# Drifter: Efficient Online Feature Monitoring for Improved Data Integrity in
Large-Scale Recommendation Systems

Blaž Škrlj<EMAIL_ADDRESS>0000-0002-9916-8756 Outbrain
Inc.LjubljanaSlovenia ,  Nir Ki-Tov<EMAIL_ADDRESS>Outbrain
Inc.NetanyaIsrael ,  Lee Edelist<EMAIL_ADDRESS>-NetanyaIsrael ,  Natalia
Silberstein<EMAIL_ADDRESS>Outbrain Inc.NetanyaIsrael ,  Hila
Weisman-Zohar<EMAIL_ADDRESS>-NetanyaIsrael ,  Blaž Mramor
<EMAIL_ADDRESS>Outbrain Inc.LjubljanaSlovenia ,  Davorin Kopič
<EMAIL_ADDRESS>Outbrain Inc.LjubljanaSlovenia and  Naama Ziporin
<EMAIL_ADDRESS>Outbrain Inc.NetanyaIsrael

(2023)

###### Abstract.

Real-world production systems often grapple with maintaining data quality in
large-scale, dynamic streams. We introduce Drifter, an efficient and
lightweight system for online feature monitoring and verification in
recommendation use cases. Drifter addresses limitations of existing methods by
delivering agile, responsive, and adaptable data quality monitoring, enabling
real-time root cause analysis, drift detection and insights into problematic
production events. Integrating state-of-the-art online feature ranking for
sparse data and anomaly detection ideas, Drifter is highly scalable and
resource-efficient, requiring only two threads and less than a gigabyte of RAM
per production deployments that handle millions of instances per minute (model
training). Drifter’s effectiveness in alerting and mitigating data quality
issues was demonstrated on a real-life system that handles up to a billion
predictions per second.

feature monitoring, recommender systems, online learning, online advertising

††copyright: acmcopyright††journalyear: 2023††doi: XXXXXXX.XXXXXXX††journal:
TOG††journalvolume: 37††journalnumber: 4††article: 111††publicationmonth:
8††ccs: Computer systems organization Real-time systems††ccs: Information
systems Data mining††ccs: Information systems Data stream mining††ccs:
Information systems Computational advertising††ccs: Information systems
Information integration

##  1\. Introduction

Designing and developing online machine learning systems is a complex
endeavour, where data quality and integrity plays a crucial role for models’
online performance. This is in particular the case for contemporary
recommender systems (see (Hoi et al., 2021)), which often rely on frequent
model updates (every few minutes or even less) and are thus subject to
especially rapid negative impact from data quality degradation. To address
potential data quality issues, such recommender systems can considerably
benefit from supporting data-monitoring systems that enable fast
alerting/responsiveness when data-related issues are present. Systems like
Greykite (Hosseini et al., 2021) enable automated forecasting and facilitate
profiling of emerging issues related to internal system behaviour. The study
of online features’ behavior is commonly referred to as _online feature
selection_. This branch of methods attempts to distill relevant features from
irrelevant ones in an online learning setting (Haug et al., 2020). Approaches
focusing on mining larger (online) data sets must be computationally efficient
and easily interpretable (Hoi et al., 2012). Online feature selection and
ranking has also been a lively research endeavour for building real-time
recommender systems. For example, (Wang et al., 2015) demonstrated that
feature groups are a possible way of efficient online feature selection since
similar features tend to behave similarly in time. Furthermore, large amounts
of data that require processing can already represent substantial
computational burden as processing and transformation of instances can be
expensive. It was shown that higher-dimensional feature spaces in online
settings require specialized approaches that are versatile enough and can
scale with real-life data sizes (Manikandan and Abirami, 2021). The field of
online feature ranking is a vibrant area of research and development; however,
it needs to extend beyond the algorithmic aspects typically associated with it
– in addition to the algorithms, there is a growing need for systems that can
effectively monitor features in _real-time_. Design of such systems also
encompasses implementing and deploying mechanisms that enable online
inspection of feature scores and other related metrics (e.g., features’
cardinalities and coverages). In order to optimize utility and effectiveness,
these monitoring systems cannot exist in isolation. They must be coupled with
a visibility layer incorporating _alerting mechanisms_ and visualizations.
This integration allows for an accessible and user-friendly interface,
enabling the study of granular details of the data consumed by real-time
models. By providing such visibility, users can gain insights into the
underlying factors influencing the performance and behaviour of these models,
and make informed decisions and analysis based on them.

The work presented in this paper builds upon recent ideas and advances in both
online feature ranking and real-time signal analysis. Drawing on these
domains, we developed a system that has been deployed in a large-scale online
cloud environment. This system handles real-life data streams for various use
cases, including click-through rate prediction, conversion rate prediction,
and item viewability prediction. Furthermore, by operating in real-world and
real-time settings, the system is actively used when handling complex, dynamic
data monitoring scenarios. Overall, the paper emphasizes the importance of
online feature monitoring and highlights the value of integrating this
functionality with a comprehensive visibility layer. Through this integration,
the presented system offers a powerful tool for inspecting and analyzing the
most granular levels of data used in real-time models, ultimately contributing
to improved decision-making and performance optimization in various
application domains.

The remainder of this paper is structured as follows. We begin by discussing
the existing use cases where online feature monitoring and verification was
used to facilitate deployment, monitoring and understanding of online
recommender systems. Second, we describe Drifter, the system used for online
feature monitoring and verification, its implementation, and the metrics
implemented for measuring feature drifts and anomalies online. Third, we
describe a use case where Drifter helped profile and identify features that
were subject to drift. Finally, we discuss the implications and lessons
learned when deploying and designing Drifter.

##  2\. Online feature monitoring overview

We describe the main use cases where online feature monitoring is a suitable
approach for understanding, mitigating and improving online learning
processes.

Introduction of new features A typical process in many online learning
workflows involves the introduction of new signals or features. However, due
to the dynamic nature of online learning, incorporating new features often
requires substantial engineering efforts that may span multiple teams.
Unfortunately, various factors, such as miscommunications, logging bugs, or
other issues can unintentionally impact these features’ distribution,
coverage, or relative importance. To mitigate these challenges, it is crucial
to have a mechanism in place to monitor and automatically raise alerts based
on predefined conditions that indicate problematic change in feature values.
By proactively detecting and addressing such issues as early as possible in
the data pipeline, we are able to prevent the deployment of models that could
negatively affect the business. Furthermore, understanding how existing
features vary in time provides an additional perspective on their behaviour.
This knowledge can be valuable for prioritizing testing of new features and
their transformations, used by the predictive model.

Measuring quality drops of existing features As soon as an online feature
monitoring system is capable of running in real-time, it can (and does) serve
as a ”ghost mode” for a given data stream (consumed by, e.g., click-through
rate or conversion rate prediction models). Furthermore, by systematically
measuring features’ properties such as cardinality, coverage, and statistics
such as quantiles and histograms, the monitoring system can, in a matter of
milliseconds, alert relevant model stakeholders that a change in the
distribution of an existing feature has occurred. Such events can occur due to
multiple reasons; examples include changes in a component that participates in
the construction or final transformation of the feature, a drop of data
quality due to an external event and feature drifts – gradual changes in a
feature’s distribution, eventually resulting in problematic behaviour (Barddal
et al., 2017, 2015).

Debugging online models When working with data streams in online learning
systems, explainability can become a challenge – the use of deep factorization
machine-based models or similar variants have made it increasingly difficult
and time-consuming to inspect problematic models directly. However, by
establishing a connection between a model’s behaviour and detectable shifts in
feature distributions, it becomes possible to conduct more systematic _post-
hoc_ evaluations of the model itself. One such example includes perturbation-
based analysis, which focuses on identifying the effects of the distribution
shifts of single or multiple features. Observing such shifts automatically
online makes it easier to link them to the model’s behaviour and gain insights
into its performance and potential issues. Further, understanding temporal
behaviour facilitates the design of follow-up offline experiments that help
identify potentially useful transformations of existing features.
Understanding feature distribution shifts thus enables a more structured
evaluation of the model, facilitating the identification and resolution of
problems that may arise during online learning.

Understanding of temporal dynamics of features The endeavour to study the
behaviour of multiple features online simultaneously is not necessarily
considered due to its time-consuming nature. However, by being able to observe
whole feature spaces’ dynamics in time, patterns related to the complementary
nature of features can arise, deepening the data scientists’ understanding of
which features fluctuate together; understanding temporal relations can help
with the creation of new features that account for this dynamics, or
facilitate exploration of alternative features that would otherwise be
ignored. For example, separate teams can introduce features in isolation, not
being aware of their complementary nature – by visualizing the joint space,
such patterns can be studied and can further simplify existing models.

Online ranking of features’ contributions Productization of new signals
(features) can be expensive and time-consuming. By simulating a feature’s
behaviour with the target space of interest, prioritization of new features
can be facilitated, saving valuable resources that would otherwise be spent on
multiple deployments, running A/B tests and other costly procedures, that were
always inevitable for testing the new features’ contribution to the target
space. We proceed with an overview of related work.

We continue the discussion with an overview of existing feature
monitoring/inspection systems and how they compare to Drifter. An overview of
how Drifter compares to existing products and tools is summarized in Table 1.
The selected tools include existing, well-established solutions for online
feature store construction and subsequent machine learning, as well as up-an-
coming solutions.

Table 1. Overview of existing online feature monitoring methods and their main
(out-of-the-box) properties.

method | Own ML engine | Feature constr. | Extensions | Streaming statistics | Sparse data | Metrics support | Drift detection | Online visualiz. | Main use case
---|---|---|---|---|---|---|---|---|---
feathr (LinkedIn) |  |  |  | – | – |  |  |  | Online store
hopsworks |  |  |  |  | – |  |  | – | End-to-end platform
RasgoQL |  | – |  |  | – |  |  |  | Scaling Pandas
Vertex AI |  |  |  |  |  | – |  | – | End-to-end platform
Tecton |  |  |  |  |  |  |  | – | End-to-end platform
Amazon SAGEMAKER Store |  | – |  | – |  |  |  | – | End-to-end platform
butterfree |  |  | – |  |  |  |  |  | End-to-end platform
ByteHub111test |  |  | – |  | – |  |  |  | Online store
Databricks feature store |  |  |  | – | – | – |  | – | Online store
Drifter (this paper) |  |  | – |  |  |  |  |  | Drift detection and alerting

The three possible table marks represent complete feature/capability (), lack
thereof () or partial compatibility (–). The categories of comparison were
selected in a way to entail different properties of this type of systems; from
their extendability, compliance with sparse data formats to less known
functionalities such as drift detection and support for computation of
streaming statistics (sketching algorithms). The more extensive solutions
include feathr222https://github.com/feathr-ai/feathr, Vertex
AI333https://cloud.google.com/vertex-ai/docs/featurestore, SAGEMAKER
Store444https://aws.amazon.com/sagemaker/feature-store,
Tecton555https://www.tecton.ai/ and Databricks feature
store666https://docs.gcp.databricks.com/machine-learning/feature-
store/index.html. Other considered systems include
ByteHub777https://github.com/bytehub-ai/bytehub,
butterfree888https://github.com/quintoandar/butterfree,
RasgoQL999https://github.com/rasgointelligence/RasgoQL and
hospworks101010https://github.com/logicalclocks/hopsworks. The capabilities
were assessed based on products’ websites and readily available documentation.
We observed that most end-to-end tools could be, with varying degrees of
effort, extended to fulfuil missing/partially missing fields of comparison (we
evaluated out-of-the-box capabilities that require minimum engineering
overhead). Finally, systems that do not employ their own engines are mostly
based around Scikit-learn library (Pedregosa et al., 2011).

##  3\. Drifter - an overview

Having established the motivating use cases which led us to build Drifter, we
continue with its overview, design and implementation choices, user
interaction with the service and future applications.

###  3.1. Overall architecture and implementation

Drifter was built as a component of the microservice architecture. It was
built to operate with a distributed data source and a metrics endpoint of
choice; in the presented work, however, it is a stand-alone service that
receives the data via Hive111111https://hive.apache.org/, and outputs the
metrics to a metrics service (Prometheus121212https://prometheus.io/). This
design choice was undertaken so that each use case (e.g., a team owning a CTR
prediction or a viewability prediction model) has ownership over the relevant
Drifter instance(s), and can modify their queries or data regime according to
their preferences. Each metrics endpoint is aware of a particular deployment,
meaning that Grafana dashboards can be built with specific Drifter instance(s)
in mind. This way, isolation of metrics is possible, but at the same time,
teams can use other teams’ metrics and information when defining their
visualizations and alerts. The service itself is deployed to an in-house cloud
platform, where each Drifter instance is monitored by default (resource-wise),
offering users insights into the amount of resources required per each Drifter
pod – this functionality is useful for profiling changes in data regimes and
their impact on the resources (which are finite for each use case). A single
Drifter instance is summarized in Figure 1(a). Each Drifter is a self-
contained unit that can be deployed on a per-demand basis in the in-house
cloud platform. Although multiple different Drifters are simultaneously
deployed (different use cases), their metrics are aggregated in a joint
endpoint (Prometheus), enabling a global overview of features being monitored
and their states. This overview is shown in Figure 1(b).

[Visualization of anomaly plots.]

(a) Cloud-based perspective of the Drifter service.

(b) Overview of a single Drifter service instance.

Figure 1. Overview of Drifter and its placement into a broader (cloud
computing grid) context. (a) The second layer of services is comprised of
individual deployments (Figure 1). Their metrics are jointly aggregated,
enabling the infrastructure team to monitor the global landscape of use cases
for potentially extreme events (e.g., missing feature values, data leaks or
similar). (b) Each instance has a scheduling system that governs the data
updates/ingestion. The ranking engine enables fast feature profiling that can
be inspected at the level of each Drifter pod or in Grafana-based dashboards
derived by the produced metrics.

We continue with a more detailed overview of each of the components that
constitute Drifter (service). Drifter is implemented as a Python-based
service, utilizing an in-house library that, out-of-the-box, enables reporting
of Prometheus-based metrics per deployed pod. Each Drifter has its internal
scheduler, enabling flexibility in terms of time zones. In the initial phase
of development, we observed two main computational bottlenecks to running
Drifter instances online at scale: Consuming production-level data volumes
comprised of up to millions of instances per ten minutes, and computing scores
between features of interest. The computationally expensive parts of feature
ranking related to score computing are written in Numba (Lam et al., 2015) (an
LLVM-based Python JIT compiler). Dockerized Drifter instances communicate with
Prometheus endpoints (metrics). A Grafana-based dashboard enables the
inspection of these metrics in real-time. The service is implemented with ease
of on-boarding in mind; when a new use case needs to be accommodated (e.g., a
novel model), a template Drifter pod is cloned and configured to operate with
a dedicated data stream specific to a given use case. This way, use cases are
separate and do not interfere with one another. Further, as they jointly push
metrics to the common endpoint, visibility at the level of all active Drifter
pods is possible and facilitates monitoring of their health by the
infrastructure team.

In order for Drifter to be available to as many use cases as possible, we
optimized the service to a point it requires less than 1GB of RAM and only two
threads. This was achieved by inspecting and optimizing Hive queries and the
ranking engine itself. Optimizations that enable such low footprint include
mini-batch feature ranking, probabilistic estimation of cardinalities
(Hyperloglog-based counting), hashing trick for clipping values to a fixed
(integer) range and randomized estimation of feature interactions – for each
mini-batch, the number of interactions computed is upper-bounded by a fixed
maximum number, ensuring consistent performance. Overview of a live benchmark
of the service on a week of production data is shown in Figure 2. Memory limit
(last plot in (a)) was set to 1GB. It can be observed that on average less
than a single CPU is used. Memory spikes observed around e.g., 3rd of July
correlate to variability in data quantity received by the service, showing
resilience of Drifters to traffic spikes and similar events. Further, it can
be observed that fluctuations in data have an impact that is within the
resource constraints of the deployed pod. Each Drifter instance, as soon as
it’s deployed, produces metrics for resource consumption (apart from the ones
related to feature space). Subfigure (b) demonstrates performance if the in-
built ranking engine, in particular Numba-based re-implementation of mutual
information. The algorithm was further extended to skip computations that are
redundant due to data sparsity - benchmark shows evaluations of input vectors
of different sizes with varying degrees if in-built sparseness (from 1% to 50%
present values). For very sparse inputs, sparsity-aware mutual information can
be substantially faster (on average it is comparable to the baseline).

[Visualization of anomaly plots.]

(a) Overview of resource utilization of Drifter instance (seven day period)
that monitors CTR-related traffic. Drifter pod is stable even during traffic
peaks (last sub-plot). CPU utilization is minimal (first plot).

(b) Performance of Numba-based mutual information implementation against the
established Scikit-learn one (different configurations ran ten times). Green
samples represent a realistic scenario where only 30% or less values are
present.

Figure 2. One colour represents one distinct feature. For this model, daily
fluctuations in coverage are apparent for some features (e.g., the pink-
coloured trace at the bottom part).

###  3.2. Visualization layer

A vital component of each Drifter instance are its resulting visualizations.
The design choice of metric-based visualizations (Prometheus and Grafana)
enabled us to generalize metric retrieval and, at the same time, facilitate
the creation of custom metrics and alerts based on them. Further, as
Prometheus comes equipped with a collection of aggregation functions, users
can create complex queries based on many existing examples while also sharing
the knowledge – resulting PromQL queries can easily be shared and tested
across use cases. Similarly, Grafana-based visualizations are a direct
extension of the metrics. Out-of-the-box capabilities suffice for most use
cases, are flexible and customizable, and are easy to maintain or change if
needed. An example visualization of feature coverage in time is shown in
Figure 3(a).

[Visualization of anomaly plots.]

(a) One week of coverage information for a given feature space (click-through
rate model).

(b) Cardinality fluctuations in time.

Figure 3. One colour represents one distinct feature. For this model, daily
fluctuations in coverage are apparent for some features (e.g., the pink-
coloured trace at the bottom part).

A similar view showing features’ cardinalities is shown in Figure 3(b).

A more complex example includes the computation of drifts – changes in the
feature’s value within a given (parametrized) time frame. Parameterization of
this aspect was necessary, as different use cases adhere to different temporal
dynamics. An example is shown in Figure 4.

[Visualization of anomaly plots.]

(a) An example visualization of a metric associated with feature drifts.
Periodic and anomalous behavior are considered.

(b) Apart from basic statistics such as coverage and cardinality, Drifter also
performs online feature ranking.

Figure 4. Overview of online feature drift-related metrics. (a) Large coverage
changes are problematic and can be directly detected by Drifter. Every colour
represents a unique feature. The green (highest) bars indicate a feature that
varies in coverage (in time).

The visualizations above are possible with out-of-the-box Grafana
capabilities. However, more custom visualizations are also possible and are
implemented at the pod level. An example includes interactive hierarchical
clusters of time series of features’ cardinalities.

##  4\. Use case example: Anomaly detection

Having discussed the overall Drifter architecture, we proceed with a
collection of production use cases where Drifter was used to facilitate and
enable data-related monitoring.

[Visualization of anomaly plots.]

(a) A feature’s coverage.

(b) Coverage drift events in real-time.

Figure 5. Different perspectives of a feature-level anomaly. Different metrics
capture the anomaly, indicating a period where the feature was missing –
negative impact on its score is complementary to drops in coverage. The lift
in standard deviation indicates that the process which generates the feature
also changed, as even though its scores normalized after the anomaly, its
deviation changed.

Models responsible for click-through and conversion rate prediction can be
comprised of hundreds of features. By monitoring each feature that is in
production’s distribution and its shifts, Drifter instances alert the users
when, e.g., a feature’s coverage or cardinality score changes beyond
expectations. We describe a use case where a feature that required effort from
multiple teams to be productized required additional inspection as it
misbehaved online – being one of the more-relevant features, impact on model
quality could be detected (with slight delay). The visualization layer of
Drifter (Grafana-based visualizations of PromQL-based queries), apart from
monitoring of the feature’s distribution, also enables comparisons to previous
time points. If the difference in a feature’s coverage is beyond a pre-
defined, acceptable threshold, Drifter logs it as an anomaly (visible in a
designated dashboard panel), and can trigger the related alert. Each
visualization considers metrics derived from the main signals outputted by
each Drifter instance. PromQL query examples are summarized in Table 2. The
examples are straightforward to implement and are easily extensible with
Prometheus’s in-built capabilities related to computation of derivatives,
linear extrapolation and similar. However, we observed that simple ”deltas”
between a quantity in a designated time frame already offer sufficient
information for a better understanding of the issue at hand. Note that this
setup does not require any external anomaly detection services (e.g.,
‘prometheus-anomaly-detector‘131313https://github.com/AICoE/prometheus-
anomaly-detector), even though it can be extended to offer such
functionalities should the need arise.

Table 2. Overview of selected metrics that enable flexible monitoring of
metric changes in time. The _metric_ part of specifications contains
information about specific deployments, time intervals and ranges and other
characteristics that enable detailed comparisons.

Metric’s use | PromQL
---|---
Sliding differences - relative | $abs((\textrm{avg by}(\textrm{feature\\_name})(metric)\textrm{ offset interval})/(\textrm{avg by}(\textrm{feature\\_name})(metric)*100>\textrm{threshold}$
Sliding differences - absolute | $abs((\textrm{avg by}(\textrm{feature\\_name})(metric)\textrm{ offset interval})-(\textrm{avg by}(\textrm{feature\\_name})(metric)*100>threshold$
Outlier detection | $\textrm{stddev by}(\textrm{feature\\_name})(metric)>1/2*\textrm{avg by}(\textrm{feature\\_name})(metric)$

##  5\. Lessons learned and conclusions

Initially, Drifter was conceived as a stand-alone service with its front-end
interface, enabling users to explore ”personalized” visualizations
independently of others. Even though part of this implementation remains
available for each Drifter pod, we realized that the users’ needs to define
novel metrics of interest over longer time ranges could not be mimicked
elegantly at the level of a custom in-house front-end solution. Furthermore,
the architectural change that came with one UI per pod included extended data
retention (per pod), causing additional disk overhead even though as soon as
metrics are computed, raw outputs are no longer needed. By adopting the
metric-push approach, we substantially reduced the computational resources
required per Drifter pod and, at the same time, facilitated visualizations.
Drifter was initially designed to incorporate all required algorithms for fast
and accurate ranking. However, by realizing that our in-house built, already
optimized solution for feature ranking, enables fast-enough ranking out-of-
the-box, we proceeded by creating an interface rather than writing a Drifter-
specific solution. This way, algorithmic improvements already present in the
proposed feature ranking engine could be further optimized and modified for
Drifter. Furthermore, should the need arise, switching the ranking engine is
straightforward, a matter of a changing interface. We finally discuss the
design choice of using the same data pipeline/processing steps as most machine
learning flows. As Drifter can operate with raw Hive 141414https://hive.com
dumps directly, there was initially no need to be entirely aligned with the
processing steps undertaken to prepare data for, e.g., CTR prediction.
However, we concluded that approximating the existing flows (data-wise) is a
mandatory capability, as Drifters are, in most use cases, utilized to help
explain anomalies or drifts of data that is fed to the prediction engine(s).
This way, alignment up to the level of input (Vowpal Wabbit format -
compressed) enabled us to approximate and enable the study of the very inputs
that are used for downstream machine learning (e.g., CTR, CVR). Finally, we
discussed the design choices made along the way, to guide similar projects in
avoiding the repetition of mistakes that were successfully addressed through
the implementation of Drifter. The feature ranking and verification engine
used in Drifter will be made public soon as an open source project.

## References

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#  Conserved quantities in STEGR and applications

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

###### Abstract

We derive conservation laws in Symmetric Teleparallel Equivalent of General
Relativity (STEGR) with direct application of Noether’s theorem. This approach
allows us to construct covariant conserved currents, corresponding
superpotentials and invariant charges. A necessary component of our
constructions is the concept of “turning off” gravity, introduced in the
framework of STEGR to define the flat and torsionless connection. By
calculating currents, one can obtain local characteristics of gravitational
field like energy density. Surface integration of superpotentials gives
charges which correspond to global quantities of the system like mass,
momentum, etc. To test our results for the obtained currents and
superpotentials, we calculate the energy density measured by freely falling
observer in the simple solutions (Friedman universe, Schwartzchild black hole)
and total mass of the Schwartzchild black hole. We find ambiguities in
obtaining the connection, which explicitly affect the values of conserved
quantities, and discuss possible solutions to this problem.

##  1 Introduction

General Relativity (GR) quite accurately describes modern astrophysical
observations. However, at the cosmological level, to satisfy the observations,
one has to introduce phenomenologically additional structures: dark matter —
to explain galaxy rotation curves, dark energy — to explain the accelerated
expansion of the universe. Theoretical explanation of dark matter and dark
energy can be made by introducing various exotic models of elementary
particles, or by different modified theories of gravity.

Most of modified gravity theories are based on the Riemannian curvature, but
one can also modify general relativity basing on its equivalent formulations,
such as Teleparallel Equivalent of General Relativity (TEGR), Symmetric
Teleparallel Equivalent of General Relativity (STEGR) and other metric-affine
theories [1]. Such modifications like $f(T)$, $f(Q)$, and others have
attracted growing interest in recent years. From a theoretical point of view,
torsion- and non-metricity-based theories are interesting because their field
equations are second-order equations.

Torsion-based teleparallel theories of gravity including TEGR itself [2], and
its various modifications, such as $f(T)$ gravity [3] and other models [4, 5]
have been more developed and studied than non-metricity-based theories
including STEGR with it’s modification. However, the latter have been
developing more and more intensively in recent years [6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19]. In the framework of symmetric teleparallel
gravity, there are considered black hole solutions [20, 21, 22, 23],
cosmological problems [24, 25, 26, 27, 28, 23, 29], PPN formalism [30] etc. It
often happens that problems which have arisen in torsion-based teleparallel
theories remain relevant in non-metricity-based theories.

In general relativity, there are problems in construction of covariant
conserved quantities: taking the Lagrangian of the first derivatives of
dynamical variables only, it is impossible to construct them without
introducing additional structures. Indeed, then one obtains either coordinate
non-covariant quantities in metric GR, or coordinate covariant but Lorentz
non-covariant quantities in tetrad GR [31]. In TEGR, this problem has been
solved. For example, in papers [32, 33], using Noether theorem, fully
covariant conserved quantities are constructed in the formalism of
differential forms, which is not so popular in astrophysical and cosmological
applications. In [34, 35] covariant Noether conserved quantities were
constructed for general metric-affine gravity including TEGR and STEGR but
their correspondence to physically expected values was not studied. In [36,
37], first, fully covariant conserved quantities in TEGR were constructed in
the tensorial formalism by direct application of the Noether theorem. And,
second, it was shown that Noether’s current describes the Einstein equivalence
principle for the freely falling observers “frozen” into the Hubble flow in
Friedmann–Lemaître–Robertson–Walker (FRLW) universe and (anti-)de Sitter
space, and Noether’s charge gives correct black hole mass for the
Schwarzschild black hole. In [38, 39], in TEGR, the Noether conserved
quantities were studied in detail in different coordinate frames of the
Schwarzschild solution, and the cases when they corresponded to physically
meaningful ones where identified. In [40, 41] it was shown that the Noether
conserved quantities give correct mass and momentum for the moving
Schwarzschild black hole.111Noether’s procedure can be useful not only for
construction of conserved quantities. For example, in [42], assuming a set of
Noether’s symmetries in the framework of $f(T)$ theory, the authors find new
solutions. The static spherically symmetric spacetime in $f(T)$ gravity is
written in terms of the well-known Schwarzschild spacetime modified by a
distortion function that depends on a characteristic radius. The obtained
solutions are new ones and cannot be obtained by the usual methods.

The purpose of this article is to begin a development of the research program
carried out the framework of TEGR [36, 37, 38, 39] within the framework of
STEGR. That is, using Noether’s theorem, we derive conservation laws and
corresponding conserved quantities in STEGR, and then study their applications
on the examples of the FRLW universe of the Schwarzschild black hole. The
basis of the constructions in [36, 37, 38, 39] was also the definition of
generalized and universal “turning off” gravity principle and generalized
concept of “gauges” in TEGR. Then the use of these constructions in
applications was presented. Even within the framework of generalized and
universal principles, problems and ambiguities have been found. These
ambiguities have been studied and their interpretation have been given. Here,
the “turning off” gravity principle, similar to that in TEGR, is introduced
and applied; the results of its application are discussed.

The paper is organized as follows. In the next section 2 we introduce the main
quantities of STEGR. In section 3 we (i) describe the Noether pricedure, (ii)
construct covariant conserved quantities with direct application of Noether’s
theorem in STEGR, (iii) introduce the “turning off” gravity principle to
obtain the flat and torsionless connection in STEGR. In section 4 we first
find that the Noether current is in a correspondence with the Einstein
equivalence principle for the freely falling observer in the FRLW universe.
Then we study if the Noether conserved quantities give physically expected
values in different coordinates of the Schwarzschild solution: total black
hole mass and a correspondance to the Einstein equivalence principle for the
freely falling observer. In section 5 we outline new results.

##  2 Elements of STEGR

The Lagrangian in STEGR has the form [1]:

$\stackrel{{\scriptstyle}}{{\cal
L}}{}=\frac{\sqrt{-g}}{2\kappa}g^{\mu\nu}(L^{\alpha}{}_{\beta\mu}L^{\beta}{}_{\nu\alpha}-L^{\alpha}{}_{\beta\alpha}L^{\beta}{}_{\mu\nu}),$
(2.1)

where $\kappa=8\pi$, and disformation is

$L^{\alpha}{}_{\mu\nu}=\frac{1}{2}Q^{\alpha}{}_{\mu\nu}-\frac{1}{2}Q_{\mu}{}^{\alpha}{}_{\nu}-\frac{1}{2}Q_{\nu}{}^{\alpha}{}_{\mu}\,.$
(2.2)

Here, Greek indices are spacetime ones: $\alpha,\beta,\ldots=(0,1,2,3)$; Latin
indices will denote only space ones: $i,j,\ldots=(1,2,3)$; the non-metricity
$Q_{\alpha\mu\nu}$ is defined as follows:

$Q_{\alpha\mu\nu}\equiv\nabla_{\alpha}g_{\mu\nu},$ (2.3)

where the covariant derivative $\nabla_{\alpha}$ is defined with the use of
the connection $\Gamma^{\alpha}{}_{\mu\nu}$ which is symmetric in lower
indexes and 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:

$R^{\alpha}{}_{\beta\mu\nu}(\Gamma)=\partial_{\mu}\Gamma^{\alpha}{}_{\nu\beta}-\partial_{\nu}\Gamma^{\alpha}{}_{\mu\beta}+\Gamma^{\alpha}{}_{\mu\lambda}\Gamma^{\lambda}{}_{\nu\beta}-\Gamma^{\alpha}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\mu\beta}=0.$
(2.4)

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

$L^{\beta}{}_{\mu\nu}\equiv\Gamma^{\beta}{}_{\mu\nu}-\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\beta}{}_{\mu\nu},$
(2.5)

where $L^{\beta}{}_{\mu\nu}$ and
$\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\beta}{}_{\mu\nu}$ are just the
disformation and the Levi-Civita connenction, respectively.

With the use of (2.2) - (2.5) one can rewrite (2.1) as

$\stackrel{{\scriptstyle}}{{\cal L}}{}=\stackrel{{\scriptstyle}}{{\cal
L}}{}_{Hilb}+\frac{\sqrt{-g}g^{\mu\nu}}{2\kappa}R_{\mu\nu}+\stackrel{{\scriptstyle}}{{\cal
L}}{}^{\prime}\equiv-\frac{\sqrt{-g}}{2\kappa}\stackrel{{\scriptstyle\circ}}{{R}}+\frac{\sqrt{-g}g^{\mu\nu}}{2\kappa}R_{\mu\nu}+\partial_{\alpha}\stackrel{{\scriptstyle}}{{\cal
D}}{}^{\alpha},$ (2.6)

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

$\stackrel{{\scriptstyle}}{{\cal
L}}^{\prime}{}=-\frac{\sqrt{-g}}{2\kappa}\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\alpha}(Q^{\alpha}-\hat{Q}^{\alpha})=\partial_{\alpha}\left(-\frac{1}{2\kappa}\sqrt{-g}(Q^{\alpha}-\hat{Q}^{\alpha})\right),$
(2.7)

and $Q_{\alpha}=g^{\mu\nu}Q_{\alpha\mu\nu}$,
$\hat{Q}_{\alpha}=g^{\mu\nu}Q_{\mu\alpha\nu}$,
$\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\alpha}$ is the Levi-Civita
covariant derivative.

Let us discuss the second term
$+\frac{\sqrt{-g}g^{\mu\nu}}{2\kappa}R_{\mu\nu}$ in (2.6). If we vary (2.1)
directly we vary this term anyway, but only not explicitly. First, because
$R^{\alpha}{}_{\mu\alpha\nu}=R_{\mu\nu}=0$ variation of this term with respect
to $g^{\mu\nu}$ does not contribute into the field equations. Thus, in this
sense, the term $+\frac{\sqrt{-g}g^{\mu\nu}}{2\kappa}R_{\mu\nu}$ is not
sensible. On the other hand, the variation of (2.6), the same (2.1) with
respect to $\Gamma^{\alpha}{}_{\mu\nu}$ gives the equation $\delta{\cal
L}/\delta\Gamma^{\alpha}{}_{\mu\nu}=0$ that determines the teleparallel
connection as
$\Gamma^{\alpha}{}_{\mu\nu}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}{}^{\alpha}{}_{\mu\nu}$.
Due to (2.5) this restriction is very radical, and therefore impermissible.
Thus, finally, under variation we consider the Lagrangian

$\stackrel{{\scriptstyle}}{{\cal
L}}{}=-\frac{\sqrt{-g}}{2\kappa}\stackrel{{\scriptstyle\circ}}{{R}}+\partial_{\alpha}\stackrel{{\scriptstyle}}{{\cal
D}}{}^{\alpha}$ (2.8)

instead of (2.6), the same (2.1), where

$\stackrel{{\scriptstyle}}{{\cal
D}}^{\alpha}\equiv-\frac{\sqrt{-g}}{2\kappa}(Q^{\alpha}-\hat{Q}^{\alpha})\,.$
(2.9)

##  3 Conservation laws for STEGR Lagrangian

Lagrangian (2.8) is a scalar density, therefore one can apply directly a
Noether theorem to obtain the related conserved current and superpotetntial.

###  3.1 Noether conserved quantities in general

The derivation of conserved quantities and related conservation laws for an
arbitrary covariant theory of fields $\psi^{A}$ with an action

$S=\int dy^{4}x{{\cal
L}}(\psi^{A};\psi^{A}{}_{,\alpha};\psi^{A}{}_{,\alpha\beta})\,.$ (3.1)

follows from diffeomorphic invariance and can be found in [43, 44, 31]. Here,
the fields $\psi^{A}$ are an arbitrary tensor density or a set of such
densities, and $A$ is a collective index. Considering invariance of the theory
(3.1) with respect to diffeomorfisms, using the variations of fields
$\psi^{A}$ in the form of Lie derivatives of the fields $\psi^{A}$ [31] one
derives the conservation law for the Noether current ${\cal
J}{}^{\alpha}(\xi)$:

$\partial_{\alpha}{\cal J}^{\alpha}(\xi)\equiv 0\,,$ (3.2)

from which follows that

${\cal J}{}^{\alpha}(\xi)=\partial_{\beta}{\cal J}{}^{\alpha\beta}(\xi),$
(3.3)

where Noether superpotential ${\cal J}{}^{\alpha\beta}(\xi)$ is an
antisymmetric tensor density.

Introducing the coefficients ${\cal U}_{\sigma}{}^{\alpha}$, ${\cal
M}_{\sigma}{}^{\alpha\tau}$, ${\cal N}_{\sigma}{}^{\alpha\tau\beta}$ expressed
through the Lagrangian derivatives in the form

$\displaystyle{\cal U}_{\sigma}{}^{\alpha}$ $\displaystyle\equiv$
$\displaystyle{{\cal L}}\delta^{\alpha}_{\sigma}+{{\delta{{\cal
L}}}\over{\delta\psi^{B}}}\left.\psi^{B}\right|^{\alpha}_{\sigma}-{{\delta{{\cal
L}}}\over{\delta\psi^{B}{}_{,\alpha}}}\partial_{\sigma}\psi^{B}-{{\partial{{\cal
L}}}\over{\partial\psi^{B}{}_{,\beta\alpha}}}\partial_{\beta\sigma}\psi^{B}\,,$
(3.4) $\displaystyle{\cal M}_{\sigma}{}^{\alpha\tau}$ $\displaystyle\equiv$
$\displaystyle{{\delta{{\cal
L}}}\over{\delta\psi^{B}{}_{,\alpha}}}\left.\psi^{B}\right|^{\tau}_{\sigma}-{{\partial{{\cal
L}}}\over{\partial\psi^{B}{}_{,\tau\alpha}}}\partial_{\sigma}\psi^{B}+{{\partial{{\cal
L}}}\over{\partial\psi^{B}{}_{,\beta\alpha}}}\partial_{\beta}(\left.\psi^{B}\right|^{\tau}_{\sigma})\,,$
(3.5) $\displaystyle{\cal N}_{\sigma}{}^{\alpha\tau\beta}$
$\displaystyle\equiv$ $\displaystyle{\frac{1}{2}}\left[{{\partial{{\cal
L}}}\over{\partial\psi^{B}{}_{,\beta\alpha}}}\left.\psi^{B}\right|^{\tau}_{\sigma}+{{\partial{{\cal
L}}}\over{\partial\psi^{B}{}_{,\tau\alpha}}}\left.\psi^{B}\right|^{\beta}_{\sigma}\right],$
(3.6)

where the operation of the vertical line is defined as follows, see [31]. We
consider the Lie derivative of $\psi^{A}$ in the form
${\pounds}_{\xi}\psi^{A}=-\xi^{\alpha}\partial_{\alpha}\psi^{A}+{\left.\psi^{A}\right|}^{\alpha}_{\beta}\partial_{\alpha}\xi^{\beta}\,,$
note that we use the opposite signs with respect to the standard ones. The
notation ${\left.\psi^{A}\right|}^{\alpha}_{\beta}$ is defined by a concrete
transformation properties of $\psi^{A}$, for example, for the vector
$\psi^{A}=\psi^{\sigma}$ one has
${\left.\psi^{\sigma}\right|}^{\alpha}_{\beta}=\delta^{\sigma}_{\beta}\psi^{\alpha}$.
Then the conserved current in (3.2) has a form

${\cal J}{}^{\alpha}(\xi)\equiv-({\cal
U}_{\sigma}{}^{\alpha}\xi^{\sigma}+{\cal
M}_{\sigma}{}^{\alpha\tau}\partial_{\tau}\xi^{\sigma}r+{\cal
N}_{\sigma}{}^{\alpha\tau\beta}\partial_{\tau\beta}\xi^{\sigma}),$ (3.7)

and the superporential in (3.3) is

${\cal J}{}^{\alpha\beta}(\xi)=-\left({\cal
M}_{\sigma}{}^{[\alpha\beta]}\xi^{\sigma}-\frac{2}{3}\partial_{\lambda}{\cal
N}_{\sigma}{}^{[\alpha\beta]\lambda}\xi^{\sigma}+\frac{4}{3}{\cal
N}_{\sigma}{}^{[\alpha\beta]\lambda}\partial_{\lambda}\xi^{\sigma}\right)\,.$
(3.8)

Finally, one defines a conserved quantity ${\cal P}(\xi)$ called as Noether
charge:

${\cal P}(\xi)=\int_{\Sigma}d^{3}x{\cal J}^{0}(\xi)\,,$ (3.9)

and using (3.3) and (3.8) it effectively reduced to a surface integral

${\cal P}(\xi)=\oint_{\partial\Sigma}ds_{i}{\cal J}^{0i}(\xi)\,,$ (3.10)

where $\partial\Sigma$ is a boundary of $\Sigma=:x^{0}=t={\rm const}$.

According to [31] chapter 7, for every additional total divergence
$div=\partial_{\alpha}{\cal D}^{\alpha}$, without considering its inner
structure, it is possible to construct the corresponding additional parts of
the current and the superpotential in the form:

${\cal J}{}_{div}^{\alpha}(\xi)=\partial_{\beta}{\cal
J}{}_{div}^{\alpha\beta}(\xi)=\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\beta}{\cal
J}{}_{div}^{\alpha\beta}(\xi).$ (3.11)

Here, the additional current ${\cal J}{}_{div}^{\alpha}$ and superpotential
${\cal J}{}_{div}^{\alpha\beta}$ for the divergence of ${\cal D}^{\alpha}$ are
defined as:

$\displaystyle{\cal J}{}_{div}^{\alpha}$ $\displaystyle=$
$\displaystyle\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\beta}(-{\cal
M}_{(div)\sigma}{}^{[\alpha\beta]}\xi^{\sigma})=-{\cal
U}_{(div)\sigma}{}^{\alpha}\xi^{\sigma}-{\cal
M}_{(div)\sigma}{}^{[\alpha\beta]}\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\beta}\xi^{\sigma},$
(3.12) $\displaystyle{\cal J}{}_{div}^{\alpha\beta}$ $\displaystyle=$
$\displaystyle-{\cal M}_{(div)\sigma}{}^{[\alpha\beta]}\xi^{\sigma},$ (3.13)

where

$\displaystyle{\cal M}_{(div)\sigma}{}^{\alpha\beta}{}$ $\displaystyle=$
$\displaystyle 2\delta_{\sigma}^{[\alpha}\stackrel{{\scriptstyle}}{{\cal
D}}^{\beta]},$ (3.14) $\displaystyle{\cal U}_{(div)\sigma}{}^{\alpha}{}$
$\displaystyle=$ $\displaystyle
2\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\beta}(\delta_{\sigma}^{[\alpha}{\cal
D}^{\beta]})\equiv 2\partial_{\beta}(\delta_{\sigma}^{[\alpha}{\cal
D}^{\beta]})\,.$ (3.15)

The additional Noether charge can be defined in analogy to (3.9), (3.10):

${\cal P}_{(div)}(\xi)=\int_{\Sigma}d^{3}x{\cal
J}_{(div)}^{0}(\xi)=\oint_{\partial\Sigma}ds_{i}{\cal J}_{(div)}^{0i}(\xi).$
(3.16)

So, despite the fact that adding total divergence to the Lagrangian does not
affect the field equations, this divergence is explicitly added to the Noether
current, superpotential, and charge.

###  3.2 STEGR conserved quantities

Because all items in (2.8) are scalar densities, it is natural to consider
each item in (2.8) separately under application of the Noether theorem.
Considering the Hilbert Lagrangian, the current and superpotential are already
derived in [43] and [31]. Thus, the Noether current for the Hilbert Lagrangian
is

${\cal J}_{GR}^{\alpha}(\xi)\equiv-({\cal
U}_{\sigma}{}^{\alpha}\xi^{\sigma}+{\cal
M}_{\sigma}{}^{\alpha\tau}\partial_{\tau}\xi^{\sigma}+{\cal
N}_{\sigma}{}^{\alpha\tau\beta}\partial_{\tau\beta}\xi^{\sigma}),$ (3.17)

where

$\displaystyle{\cal N}_{\sigma}{}^{\alpha\tau\beta}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-g}}{4\kappa}(2g^{\tau\beta}\delta^{\alpha}_{\sigma}-g^{\alpha\beta}\delta^{\tau}_{\sigma}-g^{\alpha\tau}\delta^{\beta}_{\sigma}),$
(3.18) $\displaystyle{\cal M}_{\sigma}{}^{\alpha\tau}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-g}}{2\kappa}(2\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\alpha}{}_{\sigma\omega}g^{\tau\omega}-\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\omega}{}_{\sigma\omega}g^{\alpha\tau}-\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\tau}{}_{\omega\epsilon}g^{\omega\epsilon}\delta^{\alpha}_{\sigma}),$
(3.19) $\displaystyle{\cal U}_{\sigma}{}^{\alpha}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-g}}{2\kappa}(g^{\alpha\lambda}g^{\omega\epsilon}-g^{\alpha\epsilon}g^{\omega\lambda})(g_{\lambda\sigma,\omega\epsilon}+\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\nu}{}_{\omega\epsilon}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}_{\nu|\lambda\sigma}-\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\nu}{}_{\omega\epsilon}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}_{\lambda|\nu\sigma})$
(3.20) $\displaystyle=$
$\displaystyle-\frac{\sqrt{-g}}{\kappa}(G^{\alpha}_{\sigma}+\frac{1}{2}\delta^{\alpha}_{\sigma}R+\frac{1}{2}g^{\alpha\omega}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}{}^{\rho}{}_{\rho(\omega,\sigma)}-\frac{1}{2}g^{\omega\epsilon}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}{}^{\alpha}{}_{\omega\epsilon,\sigma})\,,$

where the notation
$\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}_{\alpha|\beta\gamma}$ means
$\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}_{\alpha|\beta\gamma}=g_{\alpha\rho}\stackrel{{\scriptstyle\circ}}{{\Gamma}}^{\rho}{}_{\beta\gamma}$.

To represent the current and the superpotential in an explicitly covariant
form we use the evident identities

$\displaystyle\partial_{\beta}\xi^{\sigma}$ $\displaystyle\equiv$
$\displaystyle\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\beta}\xi^{\sigma}-\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\sigma}{}_{\lambda\beta}\xi^{\lambda},$
(3.21) $\displaystyle\partial_{\tau}\partial_{\beta}\xi^{\sigma}$
$\displaystyle\equiv$
$\displaystyle\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\tau}\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\beta}\xi^{\sigma}+\left[\delta^{\sigma}_{\lambda}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\rho}{}_{\beta\tau}-\delta^{\rho}_{\beta}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\sigma}{}_{\lambda\beta}-\delta^{\rho}_{\beta}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\sigma}{}_{\lambda\tau}\right]\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\rho}\xi^{\lambda}$
(3.22) $\displaystyle-$
$\displaystyle\left[\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\sigma}{}_{\lambda\beta,\tau}-\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\sigma}{}_{\rho\beta}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\rho}{}_{\lambda\tau}\right]\xi^{\lambda}.$

Substituting (3.21) and (3.22) into (3.17), one rewrites the GR current in a
covariant form:

${\cal J}_{GR}^{\alpha}(\xi)\equiv-({\cal
U*}_{\sigma}{}^{\alpha}\xi^{\sigma}+{\cal
M*}_{\sigma}{}^{\alpha\tau}\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\tau}\xi^{\sigma}+{\cal
N*}_{\sigma}{}^{\alpha\tau\beta}\stackrel{{\scriptstyle\circ}}{{\nabla}}_{(\tau}\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\beta)}\xi^{\sigma}),$
(3.23)

with

$\displaystyle{\cal N*}_{\sigma}{}^{\alpha\tau\beta}$ $\displaystyle=$
$\displaystyle{\cal N}_{\sigma}{}^{\alpha\tau\beta},$ (3.24)
$\displaystyle{\cal M*}_{\sigma}{}^{\alpha\tau}$ $\displaystyle=$
$\displaystyle{\cal M}_{\sigma}{}^{\alpha\tau}-2{\cal
N}_{\lambda}{}^{\alpha\tau\beta}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\lambda}{}_{\sigma\beta}+{\cal
N}_{\sigma}{}^{\alpha\lambda\beta}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\tau}{}_{\beta\lambda}=0,$
(3.25) $\displaystyle{\cal U*}_{\sigma}{}^{\alpha}$ $\displaystyle=$
$\displaystyle{\cal U}_{\sigma}{}^{\alpha}-{\cal
M}_{\lambda}{}^{\alpha\tau}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\lambda}{}_{\sigma\tau}-{\cal
N}_{\lambda}{}^{\alpha\tau\beta}\partial_{\tau}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\lambda}{}_{\sigma\beta}+{\cal
N}_{\kappa}{}^{\alpha\tau\beta}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\kappa}{}_{\lambda\beta}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\lambda}{}_{\sigma\tau}$
(3.26) $\displaystyle=$
$\displaystyle-\frac{\sqrt{-g}}{4\kappa}g^{\alpha\omega}\stackrel{{\scriptstyle\circ}}{{R}}{}_{\omega\sigma}.$

The superporential for the Hilbert Lagrangian after substituting (3.21),
(3.19) and (3.18) into (3.8) has the form

${\cal J}{}_{GR}^{\alpha\beta}=\stackrel{{\scriptstyle}}{{\cal
K}}{}^{\alpha\beta}=\frac{\sqrt{-g}}{\kappa}\stackrel{{\scriptstyle\circ}}{{\nabla}}{}^{[\alpha}\xi^{\beta]},$
(3.27)

that is well-known Komar’s superpotential [43, 31].

Applying Noether theorem to the divergence $\stackrel{{\scriptstyle}}{{\cal
L}}^{\prime}{}$ in (2.7) with (2.9), with the use of (3.15) and 3.14 one
calculates additional Noether current (3.12), superpotential (3.13) and charge
(3.16). Thus, the additional superpotential for
$\stackrel{{\scriptstyle}}{{\cal L}}^{\prime}{}$ in STEGR has an easy
expression:

${\cal
J}{}_{div}^{\alpha\beta}=\frac{\sqrt{-g}}{\kappa}\delta_{\sigma}^{[\alpha}(Q^{\beta]}-\hat{Q}^{\beta]})\xi^{\sigma}.$
(3.28)

Integrating it by (3.16) we get the additional Noether charge. Taking the
divergence of (3.28) as (3.3) we get the additional Noether current same as
(3.12).

Finally, the construction of the current and superpotential of the Lagrangian
(2.8) in STEGR is:

${\cal J}{}^{\alpha}={\cal J}{}_{GR}^{\alpha}+{\cal J}{}_{div}^{\alpha},$
(3.29) ${\cal J}{}^{\alpha\beta}={\cal J}{}_{GR}^{\alpha\beta}+{\cal
J}{}_{div}^{\alpha\beta}.$ (3.30)

By construction, the current ${\cal J}{}^{\alpha}(\xi)$ is a vector density,
the superpotential ${\cal J}{}^{\alpha\beta}(\xi)$ is an antisymmetric tensor
density, and both are covariant. The Noether current is conserved:

$\partial_{\alpha}{\cal
J}{}^{\alpha}(\xi)=\stackrel{{\scriptstyle\circ}}{{\nabla}}_{\alpha}{\cal
J}{}^{\alpha}(\xi)=0\,.$ (3.31)

Integrating each part of (3.30) one gets

${\cal P}(\xi)={\cal P}_{GR}(\xi)+{\cal P}_{div}(\xi),$ (3.32)

where ${\cal P}(\xi)$ is (3.10) and

${\cal P}_{GR}(\xi)=\oint_{\partial\Sigma}ds_{i}{\cal J}_{GR}^{0i}(\xi),$
(3.33)

where ${\cal J}_{GR}^{0i}(\xi)$ is Komar superpotential (3.27);

${\cal P}_{div}(\xi)=\oint_{\partial\Sigma}ds_{i}{\cal J}_{div}^{0i}(\xi),$
(3.34)

where ${\cal J}_{div}^{0i}(\xi)$ is (3.28).

###  3.3 Defining the connection

To define the connection which is not a dynamical quantity we adapt for STEGR
the “turning off” gravity principle we developed in TEGR [36, 37, 38, 39].
Assuming that $Q_{\alpha\mu\nu}$ and $L^{\alpha}{}_{\mu\nu}$ vanish in the
absence of gravity and taking into account that
$\stackrel{{\scriptstyle\circ}}{{R}}{}^{\alpha}{}_{\beta\mu\nu}$ in GR
vanishes in the absence of gravity too we find the connection in STEGR as
follows:

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

$\stackrel{{\scriptstyle\circ}}{{R}}{}^{\alpha}{}_{\beta\mu\nu}=\partial_{\mu}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\alpha}{}_{\beta\nu}-\partial_{\nu}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\alpha}{}_{\beta\mu}+\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\alpha}{}_{\kappa\mu}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\kappa}{}_{\beta\nu}-\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\alpha}{}_{\kappa\nu}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{\kappa}{}_{\beta\mu};$
(3.35)

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 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
$\stackrel{{\scriptstyle\circ}}{{R}}{}^{\alpha}{}_{\beta\gamma\delta}=0$.

##  4 Applications

###  4.1 Preliminary remarks

Noether currents and superpotentials depend on the vector field $\xi$ that is
not fixed a priori, so we need to find a way to determine it to obtain
physically meaningful quantities. A lot of ways to define $\xi$ exist, but
which are physically meaningful?

a) In the standard metric formulation of GR the physically preferred choice is
to use Killing fields of the reference geometry [45]. Time-like Killing
vectors coincide with observers’ proper vectors at the spatial infinity for
isolated systems and we use them to obtain global characteristics of the
object under consideration, which are total Noether charges in the studied
approach. $\Sigma$ in (3.9),(3.32) is the whole infinite hypersurface of
constant time in this case. It is expected that the result obtained by this
way coincides with the mass of the object under consideration [36, 38].

b) Obtaining local characteristics of the gravitational field we use the
observer’s proper vector (observer’s 4-velocity). It is expected that the
components of Noether current (3.29) give energy-momentum densities measured
by the chosen observer [36, 38]. The main experimental basis of general
relativity (and other metric theories) is the weak equivalence principle. Of
course, STERG as an equivalent representation of GR has to possess it. This
means that a freely falling observer feels himself locally in Minkowski space.
Thus, he does not measure any energetic characteristics. Calculating energy
densities measured by freely falling observes, we check an applicability of
suggested by us expressions for conserved quantities. In the case if the weak
equivalence principle is not satisfied this means that the presumed
interpretation of the conservation laws are invalid, and have to be changed or
specified. Note, for example, that the current explicitly depends on flat
torsionless connection, see (3.11) for (3.28) with (2.3), so we will try to
obtain such flat connections which give zero current for the freely falling
observer.

We calculate energy desity measured by the freely falling observer in FRLW
universe, energy desity measured by the freely falling observer into the
Schwarzschild black hole and Schwarzschild black hole total mass. For the
Schwarzschild solution we take static coordinates and general radially falling
coordinates introduced by us in [38, 39] which can be reduced to the Lemaitre
coordinates in a particular case.

###  4.2 FRLW universe

We take the FRLW metric in the form:

$ds^{2}=-dt^{2}+a^{2}(t)\left(\frac{1}{1-kr^{2}}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)\right)$ (4.1)

where $k=+1$ for a positively curved space, $k=0$ for a flat space and $k=-1$
for a negatively curved space.

The non-zero components of Levi-Civita connection are:

$\begin{array}[]{cccc}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{11}=\frac{a(t)a^{\prime}(t)}{1-kr^{2}};~{}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{22}=r^{2}a(t)a^{\prime}(t);~{}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{33}=r^{2}a(t)a^{\prime}(t)\sin^{2}\theta;\\\
\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{01}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{10}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{02}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{20}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{03}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{30}=\frac{a^{\prime}(t)}{a(t)};\\\
\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{11}=\frac{kr}{1-kr^{2}};~{}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{22}=r\left(kr^{2}-1\right);~{}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{33}=r\sin^{2}\theta\left(kr^{2}-1\right);\\\
\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{12}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{21}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{13}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{31}=\frac{1}{r};\\\
\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{33}=-\sin\theta\cos\theta;~{}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{23}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{32}=\cot\theta;\\\
\end{array}$ (4.2)

where numeration of coordinates is
$(x^{0},x^{1},x^{2},x^{3})\equiv(t,r,\theta,\phi)$.

Non-zero components of Riemann tensor are proportional to:

$\stackrel{{\scriptstyle\circ}}{{R}}{}^{0}{}_{i0i}\thicksim-\stackrel{{\scriptstyle\circ}}{{R}}{}^{0}{}_{ii0}\thicksim\stackrel{{\scriptstyle\circ}}{{R}}{}^{i}{}_{00i}\thicksim-\stackrel{{\scriptstyle\circ}}{{R}}{}^{i}{}_{0i0}\thicksim\frac{\ddot{a}}{a},$
(4.3)
$\stackrel{{\scriptstyle\circ}}{{R}}{}^{i}{}_{jij}\thicksim-\stackrel{{\scriptstyle\circ}}{{R}}{}^{i}{}_{jji}\thicksim\frac{k+\dot{a}^{2}}{a^{2}}$
(4.4)

where $i,j=1,2,3$. Equating these components to zero we got two equations
which “turn off” gravity: $\dot{a}^{2}+k=0$ and $\ddot{a}=0$. All solutions to
the first equation satisfy the second equation, so only

$\dot{a}^{2}+k=0$ (4.5)

makes sense. Note that “vacuum” Friedmann equation

$H^{2}=\rho_{curv}\qquad{\rm
or}\qquad\left(\frac{\dot{a}}{a}\right)^{2}=-\frac{k}{a^{2}}\,,$ (4.6)

is equivalent to (4.5). The same condition was used to “turn off” gravity in
TEGR in FRLW universe [36, 37].

Taking $a(t)=\sqrt{-k}t$ in Levi-Civita connection we get the symmetric
teleparallel connection:

$\begin{array}[]{cccc}\Gamma{}^{0}{}_{11}=\frac{kt}{kr^{2}-1};~{}\Gamma{}^{0}{}_{22}=-kr^{2}t;~{}\Gamma{}^{0}{}_{33}=-kr^{2}t\sin^{2}\theta;\\\
\Gamma{}^{1}{}_{01}=\Gamma{}^{1}{}_{10}=\Gamma{}^{2}{}_{02}=\Gamma{}^{2}{}_{20}=\Gamma{}^{3}{}_{03}=\Gamma{}^{3}{}_{30}=\frac{1}{t};\\\
\Gamma{}^{1}{}_{11}=\frac{kr}{1-kr^{2}};~{}\Gamma{}^{1}{}_{22}=r\left(kr^{2}-1\right);~{}\Gamma{}^{1}{}_{33}=r\sin^{2}\theta\left(kr^{2}-1\right);\\\
\Gamma{}^{2}{}_{12}=\Gamma{}^{2}{}_{21}=\Gamma{}^{3}{}_{13}=\Gamma{}^{3}{}_{31}=\frac{1}{r};\\\
\Gamma{}^{2}{}_{33}=-\sin\theta\cos\theta;~{}\Gamma{}^{3}{}_{23}=\Gamma{}^{3}{}_{32}=\cot\theta;\\\
\end{array}$ (4.7)

where $(x^{0},x^{1},x^{2},x^{3})\equiv(t,r,\theta,\phi)$. Then we can obtain
the non-metricity (2.3) and disformation (2.5).

We consider a freely falling observer which is static in co-moving coordinates
of the metric (4.1). Then components of his proper vector are:

$\xi^{\sigma}=(-1,0,0,0)\,.$ (4.8)

We calculate the Noether current from the Noether superpotential (3.30) using
(3.3), where GR term is Komar superpotential (3.27) and additional term is
(3.28). The same result can be obtained calculating the Noether current (3.29)
directly where GR term is obtained from (3.17) - (3.20) or (3.23), (3.25),
(3.26); and additional term is obtained from (3.12), (3.14), (3.15), (2.9).
The result for Noether current is:

$\stackrel{{\scriptstyle\bullet}}{{\cal
J}}{}^{\alpha}(\xi)=(0,~{}0,~{}0,~{}0)\,.$ (4.9)

This means that the freely falling observer with the proper vector (4.8) does
not measure energy and momentum densities what corresponds to Einstein
equivalence principle.

###  4.3 Schwarzschild black hole

The Schwarzschild metric in static coordinates has a form:

$ds^{2}=-fdt^{2}+f^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}),$
(4.10)

where

$f=f(r)=1-\frac{r_{g}}{r}=1-\frac{2M}{r}$ (4.11)

and we keep $(t,r,\theta,\phi)=(x^{0},x^{1},x^{2},x^{3})$, respectively. The
components of the timelike Killing vector in static coordinates are:

$\xi^{\alpha}=(-1,~{}0,~{}0,~{}0),$ (4.12)

After applying the coordinate transformations from the Schwarzschild static
coordinates $(t,r,\theta,\phi)$ to Lemaitre freely falling coordinates
$(\tau,\rho,\theta,\phi)$, preserving the numeration from $0$ to $3$,

$\begin{array}[]{cccc}d\rho=dt+\frac{dr}{f\sqrt{1-f}}\,,\\\
d\tau=dt+\frac{dr}{f}\sqrt{1-f},\end{array}$ (4.13)

with $f(r)$ is defined in (4.11), the Schwarzschild static metric (4.10)
transforms to

$ds^{2}=-d\tau^{2}+(1-f)d\rho^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2},$ (4.14)

where

$r=r(\tau,\rho)=[\frac{3}{2}(\rho-\tau)]^{2/3}(2M)^{1/3}.$ (4.15)

Here, we are able to write $r$ as a function of $\rho$ and $\tau$ directly.

Solving the geodesic equation in general form for a radially freely falling
observer towards the Schwarzschild black hole in the coordinates (4.10), one
obtains the 4-velocity of radially freely falling observers with arbitrary
energy parameter $e$. Written in Schwarzschild static coordinates expression
of this 4-velocity is:

$\xi_{fall}^{\alpha}=\left(-\frac{e}{f},\sqrt{e^{2}-f},0,0\right)=\left(-\frac{e}{1-2M/r},\sqrt{e^{2}-(1-2M/r)},0,0\right).$
(4.16)

The parameter $e$ characterizes the initial state of the in-falling observer.
In the case of $e>1$ the observer has a nonzero velocity directed to the black
hole at spatial infinity $r\rightarrow\infty$ and thus can reach each region
of $r$. The observer with $e=1$ is connected to Lemaitre coordinates in
(4.14), has a zero velocity at the infinity $r\rightarrow\infty$ and still
able to achieve each point of space. In the case of $e<1$ the observer has
zero velocity at some finite $r_{0}=r_{0}(e)$ and cannot achieve the space
region $r>r_{0}(e)$, and the proper vector of this observer does not exist at
$r>r_{0}(e)$.

Basing on such observers (4.16), one can construct the coordinate
transformation from Schwarzschild static coordinates to general radially
falling coordinates (which we call e-Lemaitre coordinates for brevity) — the
freely falling observers’ proper coordinates (in which the observers’
4-velocities would be $\xi_{fall}^{\alpha}=(-1,0,0,0)$):

$\displaystyle d\tau_{e}$ $\displaystyle=$ $\displaystyle
edt+\frac{\sqrt{e^{2}-1+\frac{2M}{r}}}{1-\frac{2M}{r}}dr,$ $\displaystyle
d\rho_{e}$ $\displaystyle=$ $\displaystyle
dt+\frac{e}{\left(1-\frac{2M}{r}\right)\sqrt{e^{2}-1+\frac{2M}{r}}}dr.$ (4.17)

Then, using (4.17), one transforms the Schwarzschild metric (4.10) to the
form:

$ds^{2}=-d\tau_{e}^{2}+\left(e^{2}-1+\frac{2M}{r}\right)d\rho_{e}^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)\,,$ (4.18)

where $r=r(\tau_{e},\rho_{e})$.

These coordinates contain the classical Lemaitre coordinates as a particular
case ($e=1$), being in its turn a particular example of more general
construction [46, 47]. In the e-Lemaitre metric the dependence of $r$ as a
function of $\rho_{e}$ and $\tau_{e}$ is not so simple as in the classical
Lemaitre case (4.15), and we will try to avoid to use this function directly
in what follows.

###  4.4 Calculations in static case

Non-zero Levi-Civita connection components calculated for the metric (4.10)
are:

$\begin{array}[]{cccc}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{01}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{11}=\frac{{r_{g}}}{2r^{2}}\frac{1}{1-{r_{g}}/r};~{}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{00}=\frac{{r_{g}}}{2r^{2}}(1-\frac{r_{g}}{r});~{}~{}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{12}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{13}=\frac{1}{r};\\\
\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{22}=-r\left(1-\frac{r_{g}}{r}\right);~{}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{33}=-r\left(1-\frac{r_{g}}{r}\right)\sin^{2}\theta;\\\
\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{33}=-\sin\theta\cos\theta;~{}~{}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{23}=\cot\theta.\\\
\end{array}$ (4.19)

Then only non-zero components of the Komar superpotantial of GR (3.27) for
Killing vector (4.12) are

$\stackrel{{\scriptstyle}}{{\cal K}}{}^{01}=-\stackrel{{\scriptstyle}}{{\cal
K}}{}^{10}=\frac{r_{g}}{16\pi}\sin\theta.$ (4.20)

“Turning off” gravity in (4.19) by $r_{g}\rightarrow 0$ we get:

$\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{12}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{13}={1}/{r};~{}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{22}=-r;~{}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{33}=-r\sin^{2}\theta;~{}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{33}=-\sin\theta\cos\theta;~{}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{23}=\cot\theta.$
(4.21)

Recently, see (33) in [48], the same teleparallel connection components have
been derived in $f(Q)$ theories for the static spherical solutions in the
standard coordinates by “turning off” gravity. For (4.19) and (4.21) for
01-component and 10-component of (3.13) with (3.14) and (2.9) and the Killing
vector (4.12) one has

${\cal J}_{div}^{01}(\xi_{kill})=-{\cal
J}_{div}^{10}(\xi_{kill})=\frac{r_{g}}{16\pi}\sin\theta.$ (4.22)

Summing (4.20) and (4.22) by (3.30) and providing the calculation of charge
(3.10), (3.32), we obtain for the total mass:

$E=\lim_{r\rightarrow\infty}\oint_{\partial\Sigma}{\cal
J}{}^{01}(\xi_{kill})d\theta d\phi=\frac{M}{2}+\frac{M}{2}=M,$ (4.23)

that is the acceptable result.

Let us consider a freely falling observers. Then the 01-component and
10-component of the Komar superpotantial of GR (3.27) with the freely falling
observer’s vector (4.16) are zero

$\stackrel{{\scriptstyle}}{{\cal K}}{}^{01}=-\stackrel{{\scriptstyle}}{{\cal
K}}{}^{10}=0$ (4.24)

without dependence on $e$ the same as other components
$\stackrel{{\scriptstyle}}{{\cal K}}{}^{\alpha\beta}=0$. As a result, the
related current by (3.3) is zero too.

For the additional superpotential (3.13) for (4.19) and (4.21) with the freely
falling observer’s vector (4.16) one has

${\cal J}_{div}^{01}(\xi_{fall})=-{\cal
J}_{div}^{10}(\xi_{fall})=\frac{e{r_{g}}\sin\theta}{16\pi(1-{r_{g}}/r)}$
(4.25)

with other components of ${\cal J}_{div}^{\alpha\beta}(\xi_{fall})=0$. Then,
by (3.11) we obtain the current:

${\cal
J}^{\alpha}_{div}(\xi_{fall})=\left(-\frac{e{r_{g}}^{2}\sin\theta}{16\pi
r^{2}(1-{r_{g}}/r)^{2}},0,0,0\right).$ (4.26)

Thus, the total current components are non-zero that is not in correspondence
with the equivalence principle.

Principally, the results of this subsection repeat the results obtained in
[38],[39] in the static gauge.

###  4.5 The e-Lemaitre case.

We calculate the Levi-Civita connection components for the metric (4.18):

$\begin{array}[]{cccc}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{11}=\frac{{r_{g}}}{2r^{2}}\sqrt{e^{2}-1+{r_{g}}/r};~{}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{22}=-r\sqrt{e^{2}-1+{r_{g}}/r};\\\
\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{0}{}_{33}=-r\sin^{2}\theta\sqrt{e^{2}-1+{r_{g}}/r};\\\
\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{01}=\frac{{r_{g}}}{2r^{2}}\left({\sqrt{e^{2}-1+{r_{g}}/r}}\right)^{-1};~{}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{11}=-\frac{{er_{g}}}{2r^{2}}\left({\sqrt{e^{2}-1+{r_{g}}/r}}\right)^{-1};\\\
\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{22}=-{er}\left({\sqrt{e^{2}-1+{r_{g}}/r}}\right)^{-1};~{}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{1}{}_{33}=-{er}\sin^{2}\theta\left({\sqrt{e^{2}-1+{r_{g}}/r}}\right)^{-1};\\\
\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{02}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{03}=-\frac{{1}}{r}\sqrt{e^{2}-1+{r_{g}}/r};~{}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{12}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{13}=\frac{{e}}{r}\sqrt{e^{2}-1+{r_{g}}/r};\\\
\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{2}{}_{33}=-\sin\theta\cos\theta;~{}\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{23}=\cot\theta.\\\
\end{array}$ (4.27)

“Turning off” gravity by $r_{g}\rightarrow 0$, where $r$ is an independent on
$r_{g}$ coordinate (function) we get the teleparallel connection components
with the $e$-case indices in coordinates $(\tau_{e},\rho_{e},\theta,\phi)$:

$\begin{array}[]{cccc}\Gamma{}^{0}{}_{22}=-r\sqrt{e^{2}-1};~{}\Gamma{}^{0}{}_{33}=-r\sin^{2}\theta\sqrt{e^{2}-1};\\\
\Gamma{}^{1}{}_{22}=-{er}\left(\sqrt{e^{2}-1}\right)^{-1};~{}\Gamma{}^{1}{}_{33}=-{er}\sin^{2}\theta\left(\sqrt{e^{2}-1}\right)^{-1};\\\
\Gamma{}^{2}{}_{02}=\stackrel{{\scriptstyle\circ}}{{\Gamma}}{}^{3}{}_{03}=-\frac{{1}}{r}\sqrt{e^{2}-1};\Gamma{}^{2}{}_{12}=\Gamma{}^{3}{}_{13}=\frac{{e}}{r}\sqrt{e^{2}-1};\\\
\Gamma{}^{2}{}_{33}=-\sin\theta\cos\theta;~{}\Gamma{}^{3}{}_{23}=\cot\theta.\\\
\end{array}$ (4.28)

Then transforming this connection components to the Schwarzschild coordinates
by the usual way we get for the further calculations in convenient
Schwarzschild coordinates:

$\begin{array}[]{cccc}\Gamma{}^{0}{}_{11}=-\frac{e{r_{g}}}{r^{2}(r-{r_{g}/r})^{2}}\frac{1}{\sqrt{e^{2}-1+{r_{g}}/r}};~{}\Gamma{}^{0}{}_{22}=\frac{e{r_{g}}}{1-{r_{g}}/r}\frac{1}{\sqrt{e^{2}-1}};\\\
\Gamma{}^{0}{}_{33}=\frac{e{r_{g}}\sin^{2}\theta}{1-{r_{g}}/r}\frac{1}{\sqrt{e^{2}-1}};~{}\Gamma{}^{1}{}_{11}=\frac{{r_{g}}}{2r^{2}}\frac{1}{e^{2}-1+{r_{g}}/r};\\\
\Gamma{}^{1}{}_{22}=-r\sqrt{\frac{e^{2}-1+{r_{g}}/r}{e^{2}-1}};~{}\Gamma{}^{1}{}_{33}=-r\sqrt{\frac{e^{2}-1+{r_{g}}/r}{e^{2}-1}}\sin^{2}\theta;\\\
\Gamma{}^{2}{}_{12}=\Gamma{}^{3}{}_{13}=\frac{1}{r}\sqrt{\frac{e^{2}-1}{e^{2}-1+{r_{g}}/r}};~{}\Gamma{}^{2}{}_{33}=-\sin\theta\cos\theta;~{}\Gamma{}^{3}{}_{23}=\cot\theta.\\\
\end{array}$ (4.29)

It turns out that this way of the “switching-off” is not correct because
Riemannian tensor is not zero for this connection and thus we cannot use it
anymore.

We think that the reason of this unacceptable result is that we did not “turn
off” gravity totally, i.e. we did not provide $r_{g}\to 0$ inside function
$r(\tau_{e},\rho_{e})$ because really it is not independent function, since we
made a coordinate transformation to freely falling coordinates
$(\tau_{e},\rho_{e},\theta,\phi)$. Calculating Riemann tensor (3.35) of the
connection (4.28) directly one can use partial derivatives taken from (4.17)
and see that Riemann tensor components depend on $r_{g}$.

To provide a correct “switching off” gravity setting $r_{g}=0$ we introduce
the operation $r(\tau_{e},\rho_{e})\stackrel{{\scriptstyle r_{g}\rightarrow
0}}{{\longrightarrow}}\tilde{r}(\tau_{e},\rho_{e})$ where
$\tilde{r}(\tau_{e},\rho_{e})$ totally does not depend on $r_{g}$. To define
the function $\tilde{r}(\tau_{e},\rho_{e})$ we do the following. From the
coordinate transformations (4.17) we have

$d(\tau_{e}-e\rho_{e})=\left(\frac{\sqrt{e^{2}-1+\frac{r_{g}}{r}}}{1-\frac{r_{g}}{r}}-\frac{e^{2}}{\left(1-\frac{r_{g}}{r}\right)\sqrt{e^{2}-1+\frac{r_{g}}{r}}}\right)dr=-\frac{dr}{\sqrt{e^{2}-1+\frac{r_{g}}{r}}}.$
(4.30)

After integration one has

$\displaystyle\tau_{e}-e\rho_{e}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{r\left(r(e^{2}-1)+{r_{g}}\right)}}{e^{2}-1}$ (4.31)
$\displaystyle+$
$\displaystyle\frac{r_{g}}{(e^{2}-1)^{3/2}}\left\\{\log\left[\sqrt{r(e^{2}-1)+r_{g}}+\sqrt{r(e^{2}-1)}\right]+C\right\\},$

where $C$ is the integration constant. Let us “switch off” gravity in this
relation by $r_{g}=0$, as usual. Then, we obtain $\tilde{r}$ without
dependence on $r_{g}$ as

$\tau_{e}-e\rho_{e}=-\frac{\tilde{r}}{\sqrt{e^{2}-1}}.$ (4.32)

So, comparing last two equations we get

${\tilde{r}}=\frac{\sqrt{r\left(r(e^{2}-1)+{r_{g}}\right)}}{\sqrt{e^{2}-1}}-\frac{r_{g}}{e^{2}-1}\left\\{\log\left[\sqrt{r(e^{2}-1)+r_{g}}+\sqrt{r(e^{2}-1)}\right]+C\right\\}.$
(4.33)

Substituting the last expression into (4.28), we are “turning off” gravity in
external way and in internal way inside $r$ that goes to $\tilde{r}$. Under
this paradigma, the teleparallel connection in e-falling coordinates is

$\begin{array}[]{cccc}\Gamma{}^{0}{}_{22}=-\tilde{r}\sqrt{e^{2}-1};~{}\Gamma{}^{0}{}_{33}=-\tilde{r}\sqrt{e^{2}-1}\sin^{2}\theta;~{}\Gamma{}^{1}{}_{22}=-\frac{e\tilde{r}}{\sqrt{e^{2}-1}};\\\
\Gamma{}^{1}{}_{33}=-\frac{e\tilde{r}\sin^{2}\theta}{\sqrt{e^{2}-1}};~{}\Gamma{}^{2}{}_{02}=\Gamma{}^{3}{}_{03}=-\frac{\sqrt{e^{2}-1}}{\tilde{r}};{}\\\
\Gamma{}^{2}{}_{12}=\Gamma{}^{3}{}_{13}=\frac{e\sqrt{e^{2}-1}}{\tilde{r}};~{}\Gamma{}^{2}{}_{33}=-\sin\theta\cos\theta;~{}\Gamma{}^{3}{}_{23}=\cot\theta.\\\
\end{array}$ (4.34)

So, the connection in e-falling coordinates is

$\begin{array}[]{cccc}\Gamma{}^{0}{}_{22}=-\sqrt{r\left(r\left(e^{2}-1\right)+{r_{g}}\right)}+\frac{{r_{g}}\log\left(\sqrt{r\left(e^{2}-1\right)+{r_{g}}}+\sqrt{r\left(e^{2}-1\right)}\right)+C}{\sqrt{e^{2}-1}};\\\
\Gamma{}^{0}{}_{33}=\left[-\sqrt{r\left(r\left(e^{2}-1\right)+{r_{g}}\right)}+\frac{{r_{g}}\log\left(\sqrt{r\left(e^{2}-1\right)+{r_{g}}}+\sqrt{r\left(e^{2}-1\right)}\right)+C}{\sqrt{e^{2}-1}}\right]\sin^{2}\theta;\\\
\Gamma{}^{1}{}_{22}=e\left[-\frac{\sqrt{r\left(r\left(e^{2}-1\right)+{r_{g}}\right)}}{e^{2}-1}+\frac{{r_{g}}\log\left(\sqrt{r\left(e^{2}-1\right)+{r_{g}}}+\sqrt{r\left(e^{2}-1\right)}\right)+C}{{e^{2}-1}}\right];\\\
\Gamma{}^{1}{}_{33}=e\left[-\frac{\sqrt{r\left(r\left(e^{2}-1\right)+{r_{g}}\right)}}{e^{2}-1}+\frac{{r_{g}}\log\left(\sqrt{r\left(e^{2}-1\right)+{r_{g}}}+\sqrt{r\left(e^{2}-1\right)}\right)+C}{{e^{2}-1}}\right]\sin^{2}\theta;\\\
\Gamma{}^{2}{}_{02}=\Gamma{}^{3}{}_{03}=\frac{e^{2}-1}{-\sqrt{r\left(\left(e^{2}-1\right)r+{r_{g}}\right)}+{r_{g}}\log\left(\sqrt{\left(e^{2}-1\right)r+{r_{g}}}+\sqrt{r\left(e^{2}-1\right)}\right)+C};\\\
\Gamma{}^{2}{}_{12}=\Gamma{}^{3}{}_{13}=\frac{e\left(e^{2}-1\right)}{\sqrt{r\left(\left(e^{2}-1\right)r+{r_{g}}\right)}-{r_{g}}\log\left(\sqrt{\left(e^{2}-1\right)r+{r_{g}}}+\sqrt{r\left(e^{2}-1\right)}\right)-C};\\\
\Gamma{}^{2}{}_{33}=-\sin\theta\cos\theta;~{}~{}\Gamma{}^{3}{}_{23}=\cot\theta.\\\
\end{array}$ (4.35)

Connection after transformation to static coordinates takes the form

$\begin{array}[]{cccc}\Gamma{}^{0}{}_{11}=-\frac{e{r_{g}}\sqrt{r}}{(r-{r_{g}})^{2}\sqrt{r(e^{2}-1)+{r_{g}}}};~{}~{}\Gamma{}^{1}{}_{11}=\frac{{r_{g}}}{2r\left(\left(e^{2}-1\right)r+{r_{g}}\right)};\\\
\Gamma{}^{0}{}_{22}={e{r_{g}}}\left[\frac{\sqrt{r\left(r\left(e^{2}-1\right)+{r_{g}}\right)}}{\left(e^{2}-1\right)(r-{r_{g}})}-\frac{{r_{g}}\log\left(\sqrt{\left(r\left(e^{2}-1\right)+{r_{g}}\right)}+\sqrt{r\left(e^{2}-1\right)}\right)-C}{\left(e^{2}-1\right)(r-{r_{g}})}\right];\\\
\Gamma{}^{0}{}_{33}=e{r_{g}}\left[\frac{\sqrt{r\left(\left(e^{2}-1\right)r+{r_{g}}\right)}}{\left(e^{2}-1\right)(r-{r_{g}})}-\frac{{r_{g}}\log\left(\sqrt{\left(r\left(e^{2}-1\right)+{r_{g}}\right)}+\sqrt{r\left(e^{2}-1\right)}\right)-C}{\left(e^{2}-1\right)(r-{r_{g}})}\right]\sin^{2}\theta;\\\
\Gamma{}^{1}{}_{22}=-\sqrt{e^{2}+\frac{{r_{g}}}{r}-1}\left[\frac{\sqrt{r\left(\left(e^{2}-1\right)r+{r_{g}}\right)}}{\left(e^{2}-1\right)}-\frac{{r_{g}}\log\left(\sqrt{\left(r\left(e^{2}-1\right)+{r_{g}}\right)}+\sqrt{r\left(e^{2}-1\right)}\right)-C}{\left(e^{2}-1\right)}\right];\\\
\Gamma{}^{1}{}_{33}=-\sqrt{e^{2}+\frac{{r_{g}}}{r}-1}\left[\frac{\sqrt{r\left(\left(e^{2}-1\right)r+{r_{g}}\right)}}{\left(e^{2}-1\right)}-\frac{{r_{g}}\log\left(\sqrt{\left(r\left(e^{2}-1\right)+{r_{g}}\right)}+\sqrt{r\left(e^{2}-1\right)}\right)-C}{\left(e^{2}-1\right)}\right]\sin^{2}\theta;\\\
\Gamma{}^{2}{}_{12}=\Gamma{}^{3}{}_{13}=\frac{e^{2}-1}{\sqrt{e^{2}+\frac{{r_{g}}}{r}-1}\left(\sqrt{r\left(\left(e^{2}-1\right)r+{r_{g}}\right)}-{r_{g}}\log\left(\sqrt{\left(e^{2}-1\right)r+{r_{g}}}+\sqrt{r\left(e^{2}-1\right)}\right)-C\right)};\\\
\Gamma{}^{2}{}_{33}=-\sin\theta\cos\theta;~{}~{}\Gamma{}^{3}{}_{23}=\cot\theta.\\\
\end{array}$ (4.36)

Calculating the Riemannian tensor components for this connection we find that
all of them present fractions where the numerators are equal to zero
identically. This means that despite the denominators can vanish for some
particular $r=r(C)$, the curvature equals zero for all permissible $r$. So
that, the connection (4.36), despite it is singular at the point when the
denominator vanishes, is a teleparallel connection.

For the sake of simplicity and certainty let us choose $C=-{\textstyle{1\over
2}}r_{g}\log\sqrt{e^{2}-1}$. Then, the only non-zero components of the
additional divergence superpotential with Killing vector (4.12) are

$\begin{array}[]{cccc}{\cal J}^{01}_{div}(\xi_{kill})=-{\cal
J}^{10}_{div}(\xi_{kill})=\left({r_{g}}r^{2}\sin\theta\left(2r\left(e^{2}-1\right)^{3}r+e^{2}\left(3e^{2}-4\right){r_{g}}+{r_{g}}\right)\right.\\\
+{r_{g}}\left[2\log\left(\sqrt{\left(e^{2}-1\right)r+{r_{g}}}+\sqrt{r\left(e^{2}-1\right)}\right)-\log\sqrt{e^{2}-1}\right]*\\\
\left.*\left(\left(3e^{2}+1\right)\sqrt{r(e^{2}-1)(r(e^{2}-1)+r_{g})}\right.\right.\\\
\left.\left.+\left(\left(e^{2}-1\right)r+{r_{g}}\right)\left[2\log\left(\sqrt{\left(e^{2}-1\right)r+{r_{g}}}+\sqrt{r\left(e^{2}-1\right)}\right)-\log\sqrt{e^{2}-1}\right]\right)\right)/\\\
(16\pi\left(e^{2}-1\right)r^{3}\left(2\left(e^{2}-1\right)\left(\left(e^{2}-1\right)r+{r_{g}}\right)+(r_{g}/r)\sqrt{r(e^{2}-1)(r(e^{2}-1)+r_{g})}~{}~{}*\right.\\\
\left.*\log\left[2\log\left(\sqrt{\left(e^{2}-1\right)r+{r_{g}}}+\sqrt{r\left(e^{2}-1\right)}\right)-\log\sqrt{e^{2}-1}\right]\right)).\\\
\end{array}$ (4.37)

This gives the additional $+M/2$ to Komar $M/2$ to obtain the acceptable value
$M$. This result is valid for all $C$ introduced in (4.31) as well.

The only non zero components of the additional divergence superpotential with
a free falling observer’s vector (4.16) are

$\begin{array}[]{cccc}{\cal J}^{01}_{div}(\xi_{fall})=-{\cal
J}^{10}_{div}(\xi_{fall})=\left(e{r_{g}}^{2}r^{2}\sin\theta\left(\left(e^{2}-1\right)r+{r_{g}}\right)*\right.\\\
\left[2\log\left(\sqrt{\left(e^{2}-1\right)r+{r_{g}}}+\sqrt{r\left(e^{2}-1\right)}\right)-\log\sqrt{e^{2}-1}\right]^{2}\\\
+4\sqrt{r(e^{2}-1)(r(e^{2}-1)+r_{g})}*\\\
\left.*\left[2\log\left(\sqrt{\left(e^{2}-1\right)r+{r_{g}}}+\sqrt{r\left(e^{2}-1\right)}\right)-\log\sqrt{e^{2}-1}\right]+4\left(e^{2}-1\right)\right)/\\\
(16\pi\left(e^{2}-1\right)r^{3}\left(2\left(e^{2}-1\right)\left(\left(e^{2}-1\right)r+{r_{g}}\right)+(r_{g}/r)\sqrt{r(e^{2}-1)(r(e^{2}-1)+r_{g})}~{}~{}*\right.\\\
\left.*\log\left[2\log\left(\sqrt{\left(e^{2}-1\right)r+{r_{g}}}+\sqrt{r\left(e^{2}-1\right)}\right)-\log\sqrt{e^{2}-1}\right]\right)).\\\
\end{array}$ (4.38)

So, the 0-component of current is not zero. We see that unlike the TEGR, we
got the correct black hole mass in static case like in TEGR, but, unlike the
TEGR, we couldn’t get Einstein equivalence principle for the freely falling
observer not only in static case but in the freely falling case too.

Note, that some components in (4.34) tend to infinity when $e\rightarrow+1$.
Thus, substituting this connection into the expressions for current and
superpotential which explicitly depend on the flat torsionless connection, see
(3.11) for (3.28) with (2.3), we get infinite values too, although the flat
connection components are not observable. “Turning off” gravity in Lemaitre
coordinates directly leads also to infinite connection or non-zero Riemann
tensor values, although it has to be zero for the flat connection. So,
“turning off” gravity principle might be unsuccessful in certain cases. Here,
for example, it is unsuccessful with making the use of the Lemaitre
coordinates for the Schwarzschild solution. In this particular case, the
reason is in that the transformation (4.13) is singular with respect to
$r_{g}$. This is a problem of calculation methods only, and has to be resolved
by a related regularization procedure. This suggests also that we could probe
other ways of obtaining the connection in STEGR.

##  5 Conclusions

Applying Noether theorem we have constructed in tensorial form covariant
conserved quantities in STEGR: currents, superpotentials and charges. Because
STEGR Lagrangian can be written as the Hilbert Lagrangian plus the total
divergence, these quantities consist of two components: GR term plus the
divergent term, see (3.29), (3.30), (3.32). Each term is expressed in
evidently covariant form: (3.27) and (3.28) for the Noether superpotential,
covariant derivatives of (3.27) and (3.28) give the Noether current,
integration (3.10) of (3.27) and (3.28) gives the Noether charge. All these
expressions are covariant under coordinate transformations: GR parts are
covariant due to the covariance of the Komar superpotential and covariant
derivatives of it; the superpoential (3.28) related to the divergent term and
the corresponding current ${\cal
J}{}_{div}^{\alpha}(\xi)=\partial_{\beta}{\cal J}{}_{div}^{\alpha\beta}(\xi)$
are proportional to non-metricity (2.3); the same, to the disformation tensor
(2.5) (which are coordinate-covariant due to introducing of flat torsionless
connection) and their covariant derivatives. However, the flat torsionless
connection in STEGR is undetermined — it cannot be obtained by the field
equations. To solve this problem we find the connection using the “turning
off” gravity principle, the idea for which was taken from TEGR “turning off”
gravity principle, and the method itself was then adapted for STEGR. Usually,
authors aim to find a “coincident” gauge with zero connection corresponding to
particular coordinates [49, 20]. In our case of covariant expressions, we can
obtain the necessary teleparallel connection for each coordinates, and then
make coordinate transformation to “coincident” gauge.

To test the applications for the new Noether conserved quantities we consider
FRLW and Schwartzchild solutions. For the Friedmann solution we obtained zero
Noether current for the freely falling observers “frozen” into the Hubble
flow. This means that such observer measures zero energy and momentum
densities what corresponds to the Einstein equivalence principle. The same
result was obtained in the FRLW solution in TEGR [36, 37]. Schwartzchild
solution was considered in different coordinates: static and general radially
freely falling coordinates. “Turning off” gravity in different coordinates we
obtained different connections (different when expressed at the same
coordinates). Such kind of ambiguities in “turning off” gravity principle was
also found in TEGR [38, 39]. However, in the Schwartzchild solution, in TEGR
there were more physically expected results than in STEGR. In static case the
situations in TEGR and STEGR are the same: Noether charge gives the correct
black hole mass $M$ but non-zero Noether current for the freely falling
observer that means that the Einstein equivalence principle is not satisfied.
In the Lemaitre coordinates (general radially freely falling coordinates with
$e=1$) it is impossible to “turn off” gravity in STEGR while in TEGR the
Einstein equivalence principle was obtained. In the general radially freely
falling coordinates, in STEGR, the correct black hole mass $M$ was obtained,
but not the Einstein equivalence principle for the freely falling observer;
while in TEGR both the correct black hole mass $M$ and the Einstein
equivalence principle for the freely falling observer were satisfied.

In a more complicated case of axially symmetric solutions in $f(T,B,\phi,X)$
theories our “turning off gravity” procedure elaborated in [36, 37] has been
applied to find the correct connection in [50]. The resulting tetrads and
connections solved the field equations (including antisymmetric parts of them)
for Taub-NUT solution, but not for Kerr and C-metric solutions. It could be
interesting to examine these solutions in constructing conserved quantities in
the framework of STEGR with the “turning off gravity” procedure outlined in
subsection 3.3.

Can we satisfy both the correct black hole mass $M$ and the Einstein
equivalence principle in STEGR? In fact, one can take more general set of flat
torsionless connections without the “turning off” gravity principle. Thus, the
components of such a connection in general form were found in [51], and the
most generic static spherical connections in symmetric teleparallel geometry
were worked out in [52]. Basing on the results [51, 52] one can try to solve
the equations of zero current for the freely falling observer and condition
for Noether’s charge $M$ inside an unique set of appropriate teleparallel
connections.

It is well known that, for example, cosmological and astrophysical models
considered in GR, equivalently in STEGR, differ from those considered in
$f(Q)$ theories [53]. Therefore, it would be interesting to apply the methods
developed in STEGR to $f(Q)$ gravity and compare results of their
applications. First, because $f(Q)$ theories are covariant ones we can apply
the Noether procedure to derive the conservation laws and conserved quantities
as well. At the moment, we do not know their connection with those in STEGR,
and it would be interesting to study it. Second, the definitions of a flat
torsionless connection in STEGR and in $f(Q)$ gravity differ. Recall the
situation in TEGR and $f(T)$. The inertial spin connection in TEGR is defined
by the external principle of ”turning off” gravity, without taking into
account field equations, see [36, 37]. Unlike this, in covariant $f(T)$
theories (see, for example, [54]), inertial spin connection has to follow
after solving the related system of field equations that can be not so simple.
Concerning STEGR and $f(Q)$ theories, the situation has to be analogous. For
example, it could be very difficult to find a related flat connection for the
FRLW solution in the framework of $f(Q)$ theory. We think also that in the
case of some spacetime symmetries in the process of the “turning off” gravity
one can satisfy the field equations in $f(Q)$ gravity, and it would be
interesting to find such symmetries and then study the Noether conserved
quantities for them.

We leave the above problems for a future work.

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. The authors are
grateful to Martin Krššák and Laur Järv for helpful discussions and
recommendations.

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