monthyeardate[]

# Gibbons-Hawking-York boundary terms and
the generalized geometrical trinity of gravity

Johanna Erdmenger  Institute for Theoretical Physics and Astrophysics, Julius-
Maximilians-Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany
Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum
Matter ct.qmat  Bastian Heß<EMAIL_ADDRESS>Institute for
Theoretical Physics and Astrophysics, Julius-Maximilians-Universität Würzburg,
Am Hubland, D-97074 Würzburg, Germany  Würzburg-Dresden Cluster of Excellence
on Complexity and Topology in Quantum Matter ct.qmat  Ioannis Matthaiakakis
Dipartimento di Fisica, Università di Genova, via Dodecaneso 33, I-16146,
Genova, Italy  I.N.F.N. - Sezione di Genova, via Dodecaneso 33, I-16146,
Genova, Italy  Institute for Theoretical Physics and Astrophysics, Julius-
Maximilians-Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany
Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum
Matter ct.qmat  René Meyer  Institute for Theoretical Physics and
Astrophysics, Julius-Maximilians-Universität Würzburg, Am Hubland, D-97074
Würzburg, Germany  Würzburg-Dresden Cluster of Excellence on Complexity and
Topology in Quantum Matter ct.qmat

###### Abstract

General relativity (GR) as described in terms of curvature by the Einstein-
Hilbert action is dynamically equivalent to theories of gravity formulated in
terms of spacetime torsion or non-metricity. This forms what is called the
geometrical trinity of gravity. The theories corresponding to this trinity
are, apart from GR, the teleparallel (TEGR) and symmetric teleparallel (STEGR)
equivalent theories of general relativity, respectively, and their actions are
equivalent to GR up to boundary terms $B$. We investigate how the Gibbons-
Hawking-York (GHY) boundary term of GR changes under the transition to TEGR
and STEGR within the framework of metric-affine gravity. We show that $B$ is
the difference between the GHY term of GR and that of metric-affine gravity.
Moreover, we show that the GHY term for both TEGR and STEGR must vanish for
consistency of the variational problem. Furthermore, our approach allows to
extend the equivalence between GR, TEGR and STEGR beyond the Einstein-Hilbert
action to any action built out of the curvature two-form, thus establishing
the generalized geometrical trinity of gravity. We argue that these results
will be particularly useful in view of studying gauge/gravity duality for
theories with torsion and non-metricity.

toc.sectiontoc.sectionContentsContentstoc.section

###### Contents

  1. 1 Introduction
  2. 2 Geometric setup
  3. 3 (S)TEGR action and its boundary term in differential geometry
  4. 4 Generalization to $\mathcal{L}(\mathring{\Omega}\indices{{}^{\mu}_{\nu}})$ theories of gravity
  5. 5 Conclusions and outlook
  6. 6 Acknowledgements
  7. A Tensor component formulation of the (S)TEGR results

##  1 Introduction

Einstein’s theory of general relativity (GR) is a theory with tremendous
success in describing gravity [1, 2, 3]. In standard formulations of GR,
gravity is described entirely in terms of spacetime curvature. However,
torsion as well as non-metricity are two additional geometric quantities any
given spacetime may exhibit [4]. In recent years, progress has been made
towards understanding theories of gravity including torsion and non-metricity,
as well as their implications to cosmology and black hole physics (see e.g.
[5, 6, 7, 8, 9, 10, 11]). Our motivation is to examine theories of gravity
containing torsion and non-metricity within the context of gauge/gravity
duality. Within gauge/gravity duality, a theory of gravity in an
(asymptotically local) Anti-de Sitter spacetime provides us with a dual
description of the dynamics of a strongly coupled conformal field theory (CFT)
[12, 13, 14, 15]. In addition this dual CFT is realized holographically, in
the sense that it is defined on the boundary of the spacetime in which gravity
propagates. Gauge/gravity duality has been used successfully to analyze
extensively the transport properties of strongly coupled CFTs (see e.g. [16,
17, 18, 19, 20]). However, these analyses have been performed almost entirely
in torsion-free, metric curved spacetimes.111See [21, 22, 23, 24] for some
early work on torsionful and non-metric realizations of gauge/gravity duality.
We aim at introducing torsion and non-metricity into this duality in order to
describe spin and hypermomentum transport in strongly coupled systems. In
particular, we want to understand the hydrodynamic transport of spin and
hypermomentum in strongly coupled fluids, such as Dirac and Weyl semimetals or
the quark gluon plasma [25, 26, 27, 28, 29, 30, 31, 32], by using the
fluid/gravity correspondence [33, 34, 35]. Since the spin and hypermomentum of
the dual fluid are encoded on the boundary of the spacetime by definition, it
is important to study the gravitational boundary terms involved in torsionful
and non-metric theories of gravity.

Theories in which either torsion or non-metricity are non-vanishing while
curvature vanishes identically are called teleparallel and symmetric
teleparallel theories of gravity, respectively. We focus on a subset of these
theories, the teleparallel equivalent of general relativity (TEGR) [6, 7, 8,
9, 10, 36, 37] and the symmetric teleparallel equivalent of general relativity
(STEGR) [38, 39, 40]. As the names suggest, these theories are dynamically
equivalent to GR. A particular advantage of these reformulations of GR is the
fact that their dynamics are described by Lagrangians reminiscent of gauge
theory Lagrangians. For example, model Lagrangians for (S)TEGR are

$\displaystyle\mathcal{L}_{\mathrm{TEGR}}\propto T_{\mu}\wedge\star
T^{\mu}\,,\qquad\mathcal{L}_{\mathrm{STEGR}}\propto Q^{\mu\nu}\wedge\star
Q_{\mu\nu}\,,$ (1)

where $T^{\mu}$ and $Q_{\mu\nu}$ are the differential forms of torsion and
non-metricity, respectively.222Beyond (1), we may study actions built out of
the weighted sums of the irreducible decompositions of $T^{\mu}$ and
$Q_{\mu\nu}$ under $\mathrm{SO}(1,n-1)$, see [4, 41].

There are several points in relation to actions of the form (1) that we would
like to emphasize. First, the gauge degrees of freedom of $T^{\mu}$ and
$Q_{\mu\nu}$ are defined on vector spaces that are locally isomorphic to the
underlying spacetime. As a result, the standard Hodge dual used in defining
gauge theories with internal degrees of freedom must be modified in non-
trivial ways in order to take into account their action on the gauge indices
[42, 10]. In particular, it is necessary to define a generalized Hodge dual
whose explicit expression is not known for all dimensions [43]. As a result,
it is not obvious how to write down the most general (S)TEGR Lagrangians in
dimensions different than four. Hence, the gauge theory-like Lagrangians (1)
have only limited use in arbitrary dimensions, and for explicit calculations.
A version of the (S)TEGR Lagrangian without generalized Hodge duals is
needed.333Note, however, that a way to circumvent this issue is implicitly
present in [38, 44], where the authors discuss (S)TEGR in $3+1$ dimensions.
Our analysis generalizes their approach to a broader class of Lagrangians.

The second feature of (S)TEGR we focus on is their corresponding boundary
terms. It is well-known that GR has a well-defined variational formulation in
spacetimes with boundaries only if a boundary Gibbons-Hawking-York (GHY) term
is included alongside the bulk Einstein-Hilbert Lagrangian [45, 46, 3, 5]. A
so far unsolved issue is whether a GHY equivalent term must be included
alongside the bulk actions in (1) (see e.g. [47, 8]).

Finally, we note that the geometrical trinity of gravity was established for
the Einstein-Hilbert action. However, extensions of GR, including arbitrary
polynomials of the Riemann tensor, are routinely studied in the literature,
see e.g. [15]. It is then necessary to understand whether such extensions may
be equivalently recast as extensions of (S)TEGR.

In the present paper, we use the formalism developed in [5] to address all
three of these points. In particular, we consider GR from the point of view of
metric-affine gravity (MAG) [4]. MAG describes the contributions of both
torsion and non-metricity to curvature in terms of a single geometric object,
the deformation one-form $A_{\mu\nu}$ defined in (7). This allows us to re-
derive TEGR and STEGR simultaneously. We show that the boundary term generated
when transitioning from GR to one of its equivalents, simply translates the
GHY term of GR to the GHY term of MAG generated by the curvature two-form.
Since curvature is set to zero when we enforce teleparallelism, we conclude
that (S)TEGR does not need a boundary term. In addition, our formalism is
defined naturally in any dimension. Thus, our derivation of the geometrical
trinity also provides the Lagrangian of (S)TEGR in general dimensions, see
(31).

Finally, we investigate under which conditions we may generalize the
equivalence between GR and (S)TEGR to extended theories of gravity. We show
that the geometrical trinity of gravity exists for any GR extension that may
be written in terms of the curvature two-form alone. Furthermore, as for the
ordinary geometrical trinity, the boundary term for our generalized trinity is
also found to vanish identically.

In section 2, we begin our investigation by setting up the geometric notions
necessary to understand our results. In section 3, we then derive the
equivalence between GR and (S)TEGR, as well as their corresponding GHY terms,
from the point of view of MAG via a differential geometric formalism. A
generalization of our proof to higher-curvature extensions of GR is given in
section 4. Finally, in section 5 we summarize our results and provide some
outlook for further research. We express our results in a more familiar tensor
calculus notation in appendix A.

##  2 Geometric setup

In this section we define the basic geometric objects necessary for
understanding the derivation of our main result in sections 3 and 4. For a
more thorough discussion of the formalism employed, the reader may consult
[5].

We begin by considering an $n$-dimensional manifold $\mathcal{M}$, the
spacetime where our theory of gravity lives on. We assume $\mathcal{M}$ is
equipped with a connection one-form
$\omega^{\mu}_{\nu}=\Gamma^{\mu}_{\rho\nu}\theta^{\rho}$ and a metric
$\mathrm{d}s^{2}=g_{\mu\nu}\theta^{\mu}\otimes\theta^{\nu}$, as well as a
coframe basis $\theta^{\mu}$ of the cotangent spaces of $\mathcal{M}$, which
is not necessarily holonomic ($\mathrm{d}\theta^{\mu}\neq 0$).444Greek indices
take values in $\\{0,\dots,n-1\\}$. The fields $\omega^{\mu}_{\nu}$,
$g_{\mu\nu}$ and $\theta^{\mu}$ are the dynamical fields of the theories of
gravity we consider. The gravitational Lagrangians we focus on are those built
out of the field strengths of the aforementioned dynamical fields defined as

$\begin{aligned} &\text{the curvature two-form}\vphantom{\frac{1}{2}}\\\
&\text{the torsion two-form}\vphantom{\frac{1}{2}}\\\ &\text{the non-metricity
one-form}\vphantom{\frac{1}{2}}\end{aligned}\quad\begin{aligned}
&\Omega\indices{{}^{\mu}_{\nu}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{d}\omega^{\mu}_{\nu}+\omega^{\mu}_{\rho}\wedge\omega^{\rho}_{\nu}=\frac{1}{2}R\indices{{}^{\mu}_{\nu}{}_{\rho}{}_{\sigma}}\theta^{\rho}\wedge\theta^{\sigma}\,,\\\
&T^{\mu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=D\theta^{\mu}=\mathrm{d}\theta^{\mu}+\omega^{\mu}_{\nu}\wedge\theta^{\nu}=\frac{1}{2}T\indices{{}^{\mu}_{\rho}{}_{\sigma}}\theta^{\rho}\wedge\theta^{\sigma}\,,\\\
&Q_{\mu\nu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-Dg_{\mu\nu}=-(\mathrm{d}g_{\mu\nu}-\omega^{\rho}_{\mu}g_{\rho\nu}-\omega^{\rho}_{\nu}g_{\mu\rho})=Q_{\mu\nu\rho}\theta^{\rho}\,,\vphantom{\frac{1}{2}}\end{aligned}$
(2)

where $\mathrm{d}$ denotes the exterior derivative and $D$ the exterior
covariant derivative with respect to the connection
$\omega^{\mu}_{\nu}$.555Note that we chose to write the tensor components of
(2) in terms of $\theta^{\mu}$ instead of a co-ordinate basis $dx^{I}$, in
order work with only one index type for simplicity. This is always possible
since the components of $\theta^{\mu}=\theta^{\mu}_{I}dx^{I}$ provide a
soldering/intertwiner form connecting the two bases. The components of the
curvature two-form are those of the Riemann tensor
$R\indices{{}^{\mu}_{\nu}{}_{\rho}{}_{\sigma}}$, while those of torsion and
non-metricity are the torsion tensor $T\indices{{}^{\mu}_{\rho}{}_{\sigma}}$
and non-metricity tensor $Q_{\mu\nu\rho}$, respectively.

We define GR, TEGR and STEGR in terms of $\Omega\indices{{}^{\mu}_{\nu}}$,
$T^{\mu}$ and $Q_{\mu\nu}$ as

$\begin{aligned} &\text{GR}\vphantom{\frac{1}{2}}\\\
&\text{TEGR}\vphantom{\frac{1}{2}}\\\
&\text{STEGR}\vphantom{\frac{1}{2}}\end{aligned}\quad\begin{aligned}
&\Omega\indices{{}^{\mu}_{\nu}}\neq 0\,,\leavevmode\nobreak\
T^{\mu}=0\,,\leavevmode\nobreak\ Q_{\mu\nu}=0\,,\vphantom{\frac{1}{2}}\\\
&\Omega\indices{{}^{\mu}_{\nu}}=0\,,\leavevmode\nobreak\ T^{\mu}\neq
0\,,\leavevmode\nobreak\ Q_{\mu\nu}=0\,,\vphantom{\frac{1}{2}}\\\
&\Omega\indices{{}^{\mu}_{\nu}}=0\,,\leavevmode\nobreak\
T^{\mu}=0\,,\leavevmode\nobreak\ Q_{\mu\nu}\neq
0\,.\vphantom{\frac{1}{2}}\end{aligned}$ (3)

The definition (3) also clarifies the naming of (S)TEGR as teleparallel
theories: Both TEGR and STEGR have by definition vanishing curvature, meaning
a vector parallel transported by $D$ in those theories remains parallel to
itself. In addition, STEGR is symmetric in the sense that the fundamental
dynamical field, $Q_{\mu\nu}$, is symmetric in its two indices unlike the two-
form $T^{\mu}$ in TEGR.

GR is special within this family of theories, because the vanishing of both
$T^{\mu}$ and $Q_{\mu\nu}$ constrains the connection $\omega^{\mu}_{\nu}$ to
be the Christoffel connection: If we denote the Christoffel connection by
$\mathring{\omega}^{\mu}_{\nu}$, its corresponding covariant derivative,
$\mathring{D}$, is the unique covariant derivative satisfying

$\displaystyle 0$
$\displaystyle=\mathring{T}^{\mu}=\mathring{D}\theta^{\mu}=\mathrm{d}\theta^{\mu}+\mathring{\omega}^{\mu}_{\nu}\wedge\theta^{\nu}\quad\Leftrightarrow\quad\mathrm{d}\theta^{\mu}=-\mathring{\omega}^{\mu}_{\nu}\wedge\theta^{\nu}\,,$
(4a) $\displaystyle 0$
$\displaystyle=\mathring{Q}_{\mu\nu}=-\mathring{D}g_{\mu\nu}=-\mathrm{d}g_{\mu\nu}+\mathring{\omega}^{\rho}_{\mu}g_{\rho\nu}+\mathring{\omega}^{\rho}_{\nu}g_{\mu\rho}\,.$
(4b)

In discussing (S)TEGR it is useful to isolate the part of the connection that
allows for non-trivial torsion and non-metricity. We do this by defining the
deformation one-form

$\displaystyle
A\indices{{}^{\mu}_{\nu}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\omega^{\mu}_{\nu}-\mathring{\omega}^{\mu}_{\nu}\,,$
(5)

with respect to which we may express $T^{\mu}$ and $Q_{\mu\nu}$ as

$\displaystyle A_{\mu\nu}+A_{\nu\mu}$ $\displaystyle=Q_{\mu\nu}\,,\qquad
A\indices{{}^{\mu}_{\nu}}\wedge\theta^{\nu}=T^{\mu}\,.$ (6)

The latter equation does not imply that the antisymmetric part of $A_{\mu\nu}$
consists of torsion only. In fact, we have
$(A_{\mu\nu}-A_{\nu\mu})\wedge\theta^{\nu}=2T_{\mu}-Q_{\mu\nu}\wedge\theta^{\nu}$.
This will be crucial when discussing the STEGR boundary term, as this anti-
symmetric part is how non-metricity enters the equations.

To give more intuition for the properties of $A_{\mu\nu}$, we invert (6) and
express $A_{\mu\nu}$ in terms of torsion and non-metricity [4] as

$\displaystyle A_{\mu\nu}=-\vartheta_{[\mu}\rfloor
T_{\nu]}+\frac{1}{2}\left(\vartheta_{\mu}\rfloor\vartheta_{\nu}\rfloor
T_{\rho}\right)\theta^{\rho}+\frac{1}{2}Q_{\mu\nu}+\left(\vartheta_{[\mu}\rfloor
Q_{\nu]\rho}\right)\theta^{\rho}\,,$ (7)

where $\rfloor$ is the interior product and $\vartheta_{\mu}$ is the dual
frame to $\theta^{\mu}$,
$\vartheta_{\nu}\rfloor\theta^{\mu}=\delta^{\mu}_{\nu}$. In terms of
components, we have

$\displaystyle
A_{\mu\nu\rho}=-\frac{1}{2}\left(T_{\rho\nu\mu}+T_{\nu\mu\rho}+T_{\mu\rho\nu}\right)+\frac{1}{2}\left(Q_{\rho\nu\mu}+Q_{\nu\mu\rho}+Q_{\mu\rho\nu}\right)\,.$
(8)

We observe that the torsion terms define the contorsion tensor
$K_{\mu\nu\rho}$, well-known in Einstein-Cartan theories of gravity [48, 49,
50, 51, 52, 53, 54], while the non-metricity terms define a Weyl connection in
the special case where $Q_{\mu\nu}=Qg_{\mu\nu}$ for some one-form $Q$ (see
e.g. [22]). Furthermore, from (7) we see that $A_{\mu\nu}$ captures any
deviation from GR, be it due to torsion or non-metricity. Therefore, the
deformation one-form allows us to describe both TEGR and STEGR simultaneously
within a single formalism.

To complete our discussion of geometric preliminaries, we introduce a boundary
$\partial\mathcal{M}$ in our spacetime $\mathcal{M}$, as well as the
decomposition of the dynamical fields in contributions normal and tangent to
$\partial\mathcal{M}$. Again, a thorough discussion of what follows along with
proofs may be found in [5] and the references therein. The geometry of
$\partial\mathcal{M}$ is described by tangent vectors $e^{a}_{\mu}$ and a
normal vector $n^{\mu}$, fulfilling $e^{a}_{\mu}n^{\mu}=0$. Latin indices
denote internal indices of the boundary manifold and take values in
$\\{0,\dots,n-2\\}$. We normalize $n^{\mu}$ as $n_{\mu}n^{\mu}=\varepsilon$,
where $\varepsilon=-1$ for spacelike and $\varepsilon=+1$ for timelike
boundaries. The connection one-form on the boundary is induced by the
connection in the bulk via

$\displaystyle\omega^{b}_{a}=e^{b}_{\mu}\left(\mathrm{d}e^{\mu}_{a}+\omega^{\mu}_{\nu}e^{\nu}_{a}\right)\,,$
(9)

whereas general tensors are pulled back to $\partial\mathcal{M}$ as usual by
contraction with $e^{a}_{\mu}$. In this manner,
$\phi^{a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{a}_{\mu}\theta^{\mu}$
defines a basis coframe on $\partial\mathcal{M}$ and
$\gamma_{ab}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{\mu}_{a\vphantom{b}}e^{\nu\vphantom{\mu}}_{b}g_{\mu\nu}$
a boundary metric. These pullbacks allow us to split $\theta^{\mu}$ and
$g_{\mu\nu}$ into tangent and normal contributions as

$\displaystyle\theta^{\mu}$
$\displaystyle=e^{\mu}_{a}\phi^{a}+Nn^{\mu}\phi\,,\qquad
g_{\mu\nu}=e^{a}_{\mu}e^{b}_{\nu}\gamma_{ab}+\varepsilon n_{\mu}n_{\nu}\,,$
(10)

where
$N\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=1/\sqrt{\left|n^{\mu}n_{\mu}\right|}$
is the normalization factor of $n^{\mu}$ and
$\phi\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{\varepsilon}{N}n_{\mu}\theta^{\mu}$.
The embedding of $\partial\mathcal{M}$ in $\mathcal{M}$ is characterized by
the extrinsic curvature one-forms

$\displaystyle
K^{a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{a}_{\mu}Dn^{\mu}\qquad\text{and}\qquad\tilde{K}_{a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{\mu}_{a}Dn_{\mu}=K_{a}-e^{\mu}_{a}n^{\nu}Q_{\mu\nu}\,,$
(11)

which differ only due to non-metricity. The projections of the components of
these one-forms to the boundary give the well-known expressions of extrinsic
curvature in tensor language [5].

##  3 (S)TEGR action and its boundary term in differential geometry

In the present section, we present our derivation of the GHY boundary term for
(S)TEGR. The derivation boils down to two steps, first we consider the
Einstein-Hilbert Lagrangian of GR and use (5) to express it in terms of
$A_{\mu\nu}$ and $\Omega\indices{{}^{\mu}_{\nu}}$. Second, we enforce
teleparallelism by setting $\Omega\indices{{}^{\mu}_{\nu}}=0$. During the
first part of this derivation a boundary term appears which we must treat
carefully as it is this boundary term that holds all the information regarding
the GHY term for (S)TEGR.

To begin our derivation, we insert the deformation one-form (5) into the
curvature definition (2) to obtain

$\displaystyle\Omega\indices{{}^{\mu}_{\nu}}=\mathring{\Omega}\indices{{}^{\mu}_{\nu}}+\mathring{D}A\indices{{}^{\mu}_{\nu}}+A\indices{{}^{\mu}_{\rho}}\wedge
A\indices{{}^{\rho}_{\nu}}\,,$ (12)

where
$\mathring{\Omega}\indices{{}^{\mu}_{\nu}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{d}\mathring{\omega}^{\mu}_{\nu}+\mathring{\omega}^{\mu}_{\rho}\wedge\mathring{\omega}^{\rho}_{\nu}$
is the curvature two-form built from the Christoffel connection
$\mathring{\omega}^{\mu}_{\nu}$, that is the curvature two-form of GR. For the
Einstein-Hilbert action in generic $n$ dimensions, this implies

$\displaystyle S^{\mathrm{EH}}_{\mathring{\Omega}}$
$\displaystyle=\frac{1}{2\kappa}\int\mathrm{d}^{n}x\sqrt{-g}\mathring{R}+S^{\mathrm{EH}}_{\mathring{\Omega},\mathrm{GHY}}=\frac{1}{2\kappa}\int\eta^{\mu\nu}\wedge\mathring{\Omega}_{\mu\nu}+S^{\mathrm{EH}}_{\mathring{\Omega},\mathrm{GHY}}$
(13a)
$\displaystyle=\frac{1}{2\kappa}\int\eta^{\mu\nu}\wedge\left(\Omega_{\mu\nu}-\mathring{D}A\indices{{}_{\mu}{}_{\nu}}-A\indices{{}_{\mu}{}_{\rho}}\wedge
A\indices{{}^{\rho}_{\nu}}\right)+S^{\mathrm{EH}}_{\mathring{\Omega},\mathrm{GHY}}\,,$
(13b)

where we define
$\eta^{\mu\nu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=*(\theta^{\mu}\wedge\theta^{\nu})$
in which $*$ denotes the Hodge duality. We supplement the action by the well-
known GHY term of GR, given by

$\displaystyle
S^{\mathrm{EH}}_{\mathring{\Omega},\mathrm{GHY}}=\frac{\varepsilon}{\kappa}\int_{\partial\mathcal{M}}\left.\mathring{K}_{a}\wedge\eta^{\mathbf{n}a}\right|_{\partial\mathcal{M}}=\frac{\varepsilon}{\kappa}\int_{\partial\mathcal{M}}\mathrm{d}\text{Vol}_{\partial\mathcal{M}}\sqrt{|\gamma|}\mathring{K}\indices{{}^{a}_{a}}\,,$
(14)

where the extrinsic curvature one-form was defined in (11) (for
$Q_{\mu\nu}=0$) and we abbreviate
$\eta^{\mathbf{n}a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=n_{\mu}e^{a}_{\nu}\eta^{\mu\nu}$.
Recall that the GHY term is necessary to obtain a well-defined variational
problem [45, 46, 3]. According to (3), the action in (13b) represents TEGR if
$Q_{\mu\nu}=0=\Omega_{\mu\nu}$, in which case $A\indices{{}^{\mu}_{\nu}}$ is
called the contorsion one-form, while it represents STEGR if we choose
$T_{\mu}=0$ and $\Omega_{\mu\nu}=0$ instead.

Next, we consider the $A_{\mu\nu}$ terms in (13). In particular, we focus on
$\mathring{D}A\indices{{}^{\mu}_{\nu}}$, which is the all important boundary
term mentioned earlier. To see this explicitly, we isolate this term and
consider the action

$\displaystyle
S^{\mathrm{\mathring{D}A}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-\frac{1}{2\kappa}\int_{\mathcal{M}}\eta^{\mu\nu}\wedge\mathring{D}A\indices{{}_{\mu}{}_{\nu}}=-\frac{1}{2\kappa}\int_{\mathcal{M}}A_{\mu\nu}\wedge\mathring{D}\eta^{\mu\nu}-\frac{1}{2\kappa}\int_{\partial\mathcal{M}}\left.A_{\mu\nu}\wedge\eta^{\mu\nu}\vphantom{\left(\tilde{K}_{a}\right)}\right|_{\partial\mathcal{M}}\,.$
(15)

The first term on the right-hand side of (15) vanishes identically after we
recall the definitions of the Hodge star operator $*$ and the Christoffel
exterior covariant derivative $\mathring{D}$ from (4). These definitions imply

$\displaystyle\mathring{D}\eta^{\mu\nu}=\mathring{D}\left(\frac{\sqrt{|\det
g|}}{(n-2)!}\varepsilon_{\sigma_{1}\dots\sigma_{2}\rho_{1}\dots\rho_{n-2}}g^{\sigma_{1}\mu}g^{\sigma_{2}\nu}\theta^{\rho_{1}}\wedge\dots\wedge\theta^{\rho_{n-2}}\right)=0\,,$
(16)

such that only the boundary term remains in $S^{\mathrm{\mathring{D}A}}$,

$\displaystyle
S^{\mathrm{\mathring{D}A}}=-\frac{1}{2\kappa}\int_{\partial\mathcal{M}}\left.A_{\mu\nu}\wedge\eta^{\mu\nu}\vphantom{\left(\tilde{K}_{a}\right)}\right|_{\partial\mathcal{M}}\,.$
(17)

To proceed, we need to express $S^{\mathrm{\mathring{D}A}}$ entirely in terms
of boundary data. To achieve this, we use the definition of
$A\indices{{}^{\mu}_{\nu}}=\omega^{\mu}_{\nu}-\mathring{\omega}^{\mu}_{\nu}$
as well as the $3+1$ decomposition of the connection.666 We follow convention
and call the decomposition of the connection in tangential and normal
components the connection’s $3+1$ decomposition. However, our results are
valid in any dimension. We derive said decomposition by employing

$\displaystyle\delta^{\mu}_{\nu}=e^{\mu}_{a}e^{a}_{\nu}+\varepsilon
n^{\mu}n_{\nu}$ (18)

to decompose the indices of $\omega^{\mu}_{\nu}$ into boundary normal and
tangent contributions. Namely,

$\displaystyle\omega^{\mu}_{\nu}=\delta^{\mu}_{\alpha}\delta^{\beta}_{\nu}\omega^{\alpha}_{\beta}=e^{\mu}_{a}e^{b}_{\nu}(e^{a}_{\alpha}e_{b}^{\beta}\omega^{\alpha}_{\beta})+n^{\mu}n_{\nu}(n_{\alpha}n^{\beta}\omega^{\alpha}_{\beta})+\varepsilon
n^{\mu}e^{a}_{\nu}(n_{\alpha}e_{a}^{\beta}\omega^{\alpha}_{\beta})+\varepsilon
e^{\mu}_{a}n_{\nu}(e_{\alpha}^{a}n_{\beta}\omega^{\alpha}_{\beta})\,.$ (19)

The decomposition of $\omega^{\mu}_{\nu}$ then amounts to determining the
projections in parentheses in (19). The projection of $\omega^{\mu}_{\nu}$ on
$\partial M$ in both indices is given in (9) as

$\displaystyle
e^{a}_{\mu}e^{\nu}_{b}\omega^{\mu}_{\nu}=\omega^{a}_{b}-e^{a}_{\mu}\mathrm{d}e^{\mu}_{b}\,.$
(20)

The corresponding projections in which one of the indices is projected to the
normal vector $n_{\mu}$ instead are obtained from the extrinsic curvatures
$K^{a}$ and $\tilde{K}_{a}$ in (11), which decompose as

$\displaystyle
K^{a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{a}_{\mu}Dn^{\mu}=e^{a}_{\mu}\mathrm{d}n^{\mu}+e^{a}_{\mu}\omega^{\mu}_{\nu}n^{\nu}\qquad\text{and}\qquad\tilde{K}_{a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{\mu}_{a}Dn_{\mu}=e^{\mu}_{a}\mathrm{d}n_{\mu}-e^{\mu}_{a}\omega^{\nu}_{\mu}n_{\nu}\,.$
(21)

We solve for the projections of the connection as

$\displaystyle
e^{a}_{\mu}n^{\nu}\omega^{\mu}_{\nu}=K^{a}-e^{a}_{\mu}\mathrm{d}n^{\mu}\qquad\text{and}\qquad
n_{\mu}e^{\nu}_{a}\omega^{\mu}_{\nu}=-\tilde{K}_{a}+e^{\mu}_{a}\mathrm{d}n_{\mu}\,.$
(22)

Lastly, the twice normal projected part of the connection is obtained from the
normal component of non-metricity,

$\displaystyle
Q_{\mathbf{n}\mathbf{n}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=n^{\mu}n^{\nu}Q_{\mu\nu}=2n_{\mu}Dn^{\mu}=2n_{\mu}\mathrm{d}n^{\mu}+2n_{\mu}\omega^{\mu}_{\nu}n^{\nu}\,,$
(23)

yielding

$\displaystyle
n_{\mu}n^{\nu}\omega^{\mu}_{\nu}=\frac{1}{2}Q_{\mathbf{n}\mathbf{n}}-n_{\mu}\mathrm{d}n^{\mu}\,.$
(24)

We thus obtain the $3+1$ decomposition of the connection as

$\displaystyle\begin{split}\omega^{\mu}_{\nu}=&e^{\mu}_{a}e^{b}_{\nu}(\omega^{a}_{b}-e^{a}_{\rho}\mathrm{d}e^{\rho}_{b})+n^{\mu}n_{\nu}(\frac{1}{2}Q_{\mathbf{n}\mathbf{n}}-n_{\rho}\mathrm{d}n^{\rho})\\\
&+\varepsilon
n^{\mu}e^{a}_{\nu}(e^{\rho}_{a}\mathrm{d}n_{\rho}-\tilde{K}_{a})+\varepsilon
e^{\mu}_{a}n_{\nu}(K^{a}-e^{a}_{\rho}\mathrm{d}n^{\rho})\end{split}$ (25)

which immediately implies the 3+1 decomposition of the deformation one-form
(5)777The $3+1$ decomposition of $\mathring{\omega}^{\mu}_{\nu}$ is given by
(25) after setting $Q_{\mathbf{n}\mathbf{n}}=0=\tilde{K}^{a}-K^{a}$.

$\displaystyle
A\indices{{}^{\mu}_{\nu}}=e^{\mu}_{a}e^{b}_{\nu}A\indices{{}^{a}_{b}}+\frac{1}{2}n^{\mu}n_{\nu}Q_{\mathbf{n}\mathbf{n}}+\varepsilon
n^{\mu}e^{a}_{\nu}(\mathring{K}_{a}-\tilde{K}_{a})+\varepsilon
e^{\mu}_{a}n_{\nu}(K^{a}-\mathring{K}^{a})\,,$ (26)

where
$A\indices{{}^{a}_{b}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\omega^{a}_{b}-\mathring{\omega}^{a}_{b}$
is the boundary deformation one-form. Knowing this decomposition, we come back
to the boundary action (17) which evaluates to

$\displaystyle
S^{\mathrm{\mathring{D}A}}=-\frac{1}{2\kappa}\int_{\partial\mathcal{M}}\left.A\indices{{}_{a}{}_{b}}\wedge\left(e^{a}_{\mu}e^{b}_{\nu}\eta^{\mu\nu}\right)\right|_{\partial\mathcal{M}}-\frac{\varepsilon}{2\kappa}\int_{\partial\mathcal{M}}\left.\left(2\mathring{K}_{a}-\tilde{K}_{a}-K_{a}\right)\wedge\eta^{\mathbf{n}a}\right|_{\partial\mathcal{M}}\,.$
(27)

We further simplify this boundary term by examining the Hodge dual in
$\left.A\indices{{}_{a}{}_{b}}\wedge\left(e^{a}_{\mu}e^{b}_{\nu}\eta^{\mu\nu}\right)\right|_{\partial\mathcal{M}}$.
Using the definition
$\eta^{\mu\nu}=*\left(\theta^{\mu}\wedge\theta^{\nu}\right)$ we obtain

$\displaystyle\left.A\indices{{}_{a}{}_{b}}\wedge\left(e^{a}_{\mu}e^{b}_{\nu}\eta^{\mu\nu}\right)\right|_{\partial\mathcal{M}}=\frac{\sqrt{|\det
g|}}{(n-2)!}\varepsilon_{\mu\nu\rho_{1}\dots\rho_{n-2}}e^{\mu}_{a}e^{\nu}_{b}e^{\rho_{1}}_{c_{1}}\cdots
e^{\rho_{n-2}}_{c_{n-2}}A^{ab}\wedge\phi^{c_{1}}\wedge\dots\phi^{c_{n-2}}\,,$
(28)

where $\phi^{a}=e^{a}_{\mu}\theta^{\mu}$ is the boundary coframe. Note that
all $n$ indices of the $\varepsilon$-symbol take their values on the $n-1$
dimensional boundary. As a result,
$\left.A\indices{{}_{a}{}_{b}}\wedge\left(e^{a}_{\mu}e^{b}_{\nu}\eta^{\mu\nu}\right)\right|_{\partial\mathcal{M}}$
vanishes identically. This simplifies the boundary action (27) to

$\displaystyle
S^{\mathrm{\mathring{D}A}}=-\frac{\varepsilon}{2\kappa}\int_{\partial\mathcal{M}}\left.\left(2\mathring{K}_{a}-\tilde{K}_{a}-K_{a}\right)\wedge\eta^{\mathbf{n}a}\right|_{\partial\mathcal{M}}\,.$
(29)

Inserting everything into the Einstein-Hilbert action (13), we obtain

$\displaystyle\begin{split}S^{\mathrm{EH}}_{\mathring{\Omega}}&=\frac{1}{2\kappa}\int_{\mathcal{M}}\eta^{\mu\nu}\wedge\mathring{\Omega}_{\mu\nu}+\frac{\varepsilon}{\kappa}\int_{\partial\mathcal{M}}\left.\mathring{K}_{a}\wedge\eta^{\mathbf{n}a}\right|_{\partial\mathcal{M}}\\\
&=\frac{1}{2\kappa}\int_{\mathcal{M}}\eta^{\mu\nu}\wedge\left(\Omega_{\mu\nu}-A\indices{{}_{\mu}{}_{\rho}}\wedge
A\indices{{}^{\rho}_{\nu}}\right)+\frac{\varepsilon}{\kappa}\int_{\partial\mathcal{M}}\left.\mathring{K}_{a}\wedge\eta^{\mathbf{n}a}\right|_{\partial\mathcal{M}}+S^{\mathrm{\mathring{D}A}}\\\
&=\frac{1}{2\kappa}\int_{\mathcal{M}}\eta^{\mu\nu}\wedge\left(\Omega_{\mu\nu}-A\indices{{}_{\mu}{}_{\rho}}\wedge
A\indices{{}^{\rho}_{\nu}}\right)+\frac{\varepsilon}{2\kappa}\int_{\partial\mathcal{M}}\left.\left(\tilde{K}_{a}+K_{a}\right)\wedge\eta^{\mathbf{n}a}\right|_{\partial\mathcal{M}}\,.\end{split}$
(30)

At this point, we have finished the first part of our derivation of (S)TEGR.
In the second part, we must enforce teleparallelism by setting
$\Omega_{\mu\nu}=0$. Naively, this would leave us with a bulk term quadratic
in $A_{\mu\nu}$ and a boundary term depending on the extrinsic curvature.
However, this is not correct: As shown in [5], the boundary term is in fact
the GHY term corresponding to the curvature term. That is, the first and third
term in the last line of (30) combine to form $S^{\rm EH}_{\Omega}$, the
Einstein-Hilbert action of $\Omega$ including its corresponding GHY term. This
GHY term is necessary for the well-posedness of the variational problem, since
its variation with respect to the connection cancels the variation of the
curvature term with respect to the connection. This implies that simply
setting $\Omega_{\mu\nu}=0$ in $S^{\rm EH}_{\mathring{\Omega}}$, the GHY term
would once again make the variational problem ill-defined. Instead, the
correct way to enforce teleparallelism is to set $S^{\rm EH}_{\Omega}$ to zero
all together. Therefore, the complete and well-defined action for (S)TEGR is

$\displaystyle
S^{\mathrm{(S)TEGR}}=-\frac{1}{2\kappa}\int_{\mathcal{M}}\eta^{\mu\nu}\wedge
A\indices{{}_{\mu}{}_{\rho}}\wedge A\indices{{}^{\rho}_{\nu}}\,.$ (31)

The action $S^{\rm(S)TEGR}$ coincides with the differential form (S)TEGR
Lagrangian found in [38] and the components of (31) also reproduce the (S)TEGR
bulk actions considered in [42, 10, 47]. It is particularly interesting to
note that the components of the Lagrangian in (31) are

$\displaystyle\eta^{\mu\nu}\wedge A_{\mu\rho}\wedge
A\indices{{}^{\rho}_{\nu}}=2\sqrt{|g|}\mathrm{dVol}_{\mathcal{M}}\,A\indices{{}^{\mu}_{\rho}{}_{[\mu|}}A\indices{{}^{\rho}{}^{\nu}_{|\nu]}}\,,$
(32)

which also have been found to be the components of

$\displaystyle T_{\mu}\wedge\star T^{\mu}$ (33)

in [42, 10] for 3+1d TEGR. However, $\star$ denotes a generalized Hodge dual
which is necessary for soldered bundles and a gauge interpretation of the TEGR
Lagrangian. The generalization of this Hodge dual to arbitrary dimensions is
involved and not known so far [43]. Hence, it is not immediately clear what
the component expression of (33) is in general dimensions. In contrast to
that, our expression (32) has well-defined and directly accessible components
in an arbitrary number of dimensions. Apart from the bulk dynamics of (S)TEGR,
our formalism also establishes unambiguously the (S)TEGR GHY term. We find it
to vanish identically. This is the first main result of our paper.

Our second main result is a new systematic interpretation of the boundary
term. Namely, we see that the (S)TEGR boundary term (17) is not a GHY term,
but rather the difference between the GHY term of GR and the one of metric-
affine gravity! That is,

$\displaystyle
S^{\mathrm{\mathring{D}A}}=S^{\mathrm{EH}}_{\vphantom{\mathring{Omega}}\Omega,\mathrm{GHY}}-S^{\mathrm{EH}}_{\mathring{\Omega},\mathrm{GHY}}\,.$
(34)

As a final comment we note that while our derivation is in the language of
differential forms, it may equally well be reformulated in a more traditional
way in terms of tensor components. We show this explicitly in appendix A.

##  4 Generalization to
$\mathcal{L}(\mathring{\Omega}\indices{{}^{\mu}_{\nu}})$ theories of gravity

In the present section, we will go beyond the Einstein-Hilbert theory of
gravity and consider whether general extensions of GR built out of
$\mathring{\Omega}\indices{{}^{\mu}_{\nu}}$ are equivalent to extensions of
(S)TEGR. In this way, we derive a generalized geometrical trinity (GGT) of
gravity beyond the linear Lagrangian regime.

Consider a generic action depending only on Riemannian curvature,

$\displaystyle
S_{\mathring{\Omega}}=\int_{\mathcal{M}}\mathcal{L}(\mathring{\Omega}\indices{{}^{\mu}_{\nu}})+S_{\mathring{\Omega},{\rm
GHY}}\,,$ (35)

in which we do not fix the form of the Lagrangian $\mathcal{L}$. The GHY term
corresponding to ${\cal L}$ was derived in [5] and takes the form

$\displaystyle S_{\mathring{\Omega},{\rm
GHY}}=\varepsilon\int_{\partial\mathcal{M}}\left.\mathring{K}^{a}\wedge*(\varphi_{\mathbf{n}a}-\varphi_{a\mathbf{n}})\right|_{\partial\mathcal{M}}\,,$
(36)

where $*\varphi\indices{{}_{\mu}^{\nu}}$ is defined via
$*\varphi\indices{{}_{\mu}^{\nu}}\wedge\delta\mathring{\Omega}\indices{{}^{\mu}_{\nu}}=\delta_{\mathring{\Omega}\indices{{}^{\mu}_{\nu}}}\mathcal{L}(\mathring{\Omega}\indices{{}^{\mu}_{\nu}})$.
The action built out of the curvature $\Omega\indices{{}^{\mu}_{\nu}}$ reads

$\displaystyle
S_{\vphantom{\mathring{Omega}}\Omega}=\int_{\mathcal{M}}\mathcal{L}(\Omega\indices{{}^{\mu}_{\nu}})+S_{\vphantom{\mathring{Omega}}\Omega,{\rm
GHY}}\,,$ (37a) $\displaystyle S_{\vphantom{\mathring{Omega}}\Omega,{\rm
GHY}}=-\int_{\partial\mathcal{M}}\left.\left(-\varepsilon\tilde{K}_{a}\wedge*\varphi^{\mathbf{n}a}+\varepsilon
K^{a}\wedge*\varphi_{a\mathbf{n}}+\frac{1}{2}Q_{\mathbf{n}\mathbf{n}}\wedge*\varphi_{\mathbf{n}\mathbf{n}}\right)\right|_{\partial\mathcal{M}}\,.$
(37b)

To simplify the derivation of the GGT, we use the trick of [5, 55] and
introduce Lagrange multipliers $\rho\indices{{}^{\mu}_{\nu}}$ into
$S_{\mathring{\Omega}}$. These allow us to linearize the terms in the action
depending on the curvature, namely888In (38), we have abused notation and
denote $S_{\mathring{\Omega}}$ as well as its version including Lagrange
multipliers by the same symbol. The actions are only equal, however, when the
equations of motion of $*\varphi\indices{{}_{\mu}^{\nu}}$ are enforced.

$\displaystyle\begin{split}S_{\mathring{\Omega}}&=\int_{\mathcal{M}}\left(\mathcal{L}(\mathring{\rho}\indices{{}^{\mu}_{\nu}})+*\varphi\indices{{}_{\mu}^{\nu}}\wedge(\mathring{\Omega}\indices{{}^{\mu}_{\nu}}-\mathring{\rho}\indices{{}^{\mu}_{\nu}})\right)+S_{\mathring{\Omega},\mathrm{GHY}}\\\
&=\int_{\mathcal{M}}\left(\mathcal{L}(\mathring{\rho}\indices{{}^{\mu}_{\nu}})+*\varphi\indices{{}_{\mu}^{\nu}}\wedge(\Omega\indices{{}^{\mu}_{\nu}}-\mathring{D}A\indices{{}^{\mu}_{\nu}}-A\indices{{}^{\mu}_{\rho}}\wedge
A\indices{{}^{\rho}_{\nu}}-\mathring{\rho}\indices{{}^{\mu}_{\nu}})\right)+S_{\mathring{\Omega},\mathrm{GHY}}\,,\end{split}$
(38)

where in the last equality we used the curvature decomposition (12). Given the
linearization of the action, the derivation of the GGT follows as in the
previous section with $\eta^{\mu\nu}$ replaced by
$*\varphi\indices{{}_{\mu}^{\nu}}$. The equivalent of $S^{\mathring{D}A}$ in
(15) in the present case yields

$\displaystyle\begin{split}S^{\mathring{D}A}_{\varphi}=-\int_{\mathcal{M}}*\varphi\indices{{}_{\mu}^{\nu}}\wedge\mathring{D}A\indices{{}^{\mu}_{\nu}}&=-\int_{\partial\mathcal{M}}\left.A\indices{{}^{\mu}_{\nu}}\wedge*\varphi\indices{{}_{\mu}^{\nu}}\right|_{\partial\mathcal{M}}-\int_{\mathcal{M}}A\indices{{}^{\mu}_{\nu}}\wedge\mathring{D}*\varphi\indices{{}_{\mu}^{\nu}}\,.\end{split}$
(39)

We expand the $\partial\mathcal{M}$ term in $S^{\mathring{D}A}_{\varphi}$ in
terms of boundary components by using the index split (18), as well as the
decomposition (26) of $A\indices{{}^{\mu}_{\nu}}$. We arrive at

$\displaystyle\begin{split}S^{\mathring{D}A}_{\varphi}&=-\varepsilon\int_{\partial\mathcal{M}}\left.\mathring{K}^{a}\wedge*(\varphi_{\mathbf{n}a}-\varphi_{a\mathbf{n}})\right|_{\partial\mathcal{M}}\\\
&\hskip
13.0pt-\int_{\partial\mathcal{M}}\left.\left(-\varepsilon\tilde{K}_{a}\wedge*\varphi^{\mathbf{n}a}+\varepsilon
K^{a}\wedge*\varphi_{a\mathbf{n}}+\frac{1}{2}Q_{\mathbf{n}\mathbf{n}}\wedge*\varphi_{\mathbf{n}\mathbf{n}}\right)\right|_{\partial\mathcal{M}}\\\
&\hskip
13.0pt-\int_{\partial\mathcal{M}}\left.A^{ab}\wedge*\varphi_{ab}\right|_{\partial\mathcal{M}}-\int_{\mathcal{M}}A\indices{{}^{\mu}_{\nu}}\wedge\mathring{D}*\varphi\indices{{}_{\mu}^{\nu}}\\\
&=-S_{\mathring{\Omega},{\rm GHY}}+S_{\vphantom{\mathring{Omega}}\Omega,{\rm
GHY}}-\int_{\partial\mathcal{M}}\left.A^{ab}\wedge*\varphi_{ab}\right|_{\partial\mathcal{M}}-\int_{\mathcal{M}}A\indices{{}^{\mu}_{\nu}}\wedge\mathring{D}*\varphi\indices{{}_{\mu}^{\nu}}\,.\end{split}$
(40)

We observe that $S^{\mathring{D}A}_{\varphi}$ contains the difference of GHY
terms in addition to two additional terms, which in general depend on
$\mathring{\Omega}\indices{{}^{\mu}_{\nu}}$. Let us consider the latter terms
in more detail. First, the term
$A^{ab}\wedge*\varphi_{ab}|_{\partial\mathcal{M}}$ may be evaluated
analogously to (28). Hence, by the same index counting argument this term
vanishes identically. The remaining term, proportional to
$\mathring{D}*\varphi\indices{{}_{\mu}^{\nu}}$, vanishes identically as well:
Recall from [5] that $\mathcal{L}(\mathring{\Omega}\indices{{}^{\mu}_{\nu}})$
is allowed to be a polynomial999This includes Taylor expandable functions of
curvature, see [5]. Derivatives of curvature are evaluated using the Bianchi
identities. in the Riemannian curvature
$\mathring{\Omega}\indices{{}^{\mu}_{\nu}}$, the frame $\theta^{\mu}$ and the
metric $g_{\mu\nu}$. Hence, $*\varphi\indices{{}_{\mu}^{\nu}}$, evaluated via
$*\varphi\indices{{}_{\mu}^{\nu}}\wedge\delta\mathring{\Omega}\indices{{}^{\mu}_{\nu}}=\delta_{\mathring{\Omega}\indices{{}^{\mu}_{\nu}}}\mathcal{L}(\mathring{\Omega}\indices{{}^{\mu}_{\nu}})$,
is a polynomial in these fields as well. From (4) we recall that by definition
the Christoffel connection satisfies $\mathring{D}\theta^{\mu}=0$ and
$\mathring{D}g_{\mu\nu}=0$. In addition, the covariant derivative of the
Riemannian curvature two-form also vanishes due to the Bianchi identity
$\mathring{D}\mathring{\Omega}\indices{{}^{\mu}_{\nu}}=0$. Hence,
$\mathring{D}*\varphi\indices{{}_{\mu}^{\nu}}=0$.101010This holds true for
Hodge duals of curvature included in the Lagrangian, since these are again
polynomials in curvature, frame and metric. Apart from a mathematical
argument, there is also a physical reason why
$\mathring{D}*\varphi\indices{{}_{\mu}^{\nu}}=0$. In particular,
$\mathring{D}*\varphi\indices{{}_{\mu}^{\nu}}=0$ are the equations of motion
of the connection $\mathring{\omega}^{\mu}_{\nu}$ in a first order formalism
of GR [56]. Physically, these equations enforce the conservation of orbital
angular momentum. Therefore, for any theory of gravity without spinful
dynamical fields, $\mathring{D}*\varphi\indices{{}_{\mu}^{\nu}}=0$. As a
result the boundary term is once again the difference of GHY terms,
$S^{\mathring{D}A}_{\varphi}=S_{\vphantom{\mathring{\Omega}}\Omega,{\rm
GHY}}-S_{\mathring{\Omega},{\rm GHY}}$. With this result at hand, we come back
to (38) to find

$\displaystyle\begin{split}S_{\mathring{\Omega}}&=\int_{\mathcal{M}}\left(\mathcal{L}(\mathring{\rho}\indices{{}^{\mu}_{\nu}})+*\varphi\indices{{}_{\mu}^{\nu}}\wedge(\Omega\indices{{}^{\mu}_{\nu}}-A\indices{{}^{\mu}_{\rho}}\wedge
A\indices{{}^{\rho}_{\nu}}-\mathring{\rho}\indices{{}^{\mu}_{\nu}})\right)+S_{\vphantom{\mathring{Omega}}\Omega,{\rm
GHY}}\\\
&=\int_{\mathcal{M}}\mathcal{L}(\Omega\indices{{}^{\mu}_{\nu}}-A\indices{{}^{\mu}_{\rho}}\wedge
A\indices{{}^{\rho}_{\nu}})+S_{\vphantom{\mathring{Omega}}\Omega,{\rm
GHY}}\,,\end{split}$ (41)

where we used the equations of motion of $\varphi$ to arrive at the final
equality. Once again, $S_{\Omega,{\rm GHY}}$ is the GHY term necessary to make
the variations with respect to $\Omega\indices{{}^{\mu}_{\nu}}$ well-defined.
Thus, if we wish to enforce teleparallelism consistently, we must eliminate
$S_{\Omega,{\rm GHY}}$ from $S_{\mathring{\Omega}}$ as well as set
$\Omega\indices{{}^{\mu}_{\nu}}=0$. Hence, we conclude that extensions of GR
are equivalent to extensions of (S)TEGR with action

$\displaystyle
S_{\mathring{\Omega}}=\int_{\mathcal{M}}\mathcal{L}(\mathring{\Omega}\indices{{}^{\mu}_{\nu}})+\varepsilon\int_{\partial\mathcal{M}}\left.\mathring{K}^{a}\wedge*(\varphi_{\mathbf{n}a}-\varphi_{a\mathbf{n}})\right|_{\partial\mathcal{M}}=\int_{\mathcal{M}}\mathcal{L}(-A\indices{{}^{\mu}_{\rho}}\wedge
A\indices{{}^{\rho}_{\nu}})\equiv S_{A}\,.$ (42)

The result (42) is the generalization of the geometrical trinity of gravity
advertised in the beginning of this section. In this sense, any theory build
in terms of the curvature two-form may be formulated solely through torsion or
non-metricity by simply replacing
$\mathring{\Omega}\indices{{}^{\mu}_{\nu}}\mapsto-A\indices{{}^{\mu}_{\rho}}\wedge
A\indices{{}^{\rho}_{\nu}}$ and removing the GHY term. In tensor components,
the corresponding replacement reads
$\mathring{R}\indices{{}^{\mu}_{\nu}{}_{\alpha}{}_{\beta}}\mapsto
A\indices{{}^{\mu}_{\rho}{}_{\beta}}A\indices{{}^{\rho}_{\nu}{}_{\vphantom{\beta}}{}_{\alpha}}-A\indices{{}^{\mu}_{\rho}{}_{\vphantom{\beta}}{}_{\alpha}}A\indices{{}^{\rho}_{\nu}{}_{\beta}}$,
where the components of the deformation one-form are given by (8).

Note that we have shown any theory built from curvature is equivalent to one
built in terms of torsion and non-metricity. However, one may consider
theories of gravity built on the irreducible components of curvature, torsion
or non-metricity with respect to $\mathrm{SO}(1,n-1)$ [4]. In general, these
theories have different symmetries and, hence, different degrees of freedom
and there is a priori no reason to expect any equivalence relation between
them. In this sense, our work may be used as guide on building (S)TEGR
theories that may, even in principle, make predictions differing from those of
GR. As a final comment, we note that (42) obviously reproduces the results of
section 3 if ${\cal L}$ is the Einstein-Hilbert Lagrangian.

##  5 Conclusions and outlook

In this work, we have employed the 3+1 decomposition of the connection within
metric-affine gravity to elucidate the precise role of boundary terms
appearing in the derivation of the geometrical trinity of gravity. We have
further established that the actions of TEGR and STEGR are in no need of GHY
terms in order to admit a well-defined variational formulation. In particular,
we showed that the boundary term known in the literature for TEGR and STEGR is
a difference of GHY terms. Furthermore, we have generalized the geometrical
trinity of gravity by showing that any extension of GR built on the curvature
two-form $\mathring{\Omega}\indices{{}^{\mu}_{\nu}}$ is equivalent to an
extension of (S)TEGR which we explicitly defined.

In view of our results, it will be interesting to re-evaluate existing
results, which were derived including the GHY term of GR to the (S)TEGR
Lagrangian. This will influence how we understand the thermodynamic properties
of black holes within (S)TEGR, which were previously shown to be equivalent to
those of GR precisely because of the included GHY term [8]. This is
particularly important in view of applying the fluid/gravity correspondence to
torsionful spacetimes, since the zeroth order contribution in the hydrodynamic
derivative expansion is defined entirely in terms of thermal equilibrium [57,
58]. As a precursor to establishing fluid/gravity duality for spin currents,
it is necessary to establish holographic renormalization [59, 60, 61] for
gravity theories with torsion, for which the results presented here are
expected to be useful. In addition, it will be of interest to re-examine
theories of gravity which consider generalized Lagrangians depending on the
torsion/non-metricity scalar as well as the GHY term in non-linear fashion. In
particular, it will also be of interest to check whether the reformulation of
modified theories of gravity in [11] in terms of a non-diffeomorphism
invariant decomposition of curvature is equivalent to the construction
presented here. As a final point in this regard, we can use the (S)TEGR action
(31) in order to derive a closed form for the generalized Hodge duals of
torsion and non-metricity, defined in [42, 43, 10], for arbitrary dimension.

Furthermore, the generalized geometrical trinity of gravity has interesting
implications for string theory. Namely, our proof of the equivalence may be
applied mutatis mutandi to theories of gravity obtained from string theoretic
constructions, after the matter fields have been consistently integrated out
of the low-energy spectrum. This suggests we may reformulate the graviton
dynamics withing string theory in terms of torsion or non-metricity degrees of
freedom. It would be interesting to carry out this reformulation explicitly in
textbook string and supergravity theories [62, 63, 64, 56].

An additional interesting venue of research are topological theories of
gravity. In view of our generalized geometrical trinity of gravity, the
existence of these topological theories of gravity expressed in terms of
curvature implies the existence of topological theories of torsion or non-
metricity. It would be interesting to derive and examine the properties of
such topological theories. We note that a first step in this direction has
already been taken for Gauss-Bonnet gravity, within the setting of
teleparallel gravity. Namely, [65] have used the Gauss-Bonnet Lagrangian to
derive a topological invariant based entirely on torsion. Similarly, such
invariants should exist for symmetric teleparallel gravity, as well as for
topological actions containing higher powers of the curvature two-form.

Finally, note we considered the cases of spacelike and timelike boundaries.
However, spacetimes may also have boundaries with a lightlike normal vector,
$n_{\mu}n^{\mu}=0$. These are particularly important from the point of view of
gauge/gravity duality, since hypersurfaces with lightlike boundaries are used
to compute quantum complexity [66, 67, 68]. Furthermore, lightlike
hypersurfaces are important in spacetimes containing black holes, since the
black hole horizons may be interpreted as a lightlike boundary. Hence, it
would be interesting to extend our analysis to the case of lightlike
boundaries based on the formalisms in [69, 70].

##  6 Acknowledgements

We are grateful for the funding by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) through Project-ID 258499086—SFB 1170
?ToCoTronics?, through the Würzburg-Dresden Cluster of Excellence on
Complexity and Topology in Quantum Matter – ct.qmat Project-ID 390858490—EXC
2147, and in part through the German-Israeli Project Cooperation (DIP) grant
”Holography and the Swampland”. The research of B.H. is funded by the
Bundesministerium für Bildung und Forschung (BMBF, German Federal Ministry of
Education and Research) through the Cusanuswerk - Bischöfliche
Studienförderung. I.M.’s research is supported by the ?Curiosity Driven Grant
2020? of the University of Genova and by the INFN Scientific Initiatives SFT:
?Statistical Field Theory, Low-Dimensional Systems, Integrable Models and
Applications?.

##  Appendix A Tensor component formulation of the (S)TEGR results

The results of section 3 translate to the familiar tensor component
formulations of (S)TEGR qualitatively. The transformation of the Einstein-
Hilbert action to (S)TEGR may be found in many references, see [7, 39] for
example. We will therefore not repeat the derivation at this point. However,
the connection of the (S)TEGR boundary term to the GHY term proceeds different
as in differential form language. Since the qualitative results coincide with
those presented in section 3, we only give their derivation in tensorial
language for TEGR. They generalize to STEGR similarly.

We adapt the notations from [7]. Recall that the Riemannian Ricci tensor
$\mathring{R}$ transforms to the torsion scalar $T$ and the TEGR boundary term
$B\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=2\mathring{\nabla}_{\mu}T\indices{{}^{\nu}_{\nu}{}^{\mu}}$
as

$\displaystyle\mathring{R}=R-T+B\,.$ (43)

To convert the integral of $B$ into a boundary term, we invoke Stokes’ theorem
and obtain

$\displaystyle\int_{\mathcal{M}}\mathrm{d}\mathrm{Vol}_{\mathcal{M}}\sqrt{|g|}B=\int_{\partial\mathcal{M}}\mathrm{d}\mathrm{Vol}_{\partial\mathcal{M}}\sqrt{|\gamma|}2\varepsilon
n^{\mu}T\indices{{}^{\nu}_{\nu}{}_{\mu}}\,.$ (44)

Showing the connection of the (S)TEGR boundary term and the GHY term thus is
the same as performing the 3+1 decomposition of
$n_{\mu}T\indices{{}^{\nu}_{\nu}{}^{\mu}}$ which we do next. Decomposing the
indices of the torsion tensor by means of (18) yields

$\displaystyle
n^{\mu}T\indices{{}^{\nu}_{\nu}{}_{\mu}}=e^{a}_{\nu}e_{a}^{\rho}n^{\mu}T\indices{{}^{\nu}_{\rho}{}_{\mu}}$
(45)

since $n_{\mu}n^{\nu}n^{\rho}T\indices{{}^{\mu}_{\nu}{}_{\rho}}$ vanishes due
to the antisymmetry of $T$. From [5] we use the 3+1 decomposition of torsion
in the form

$\displaystyle e^{a}_{\mu}T^{\mu}=T^{a}+NK^{a}\wedge\phi\,,$ (46)

where $T^{a}\equiv D\phi^{a}$ is the boundary torsion two-form and $\phi$ and
$N$ have been defined in (10). This decomposition implies that its tensor
components transform to the boundary as

$\displaystyle
e^{a}_{\nu}e_{b}^{\rho}n^{\mu}T\indices{{}^{\nu}_{\rho}{}_{\mu}}=e^{\rho}_{b}n^{\mu}T\indices{{}^{a}_{\rho}{}_{\mu}}+K\indices{{}^{a}_{b}}\,.$
(47)

The Riemannian limit of the latter equation yields

$\displaystyle
e^{\rho}_{b}n^{\mu}T\indices{{}^{a}_{\rho}{}_{\mu}}=-\mathring{K}\indices{{}^{a}_{b}}\,.$
(48)

We insert these results into (44) to obtain

$\displaystyle\int_{\mathcal{M}}\mathrm{d}\mathrm{Vol}_{\mathcal{M}}\sqrt{|g|}B=2\varepsilon\int_{\partial\mathcal{M}}\mathrm{d}\mathrm{Vol}_{\partial\mathcal{M}}\sqrt{|\gamma|}\left(K\indices{{}^{a}_{a}}-\mathring{K}\indices{{}^{a}_{a}}\right)\,.$
(49)

This is the transition of the GHY term of the Lagrangian $\mathring{R}$ to the
one of $R$ in components. Analogous to the differential geometric formulation
in section 3 we transition from the GR action to the TEGR action in components
by setting $R=0$. Since the GHY term is there to cancel the boundary terms
which are non-vanishing under variation of the action, it introduces non-
vanishing boundary terms in the variational calculation if $R=0$ is imposed.
In this way it makes the variational problem ill-defined again. Thus, we need
to cancel the GHY term in the transition to TEGR additional to imposing $R=0$.
Hence, we find the same behavior as in the differential form calculation in
section 3. Of course, this result also implies that a GHY term must be
included in the Riemannian Einstein-Hilbert action before transforming to
(S)TEGR. Hence, the transformed Einstein-Hilbert action in tensor components
reads

$\displaystyle\begin{split}S^{\mathrm{EH}}&=\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{dVol}_{\mathcal{M}}\sqrt{|g|}\mathring{R}+\frac{\varepsilon}{\kappa}\int_{\partial\mathcal{M}}\mathrm{dVol}_{\partial\mathcal{M}}\sqrt{|\gamma|}\mathring{K}\indices{{}^{a}_{a}}\\\
&=\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{dVol}_{\mathcal{M}}\sqrt{|g|}(R-T+B)+\frac{\varepsilon}{\kappa}\int_{\partial\mathcal{M}}\mathrm{dVol}_{\partial\mathcal{M}}\sqrt{|\gamma|}\mathring{K}\indices{{}^{a}_{a}}\\\
&=\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{dVol}_{\mathcal{M}}\sqrt{|g|}(R-T)+\frac{\varepsilon}{\kappa}\int_{\partial\mathcal{M}}\mathrm{dVol}_{\partial\mathcal{M}}\sqrt{|\gamma|}K\indices{{}^{a}_{a}}\,.\end{split}$
(50)

TEGR is therefore completely described by the action

$\displaystyle
S^{\mathrm{TEGR}}=-\frac{1}{2\kappa}\int_{\mathcal{M}}\mathrm{dVol}_{\mathcal{M}}\sqrt{|g|}\,T\,.$
(51)

This result proves that (51) is the full TEGR action. Schematically, (51)
means that the GHY term needs to be canceled in the transition from general
relativity to TEGR while no TEGR boundary term $B$ is needed. The results of
the tensor component formulation of TEGR thus coincide with the differential
form formulation in section (3).

## References

  * [1] Steven Weinberg  “Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity”  New York: John WileySons, 1972
  * [2] S.. Hawking and G… Ellis  “The Large Scale Structure of Space-Time”, Cambridge Monographs on Mathematical Physics  Cambridge University Press, 1973  DOI: 10.1017/CBO9780511524646
  * [3] Robert M. Wald  “General Relativity”  Chicago, USA: Chicago Univ. Pr., 1984  DOI: 10.7208/chicago/9780226870373.001.0001
  * [4] Friedrich W. Hehl, J.Dermott McCrea, Eckehard W. Mielke and Yuval Ne’eman  “Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance”  In _Physics Reports_ 258.1, 1995, pp. 1–171  DOI: 10.1016/0370-1573(94)00111-F
  * [5] Johanna Erdmenger, Bastian Heß, Ioannis Matthaiakakis and René Meyer  “Universal Gibbons-Hawking-York term for theories with curvature, torsion and non-metricity”  In _accepted for publication in SciPost Physics_ , 2022  arXiv:2211.02064 [hep-th]
  * [6] Sebastian Bahamonde et al.  “Teleparallel gravity: from theory to cosmology”  In _Rept. Prog. Phys._ 86.2, 2023  DOI: 10.1088/1361-6633/ac9cef
  * [7] Sebastian Bahamonde et al.  “Cosmological perturbations in modified teleparallel gravity models: Boundary term extension”  In _Eur. Phys. J. C_ 81.1, 2021, pp. 53  DOI: 10.1140/epjc/s10052-021-08833-2
  * [8] Gregorio Paci  “Black Hole Thermodynamics and Boundary Terms in Teleparallel Gravity”  University of Bologna, 2021
  * [9] E. Saridakis et al.  “Modified Gravity and Cosmology: An Update by the CANTATA Network: An Update by the CANTATA Network”, 2021  DOI: 10.1007/978-3-030-83715-0
  * [10] Ruben Aldrovandi and José Geraldo Pereira  “Teleparallel Gravity: An Introduction”  Springer, 2013  DOI: 10.1007/978-94-007-5143-9
  * [11] Christian G. Boehmer and Erik Jensko  “Modified gravity: A unified approach”  In _Phys. Rev. D_ 104.2, 2021, pp. 024010  DOI: 10.1103/PhysRevD.104.024010
  * [12] Juan Maldacena  “The Large-N Limit of Superconformal Field Theories and Supergravity”  In _International Journal of Theoretical Physics_ 38.4 Springer ScienceBusiness Media LLC, 1999, pp. 1113–1133  DOI: 10.1023/a:1026654312961
  * [13] Edward Witten  “Anti-de Sitter space and holography”  In _Adv. Theor. Math. Phys._ 2, 1998, pp. 253–291  DOI: 10.4310/ATMP.1998.v2.n2.a2
  * [14] S.S. Gubser, Igor R. Klebanov and Alexander M. Polyakov  “Gauge theory correlators from noncritical string theory”  In _Phys. Lett. B_ 428, 1998, pp. 105–114  DOI: 10.1016/S0370-2693(98)00377-3
  * [15] Martin Ammon and Johanna Erdmenger  “Gauge/gravity duality: Foundations and applications”  Cambridge: Cambridge University Press, 2015  DOI: 10.1017/CBO9780511846373
  * [16] P.. Kovtun, D.. Son and A.. Starinets  “Viscosity in Strongly Interacting Quantum Field Theories from Black Hole Physics”  In _Phys. Rev. Lett._ 94 American Physical Society, 2005, pp. 111601  DOI: 10.1103/PhysRevLett.94.111601
  * [17] Christopher P. Herzog, Pavel Kovtun, Subir Sachdev and Dam Thanh Son  “Quantum critical transport, duality, and M theory”  In _Phys. Rev. D_ 75 American Physical Society, 2007, pp. 085020  DOI: 10.1103/PhysRevD.75.085020
  * [18] Christos Charmousis et al.  “Effective holographic theories for low-temperature condensed matter systems”  In _Journal of High Energy Physics_ 2010.11, 2010, pp. 151  DOI: 10.1007/JHEP11(2010)151
  * [19] Johanna Erdmenger, Ioannis Matthaiakakis, René Meyer and David Rodriguez Fernández  “Strongly coupled electron fluids in the Poiseuille regime”  In _Phys. Rev. B_ 98 American Physical Society, 2018, pp. 195143  DOI: 10.1103/PhysRevB.98.195143
  * [20] Domenico Di Sante et al.  “Turbulent hydrodynamics in strongly correlated Kagome metals”, 2019  arXiv:1911.06810 [cond-mat.str-el]
  * [21] Robert G. Leigh, Nam N. Hoang and Anastasios C. Petkou  “Torsion and the Gravity Dual of Parity Symmetry Breaking in AdS(4) / CFT(3) Holography”  In _JHEP_ 03, 2009, pp. 033  DOI: 10.1088/1126-6708/2009/03/033
  * [22] Luca Ciambelli and Robert G. Leigh  “Weyl Connections and their Role in Holography”  In _Phys. Rev. D_ 101.8, 2020  DOI: 10.1103/PhysRevD.101.086020
  * [23] Damianos Iosifidis, Christos G. Tsagas and Anastasios C. Petkou  “Raychaudhuri equation in spacetimes with torsion and nonmetricity”  In _Phys. Rev. D_ 98.10, 2018, pp. 104037  DOI: 10.1103/PhysRevD.98.104037
  * [24] A.. Gallegos and U. Gürsoy  “Holographic spin liquids and Lovelock Chern-Simons gravity”  In _JHEP_ 11, 2020, pp. 151  DOI: 10.1007/JHEP11(2020)151
  * [25] Johanna Erdmenger, Michael Haack, Matthias Kaminski and Amos Yarom  “Fluid dynamics of R-charged black holes”  In _Journal of High Energy Physics_ 2009.01 Springer ScienceBusiness Media LLC, 2009  DOI: 10.1088/1126-6708/2009/01/055
  * [26] Martin Ammon et al.  “Quasinormal modes of charged magnetic black branes & chiral magnetic transport”  In _JHEP_ 04, 2017, pp. 067  DOI: 10.1007/JHEP04(2017)067
  * [27] Yago Ferreiros, Yaron Kedem, Emil J. Bergholtz and Jens H. Bardarson  “Mixed Axial-Torsional Anomaly in Weyl Semimetals”  In _Phys. Rev. Lett._ 122 American Physical Society, 2019, pp. 056601  DOI: 10.1103/PhysRevLett.122.056601
  * [28] Martin Ammon et al.  “Chiral hydrodynamics in strong external magnetic fields”  In _JHEP_ 04, 2021, pp. 078  DOI: 10.1007/JHEP04(2021)078
  * [29] Casey Cartwright et al.  “Convergence of hydrodynamics in rapidly spinning strongly coupled plasma”, 2021  DOI: 10.48550/ARXIV.2112.10781
  * [30] Martin Ammon et al.  “Pseudo-spontaneous $U(1)$ symmetry breaking in hydrodynamics and holography”  In _JHEP_ 03, 2022, pp. 015  DOI: 10.1007/JHEP03(2022)015
  * [31] Masaru Hongo et al.  “Spin relaxation rate for heavy quarks in weakly coupled QCD plasma”  In _JHEP_ 08, 2022, pp. 263  DOI: 10.1007/JHEP08(2022)263
  * [32] Zhuo-Yu Xian et al.  “Hall viscosity and hydrodynamic inverse Nernst effect in graphene”, 2022  arXiv:2207.10528 [cond-mat.mes-hall]
  * [33] Veronika E. Hubeny, Shiraz Minwalla and Mukund Rangamani  “The fluid/gravity correspondence”  In _Black holes in higher dimensions_ , 2012, pp. 348–383  DOI: 10.48550/ARXIV.1107.5780
  * [34] Sayantani Bhattacharyya, Veronika E Hubeny, Shiraz Minwalla and Mukund Rangamani  “Nonlinear Fluid Dynamics from Gravity”  In _JHEP_ 02, 2008, pp. 045  DOI: 10.1088/1126-6708/2008/02/045
  * [35] Mukund Rangamani  “Gravity and hydrodynamics: lectures on the fluid-gravity correspondence”  In _Classical and Quantum Gravity_ 26.22 IOP Publishing, 2009  DOI: 10.1088/0264-9381/26/22/224003
  * [36] Lavinia Heisenberg, Manuel Hohmann and Simon Kuhn  “Homogeneous and isotropic cosmology in general teleparallel gravity”, 2022  arXiv:2212.14324 [gr-qc]
  * [37] Manuel Hohmann  “Teleparallel gravity”, 2022  arXiv:2207.06438 [gr-qc]
  * [38] James M. Nester and Hwei-Jang Yo  “Symmetric teleparallel general relativity”  In _Chin. J. Phys._ 37, 1999, pp. 113  arXiv:gr-qc/9809049
  * [39] Laur Järv, Mihkel Rünkla, Margus Saal and Ott Vilson  “Nonmetricity formulation of general relativity and its scalar-tensor extension”  In _Phys. Rev. D_ 97.12, 2018  DOI: 10.1103/PhysRevD.97.124025
  * [40] Manuel Hohmann  “Gauge-Invariant Post-Newtonian Perturbations in Symmetric Teleparallel Gravity”  In _Astron. Rep._ 65.10, 2021, pp. 952–956  DOI: 10.1134/S1063772921100140
  * [41] Muzaffer Adak  “The Symmetric teleparallel gravity”  In _Turk. J. Phys._ 30, 2006, pp. 379–390  arXiv:gr-qc/0611077
  * [42] Tiago Gribl Lucas and J.. Pereira  “Hodge Dual for Soldered Bundles”  In _J. Phys. A_ 42, 2009  DOI: 10.1088/1751-8113/42/3/035402
  * [43] Peng Huang and Fang-Fang Yuan  “Generalized Hodge dual for torsion in teleparallel gravity”  In _Int. J. Geom. Meth. Mod. Phys._ 13.09, 2016  DOI: 10.1142/S0219887816501061
  * [44] Jose Beltrán Jiménez et al.  “General teleparallel quadratic gravity”  In _Physics Letters B_ 805, 2020, pp. 135422  DOI: https://doi.org/10.1016/j.physletb.2020.135422
  * [45] James W. York  “Role of Conformal Three-Geometry in the Dynamics of Gravitation”  In _Phys. Rev. Lett._ 28 American Physical Society, 1972, pp. 1082–1085  DOI: 10.1103/PhysRevLett.28.1082
  * [46] G.. Gibbons and S.. Hawking  “Action integrals and partition functions in quantum gravity”  In _Phys. Rev. D_ 15 American Physical Society, 1977, pp. 2752–2756  DOI: 10.1103/PhysRevD.15.2752
  * [47] Jose Beltrán Jiménez, Lavinia Heisenberg and Tomi S. Koivisto  “The Geometrical Trinity of Gravity”  In _Universe_ 5.7, 2019  DOI: 10.3390/universe5070173
  * [48] Élie Cartan  “Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion”  In _Comptes Rendus, Ac. Sc. Paris_ 174, 1922, pp. 593–595
  * [49] Élie Cartan  “Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie)”  In _Annales scientifiques de l’École Normale Supérieure_ 3e série, 40 Elsevier, 1923, pp. 325–412  DOI: 10.24033/asens.751
  * [50] Élie Cartan  “Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie) (Suite)”  In _Annales scientifiques de l’École Normale Supérieure_ 3e série, 41 Elsevier, 1924, pp. 1–25  DOI: 10.24033/asens.753
  * [51] Élie Cartan  “Sur les variétés à connexion affine, et la théorie de la relativité généralisée (deuxième partie)”  In _Annales scientifiques de l’École Normale Supérieure_ 3e série, 42 Elsevier, 1925, pp. 17–88  DOI: 10.24033/asens.761
  * [52] D.. Sciama  “The Physical Structure of General Relativity”  In _Rev. Mod. Phys._ 36 American Physical Society, 1964, pp. 463–469  DOI: 10.1103/RevModPhys.36.463
  * [53] T… Kibble  “Lorentz Invariance and the Gravitational Field”  In _Journal of Mathematical Physics_ 2.2, 1961, pp. 212–221  DOI: 10.1063/1.1703702
  * [54] Andrzej Trautman  “Einstein-Cartan theory”  In _Encyclopedia of Mathematical Physics_ 2 Oxford: Elsevier, 2006, pp. 189–195  DOI: 10.48550/ARXIV.GR-QC/0606062
  * [55] Nathalie Deruelle, Misao Sasaki, Yuuiti Sendouda and Daisuke Yamauchi  “Hamiltonian formulation of f(Riemann) theories of gravity”  In _Prog. Theor. Phys._ 123, 2010, pp. 169–185  DOI: 10.1143/PTP.123.169
  * [56] Daniel Z. Freedman and Antoine Van Proeyen  “Supergravity”  Cambridge, UK: Cambridge University Press, 2012  DOI: 10.1017/CBO9781139026833
  * [57] P.. Kovtun  “Lectures on hydrodynamic fluctuations in relativistic theories”  In _J. Phys. A_ 45, 2012  DOI: 10.1088/1751-8113/45/47/473001
  * [58] Ioannis Matthaiakakis  “Hydrodynamics in Solid State Systems and the AdS/CFT correspondence”, 2021  DOI: 10.25972/OPUS-24439
  * [59] Massimo Bianchi, Daniel Z. Freedman and Kostas Skenderis  “Holographic renormalization”  In _Nucl. Phys. B_ 631, 2002, pp. 159–194  DOI: 10.1016/S0550-3213(02)00179-7
  * [60] Sebastian Haro, Sergey N. Solodukhin and Kostas Skenderis  “Holographic reconstruction of space-time and renormalization in the AdS / CFT correspondence”  In _Commun. Math. Phys._ 217, 2001, pp. 595–622  DOI: 10.1007/s002200100381
  * [61] Kostas Skenderis  “Lecture notes on holographic renormalization”  In _Class. Quant. Grav._ 19, 2002, pp. 5849–5876  DOI: 10.1088/0264-9381/19/22/306
  * [62] Joseph Polchinski  “String Theory” 1, Cambridge Monographs on Mathematical Physics  Cambridge University Press, 1998  DOI: 10.1017/CBO9780511816079
  * [63] Katrin Becker, Melanie Becker and John H. Schwarz  “String Theory and M-Theory: A Modern Introduction”  Cambridge University Press, 2006  DOI: 10.1017/CBO9780511816086
  * [64] Elias Kiritsis  “String theory in a nutshell”  USA: Princeton University Press, 2019
  * [65] Georgios Kofinas and Emmanuel N. Saridakis  “Teleparallel equivalent of Gauss-Bonnet gravity and its modifications”  In _Phys. Rev. D_ 90, 2014, pp. 084044  DOI: 10.1103/PhysRevD.90.084044
  * [66] Adam R. Brown et al.  “Holographic Complexity Equals Bulk Action?”  In _Phys. Rev. Lett._ 116 American Physical Society, 2016, pp. 191301  DOI: 10.1103/PhysRevLett.116.191301
  * [67] Josiah Couch, Willy Fischler and Phuc H. Nguyen  “Noether charge, black hole volume, and complexity”  In _Journal of High Energy Physics_ 2017.3, 2017, pp. 119  DOI: 10.1007/JHEP03(2017)119
  * [68] Alexandre Belin et al.  “Complexity equals anything II”  In _JHEP_ 01, 2023, pp. 154  DOI: 10.1007/JHEP01(2023)154
  * [69] Eric Gourgoulhon and Jose Luis Jaramillo  “A 3+1 perspective on null hypersurfaces and isolated horizons”  In _Phys. Rept._ 423, 2006, pp. 159–294  DOI: 10.1016/j.physrep.2005.10.005
  * [70] Luis Lehner, Robert C. Myers, Eric Poisson and Rafael D. Sorkin  “Gravitational action with null boundaries”  In _Phys. Rev. D_ 94.8, 2016, pp. 084046  DOI: 10.1103/PhysRevD.94.084046