelike $3$-surfaces, $\Sigma$, such that each surface is a level surface of
a scalar function ”$t$”. This implies that these slices are purely spatial
slices each at their own instance in time. In this case, this function can be
interpreted as a global time function.

The one-form $\Omega_{\nu}$ is now defined as the covariant derivative of this
time function, $\nabla_{\nu}{t}$. By contracting this one-form twice with the
metric of the manifold, we obtain the norm of $\Omega$ baumgarte ,

$\displaystyle{|\Omega|}^{2}$
$\displaystyle=g^{\mu\nu}\nabla_{\mu}{t}\nabla_{\nu}{t}$
$\displaystyle=g^{00}$ $\displaystyle=-\frac{1}{\alpha^{2}}\,,$ (32)

where $\alpha$ is the lapse function and $\nabla_{\mu}$ is the general
covariant derivative. The lapse function can be interpreted as a measure of
how much time elapses between one time slice and the next. It is also assumed
that $\alpha$ is positive making $\Omega_{\nu}$ timelike and the
hypersurfaces, $\Sigma$, everywhere spacelike.

The used general covariant derivative is associated with the general affine
connection. This derivative also adheres to nonmetricity Teleparallel_Palatini
; Victor_stg

$\displaystyle\nabla_{\lambda}g_{\mu\nu}=$ $\displaystyle
Q_{\lambda\mu\nu}\,,$ (33) $\displaystyle\nabla_{\lambda}g^{\mu\nu}=$
$\displaystyle-{Q_{\lambda}}^{\mu\nu}\,.$ (34)

In general, the following convention for the covariant derivative of any mixed
tensor $A^{a\nu}_{\;\;\;\;b\mu}$ will be followed throughout this work

$\displaystyle\nabla_{\lambda}A^{A\nu}_{\;\;\;\;b\mu}=A^{C\rho}_{\;\;\;\;B\mu}$
$\displaystyle\omega^{A}_{\;\;C\lambda}+A^{C\rho}_{\;\;\;\;B\mu}\Gamma^{\nu}_{\rho\lambda}$
(35)
$\displaystyle-A^{A\rho}_{\;\;\;\;C\mu}\omega^{C}_{\;\;B\lambda}-A^{A\nu}_{\;\;\;\;B\rho}\Gamma^{\rho}_{\mu\lambda}\,.$

Here $\omega^{A}_{\;\;C\lambda}$ is the local spin connection and
$\Gamma^{\nu}_{\rho\lambda}$ is a global general affine connection. By
normalizing $\Omega$, the unit normal to the foliations and its inverse are
defined as

$\displaystyle n_{\nu}$ $\displaystyle:=-\alpha\Omega_{\nu},$ (36)
$\displaystyle n^{\nu}$ $\displaystyle:=-g^{\mu\nu}(\alpha\Omega_{\mu})$ (37)
$\displaystyle=-g^{\mu\nu}(\alpha\nabla_{\mu}{\left(t\right)})\,,$

such that

$n^{\nu}n_{\nu}=-1\,.$ (38)

The normal tensors are built to generate this negative sign so as to point the
normal $n^{\nu}$ towards increasing time $t$.

We can now set our normal vector and co-vector in terms of the lapse function
$\alpha$ and the shift vector $\beta$ as in standard gravity as baumgarte ;
gourgoulhon ,

$\displaystyle n_{\mu}=$ $\displaystyle\left(-\alpha,0,0,0\right)\,,$ (39)
$\displaystyle n^{\mu}=$
$\displaystyle\left(\frac{1}{\alpha},-\frac{1}{\alpha}\beta^{i}\right)\,.$
(40)

Having thoroughly defined the normal to the foliations, the $3$-metric on the
foliation $\Sigma$ can now be set to

$\displaystyle\gamma^{\mu\nu}=$ $\displaystyle g^{\mu\nu}+n^{\mu}n^{\nu}\,,$
(41) $\displaystyle\gamma_{\mu\nu}=$ $\displaystyle
g_{\mu\nu}+n_{\mu}n_{\nu}\,.$ (42)

In order to break up arbitrary 4-dimensional tensors on the manifold
($M$,$g_{\mu\nu}$) into their spatial and temporal parts, a variant
decomposition of this metric, $g_{\mu\nu}\gamma^{\lambda\mu}$, can be used.
This is defined as follows

$\gamma^{\mu}_{\;\;\nu}=\delta^{\mu}_{\nu}+n^{\mu}n_{\nu}\,.$ (43)

With this definition in hand, $\gamma^{\mu}_{\;\;\nu}$ can be thought as the
$3$-delta. While this form of the delta function exhibits all the properties
of a spatial delta on spatial vectors and tensors, this is not the spatial
identity matrix. This tensor has off-diagonal temporal components similarly to
the inverse spatial metric.

$\displaystyle\gamma^{\mu\nu}=$ $\displaystyle\begin{pmatrix}0&0\\\
0&\gamma^{ij}\end{pmatrix}\,,$ (44) $\displaystyle\gamma_{\mu\nu}=$
$\displaystyle\begin{pmatrix}\beta^{k}\beta_{k}&\beta_{i}\\\
\beta_{j}&\gamma_{ij}\end{pmatrix}\,,$ (45)
$\displaystyle{\gamma^{\mu}}_{\nu}=$
$\displaystyle\begin{pmatrix}0&\beta^{i}\\\
0&{\gamma^{i}}_{j}\end{pmatrix}\,.$ (46)

As a result of this and Eqs. (41, 42, 43), we get

$\displaystyle g^{\mu\nu}=$
$\displaystyle\begin{pmatrix}-\frac{1}{\alpha^{2}}&\frac{1}{\alpha^{2}}\beta^{i}\\\
\frac{1}{\alpha^{2}}\beta^{i}&\gamma^{ij}-\frac{1}{\alpha^{2}}\beta^{i}\beta^{j}\end{pmatrix}\,,$
(47) $\displaystyle g_{\mu\nu}=$
$\displaystyle\begin{pmatrix}-\alpha^{2}+\beta^{k}\beta_{k}&\beta_{i}\\\
\beta_{j}&\gamma_{ij}\end{pmatrix}\,.$ (48)

The following relations are some useful consequences of the previous
definitions

$\displaystyle\gamma^{\mu\lambda}\gamma_{\lambda\nu}$
$\displaystyle={\gamma^{\mu}}_{\nu}\,,$ (49)
$\displaystyle\gamma^{\mu}_{\;\;\nu}V^{(3)}_{\mu}$
$\displaystyle=V^{(3)}_{\nu}\,,$ (50) $\displaystyle n^{\mu}V^{(3)}_{\mu}$
$\displaystyle=0\,,$ (51) $\displaystyle n^{\mu}\gamma_{\mu\nu}$
$\displaystyle=0\,.$ (52)

Similar to the the global manifold defined above, we now specify an inertial
manifold $\tilde{M}$ along with its metric, the Minkowski metric $\eta_{ab}$,
and non-intersecting spacelike $3$-surfaces $\tilde{\Sigma}$ are defined. The
normal $\tilde{n}^{a}$ to the foliations and the inertial $3$-metric,
$\tilde{\gamma}_{ab}$, can also be similarly defined.

It should be noted that since we are assuming nonmetricity, the covariant
derivative of the Minkowski metric is non-zero in this setting, and it results
in

$\displaystyle\nabla_{\lambda}\eta_{AB}$
$\displaystyle=\partial_{\lambda}\eta_{AB}-\eta_{CB}{\omega^{C}}_{A\lambda}-\eta_{AC}{\omega^{C}}_{B\lambda}$
(53)
$\displaystyle=-\eta_{CB}{\omega^{C}}_{A\lambda}-\eta_{AC}{\omega^{C}}_{B\lambda}\,.$

In theories with metricity, this would result in an antisymmetry in the first
two indices of the spin connection.

By choosing the Minkowski metric as our local metric, we can show that the
inertial normal $\tilde{n}$ and its inverse are constant vectors

$\displaystyle\tilde{n}_{A}$
$\displaystyle=\begin{pmatrix}-1&0&0&0\end{pmatrix}\,,$ (54)
$\displaystyle\tilde{n}^{A}$
$\displaystyle=\begin{pmatrix}1&0&0&0\end{pmatrix}\,,$ (55)

since the lapse function and the shift vector are unity and a zero vector
respectively, in this case. If a non-diagonal local metric were chosen, the
elements of $\tilde{n}^{a}$ would become dependent on the spatial shift vector
$\tilde{\beta}^{\tilde{i}}$, the same as the standard global one baumgarte .

At this point we reintroduce the tetrad and attempt to find a form for it
equivalent to the metric ones in Eqs. (47,48). Starting from Eq. (31) we get
that

$g_{\mu\nu}=\begin{pmatrix}{e^{\tilde{0}}}_{0}{e^{\tilde{0}}}_{0}\eta_{\tilde{0}\tilde{0}}+{e^{\tilde{i}}}_{0}{e^{\tilde{j}}}_{0}\eta_{\tilde{i}\tilde{j}}&{e^{\tilde{0}}}_{i}{e^{\tilde{0}}}_{0}\eta_{\tilde{0}\tilde{0}}+{e^{\tilde{i}}}_{i}{e^{\tilde{j}}}_{0}\eta_{\tilde{i}\tilde{j}}\\\
{e^{\tilde{0}}}_{0}{e^{\tilde{0}}}_{j}\eta_{\tilde{0}\tilde{0}}+{e^{\tilde{i}}}_{0}{e^{\tilde{j}}}_{j}\eta_{\tilde{i}\tilde{j}}&{e^{\tilde{0}}}_{i}{e^{\tilde{0}}}_{j}\eta_{\tilde{0}\tilde{0}}+{e^{\tilde{i}}}_{i}{e^{\tilde{j}}}_{j}\eta_{\tilde{i}\tilde{j}}\end{pmatrix}\,.$
(56)

Comparing this to Eq.(48), we get that

${e^{\tilde{0}}}_{i}{e^{\tilde{0}}}_{j}\eta_{\tilde{0}\tilde{0}}+{e^{\tilde{i}}}_{i}{e^{\tilde{j}}}_{j}\eta_{\tilde{i}\tilde{j}}=\gamma_{ij}\,.$
(57)

Now since we know that
${e^{\tilde{i}}}_{i}{e^{\tilde{j}}}_{j}\eta_{\tilde{i}\tilde{j}}=\gamma_{ij}$
and that $\eta_{\tilde{0}\tilde{0}}=-1$, we conclude that
${e^{\tilde{0}}}_{i}=0$.

Using this we also get that
$\beta_{j}={e^{\tilde{i}}}_{i}{e^{\tilde{j}}}_{0}\eta_{\tilde{i}\tilde{j}}$,
from which we can conclude that
${e^{\tilde{i}}}_{0}={e^{\tilde{i}}}_{m}\gamma^{mn}\beta_{n}$. Substituting
this in the second term of $g_{00}$ component of Eq. (56), we get that
${e^{\tilde{i}}}_{0}{e^{\tilde{j}}}_{0}\eta_{\tilde{i}\tilde{j}}=\beta^{k}\beta_{k}$
implying that ${e^{\tilde{0}}}_{0}=\pm\alpha$. Noticing that
$e_{A}^{\;\;\nu}=\eta_{AB}g^{\mu\nu}e^{B}_{\;\;\mu}$, while using the above
result, we get

$\displaystyle{e^{A}}_{\nu}=$
$\displaystyle\begin{pmatrix}\pm\alpha&{e^{\tilde{i}}}_{k}\beta^{k}\\\
0&{e^{\tilde{i}}}_{j}\end{pmatrix},$ (58) $\displaystyle{e_{A}}^{\nu}=$
$\displaystyle\begin{pmatrix}\pm\frac{1}{\alpha}&0\\\
\mp\frac{1}{\alpha}\beta^{j}&{e_{\tilde{i}}}^{j}\end{pmatrix}\,.$ (59)

At this point, it is worth noticing that the two defined normal vectors, the
local one and the global one, are not necessarily related through the tetrad,
namely

$\tilde{n}_{A}=e_{A}^{\phantom{A}\nu}n_{\nu}\,,$ (60)

since this is only the case if ${e_{\tilde{i}}}^{0}=0$. From the above
considerations, we conclude that, for the current choice of metric Eq. (47),
this is the case and so, using this relation and its variant decompositions,
we obtain the correct signs for the tetrad components resulting in

$\displaystyle{e^{A}}_{\nu}=$
$\displaystyle\begin{pmatrix}\alpha&{e^{\tilde{i}}}_{k}\beta^{k}\\\
0&{e^{\tilde{i}}}_{j}\end{pmatrix}\,,$ (61) $\displaystyle{e_{A}}^{\nu}=$
$\displaystyle\begin{pmatrix}\frac{1}{\alpha}&0\\\
-\frac{1}{\alpha}\beta^{j}&{e_{\tilde{i}}}^{j}\end{pmatrix}\,.$ (62)

It has to be noted that these agree with the results obtained in Refs.
Blixt:2019mkt ; Blixt:2020ekl  at least up to the local index $3+1$ split.
Having the tatrads in this form, we can now move on to find an equivalent to
$\gamma_{\mu\nu}$, say ${\theta^{A}}_{\mu}$, such that

${\theta^{A}}_{\mu}={e^{A}}_{\mu}+{U^{A}}_{\mu}\,,$ (63)

where ${U^{A}}_{\mu}$ is some tensor that will embody the temporal part of the
tetrad. The only property we need for ${\theta^{A}}_{\mu}$ is that it is
orthogonal to the normal vectors. Since we have two unknowns,
${\theta^{A}}_{\mu}$ and ${U^{A}}_{\mu}$, there is nothing preventing us from
defining ${U^{A}}_{\mu}$ as any tensor we want so long as it preserves the
orthogonality of ${\theta^{A}}_{\mu}$ with the normal vectors. Thus, we set
this statement as

${U^{A}}_{\mu}=\tilde{n}^{A}n_{\mu}\,.$ (64)

Below, we show that this property preserves indeed the orthogonality of
${\theta^{A}}_{\mu}$ in that we get

$\displaystyle n^{\mu}{\theta^{A}}_{\mu}=$ $\displaystyle
n^{\mu}{e^{A}}_{\mu}+n^{\mu}n_{\mu}\tilde{n}^{A}$ (65) $\displaystyle=$
$\displaystyle\tilde{n}^{A}-\tilde{n}^{A}$ $\displaystyle=$ $\displaystyle
0\,,$ $\displaystyle n_{A}{\theta^{A}}_{\mu}=$
$\displaystyle\tilde{n}_{A}{e^{A}}_{\mu}+n^{\mu}\tilde{n}_{A}\tilde{n}^{A}$
(66) $\displaystyle=$ $\displaystyle n^{\mu}-n^{\mu}$ $\displaystyle=$
$\displaystyle 0\,.$

The same exact procedure can be applied when splitting the inverse tetrad,
giving

$\displaystyle{\theta^{A}}_{\mu}={e^{A}}_{\mu}+{\tilde{n}^{A}}n_{\mu}\,,$ (67)
$\displaystyle{\theta_{A}}^{\mu}={e_{A}}^{\mu}+{\tilde{n}_{A}}n^{\mu}\,.$ (68)

Using these equations, we can now determine the components of the $\theta$
tensors to be

$\displaystyle{\theta^{A}}_{\nu}=$
$\displaystyle\begin{pmatrix}0&{e^{\tilde{i}}}_{k}\beta^{k}\\\
0&{e^{\tilde{i}}}_{j}\end{pmatrix}\,,$ (69) $\displaystyle{\theta_{A}}^{\nu}=$
$\displaystyle\begin{pmatrix}0&0\\\ 0&{e_{\tilde{i}}}^{j}\end{pmatrix}\,.$
(70)

Among others, some convenient properties of this form of ${\theta^{A}}_{\mu}$
that one can easily show are

$\displaystyle{\theta^{A}}_{\mu}{\theta_{A}}^{\nu}$
$\displaystyle={\gamma^{\nu}}_{\mu}\,,$ (71)
$\displaystyle{\theta^{A}}_{\mu}{\theta_{B}}^{\mu}$
$\displaystyle={\tilde{\gamma}^{A}}_{B}\,,$ (72)
$\displaystyle\tilde{\gamma}_{AB}\theta^{A}_{\;\;\mu}\theta^{B}_{\;\;\nu}$
$\displaystyle=\gamma_{\mu\nu}\,,$ (73)
$\displaystyle\tilde{\gamma}^{AB}\theta_{A}^{\;\;\mu}\theta_{B}^{\;\;\nu}$
$\displaystyle=\gamma^{\mu\nu}\,.$ (74)

Through these properties, we can then derive a number of other relations as a
consequence.

While $\theta^{A}_{\;\;\mu}$ should not be confused with
${e^{\tilde{i}}}_{j}$, it has to be noted that from this point onward we will
be re-labeling ${\theta^{A}}_{\mu}$ as ${e^{A}}^{(3)}_{\mu}$ since it
effectively embodies the role of the spatial tetrad.

Let us now define the spatial covariant derivative of a general spatial tensor
$A$ as

$D_{\nu}A^{A}_{\;\;\mu}=\gamma^{\lambda}_{\;\;\mu}\tilde{\gamma}^{A}_{\;\;B}\gamma^{\rho}_{\;\;\nu}\nabla_{\rho}A^{B}_{\;\;\lambda}\,.$
(75)

This derivative can be shown to follow the Leibniz rule only when applied to
spatial tensors thus resulting in additional terms if applied to a spacetime
tensor gourgoulhon . Two results of note from this definition are the
following

$\displaystyle D_{\nu}n^{\mu}$
$\displaystyle=\partial^{(3)}_{\nu}n^{\mu}+n^{\lambda}\Gamma^{\mu(3)}_{\lambda\nu}$
(76) $\displaystyle=\partial^{(3)}_{\nu}n^{\mu}\,,$ $\displaystyle
D_{\nu}\tilde{n}^{A}$
$\displaystyle=\partial^{(3)}_{\nu}\tilde{n}^{A}+\tilde{n}^{B}\omega^{A(3)}_{B\nu}$
(77) $\displaystyle=\partial^{(3)}_{\nu}\tilde{n}^{A}$ $\displaystyle=0\,.$

The spatial covariant derivative can also be shown to inherit the nonmetricity
property when this is also the case for the spacetime variant and that it
similarly inherits metricity when not.

Before continuing with the development of the $3+1$ formalism, it is useful to
present the following relations as a consequence of nonmetricity. They are

$\displaystyle n^{\lambda}\nabla_{\rho}n_{\lambda}$
$\displaystyle=\frac{1}{2}n^{\lambda}n^{\nu}Q_{\rho\lambda\nu}\,,$ (78)
$\displaystyle n_{\lambda}\nabla_{\rho}n^{\lambda}$
$\displaystyle=-\frac{1}{2}n^{\lambda}n^{\nu}Q_{\rho\lambda\nu}\,.$ (79)

Defining the acceleration $a_{\lambda}$ as
$n^{\sigma}\nabla_{\sigma}n_{\lambda}$, we get

$\displaystyle a^{\epsilon}$
$\displaystyle=n^{\sigma}\nabla_{\sigma}n^{\epsilon}+n^{\sigma}n_{\lambda}{Q_{\sigma}}^{\lambda\epsilon}\,,$
(80) $\displaystyle n^{\lambda}a_{\lambda}$
$\displaystyle=\frac{1}{2}n^{\sigma}n^{\lambda}n^{\epsilon}Q_{\sigma\lambda\epsilon}\,.$
(81)

It is now possible to define the extrinsic curvature $k$ as

$\displaystyle k_{\alpha\beta}$
$\displaystyle=-\gamma^{\nu}_{\;\;\alpha}\gamma^{\mu}_{\;\;\beta}\nabla_{\nu}n_{\mu}\,.$
(82)

From the definition of $k_{\alpha\beta}$, one can easily show that it is a
purely spatial tensor. The extrinsic curvature can be thought of as the change
of direction of the normal vector $n$ as it is moved along the foliation
$\Sigma$ baumgarte . From this definition, the following alternate expression
for the extrinsic curvature can be derived

$\displaystyle k_{\nu\mu}$
$\displaystyle=-\nabla_{\nu}n_{\mu}-n_{\nu}a_{\mu}-\frac{1}{2}n^{\rho}n^{\beta}n_{\mu}{\gamma^{\alpha}}_{\nu}Q_{\alpha\rho\beta}\,.$
(83)

It should be noted that the covector extrinsic curvature is defined as

$\displaystyle k^{\alpha\beta}$
$\displaystyle=-\gamma^{\nu\alpha}\gamma^{\mu\beta}\nabla_{\nu}n_{\mu}\,,$

and that, due to nonmetricity, it is

$\displaystyle\gamma^{\nu\alpha}\gamma^{\beta}_{\;\;\mu}\nabla_{\nu}n^{\mu}=-n_{\lambda}\gamma^{\alpha\nu}{\gamma^{\beta}}_{\mu}{Q_{\nu}}^{\lambda\mu}-k^{\alpha\beta}\,.$
(84)

Having now defined all the basic building blocks of a generalized $3+1$
formalism, we consider the terms which can be used to characterize gravitation
in such a theory. We define the Riemann and the torsion tensors in terms of
the general affine connection $\Gamma^{\rho}_{\lambda\mu}$ as follows
PereiraBook  and the nonmetricity term once again for convenience

$\displaystyle R^{\;\rho}_{\;\;\lambda\nu\mu}$
$\displaystyle=\partial_{\nu}{\Gamma^{\rho}_{\lambda\mu}}-\partial_{\mu}{\Gamma^{\rho}_{\lambda\nu}}\,,+\Gamma^{\rho}_{\alpha\nu}\Gamma^{\alpha}_{\lambda\mu}-\Gamma^{\rho}_{\alpha\mu}{\Gamma^{\alpha}_{\lambda\nu}}\,,$
(85) $\displaystyle T^{\lambda}_{\;\;\mu\nu}$
$\displaystyle=\Gamma^{\lambda}_{\nu\mu}-\Gamma^{\lambda}_{\mu\nu}\,,$ (86)
$\displaystyle Q_{\lambda\mu\nu}$
$\displaystyle=\nabla_{\lambda}g_{\mu\nu}\,,$ (87)

where we note that, for a general affine connection, the Riemann tensor only
possesses the antisymmetry $R^{\chi}_{\;\;\lambda[\rho\pi]}$.

Through the definition of the torsion tensor it is also possible to show that
it is related to the antisymmetry of the extrinsic curvature through

$\displaystyle k_{[\alpha\beta]}$
$\displaystyle=\;\;\gamma^{\nu}_{\;\;\alpha}\gamma^{\mu}_{\;\;\beta}n_{\sigma}T^{\sigma}_{\;\;\nu\mu}\,.$
(88)

Using the above definitions, the commutator of the spacetime covariant
derivative of any vector $V^{\lambda}$ and covector $V_{\lambda}$ can be shown
to be carroll2004spacetime

$\displaystyle\nabla_{[\nu}\nabla_{\mu]}V^{\lambda}=T^{\sigma}_{\;\;\mu\nu}\nabla_{\sigma}V^{\lambda}+V^{\sigma}R^{\lambda}_{\;\;\sigma\nu\mu}\,,$
(89)
$\displaystyle\nabla_{[\nu}\nabla_{\mu]}V_{\lambda}=T^{\sigma}_{\;\;\mu\nu}\nabla_{\sigma}V_{\lambda}+V_{\sigma}R^{\sigma}_{\;\;\lambda\mu\nu}\,.$
(90)

Here we note that the commutator is independent of nonmetricity.

Taking the commutator of the purely spatial covariant derivative applied to a
general purely spatial vector $V^{\beta}_{(3)}$ and spatial co-vector
$V_{\beta}^{(3)}$, one obtains the following equations

$\displaystyle D_{[\sigma}D_{\alpha]}V^{\beta}$
$\displaystyle=T^{\lambda(3)}_{\;\;\alpha\sigma}D_{\lambda}V^{\beta}_{(3)}+V^{\lambda}_{(3)}R^{\beta(3)}_{\;\;\lambda\sigma\alpha}$
(91)
$\displaystyle=\;\gamma^{\rho}_{\;\;\sigma}\gamma^{\pi}_{\;\;\alpha}\gamma^{\beta}_{\;\;\chi}\left(T^{\lambda(4)}_{\;\;\pi\rho}D_{\lambda}V^{\chi}_{(3)}+V^{\lambda}_{(3)}R^{\chi(4)}_{\;\;\lambda\rho\pi}\right)$
$\displaystyle-V^{\lambda}_{(3)}k_{[\alpha|\lambda}{k_{|\sigma]}}^{\beta}-\gamma^{\rho}_{\;\;[\sigma|}\gamma^{\beta}_{\;\;\chi}V^{\lambda}_{(3)}k_{|\alpha]\lambda}n_{\mu}{Q_{\rho}}^{\mu\chi}\,,$

and

$\displaystyle D_{[\sigma}D_{\alpha]}V_{\beta}$
$\displaystyle=T^{\lambda(3)}_{\;\;\alpha\sigma}D_{\lambda}V_{\beta}^{(3)}+V_{\lambda}^{(3)}R^{\lambda(3)}_{\;\;\beta\alpha\sigma}$
(92)
$\displaystyle=\;\gamma^{\rho}_{\;\;\sigma}\gamma^{\pi}_{\;\;\alpha}\gamma^{\chi}_{\;\;\beta}\left(T^{\lambda(4)}_{\;\;\pi\rho}D_{\lambda}V_{\chi}^{(3)}+V_{\lambda}^{(3)}R^{\lambda(4)}_{\;\;\chi\pi\rho}\right)$
$\displaystyle-V^{\lambda}_{(3)}k_{[\alpha|\lambda}{k_{|\sigma]\beta}}-\gamma^{\pi}_{\;\;[\alpha|}n_{\lambda}V_{\nu}^{(3)}{Q_{\pi}}^{\lambda\nu}k_{|\sigma]\beta}\,.$

Once a particular connection is chosen, this equation will play a pivotal role
in determining what is called the Gauss equation of the theoretical framework
in question. The Gauss equation shows the relation between the four
dimensional tensor used to characterize gravity like the Riemann or the
torsion tensors and their three dimensional counterparts. With regards to the
nonmetricity term $Q_{\alpha\mu\nu}$, it is not necessary for a connection, or
theory, to be chosen to derive its Gauss equation. Starting from the spatial
covariant derivative of the spacetime metric, we get

$\displaystyle D_{\rho}g_{\mu\nu}$
$\displaystyle=\gamma^{\lambda}_{\;\;\rho}\gamma^{\beta}_{\;\;\mu}\gamma^{\alpha}_{\;\;\nu}\nabla_{\lambda}g_{\beta\alpha}=\gamma^{\lambda}_{\;\;\rho}\gamma^{\beta}_{\;\;\mu}\gamma^{\alpha}_{\;\;\nu}Q_{\lambda\beta\alpha}$
(93)
$\displaystyle=\gamma^{\lambda}_{\;\;\rho}\gamma^{\beta}_{\;\;\mu}\gamma^{\alpha}_{\;\;\nu}\nabla_{\lambda}\left(\gamma_{\beta\alpha}-n_{\beta}n_{\alpha}\right)$
$\displaystyle=D_{\rho}\gamma_{\mu\nu}=Q_{\rho\mu\nu}^{\;(3)}\,.$

From this, we determine that, independent of the theory being considered, the
Gauss equation for the nonmetricity term is

$\gamma^{\lambda}_{\;\;\rho}\gamma^{\beta}_{\;\;\mu}\gamma^{\alpha}_{\;\;\nu}Q_{\lambda\beta\alpha}=Q_{\rho\mu\nu}^{\;(3)}\,.$
(94)

###  III.2 Lie derivatives and generalized evolution equations

To obtain the generalized evolution equations for the formalisms, we first
define a general Lie derivative and explore some of its properties. For a
general global 4-vector, $V^{\nu}$, and $4$-covector, $V_{\nu}$, the Lie
derivative along a general vector $\chi^{\mu}$ is LieAsPartial1 ;
LieAsPartial2 ; baumgarte

$\displaystyle\mathcal{L}_{\chi}V^{\nu}$
$\displaystyle=\chi^{\mu}\partial_{\mu}V^{\nu}-V^{\mu}\partial_{\mu}\chi^{\nu}$
(95)
$\displaystyle=\chi^{\mu}\nabla_{\mu}V^{\nu}-V^{\mu}\nabla_{\mu}\chi^{\nu}+V^{\mu}\chi^{\lambda}T^{\nu}_{\;\;\lambda\mu}\,,$
$\displaystyle\mathcal{L}_{\chi}V_{\nu}$
$\displaystyle=\chi^{\mu}\partial_{\mu}V_{\nu}+V_{\mu}\partial_{\nu}\chi^{\mu}$
(96)
$\displaystyle=\chi^{\mu}\nabla_{\mu}V_{\nu}+V_{\mu}\nabla_{\nu}\chi^{\mu}+V_{\lambda}\chi^{\mu}T^{\lambda}_{\;\;\mu\nu}\,.$

Visually this can be understood as the change in pointing direction and
magnitude of the vector $V^{\nu}$ as it is infinitesimally dragged along the
vector $\chi^{\mu}$ on the manifold.

In the case of a tensor with mixed global and local indices, a Lie derivative
along a global vector is blind to the local indices and acts only on the
global indices. This is understandable as locally no gravitational effect can
be felt and there is nothing to evolve on the Minkowski metric. In a more
general, purely geometric setting, going from one metric to another, it is
simply a transformation of the metric. The secondary background metric does
not have to change when the primary changes, only the transformation or
mapping tensor must change. In our case, the mapping tensor is the tetrad and
as such it evolves as the metric evolves while conserving the secondary
metric. Given this fact, tensors with mixed indices will be treated as such

$\displaystyle\mathcal{L}_{\chi}V^{\nu}_{\;\;A}$
$\displaystyle=\chi^{\mu}\partial_{\mu}V^{\nu}_{\;\;A}-V^{\mu}_{\;\;A}\partial_{\mu}\chi^{\nu}\,.$
(97)

Let us now consider the Lie derivatives along the global normal $n_{\mu}$. We
note that spatial-ness is conserved when such a derivative is applied to a
general $3$-covector but not when applied to a general $3$-vector giving

$\displaystyle n_{\nu}\mathcal{L}_{n}V^{\nu}_{(3)}$
$\displaystyle=-\frac{1}{\alpha}\partial_{\mu}(\alpha)V^{\mu}_{(3)}=\partial_{\mu}(Ln(\alpha^{-1}))V^{\mu}_{(3)}\,,$

The Lie derivative along the normal and its double action are used instead of
any other derivative to obtain the evolution equations as they give the
evolution of the fundamental variables of our formalisms, as they iterate
through time. While a form of the evolution equations for a general affine
connection can be obtained, it is necessary to choose a specific connection
and Lagrangian, thus a theory, in order to obtain the final form of the
equations that can actually be solved numerically to find the fundamental
variables. The reason is that one of the final steps involves substituting the
field equations of whatever theory one is considering into the evolution
equations.

Noting that the Lie derivative of the upper local index spatial tensor along
any vector is spatial on that local index since

$\displaystyle\bar{n}_{A}\mathcal{L}_{\chi}U^{A(3)}_{\;\;\nu}=$
$\displaystyle\bar{n}_{A}\chi^{\mu}\partial_{\mu}U^{A(3)}_{\;\;\nu}+\bar{n}_{A}U^{A(3)}_{\;\;\mu}\partial_{\nu}\chi^{\mu}$
(98) $\displaystyle=$
$\displaystyle-U^{A(3)}_{\;\;\nu}\chi^{\mu}\partial_{\mu}\bar{n}_{A}$
$\displaystyle=$ $\displaystyle 0\,.$

That being said, this is only true due to our choice of local metric, i.e.,
the Minkowski metric. As such, the Lie derivative of the tetrad is fully
spatial. This result similarly applies for the second order Lie derivative of
the tetrad and so on.

Thus, we present the first and second order Lie derivatives of the spatial
metric and spatial tetrad along the normal vector for a general affine
connection with nonmetricity

$\displaystyle\mathcal{L}_{n}\gamma_{\mu\nu}$
$\displaystyle=\gamma_{\sigma\left(\nu|\right.}\gamma^{\beta}_{\;\;\left.|\mu\right).}n^{\lambda}T^{\sigma}_{\;\;\lambda\beta}+2\gamma^{\alpha}_{\;\;\nu}\gamma^{\beta}_{\;\;\mu}n_{\lambda}{L^{\lambda}}_{\alpha\beta}-k_{\left(\mu\nu\right)}$
(99) $\displaystyle=A_{(\mu\nu)}+B_{(\mu\nu)}-k_{\left(\mu\nu\right)}\,,$

$\displaystyle\mathcal{L}_{n}A_{\mu\nu}$
$\displaystyle=-\gamma^{\alpha}_{\;\;\nu}\gamma^{\beta}_{\;\;\mu}\nabla_{\lambda}{{T_{\alpha}}^{\lambda}}_{\beta}+D_{\lambda}{{T_{\nu}}^{\lambda}}_{\mu}$
(100)
$\displaystyle+\gamma^{\alpha}_{\;\;\nu}\gamma^{\beta}_{\;\;\mu}a_{\sigma}{{T_{\alpha}}^{\sigma}}_{\beta}+\gamma^{\beta}_{\;\;\mu}n_{\sigma}n^{\epsilon}{{T_{\epsilon}}^{\sigma}}_{\beta}\gamma^{\alpha}_{\;\;\nu}a_{\alpha}$
$\displaystyle-
A_{\lambda\mu}\left(\gamma^{\lambda}_{\;\;\sigma}\gamma^{\alpha}_{\;\;\nu}n_{\epsilon}{Q_{\alpha}}^{\epsilon\sigma}+{k_{\nu}}^{\lambda}\right)$
$\displaystyle-
A_{\nu\lambda}\left(\gamma^{\lambda}_{\;\;\sigma}\gamma^{\beta}_{\;\;\mu}n_{\epsilon}{Q_{\beta}}^{\epsilon\sigma}+{k_{\mu}}^{\lambda}\right)$
$\displaystyle+A_{\sigma\mu}{A^{\sigma}}_{\nu}+A_{\nu\sigma}{A^{\sigma}}_{\mu}\,,$

$\displaystyle\mathcal{L}_{n}B_{\mu\nu}$
$\displaystyle=-\gamma^{\alpha}_{\;\;\nu}\gamma^{\beta}_{\;\;\mu}\nabla_{\lambda}{L^{\lambda}}_{\alpha\beta}+D_{\lambda}{L^{\lambda}}_{\mu\nu}$
(101)
$\displaystyle+\gamma^{\alpha}_{\;\;\nu}\gamma^{\beta}_{\;\;\mu}\left(a_{\sigma}{L^{\sigma}}_{\alpha\beta}+a_{\left(\beta\right|}{L^{\sigma}}_{\epsilon\left|\alpha\right)}n_{\sigma}n^{\epsilon}\right)$
$\displaystyle-\gamma^{\epsilon}_{\;\;\left(\nu\right|}\gamma^{\chi}_{\;\;\left|\mu\right)}B_{\alpha\chi}\left(n_{\sigma}{Q_{\epsilon}}^{\sigma\alpha}+{k_{\epsilon}}^{\alpha}\right)$
$\displaystyle+B_{\sigma\left(\mu\right|}{A^{\sigma}}_{\left|\nu\right)}\,,$

$\displaystyle\mathcal{L}_{n}k_{\mu\nu}$
$\displaystyle=\gamma^{\alpha}_{\;\;\nu}\gamma^{\beta}_{\;\;\mu}n^{\lambda}\nabla_{\lambda}\nabla_{\alpha}n_{\beta}$
(102)
$\displaystyle+\gamma^{\alpha}_{\;\;\nu}\gamma^{\beta}_{\;\;\mu}\left(\frac{1}{2}n^{\chi}n^{\epsilon}Q_{\alpha\chi\epsilon}a_{\beta}+a_{\beta}a_{\alpha}\right)$
$\displaystyle-
k_{\lambda\mu}\left(\gamma^{\lambda}_{\;\;\sigma}\gamma^{\alpha}_{\;\;\nu}n_{\epsilon}{Q_{\alpha}}^{\epsilon\sigma}+{k_{\nu}}^{\lambda}\right)$
$\displaystyle-
k_{\nu\lambda}\left(\gamma^{\lambda}_{\;\;\sigma}\gamma^{\beta}_{\;\;\mu}n_{\epsilon}{Q_{\beta}}^{\epsilon\sigma}+{k_{\mu}}^{\lambda}\right)$
$\displaystyle+k_{\sigma\mu}{A^{\sigma}}_{\nu}+k_{\nu\sigma}{A^{\sigma}}_{\mu}\,,$

$\displaystyle\mathcal{L}_{n}e^{A(3)}_{\;\;\nu}$
$\displaystyle=n^{\mu}\partial_{\mu}e^{A(3)}_{\;\;\nu}+e^{A(3)}_{\;\;\mu}\partial_{\nu}n^{\mu}$
(103)
$\displaystyle=n^{\mu}\nabla_{\mu}e^{A(3)}_{\;\;\nu}+e^{A(3)}_{\;\;\mu}\nabla_{\nu}n^{\mu}$
$\displaystyle\;\;\;\;+n^{\mu}e^{A(3)}_{\;\;\lambda}\left(\Gamma^{\lambda}_{\nu\mu}-\Gamma^{\lambda}_{\mu\nu}\right)-n^{\mu}e^{B(3)}_{\;\;\nu}\omega^{A}_{\;\;B\mu}$
$\displaystyle={\tilde{\gamma}^{A}}_{C}{\gamma^{\alpha}}_{\nu}n^{\mu}\nabla_{\mu}{e^{C}}_{\alpha}-e^{A(3)}_{\;\;\mu}n_{\epsilon}{{\gamma}^{\mu}}_{\sigma}{\gamma^{\alpha}}{Q_{\alpha}}^{\epsilon\sigma}$
$\displaystyle\;\;\;\;-e^{A(3)}_{\;\;\mu}\gamma_{\alpha\nu}k^{\alpha\mu}+e^{A(3)}_{\;\;\lambda}{\gamma^{\lambda}}_{\epsilon}{\gamma^{\alpha}}_{\nu}n^{\mu}T^{\epsilon}_{\;\;\mu\alpha}$
$\displaystyle\;\;\;\;-e^{F(3)}_{\;\;\nu}{\tilde{\gamma}^{B}}_{F}{\tilde{\gamma}^{A}}_{C}n^{\mu}\omega^{C}_{\;\;B\mu},$
$\displaystyle={\mathcal{Q}^{A}}_{\nu}-{\mathcal{B}^{A}}_{\nu}-{k_{\nu}}^{A}+{A^{A}}_{\nu}-{\mathcal{C}^{A}}_{\nu}\,,$

$\displaystyle\mathcal{L}_{n}U^{A}_{\;\;\nu}$
$\displaystyle={\tilde{\gamma}^{A}}_{C}{\gamma^{\alpha}}_{\nu}n^{\lambda}\left(\nabla_{\lambda}{U^{C}}_{\alpha}-\nabla_{\alpha}{U^{C}}_{\lambda}\right)$
(104)
$\displaystyle\;\;\;\;-{U^{B}}_{\nu}{\mathcal{C}^{A}}_{B}+{U^{A}}_{\lambda}{A^{\lambda}}_{\nu}\,,$

where ${{U}^{A}}_{\nu}$ is a general spatial tensor that can be substituted by
each of the terms in the first Lie derivative of the tetrad to obtain a final
general form for the second Lie derivative of the tetrad similar to that of
the metric.

We end this section by presenting the Lie derivative of a general spatial
tensor along $\alpha n^{\mu}$. It can be shown that, even for a general
connection, the Lie derivative along $\alpha n^{\mu}$ of such a tensor is
itself spatial. Taking this Lie derivative of some tensor
${X^{\epsilon_{1}...\epsilon_{i}}}_{\sigma_{1}...\sigma_{j}}$, where $i$ and
$j$ are integers, it results in

$\displaystyle\mathcal{L}_{\alpha
n}{X^{\epsilon_{1}...\epsilon_{i}}}_{\sigma_{1}...\sigma_{j}}=$
$\displaystyle\alpha
n^{\lambda}\partial_{\lambda}{X^{\epsilon_{1}...\epsilon_{i}}}_{\sigma_{1}...\sigma_{j}}$
(105)
$\displaystyle\;\;-\sum_{p=1}^{i}{X^{\epsilon_{1}..\lambda_{p}..\epsilon_{i}}}_{\sigma_{1}...\sigma_{j}}\partial_{\lambda_{p}}\left(\alpha
n^{\epsilon_{p}}\right)$
$\displaystyle\;\;\;\;+\sum_{q=1}^{j}{X^{\epsilon_{1}..\epsilon_{i}}}_{\sigma_{1}..\lambda_{q}..\sigma_{j}}\partial_{\sigma_{q}}\left(\alpha
n^{\lambda_{q}}\right)\,.$

Thus, taking
$n^{\sigma_{k}}\mathcal{L}_{n}{X^{\epsilon_{1}...\epsilon_{i}}}_{\sigma_{1}...\sigma_{j}}$
for $1\leq k\leq j$ will result in the second term of Eq. (105) and all of the
third term except for when $k=q$ to become zero. What remains of Eq. (105) is

$\displaystyle n^{\sigma_{k}}\mathcal{L}_{\alpha
n}{X^{\epsilon_{1}...\epsilon_{i}}}_{\sigma_{1}...\sigma_{j}}=$
$\displaystyle\alpha
n^{\sigma_{k}}n^{\lambda}\partial_{\lambda}{X^{\epsilon_{1}...\epsilon_{i}}}_{\sigma_{1}...\sigma_{j}}$
(106)
$\displaystyle\;\;\;\;+n^{\sigma_{k}}{X^{\epsilon_{1}..\epsilon_{i}}}_{\sigma_{1}..\lambda..\sigma_{j}}\partial_{\sigma_{k}}\left(\alpha
n^{\lambda}\right)$ $\displaystyle=$ $\displaystyle\alpha
n^{\sigma_{k}}n^{\lambda}\left(\partial_{\lambda}{X^{\epsilon_{1}...\epsilon_{i}}}_{\sigma_{1}..\sigma_{k}..\sigma_{j}}\right.$
$\displaystyle\;\;\;\;\left.-\partial_{\sigma_{k}}{X^{\epsilon_{1}..\epsilon_{i}}}_{\sigma_{1}..\lambda..\sigma_{j}}\right)$
$\displaystyle=$ $\displaystyle 0\,.$

Taking
$n_{\epsilon_{k}}\mathcal{L}_{n}{X^{\epsilon_{1}...\epsilon_{i}}}_{\sigma_{1}...\sigma_{j}}$
for $1\leq k\leq i$, it will result in the third term of Eq. (105) and all of
the second term except for when $k=p$ to become zero. what remains of Eq.
(105) is

$\displaystyle n_{\epsilon_{k}}\mathcal{L}_{\alpha
n}{X^{\epsilon_{1}...\epsilon_{i}}}_{\sigma_{1}...\sigma_{j}}=$
$\displaystyle\alpha
n_{\epsilon_{k}}n^{\lambda}\partial_{\lambda}{X^{\epsilon_{1}...\epsilon_{i}}}_{\sigma_{1}...\sigma_{j}}$
(107) $\displaystyle-
n_{\epsilon_{k}}{X^{\epsilon_{1}..\lambda..\epsilon_{i}}}_{\sigma_{1}...\sigma_{j}}\partial_{\lambda}\left(\alpha
n^{\epsilon_{k}}\right)$ $\displaystyle=$ $\displaystyle
n_{\epsilon_{k}}\partial_{\lambda}\left(\alpha\right)\left(n^{\epsilon_{k}}{X^{\epsilon_{1}..\lambda..\epsilon_{i}}}_{\sigma_{1}...\sigma_{j}}\right.$
$\displaystyle\left.-n^{\lambda}{X^{\epsilon_{1}...\epsilon_{i}}}_{\sigma_{1}...\sigma_{j}}\right)$
$\displaystyle+{X^{\epsilon_{1}..\lambda..\epsilon_{i}}}_{\sigma_{1}...\sigma_{j}}\partial_{\lambda}\left(\alpha\right)$
$\displaystyle=$ $\displaystyle 0\,,$

where

$\displaystyle\alpha\partial_{\lambda}\left(n_{\epsilon_{k}}\right)=$
$\displaystyle-\alpha\partial_{\lambda}\left(\alpha\partial_{\epsilon_{k}}\left\\{t\right\\}\right)$
(108) $\displaystyle=$ $\displaystyle
n_{\epsilon_{k}}\partial_{\lambda}\left(\alpha\right)\,,$

was used.

##  IV Gauss Constraint and Evolution equations in Specific theories

In this section, the generalized $3+1$ metric and tetrad formalisms set up in
Sec. III will be applied to three actual theories of interest by choosing
their respective connections. In the first case, the Levi-Civita connection is
chosen specifying GR as the theory under consideration.

###  IV.1 Gauss constraint and evolution equations in general relativity

GR is built on the affine Levi-Civita connection which can be defined through
the $4$-metric as carroll2004spacetime

$\displaystyle\accentset{\circ}{\Gamma}^{\sigma}_{\mu\nu}$
$\displaystyle:=\frac{1}{2}g^{\sigma\rho}\left(\partial_{\mu}g_{\rho\nu}+\partial_{\nu}g_{\nu\rho}-\partial_{\rho}g_{\mu\nu}\right)\,.$
(109)

From this definition, one can easily determine that this connection is
symmetric in the bottom two indices. It can also be shown that this connection
is not a tensor as indicated in the placement of the indices in Eq. (109). It
is this connection which characterizes curvature rather than the metric
itself.

It should be noted that, unless otherwise specified, throughout this
subsection all covariant derivatives, Lie derivatives and tensors are derived
through the Levi-Civita connection.

Choosing to work with the Levi-Civita connection, it is important to point out
some natural consequences of the symmetry of this connection both in general
and more specifically to our $3+1$ decomposition approach. The Riemann tensor,
as defined in Eq. (85), acquires two additional symmetries apart from the
antisymmetry in the last two indices. These are given below for ease of
reference carroll2004spacetime ; Boskoff

$\displaystyle\accentset{\circ}{R}_{\rho\sigma\mu\nu}=-\accentset{\circ}{R}_{\sigma\rho\mu\nu}\,,$
(110)
$\displaystyle\accentset{\circ}{R}_{\rho\sigma\mu\nu}=-\accentset{\circ}{R}_{\mu\nu\sigma\rho}\,.$
(111)

The Riemann tensor also acquires the following cyclic property

$\accentset{\circ}{R}_{\rho\sigma\mu\nu}+\accentset{\circ}{R}_{\rho\mu\nu\sigma}+\accentset{\circ}{R}_{\rho\nu\sigma\mu}=0\,.$
(112)

Other major consequences are that the torsion tensor, by definition, is now a
zero tensor and the theory assumes metricity so all nonmetricity and
disformation tensors are zero. This leads to another important result which is
that the definition of the Lie derivative can be equivalently written using
partial derivatives and covariant derivatives for purely global tensors. This
is the case as the difference between using a partial and a covariant
derivative consists only of torsion tensor terms. Thus, in GR, when taking the
Lie derivative of any global tensor along any vector, the following
definitions are equivalent

$\displaystyle\mathcal{L}_{\chi}V^{\nu}_{\;\;a\mu}$
$\displaystyle=\chi^{\sigma}\partial_{\sigma}V^{\nu}_{\;\;a\mu}-V^{\sigma}_{\;\;a\mu}\partial_{\sigma}\chi^{\nu}+V^{\nu}_{\;\;a\sigma}\partial_{\mu}\chi^{\sigma}$
(113)
$\displaystyle=\chi^{\sigma}\accentset{\circ}{\nabla}_{\sigma}V^{\nu}_{\;\;a\mu}-V^{\sigma}_{\;\;a\mu}\accentset{\circ}{\nabla}_{\sigma}\chi^{\nu}+V^{\nu}_{\;\;a\sigma}\accentset{\circ}{\nabla}_{\mu}\chi^{\sigma}\,.$

In the case of a tensor with local indices, this does not apply as spin
connection terms will still persist. A zero torsion tensor also results in a
completely symmetric extrinsic curvature. Finally, it can be shown that the
covariant derivative of the one-form $\Omega_{\nu}=\nabla_{\nu}t$ is zero as
it is also composed by just a torsion term baumgarte .

Having now defined the basic consequences of the symmetry of Levi-Civita
connection, we derive the Gauss-Codazzi constraint and the evolution equations
for a $3+1$ formalism in GR based on the metric baumgarte .

Given the generality of the $3$-vector $V^{\lambda}$, we retrieve the Riemann
Gauss equation from the generalized relation in Eq. (91) between the
antisymmetry of the purely spatial and space time covariant derivatives
baumgarte

$\displaystyle\accentset{\circ}{R}^{\beta(3)}_{\;\;\lambda\sigma\alpha}=\;\gamma^{\rho}_{\;\;\sigma}\gamma^{\pi}_{\;\;\alpha}\gamma^{\nu}_{\;\;\lambda}\gamma^{\beta}_{\;\;\chi}\accentset{\circ}{R}^{\chi}_{\;\;\nu\rho\pi}-\accentset{\circ}{k}_{[\alpha|\lambda}{\accentset{\circ}{k}_{|\sigma]}}{}^{\beta}\,.$
(114)

This equation relates the fully spatial Riemann tensor to its $4$-dimensional
counterpart. Contracting once and then twice, we obtain the equivalent
relation for the Ricci tensor and the Ricci scalar

$\accentset{\circ}{R}^{(3)}_{\lambda\alpha}=\;\gamma^{\pi}_{\;\;\alpha}\gamma^{\nu}_{\;\;\lambda}\gamma^{\rho}_{\;\;\chi}\accentset{\circ}{R}^{\chi}_{\;\;\nu\rho\pi}-\accentset{\circ}{k}_{\alpha\lambda}\accentset{\circ}{k}+\accentset{\circ}{k}_{\beta\lambda}{\accentset{\circ}{k}_{\alpha}}{}^{\beta}\,,$
(115)

$\accentset{\circ}{R}^{(3)}=\;\gamma^{\pi\nu}\gamma^{\rho}_{\;\;\chi}\accentset{\circ}{R}^{\chi}_{\;\;\nu\rho\pi}-\accentset{\circ}{k}^{2}+\accentset{\circ}{k}_{\beta\lambda}\accentset{\circ}{k}{}^{\lambda\beta}\,.$
(116)

Finally the Codazzi equation can also be shown to be given by

$\displaystyle\accentset{\circ}{D}_{\left[{\mu}\right.}\accentset{\circ}{k}_{\left.\nu\right]\lambda}$
$\displaystyle=\;\gamma^{\rho}_{\;\;\nu}\gamma^{\pi}_{\;\;\mu}\gamma^{\beta}_{\;\;\lambda}n^{\sigma}\accentset{\circ}{R}_{\rho\pi\beta\sigma}\,.$
(117)

We note that these equations fully agree with those obtained in literature
baumgarte ; gourgoulhon .

Another consequence of the Levi-Civita connection is that by putting the spin
connection as subject of the formula of the covariant derivative of the
tetrad, and applying the tetrad metricity property and substituting Eq. (109),
we can write the spin connection purely in terms of tetrads and the Minkowski
metric, that is

$\displaystyle\omega^{A}_{\;\;C\nu}=$
$\displaystyle\frac{1}{2}\left(\bar{e}^{A\alpha}e^{B}_{\;\;\nu}e_{C}^{\;\;\beta}\left(\partial_{\beta}\bar{e}_{B\alpha}-\partial_{\alpha}\bar{e}_{B\beta}\right)\right.$
$\displaystyle\left.+e_{C}^{\;\;\alpha}\left(\partial_{\alpha}e^{A}_{\;\;\nu}-\partial_{\nu}e^{A}_{\;\;\alpha}\right)+\bar{e}^{A\alpha}\left(\partial_{\nu}\bar{e}_{C\alpha}-\partial_{\alpha}\bar{e}_{C\nu}\right)\right)\,.$
(118)

It should be noted that, for the sake of simplicity, terms like
$\eta^{AB}e_{B}^{\;\;\alpha}$ are written as $\bar{e}^{A\alpha}$. This would
be useful if a tetrad $3+1$ is considered for GR. An interesting by-product of
this form of spin connection is that the antisymmetry in the first two indices
of the connection can be very clearly observed.

Finally, we consider the Ricci equation. This equation is understood to be the
second order Lie derivative of the fundamental variable. The $3+1$ formalism
is being based on. Otherwise one may look at it as an extra evolution equation
of some resulting variable from the first order Lie derivative of the
fundamental variable for which an expression consisting only of such a
fundamental variable is not know. This is effectively the concept of requiring
two equations for two unknowns.

Through simplifying the general form of this equation for the metric
formalism, Eqs. (100-102), using the Levi-Civita properties, assuming
metricity and substituting in the Einstein field equations, we find

$\displaystyle\mathcal{L}_{n}\gamma_{\mu\nu}$
$\displaystyle=-2\accentset{\circ}{k}_{\mu\nu}\,,$ (119)
$\displaystyle\mathcal{L}_{n}\accentset{\circ}{k}_{\mu\nu}$
$\displaystyle=-\frac{1}{2}\mathcal{L}_{n}\mathcal{L}_{n}\gamma_{\mu\nu}$
(120)
$\displaystyle=\;n^{\rho}\gamma^{\alpha}_{\;\;\nu}\gamma^{\beta}_{\;\;\mu}n_{\chi}\accentset{\circ}{R}^{\chi}_{\;\;\alpha\rho\beta}-\accentset{\circ}{D}_{\mu}\accentset{\circ}{a}_{\nu}-\accentset{\circ}{a}_{\mu}\accentset{\circ}{a}_{\nu}-{\accentset{\circ}{k}_{\mu}}{}^{\rho}\accentset{\circ}{k}_{\nu\rho}$
$\displaystyle=\;n^{\rho}\gamma^{\alpha}_{\;\;\nu}\gamma^{\beta}_{\;\;\mu}n_{\chi}\accentset{\circ}{R}^{\chi}_{\;\;\alpha\rho\beta}-\frac{1}{\alpha}\accentset{\circ}{D}_{\mu}\accentset{\circ}{D}_{\nu}\alpha-{\accentset{\circ}{k}_{\mu}}{}^{\rho}\accentset{\circ}{k}_{\nu\rho}\,,$

where
$\accentset{\circ}{D}_{\mu}\accentset{\circ}{a}_{\nu}+\accentset{\circ}{a}_{\mu}\accentset{\circ}{a}_{\nu}$
can be shown to be equivalent to
$\frac{1}{\alpha}\accentset{\circ}{D}_{\mu}\accentset{\circ}{D}_{\nu}\alpha$
given that the torsion tensor is zero.

Finally, the momentum and Hamiltonian constraints are derived to be

$\displaystyle\accentset{\circ}{D}_{\lambda}\accentset{\circ}{k}-\accentset{\circ}{D}_{\chi}{\accentset{\circ}{k}_{\lambda}}^{\chi}$
$\displaystyle=8\pi S_{\lambda}\,,$ (121)
$\displaystyle\accentset{\circ}{R}^{(3)}+\accentset{\circ}{k}^{2}-\accentset{\circ}{k}_{\beta\lambda}\accentset{\circ}{k}^{\lambda\beta}$
$\displaystyle=16\pi\rho\,.$ (122)

It should be noted that the metric evolution equations and the constraint
equation agree perfectly with those found in literature baumgarte ;
gourgoulhon .

###  IV.2 Gauss constraint and evolution equations in the teleparallel
equivalent of general relativity

Here we consider TG, specifically the TEGR and perform a $3+1$ decomposition
in the Weitzenböck gauge. In this case, all curvature terms are identically
zero and metricity is assumed. Similarly to what was done in the case of GR, a
tetrad formulation of $3+1$ decomposition will be derived through an
investigation starting with the connection in this approach to gravity. In TG,
the teleparallel connection defined in Eq. (3) is used in terms of the
$4$-tetrad and the spin connection

$\accentset{\land}{\Gamma}^{\lambda}_{\mu\nu}:={e_{A}}^{\lambda}\partial_{\nu}{e^{A}}_{\mu}+{e_{A}}^{\lambda}{e^{B}}_{\mu}\omega^{a}_{B\nu}\,.$
(123)

Throughout this section, all covariant derivatives, Lie derivatives and
tensors are derived through the Weitzenböck connection (which is the
teleparallel connection in the Weitzenböck gauge) and as such they are denoted
by a hat accent, such as in $\accentset{\land}{A}$. It is important to point
out that even spatial components are denoted in this way.

Unlike in the GR case, the TG spin connection has not an explicit expression
in terms of fundamental variables. For this reason, we only consider tetrads
in their Weitzenböck gauge, simplifying the resulting expressions. In the
Weitzenböck gauge, the $4$-covariant derivative of the normal vector is zero
resulting in a zero covariant derivative of the $3$-metric and the $3$-tetrad,
in a zero extrinsic curvature tensor and in a zero acceleration vector.

After applying all of these results to Eq. (91), we obtain the following
equation

$\displaystyle\accentset{\land}{D}_{\lambda}V^{\beta}_{(3)}\left(\accentset{\land}{T}^{\lambda(3)}_{\;\;\alpha\sigma}-\gamma^{\rho}_{\;\;\sigma}\gamma^{\pi}_{\;\;\alpha}\gamma^{\lambda}_{\;\;\nu}\accentset{\land}{T}^{\nu(4)}_{\;\;\pi\rho}\right)=0\,.$
(124)

There are three cases which would result in this dot product being zero. The
first is that $\accentset{\land}{D}_{\lambda}V^{\beta}_{(3)}=0$, which is not
possible since $V^{\beta}_{(3)}$ is a general $3$-vector. The second is that
$\left(\accentset{\land}{T}^{\lambda(3)}_{\;\;\alpha\sigma}-\gamma^{\rho}_{\;\;\sigma}\gamma^{\pi}_{\;\;\alpha}\gamma^{\lambda}_{\;\;\nu}\accentset{\land}{T}^{\nu(4)}_{\;\;\pi\rho}\right)$
is perpendicular to $\accentset{\land}{D}_{\lambda}V^{\beta}_{(3)}$ for all
$3$-vectors $V^{\beta}_{(3)}$. However, the bracket term is in no-way
dependent on the general $3$-vector $V^{\beta}_{(3)}$, then the bracket term
would have to be purely temporal. This is a contradiction since applying a dot
product on any of the free indices of the bracket term with a normal vector
would result in zero

$\displaystyle
n_{\lambda}\left(\accentset{\land}{T}^{\lambda(3)}_{\;\;\alpha\sigma}-\gamma^{\rho}_{\;\;\sigma}\gamma^{\pi}_{\;\;\alpha}\gamma^{\lambda}_{\;\;\nu}\accentset{\land}{T}^{\nu(4)}_{\;\;\pi\rho}\right)$
$\displaystyle=0\,,$ (125) $\displaystyle
n^{\alpha}\left(\accentset{\land}{T}^{\lambda(3)}_{\;\;\alpha\sigma}-\gamma^{\rho}_{\;\;\sigma}\gamma^{\pi}_{\;\;\alpha}\gamma^{\lambda}_{\;\;\nu}\accentset{\land}{T}^{\nu(4)}_{\;\;\pi\rho}\right)$
$\displaystyle=0\,,$ $\displaystyle
n^{\sigma}\left(\accentset{\land}{T}^{\lambda(3)}_{\;\;\alpha\sigma}-\gamma^{\rho}_{\;\;\sigma}\gamma^{\pi}_{\;\;\alpha}\gamma^{\lambda}_{\;\;\nu}\accentset{\land}{T}^{\nu(4)}_{\;\;\pi\rho}\right)$
$\displaystyle=0\,,$

implying that the bracket term is purely spatial. The only option left is that
$\left(\accentset{\land}{T}^{\lambda(3)}_{\;\;\alpha\sigma}-\gamma^{\rho}_{\;\;\sigma}\gamma^{\pi}_{\;\;\alpha}\gamma^{\lambda}_{\;\;\nu}\accentset{\land}{T}^{\nu(4)}_{\;\;\pi\rho}\right)$
is in fact zero itself producing the Gauss equation for torsion

$\accentset{\land}{T}^{\lambda(3)}_{\;\;\alpha\sigma}=\gamma^{\rho}_{\;\;\sigma}\gamma^{\pi}_{\;\;\alpha}\gamma^{\lambda}_{\;\;\nu}\accentset{\land}{T}^{\nu(4)}_{\;\;\pi\rho}\,.$
(126)

Through this relation, similar relationships can be obtained for the torsion
vector and scalar

$\displaystyle\accentset{\land}{T}^{(3)}_{\alpha}$
$\displaystyle=\gamma^{\rho}_{\;\;\lambda}\gamma^{\pi}_{\;\;\alpha}\gamma^{\lambda}_{\;\;\nu}\accentset{\land}{T}^{\nu(4)}_{\;\;\pi\rho}$
$\displaystyle=\accentset{\land}{T}^{(4)}_{\alpha}+n_{\alpha}n^{\pi}\accentset{\land}{T}^{(4)}_{\pi}+n_{\nu}n^{\rho}\accentset{\land}{T}^{\nu(4)}_{\;\;\alpha\rho}$
$\displaystyle=\gamma^{\lambda}_{\;\;\alpha}\accentset{\land}{T}^{(4)}_{\lambda}+n_{\nu}n^{\rho}\accentset{\land}{T}^{\nu(4)}_{\;\;\alpha\rho}\,,$
(127) $\displaystyle\accentset{\land}{T}^{(3)}=$
$\displaystyle\tfrac{1}{4}\accentset{\land}{T}_{\lambda}^{(3)\mu\nu}\accentset{\land}{T}^{\lambda(3)}_{\;\;\mu\nu}+\tfrac{1}{2}\accentset{\land}{T}^{\lambda(3)}_{\;\;\mu\nu}\accentset{\land}{T}^{\nu\mu(3)}_{\;\;\;\;\lambda}-\accentset{\land}{T}^{\lambda(3)}_{\;\;\mu\lambda}\accentset{\land}{T}^{\nu\mu(3)}_{\;\;\;\;\nu}$
$\displaystyle=$
$\displaystyle\accentset{\land}{T}^{(4)}+2n^{\lambda}n^{\mu}\accentset{\land}{T}^{\alpha}{}_{\nu\alpha}\accentset{\land}{T}_{\lambda\mu}{}^{\nu}+\tfrac{1}{4}n^{\lambda}n^{\mu}\accentset{\land}{T}_{\lambda}{}^{\nu\alpha}\accentset{\land}{T}_{\mu\nu\alpha}$
$\displaystyle+n^{\lambda}n^{\mu}\accentset{\land}{T}_{\lambda}{}^{\nu\alpha}\accentset{\land}{T}_{\nu\mu\alpha}+\tfrac{1}{2}n^{\lambda}n^{\mu}\accentset{\land}{T}_{\alpha\mu\nu}\accentset{\land}{T}^{\nu}{}_{\lambda}{}^{\alpha}$
$\displaystyle+\tfrac{1}{2}n^{\lambda}n^{\mu}\accentset{\land}{T}_{\nu\mu\alpha}\accentset{\land}{T}^{\nu}{}_{\lambda}{}^{\alpha}-n^{\lambda}n^{\mu}\accentset{\land}{T}^{\alpha}{}_{\mu\alpha}\accentset{\land}{T}^{\nu}{}_{\lambda\nu}$
$\displaystyle=$
$\displaystyle\accentset{\land}{T}^{(4)}+\frac{2}{\alpha}\accentset{\land}{T}^{\nu(3)}\partial^{(3)}_{\nu}\left(\alpha\right)+\frac{1}{2}\accentset{\land}{A}^{\nu}_{\lambda}\left(\accentset{\land}{A}_{\nu}^{\lambda}+\accentset{\land}{A}^{\lambda}_{\nu}\right)-\accentset{\land}{A}^{2}\,,$
(128)

where $A$ is the first order Lie derivative of the spatial tetrad along the
normal vector to the foliations $n$.

We now move on to the evolution equations by considering the TEGR field
equations fieldequasions ; Krssak:2015oua  as given in Eq. (11). After setting
the Weitzenböck gauge and some minor restructuring, these field equations can
be written as

$\displaystyle\accentset{\land}{S}_{\alpha\sigma}{}^{\rho}\accentset{\land}{T}^{\lambda}{}_{\rho\lambda}-\accentset{\land}{S}^{\rho\lambda}{}_{\sigma}$
$\displaystyle\accentset{\land}{T}_{\rho\lambda\alpha}-\tfrac{1}{2}\accentset{\land}{S}_{\alpha}{}^{\rho\lambda}\accentset{\land}{T}_{\sigma\lambda\rho}$
$\displaystyle+\tfrac{1}{2}g_{\alpha\sigma}\accentset{\land}{T}-\accentset{\land}{\nabla}^{\lambda}\accentset{\land}{S}_{\alpha\lambda\sigma}=\Theta_{\alpha\sigma}\,.$
(129)

Contracting these equations, we obtain an alternative expression for the
torsion scalar in terms of the energy-momentum scalar

$\displaystyle\accentset{\land}{T}$
$\displaystyle=\Theta+2\accentset{\land}{T}^{\alpha}{}_{\rho\alpha}\accentset{\land}{T}^{\lambda}{}_{\lambda}{}^{\rho}+2\accentset{\land}{\nabla}_{\rho}\accentset{\land}{T}^{\lambda}{}_{\lambda}{}^{\rho}\,.$
(130)

The generalized tetrad evolution equations in Eq.(103) and Eq.(104) then
reduce to

$\displaystyle\mathcal{L}_{n}e^{A(3)}_{\;\;\nu}$
$\displaystyle=n^{\lambda}e^{A(3)}_{\;\;\rho}\gamma^{\sigma}_{\;\;\nu}\accentset{\land}{T}^{\rho}_{\;\;\lambda\sigma}$
$\displaystyle=\accentset{\land}{A}^{A}_{\;\;\nu}\,,$ (131)
$\displaystyle\mathcal{L}_{n}\accentset{\land}{A}^{A}_{\;\;\nu}$
$\displaystyle=n^{\rho}n^{\mu}e^{A(3)}_{\;\;\lambda}\gamma^{\alpha}_{\;\;\nu}\left(\accentset{\land}{\nabla}_{\rho}\accentset{\land}{T}^{\lambda}_{\;\;\mu\alpha}+\accentset{\land}{T}^{\lambda}_{\;\;\mu\pi}\accentset{\land}{T}^{\pi}_{\;\;\rho\alpha}\right)$
$\displaystyle=\accentset{\land}{D}_{\rho}\accentset{\land}{T}^{A\rho}_{(3)\nu}+\accentset{\land}{A}^{A}_{\;\;\rho}\accentset{\land}{A}^{\rho}_{\;\;\nu}+e^{A(3)}_{\;\;\sigma}\gamma^{\sigma}_{\;\;\lambda}\gamma^{\alpha}_{\;\;\nu}\accentset{\land}{\nabla}_{\rho}\accentset{\land}{T}^{\lambda\;\;\rho}_{\;\;\alpha}\,.$
(132)

In order to find the final from of the TEGR evolution equations, we now
substitute the field equations into the evolution equations

$\displaystyle\mathcal{L}_{n}e^{A(3)}_{\;\;\nu}$
$\displaystyle=n^{\lambda}e^{A(3)}_{\;\;\rho}\gamma^{\sigma}_{\;\;\nu}\accentset{\land}{T}^{\rho}_{\;\;\lambda\sigma}$
$\displaystyle=\accentset{\land}{A}^{A}_{\;\;\nu}\,,$ (133)
$\displaystyle\mathcal{L}_{n}\accentset{\land}{A}^{a}_{\;\;\nu}$
$\displaystyle=\accentset{\land}{D}_{\rho}\accentset{\land}{T}^{A\rho}_{(3)\nu}+\accentset{\land}{A}^{A}_{\;\;\rho}\accentset{\land}{A}^{\rho}_{\;\;\nu}+e^{A(3)}_{\;\;\sigma}\left[{\accentset{\land}{A}_{\nu}}^{\rho}{\accentset{\land}{A}^{\sigma}}_{\rho}\right.$
$\displaystyle-{\accentset{\land}{A}^{\rho}}_{\nu}{\accentset{\land}{A}_{\rho}}^{\sigma}-\frac{1}{2}\accentset{\land}{T}_{\;\;\nu}^{(3)\rho\chi}\accentset{\land}{T}^{\sigma(3)}_{\;\;\rho\chi}+\accentset{\land}{T}^{\rho(3)\chi}_{\;\;\;\;\nu}\left(\accentset{\land}{T}_{\rho\;\;\chi}^{\;\;\sigma(3)}+\accentset{\land}{T}_{\chi\;\;\rho}^{\;\;\sigma(3)}\right)$
$\displaystyle+\frac{2}{\alpha^{2}}\partial_{\nu}^{(3)}(\alpha)\partial^{\sigma}_{(3)}(\alpha)-\accentset{\land}{T}^{\chi(3)}_{\;\;\rho\chi}\left(\accentset{\land}{T}_{\;\;\nu}^{(3)\sigma\rho}+\accentset{\land}{T}^{\sigma(3)\rho}_{\;\;\;\;\nu}\right)$
$\displaystyle-\frac{1}{\alpha}\partial_{\rho}^{(3)}(\alpha)\left(\accentset{\land}{T}_{\;\;\nu}^{(3)\sigma\rho}+\accentset{\land}{T}^{\sigma(3)\rho}_{\;\;\;\;\nu}\right)-\accentset{\land}{A}\left({\accentset{\land}{A}_{\nu}}^{\sigma}+{\accentset{\land}{A}^{\sigma}}_{\nu}\right)$
$\displaystyle+\accentset{\land}{D}_{\nu}\accentset{\land}{T}^{\rho\sigma}_{(3)\rho}+\accentset{\land}{D}^{\sigma}\accentset{\land}{T}^{\rho(3)}_{\;\;\nu\rho}+\accentset{\land}{D}_{\nu}\left(\frac{1}{\alpha}\partial^{\sigma}_{(3)}\\{\alpha\\}\right)$
$\displaystyle+{\accentset{\land}{D}^{\sigma}}\left(\frac{1}{\alpha}\partial^{(3)}_{\nu}\\{\alpha\\}\right)+8\pi
G\left(2{S_{\nu}}^{\sigma}-{\gamma^{\sigma}}_{\nu}\left\\{S-\rho\right\\}\right)\left.\right]$
$\displaystyle-e^{A(3)}_{\;\;\sigma}\gamma^{\sigma}_{\;\;\lambda}\gamma^{\alpha}_{\;\;\nu}\accentset{\land}{\nabla}_{\rho}\accentset{\land}{T}_{\alpha}^{\;\;\lambda\rho}\,,$
(134)

where ${S_{\nu}}^{\sigma}$ and $S$ are the spatial stress and scalar stress
while $\rho$ is the density. At this point, we note an important issue with
this $3+1$ decomposition of TEGR. Specifically we note that the final term
$e^{A(3)}_{\;\;\sigma}\gamma^{\sigma}_{\;\;\lambda}\gamma^{\alpha}_{\;\;\nu}\accentset{\land}{\nabla}_{\rho}\accentset{\land}{T}_{\alpha}^{\;\;\lambda\rho}$
cannot be converted into a purely spatial form which is necessary to obtain a
valid form for the evolution equations. Such terms are normally substituted
out through the field equations, however, due to the symmetry of the field
equations and asymmetry of the second evolution equation, this term is not
eliminated. If instead of considering the tetrad evolution equations, one
considers the metric ones, the problem does not feature due to the symmetry of
the metric. However, this creates a new issue since it is not possible to
write torsion tensors purely in terms of metrics making the equations
insoluble. For completeness, we finalize the remainder of the $3+1$
decomposition.

An interesting relationship between GR and TEGR comes out from manipulating
the definition of the first Lie derivative of the spatial tetrad or spatial
metric. Taking the symmetry of $\accentset{\land}{A}^{A}_{\;\;\nu}$, one gets
the following

$\displaystyle\accentset{\land}{A}^{A}_{\;\;\nu}+\accentset{\land}{A}_{\nu}^{\;\;A}=$
$\displaystyle
n^{\lambda}e^{A(3)}_{\;\;\rho}\gamma^{\sigma}_{\;\;\nu}\left(\accentset{\land}{T}^{\rho}_{\;\;\lambda\sigma}+\accentset{\land}{T}^{\;\;\;\;\rho}_{\sigma\lambda}\right)$
$\displaystyle=$ $\displaystyle
n_{\lambda}e^{A(3)}_{\;\;\rho}\gamma^{\sigma}_{\;\;\nu}\gamma^{\rho\epsilon}\left(2\accentset{\land}{K}^{\lambda}_{\;\;\epsilon\sigma}+\accentset{\land}{T}^{\lambda}_{\;\;\epsilon\sigma}\right)$
$\displaystyle=$ $\displaystyle
n_{\lambda}e^{A(3)}_{\;\;\rho}\gamma^{\sigma}_{\;\;\nu}\gamma^{\rho\epsilon}2\left(\accentset{\land}{\Gamma}^{\lambda}_{\;\;\epsilon\sigma}-\accentset{\circ}{\Gamma}^{\lambda}_{\;\;\epsilon\sigma}\right)$
$\displaystyle=$ $\displaystyle
e^{A(3)}_{\;\;\rho}\gamma^{\sigma}_{\;\;\nu}\gamma^{\rho\epsilon}2\left(\partial_{\sigma}n_{\epsilon}-\partial_{\sigma}n_{\epsilon}+\accentset{\circ}{\nabla}_{\sigma}n_{\epsilon}\right)$
$\displaystyle=$ $\displaystyle-2\accentset{\circ}{k}_{\nu}^{\;\;a}\,.$ (135)

This confirms that the Lie derivative of the spatial metric and spatial tetrad
for the Levi-Civita in GR and for the Weitzenböck gauge in TEGR are in fact
directly related.

Let us now derive the momentum and Hamiltonian constraints for TEGR. These are
given here in their final form

$\displaystyle\accentset{\land}{T}^{(3)}_{\rho}\left(\accentset{\land}{A}^{\rho}_{\;\;\chi}+\accentset{\land}{A}_{\chi}^{\;\;\rho}\right)$
$\displaystyle-\accentset{\land}{A}^{\lambda\rho}\left(\accentset{\land}{T}^{(3)}_{\lambda\chi\rho}+\accentset{\land}{T}^{(3)}_{\rho\chi\lambda}\right)$
$\displaystyle-2\accentset{\land}{D}_{\chi}\accentset{\land}{A}-\accentset{\land}{D}_{\lambda}\accentset{\land}{A}^{\lambda}_{\;\;\chi}=16\pi
GS_{\lambda}\,,$ (136)

$\displaystyle
2\accentset{\land}{T}^{\lambda}_{(3)}\accentset{\land}{T}_{\lambda}^{(3)}+$
$\displaystyle\frac{1}{2}\accentset{\land}{A}^{\rho\sigma}\left(\accentset{\land}{A}_{\rho\sigma}+\accentset{\land}{A}_{\sigma\rho}\right)-\accentset{\land}{A}^{2}+2\accentset{\land}{D}_{\lambda}\accentset{\land}{T}^{\lambda}_{(3)}$
$\displaystyle+4\accentset{\land}{D}_{\rho}\left(\frac{1}{\alpha}\partial^{\rho}_{(3)}(\alpha)\right)+\accentset{\land}{T}^{(3)}=-16\pi
G\rho\,.$ (137)

We finally consider the time vector $t^{\lambda}$ which can be written as
baumgarte

$t^{\lambda}=\alpha n^{\lambda}+\beta^{\lambda}\,,$ (138)

where $\beta^{\lambda}$ is the shift vector. Taking the Lie derivative of
$\accentset{\land}{A}^{A}_{\;\;\nu}$ along the time vector gives

$\mathcal{L}_{t}\accentset{\land}{A}^{A}_{\;\;\nu}=\alpha\mathcal{L}_{n}\accentset{\land}{A}^{A}_{\;\;\nu}+\mathcal{L}_{\beta}\accentset{\land}{A}^{A}_{\;\;\nu}\,,$
(139)

where general Lie derivative properties have been used yano1957theory ;
baumgarte . What is left is to consider the second term of Eq. (139).
Expanding this term, one obtains the following

$\displaystyle\mathcal{L}_{\beta}\accentset{\land}{A}^{A}_{\;\;\nu}=\beta^{\lambda}\partial_{\lambda}\left(\accentset{\land}{A}^{A}_{\;\;\nu}\right)+\accentset{\land}{A}^{A}_{\;\;\lambda}\partial_{\nu}\left(\beta^{\lambda}\right)\,.$
(140)

Similarly the spatial tetrad evolution equation becomes

$\displaystyle\mathcal{L}_{t}e^{A(3)}_{\;\;\nu}$
$\displaystyle=\alpha\accentset{\land}{A}^{A}_{\;\;\nu}+\beta^{\lambda}\left(\partial_{\lambda}\left\\{e^{A(3)}_{\;\;\nu}\right\\}-\partial_{\nu}\left\\{e^{A(3)}_{\;\;\lambda}\right\\}\right)\,.$
(141)

###  IV.3 Gauss constraint and evolution equations in symmetric teleparallel
gravity with the coincident gauge

We now consider STG within the coincident gauge which was described in Sec.
II.2. This theory is based on two field equations, one obtained by varying the
Lagrangian with respect to the metric in Eq. (28), and another by varying it
with respect to the connection in Eq. (30). The coincident gauge for STEGR is
important since it identically satisfies the connection field equations by
setting the connection to be zero. This fact has a number of important
implications. The most important of these is that the covariant derivative and
the partial derivative become the same for this formalism.

The main Gauss equation for this construction is that for the metricity
tensor. This was derived in Sec. III and it is given again here for ease of
reference along with the Gauss constraint of disformation tensor that is its
direct consequence,

$\displaystyle\gamma^{\lambda}_{\;\;\rho}\gamma^{\beta}_{\;\;\mu}\gamma^{\alpha}_{\;\;\nu}\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\lambda\beta\alpha}=\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\rho\mu\nu}^{\;(3)}\,,$
(142)
$\displaystyle\gamma^{\lambda}_{\;\;\rho}\gamma^{\beta}_{\;\;\mu}\gamma^{\alpha}_{\;\;\nu}\accentset{\scaleto{\diamond}{3.5pt}}{L}_{\lambda\beta\alpha}=\accentset{\scaleto{\diamond}{3.5pt}}{L}_{\rho\mu\nu}^{(3)}\,.$
(143)

Let us now move on to the evolution equations for STEGR. We start by
introducing a modified version of the field equations in STEGR based on the
fact that the evolution equations, in this case, are purely based on the
disformation tensor, that is

$\displaystyle\partial_{\alpha}{\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\alpha}}_{\mu\nu}=$
$\displaystyle\;\Theta_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\Theta+\accentset{\scaleto{\diamond}{3.5pt}}{L}_{\sigma\;\;\alpha}^{\;\;\sigma}{\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\alpha}}_{\mu\nu}$
$\displaystyle+\frac{1}{2}\partial_{(\mu|}{\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\epsilon}}_{\epsilon|\nu)}+{\accentset{\scaleto{\diamond}{3.5pt}}{Q}^{\sigma\epsilon}}_{\nu}\accentset{\scaleto{\diamond}{3.5pt}}{L}_{\sigma\epsilon\mu}-\frac{1}{2}\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\mu\nu\epsilon}{\accentset{\scaleto{\diamond}{3.5pt}}{L}_{\nu}}^{\sigma\epsilon}\,.$
(144)

Here, the scalar metricity is replaced by a combination of its definition and
an expression obtained through the contraction of the metric filed equations

$\displaystyle\accentset{\scaleto{\diamond}{3.5pt}}{Q}=$
$\displaystyle\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\;\;\lambda}_{\lambda\;\;\alpha}\left({\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\alpha\epsilon}}_{\epsilon}-{\accentset{\scaleto{\diamond}{3.5pt}}{L}_{\epsilon}}^{\epsilon\alpha}\right)+\partial_{\alpha}\left({\accentset{\scaleto{\diamond}{3.5pt}}{L}_{\epsilon}}^{\epsilon\alpha}-{\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\alpha\epsilon}}_{\epsilon}\right)-\Theta\,.$
(145)

The next step is to consider the evolution equations for STEGR within the
coincident gauge. The generalized evolution equations Eqs. (99,101) thus
become

$\displaystyle\mathcal{L}_{n}\gamma_{\mu\nu}=$
$\displaystyle\accentset{\scaleto{\diamond}{3.5pt}}{B}_{(\mu\nu)}$
$\displaystyle=$ $\displaystyle
2\accentset{\scaleto{\diamond}{3.5pt}}{B}_{\mu\nu}\,,$ (146)
$\displaystyle\mathcal{L}_{n}\accentset{\scaleto{\diamond}{3.5pt}}{B}_{\mu\nu}=$
$\displaystyle-\gamma^{\alpha}_{\;\;\nu}\gamma^{\beta}_{\;\;\mu}\partial_{\lambda}{\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\lambda}}_{\alpha\beta}+\partial^{(3)}_{\lambda}\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\lambda}_{\;\;\mu\nu}$
$\displaystyle+\gamma^{\alpha}_{\;\;\nu}\gamma^{\beta}_{\;\;\mu}\accentset{\scaleto{\diamond}{3.5pt}}{a}_{\sigma}{\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\sigma}}_{\alpha\beta}-\gamma^{\epsilon}_{\;\;\left(\nu\right|}\gamma^{\chi}_{\;\;\left|\mu\right)}\accentset{\scaleto{\diamond}{3.5pt}}{B}_{\alpha\chi}n_{\sigma}{\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\epsilon}}^{\sigma\alpha}\,.$
(147)

After substituting the metric field equations into the second evolution
equation, expressing all of the right hand side in spatial tensors, and
writing the energy-momentum tensor and scalar in terms of the spatial stress
$S_{\mu\nu}$, its trace $S$ and the density $\rho$, we obtain the final form
of the second evolution equation in STEGR, that is

$\displaystyle\mathcal{L}_{n}\accentset{\scaleto{\diamond}{3.5pt}}{B}_{\mu\nu}=$
$\displaystyle-8\pi
G\left[S_{\mu\nu}-\frac{1}{2}\gamma_{\mu\nu}\left(S-\rho\right)\right]$
$\displaystyle+\frac{1}{2}\accentset{\scaleto{\diamond}{3.5pt}}{Q}_{\epsilon\;\;\sigma}^{\;\;\sigma(3)}\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\epsilon(3)}_{\;\mu\nu}+\frac{2}{\alpha}\partial^{(3)}_{\epsilon}\left(\alpha\right)\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\epsilon(3)}_{\;\mu\nu}+\partial^{(3)}_{\sigma}\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\sigma(3)}_{\;\mu\nu}$
$\displaystyle-\accentset{\scaleto{\diamond}{3.5pt}}{B}_{\mu\nu}\accentset{\scaleto{\diamond}{3.5pt}}{B}-\frac{1}{2}\partial_{(\nu|}^{(3)}\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\sigma(3)}_{\;\sigma|\mu)}+\frac{1}{2}\partial_{(\nu|}^{(3)}\left(\frac{1}{\alpha}\partial_{|\mu)}^{(3)}\alpha\right)$
$\displaystyle-\frac{1}{2}\accentset{\scaleto{\diamond}{3.5pt}}{Q}^{\lambda\sigma(3)}_{\;\;\;\;\;\nu}\accentset{\scaleto{\diamond}{3.5pt}}{Q}^{(3)}_{[\lambda\sigma]\mu}-\frac{1}{2}Q^{(3)}_{\mu\lambda\sigma}\accentset{\scaleto{\diamond}{3.5pt}}{Q}^{(3)\lambda\sigma}_{\;\nu}+{\accentset{\scaleto{\diamond}{3.5pt}}{B}^{\sigma}}_{(\nu|}\accentset{\scaleto{\diamond}{3.5pt}}{B}_{\sigma|\mu)}$
$\displaystyle-\frac{1}{\alpha^{2}}\partial^{(3)}_{\nu}\alpha\;\partial^{(3)}_{\mu}\alpha\,.$
(148)

The Hamiltonian and momentum constraints for STEGR can then be derived through
contracting Eq. (IV.3) in the standard way, and simplifying the results to
obtain the respective final forms of the constraints

$\displaystyle 16\pi G\rho=$
$\displaystyle-\frac{2}{\alpha^{2}}\partial^{(3)}_{\nu}(\alpha)\partial^{\nu}_{(3)}(\alpha)+\frac{1}{2}\accentset{\scaleto{\diamond}{3.5pt}}{Q}^{\;\;\sigma(3)}_{\epsilon\;\;\sigma}\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\epsilon\;\;\nu}_{\;\nu(3)}+\gamma^{\mu\nu}\partial^{(3)}_{\epsilon}\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\epsilon(3)}_{\;\;\mu\nu}-\accentset{\scaleto{\diamond}{3.5pt}}{B}^{2}-\partial^{\nu}_{(3)}\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\epsilon(3)}_{\;\;\epsilon\nu}$
$\displaystyle\;-\frac{1}{2}\accentset{\scaleto{\diamond}{3.5pt}}{Q}^{\lambda\sigma\nu}_{(3)}\accentset{\scaleto{\diamond}{3.5pt}}{Q}^{(3)}_{\left[\lambda\sigma\right]\nu}-\frac{1}{2}\accentset{\scaleto{\diamond}{3.5pt}}{Q}^{\lambda\sigma\nu}_{(3)}\accentset{\scaleto{\diamond}{3.5pt}}{Q}^{(3)}_{\lambda\sigma\nu}+\accentset{\scaleto{\diamond}{3.5pt}}{B}^{\sigma\nu}\accentset{\scaleto{\diamond}{3.5pt}}{B}_{\sigma\nu}\,,$
(149) $\displaystyle 8\pi GS_{\beta}=$
$\displaystyle\;\frac{1}{2}\accentset{\scaleto{\diamond}{3.5pt}}{B}^{\epsilon\lambda}\accentset{\scaleto{\diamond}{3.5pt}}{Q}^{(3)}_{\beta\epsilon\lambda}+\accentset{\scaleto{\diamond}{3.5pt}}{L}^{\theta\rho(3)}_{\;\;\rho}\accentset{\scaleto{\diamond}{3.5pt}}{B}_{\beta\theta}-\gamma^{\sigma\rho}\partial_{\beta}^{(3)}\left(\accentset{\scaleto{\diamond}{3.5pt}}{B}_{\rho\sigma}\right)+\gamma^{\sigma\rho}\partial_{\rho}^{(3)}\left(\accentset{\scaleto{\diamond}{3.5pt}}{B}_{\beta\sigma}\right)\,.$
(150)

Similar to the torsional case, we now consider the time vector and take the
Lie derivative of $B_{\mu\nu}$ along the time vector, so that

$\mathcal{L}_{t}\accentset{\scaleto{\diamond}{3.5pt}}{B}_{\mu\nu}=\alpha\mathcal{L}_{n}\accentset{\scaleto{\diamond}{3.5pt}}{B}_{\mu\nu}+\mathcal{L}_{\beta}\accentset{\scaleto{\diamond}{3.5pt}}{B}_{\mu\nu}\,,$
(151)

where general Lie derivative properties have been used yano1957theory ;
baumgarte . Expanding the second term of Eq. (151), one obtains the following

$\displaystyle\mathcal{L}_{\beta}\accentset{\scaleto{\diamond}{3.5pt}}{B}_{\mu\nu}=\beta^{\lambda}\partial^{(3)}_{\lambda}\left(\accentset{\scaleto{\diamond}{3.5pt}}{B}_{\mu\nu}\right)+\accentset{\scaleto{\diamond}{3.5pt}}{B}_{\mu\lambda}\partial^{(3)}_{\nu}\left(\beta^{\lambda}\right)+\accentset{\scaleto{\diamond}{3.5pt}}{B}_{\lambda\nu}\partial^{(3)}_{\mu}\left(\beta^{\lambda}\right)\,.$
(152)

Similarly the spatial metric evolution equation becomes

$\displaystyle\mathcal{L}_{t}\gamma_{\mu\nu}$ $\displaystyle=\alpha
2\accentset{\scaleto{\diamond}{3.5pt}}{B}_{\mu\nu}+\beta^{\lambda}\left(\partial^{(3)}_{\lambda}\left\\{\gamma_{\mu\nu}\right\\}-\partial^{(3)}_{\nu}\left\\{\gamma_{\mu\lambda}\right\\}-\partial^{(3)}_{\mu}\left\\{\gamma_{\lambda\nu}\right\\}\right)\,.$
(153)

Thus, this completes the transformation of the field equations into a set of
four equations, two of which are evolution equations while the other two are
constraint equations.

##  V Discussion and Conclusions

In this paper, we have further probed the torsional and nonmetrical sides of
GR through TEGR and STEGR, respectively. While TG and STG rely on different
constructions of gravity, as described in Sec. II, they both produce field
equations that are dynamically equivalent to standard GR. For this reason it
is crucial to study the entire trinity of ways in which GR can be written
since different formalisms may offer better solutions for different issues of
the theory.

Specifically, we presented a $3+1$ decomposition of both TEGR and STEGR in
addition to the well-known GR decomposition. We first reconsider the $3+1$
decomposition strategy without assuming a connection by which a particular
geometry can centre into the construction. We do this in Sec. III where we
take a general decomposition of the metric and the tetrad and lay out the
general strategy of this approach by describing the Lie derivatives and how
they will lead to the evolution and constraints equations.

Then, in Sec. IV, we describe our main results where we first present the well
known $3+1$ decomposition within GR. See Sec. IV.1. This analysis leads to two
evolution equations and two constraint equations which are a set of equations
that can fully determine any ensuing system analogous to the actual field
equations. In Sec. IV.2, we extend this strategy to the TEGR version of GR in
which the fundamental dynamical variable becomes the tetrad (in the
Weitzenböck gauge) which adds an additional layer of difficulty to the issue
since this acts both on the general manifold and on the associated Minkowski
space. In particular, we have considered both holonomic and nonholonomic
coordinates and transformations. In this part of the work, we showed that this
decomposition can take place and lay out the key elements of how this can be
done for the evolution equations and highlighted the symmetry-asymmetry issue
within them. We also present momentum and Hamiltonian constraint equations
which complete the system. Altogether this gives a construction of TEGR in
which numerical analyses, using this $3+1$ decomposition, can be carried out.

The STG version of GR is then probed in Sec. IV.3 where STEGR is fully
decomposed using a $3+1$ approach. We do this in the coincident gauge so that
the only dynamical variable is the metric tensor. Since we are again reliant
on the metric tensor, the approach has some resemblance to the approach in
standard GR, meaning that we can use some analogous relations in this setting.
Indeed, we derive the evolution equations for STEGR in Eqs. (152,153), while
the evolution equations are written explicitly in Eqs. (149,150). Thus, as in
GR, we can rewrite the field equations in a genuine $3+1$ decomposed structure
which will be more amenable to numerical analyses.

The $3+1$ decompositions presented here can be further developed for
implementation in numerical settings for specific scenarios. It may turn out
that the novel approaches of TEGR and STEGR may provide some advantage over
the traditional method by which this formalism is implemented. One would
expect this since both TEGR and STEGR remove boundary elements from their
action whereas GR retains these elements. Since these three constructions of
gravity form a trinity of the same underlying dynamical equations, they are
guaranteed to be equivalent. These decomposition approaches are also
interesting because they may provide a way in which modified TG or STG gravity
theories may be decomposed in a similar fashion in future works. Furthermore,
the ADM couterparts for TG and STG can be extremely useful for achieving the
canonical quantization of these theories also in view of Quantum Cosmology
applications Book .

###### Acknowledgements.

JLS would like to acknowledge support from Cosmology@MALTA which is supported
by the University of Malta. SC acknowledges the support of Istituto Nazionale
di Fisica Nucleare (INFN), iniziative specifiche QGSKY and MOONLIGHT2. The
authors also acknowledge also funding from “The Malta Council for Science and
Technology” in project IPAS-2020-007.

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