#  Conserved quantities for the plane waves in TEGR and STEGR

E. D. Emtsova Electronic address<EMAIL_ADDRESS>Kazan Federal
University, Kremlevskaya 18, Kazan, 420008, Russia  A. N. Petrov Electronic
address<EMAIL_ADDRESS>Sternberg Astronomical institute, MV
Lomonosov State university
Universitetskii pr., 13, Moscow, 119992, Russia  A. V. Toporensky Electronic
address<EMAIL_ADDRESS>Kazan Federal University, Kremlevskaya 18, Kazan,
420008, Russia  Sternberg Astronomical institute, MV Lomonosov State
university
Universitetskii pr., 13, Moscow, 119992, Russia

(August 28, 2024)

###### Abstract

We study the energy-momentum characteristics of the plane “+”-polarised
gravitational wave solution of general relativity in the Teleparallel
Equivalent of General Relativity (TEGR) and the Symmetric Teleparallel
Equivalent of General Relativity (STEGR) using the previously constructed
Noether currents. These currents can describe locally measured by observer
energy-momentum if the displacement vector $\xi$ is equal to the observer’s
4-velocity. To determine the non-dynamical connection in these theories we use
the unified “turning off” gravity principle. For a constructive analysis of
the values of Noether currents and superpotentials in TEGR and STEGR, we use
the concept of “gauges”. The gauge changing can affect the Noether current
values. We study under what conditions the Noether current for the freely
falling observer is zero because this can be interpreted as the equivalence
principle. We highlight two important cases with positive and zero energy,
which reproduce the results of previous works with a different approach to
determine gravitational energy-momentum in TEGR, and give their
interpretation.

##  1 Introduction

Teleparallel theories of gravity have been actively developed in recent years.
A main feature of these theories is the use of connection with zero Riemann
curvature. These theories include the Teleparallel Equivalent of General
Relativity (TEGR), the Symmetric Teleparallel Equivalent of General Relativity
(STEGR) and modifications of these theories [1, 2, 3, 4, 5, 6]. In TEGR and
its modifications, a flat metric compatible connection is used. In STEGR and
its modifications, a flat connection with zero torsion is used. TEGR and STEGR
are fully equivalent to GR (General Relativity) at the level of field
equations, thus, the solutions of the field equations in TEGR and STEGR are
exactly the same as those in GR. Modifications of TEGR and STEGR have the
advantage that their field equations are of the second-order, what gives
similarities with gauge field theories and potentially links gravity to other
theories of fundamental interactions in nature.

The teleparallel connections in TEGR and STEGR are not dynamical quantities
and cannot be determined by field equations. This fact does not influence the
dynamics of gravitational interacting objects but gives ambiguities to values
of main quantities which define gravitational energy-momentum [7, 8].

Despite the fact that now the main attention is paid to how accurate the
modified teleparallel theories can describe the observed phenomena, not all
the issues have been resolved in TEGR and STEGR themselves. One of such issues
is definition of gravitational energy-momentum. There are different approaches
to this problem. Various approaches in constructing conserved quantities both
in TEGR and in STEGR already have been tested to construct them for the
Schwarzschild solution and cosmological models [9, 10, 11, 12, 13, 14, 15, 16,
17].

However, for the best of our knowledge, construction of conserved quantities
for the gravitational waves in TEGR did not attract much attention of
researches. We could cite only the central papers [18, 19, 20, 21, 22] with a
few references therein. In calculations the energy-momentum tensor of
gravitational field [3] is applied. By these studies, authors sometimes got
unexpected results. Thus, for example, in [19] non-positive energy for
gravitational waves is obtained; on the other side, in [20] zero energy for
gravitational waves is obtained. Such results are not acceptable; indeed, the
modern cosmological and astrophysical observable data supports the textbook
predictions that gravitational waves bring a positive energy [23, 24, 25] At
last, we have not find in a literature any works where gravitational waves
have been considered in STEGR.

Thus, the main purpose of this paper is to discuss the aforementioned
questions and problems, and try to close the remarked gaps. We use the totally
covariant formalism developed by us [26, 7, 8]. The paper is organized as
follows:

In section 2, we give brief introduction into TEGR and STEGR theories,
introduce the Noether currents and superpotentials for these theories and
describe the unified “turning off” gravity principle to define the
teleparallel connection.

In section 3, we find energy and momentum of the plane +polarised wave
measured by the freely falling observers in TEGR and STEGR in the simplest
frames.

In section 4, we find the conditions when Noether currents measured by the
freely falling observer in TEGR and STEGR are in a correspondence with the
equivalence principle.

In section 5, we take another coordinates for the plane wave, reproduce the
results of [20] and equivalence principle for the freely falling observer.

In section 6, we discuss the results

All definitions in TEGR correspond to [3]. All definitions in STEGR correspond
to [1].

##  2 Elements of teleparallel gravity

In this section, we briefly introduce the elements of teleparallel equivalents
of GR (TEGR and STEGR) and, following [27, 7, 8] give totally covariant
expressions for conserved quantities, which are necessary for our
calculations.

###  2.1 Elements of TEGR

The Lagrangian in TEGR has a form [3]

$\stackrel{{\scriptstyle\bullet}}{{{{\cal
L}}}}=\frac{h}{{2}\kappa}{\stackrel{{\scriptstyle\bullet}}{{T}}}{}\,\equiv\frac{h}{2\kappa}\left(\frac{1}{4}{{\stackrel{{\scriptstyle\bullet}}{{T}}}{}}{}^{\rho}{}_{\mu\nu}{{\stackrel{{\scriptstyle\bullet}}{{T}}}{}}_{\rho}{}^{\mu\nu}+\frac{1}{2}{{\stackrel{{\scriptstyle\bullet}}{{T}}}{}}{}^{\rho}{}_{\mu\nu}{{\stackrel{{\scriptstyle\bullet}}{{T}}}{}}{}^{\nu\mu}{}_{\rho}-{{\stackrel{{\scriptstyle\bullet}}{{T}}}{}}{}^{\rho}{}_{\mu\rho}{{\stackrel{{\scriptstyle\bullet}}{{T}}}{}}{}^{\nu\mu}{}_{\nu}\right),$
(2.1)

where ${\stackrel{{\scriptstyle\bullet}}{{T}}}{}$ is called as the torsion
scalar, $\kappa=8\pi G$ (in $c=1$ units), and $h=\det h^{a}{}_{\nu}$. The
torsion tensor ${{\stackrel{{\scriptstyle\bullet}}{{T}}}{}}{}^{a}{}_{\mu\nu}$
is defined as

${{\stackrel{{\scriptstyle\bullet}}{{T}}}{}}{}^{a}{}_{\mu\nu}=\partial_{\mu}h^{a}{}_{\nu}-\partial_{\nu}h^{a}{}_{\mu}+{{\stackrel{{\scriptstyle\bullet}}{{A}}}{}}{}^{a}{}_{c\mu}h^{c}{}_{\nu}-{{\stackrel{{\scriptstyle\bullet}}{{A}}}{}}{}^{a}{}_{c\nu}h^{c}{}_{\mu},$
(2.2)

where $h^{a}{}_{\nu}$ are the tetrad components (we denote tetrad indexes of a
quantity by Latin letters and spacetime indexes by Greek letters), connected
to metric by

$g_{\mu\nu}=\eta_{ab}h^{a}{}_{\mu}h^{b}{}_{\nu}.$ (2.3)

The inertial spin connection (ISC)
${{\stackrel{{\scriptstyle\bullet}}{{A}}}{}}{}^{a}{}_{c\nu}$ is not included
into the Lagrangian (2.1) and related field equations, but it covariantizes
the expression for the torsion tensor (2.2) and other expressions. It can be
suppressed by a related local Lorentz transformation.

From now and below, we denote by $\bullet$ all teleparallel quantities, which
are constructed using the teleparallel connection
$\stackrel{{\scriptstyle\bullet}}{{\Gamma}}{}{}^{\alpha}{}_{\kappa\lambda}$.
Thus, by definition

${\stackrel{{\scriptstyle\bullet}}{{A}}}{}{}^{a}{}_{b\mu}=-h_{b}{}^{\nu}\stackrel{{\scriptstyle\bullet}}{{\nabla}}_{\mu}h^{a}{}_{\nu},$
(2.4)

where the covariant derivative
$\stackrel{{\scriptstyle\bullet}}{{\nabla}}_{\mu}$ is defined by
$\stackrel{{\scriptstyle\bullet}}{{\Gamma}}{}{}^{\alpha}{}_{\kappa\lambda}$.
The connection
$\stackrel{{\scriptstyle\bullet}}{{\Gamma}}{}{}^{\alpha}{}_{\kappa\lambda}$ is
flat, for which the corresponding curvature is equal to zero:

$\stackrel{{\scriptstyle\bullet}}{{R}}{}{}^{\alpha}{}_{\beta\mu\nu}\equiv\partial_{\mu}\stackrel{{\scriptstyle\bullet}}{{\Gamma}}{}^{\alpha}{}_{\beta\nu}-\partial_{\nu}\stackrel{{\scriptstyle\bullet}}{{\Gamma}}{}^{\alpha}{}_{\beta\mu}+\stackrel{{\scriptstyle\bullet}}{{\Gamma}}{}^{\alpha}{}_{\kappa\mu}\stackrel{{\scriptstyle\bullet}}{{\Gamma}}{}^{\kappa}{}_{\beta\nu}-\stackrel{{\scriptstyle\bullet}}{{\Gamma}}{}^{\alpha}{}_{\kappa\nu}\stackrel{{\scriptstyle\bullet}}{{\Gamma}}{}^{\kappa}{}_{\beta\mu}=0,$
(2.5)

$\stackrel{{\scriptstyle\bullet}}{{\Gamma}}{}{}^{\alpha}{}_{\kappa\lambda}$ is
also compatible with the physical metric, that is, the corresponding non-
metricity is zero:

$\stackrel{{\scriptstyle\bullet}}{{Q}}{}_{\mu\alpha\beta}\equiv\stackrel{{\scriptstyle\bullet}}{{\nabla}}{}_{\mu}g_{\alpha\beta}=0.$
(2.6)

In our notations, following to [3], quantities denoted by a $\circ$ are
constructed with the use of the Levi-Civita connection
$\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}{}^{\alpha}{}_{\kappa\lambda}$.
Thus, ${\stackrel{{\scriptstyle\circ}}{{A}}}{}^{a}{}_{b\rho}$ is the usual
Levi-Civita spin connection (L-CSC) defined by

${\stackrel{{\scriptstyle\circ}}{{A}}}{}{}^{a}{}_{b\mu}=-h_{b}{}^{\nu}\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\mu}h^{a}{}_{\nu},$
(2.7)

where the covariant derivative
$\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\mu}$ is constructed with
$\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}{}^{\alpha}{}_{\kappa\lambda}$. Now
it is useful to introduce a contortion tensor defined as:

${\stackrel{{\scriptstyle\bullet}}{{K}}}{}^{a}{}_{b\rho}={\stackrel{{\scriptstyle\bullet}}{{A}}}{}^{a}{}_{b\rho}-{\stackrel{{\scriptstyle\circ}}{{A}}}{}^{a}{}_{b\rho}.$
(2.8)

Besides, it is useful to recall that the transformation of tetrad indices into
spacetime ones and vice versa is performed by contraction with tetrad vectors,
for example,
$\stackrel{{\scriptstyle\circ}}{{R}}{}^{a}{}_{b\mu\nu}=h^{a}{}_{\alpha}h^{\beta}{}_{b}\stackrel{{\scriptstyle\circ}}{{R}}{}^{\alpha}{}_{\beta\mu\nu}$;
${\stackrel{{\scriptstyle\bullet}}{{T}}}{}{}^{\alpha}{}_{\mu\nu}=h^{\alpha}{}_{a}{\stackrel{{\scriptstyle\bullet}}{{T}}}{}{}^{a}{}_{\mu\nu}$,
etc.

Thus, the contortion tensor can be rewritten in the convenient form:

${\stackrel{{\scriptstyle\bullet}}{{K}}}{}^{\rho}{}_{\mu\nu}=\frac{1}{2}({\stackrel{{\scriptstyle\bullet}}{{T}}}{}_{\mu}{}^{\rho}{}_{\nu}+{\stackrel{{\scriptstyle\bullet}}{{T}}}{}_{\nu}{}^{\rho}{}_{\mu}-{\stackrel{{\scriptstyle\bullet}}{{T}}}{}^{\rho}{}_{\mu\nu}).$
(2.9)

The torsion scalar in (2.1) can be rewritten in the form:

${\stackrel{{\scriptstyle\bullet}}{{T}}}{}=\frac{1}{2}\,{{\stackrel{{\scriptstyle\bullet}}{{S}}}{}}_{a}{}^{\rho\sigma}{{\stackrel{{\scriptstyle\bullet}}{{T}}}{}}{}^{a}{}_{\rho\sigma},$
(2.10)

where the teleparallel superpotential
${{\stackrel{{\scriptstyle\bullet}}{{S}}}{}}_{a}{}^{\rho\sigma}$ defined as

${{\stackrel{{\scriptstyle\bullet}}{{S}}}{}}_{a}{}^{\rho\sigma}={\stackrel{{\scriptstyle\bullet}}{{K}}}{}{}^{\rho\sigma}{}_{a}+h_{a}{}^{\sigma}{\stackrel{{\scriptstyle\bullet}}{{K}}}{}{}^{\theta\rho}{}_{\theta}-h_{a}{}^{\rho}{\stackrel{{\scriptstyle\bullet}}{{K}}}{}{}^{\theta\sigma}{}_{\theta}\,$
(2.11)

is an antisymmetric tensor in the last two indices.

All tensors ${{\stackrel{{\scriptstyle\bullet}}{{T}}}{}}{}^{a}{}_{\mu\nu}$,
${\stackrel{{\scriptstyle\bullet}}{{K}}}{}^{\rho\sigma}{}_{a}$, and
${{\stackrel{{\scriptstyle\bullet}}{{S}}}{}}_{a}{}^{\rho\sigma}$, are
covariant with respect to both coordinate transformations and local Lorentz
transformations. (For tensors without Lorentz indexes or scalars we say
“invariant with respect to Lorentz transformations”.) Note, that working in
the covariant formulation of teleparallel gravity, the local Lorentz
covariance means that the tensorial quantities are transformed covariantly
under the simultaneous transformation of both the tetrad and the ISC:

$h^{\prime a}{}_{\mu}=\Lambda^{a}{}_{b}(x)h^{b}{}_{\mu}\,,$ (2.12)
${\stackrel{{\scriptstyle\bullet}}{{A}}}{}{}^{\prime}{}^{a}{}_{b\mu}=\Lambda{}^{a}{}_{c}(x){\stackrel{{\scriptstyle\bullet}}{{A}}}{}{}{}^{c}{}_{d\mu}\Lambda{}_{b}{}^{d}(x)+\Lambda{}^{a}{}_{c}(x)\partial_{\mu}\Lambda{}_{b}{}^{c}(x),$
(2.13)

where $\Lambda{}^{a}{}_{c}(x)$ is the matrix of a local Lorentz rotation, and
$\Lambda{}_{a}{}^{c}(x)$ is an inverse matrix of the latter. The operation at
the right hand side of (2.13) tells us that ISC can be equalized to zero by an
appropriate local Lorentz transformation. Then, by another local Lorentz
rotation it can be represented in the form:

${{\stackrel{{\scriptstyle\bullet}}{{A}}}{}}{}^{a}{}_{c\nu}=\Lambda^{a}{}_{b}\partial_{\nu}(\Lambda^{-1}){}^{b}{}_{c}.$
(2.14)

Since the main goal of the present paper is to consider gravitational waves we
restrict ourselves by vacuum solutions only. Then, varying the action with the
Lagrangian (2.1), one obtains the Euler-Lagrange equation

$E_{a}{}^{\rho}=\frac{\partial\stackrel{{\scriptstyle\bullet}}{{{{\cal
L}}}}}{\partial
h^{a}{}_{\rho}}-\partial_{\sigma}\left(\frac{\partial\stackrel{{\scriptstyle\bullet}}{{{{\cal
L}}}}}{\partial h^{a}{}_{\rho,\sigma}}\right)=0.$ (2.15)

It can be rewritten as

$\kappa
h\stackrel{{\scriptstyle\bullet}}{{J}}_{a}{}^{\rho}=\partial_{\sigma}\Big{(}h{\stackrel{{\scriptstyle\bullet}}{{S}}}{}_{a}{}^{\rho\sigma}\Big{)},$
(2.16)

where

$\stackrel{{\scriptstyle\bullet}}{{J}}_{a}{}^{\rho}=\frac{1}{\kappa}h_{a}{}^{\mu}{\stackrel{{\scriptstyle\bullet}}{{S}}}{}_{c}{}^{\nu\rho}{\stackrel{{\scriptstyle\bullet}}{{T}}}{}^{c}{}_{\nu\mu}-\frac{h_{a}{}^{\rho}}{h}\stackrel{{\scriptstyle\bullet}}{{{{\cal
L}}}}+\frac{1}{\kappa}{\stackrel{{\scriptstyle\bullet}}{{A}}}{}^{c}{}_{a\sigma}{\stackrel{{\scriptstyle\bullet}}{{S}}}{}_{c}{}^{\rho\sigma}$
(2.17)

is not Lorentz covariant gravitational energy-momentum, in [3] it is called as
a gravitational current as well.

###  2.2 Elements of STEGR

The Lagrangian in STEGR has the form [1]:

$\stackrel{{\scriptstyle}}{{\cal
L}}{}=\frac{\sqrt{-g}}{2\kappa}g^{\mu\nu}(L^{\alpha}{}_{\beta\mu}L^{\beta}{}_{\nu\alpha}-L^{\alpha}{}_{\beta\alpha}L^{\beta}{}_{\mu\nu}),$
(2.18)

where $\kappa=8\pi$, and $L^{\alpha}{}_{\mu\nu}$ is disformation tensor:

$L^{\alpha}{}_{\mu\nu}=\frac{1}{2}Q^{\alpha}{}_{\mu\nu}-\frac{1}{2}Q_{\mu}{}^{\alpha}{}_{\nu}-\frac{1}{2}Q_{\nu}{}^{\alpha}{}_{\mu}\,,$
(2.19)

and the non-metricity tensor $Q_{\alpha\mu\nu}$ is defined as follows:

$Q_{\alpha\mu\nu}\equiv\nabla_{\alpha}g_{\mu\nu}.$ (2.20)

Here, the covariant derivative $\nabla_{\alpha}$ is defined with the use of
the teleparallel connection $\Gamma^{\alpha}{}_{\mu\nu}$ which is symmetric in
lower indexes, thus, the corresponding torsion is zero:
$T{}^{\alpha}{}_{\mu\nu}\equiv\Gamma{}^{\alpha}{}_{\mu\nu}-\Gamma{}^{\alpha}{}_{\nu\mu}=0$.
The curvature tensor of this connection is zero as well:

$R^{\alpha}{}_{\beta\mu\nu}(\Gamma)=\partial_{\mu}\Gamma^{\alpha}{}_{\nu\beta}-\partial_{\nu}\Gamma^{\alpha}{}_{\mu\beta}+\Gamma^{\alpha}{}_{\mu\lambda}\Gamma^{\lambda}{}_{\nu\beta}-\Gamma^{\alpha}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\mu\beta}=0.$
(2.21)

One can easily verify that the decomposition of a general connection
$\Gamma^{\beta}{}_{\mu\nu}$ into the Levi-Civita connection, contortion and
the disformation terms, see [1], reduces to

$L^{\beta}{}_{\mu\nu}\equiv\Gamma^{\beta}{}_{\mu\nu}-\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\beta}{}_{\mu\nu}.$
(2.22)

With the use of (2.19) - (2.22) one can rewrite (2.18) as

$\stackrel{{\scriptstyle}}{{\cal L}}{}=\stackrel{{\scriptstyle}}{{\cal
L}}{}_{Hilb}+\frac{\sqrt{-g}g^{\mu\nu}}{2\kappa}R_{\mu\nu}+\stackrel{{\scriptstyle}}{{\cal
L}}{}^{\prime},$ (2.23)

where the first term is the Hilbert Lagrangian and the third term
$\stackrel{{\scriptstyle}}{{\cal L}}^{\prime}$ is a total divergence:

$\stackrel{{\scriptstyle}}{{\cal
L}}^{\prime}{}=-\frac{\sqrt{-g}}{2\kappa}\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\alpha}(Q^{\alpha}-\hat{Q}^{\alpha})=\partial_{\alpha}\left(-\frac{1}{2\kappa}\sqrt{-g}(Q^{\alpha}-\hat{Q}^{\alpha})\right)=\partial_{\alpha}\stackrel{{\scriptstyle}}{{\cal
D}}{}^{\alpha},$ (2.24)

and $Q_{\alpha}=g^{\mu\nu}Q_{\alpha\mu\nu}$,
$\hat{Q}_{\alpha}=g^{\mu\nu}Q_{\mu\alpha\nu}$. According to [8] we neglect the
second term in (2.23) and, thus, under variation we consider the Lagrangian
(instead of (2.18) or (2.23)):

$\stackrel{{\scriptstyle}}{{\cal
L}}{}=-\frac{\sqrt{-g}}{2\kappa}\stackrel{{\scriptstyle\circ}}{{R}}+\partial_{\alpha}\stackrel{{\scriptstyle}}{{\cal
D}}{}^{\alpha},$ (2.25)

where

$\stackrel{{\scriptstyle}}{{\cal
D}}^{\alpha}\equiv-\frac{\sqrt{-g}}{2\kappa}(Q^{\alpha}-\hat{Q}^{\alpha})\,.$
(2.26)

###  2.3 Noether conserved quantities

The Noether conserved quantities were derived for TEGR Lagrangian (2.1) in
[26, 27] and for STEGR Lagrangian (2.25) in [8]. In both theories Noether
current ${\cal I}{}^{\alpha}(\xi)$ is a vector density of the weight +1, and
the Noether superpotential ${\cal I}{}^{\alpha\beta}(\xi)$ is an antisymmetric
tensor density of the weight +1. For such quantities
$\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\mu}\equiv\partial_{\mu}$, and,
thus, the conservation laws have evidently covariant form:

$\partial_{\alpha}{\cal
I}{}^{\alpha}(\xi)\equiv\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\alpha}{\cal
I}{}^{\alpha}(\xi)=0\,.$ (2.27) ${\cal
I}{}^{\alpha}(\xi)=\partial_{\alpha}{\cal
I}{}^{\alpha\beta}(\xi)\equiv\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\alpha}{\cal
I}{}^{\alpha\beta}(\xi)\,.$ (2.28)

It is important to give a physical interpretation of the quantities presented
above. We follow the prescription in [27, 8]. First, one has to choose the
displacement vector $\xi^{\alpha}$, it can be Killing vector, proper vector of
observer, etc. Second, setting a time coordinate as $t=x^{0}$ and choosing a
space section as $\Sigma:=t=\rm const$, one can interpret ${\cal I}^{0}(\xi)$
as a density on the section $\Sigma$ of the quantity related to a chosen
$\xi^{\alpha}$. For example, if $\xi^{\alpha}$ is a time-like Killing vector
it is interpreted as the energy density on $\Sigma$. On the other hand, if
$\xi^{\alpha}$ is an observer’s proper vector, then the components ${\cal
I}^{\alpha}(\xi)$ can be interpreted as components of the energy-momentum
vector measured by such an observer.

The Noether current ${\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{\alpha}(\xi)$ in TEGR in vacuum case when the field equations (2.15)
hold is [26, 27]:

${\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{\alpha}(\xi)=h\stackrel{{\scriptstyle\bullet}}{{\theta}}_{\sigma}{}^{\alpha}\xi^{\sigma}+\frac{h}{\kappa}{\stackrel{{\scriptstyle\bullet}}{{S}}}{}_{\sigma}{}^{\alpha\rho}\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\rho}\xi^{\sigma}\,,$
(2.29)

where the gravitational Noether energy-momentum tensor
$\stackrel{{\scriptstyle\bullet}}{{\theta}}_{\sigma}{}^{\alpha}$ is

$\stackrel{{\scriptstyle\bullet}}{{\theta}}_{\sigma}{}^{\alpha}\equiv\frac{1}{\kappa}{\stackrel{{\scriptstyle\bullet}}{{S}}}{}_{a}{}^{\alpha\rho}{{\stackrel{{\scriptstyle\bullet}}{{K}}}{}}^{a}{}_{\sigma\rho}-\frac{1}{h}\stackrel{{\scriptstyle\bullet}}{{{{\cal
L}}}}\delta^{\alpha}_{\sigma}\,.$ (2.30)

The related Noether superpotential in TEGR is

${\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{\alpha\beta}(\xi)=\frac{h}{\kappa}{{\stackrel{{\scriptstyle\bullet}}{{S}}}{}}_{a}{}^{\alpha\beta}h^{a}{}_{\sigma}\xi^{\sigma}\,.$
(2.31)

In correspondence with (2.28) one can derive the current from the
superpotential:

${\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{\alpha}(\xi)=\partial_{\beta}{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{\alpha\beta}(\xi).$ (2.32)

As one can see, both ${\stackrel{{\scriptstyle\bullet}}{{J}}}{}^{\alpha}(\xi)$
and ${\stackrel{{\scriptstyle\bullet}}{{J}}}{}^{\alpha\beta}(\xi)$ are
explicitly spacetime covariant and Lorentz invariant. Indeed, the Lorentz
invariance of the Noether conserved quantities and the Lorentz covariance of
the superpotential
${{\stackrel{{\scriptstyle\bullet}}{{S}}}{}}_{a}{}^{\alpha\beta}$ is satisfied
only under the simultaneous transformation of the tetrad (2.12) and the ISC
(2.13). However, as we see below, one can define different ISCs for one
tetrad, what gives us different values of the Noether conserved quantities.
For a detailed analysis of this situation, in [7, 28], we generalise the
definition of “gauges” in TEGR. A gauge is defined as a set of pairs (tetrad,
inertial spin connection) which can be obtained from a given combination of
the tetrad and the spin connection by an arbitrary covariant Lorentz
transformations where tetrad transforms as (2.12) and inertial spin connection
transforms as (2.13) simultaneously, and/or arbitrary coordinate
transformations. The case in which zero ISC corresponds to some tetrad is
called traditionally as the Wietzenböck gauge [3]. Note, that the word “gauge”
is used in different senses: when we say “Wietzenböck gauge” we mean the only
one pair - tetrad and zero ISC; and when we say “gauge” in our definition we
mean the whole equivalence class of pairs of tetrads and ISCs in which two
pairs are equivalent if and only if they are connected as defined above.

For the STEGR with the Lagrangian (2.25) the conserved quantities were derived
in [8] for each term separately. The superpotential ${\cal
J}{}_{GR}^{\alpha\beta}$ for the Hilbert term
$-\frac{\sqrt{-g}}{2\kappa}\stackrel{{\scriptstyle\circ}}{{R}}$ is the Komar
superpotential [29, 30]

${\cal J}{}_{GR}^{\alpha\beta}=\stackrel{{\scriptstyle}}{{\cal
K}}{}^{\alpha\beta}=\frac{\sqrt{-g}}{\kappa}\stackrel{{\scriptstyle\circ}}{{\nabla}}{}^{[\alpha}\xi^{\beta]},$
(2.33)

For the divergent term $\stackrel{{\scriptstyle}}{{\cal
L}}^{\prime}{}=\partial_{\alpha}\stackrel{{\scriptstyle}}{{\cal
D}}{}^{\alpha}$, see (2.24) and (2.26), the Noether superpotential is:

${\cal
J}{}_{div}^{\alpha\beta}=\frac{\sqrt{-g}}{\kappa}\delta_{\sigma}^{[\alpha}(Q^{\beta]}-\hat{Q}^{\beta]})\xi^{\sigma}.$
(2.34)

The total Noether superpotential of the Lagrangian (2.25) in STEGR is:

${\cal J}{}^{\alpha\beta}={\cal J}{}_{GR}^{\alpha\beta}+{\cal
J}{}_{div}^{\alpha\beta}.$ (2.35)

Taking the divergence of each term of (2.35) in correspondence with (2.28) one
gets the total Noether current

${\cal J}{}^{\alpha}={\cal J}{}_{GR}^{\alpha}+{\cal J}{}_{div}^{\alpha}.$
(2.36)

As one can see ${\cal J}{}^{\alpha}(\xi)$ and ${\cal J}{}^{\alpha\beta}(\xi)$
are explicitly spacetime covariant.

In [8], the concept of gauges in STEGR has not been introduced. In the present
paper we want to have the same terminology in STEGR as in TEGR, so formally,
we define a pair of coordinates and connection $\Gamma{}^{\alpha}{}_{\mu\nu}$,
with the set of pairs which are connected to it by the transformation

$\Gamma{}^{\alpha}{}_{\mu\nu}=\frac{\partial
x^{\alpha}}{\partial\bar{x}{}^{\bar{\alpha}}}\frac{\partial\bar{x}^{\bar{\mu}}}{\partial
x^{\mu}}\frac{\partial\bar{x}^{\bar{\nu}}}{\partial
x^{\nu}}\bar{\Gamma}{}^{\bar{\alpha}}{}_{\bar{\mu}\bar{\nu}}+\frac{\partial
x^{\alpha}}{\partial\bar{x}{}^{\bar{\lambda}}}\frac{\partial}{\partial
x^{\mu}}\left(\frac{\partial\bar{x}{}^{\bar{\lambda}}}{\partial
x^{\nu}}\right)$ (2.37)

as a “gauge”. Usually, the case of zero teleparallel connection is called as
“coincident gauge” [31, 32]. Here, we note again, that in the therm
“coincident gauge” we mean the only case of coordinates in which the
teleparallel connection in zero; and when we say “gauge” we mean the set of
all possible coordinates and the values of teleparallel connections in them,
such that relation (2.37) is satisfied under all possible coordinate
transformations.

###  2.4 Defining the connection: “turning off” gravity principle

Teleparallel connections in TEGR and STEGR are not dynamical quantities and
left undetermined [33, 1] — the choice of them is not unique.

To determine the ISC in TEGR the “turning off” gravity principle was
introduced in [26, 27]. This principle is based on the assumption that
Noether’s current and superpotential are proportional to contortion components
$\stackrel{{\scriptstyle\bullet}}{{K}}{}^{a}{}_{c\mu}$ (or, alternatively
$\stackrel{{\scriptstyle\bullet}}{{T}}{}^{\alpha}{}_{\mu\nu}$ or
$\stackrel{{\scriptstyle\bullet}}{{S}}{}_{a}{}^{\mu\nu}$) and they have to
vanish in absence of gravity. Thus, to determine
$\stackrel{{\scriptstyle\bullet}}{{A}}{}^{a}{}_{c\mu}$ in correspondence of
this requirement we turn to the formula (2.8). It was suggested:

1) for known GR solution, to choose a convenient tetrad and define
$\stackrel{{\scriptstyle\circ}}{{A}}{}^{a}{}_{c\mu}=-h_{b}{}^{\nu}\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\mu}h^{a}{}_{\nu}$;

2) to construct related curvature of Levi-Civita spin connection
$\stackrel{{\scriptstyle\circ}}{{R}}{}^{i}{}_{j\mu\nu}=\partial_{\mu}{\stackrel{{\scriptstyle\circ}}{{A}}}{}{}^{i}{}_{j\nu}-\partial_{\nu}{\stackrel{{\scriptstyle\circ}}{{A}}}{}{}^{i}{}_{j\mu}+{\stackrel{{\scriptstyle\circ}}{{A}}}{}{}^{i}{}_{k\mu}{\stackrel{{\scriptstyle\circ}}{{A}}}{}{}^{k}{}_{j\nu}-{\stackrel{{\scriptstyle\circ}}{{A}}}{}{}^{i}{}_{k\nu}{\stackrel{{\scriptstyle\circ}}{{A}}}{}{}^{k}{}_{j\mu}$;

3) to “switch off” gravity solving the absent gravity equation
$\stackrel{{\scriptstyle\circ}}{{R}}{}^{a}{}_{b\gamma\delta}=0$ for parameters
of the chosen GR solution;

4) when the parameters satisfying
$\stackrel{{\scriptstyle\circ}}{{R}}{}^{a}{}_{b\gamma\delta}=0$ are found, we
take
$\stackrel{{\scriptstyle\circ}}{{A}}{}^{a}{}_{c\mu}=\stackrel{{\scriptstyle\bullet}}{{A}}{}^{a}{}_{c\mu}$
for the found parameter values.

To define the undetermined connection in STEGR we use the adapted for STEGR
“turning off” gravity principle [8]. It is based on the assumption that
$Q_{\alpha\mu\nu}$ and $L^{\alpha}{}_{\mu\nu}$ vanish in the absence of
gravity and $\stackrel{{\scriptstyle\circ}}{{R}}{}^{\alpha}{}_{\beta\mu\nu}$
in GR vanishes in the absence of gravity too. To find the connection in STEGR
there are the following steps:

1) for known GR solution, to construct related Riemann curvature tensor of the
Levi-Civita connection:

$\stackrel{{\scriptstyle\circ}}{{R}}{}^{\alpha}{}_{\beta\mu\nu}=\partial_{\mu}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\alpha}{}_{\beta\nu}-\partial_{\nu}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\alpha}{}_{\beta\mu}+\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\alpha}{}_{\kappa\mu}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\kappa}{}_{\beta\nu}-\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\alpha}{}_{\kappa\nu}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\kappa}{}_{\beta\mu};$

2) to “switch off” gravity solving the absent gravity equation
$\stackrel{{\scriptstyle\circ}}{{R}}{}^{\alpha}{}_{\beta\mu\nu}=0$ for
parameters of the chosen GR solution;

3) when the parameters satisfying
$\stackrel{{\scriptstyle\circ}}{{R}}{}^{\alpha}{}_{\beta\mu\nu}=0$ are found,
we take
$\Gamma{}^{\alpha}{}_{\mu\nu}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\alpha}{}_{\mu\nu}$
for the found parameter values.
Torsion of the found connection should be zero automatically because we take
it from the Levi-Civita connection for some parameter values, and Levi-Civita
connection is always symmetric. Curvature of the found connection should be
zero too because we found it from the equation
$\stackrel{{\scriptstyle\circ}}{{R}}{}^{\alpha}{}_{\beta\gamma\delta}=0$.

It turns out, that the described above “turning off” gravity method gives
ambiguities both in TEGR [7, 28] and in STEGR [8]. As a rule, “turning off”
gravity in TEGR, the result depends on the tetrad that we choose. For each
tetrad where we “turn off” gravity we obtain different gauge (pairs of tetrad
and ISC). In general, these pairs are not connected by (2.12) and (2.13)
applied simultaneously. Thus, torsion, contortion and superpotential being
expressed in the same tetrad and coordinates are completely different in each
case. This gives us different values of conserved quantities. But, sometimes,
“turning off” gravity we can obtain the same gauge [34, 35]. The same way,
when “turning off” gravity in STEGR, the result depends on the coordinates
that we choose. Thus, disformation and non-metricity being expressed in the
same coordinates are different and this gives us different values of conserved
quantities in each case. One of the main purposes of this work is to find the
gauges in TEGR and connections in STEGR in which we would have physically
meaningful results for the concrete solutions - gravitational waves in a
vacuum.

###  2.5 Plane gravitational wave under consideration

In this paper, we consider only one polarization of the plane gravitational
wave. The simplest form of the metric for the exact (strong) plane
gravitational wave has the form of [21]:

$g_{\mu\nu}=\left(\begin{array}[]{cccc}-1&0&0&0\\\ 0&f^{2}(t-z)&0&0\\\
0&0&g^{2}(t-z)&0\\\ 0&0&0&1\\\ \end{array}\right).$ (2.38)

Here and below the numeration of the coordinates are $t=x^{0}$, $x=x^{1}$,
$y=x^{2}$, and $z=x^{3}$.

Because the wave-solutions in GR are time dependent ones there are no time-
like Killing vectors for them, therefore we do not expect to obtain any
conserved charges in this case. So, our goal is to calculate energy-momentum
density (current components) measured by the freely falling observer in wave
solution taking his proper vector (4-velocity) and interpret the result.

##  3 Simplest gauges

###  3.1 Diagonal polarisation in TEGR

The simplest diagonal tetrad that can be introduced for (2.38) is

$h^{A}{}_{\mu}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&f(t-z)&0&0\\\
0&0&g(t-z)&0\\\ 0&0&0&1\\\ \end{array}\right).$ (3.1)

To make the formulae shorter, we won’t write the argument $(t-z)$ of functions
$f$ and $g$ and their derivatives of each order. For this tetrad, we calculate
L-CSC (2.7) which has the non-zero components:

$\begin{array}[]{cccc}{\stackrel{{\scriptstyle\circ}}{{A}}}{}{}^{\hat{0}}{}_{\hat{1}1}={\stackrel{{\scriptstyle\circ}}{{A}}}{}{}^{\hat{1}}{}_{\hat{0}1}=-{\stackrel{{\scriptstyle\circ}}{{A}}}{}{}^{\hat{1}}{}_{\hat{3}1}={\stackrel{{\scriptstyle\circ}}{{A}}}{}{}^{\hat{3}}{}_{\hat{1}1}=f^{\prime},\\\
{\stackrel{{\scriptstyle\circ}}{{A}}}{}{}^{\hat{0}}{}_{\hat{2}2}={\stackrel{{\scriptstyle\circ}}{{A}}}{}{}^{\hat{2}}{}_{\hat{0}2}={\stackrel{{\scriptstyle\circ}}{{A}}}{}{}^{\hat{3}}{}_{\hat{2}2}=-{\stackrel{{\scriptstyle\circ}}{{A}}}{}{}^{\hat{2}}{}_{\hat{3}2}=g^{\prime},\\\
\end{array}$ (3.2)

where prime means a derivative with respect to $u=(t-z)$, and the index with
“hat” is the tetrad index.

Non-zero components of Riemann tensor are proportional to $f^{\prime\prime}$
or $g^{\prime\prime}$, so turning off gravity both in STEGR and TEGR means
that:

$\begin{array}[]{cccc}f^{\prime\prime}=0,\\\ g^{\prime\prime}=0,\end{array}$
(3.3)

i.e

$\begin{array}[]{cccc}f={c_{1}}(t-z)+{d_{1}},\\\
g={c_{2}}(t-z)+{d_{2}},\end{array}$ (3.4)

where $c_{1}$, $c_{2}$, $d_{1}$ and $d_{2}$ are integration constants. Turning
off gravity by (3.4) we get ISC with non-zero components:

$\begin{array}[]{cccc}{\stackrel{{\scriptstyle\bullet}}{{A}}}{}{}^{\hat{0}}{}_{\hat{1}1}={\stackrel{{\scriptstyle\bullet}}{{A}}}{}{}^{\hat{1}}{}_{\hat{0}1}=-{\stackrel{{\scriptstyle\bullet}}{{A}}}{}{}^{\hat{1}}{}_{\hat{3}1}={\stackrel{{\scriptstyle\bullet}}{{A}}}{}{}^{\hat{3}}{}_{\hat{1}1}={c_{1}},\\\
{\stackrel{{\scriptstyle\bullet}}{{A}}}{}{}^{\hat{0}}{}_{\hat{2}2}={\stackrel{{\scriptstyle\bullet}}{{A}}}{}{}^{\hat{2}}{}_{\hat{0}2}={\stackrel{{\scriptstyle\bullet}}{{A}}}{}{}^{\hat{3}}{}_{\hat{2}2}=-{\stackrel{{\scriptstyle\bullet}}{{A}}}{}{}^{\hat{2}}{}_{\hat{3}2}={c_{2}}.\\\
\end{array}$ (3.5)

Then we calculate superpotential (2.11) with (2.8) definitions which has the
non-zero components:

$\begin{array}[]{cccc}{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{0}}{}^{03}=-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{0}}{}^{30}=-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{3}}{}^{03}={\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{3}}{}^{30}=\frac{{c_{1}}-f^{\prime}}{f}+\frac{{c_{2}}-g^{\prime}}{g},\\\
{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{1}}{}^{01}={\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{1}}{}^{31}=-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{1}}{}^{10}=-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{1}}{}^{13}=\frac{{g^{\prime}}-c_{2}}{fg},\\\
{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{2}}{}^{02}={\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{2}}{}^{32}=-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{2}}{}^{20}=-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{2}}{}^{23}=\frac{{f^{\prime}-c_{1}}}{fg}.\\\
\end{array}$ (3.6)

We assume that superpotential, torsion, etc should be zero in absence of
waves, thus, $c_{1}=c_{2}=0$. Then, the ISC (3.5) becomes

${\stackrel{{\scriptstyle\bullet}}{{A}}}{}{}^{a}{}_{b\mu}=0.$ (3.7)

Our next goal is to study the Noether superpotential for this gauge. First, we
consider the simplest case of a freely falling observer which is static in co-
moving coordinates of the metric (2.38). Then components of his proper vector
are:

$\xi^{\sigma}=(-1,0,0,0)\,.$ (3.8)

Then one gets for the Noether superpotential (2.31) non-zero components:

${\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{03}=-{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{30}=\frac{gf^{\prime}+fg^{\prime}}{8\pi}.$ (3.9)

By (2.32), taking the divergence of superpotential and using the Einstein
equations:

$f^{\prime\prime}/f+g^{\prime\prime}/g=0$ (3.10)

we get Noether current

${\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{\mu}=\left\\{-\frac{f^{\prime}g^{\prime}}{4\pi},0,0,-\frac{f^{\prime}g^{\prime}}{4\pi}\right\\}.$
(3.11)

Let us generalize (3.8). Directly solving the geodesic equation we derive
general freely falling observer’s 4-velocity $\xi^{\mu}$:

$\begin{array}[]{cccc}\xi^{0}=-\frac{{C_{1}}^{2}e^{\alpha_{0}}}{2f^{2}}-\frac{{C_{2}}^{2}e^{\alpha_{0}}}{2g^{2}}-\cosh{\alpha_{0}},\\\
\xi^{1}=-\frac{{C_{1}}}{f^{2}},\\\ \xi^{2}=-\frac{{C_{2}}}{g^{2}},\\\
\xi^{3}=-\frac{{C_{1}}^{2}e^{\alpha_{0}}}{2f^{2}}-\frac{{C_{2}}^{2}e^{\alpha_{0}}}{2g^{2}}-\sinh{\alpha_{0}},\end{array}$
(3.12)

where $C_{1}$, $C_{2}$, $\alpha_{0}$ are constants of integration. One can see
that components (3.12) go to the ones in (3.8) when
$C_{1}=C_{2}=\alpha_{0}=0$.

Taking this general form of observer’s proper vector one gets Noether
superpotential non-zero components:

$\begin{array}[]{cccc}{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{01}={\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{31}=-{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{10}=-{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{13}=-\frac{{C_{1}}g^{\prime}}{8\pi f},\\\
{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{02}={\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{32}=-{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{20}=-{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{23}=-\frac{{C_{2}}f^{\prime}}{8\pi g},\\\
{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{03}=-{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{30}=\frac{e^{-\alpha_{0}}\left(gf^{\prime}+fg^{\prime}\right)}{8\pi}.\end{array}$
(3.13)

Taking the divergence of superpotential and using the the Einstein equations
(3.10) we get Noether current

${\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{\mu}=\left\\{-\frac{e^{-\alpha_{0}}f^{\prime}g^{\prime}}{4\pi},0,0,-\frac{e^{-\alpha_{0}}f^{\prime}g^{\prime}}{4\pi}\right\\}.$
(3.14)

Note, first, that the components of the current (3.14) do not depend on
$C_{1}$ and $C_{2}$; and second, that for $\alpha_{0}=0$ the components (3.14)
coincide with (3.11).

###  3.2 Diagonal polarisation in STEGR

The non-zero components of Levi-Civita connection for the metric (2.38) are:

$\begin{array}[]{cccc}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{11}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{11}=ff^{\prime},\\\
\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{22}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{22}=gg^{\prime},\\\
\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{01}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{10}=-\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{13}=-\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{31}=\frac{f^{\prime}}{f},\\\
\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{02}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{20}=-\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{23}=-\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{32}=\frac{g^{\prime}}{g}.\\\
\end{array}$ (3.15)

Taking (3.4) in Levi-Civita connection we get the symmetric teleparallel
connection (flat and torsionless):

$\begin{array}[]{cccc}\Gamma{}^{0}{}_{11}=\Gamma{}^{3}{}_{11}={c_{1}}({c_{1}}(t-z)+{d_{1}}),\\\
\Gamma{}^{0}{}_{22}=\Gamma{}^{3}{}_{22}={c_{2}}({c_{2}}(t-z)+{d_{2}}),\\\
\Gamma{}^{1}{}_{01}=\Gamma{}^{1}{}_{10}=-\Gamma{}^{1}{}_{13}=-\Gamma{}^{1}{}_{31}=\frac{{c_{1}}}{{c_{1}}(t-z)+{d_{1}}},\\\
\Gamma{}^{2}{}_{02}=-\Gamma{}^{2}{}_{23}=-\Gamma{}^{2}{}_{32}=\Gamma{}^{2}{}_{20}=\frac{{c_{2}}}{{c_{2}}(t-z)+{d_{2}}}.\\\
\end{array}$ (3.16)

Then we can obtain the non-metricity (2.20) and disformation (2.22).

Non-metricity (2.20) in absence of waves (i.e for Minkowski metric) is:

$\begin{array}[]{cccc}Q{}_{011}=-Q{}_{311}=-\frac{2{c_{1}}}{{c_{1}}(t-z)+{d_{1}}},\\\
Q{}_{022}=-Q{}_{322}=-\frac{2{c_{2}}}{{c_{2}}(t-z)+{d_{2}}},\\\
Q{}_{101}=Q{}_{110}=-Q{}_{113}=-Q{}_{131}=-\frac{{c_{1}}}{{c_{1}}(t-z)+{d_{1}}}+{c_{1}}({c_{1}}(t-z)+{d_{1}}),\\\
Q{}_{202}=\par
Q{}_{220}=-Q{}_{223}=-Q{}_{232}=-\frac{{c_{2}}}{{c_{2}}(t-z)+{d_{2}}}+{c_{2}}({c_{2}}(t-z)+{d_{2}}).\\\
\end{array}$ (3.17)

We assume that non-metricity should be zero in absence of waves. Thus, we have
$c_{1}=c_{2}=0$ and (3.16) becomes

$\Gamma{}^{\alpha}{}_{\mu\nu}=0.$ (3.18)

Proceeding further in this gauge to get the Noether superpotential we choose
the observer’s proper vector as (3.8). Komar superpotential (2.33) becomes
zero. After calculation of the divergent part of the superpotential (2.34) we
get the total superpotential with non-zero components:

${\cal J}{}^{03}=-{\cal J}{}^{30}={\cal J}_{div}^{03}=-{\cal
J}_{div}^{30}=\frac{{f^{\prime}}{g}+{g^{\prime}}{f}}{8\pi}.$ (3.19)

Taking the divergence of the Noether superpotential and using the Einstein
equations (3.10) we get the Noether current

${\cal
J}{}^{\mu}=\left\\{-\frac{f^{\prime}g^{\prime}}{4\pi},0,0,-\frac{f^{\prime}g^{\prime}}{4\pi}\right\\}.$
(3.20)

Note that it exactly coincides with (3.11).

Second, to get another Noether superpotential we choose the observer’s proper
vector as in (3.12). Komar superpotential (2.33) is zero again. Calculating
the divergent part (2.34) we get the total superpotential

$\begin{array}[]{cccc}{\cal J}{}^{01}={\cal J}{}^{31}=-{\cal J}{}^{10}=-{\cal
J}{}^{13}=-{C_{1}}\frac{{f^{\prime}}{g}+{g^{\prime}}{f}}{8\pi f^{2}},\\\ {\cal
J}{}^{02}={\cal J}{}^{32}=-{\cal J}{}^{20}=-{\cal
J}{}^{23}=-{C_{2}}\frac{{f^{\prime}}{g}+{g^{\prime}}{f}}{8\pi g^{2}},\\\ {\cal
J}{}^{03}=-{\cal
J}{}^{30}=\frac{e^{-\alpha_{0}}\left({f^{\prime}}{g}+{g^{\prime}}{f}\right)}{8\pi}.\end{array}$
(3.21)

And then taking the divergence and using the Einstein equations (3.10) we get
the Noether current

${\cal
J}{}^{\mu}=\left\\{-\frac{e^{-\alpha_{0}}f^{\prime}g^{\prime}}{4\pi},0,0,-\frac{e^{-\alpha_{0}}f^{\prime}g^{\prime}}{4\pi}\right\\}.$
(3.22)

Note that it exactly coincides with (3.14).

###  3.3 A linear approximation

Turn to the simplest case of freely falling observer (3.8). The Noether
currents both in TEGR (3.11) and in STEGR (3.20) are the same. For the weak
wave in Minkowski space we choose the simplest presentation [36]:

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},$ (3.23)

where $\eta_{\mu\nu}=\rm diag[-1,\leavevmode\nobreak\ +1,\leavevmode\nobreak\
+1,\leavevmode\nobreak\ +1]$. Thus, in the case of the metric (2.38) in TT-
gauge [36] one has

$f=1+\frac{1}{2}h_{xx}(t-z),\leavevmode\nobreak\ \leavevmode\nobreak\
g=1-\frac{1}{2}h_{xx}(t-z).$ (3.24)

Then, the current (3.11), the same (3.20), becomes

${\cal
J}{}^{\mu}_{lin}=\left\\{\frac{{h^{\prime}}_{xx}^{2}}{16\pi},\leavevmode\nobreak\
0,\leavevmode\nobreak\ 0,\leavevmode\nobreak\
\frac{{h^{\prime}}_{xx}^{2}}{16\pi}\right\\}.$ (3.25)

The component ${\cal J}{}^{0}_{lin}$ represents the energy density, whereas
${\cal J}{}^{3}_{lin}$ \- the energy density flux of the weak flat
gravitational wave along $z$ axis. It exactly coincides with the result in
[36] for the case of the only one polarization. Recall that these quantities
are positively defined.

It is interesting to consider a more general case of freely falling observers
(3.12) in the linear approximation (3.23) and (3.24). Then the current (3.14),
the same (3.22), becomes

${\cal
J}{}^{\mu}_{lin}=\left\\{\frac{e^{-\alpha_{0}}{h^{\prime}}_{xx}^{2}}{16\pi},\leavevmode\nobreak\
0,\leavevmode\nobreak\ 0,\leavevmode\nobreak\
\frac{e^{-\alpha_{0}}{h^{\prime}}_{xx}^{2}}{16\pi}\right\\},$ (3.26)

that is, at least, positively defined like (3.25).

Because vector (3.8) is a proper vector of a freely falling observer one could
expect that due to the equivalence principle the components of the conserved
current are to be equal to zero. So, from one side, one meets the evidently
undesirable result. From the other side, however, this coincides with the
well-known result from the Landau-Lifshitz pseudotensor formula. The same
result have been obtained for the gravitational wave (2.38) in [21] where this
fact is considered as a criterion that supports correctness of the result. In
the following paper [22], this coincidence is supported by a relation to a so-
called “ideal frame”.

From our side, we should interpret the discrepancy in interpretation of the
equivalence principle in the framework of our approach. In the formalism [26,
8] applied here, unlike other approaches, observers are connected with their
proper vectors. We remind a reader that studying conserved quantities for the
Schwarzschild black hole [34, 35] we have met with the evidence that
constructing an appropriate global mass requires the one related gauge,
whereas a correspondence to the equivalence principle requires another gauge.
In such a paradigm we need to claim that here the gauges, the pair of tetrad
(3.1) and ISC (3.7) in TEGR and the pair of metric (2.38) and teleparallel
connection (3.18) in STEGR, do not correspond to the equivalence principle. In
the next sections we present gauges for the solution (2.38) which give a
correspondence to the equivalence principle in that old sense.

However, we need to explain the quite acceptable result (3.25) in comparison
with the textbook [36] result, where components are explained as energetic
characteristics of gravitational wave. Thus, we need deeper to consider the
Landau and Lifshitz prescription, which is based on their pseudotensor. As all
pseudotensors, it is non-covariant – it is not transformed as tensors under
arbitrary coordinate transformations which makes the physical interpretation
of the conserved quantities more difficult. One of the ways to improve the
situation is a possibility to covariantize pseudotensors by introducing to the
dynamical spacetime manifold with the metric tensor $g_{\mu\nu}$ a fixed
Minkowskian background with the Minkowski metric, see book [30]. Then, the
Landau-Lifshitz conserved pseudotensor $t^{\mu\nu}_{LL}$, which has the
mathematical weight +2, permits to construct us a conserved covariant
current111The analogous method of covariantization can be applied to an
arbitrary pseudotensor.

${\cal
J}{}^{\mu}_{LL}=\left(\sqrt{-\det\eta_{\alpha\beta}}\right)^{-1}t^{\mu\nu}_{LL}\bar{\xi}{}_{\nu}.$
(3.27)

From the beginning it is assumed that (3.27) is written in the Lorentzian
coordinates, of course, $-\det\eta_{\alpha\beta}=1$. Then, ${\cal
J}{}^{\mu}_{LL}$ is thought as a vector density of the weight +1 and can be
represented in arbitrary coordinates in an ordinary way if particular
derivatives in $t^{\mu\nu}_{LL}$ are replaced by covariant ones. Vector
$\bar{\xi}{}^{\mu}$ is a Killing vector of the Minkowski space. Thus, if

$\bar{\xi}{}^{\mu}=(-1,\leavevmode\nobreak\ 0,\leavevmode\nobreak\
0,\leavevmode\nobreak\ 0),$ (3.28)

it is a timelike Killing vector of the Minkowski space. Then ${\cal
J}{}^{\mu}_{LL}=(\leavevmode\nobreak\ t^{00}_{LL},\leavevmode\nobreak\
t^{10}_{LL},\leavevmode\nobreak\ t^{20}_{LL},\leavevmode\nobreak\
t^{30}_{LL})$, that is the interpretation of ${\cal J}{}^{0}_{LL}$ and ${\cal
J}{}^{3}_{LL}$ as energy density and energy density flux coincides with a
related interpretation in $t^{00}_{LL}$ and $t^{30}_{LL}$ in the book [36].

Now let us turn to the linear approximation. The decompositions (3.23) and
(3.24), where perturbations are considered on a flat background with the
Minkowski metric, are quite appropriate to turn to the paradigm of
covariantization of (3.27) on the Minkowskian background. Besides, for the
result (3.25) we anyway use vector (3.8) that is just coincides with the
Killing vector (3.28) of the Minkowski space. Thus, the coincidence of the
result (3.25) with the Landau-Lifshitz one becomes clear.

Let us consider (3.27) without an approximation. Thus, $t^{\mu\nu}_{LL}$ for
the metric (2.38) is

${\cal J}{}^{\mu}_{LL}=\left\\{-\frac{4fgf^{\prime}g^{\prime}+g^{2}f^{\prime
2}+f^{2}g^{\prime 2}}{8\pi},\leavevmode\nobreak\ 0,\leavevmode\nobreak\
0,\leavevmode\nobreak\ -\frac{4fgf^{\prime}g^{\prime}+g^{2}f^{\prime
2}+f^{2}g^{\prime 2}}{8\pi}\right\\}.$ (3.29)

One can see that it differs from (3.11) and (3.20), although its linear
approximation gives (3.25). We stress that our results (3.11) and (3.20) are
obtained in the framework of the initially covariant method, whereas (3.29) is
obtained after covariantization (an additional procedure) of the Landau-
Lifshitz pseudotensor.

Finally, it is necessary to discuss the results (3.14), (3.22) and (3.26). The
vector (3.12) is not a Killing vector of Minkowski space. However, without
changing the results (3.14), (3.22) and (3.26) one can set $C_{1}=C_{2}=0$. By
this, we can claim that we use vector

$\xi^{\mu}=\left\\{-\cosh\alpha_{0},\leavevmode\nobreak\
0,\leavevmode\nobreak\ 0,\leavevmode\nobreak\ -\sinh\alpha_{0}\right\\},$
(3.30)

which is a timelike Killing vector of Minkowski space. Indeed, boosting (3.28)
by global Lorentz transformation in Minkowski space one gets (3.30). This
transformation has a form:
$\xi^{\mu}_{boosted}=\lambda{}^{\mu}{}_{\nu}\xi_{kill}^{\nu}$, where
$\xi_{kill}^{\nu}$ is (3.8), $\xi^{\mu}_{boosted}$ is (3.12),

$\lambda{}^{\mu}{}_{\nu}=\left(\begin{array}[]{cccc}\cosh\alpha_{0}&0&0&\sinh\alpha_{0}\\\
0&0&0&0\\\ 0&0&0&0\\\
\sinh\alpha_{0}&0&0&\cosh\alpha_{0}\end{array}\right)=\left(\begin{array}[]{cccc}\frac{1}{\sqrt{1-v^{2}}}&0&0&\frac{v}{\sqrt{1-v^{2}}}\\\
0&0&0&0\\\ 0&0&0&0\\\
\frac{v}{\sqrt{1-v^{2}}}&0&0&\frac{1}{\sqrt{1-v^{2}}}\end{array}\right),$
(3.31)

where $v$ is 3-dimensional velocity in Minkowski spacetime. Then
interpretation of (3.14), (3.22) and (3.26) becomes clear: they are obtained
by boosting the vector $\xi^{\mu}$ by (3.31) in (3.11), (3.20) and (3.25).

The result of linear approximation is an acceptable result not only because it
coincides with the Landau-Lifshitz prediction (and other pseudotensor
approaches as well), but mainly because it is checked observationally. This
possibly means that the equivalence principle itself may not require necessary
zero current in this case, and non-zero values can have a physical meaning. On
the other hand, due to lack of observational evidence for a strong
gravitational wave regime, we can not a priory say if the general form of the
current has the same meaning, regarding its difference from the pseudo-tensor
result for a strong wave.

##  4 Gauges compatible with the equivalence principle in the old sense

In this section, as it was announced above, we are searching gauges which give
zero current for a free falling observer.

###  4.1 Gauge changing in TEGR

Let us try to obtain zero Noether current in TEGR by changing a gauge. To do
this, we use the ISC changes as usual (2.13):

$A^{\prime
a}{}_{b\mu}=\Lambda^{a}{}_{c}(x^{\nu})A^{c}{}_{d\mu}\Lambda^{b}{}_{d}(x^{\nu})+\Lambda^{a}{}_{c}(x^{\nu})\partial_{\mu}\Lambda_{b}{}^{c}(x^{\nu})\,.$
(4.1)

Because in the previous section we had zero ISC (3.7), our new ISC has a form
(2.14)

$A^{\prime
a}{}_{b\mu}=\Lambda^{a}{}_{c}(x^{\nu})\partial_{\mu}\Lambda_{b}{}^{c}(x^{\nu})\,,$
(4.2)

while the tetrad remains the same (3.1). Or, another way is as follows: the
tetrad changes as (2.12)

$h^{\prime a}{}_{\mu}=\Lambda^{a}{}_{b}(x^{\nu})h^{b}{}_{\mu}\,,$ (4.3)

while ISC (3.7) remains zero. In this section, we also assume that our affine
connection $\stackrel{{\scriptstyle\bullet}}{{\Gamma}}{}^{\alpha}{}_{\mu\nu}$
should have the same symmetries as a solution (2.38). This means that
displacements in $x$, in $y$ and in $t+z$ directions do not change it. Such a
proposal was applied to the affine connection in modified teleparallel
theories with other symmetries in [37]. Because in modified teleparallel
theories the connection is dynamical, both the metric and the connection
should have the same symmetries as solution. In TEGR, the ISC and affine
connection are non dynamical, so, at the level of field equations, the
requirement for them to have the same symmetries as metric might be too
strong. Nevertheless, we consider Noether’s current (2.29) and superpotential
(2.31) and can require for them the same symmetries as metric has. For this
purpose it is sufficient to require the same symmetries for the teleparallel
superpotential, contortion or torsion. For example we can require for
contortion:

${\pounds}_{\xi}{\stackrel{{\scriptstyle\bullet}}{{K}}}{}{}^{\alpha}{}_{\mu\nu}=0,$
(4.4)

where ${\pounds}_{\xi}$ is Lie derivative and $\xi$ is a Killing vector of the
solution. For the metric (2.38), the symmetry holds along the direction
$\Delta x$: $\xi^{\mu}=(0,\leavevmode\nobreak\ 1,\leavevmode\nobreak\
0,\leavevmode\nobreak\ 0)$, $\Delta y$: $\xi^{\mu}=(0,\leavevmode\nobreak\
0,\leavevmode\nobreak\ 1,\leavevmode\nobreak\ 0)$, and $\Delta(t+z)$:
$\xi^{\mu}=(1,\leavevmode\nobreak\ 0,\leavevmode\nobreak\
0,\leavevmode\nobreak\ 1).$ The contortion is

${\stackrel{{\scriptstyle\bullet}}{{K}}}{}{}^{\alpha}{}_{\mu\nu}=\stackrel{{\scriptstyle\bullet}}{{\Gamma}}{}^{\alpha}{}_{\mu\nu}-\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\alpha}{}_{\mu\nu}=0,$
(4.5)

where the Levi-Civita connection has the same symmetries as metric (2.38). Now
let’s go back to definitions (2.4) and (2.14). The tetrad (3.1) is already
symmetric as metric (2.38). Therefore,
$\stackrel{{\scriptstyle\bullet}}{{\Gamma}}{}^{\alpha}{}_{\mu\nu}$ is
symmetric when the ISC (2.14) is symmetric. This requirement is fulfilled when
$\Lambda^{a}{}_{b}(x^{\nu})=\Lambda^{a}{}_{b}(t-z)$ in (2.14) is an arbitrary
Lorentz rotation which depends on $t-z$ only.

Keeping in mind matrices of local Lorentz rotations obtained by this
prescription and given in Appendix A, we have found that for the related
gauges (pairs of tetrad (3.1) and ISC (2.14), where $\Lambda^{a}{}_{b}(t-z)$
is a composition of matrices given in Appendix A, or transformed tetrad (3.1)
by $\Lambda^{a}{}_{b}(t-z)h^{b}{}_{\mu}$ and zero ISC (3.7)) the Noether
current does not change! Thus one cannot find a gauge for which the current
vanishes for a free moving observer with such a restrictive condition.

Now let’s assume that $\Lambda{}^{a}{}_{b}$ can be non-symmetrical, for
example, along the direction $dx$, thus, it can depend on $x$ and $(t-z)$. The
motivation for this proposal is that the connection in TEGR is not dynamical
and thus cannot be felt by observers like symmetrical metric can, and, thus,
we don’t need $\Lambda{}^{a}{}_{b}$ to be symmetrical. When we assume the
dependence on $x$, the Noether superpotential can depend on $x$. However,
making the Noether current be equal to zero, such a current will automatically
satisfy the symmetries of the solution. One of the most simple Lorentz
rotations $\Lambda{}^{a}{}_{b}$ which depend on $x$ and $(t-z)$ and can change
the Noether current is

$\Lambda{}^{a}{}_{b}=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&\cos(x{\psi}(t-z))&0&\sin(x{\psi}(t-z))\\\ 0&0&1&0\\\
0&-\sin(x{\psi}(t-z))&0&\cos(x{\psi}(t-z))\\\ \end{array}\right).$ (4.6)

The ISC (2.14)
${\stackrel{{\scriptstyle\bullet}}{{A}}}{}{}^{a}{}_{c\mu}=\Lambda{}_{b}{}^{c}\partial_{\mu}\Lambda{}^{a}{}_{b}$
calculated with (4.6) is

$\begin{array}[]{cccc}{\stackrel{{\scriptstyle\bullet}}{{A}}}{}{}^{\hat{1}}{}_{\hat{3}1}=-{\stackrel{{\scriptstyle\bullet}}{{A}}}{}{}^{\hat{3}}{}_{\hat{1}1}={\psi}(t-z);\\\
{\stackrel{{\scriptstyle\bullet}}{{A}}}{}{}^{\hat{1}}{}_{\hat{3}0}={\stackrel{{\scriptstyle\bullet}}{{A}}}{}{}^{\hat{3}}{}_{\hat{1}3}=-{\stackrel{{\scriptstyle\bullet}}{{A}}}{}{}^{\hat{1}}{}_{\hat{3}3}=-{\stackrel{{\scriptstyle\bullet}}{{A}}}{}{}^{\hat{3}}{}_{\hat{1}0}=x{\psi}^{\prime}(t-z).\\\
\end{array}$ (4.7)

Then one can calculate the contortion (2.8) with (4.6) and (3.2). Then, the
teleparallel superpotential (2.11) is

$\begin{array}[]{cccc}{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{0}}{}^{01}={\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{0}}{}^{31}=-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{0}}{}^{10}=-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{0}}{}^{13}=-\frac{x{\psi}^{\prime}(t-z)}{f};\\\
{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{0}}{}^{03}=-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{0}}{}^{30}=-\frac{f^{\prime}+{\psi}(t-z)}{f}-\frac{g^{\prime}}{g};\\\
{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{1}}{}^{01}={\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{1}}{}^{31}=-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{1}}{}^{10}=-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{1}}{}^{13}=\frac{g^{\prime}}{fg};\\\
{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{2}}{}^{02}=-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{2}}{}^{20}=\frac{f^{\prime}}{fg};\\\
{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{2}}{}^{12}=-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{2}}{}^{21}=\frac{x{\psi}^{\prime}(t-z)}{fg};\\\
{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{2}}{}^{23}=-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{2}}{}^{32}=-\frac{f^{\prime}+{\psi}(t-z)}{fg};\\\
{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{3}}{}^{03}=-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{\hat{3}}{}^{30}=\frac{f^{\prime}}{f}+\frac{g^{\prime}}{g}.\end{array}$
(4.8)

Taking (3.8) we have the Noether superpotential in TEGR (2.31):

$\begin{array}[]{cccc}{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{01}={\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{31}=-{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{10}=-{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{13}=\frac{xg{\psi}^{\prime}(t-z)}{8\pi};\\\
{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{03}=-{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{30}=\frac{g\left(f^{\prime}+{\psi}(t-z)\right)+fg^{\prime}}{8\pi};\end{array}$
(4.9)

Then, taking the divergence (2.32) of (4.9) we get the Noether current in
TEGR:

${\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{\mu}=\left\\{-\frac{gf^{\prime\prime}+2f^{\prime}g^{\prime}+fg^{\prime\prime}+{\psi}(t-z)g^{\prime}}{8\pi},\leavevmode\nobreak\
0,\leavevmode\nobreak\ 0,\leavevmode\nobreak\
-\frac{gf^{\prime\prime}+2f^{\prime}g^{\prime}+fg^{\prime\prime}+{\psi}(t-z)g^{\prime}}{8\pi}\right\\}.$
(4.10)

Applying here the Einstein equation (3.10) we get:

${\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{\mu}=\left\\{-\frac{2f^{\prime}g^{\prime}+{\psi}(t-z)g^{\prime}}{8\pi},\leavevmode\nobreak\
0,\leavevmode\nobreak\ 0,\leavevmode\nobreak\
-\frac{2f^{\prime}g^{\prime}+{\psi}(t-z)g^{\prime}}{8\pi}\right\\}.$ (4.11)

Then, the condition for zero Noether current is

${\psi}(t-z)=-2f^{\prime}.$ (4.12)

Thus a gauge compatible with the equivalence principle in the old sense is
constructed. Analogously, one can permit a dependence on $y$ and $(t-z)$ with
the same result. More complicated gauge constructed with the local Lorentz
rotations depending simultaneously on $x$, $y$ and $(t-z)$ is considered later
on the basis of work [20].

###  4.2 Gauge changing in STEGR

In this subsection, from the start we restrict ourselves by the simplest
requirement as well. We assume that changed Noether conserved quantities have
to depend on $(t-z)$ only. Komar superpotential obtained for the metric (2.38)
and vector (3.8) is left zero independently on transformations which change a
gauge. Now, consider the additional part (2.34). Because the metric (2.38)
depend on $(t-z)$ only, the teleparallel connection
$\Gamma{}^{\alpha}{}_{\mu\nu}$ included into the additional part of Noether
superpotential (2.34) through the non-metricity (2.20) should depend on
$(t-z)$ only too. Then we check if such Noether superpotentials (depending on
$(t-z)$ only) can satisfy the equivalence principle.

We assume that there exist some new coordinates $(T,X,Y,Z)$ in which the
teleparallel connection $\Gamma{}^{\alpha}{}_{\mu\nu}=0$. New coordinates
$(T,X,Y,Z)$ depend on the coordinates $(t,x,y,z)$ in a general way as:

$T=t+\Delta T,\leavevmode\nobreak\ \leavevmode\nobreak\ X=x+\Delta
X,\leavevmode\nobreak\ \leavevmode\nobreak\ Y=y+\Delta Y,\leavevmode\nobreak\
\leavevmode\nobreak\ Z=z+\Delta Z.$ (4.13)

When (4.13) is expanded in a Taylor series, the functions $\Delta T$, $\Delta
X$, $\Delta Y$, $\Delta Z$, depend on the derivatives of $(T,X,Y,Z)$ with
respect to $(t,x,y,z)$. These derivatives are included in the formula of the
transformed STEGR connection, which (after applying the coordinate
transformation from $(T,X,Y,Z)$ to $(t,x,y,z)$) is calculated as

$\Gamma{}^{\alpha}{}_{\mu\nu}=\frac{\partial x^{\alpha}}{\partial
X^{\lambda}}\frac{\partial}{\partial x^{\mu}}\left(\frac{\partial
X^{\lambda}}{\partial x^{\nu}}\right),$ (4.14)

where $x^{\mu}\equiv(t,x,y,z)$, $X^{\mu}\equiv(T,X,Y,Z)$. To make the
teleparallel connection (4.14) depend only on $(t-z)$, the derivatives of
$(T,X,Y,Z)$ with respect to $(t,x,y,z)$ defining (4.14) and, thus, the
functions $\Delta T$, $\Delta X$, $\Delta Y$, $\Delta Z$ should depend only on
$(t-z)$. Thus, assuming that the functions $\Delta T$, $\Delta X$, $\Delta Y$,
$\Delta Z$ depend on $(t-z)$ only we get for the teleparallel connection
(4.14) non-zero components

$\begin{array}[]{cccc}\Gamma{}^{0}{}_{00}=\Gamma{}^{0}{}_{33}=-\Gamma{}^{0}{}_{03}=-\Gamma{}^{0}{}_{30}=\frac{{\Delta
T}^{\prime}{\Delta Z}^{\prime\prime}-{\Delta T}^{\prime\prime}\left({\Delta
Z}^{\prime}-1\right)}{{\Delta T}^{\prime}-{\Delta Z}^{\prime}+1};\\\
\Gamma{}^{1}{}_{00}=\Gamma{}^{1}{}_{33}=\frac{{\Delta X}^{\prime}\left({\Delta
Z}^{\prime\prime}-{\Delta T}^{\prime\prime}\right)}{{\Delta
T}^{\prime}-{\Delta Z}^{\prime}+1}+{\Delta X}^{\prime\prime};\\\
\Gamma{}^{1}{}_{03}=\Gamma{}^{1}{}_{30}=\frac{{\Delta X}^{\prime}\left({\Delta
T}^{\prime\prime}-{\Delta Z}^{\prime\prime}\right)+{\Delta
X}^{\prime\prime}\left(-{\Delta T}^{\prime}+{\Delta
Z}^{\prime}-1\right)}{{\Delta T}^{\prime}-{\Delta Z}^{\prime}+1};\\\
\Gamma{}^{2}{}_{00}=\Gamma{}^{2}{}_{33}=\frac{{\Delta Y}^{\prime}\left({\Delta
Z}^{\prime\prime}-{\Delta T}^{\prime\prime}\right)}{{\Delta
T}^{\prime}-{\Delta Z}^{\prime}+1}+{\Delta Y}^{\prime\prime};\\\
\Gamma{}^{2}{}_{03}=\Gamma{}^{2}{}_{30}=\frac{{\Delta Y}^{\prime}\left({\Delta
T}^{\prime\prime}-{\Delta Z}^{\prime\prime}\right)+{\Delta
Y}^{\prime\prime}\left(-{\Delta T}^{\prime}+{\Delta
Z}^{\prime}-1\right)}{{\Delta T}^{\prime}-{\Delta Z}^{\prime}+1};\\\
\Gamma{}^{3}{}_{00}=\Gamma{}^{3}{}_{33}=-\Gamma{}^{3}{}_{03}=-\Gamma{}^{3}{}_{30}=\frac{\left({\Delta
T}^{\prime}+1\right){\Delta Z}^{\prime\prime}-{\Delta T}^{\prime\prime}{\Delta
Z}^{\prime}}{{\Delta T}^{\prime}-{\Delta Z}^{\prime}+1},\end{array}$ (4.15)

where the functions $\Delta T$, $\Delta X$, $\Delta Y$, $\Delta Z$, depend on
$(t-z)$ only.

We take again the observer’s proper vector (3.8). Because Komar superpotential
(2.33) remains zero the total Noether superpotential (2.35) is determined only
by additional part (2.34) which has non-zero components:

${\cal J}{}^{03}=-{\cal J}{}^{30}=\frac{g\left(2f^{\prime}\left(\Delta
T^{\prime}-\Delta Z^{\prime}+1\right)+f\left(\Delta Z^{\prime\prime}-\Delta
T^{\prime\prime}\right)\right)+2fg^{\prime}\left(\Delta T^{\prime}-\Delta
Z^{\prime}+1\right)}{16\pi\left(\Delta T^{\prime}-\Delta
Z^{\prime}+1\right)}.$ (4.16)

To make the Noether current zero, the Noether superpotential (following
(2.28)) should be constant ${\cal J}{}^{03}=-{\cal J}{}^{30}=A_{0}$. Then we
easily find that

$\frac{(\Delta T-\Delta Z)^{\prime\prime}}{(\Delta T-\Delta
Z)^{\prime}}=2\left(-\frac{8\pi{A_{0}}}{fg}+\frac{f^{\prime}}{f}+\frac{g^{\prime}}{g}\right).$
(4.17)

Then

$(\Delta T-\Delta
Z)^{\prime}={A}_{1}f^{2}g^{2}\exp\left(-\int\left(\frac{16\pi{A_{0}}}{fg}\right)\,dt\right),$
(4.18)

where $A_{1}$ is a constant of integration. If we assume $A_{0}=0$, in this
case the Noether superpotential is zero and

$\Delta T=\Delta Z+A_{1}\int f^{2}g^{2}dt+A_{2},$ (4.19)

where $A_{1}$ and $A_{2}$ are constants of integration. $\Delta Z$, $\Delta
X$, $\Delta Y$, can be arbitrary functions and $\Delta T$ in connected with
$\Delta Z$ by (4.18) or by (4.19) in the case of zero Noether current.

In STEGR, in contrast to TEGR, even with a very strong restriction on the
connection and Noether superpotential, requiring dependence only on $t-z$, we
have reached the goal, that is, we have found a gauge with zero current for a
free moving observer.

##  5 Obukhov-Pereira-Rubilar gauge

In [20] and [19] the authors study the problem of the energy that can be
brought by flat fronted gravitational wave. They use different set of
coordinates $(T,X,Y,Z)$, for which in $(-,+,+,+)$ signature the metric for the
gravitational wave solution has a form

$g_{\mu\nu}=\left(\begin{array}[]{cccc}-H(T-Z,X,Y)-1&0&0&H(T-Z,X,Y)\\\
0&1&0&0\\\ 0&0&1&0\\\ H(T-Z,X,Y)&0&0&-H(T-Z,X,Y)+1\\\ \end{array}\right).$
(5.1)

The Einstein equations in vacuum acquire the simple form:

$\frac{\partial^{2}H(T-Z,X,Y)}{\partial
X^{2}}+\frac{\partial^{2}H(T-Z,X,Y)}{\partial Y^{2}}=0.$ (5.2)

We discuss the results of [20] in Appendix B in detail. Here, we consider the
frame suggested in [20] in the framework of our formalism. The reason is that
result of [20] with zero energetic characteristics cloud be interpreted as
having a relation to the equivalence principle. Thus, Obukhov et. al. [20] use
a tetrad for the metric (5.1):

$h^{a}{}_{\mu}=\left(\begin{array}[]{cccc}\frac{1}{2}H(T-Z,X,Y)+1&0&0&-\frac{1}{2}H(T-Z,X,Y)\\\
0&1&0&0\\\ 0&0&1&0\\\ -\frac{1}{2}H(T-Z,X,Y)&0&0&\frac{1}{2}H(T-Z,X,Y)-1\\\
\end{array}\right).$ (5.3)

Here, we need to use the coordinates of (2.38). To transform the metric (5.1)
to the metric (2.38) under consideration we use the Formiga’s [22] Eq(2.32):

$\begin{array}[]{cccc}T=t+\frac{1}{2}\left(x^{2}f(U)\frac{df}{dU}+y^{2}g(U)\frac{dg}{dU}\right),\\\
Z=z+\frac{1}{2}\left(x^{2}f(U)\frac{df}{dU}+y^{2}g(U)\frac{dg}{dU}\right),\\\
X=f(U)x,\leavevmode\nobreak\ \leavevmode\nobreak\ Y=g(U)y\\\ U=T-Z=t-z,\\\
H(U,X,Z)=-\frac{1}{f}\frac{d^{2}f}{dU^{2}}(X^{2}-Y^{2}).\end{array}$ (5.4)

###  5.1 Zero current in TEGR

After the coordinate transformation (5.4) $(T,X,Y,Z)\rightarrow(t,x,y,z)$
metric (5.1) transforms to diagonal form (2.38). Tetrad (5.3) transforms to
the form

$h^{a}{}_{\mu}=\left(\begin{array}[]{cccc}\frac{1}{2}\left(x^{2}f^{\prime
2}+y^{2}g^{\prime
2}+2\right)&xff^{\prime}&ygg^{\prime}&\frac{1}{2}\left(-x^{2}f^{\prime
2}-y^{2}g^{\prime 2}\right)\\\ xf^{\prime}&f&0&-xf^{\prime}\\\
yg^{\prime}&0&g&-yg^{\prime}\\\ \frac{1}{2}\left(-x^{2}f^{\prime
2}-y^{2}g^{\prime
2}\right)&-xff^{\prime}&-ygg^{\prime}&\frac{1}{2}\left(x^{2}f^{\prime
2}+y^{2}g^{\prime 2}-2\right)\\\ \end{array}\right).$ (5.5)

As is was considered in [20], the corresponding ISC to the tetrad (5.3) was
zero. Because ISC transforms as spacetime vector under spacetime
transformations, corresponding to (5.5) ISC remains zero. Besides, the tetrad
(5.5) is connected to the diagonal tetrad (3.1) by
$h^{a}{}_{\mu}=\Lambda^{a}{}_{b}h_{(diag)}^{b}{}_{\mu}$, where the Lorentz
rotation $\Lambda^{a}{}_{b}$ is

$\Lambda^{a}{}_{b}=\left(\begin{array}[]{cccc}\frac{1}{2}\left(x^{2}f^{\prime
2}+y^{2}g^{\prime
2}+2\right)&xf^{\prime}&yg^{\prime}&\frac{1}{2}\left(-x^{2}f^{\prime
2}-y^{2}g^{\prime 2}\right)\\\ xf^{\prime}&1&0&-xf^{\prime}\\\
yg^{\prime}&0&1&-yg^{\prime}\\\ \frac{1}{2}\left(-x^{2}f^{\prime
2}-y^{2}g^{\prime
2}\right)&-xf^{\prime}&-yg^{\prime}&\frac{1}{2}\left(x^{2}f^{\prime
2}+y^{2}g^{\prime 2}-2\right)\\\ \end{array}\right).$ (5.6)

Thus, if one preserves zero ISC (3.7) with the tetrad (5.5) one obtains the
pair presenting the new gauge which differs from the gauge of the tetrad (3.1)
and zero ISC (3.7). We call it as the Obukhov-Pereira-Rubilar gauge.

For the new gauge, by (2.8) and (2.11), we get the teleparallel superpotential
which in all coordinate indexes has non-zero components:

$\begin{array}[]{cccc}{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{0}{}^{01}=\par{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{0}{}^{31}=\par{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{3}{}^{10}=\par{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{3}{}^{13}=-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{0}{}^{10}=\par-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{0}{}^{13}=\par-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{3}{}^{01}=\par-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{3}{}^{31}=\frac{xf^{\prime\prime}}{f};\par\\\
{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{0}{}^{02}=\par{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{0}{}^{32}=\par{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{3}{}^{20}=\par{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{3}{}^{23}=-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{0}{}^{20}=\par-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{0}{}^{23}=\par-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{3}{}^{02}=\par-{\stackrel{{\scriptstyle\bullet}}{{S}}}{}{}_{3}{}^{32}=\frac{yg^{\prime\prime}}{g}.\end{array}$
(5.7)

Taking the freely falling observer’s proper vector (3.8) in $(t,x,y,z)$
coordinates we get the Noether superpotential (2.31):

$\begin{array}[]{cccc}{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{01}={\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{31}=-{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{10}=-{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{13}=\frac{xgf^{\prime\prime}}{8\pi};\par\\\
{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{02}={\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{32}=-{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{20}=-{\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{23}=\frac{yfg^{\prime\prime}}{8\pi}.\end{array}$ (5.8)

Taking the divergence of the Noether superpotential we get the Noether current
(2.32):

${\stackrel{{\scriptstyle\bullet}}{{\cal
J}}}{}^{\mu}=\left\\{\frac{gf^{\prime\prime}+fg^{\prime\prime}}{8\pi},0,0,\frac{gf^{\prime\prime}+fg^{\prime\prime}}{8\pi}\right\\}.$
(5.9)

It is zero due to the Einstein equation (3.10).

###  5.2 Zero current in STEGR

Switching off gravity for the coordinates in (2.38) in STEGR gives a gauge
with the metric (2.38) and zero teleparallel connection (3.18). This gives
nonzero current (3.20), or (3.22). In linear approximation it rather
corresponds to the accepted energy and energy flux in (3.25), or (3.26). Here,
we switch off gravity for the metric (5.1) in the coordinates $(T,X,Y,Z)$. It
particularly gives $H(U,X,Y)=0$ and, correspondingly, zero teleparallel
connection. The transformed by (2.37) metric (5.1) goes to (2.38), and zero
teleparallel connection is transformed as well and should be calculated as

$\Gamma{}^{\alpha}{}_{\mu\nu}=\frac{\partial x^{\alpha}}{\partial
X^{\lambda}}\frac{\partial}{\partial x^{\mu}}\left(\frac{\partial
X^{\lambda}}{\partial x^{\nu}}\right),$ (5.10)

where $x^{\mu}\equiv(t,x,y,z)$, $X^{\mu}\equiv(T,X,Y,Z)$. The non-zero
components of the transformed connection are

$\begin{array}[]{cccc}\Gamma{}^{0}{}_{00}=\Gamma{}^{0}{}_{33}=\Gamma{}^{3}{}_{00}=\Gamma{}^{3}{}_{33}=-\Gamma{}^{0}{}_{03}=-\Gamma{}^{0}{}_{30}=-\Gamma{}^{3}{}_{03}=-\Gamma{}^{3}{}_{30}=\frac{1}{2}\left(x^{2}ff^{\prime\prime\prime}+x^{2}f^{\prime}f^{\prime\prime}+y^{2}gg^{\prime\prime\prime}+y^{2}g^{\prime}g^{\prime\prime}\right);\\\
\Gamma{}^{0}{}_{01}=\Gamma{}^{0}{}_{10}=\Gamma{}^{3}{}_{01}=\Gamma{}^{3}{}_{10}=-\Gamma{}^{0}{}_{13}=-\Gamma{}^{0}{}_{31}=-\Gamma{}^{3}{}_{13}=-\Gamma{}^{3}{}_{31}=xff^{\prime\prime};\\\
\Gamma{}^{0}{}_{02}=\Gamma{}^{0}{}_{20}=\Gamma{}^{3}{}_{02}=\Gamma{}^{3}{}_{20}=-\Gamma{}^{0}{}_{23}=-\Gamma{}^{0}{}_{32}=-\Gamma{}^{3}{}_{23}=-\Gamma{}^{3}{}_{32}=ygg^{\prime\prime};\\\
\Gamma{}^{0}{}_{11}=\Gamma{}^{3}{}_{11}=ff^{\prime};\\\
\Gamma{}^{0}{}_{22}=\Gamma{}^{3}{}_{22}=gg^{\prime};\\\
\Gamma{}^{1}{}_{00}=\Gamma{}^{1}{}_{33}=-\Gamma{}^{1}{}_{03}=-\Gamma{}^{1}{}_{30}=\frac{xf^{\prime\prime}}{f};\\\
\Gamma{}^{1}{}_{01}=\Gamma{}^{1}{}_{10}=-\Gamma{}^{1}{}_{13}=-\Gamma{}^{1}{}_{31}=\frac{f^{\prime}}{f};\\\
\Gamma{}^{2}{}_{00}=\Gamma{}^{2}{}_{33}=-\Gamma{}^{2}{}_{03}=-\Gamma{}^{2}{}_{30}=\frac{yg^{\prime\prime}}{g};\\\
\Gamma{}^{2}{}_{02}=\Gamma{}^{2}{}_{20}=-\Gamma{}^{2}{}_{23}=-\Gamma{}^{2}{}_{32}=\frac{g^{\prime}}{g}.\end{array}$
(5.11)

Thus, the new STEGR gauge is presented by the pair of the coordinates in
(2.38) with the connection (5.11). Using (5.11) we get the non-metricity
(2.20):

$\begin{array}[]{cccc}Q{}_{000}=\par Q{}_{033}=\par Q{}_{303}=\par
Q{}_{330}=-Q{}_{003}=\par-Q{}_{030}=\par-Q{}_{300}=\par-Q{}_{333}=\par
x^{2}ff^{\prime\prime\prime}+x^{2}f^{\prime}f^{\prime\prime}+y^{2}gg^{\prime\prime\prime}+y^{2}g^{\prime}g^{\prime\prime};\par\\\
Q{}_{100}=\par
Q{}_{133}=\par-Q{}_{103}=\par-Q{}_{130}=2xff^{\prime\prime};\par\\\
Q{}_{200}=\par Q{}_{233}=\par-Q{}_{203}=\par-Q{}_{230}=\par
2ygg^{\prime\prime}.\end{array}$ (5.12)

Taking the freely falling observer’s proper vector (3.8) in $(t,x,y,z)$
coordinates we get: zero Komar superpotential (2.33) which was found in
previous sections; by (2.24) and (2.34) one has zero additional part of
Noether superpotential. As a result, one has zero total Noether superpotential
in STEGR, what gives zero Noether current.

##  6 Concluding remarks

In article, using the developed earlier our formalism [7, 8, 26, 27, 28] in
TEGR and in STEGR, we have studied the problem of constructing conserved
quantities (energy density and energy density flux) for a flat fronted exact
(strong) gravitational wave of the only one polarization “+”. Concerning TEGR,
the applied formalism is a tensorial one and gives a possibility to construct
both local and integral conserved quantities covariant both to coordinate
transformations and to local Lorentz rotations. For the best of our knowledge,
STEGR has not still been used for construction of conserved quantities for
gravitational waves. The crucial property of the formalism both in TEGR and in
STEGR is that a displacement vector $\xi^{\alpha}$ is included. Namely,
interpretation of conserved quantities is defined by $\xi^{\alpha}$ that can
be chosen as a Killing vector of spacetime, a proper vector of an observer,
etc. Another important notion in the formalism [7, 8, 26, 27, 28] is a gauge,
that is classes of equivalence of pairs (tetrad, ISC) in TEGR, or
(coordinates, flat connections) in STEGR. To construct a physically sensible
conserved quantity one has to find out a corresponding gauge [7, 8, 28]. In
the present paper, we have found gauges when energetic characteristics of flat
gravitational wave measured by a freely falling observer are zero that is in a
correspondence with the equivalence principle. However, because the wave
solution is not stationary there is no timelike Killing vectors, formally it
means that it is impossible to construct energy density. What is interesting
is that the simplest gauge used gives non-zero results, which in the limit of
a weak wave coincides with the related pseudotensor formulae. Analogous
results obtained in the non-covariant formalism are discussed in [21, 22]. We
should stress here that despite free falling masses can be used for detecting
non-zero energy of gravitational waves, at least two masses separated by non-
zero distance are needed. Even for a simpler case of Friedmann - Robertson -
Walker cosmological metric consideration of distant masses makes energy issues
more subtle, as it, for example, have been shown in the Harrison paper with a
remarkable title “Mining energy in an expanding Universe” [38]. However, in
our studied here we consider only one point moving geodesically in a
gravitational wave metric, nevertheless, the obtained non-zero result has a
physical sense at least for a weak wave. This means that the connection
between the equivalence principle and the energy-momentum characteristics of a
gravitational field studied here needs a deeper investigation.

##  Appendix A Arbitrary Lorentz rotations

Here, we formally derive the arbitrary Lorentz rotation dependent on $(t-z)$
only, which can be applied only to the tetrad as (2.12) preserving ISC or only
to the ICS as (2.13) preserving tetrad. First, we present the compositions of
simple Lorentz rotations dependent on $(t-z)$. It was noted in subsection 4.1,
that no compositions such Lorentz rotations of the tetrad only preserving ISC
(or of ISC only preserving tetrad) can change the Noether current. Then,
connect the compositions of simple Lorentz rotations dependent on $(t-z)$ only
with arbitrary Lorentz rotation dependent on $(t-z)$ only and conclude that if
no such compositions can change the Noether current, no such arbitrary Lorentz
rotation can change the Noether current.

Arbitrary Lorentz rotation matrix $\Lambda^{a}{}_{b}(t-z)$ of $SO(1,3)$ group
can be expressed trough $so(1,3)$ algebra as

$\Lambda^{a}{}_{b}(t-z)=\exp(J_{1}\alpha_{1}(t-z)+J_{2}\alpha_{2}(t-z)+J_{3}\alpha_{3}(t-z)+K_{1}\beta_{1}(t-z)+K_{2}\beta_{2}(t-z)+K_{3}\beta_{3}(t-z)),$
(A.13)

where $\alpha_{i}(t-z),\beta_{i}(t-z)$ ($i=1,2,3$) are arbitrary functions and
the generators of algebra $so(1,3)$ are

$\begin{array}[]{cccc}J_{1}=i\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\
0&0&0&-1\\\ 0&0&1&0\\\
\end{array}\right),&J_{2}=i\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&1\\\
0&0&0&0\\\ 0&-1&0&0\\\
\end{array}\right),&J_{3}=i\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&-1&0\\\
0&1&0&0\\\ 0&0&0&0\\\ \end{array}\right),\\\
K_{1}=i\left(\begin{array}[]{cccc}0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
\end{array}\right),&K_{2}=i\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&0\\\
1&0&0&0\\\ 0&0&0&0\\\
\end{array}\right),&K_{3}=i\left(\begin{array}[]{cccc}0&0&0&1\\\ 0&0&0&0\\\
0&0&0&0\\\ 1&0&0&0\\\ \end{array}\right).\\\ \end{array}$ (A.14)

It is practically difficult to calculate directly the matrix exponent (A.13).
Therefore, we assume that there exist functions $\bar{\alpha}_{1}(t-z)$,
$\bar{\alpha}_{2}(t-z)$, $\bar{\alpha}_{3}(t-z)$, $\bar{\beta}_{1}(t-z)$,
$\bar{\beta}_{2}(t-z)$, $\bar{\beta}_{3}(t-z)$ such that

$\exp(J_{1}\alpha_{1}+J_{2}\alpha_{2}+J_{3}\alpha_{3}+K_{1}\beta_{1}+K_{2}\beta_{2}+K_{3}\beta_{3})=\exp(J_{1}\bar{\alpha}_{1})\exp(J_{2}\bar{\alpha}_{2})\exp(J_{3}\bar{\alpha}_{3})\exp(K_{1}\bar{\beta}_{1})\exp(K_{2}\bar{\beta}_{2})\exp(K_{3}\bar{\beta}_{3}),$
(A.15)

where $\bar{\alpha}_{i}\equiv\bar{\alpha}_{i}(t-z),\leavevmode\nobreak\
\bar{\beta}_{i}\equiv\bar{\beta}_{i}(t-z),\leavevmode\nobreak\
{\alpha}_{i}\equiv{\alpha}_{i}(t-z),\leavevmode\nobreak\
{\beta}_{i}\equiv{\beta}_{i}(t-z)$, $i=1,2,3$.
$\bar{\alpha}_{i},\leavevmode\nobreak\ \bar{\beta}_{i}$ can be connected with
${\alpha}_{i},\leavevmode\nobreak\ {\beta}_{i}$ using the
Baker–Campbell–Hausdorff formula [39]. This formula gives direct expression
for matrix $Z$ in $\exp(X)\exp(Y)=\exp(Z)$, as a function $Z=Z(X,Y)=\log(\exp
X\exp Y)$. Applying this formula to each term under the exponent in (A.15)
sequentially, we can get the exact expressions of matrices
$\bar{\alpha}_{i},\bar{\beta}_{i}$ as functions of matrices
${\alpha}_{i},{\beta}_{i}$. We don’t need make these calculations directly
here. It is enough for us to know that $\bar{\alpha}_{i},\bar{\beta}_{i}$ can
be expressed explicitly through ${\alpha}_{i},{\beta}_{i}$ and then take the
Lorentz rotation $\Lambda^{a}{}_{b}(t-z)$ as a composition of simple Lorentz
rotations:

$\begin{array}[]{cccc}\Lambda^{a}{}_{b}(t-z)=Exp(J_{1}\bar{\alpha}_{1})\exp(J_{2}\bar{\alpha}_{2})Exp(J_{3}\bar{\alpha}_{3})Exp(K_{1}\bar{\beta}_{1})Exp(K_{2}\bar{\beta}_{2})Exp(K_{3}\bar{\beta}_{3})=\\\
\Lambda_{(\bar{\alpha}_{1})}(t-z)\Lambda_{(\bar{\alpha}_{2})}(t-z)\Lambda_{(\bar{\alpha}_{3})}(t-z)\Lambda_{(\bar{\beta}_{1})}(t-z)\Lambda_{(\bar{\beta}_{2})}(t-z)\Lambda_{(\bar{\beta}_{3})}(t-z)\end{array}$
(A.16)

where $\bar{\alpha}_{i}$, $\bar{\beta}_{i}$, $i=1,2,3$ are arbitrary functions
of $(t-z)$ and

$\begin{array}[]{cccc}\Lambda_{(\bar{\alpha}_{1})}(t-z)=Exp(J_{1}\bar{\alpha}_{1}(t-z))=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&0\\\ 0&0&\cos[\bar{\alpha}_{1}(t-z)]&-\sin[\bar{\alpha}_{1}(t-z)]\\\
0&0&\sin[\bar{\alpha}_{1}(t-z)]&\cos[\bar{\alpha}_{1}(t-z)]\\\
\end{array}\right),\\\
\Lambda_{(\bar{\alpha}_{2})}(t-z)=Exp(J_{2}\bar{\alpha}_{2}(t-z))=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&\cos[\bar{\alpha}_{2}(t-z)]&0&\sin[\bar{\alpha}_{2}(t-z)]\\\ 0&0&0&0\\\
0&-\sin[\bar{\alpha}_{2}(t-z)]&0&\cos[\bar{\alpha}_{2}(t-z)]\\\
\end{array}\right),\\\
\Lambda_{(\bar{\alpha}_{3})}(t-z)=Exp(J_{3}\bar{\alpha}_{3}(t-z))=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&\cos[\bar{\alpha}_{3}(t-z)]&-\sin[\bar{\alpha}_{3}(t-z)]&0\\\
0&\sin[\bar{\alpha}_{3}(t-z)]&\cos[\bar{\alpha}_{3}(t-z)]&0\\\ 0&0&0&0\\\
\end{array}\right),\\\
\Lambda_{(\bar{\beta}_{1})}(t-z)=Exp(K_{1}\bar{\beta}_{1}(t-z))=\left(\begin{array}[]{cccc}\cosh[\bar{\beta}_{1}(t-z)]&\sinh[\bar{\beta}_{1}(t-z)]&0&0\\\
\sinh[\bar{\beta}_{1}(t-z)]&\cosh[\bar{\beta}_{1}(t-z)]&0&0\\\ 0&0&0&0\\\
0&0&0&0\\\ \end{array}\right),\\\
\Lambda_{(\bar{\beta}_{2})}(t-z)=Exp(K_{2}\bar{\beta}_{2}(t-z))=\left(\begin{array}[]{cccc}\cosh[\bar{\beta}_{2}(t-z)]&0&\sinh[\bar{\beta}_{2}(t-z)]&0\\\
0&0&0&0\\\ \sinh[\bar{\beta}_{2}(t-z)]&0&\cosh[\bar{\beta}_{2}(t-z)]&0\\\
0&0&0&0\\\ \end{array}\right),\\\
\Lambda_{(\bar{\beta}_{3})}(t-z)=Exp(K_{3}\bar{\beta}_{3}(t-z))=\left(\begin{array}[]{cccc}\cosh[\bar{\beta}_{3}(t-z)]&0&0&\sinh[\bar{\beta}_{3}(t-z)]\\\
0&0&0&0\\\ 0&0&0&0\\\
\sinh[\bar{\beta}_{3}(t-z)]&0&0&\cosh[\bar{\beta}_{3}(t-z)]\\\
\end{array}\right).\\\ \end{array}$ (A.17)

Arbitrary compositions of Lorentz rotations
$\Lambda_{(\bar{\alpha}_{i})}(t-z)$ and $\Lambda_{(\bar{\beta}_{i})}(t-z)$
($i=1,2,3$) are used in calculations.

##  Appendix B Not freely falling tetrad

Here, we show that the tetrad (5.3) taken in [20] is not a freely falling
tetrad and, thus, the observer that is rest in the frame (5.3) is not a freely
falling.

The inversed tetrad (5.3) is

$h_{a}{}^{\mu}=\left(\begin{array}[]{cccc}-1+\frac{1}{2}H(T-Z,X,Y)&0&0&\frac{1}{2}H(T-Z,X,Y)\\\
0&1&0&0\\\ 0&0&1&0\\\ \frac{1}{2}H(T-Z,X,Y)&0&0&\frac{1}{2}H(T-Z,X,Y)+1\\\
\end{array}\right).$ (B.1)

The time-like tetrad vector (which is taken as the observer’s 4-velocity) is

$h_{\hat{0}}{}^{\mu}=\left\\{-1+\frac{1}{2}H(T-Z,X,Y),0,0,\frac{1}{2}H(T-Z,X,Y)\right\\},$
(B.2)

and, in [20], it was assumed that (B.2) is equal to the observer’s 4-velocity,
i.e. the observer is rest in the frame (5.3).

Levi-Civita connection for (5.1) is

$\begin{array}[]{cccc}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{00}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{33}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{00}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{33}=\frac{1}{2}H^{(1,0,0)}(T-Z,X,Y);\\\
\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{03}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{30}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{30}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{03}=-\frac{1}{2}H^{(1,0,0)}(T-Z,X,Y);\\\
\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{01}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{00}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{10}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{33}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{01}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{10}=\frac{1}{2}H^{(0,1,0)}(T-Z,X,Y);\\\
\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{13}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{31}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{03}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{30}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{13}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{31}=-\frac{1}{2}H^{(0,1,0)}(T-Z,X,Y);\\\
\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{02}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{20}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{00}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{33}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{02}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{20}=\frac{1}{2}H^{(0,0,1)}(T-Z,X,Y);\\\
\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{32}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{23}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{03}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{30}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{32}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{23}=-\frac{1}{2}H^{(0,0,1)}(T-Z,X,Y),\\\
\end{array}$ (B.3)

where $U\equiv T-Z$, $H^{(1,0,0)}(U,X,Y)=\frac{\partial H(U,X,Y)}{\partial
U}$, $H^{(0,1,0)}(U,X,Y)=\frac{\partial H(U,X,Y)}{\partial X}$,
$H^{(0,0,1)}(U,X,Y)=\frac{\partial H(U,X,Y)}{\partial Y}$. The geodesic
equation has a form:

$l^{\mu}\equiv\frac{d^{2}x^{\mu}}{d\tau^{2}}+\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\mu}{}_{\alpha\beta}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau}\equiv\frac{du^{\mu}}{d\tau}+\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\mu}{}_{\alpha\beta}{u^{\alpha}}{u^{\beta}}\equiv\frac{\partial
u^{\mu}}{\partial
x^{\kappa}}u^{\kappa}+\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\mu}{}_{\alpha\beta}{u^{\alpha}}{u^{\beta}}=0,$
(B.4)

where $u^{\alpha}$ is the observer’s 4-velocity and the parameter $\tau$ can
be taken as his proper time because he moves along the time-like curves. Let’s
substitute (B.2) identified with $u^{\alpha}$ and (B.3) into the left hand
side of the equation (B.4). If we obtain zero, the observer would be freely
falling, but for general $H(U,X,Y)$ we obtain non zero result:

$l^{\mu}=\left\\{0,\frac{1}{2}H^{(0,1,0)}(U,X,Y),\frac{1}{2}H^{(0,0,1)}(U,X,Y),0\right\\}.$
(B.5)

So, the observer in [20] is not freely falling. Therefore, zero energetic
characteristics of the wave obtained in [20] with the use of (2.16) (or
(2.17)) cannot be interpreted as a correspondence with the equivalence
principle.

Acknowledgments AP has been supported by the Interdisciplinary Scientific and
Educational School of Moscow University “Fundamental and Applied Space
Research”; EE and AT are supported by RSF grant 21-12-00130. AT also thanks
the Russian Government Program of Competitive Growth of Kazan Federal
University. The authors are grateful to Alexei Starobinsky for the idea to
consider gravitational wave solution in teleparallel gravity under the
application of the Noether formalism.

## References

  * [1] Jose Beltrán Jiménez, Lavinia Heisenberg, and Tomi S. Koivisto.  The Geometrical Trinity of Gravity.  Universe, 5(7):173, 2019.
  * [2] Lavinia Heisenberg.  A systematic approach to generalisations of General Relativity and their cosmological implications.  Phys. Rept., 796:1–113, 2019.
  * [3] R. Aldrovandi and J. G. Pereira.  Teleparallel Gravity: An Introduction.  Springer, Dordrechts, 2012.
  * [4] Jose Beltrán Jiménez, Lavinia Heisenberg, Damianos Iosifidis, Alejandro Jiménez-Cano, and Tomi S. Koivisto.  General teleparallel quadratic gravity.  Phys. Lett. B, 805:135422, 2020.
  * [5] Sebastian Bahamonde, Konstantinos F. Dialektopoulos, Celia Escamilla-Rivera, Gabriel Farrugia, Viktor Gakis, Martin Hendry, Manuel Hohmann, Jackson Levi Said, Jurgen Mifsud, and Eleonora Di Valentino.  Teleparallel gravity: from theory to cosmology.  Rept. Prog. Phys., 86(2):026901, 2023.
  * [6] Muzaffer Adak, Tekin Dereli, Tomi S. Koivisto, and Caglar Pala.  General teleparallel metrical geometries.  Int. J. Geom. Meth. Mod. Phys., 20:2350215, 2023.
  * [7] E. D. Emtsova, M. Krššák, A. N. Petrov, and A. V. Toporensky.  On conserved quantities for the Schwarzschild black hole in teleparallel gravity.  Eur. Phys. J. C, 81(8):743, 2021.
  * [8] E. D. Emtsova, A. N. Petrov, and A. V. Toporensky.  Conserved quantities in STEGR and applications.  Eur. Phys. J. C, 83(5):366, 2023.
  * [9] Jose Wadih Maluf.  Accelerated observers and gravitational radiation.  Grav. Cosmol., 11:284–288, 2005.
  * [10] J. W. Maluf, F. F. Faria, and S. C. Ulhoa.  On reference frames in spacetime and gravitational energy in freely falling frames.  Class. Quant. Grav., 24:2743–2754, 2007.
  * [11] Tiago Gribl Lucas, Yuri N. Obukhov, and J. G. Pereira.  Regularizing role of teleparallelism.  Phys. Rev., D80:064043, 2009.
  * [12] Yuri N. Obukhov and Guillermo F. Rubilar.  Covariance properties and regularization of conserved currents in tetrad gravity.  Phys. Rev. D, 73:124017, 2006.
  * [13] Yuri N. Obukhov, Guillermo F. Rubilar, and J. G. Pereira.  Conserved currents in gravitational models with quasi-invariant Lagrangians: Application to teleparallel gravity.  Phys. Rev. D, 74:104007, 2006.
  * [14] Sebastian Bahamonde and Laur Järv.  Coincident gauge for static spherical field configurations in symmetric teleparallel gravity.  Eur. Phys. J. C, 82(10):963, 2022.
  * [15] Jose Beltrán Jiménez, Lavinia Heisenberg, and Tomi S. Koivisto.  The canonical frame of purified gravity.  Int. J. Mod. Phys. D, 28(14):1944012, 2019.
  * [16] Débora Aguiar Gomes, Jose Beltrán Jiménez, and Tomi S. Koivisto.  Energy and entropy in the Geometrical Trinity of gravity. Arxiv: 2205.09716 [gr-qc].  5 2022.
  * [17] Débora Aguiar Gomes, Jose Beltrán Jiménez, and Tomi S. Koivisto.  General Parallel Cosmology. Arxiv: 2309.08554 [gr-qc].  9 2023.
  * [18] J. W. Maluf, F. F. Faria, and K. H. Castello-Branco.  The Gravitational energy momentum flux.  Class. Quant. Grav., 20:4683–4694, 2003.
  * [19] J. W. Maluf and S. C. Ulhoa.  The Energy-momentum of plane-fronted gravitational waves in the teleparallel equivalent of GR.  Phys. Rev. D, 78:047502, 2008.  [Erratum: Phys.Rev.D 78, 069901 (2008)].
  * [20] Yuri N. Obukhov, J. G. Pereira, and Guillermo F. Rubilar.  On the energy transported by exact plane gravitational-wave solutions.  Class. Quant. Grav., 26:215014, 2009.
  * [21] J. B. Formiga.  The energy-momentum tensor of gravitational waves, Wyman spacetime and freely falling observers.  Annalen Phys., 530:0320, 2018.
  * [22] J. B. Formiga.  The meaning of torsion in teleparallel theories in three parts: II. On the teleparallel frame problem. Arxiv: 2004.10788v5 [gr-qc].  2020\.
  * [23] R. N. Manchester.  Pulsars and Gravity.  Int. J. Mod. Phys. D, 24(06):1530018, 2015.
  * [24] John Antoniadis.  Gravitational Radiation from Compact Binary Pulsars.  Astrophys. Space Sci. Proc., 40:1–22, 2015.
  * [25] Gilles Esposito-Farese.  Binary pulsar tests of strong field gravity and gravitational radiation damping.  In 10th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories (MG X MMIII), pages 647–666, 2 2004.
  * [26] E. D. Emtsova, A. N. Petrov, and A. V. Toporensky.  Conserved currents and superpotentials in teleparallel equivalent of GR.  Class. Quant. Grav., 37(9):095006, 2020.
  * [27] E. D. Emtsova, A. N. Petrov, and A. V. Toporensky.  On conservation laws in teleparallel gravity.  J. Phys. Conf. Ser., 1557(1):012017, 2020.
  * [28] E. D. Emtsova, M. Krššák, A. N. Petrov, and A. V. Toporensky.  On the Schwarzschild solution in TEGR.  J. Phys. Conf. Ser., 2081(1):012017, 2021.
  * [29] N. V. Mitskevich.  Physical Fields in General Relativity Theory.  Nauka, Moscow, 1969.
  * [30] Alexander N. Petrov, Sergei M. Kopeikin, Robert R. Lompay, and Bayram Tekin.  Metric Theories of Gravity: Perturbations and Conservation Laws, volume 38 of De Gruyter Studies in Mathematical Physics.  De Gruyter, 4 2017.
  * [31] Muzaffer Adak and Caglar Pala.  A novel approach to autoparallels for the theories of symmetric teleparallel gravity.  J. Phys. Conf. Ser., 2191(1):012017, 2022.
  * [32] Jose Beltrán Jiménez and Tomi S. Koivisto.  Lost in translation: The Abelian affine connection (in the coincident gauge).  Int. J. Geom. Meth. Mod. Phys., 19(07):2250108, 2022.
  * [33] Alexey Golovnev, Tomi Koivisto, and Marit Sandstad.  On the covariance of teleparallel gravity theories.  Class. Quant. Grav., 34(14):145013, 2017.
  * [34] E. D. Emtsova and A. N. Petrov.  A moving black hole in TEGR as a moving matter ball.  Space, Time and Fundamantal Interactions., (39):18–25, 2022.
  * [35] E. D. Emtsova and A. N. Petrov.  On gauges for a moving black hole in TEGR.  Gen. Rel. Grav., 54(10):114, 2022.
  * [36] L. D. Landau and E. M. Lifschits.  The Classical Theory of Fields.  Pergamon Press, 1975.
  * [37] Manuel Hohmann, Laur Järv, Martin Krššák, and Christian Pfeifer.  Modified teleparallel theories of gravity in symmetric spacetimes.  Phys. Rev. D, 100(8):084002, 2019.
  * [38] E. Harrison.  Mining Energy in an Expanding Universe, volume 446.  Astrophysical Journal, 1995.
  * [39] N. Jacobson.  Enveloping Algebras of Semi-Simple Lie Algebras, pages 77–86.  Birkhäuser Boston, Boston, MA, 1989.