# Neurosymbolic AI for Reasoning on Biomedical Knowledge Graphs

Lauren Nicole DeLong  Ramon Fernández Mir  Zonglin Ji  Fiona Niamh Coulter
Smith  Jacques D. Fleuriot

###### Abstract

Biomedical datasets are often modeled as knowledge graphs (KGs) because they
capture the multi-relational, heterogeneous, and dynamic natures of biomedical
systems. KG completion (KGC), can, therefore, help researchers make
predictions to inform tasks like drug repositioning. While previous approaches
for KGC were either rule-based or embedding-based, hybrid approaches based on
neurosymbolic artificial intelligence are becoming more popular. Many of these
methods possess unique characteristics which make them even better suited
toward biomedical challenges. Here, we survey such approaches with an emphasis
on their utilities and prospective benefits for biomedicine.

Neurosymbolic AI, knowledge graphs, bimoedicine, representation learning,
hybrid AI, graph neural networks.

, , , ,

##  1 Introduction

Often, biomedical concepts can be understood as part of a system comprising
highly multi-relational, heterogeneous, and interconnected entities. For
instance, molecular interactions are observed to engage in dynamic, condition-
dependent networks (Chang & Karin, 2001). Graph structures are a popular
choice for representing biomedical systems as they facilitate the storage of
such data (Li et al., 2022).

Recently, several artificial intelligence (AI) and machine learning (ML)
approaches for analyzing biomedical data (Esteva et al., 2017; Savoy, 2020)
and assisting clinical treatment planning (Wang et al., 2019) have been
reported. Hence, we anticipate that AI will become an integral part of modern
healthcare processes. As this becomes more of a reality, considering the
ethical and practical benefits of model interpretability, the inherent ability
to understand how and why predictions were made, could significantly improve
the resultant quality of care. However, facilitating model interpretability is
a challenge as state-of-the-art methods tend to lack it (Chakraborty et al.,
2017). Furthermore, while patterned domain knowledge between general concepts
in biology is ample (Ashburner et al., 2000; Guvench, 2016), few data-driven
ML approaches explicitly encode the high-level relationships from those
knowledge bases.

Recently, however, neurosymbolic methods have managed to leverage the benefits
of both rule-based inference and deep learning. These methods possess unique
characteristics which are suited to address the aforementioned challenges. To
facilitate a better understanding of these characteristics, we survey several
of such approaches that were used on biomedical graphs. In our previous survey
(DeLong et al., 2023), we outlined three major taxonomic categories for
neurosymbolic reasoning on graph structures:

  1. 1.

logically-informed embedding approaches,

  2. 2.

embedding approaches with logical constraints, and

  3. 3.

rule-learning approaches.

Using these categories as guides, we classify and compare the approaches with
a novel concentration on their utilities toward biomedical challenges.
Specifically, we focus on and refer back to the interpretability,
incorporation of domain knowledge, and sparsity handling abilities of each
approach since these traits are particularly interesting for biomedical
applications (Section 4). Additionally, we propose several prospective
directions within biology and medicine which could benefit from this area of
work.

##  2 Background

###  2.1 Knowledge Graphs

A _graph_ or _network_ structure is composed of a finite set of vertices or
nodes, _V_ , and a finite set of edges, E, connecting the vertices. Each edge
$e=(u,v)\in E$ connects two adjacent or _neighbouring_ vertices, creating a
triple (Rahman et al., 2017). A _knowledge graph_ (KG), in particular, uses a
graph structure to represent a knowledge base for some specific domain.

KGs are a practical way to represent datasets as they capture multi-relational
data well, are fast and easy to query, and allow us to store many different
types of data in a similar format (a “universal language”) (Nickel et al.,
2015; Ehrlinger & Wöß, 2016). Notably, KGs can represent growing or changing
knowledge bases effectively as they permit flexible storage schemas (Vicknair
et al., 2010). As biomedical systems are diverse, tightly interconnected,
complex, and frequently updated, the KG structure is a popular choice to store
biomedical knowledge (Li et al., 2022).

###  2.2 Knowledge Graph Completion

Many KGs are inherently incomplete as they are limited to the knowledge we, as
humans, possess (Chen et al., 2020). Biomedical KGs are no exception; our
knowledge of biological macromolecules and their interactions is still only a
fraction of what is predicted to exist (Povolotskaya & Kondrashov, 2010; Jiang
et al., 2014). Therefore, KG completion (KGC), the practice of adding new
information to a graph, is a popular category of reasoning tasks on KGs. One
prominent form of KGC is link prediction, where the goal is to predict whether
an edge exists between two nodes (Zitnik et al., 2018; Chen et al., 2020;
Zhang et al., 2021). Here, we contrast symbolic methods for KGC (particularly
link prediction) against corresponding embedding-based methods.

####  2.2.1 Symbolic Methods for KGC

Some of the simplest and most intuitive methods toward KGC utilize a set of
rules, which often use logical inference to perform link prediction. For
example, consider the following background knowledge regarding the three
molecular entities, Raf, MEK, and ERK, used in a common cell signaling
mechanism (Chang et al., 2003):

$\small\text{activates}(Raf,MEK),$ $\small\text{activates}(MEK,ERK),$
$\small\text{increases\\_activity}(X,Z)\leftarrow\text{activates}(X,Y)\land\text{activates}(Y,Z).$

From this, we deduce that $\text{increases\\_activity}(Raf,ERK)$. The source
of said rules varies between approaches. Ontologies, for example, are
formalisations of knowledge base semantics which often represent the unique
patterns and hierarchies within a specific knowledge domain (Dou et al.,
2015). From such patterns, one can generate rules to infer new knowledge.
Ontologies, such as those discussed in Section 3.1, are ample in biomedicine
(Ashburner et al., 2000; Schriml et al., 2022). However, they do not exist for
every domain, and existing ones sometimes impose limitations on the type of
predictions made. Alternatively, other methods mine rules directly from the KG
(Galárraga et al., 2013). In these cases, rules are based on the association
patterns that are found to exist within the KG rather than within the
application domain.

The major benefit of rule-based approaches is that they are inherently
interpretable. While interpretability is defined in various ways across the
literature, it is generally viewed as the ability to be understood by a human
(Marcinkevičs & Vogt, 2020). For instance, one could refer back to the rules
which governed the algorithm to get a human-readable understanding of how and
why certain predictions were made. Unfortunately, compared to other
methodologies, such as those discussed in the next section, rule-based methods
do not always achieve the best performance (Gema et al., 2023). Additionally,
when faced with large KGs, they often suffer from scalability issues such as
exponential computational complexity (Zhou, 2021). This is an increasingly
pressing issue with the rise of big data and digitalization (Ji et al., 2021).

####  2.2.2 Embedding Methods for KGC

In contrast to rule-based methods, KG embedding (KGE) methods often scale well
to large datasets. In general, such methods aim to compute embeddings, low-
dimensional vector representations of the graph constituents, so that
similarity in the embedding space should approximate some sort of similarity
in the original graph (Zhang & Chen, 2018; Li & Pi, 2020). They have recently
increased in popularity due to their competitive performances in fields such
as biochemistry (Zitnik et al., 2018) or disease state prediction (Ravindra et
al., 2020). While some KGE methods use a pre-determined function to generate
embeddings, others, such as a Graph Neural Network (GNN) use neural networks
and deep learning to learn the best function.

Despite their competitive performances, KGE approaches are often black-box
models, so they lack interpretability (Chakraborty et al., 2017). This can be
problematic ethically and practically in areas like medicine, as an end-user
cannot see whether predictions are reliably generated or understand the
grounds that justify novel predictions (Norori et al., 2021; Olejarczyk &
Young, 2022).

##  3 Biomedical Concepts and KGs

To appreciate the usefulness of methodological differences between the
surveyed approaches, it is critical to understand their shared biomedical
goals and the corresponding datasets on which they operated. In the following
sections, we explore what these KGs comprise and explain the biomedical
prediction tasks addressed.

###  3.1 Common Biomedical Concepts

To understand the concepts modeled within biomedical KGs, we briefly describe
the most common biomedical terms used within the literature. Fields of biology
are organized on varying levels of complexity. From simplest (or smallest) to
most complex, biology is studied on the molecular, cellular, tissue,
anatomical, individual, or population levels, among others (Motofei, 2022;
Garland Jr et al., 2022). Generally, one can understand each subsequent level
as a system which envelopes the previous level. While clinical studies that
focus on populations of patients often translate well to treatment on the
individual level, studies on the former levels, especially the molecular
level, help researchers uncover disease etiologies and regular bodily
processes through the mechanisms and interactions of the body’s most
elementary units (Funkhouser, 2020).

Here, we broadly define key biomedical concepts. Genes are pieces of genetic
material (DNA) which instruct the cell as to which functional products should
be made. Proteins, such as Raf, MEK, and ERK introduced in Section 2.2.1, are
the resultant functional products that perform important activities within the
cells (Cooper & Adams, 2022). Together, these activities work to create
biological processes (BPs) (Ashburner et al., 2000). Compounds typically
describe small chemicals found naturally or supplemented. In contrast, drugs
are compounds introduced into the body deliberately to interact with BPs and
achieve some modified, often medicinal, effect. Compounds and drugs are
sometimes referred to interchangeably throughout the literature (Tatonetti et
al., 2012; Coleman & Pontefract, 2016). A drug indication describes the
condition that a drug is intended to treat, while a side effect is an
unintended and often adverse effect induced by a drug (Coleman & Pontefract,
2016). These components interact in a complex and highly dynamic fashion.
Different individuals vary in genetic makeup, and within individuals, various
tissues and cell types carry out distinct BPs. To accomplish those unique BPs,
therefore, different cell types express, or read, different genes to create
specific proteins. Because the proteins generated vary across cell and tissue
types, the ways in which compounds interact vary across those types, leading
to altered indications and side effects (Mauger et al., 2019; Buccitelli &
Selbach, 2020; Ross et al., 2021).

###  3.2 Biomedical KGs Used

All of the approaches surveyed here were used on a biomedical KG. Notably,
many of them used a particular KG containing information from the unified
medical language system (UMLS) (McCray, 2003; Bodenreider, 2004) because it is
openly available and a commonly used benchmark dataset (Kok & Domingos, 2007;
Dai et al., 2020). Other approaches deviate from this trend, however, and
choose a KG more specific to biomedical research goals. All openly available
biomedical KGs mentioned in this survey are summarized in Table 1.

Table 1: Knowledge Graphs Commonly Used Across Surveyed Approaches KG |  Origin |  Nodes |  Node Types |  Triples |  Edge Types |  Comprises
---|---|---|---|---|---|---
UMLS |  https://www.nlm.nih.gov/research/umls/index.html |  893,025 |  135 |  5,960 |  49 |  ontology-derived biomedical concepts
SNAP |  https://github.com/williamleif/graphqembed |  97k |  UNK |  >8 mil. |  42 |  drugs, diseases, proteins, side effects, and BPs
kg-covid-19 |  https://github.com/Knowledge-Graph-Hub/kg-covid-19 |  574,778 |  20 |  24,145,562 |  41 |  drug, protein, gene interactions relevant to COVID-19
Hetionet |  https://github.com/hetio/hetionet |  47,031 |  11 |  2,250,197 |  24 |  genes, compounds, and diseases
OREGANO |  https://gitub.u-bordeaux.fr/erias/oregano |  152,086 |  10 |  680,757 |  14 |  drugs, proteins, genes, and diseases

###  3.3 Biomedical Prediction Tasks

For the most part, studies which use the UMLS KG use it as a benchmark
dataset. In contrast, studies with a more specific biomedical research goal
tend to use other KGs better tailored toward that challenge. For example,
because the COVID-19 pandemic accelerated biomedical publication output by
$\sim$10 000 new articles per month (Chen et al., 2023), the creators of
BioGRER (Section 4.3.1) (Zhao et al., 2020) aim to refine a COVID-19 KG built
by automatic information extraction from literature. While KG refinement is
not unique to biomedicine, it is highly important, especially after the surge
of pandemic-based literature.

A more prominent challenge in ML-based biomedical studies, including those
within this survey, is drug repositioning, or repurposing (DR). DR is the use
of a clinically approved drug to treat another condition for which it was not
originally intended (Ashburn & Thor, 2004). Because a drug must pass expensive
and time-consuming clinical trials before it gains clinical approval, using an
approved drug for a novel indication saves time and money (Jayasundara et al.,
2019). This process was especially important for finding potential therapies
quickly during the COVID-19 pandemic (Schultz et al., 2023). DR is often
formatted as a link prediction task between a node representing a drug and
another representing some condition or treatment (Schultz et al., 2023).

As the DR challenge has existed for several decades (Ashburn & Thor, 2004),
approaches varying from ontology-derived semantic networks to deep learning
methods have been developed to address it (Xue et al., 2018; Luo et al.,
2021). Since approaches to DR already reflect the previously described
methodological dichotomy from Section 2.2, it is an ideal area to develop
neurosymbolic, hybrid approaches. Furthermore, as Luo et al. point out, many
previous approaches do not address the sparsity of verified indication edges,
so the integration of biomedical domain knowledge into state-of-the-art models
could ameliorate that issue (Luo et al., 2021). Finally, since molecular
interactions are both spatially and temporally dynamic (Mauger et al., 2019;
Buccitelli & Selbach, 2020; Ross et al., 2021), drug indications discovered in
one cell or tissue model do not always translate to others. As neurosymbolic
approaches often enhance interpretability, they could help to mitigate this
uncertainty. In Section 4, the utilities of neurosymbolic AI for the DR
challenge are demonstrated through some of the surveyed approaches, namely
Walking RDF and OWL (Alshahrani et al., 2017), PoLo (Liu et al., 2021), and
graph query embeddings (Hamilton et al., 2018).

As neurosymbolic methods for reasoning on graphs are still growing in
popularity, there are many outstanding challenges, common for reasoning on
biomedical KGs (Li et al., 2022), that are not addressed by the approaches
discussed here. This includes the prediction of side effects (Deac et al.,
2019; Krix et al., 2022), polypharmacy side effects (involving interactions
between two or more drugs) (Zitnik et al., 2018; Gema et al., 2023), and
protein function prediction (Xia & Ku, 2021). In addition to these, we suggest
other prospective directions later in Section 5.

mytree/.style= baseline, for tree=mynode, l sep+=3pt, edge=gray, thick, parent
anchor=south, child anchor=north, align=left, edge path= [draw, edge]
(!u.parent anchor) – +(0,-5pt) -— (.child anchor)edge label;  mytree,
[,phantom [Neurosymbolic Methods on Biomedical KGs [Logically-Informed
Embedding Approaches
• Walking RDF and OWL

• Walking RDF and OWL

improvement ] [KGE with Logical Constraints
• R2N

• QLogicE

• GQE ] [Rule Learning
Methods
,l sep+=20pt, [Iterative, EM Methods
• RNNLogic

• BioGRER , edge path= [draw, edge] (!u.parent anchor) – +(0,-20pt) -— (.child
anchor)edge label; ], [New Rules: Bridging the Discrete
and Continuous Spaces
• NeuralLP

• DRUM

• LPRules , edge path= [draw, edge] (!u.parent anchor) – +(0,-20pt) -— (.child
anchor)edge label; ], [Path-Based Rule Learning
• PoLo

• LNN-MP , edge path= [draw, edge] (!u.parent anchor) – +(0,-20pt) -— (.child
anchor)edge label; ] ] ] ]

Figure 1: Taxonomy of neurosymbolic approaches for reasoning on biomedical KGs
(DeLong et al., 2023).

##  4 Neurosymbolic Approaches on Biomedical KGs

As established, we observe a dichotomy between methods for reasoning on graph
structures using symbolic, rule-based methods (Section 2.2.1) versus state-of-
the-art KGE approaches (Section 2.2.2). Recent studies in _neurosymbolic AI_
often combine aspects from deep learning and symbolic reasoning to mitigate
their respective weaknesses and bridge such chasms (Tsamoura & Michael, 2020;
Garcez & Lamb, 2020; Susskind et al., 2021). Throughout this survey, we
discuss three major characteristics of these neurosymbolic approaches which
benefit biomedical research:

  1. 1.

Interpretability: As mentioned, we often see a tradeoff between
interpretability, which symbolic AI naturally possesses, and performance, in
which black-box, deep learning methods seem to dominate. In particular,
enforcing interpretability often comes with a drop in predictive performance
(Dziugaite et al., 2020; Molnar, 2022) and might lead to infeasible
computational complexity (Bertsimas et al., 2019; Dziugaite et al., 2020).
Some of the surveyed neurosymbolic approaches, therefore, foster
interpretability without sacrificing performance. As mentioned in the previous
section, interpretability is valuable for biomedical applications: it could
grant both healthcare providers and patients clarity as to how a decision was
reached, as well as potentially expose reasoning processes derived from biases
in the training data, such as imbalanced race or gender distributions
(Olejarczyk & Young, 2019; Tat et al., 2020; Norori et al., 2021).

  2. 2.

Integration of Domain Knowledge: Biology is rich in patterned domain knowledge
which is typically only implicitly represented in data-driven approaches. As
mentioned in Section 2.2.1, many biomedical ontologies exist to represent
patterns of relationships and hierarchies. These include the Gene Ontology
(GO) (Ashburner et al., 2000), the Disease Ontology (DO) (Schriml et al.,
2022), & the Human Phenotype Ontology (HPO) (Köhler et al., 2021). There is
also a widely accepted but manually designed classification system for
macromolecules such as proteins (Lo Conte et al., 2000; Pearl et al., 2003)
and their activities (Tipton & McDonald, 2018), as well as a developing system
for classifying functional groups, or parts, on drugs (Guvench, 2016; Peng et
al., 2020). The ability of some neurosymbolic approaches to integrate this
knowledge into an otherwise data-driven approach is powerful. First, it
bypasses the need for the model to learn patterns which are already known.
Furthermore, it guides the learning toward more biologically accurate and
relevant patterns.

  3. 3.

Sparsity Handling: As stated in Section 2.2, knowledge within a biomedical KG
is likely only a fraction of the what is discoverable. Furthermore, many
clinical datasets have smaller proportions of certain genders and races (Tat
et al., 2020; Puyol-Antón et al., 2021) and infrequently contain instances of
rare conditions (Zhao et al., 2018). Data sparsity and imbalance are,
therefore, major challenges within biomedical data. Through KG augmentation,
some neurosymbolic approaches pose novel ways to address this.

Within the following subsections, we refer back to these three characteristics
and discuss the varied strategies by which different approaches cultivate
them. Notably, to facilitate easy comparison, we group the approaches
according to the taxonomy in Fig. 1, based on categories from our previous
survey (DeLong et al., 2023). Finally, we highlight unique features of each
approach that are particularly beneficial for biomedical prediction tasks. To
facilitate accessibility, we curated a repository of available code for each
work on GitHub111https://github.com/NeSymGraphs.

###  4.1 Logically-Informed Embedding Approaches

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&[e_{00},e_{01},\dots,e_{0n}]\\\ &[e_{10},e_{11},\dots,e_{1n}]\\\
&[\dots]\end{aligned}$ Logical inferenceKG embedding

Figure 2: Logically-Informed Embedding Approaches (DeLong et al., 2023).

The most straightforward way to combine the benefits of symbolic and KGE
methods is to use them in succession. Some neurosymbolic approaches on graphs
modularize the two, then feed the results from the former into the latter.
Because symbolic approaches are often based on expert-defined rules, they can
be viewed as methods to extend the ground truth and are therefore used as a
preliminary step (the logic module) which feeds into the KGE step (the neural
module). One approach (Alshahrani et al., 2017), for example, employed an
ontological reasoner to infer new edges in their KG, followed by a common KGE
method called DeepWalk (Perozzi et al., 2014). Consequently, the generated
embeddings contained richer information as they were based on an augmented KG.
Their method, Walking RDF and OWL, is a specific example of the more general
pattern illustrated in Fig. 2. Using these semantic-aware embeddings, they
performed link prediction for a DR task. Not only did prediction performance
improve with the augmented KG, but their model could also accurately predict
other links, like side effects. Another group (Agibetov & Samwald, 2018)
improved upon this approach with an alternative, log-linear embedding method.

These studies effectively demonstrate several ways in which biomedical domain
knowledge can be integrated into a KGE approach. First, Alshahrani et al.
develop their own KG from various sources, including both evidence-based
interactions from databases and higher-level relationship patterns from
biomedical ontologies, including the GO, DO, and HPO. By mapping individual
biomedical entities such as proteins and drugs to their respective
classifications in these ontologies, they expand the dimensionality of the KG
through those higher-level relationships. Furthermore, while Walking RDF and
OWL uses an ontological reasoner nonspecific to biology, this general approach
presents the ideal opportunity to use biomedical ontologies or rules in the KG
augmentation step.

The KG augmentation step also helps with sparsity handling. As mentioned, some
clinical datasets have fewer instances of rare diseases, so there will be less
known indications, side effects, and processes associated with those diseases.
On the molecular level, we also see disparities in the volume of data
surrounding certain relations; particular BPs are especially difficult to
study as the involved macromolecules are harder to isolate in vitro or degrade
easily (Lee, 2017). It is possible that, while there is expert knowledge about
the way these rare relations occur, a purely data-driven model might not
discover these patterns due to imbalance in the training data. If the KG is
initially augmented using expert-defined or domain-specific rules, this may
increase the number of these rare relations on which the KGE model is trained.
As their application is DR, this concept is particularly important for the
discovery of novel treatments: the prediction of indications for uncommon
diseases or drugs is likely more impactful than that for more common ones.

###  4.2 KGE with Logical Constraints

In contrast to integrating biomedical knowledge through KG augmentation, other
approaches use it to impose logical or rule-based constraints to the KGE
pipeline. In this case, the main focus is typically on training the KGE
method, but rules limit the scope of predictions possible. These approaches
fundamentally differ based on where logical constraints are imposed in the
learning pipeline.

For example, logical constraints can be applied directly to the latent space
embeddings, as shown in Figure 3. The Relational Reasoning Network (R2N)
(Marra et al., 2021), for instance, uses structural relational information as
constraints on GNN training. This transforms the latent space embeddings in a
way that is dependent upon each node’s neighbors in the original space so that
learned KGEs also encode information about each node’s surrounding graph
topology. This is ideal for biomedical networks as many interactions are
dependent upon some adjacent interaction. For example, a protein called a
GTPase alters a molecule called Guanosine triphosphate (GTP), and this
reaction is highly involved in the cell death BP (He et al., 2021). However, a
GTPase is regulated by interaction with another protein, the GTPase-activating
protein (He et al., 2021). Accounting for neighboring graph topology,
therefore, helps capture such relationships. Consequently, the R2N encodes
high-level relationships like the approaches in Section 4.1, but in R2N, the
corresponding truth values for these relationships are incrementally adjusted.
Furthermore, while such relationships could be encoded into R2N explicitly
like they were in Walking RDF and OWL, their particular use on the UMLS KG
relies, instead, on implicit logic rules which are not made available to the
user. Unfortunately, this is a missed opportunity regarding characteristics 1
and 2 described at the start of Section 4. We see the same traits in the
Quantum Logic Empowered Embedding (QLogicE) (Chen et al., 2022) method which
computes logic embeddings and KGE in parallel. The scoring function for each
possible triple is a weighted sum of the scores for the two embedding methods,
and the loss is determined as the weighted sum of the loss of the two
embedding methods. Like R2N, high-level relationships between general concepts
are determined in the QLogicE model through the logic embedding module, but
they are not available for the user to explore. While QLogicE achieves
competitive performance on the UMLS KG, the lack of interpretability limits
the utility in biomedicine.

Also like R2N, another study (Hamilton et al., 2018) provides a different way
to incorporate logical constraints encoding dependencies between adjacent
edges. Here, they motivate the need for conjunctive queries on a biomedical KG
through the DR task: they pose the idea that one might query for a drug that
simultaneously treats two conditions via a commonly involved protein. This
query involves the conjunction of three concepts: that a drug targets some
protein, and that the protein is associated with both diseases. Here, we see
again that the sought indications rely on the adjacent relationships between
the protein and the diseases. Their method, graph query embeddings (GQE),
combines logic and node embedding spaces in a way that the logic embeddings
represent a query. It does so by learning both node embeddings and trainable
parameter matrices which represent logical operators. Together, they use these
to compute query embeddings. Thereafter, the likelihood that a set of nodes
satisfy a query is calculated as the cosine similarity between the query
embeddings and the respective node embeddings. Similarly to R2N, GQE imposes
logical constraints onto the embedding space. However, GQE is arguably better
at integrating domain knowledge because the structure of the conjunctive query
is a more task-specific way to make predictions.

In general, while imposing logical constraints allows multiple ways to
incorporate domain-specific information, it does not necessarily add any
aspects of interpretability. To take full advantage of the neurosymbolic
concept of hybridizing performance with interpretability, the logic and neural
components of an architecture should ideally be combined in a way that user-
level explanations are generated. In the next section, we review several
approaches which do just that.

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&[e_{00},e_{01},\dots,e_{0n}]\\\ &[e_{10},e_{11},\dots,e_{1n}]\\\
&[\dots]\end{aligned}$ KG embeddingConstraints

Figure 3: KGE with Logical Constraints (DeLong et al., 2023).

###  4.3 Rule Learning Methods

In the previous sections, all of the described methodologies use predefined
rules. However, many other approaches attempt to learn rules and rule
confidences for KGC.

####  4.3.1 Iterative, EM Methods

Many rule-learning methods claim to take an Expectation-Maximization algorithm
(EM) based approach, alternating between two modules whose outcomes inform
each other. The EM algorithm is used to estimate the maximum likelihood of
model parameters for situations in which there is incomplete data (Gupta &
Chen, 2011; Ng et al., 2012). KGC, then, is an ideal and obvious setting for
such methods. The iterative nature of the EM algorithm between the prediction
of missing components and the optimization of model parameters to account for
such predictions is the core idea here. Similar to approaches described in
Section 4.1, these methods use two complementary modules to inform one
another, but the logic module is now dynamic, adjusting the rule base at each
iteration. This bidirectional iteration is shown in Fig. 4. These methods
vary, however, in their interpretations of how an EM-based algorithm is
executed.

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&[e_{00},e_{01},\dots,e_{0n}]\\\ &[e_{10},e_{11},\dots,e_{1n}]\\\
&[\dots]\end{aligned}$ $\begin{aligned} &(p_{1})&&\text{Rule 1}\\\
&(p_{2})&&\text{Rule 2}\\\ &&&\cdots\\\ &&&\text{new rules?}\end{aligned}$
Logical inferenceDecodingE-stepM-step

Figure 4: Iterative, EM Algorithm Methods (DeLong et al., 2023).

RNNLogic (Qu et al., 2020), unlike other KGC methods, employs an EM-based
algorithm without any sort of KGE. Instead, RNNLogic iterates between training
a rule generator (the “M-step”) and an inference step (the “E-step”). The two
inform one another to reduce the search space of possible logical rules. In
contrast, Zhao et al.’s method, Biomedical KG refinement with Embedding and
Rules (BioGRER) (Zhao et al., 2020) incorporates a KGE method. During the
“E-step”, logical rules are used to infer novel triples, and those inferred
triples are used to train the KGE method. Thereafter, the predicted triples
from the KGE method are used to inform and update the corresponding rule
weights in the logic module as the “M-step”.

Both methods foster interpretability in a way that the previous approaches do
not: they make both the rules and the corresponding weights available to the
end-user. This means that the user can see which reasoning processes
contributed to the prediction process and which were most influential. Zhao et
al., who used kg-covid-19 (see Table 1), give specific examples of learned
rules, such as,
$\textit{interacts\\_with}(x,y)\Leftrightarrow\textit{interacts\\_with}(y,x).$
Because BioGRER uses generic rule models as templates for learned rules, a
user can encode domain-specific knowledge into these rule models. For example,
the transitive rule written as
$\textit{r${}_{i}$}(x,y)\land\textit{r${}_{i}$}(y,z)\rightarrow\textit{r${}_{i}$}(x,z)$
may not be as useful in the biomedical context. Consider, for instance, the
binds relation between two proteins, indicating that they physically and
chemically connect. Two binds relations may constitute adjacent edges but not
indicate transitivity in that sense. Transitivity, however, can be extended to
two different but adjacent relations which result in a third, overarching
relation. Zhao et al. give the clinical example:
$\textit{tributary\\_of}(x,y)\land\textit{drains}(y,z)\rightarrow\textit{part\\_of}(x,z)$.
BioGRER, therefore, uses an extended version of the transitive rule as it
suits the application domain better:
$\textit{r${}_{i}$}(x,y)\land\textit{r${}_{j}$}(y,z)\rightarrow\textit{r${}_{k}$}(x,z)$.
Thus, BioGRER satisfies characteristics 1 and 2 at the start of Section 4. In
fact, both RNNLogic and BioGRER satisfy the third characteristic of sparsity
handling in the same way that the approaches from Section 4.1 do. Because they
use rule-based inference in their logic modules, they can increase the
instances of rare relations in the data on which the neural module is
subsequently trained. Within the next section, we survey approaches which take
the rule-learning strategy a step further, generating new rules for the logic
module.

####  4.3.2 New Rules: Bridging the Discrete and Continuous Spaces

The previous methods assigned weights to rules which were mined or constructed
according to a template, but other approaches use the learning process to
drive the construction of new rules. To learn and generate novel rules, an
algorithm must learn both structural information in a discrete space and
respective weight parameters in a continuous space. The creation of a method
which can do both and train in an end-to-end fashion is inherently difficult
(Yang et al., 2017; Sadeghian et al., 2019). The benefit, though, is that the
resultant rules are more descriptive of the reasoning process, thereby
enhancing interpretability and improving upon point 1 from the start of
Section 4.

LPRules (Dash & Goncalves, 2021), for example, repeatedly augments a small
starter pool of candidate rules, optimizing predictive capability over
iterations. By doing so, they minimize the search space in comparison to
enumerating and testing all possible rules. Alternatively, Neural LP (Yang et
al., 2017) avoids rule enumeration by learning confidence values for every
relation involved in every rule body, even accounting for differences in the
varying lengths. This makes the rule bodies dynamic with the learning process.
DRUM (Sadeghian et al., 2019), is a refined version of Neural LP in which the
authors point out that Neural LP learns high confidences for incorrect rules.
DRUM additionally encodes information about how feasibly two relations can co-
exist within a rule body. This is particularly interesting for capturing
domain-specific knowledge. For example, on the molecular scale, certain
functions can be highly specialized to a subcellular location like the nucleus
or mitochondria (Lu & Hunter, 2005); these functions are unlikely to co-occur
in a chained pattern. However, while learned rules are, in theory, more
expressive, there is a major caveat to these methods. Neural LP and DRUM only
train on positive examples and have yet to be tried on a dataset with negative
ones. A lack of negative examples in the training data increases the risk of
false positive predictions, which, in contexts such as such as DR, could lead
to a waste of time, resources, or potentially even health hazards.

####  4.3.3 Path-based Rule Learning

Most of the previous approaches treat rules as statements involving individual
relation types. Path-based methods, however, learn and make inferences from
chains of edges or relations. Path-based rules are not only interpretable but
often more expressive, as they facilitate understanding of relationships
between nodes that are several hops away from each other, a trait which we
call long-range dependency (Liu et al., 2021).

Long-range dependency is particularly important in biology in which we
frequently see chains of interactions between macromolecules. For instance,
molecular signaling cascades describe a chain in which the interaction between
two proteins is dependent upon one’s contact with another molecule upstream
(Chang & Karin, 2001; Crino, 2016). In fact, there are several databases
storing such pathway information (D’Eustachio, 2013). This is why one study,
called Policy-guided walks with Logical Rules (PoLo) (Liu et al., 2021), uses
a path-based neurosymbolic method for DR. By exploring indirect drug
indications that might operate through other entities, such as genes, they
expand the set of novel discoveries possible. PoLo encodes general path
patterns, or metapaths, as logical rules with pre-computed confidence scores.
While metapaths can be manually selected from expert knowledge, they can also
be mined, such as in a follow-up study (Drancé et al., 2021). Therefore, these
path-based methods are not only interpretable, but they also hold potential to
include domain knowledge.

Sometimes, less frequent path sequences might be less relevant to the end-
goal. Although molecular components interact in complex, multi-relational
webs, we still see both physical (Lu & Hunter, 2005) and functional
compartmentalization (Ashburner et al., 2000). For example, if a study is
searching for a drug which aids in the cellular detoxification process, the
paths involved in cell replication might be less relevant. Another group (Sen
et al., 2021) addresses such a situation with their LNN-MP method by
generating all possible length-$k$ path-based rules, then using them on KGEs
pre-trained with bias toward more prevalent relation sequences. By doing so,
LNN-MP learns to generate rules most relevant to the paths present in the KG.

Of the three major categories, the rule-learning methods most often cover the
three characteristics mentioned at the start of Section 4: their rule
production adds interpretability to otherwise black-box models, some admit the
use of domain-specific rules, and their use of logical inference can augment
the instances of rare relations. Next, we explore the prospective
opportunities to exploit these characteristics.

##  5 Prospective Directions

As mentioned, there are plenty of technical and practical areas yet to be
fully explored, especially in biomedicine. Here, we suggest a number of
prospective directions, which we hope will cultivate a greater interest in
this domain.

###  5.1 Mechanism of Action Discovery

As mentioned in Section 4.3.3, many molecular interactions rely on long-range
dependencies. In particular, some drug indications rely on a chain of events
(Schenone et al., 2013). Unfortunately, the path by which a drug compound
induces a therapeutic effect, which we call the mechanism of action (MOA)
(Schenone et al., 2013), is not always straightforward or understood.
Furthermore, cellular interactions can be nonspecific, so side effects may
occur as a result (Schenone et al., 2013). By using the rules learned by one
of the path-based methods from Section 4.3.3, one could potentially uncover
unknown MOAs of drugs. Revealing the MOA would facilitate a deeper
understanding of how directly or indirectly a given drug achieves some
therapeutic effect, what additional therapeutic or adverse effects can be
expected from usage of said compound, and how such effects may vary across
cell and tissue types.

###  5.2 Spatiotemporal Reasoning

With respect to understanding differences across cell and tissue types,
biomedical networks also frequently have spatiotemporal dependencies, as
mentioned in Section 3.1. Previous approaches for spatiotemporal-conscious
reasoning on graph structures were based on deep learning architectures (Yu et
al., 2017; Chen et al., 2018; Li et al., 2019). However, we have yet to
observe many neurosymbolic approaches. For example, methods from Section 4.3
could learn rules which describe the relationships between the various edge
types and time. Alternatively, rules could be learned for each specific time
range or spatial location. As mentioned, the types and abundance of certain
proteins vary across tissue types, so the levels to which proteins affect one
another are spatially dependent (Buccitelli & Selbach, 2020). Learning rules
specific to tissue types could better inform clinical researchers as to how
processes differ between tissues.

Alternatively, instead of learning rules, methods like those in Section 4.2
could encode time- or space-dependent rules as constraints. For example, in a
biological network, the half-lives of molecules such as messenger ribonucleic
acid not only act as a time limit on potential interactions but also determine
the levels to which various proteins are expressed, and therefore the extent
to which those proteins can have effects on other players in the network
(Mauger et al., 2019; Buccitelli & Selbach, 2020). Such constraints can more
realistically model spatiotemporal dependencies by adding logic or domain-
specific rules.

###  5.3 Conditional or Interdependent Edge Types

In some cases, certain relations or edge types might be dependent upon one
another in a way that one edge does not occur unless other edges exist. An
example of this can be found in the case of molecular signaling cascades in
which one protein will not interact with another unless contact is made with
another protein or molecule upstream (Chang & Karin, 2001; Crino, 2016).
Alternatively, such dependence might also exist in a way that more frequent
occurrences of a given relation have some effect on another relation, such as
increasing or decreasing the frequency. Taking again the signaling cascade
example, the more an inhibitor interacts with some protein, the less that
protein might be able to take part in other interactions (Gubb et al., 2010;
Pleschka et al., 2001). While such dependencies and conditional scenarios are
not unique to neurosymbolic reasoning, it might be ideal for addressing them.
With the incorporation of rules, neurosymbolic methods could provide novel
ways to encode dependency information between relation types or individual
relations. Potentially, methods from Section 4.2, which use logic to assess
the probability that an edge is implied from its adjacent edges, could be
ideal for interdependent edge types. Furthermore, since methods from Section
4.3 learn confidences for rules, there is also potential to learn weights
between rules that coexist or influence one another. Neurosymbolic methods for
graphs could, therefore, provide unique ways to interpret such dynamic
biomedical KGs and the relationships between interdependent molecular
interactions.

###  5.4 Few Shot Learning

As already mentioned, methods such as the approaches in Sections 4.1 and 4.3
increase the number of rare relation types through KG augmentation. Such
approaches could, therefore, be used as a solution for few shot learning
problems. In few shot learning, there exist a small number of instances of
some class within the training data (Yue et al., 2020). This could promote
research which is more inclusive of rare diseases or BPs.

##  6 Conclusion

In this survey, we focus on how the unique traits of neurosymbolic methods for
graphs are fitting for a multitude of challenges in biomedicine, condensing
some of the concepts covered by our previous survey (DeLong et al., 2023). We
provide a methodological taxonomy by which to classify these approaches, and
we discuss the ways in which they facilitate interpretability, knowledge-
integration, and sparsity handling within the context of biomedical KGC.
Additionally, we curated a repository of available code for each work on
GitHub222https://github.com/NeSymGraphs. Finally, with hopes to guide future
research, we propose prospective biomedical directions toward which
neurosymbolic methods for reasoning on graphs might steer.

## Acknowledgements

LND is funded by the University of Edinburgh (UoE) Informatics Graduate School
through the Global Informatics Scholarship. RFM is funded by the UoE Institute
for Academic Development through the Principal’s Career Development PhD
Scholarship. FNCS is supported by the United Kingdom Research and Innovation
(grant EP/S02431X/1), UKRI Centre for Doctoral Training in Biomedical AI at
the UoE, School of Informatics. For the purpose of open access, the author has
applied a creative commons attribution (CC BY) licence to any author accepted
manuscript version arising. We thank these institutions for their support.

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###### Abstract

Virial identities are a useful mathematical tool in General Relativity. Not
only have they been used as a numerical accuracy tool, but they have also
played a significant role in establishing no-go and no-hair theorems while
giving some physical insight into the considered system from an energy balance
perspective. While the calculation of these identities tends to be a
straightforward application of Derrick’s scaling argument [1], the complexity
of the resulting identity is system dependent. In particular, the contribution
of the Einstein-Hilbert action, due to the presence of second-order
derivatives of the metric functions, becomes increasingly complex for generic
metrics. Additionally, the Gibbons-Hawking-York term needs to be taken into
account [2]. Thankfully, since the gravitational action only depends on the
metric, it is expected that a “convenient” gauge that trivializes the
gravitational action contribution exists. While in spherical symmetry such a
gauge is known (the $m-\sigma$ parametrization), such has not been found for
axial symmetry. In this letter, we propose a “convenient” gauge for axial
symmetry and use it to compute an identity for Kerr black holes with scalar
hair.

A convenient gauge for virial identities in axial symmetry

João M. S. Oliveira1, Alexandre M. Pombo2,3

1Centro de Matemática, Universidade do Minho, 4710-057 Braga, Portugal

2Departamento de Matemática da Universidade de Aveiro and

Centre for Research and Development in Mathematics and Applications (CIDMA),

Campus de Santiago, 3810-183 Aveiro, Portugal

3Centro de Astrofísica e Gravitação - CENTRA,

Departamento de Física, Instituto Superior Técnico - IST, Universidade de
Lisboa - UL,

Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal

##  1 Introduction

In two previous papers, hereafter referred to as Paper I [2] and Paper II [3],
a generalized framework of virial identities in field theory was presented.
Such identities serve a variety of purposes, including establishing no-go
theorems for both solitonic and black hole (BH) solutions (aka no-hair
theorems), as well as serving as guides to finding new solutions to the
equations of motion and checking the accuracy of numerical solutions.

As a brief summary, consider the following one-dimensional (1D) effective
action (EA):

$\mathcal{S}^{\rm
eff}[q_{j}(r),q^{\prime}_{j}(r),q^{\prime\prime}_{j}(r),r]=\int_{r_{i}}^{\infty}\hat{\mathcal{L}}\left(q_{j},q^{\prime}_{j},r\right)dr\
,$ (1)

where $q_{j}$ are a set of $N$ parameterizing functions of some configuration,
$j=1\dots N$, depending on a single “radial” coordinate $r$,
$q_{j}^{\prime}\equiv dq_{j}/dr$, $r_{i}$ is some appropriately chosen
constant111With $r_{i}=0$ for solitons and $r_{i}=r_{H}$ for BHs, where
$r_{H}$ is the event horizon radial coordinate. and $\hat{\mathcal{L}}$
corresponds to the effective Lagrangian. Where $\hat{\mathcal{L}}$ should
depend on $q^{\prime\prime}_{i}$, the idea is that we can always absorb the
second derivatives into a function $f$ as

$\hat{\mathcal{L}}\left(q_{i},q^{\prime}_{i},q^{\prime\prime}_{i},r\right)=\mathcal{L}\left(q_{i},q^{\prime}_{i},r\right)+\frac{d}{dr}f\left(q_{i},q^{\prime}_{i},r\right)\
,$ (2)

with $f$ being some function that depends on the same variables as the non-
total derivative piece of the effective Lagrangian, $\mathcal{L}$. Following
Derrick’s scaling argument [1], the scaling transformation
$(\tilde{r}-r_{i})\to\lambda(r-r_{i})$ yields

$r\rightarrow\tilde{r}=r_{i}+\lambda(r-r_{i})\ ,$ (3)

which induces a variation of any fiducial configuration $q_{j}(r)$, as
$q_{j}(r)\rightarrow q_{\lambda j}(r)=q_{j}(\tilde{r})$. The EA of the scaled
configuration $q_{\lambda j}(r)$ becomes a function of $\lambda$, denoted as
$\mathcal{S}_{\lambda}^{\rm eff}$, and the true profile obeys the stationarity
condition

$\frac{\partial\mathcal{S}^{\rm
eff}_{\lambda}}{\partial\lambda}\bigg{|}_{\lambda=1}=0\ ,$ (4)

the resulting identity is known as the virial identity

$\int_{r_{i}}^{\infty}\left[\sum_{j}\frac{\partial\mathcal{L}}{\partial
q^{\prime}_{j}}q^{\prime}_{j}-\mathcal{L}-\frac{\partial\mathcal{L}}{\partial
r}(r-r_{i})\right]dr=\left[\frac{\partial f}{\partial
r}(r-r_{i})-\sum_{i}\frac{\partial f}{\partial
q^{\prime}_{i}}q^{\prime}_{i}\right]^{+\infty}_{r_{i}}\ .$ (5)

For spherically symmetric configurations, (5) can be readily applied to field
theory models yielding their virial identity. In the case of General
Relativity (GR), the fact that the Einstein-Hilbert (EH) action contains
second derivatives of the metric, implies that the total derivative term in
(2), defined by $f$, is non-zero. One of the key messages of Paper I was the
importance of considering the Gibbons-Hawking-York (GHY) [4, 5] term as part
of the gravitational action, to obtain the correct virial identity. Whereas
this is certainly the case in general, there is a special “gauge” choice (by
which we mean a choice of coordinates, plus a choice of metric functions) that
simplifies the computation of the virial identity by trivializing the full
gravitational contribution, $i.e.$ both EH action and the GHY term. In this
case, the full virial identity is computed from an EA that comes completely
from the “matter” (rather than gravitational) action. An example of such a
gauge choice is what was called the $m-\sigma$ parametrization in Paper I:

$ds^{2}=-\sigma^{2}\left(1-\frac{2m(r)}{r}\right)dt^{2}+\frac{dr^{2}}{1-\frac{2m(r)}{r}}+r^{2}\big{(}d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\big{)}\ .$ (6)

For this case, the total derivative term contribution to the virial identity
in (5) cancels out with the Gibbons-Hawking-York term contribution, and we
only need to consider the matter part of the effective action to calculate the
identity. Assuming that $\mathcal{L}$ is composed of two terms, gravitational
$\mathcal{L}_{grav}$ and matter $\mathcal{L}_{mat}$ terms, we have the
following virial identity for the $m-\sigma$ ansatz

$\int_{r_{i}}^{\infty}\bigg{[}\sum_{j}\frac{\partial\mathcal{L}_{mat}}{\partial
q^{\prime}_{j}}q^{\prime}_{j}-\mathcal{L}_{mat}-\frac{\partial\mathcal{L}_{mat}}{\partial
r}(r-r_{i})\bigg{]}dr=0\ .$ (7)

As one might expect, a “gauge” choice like this significantly simplifies
calculations, as the contribution of the gravitational part of the action
tends to be very complicated. The simplicity associated with the lack of the
gravitational part has allowed the use of these identities to establish no-
hair [6, 7, 8], no-go theorems [1] and to be widely used as a numerical
accuracy tool [9, 10, 11, 12, 13, 14]. This is especially true for higher
dimensional EAs, like stationary and axisymmetric spacetimes.

In an attempt to tackle the latter, two requirements seem to be necessary: i)
the complete procedure to compute virial identities for a generic metric
ansatz in GR and; ii) a metric parametrization that trivializes the
gravitational contribution. While the former was dealt with in Paper I and
Paper II, the latter piece is still absent.

What we propose in this paper is the existence of a "convenient" gauge choice
for stationary and axisymmetric spacetimes, which allows one to compute the
complete virial identity from an EA that solely contains the "matter" action.
This new gauge choice significantly simplifies both the computation and the
overall identities, similar to the $m-\sigma$ gauge (6). We hope this paper
allows authors of future works to streamline the process of obtaining virial
identities for more general spacetimes. A new mathematical tool that could be
useful for a myriad of purposes, from establishing theorems to numerical
accuracy.

This paper is organized as follows. First we present the convenient gauge
metric ansatz in Section 2. Then, in the subsections 2.1 and 2.2, we present
two examples of the calculation of the virial identity: the electro-vacuum and
Kerr black holes with scalar hair cases. Finally, in Section 3 we present our
conclusions and a discussion. Throughout this paper we use units with $G=1=c$
and, to simplify notation, we use $\frac{dX}{dr}\equiv X^{\prime}$ and
$\frac{dX}{d\theta}\equiv\hat{X}$, with $X$ being a generic function that is
$(r,\theta)-$dependent.

##  2 An axially symmetric convenient gauge

The gravitational action contains two terms:

$\mathcal{S}_{grav}=\mathcal{S}_{EH}+\mathcal{S}_{GHY}=\frac{1}{16\pi}\int_{\mathcal{M}}d^{4}x\sqrt{-g}R+\frac{1}{8\pi}\int_{\partial\mathcal{M}}d^{3}x\sqrt{-\gamma}(K-K_{0})\
,$ (8)

where $K=\nabla_{\mu}n^{\mu}$ is the extrinsic curvature of the boundary
$\partial\mathcal{M}$ with normal $n^{\mu}$, and $\gamma$ is the associated
3-metric of the boundary. The extra $K_{0}$ term corresponds to the extrinsic
curvature in flat spacetime (the background metric), necessary to obtain a
finite result.

The GHY boundary term is necessary for the well posedness of the gravitational
action. The resulting virial identity can, in general, be expressed as

$\int_{r_{i}}^{+\infty}dr\int_{0}^{\pi}d\theta\ I_{R}=I_{GHY}\ .$ (9)

The standard metric ansatz used to describe a rotating gravitating object
gives rise to a non-trivial contribution to the virial identities from the
gravitational part of the action222A computation of the virial identity of the
Kerr-Newman BH solution in Boyer-Lindquist and isotropic coordinates was
performed and can be seen in Appendix A.. As discussed in Paper I, for the
spherically symmetric case there is a convenient gauge choice which
trivializes the gravitational action contribution, making the virial identity
originate purely from the matter action. In this section we discuss an ansatz
for the axially symmetric case that produces the same effect: it trivializes
the contribution from the gravitational part of the action.

Consider the following metric ansatz:

$ds^{2}=-F_{0}^{2}\,dt^{2}+H(r)F_{1}^{2}\,dr^{2}+(r-r_{H})^{2}F_{1}^{2}\,d\theta^{2}+F_{2}^{2}\Big{(}d\varphi-
F_{W}\,dt\Big{)}^{2}\ ,$ (10)

where $\mathcal{F}_{i}\equiv\mathcal{F}_{i}(r,\theta)$ with $i=\\{0,1,2,W\\}$
are the four parameterizing functions and the radial coordinate
$r\in[r_{H},\,+\infty)$ while $r_{i}\equiv r_{H}$ corresponds to the radius of
the horizon – if it exist, which in these coordinates will always be a
topological sphere 333Observe that we are only interested in integrations
outside the horizon $r>r_{H}$, where the metric is regular.. Note that, for
this metric ansatz, the radial scaling argument

$r\to\tilde{r}=r_{H}+\lambda(r-r_{H})\ .$ (11)

The remarkable property of ansatz (10) is that, after scaling the metric
functions (11) and respective derivatives
$\big{(}\mathcal{F}_{i}^{\prime}(r,\theta)\rightarrow\mathcal{F}_{i\,\lambda}^{\prime}(r,\theta)/\lambda$
and $H^{\prime}(r)\to H^{\prime}_{\lambda}(r)/\lambda\big{)}$, it trivializes
the gravitational contribution to the virial identity.

To show that, observe that the Ricci scalar in ansatz (10) comes as:

$\displaystyle\sqrt{-g}R=\frac{1}{2F_{0}F_{1}^{2}H^{3/2}(r-r_{H})}\bigg{\\{}F_{1}^{2}F_{2}^{2}H\Big{(}H\hat{F}_{W}^{2}+(r-r_{H})^{2}F_{W}^{\prime
2}\Big{)}+2F_{0}F_{1}^{2}\bigg{[}-2H^{2}\partial_{\theta}(\hat{F}_{0}F_{2})$
$\displaystyle+(r-r_{H})\Big{(}-2\partial_{r}[F^{\prime}_{0}F_{2}H(r-r_{H})]+3F^{\prime}_{0}F_{2}H^{\prime}(r-r_{H})\Big{)}\bigg{]}+2F_{0}^{2}\bigg{[}2F_{2}H\big{(}H\hat{F}_{1}^{2}+(r-r_{H})F_{1}^{\prime
2}\big{)}$
$\displaystyle-2H^{2}F_{1}(\hat{\hat{F}}_{1}F_{2}+F_{1}\hat{\hat{F}}_{2})+(r-r_{H})\bigg{(}\big{[}(r-r_{H})H^{\prime}-2H\big{]}F_{1}\partial_{r}(F_{1}F_{2})-2F_{1}H(r-r_{H})(F_{2}\hat{\hat{F}}_{2}+F_{1}F^{\prime\prime}_{2})+F_{2}F_{1}^{2}H^{\prime}\bigg{)}\bigg{]}\bigg{\\}}\
.$ (12)

Note that, for each radial derivative $\mathcal{F}_{i}^{\prime}$ there is an
$(r-r_{H})$ term accompanying – with exception to the first
$\frac{1}{r-r_{H}}$. Hence after scaling, the integral term associated with
the Ricci scalar, $I_{R}$, becomes

$\sqrt{-g_{\lambda}}R_{\lambda}=\frac{1}{\lambda}\sqrt{-g}R\ ,$ (13)

and therefore, the transformations $dr\rightarrow\lambda dr$ will keep the
action invariant under scaling transformations. Thus, the EH action for this
ansatz becomes scale invariant; the EH action does not contribute to the
virial identity ($I_{R}=0$).

Let us now show that the GHY boundary term also vanishes for this ansatz. In
the metric parameterization (10), the normal vector is
$n^{\mu}\partial_{\mu}=\frac{1}{\sqrt{H}F_{1}}\partial_{r}$ and
$K_{0}=\frac{2}{(r-r_{H})F_{1}}$, the extra total derivative from the GHY
boundary term is

$\displaystyle f^{GHY}=$
$\displaystyle\frac{F_{0}F_{2}(r-r_{H})}{\sqrt{H}}\left[\frac{F_{0}^{\prime}}{F_{0}}+\frac{F^{\prime}_{1}}{F_{1}}+\frac{F^{\prime}_{2}}{F_{2}}-\frac{1}{r-r_{H}}\Big{(}2\sqrt{H}-1\Big{)}\right]\
.$ (14)

The resulting GHY contribution from $f^{GHY}$ to the virial identity is

$\displaystyle I_{GHY}=\int_{0}^{\pi}d\theta\ 0\Big{|}_{r_{H}}^{\infty}\ =0\
.$ (15)

Which is metric function independent. The GHY term is automatically null due
to the form of the metric ansatz, meaning that this result is spacetime
independent. The only remaining term of the action necessary to calculate the
virial identity is the matter term. Note that, if we want to compute the
virial identity for other ansatzë, one only needs to do a coordinate
transformation after obtaining the identity using ansatz (10) (see Appendix B
for an example).

###  2.1 Examples: Electro-vacuum

The most generic BH solution containing both a spin and an electric charge is
given by the Kerr-Newman family of solutions. In a generic coordinate system,
the action that describes such an electro-vacuum solution is

$\mathcal{S}=\mathcal{S}_{grav}+\mathcal{S}_{Maxwell}\ ,$ (16)

where $\mathcal{S}_{grav}$ is the gravitational action given by (8), and
$\mathcal{S}_{Maxwell}$ is the electromagnetic contribution:

$\displaystyle\mathcal{S}_{Maxwell}=-\int_{\mathcal{M}}d^{4}x\
\frac{F_{\mu\nu}F^{\mu\nu}}{4}\ ,$ (17)

Since we have already observed that the $\mathcal{S}_{grav}$ contribution to
the virial identity is zero. The resulting virial identity is

$\int_{r_{H}}^{+\infty}dr\int_{0}^{\pi}d\theta\ I_{M}=0\ ,$ (18)

where $I_{M}$ is the Maxwell contribution to the virial identity. The 4-vector
potential ansatz comes as

$A_{\mu}=\big{(}dt-a\sin^{2}\theta\,d\varphi\big{)}V(r,\theta)\ .$ (19)

The resulting Maxwell invariant computed with (19) and (10) comes as

$\displaystyle\sqrt{-g}F_{\mu\nu}F^{\mu\nu}=2\frac{F_{0}F_{2}\sqrt{H}}{r-r_{H}}$
$\displaystyle\times\bigg{\\{}-\frac{\hat{V}^{2}}{F_{0}^{2}}+a^{2}\sin^{2}\theta\bigg{[}\bigg{(}\frac{1}{F_{2}^{2}}-\frac{F_{W}^{2}}{F_{0}^{2}}\bigg{)}\big{(}2\cos\theta
V+\sin\theta\hat{V}\big{)}^{2}+\frac{(r-r_{H})^{2}}{H}\bigg{[}a^{2}\sin^{4}\theta\bigg{(}\frac{1}{F_{2}^{2}}-\frac{F_{W}^{2}}{F_{0}^{2}}\bigg{)}-\frac{1}{F_{0}^{2}}\bigg{]}V^{\prime
2}\bigg{\\}}\ .$ (20)

Again, note that, with exception to the first $\frac{1}{r-r_{H}}$ term, each
radial derivative of the vector function $V$ is accompanied by an $(r-r_{H})$
term. Therefore, after implementing Derrick’s scaling argument, the integral
term associated with the Maxwell contribution, $I_{M}$, becomes

$\sqrt{-g_{\lambda}}\big{(}F_{\mu\nu}F^{\mu\nu}\big{)}_{\lambda}=\frac{\sqrt{-g}}{\lambda}\big{(}F_{\mu\nu}F^{\mu\nu}\big{)}$
(21)

and therefore the radial transformation $dr\to\lambda dr$ will keep the action
invariant under the scaling transformation and the identity (18) is trivially
satisfied.

###  2.2 Examples: Kerr BHs with scalar hair

As a specific example, let us consider a non-vacuum case where the Kerr BH can
be embedded in a larger family of regular (on and outside an event horizon) BH
solutions, namely, Kerr BHs with synchronised scalar hair [15, 16]. The latter
consists of a complex scalar field $\Phi$, minimally coupled to Einstein’s
gravity and under a self-interaction potential $U(|\Phi|)$, which can be seen
as a generalization of spinning scalar boson star [17, 18, 19, 20]. Consider
the action that describes a self-gravitating complex scalar field, described
by the ansatz (10) in a model with a a self-interaction potential $U(\Phi)$

$\mathcal{S}=\mathcal{S}_{grav}-\frac{1}{4\pi}\int
d^{4}x\sqrt{-g}\bigg{[}\frac{1}{2}g^{\mu\nu}\big{(}\partial_{\mu}\Phi\partial_{\nu}\Phi^{*}+\partial_{\mu}\Phi^{*}\partial_{\nu}\Phi\big{)}+U(|\Phi|)\bigg{]}\
.$ (22)

Here ∗ denotes complex conjugate. For the scalar field, one considers an
ansatz with an harmonic azimuthal and time dependence while the scalar field
amplitude $\phi$ is $(r,\theta)$-dependent

$\Phi(t,r,\theta,\varphi)=\phi(r,\theta)e^{-i\omega t+im\varphi}\ ,$ (23)

with $\omega\in\mathbb{R}^{+}$ the frequency defining the harmonic time-
dependence and $m\in\mathbb{Z}$ the azimuthal harmonic index. The resulting
matter effective Lagrangian is

$\mathcal{L}(\phi;\phi^{\prime},\hat{\phi};r,\theta)=(r-r_{H})\sqrt{H}F_{0}F_{1}^{2}F_{2}\bigg{[}\bigg{(}\frac{m^{2}}{F_{2}^{2}}-\frac{(\omega-
mW)^{2}}{F_{0}^{2}}\bigg{)}\phi^{2}+\frac{1}{F_{1}^{2}}\bigg{(}\frac{\phi^{\prime
2}}{H}+\frac{\hat{\phi}^{2}}{(r-r_{H})^{2}}\bigg{)}+U(\phi)\bigg{]}\ .$ (24)

Which, after applying Derrick’s argument, gives the Virial identity

$\int_{r_{H}}^{+\infty}dr\
\int_{0}^{\pi}d\theta\Big{[}I_{\Phi}(\omega,m,r_{H})+I_{U}^{[\Phi]}(r_{H})\Big{]}=0\
,$ (25)

where the virial terms are

$\displaystyle I_{\Phi}(\omega,m,r_{H})$ $\displaystyle=$
$\displaystyle\frac{2(r-r_{H})F_{1}^{2}\sqrt{H}}{F_{0}F_{2}}\bigg{[}m^{2}F_{0}^{2}-F_{2}^{2}\big{(}\omega-
mF_{W}^{2}\big{)}\bigg{]}\phi^{2}\ ,$ (26) $\displaystyle
I_{U}^{[\Phi]}(r_{H})$ $\displaystyle=$ $\displaystyle
2(r-r_{H})F_{0}F_{1}^{2}F_{2}\ U(\phi)\ .$ (27)

Note that, by setting $r_{H}=0$, one obtains the virial identity for the
specific case of Boson Stars. To test these solutions we have used three Kerr
BH with scalar hair configurations, all with a simple mass potential
$U(\Phi)=\mu^{2}\Phi^{2}$; and two spinning scalar boson star configurations –
corresponding to the horizonless limit $r_{H}=0$ –, one with a simple mass
potential $U(\phi)=\mu^{2}\phi^{2}$ and one with an axionic-like potential
$U(\phi)=\frac{2\alpha^{2}}{B}\left(1-\sqrt{1-4B\sin^{2}\left(\phi/\alpha\right)}\right)$
[21, 22]. In all cases, we obtained an absolute (relative) difference for the
virial identity of $\sim 10^{-3}$ $(\sim 10^{-3})$.

####  2.2.1 Stress-energy tensor decomposition

Following the work done in Paper II, we were able to reduce the virial
identity of the previous subsection as a sum of the stress energy-tensor
components. Since the gravitational part is zero, the virial identities reduce
to

$\int
d^{3}x\sqrt{-g}\bigg{\\{}2\big{(}T_{t}^{t}-T_{r}^{r}-T_{\theta}^{\theta}\big{)}+\big{(}T_{t}^{t}-T_{\varphi}^{\varphi}\big{)}\bigg{[}1-\frac{4\omega
F_{2}^{2}(\omega-
mF_{W})}{m^{2}F_{0}^{2}+F_{2}^{2}\left(\omega^{2}-m^{2}F_{W}^{2}\right)}\bigg{]}\bigg{\\}}=0\
,$ (28)

which, unfortunately, has coefficients that are matter dependent, however it
seems to support the preposition that virial identities can be decomposed in a
sum of the stress-energy tensor terms.

Expression (28) also makes it so that the coordinate transformation of the
virial identity to another metric is very simple. An example is given in
Appendix B.

##  3 Conclusions and discussion

In this work we have introduced an ansatz parametrization for axial symmetry
that trivializes the gravitational contribution to the virial identity in GR.
In particular, we show that in vacuum, the metric ansatz (10) has a zero
contribution from the Einstein-Hilbert term and from the Gibbons-Hawking-York
boundary term. Concerning the latter, we observed that it is spacetime
independent since the GHY term is immediately zero and does not depend on the
asymptotics of the metric functions.

To summarize, the general method to calculate the virial identity for any
metric using ansatz (10) goes as follows:

  * •

Write the matter action $\mathcal{S}_{mat}$ using the convenient ansatz (10).

  * •

Apply the one-parameter radial scaling (11) while considering the one-
parameter family of functions
$\mathcal{F}_{i}^{\lambda}(r,\theta)=\mathcal{F}_{i}(\tilde{r},\theta)$ and
$H^{\lambda}(r)=H(\tilde{r})$.

  * •

Apply Hamilton’s principle by differentiating the resulting scaled action
$\mathcal{S}_{mat}^{\lambda}$ with respect to $\lambda$ and set $\lambda=1$.
The resulting virial identity is:

$\frac{\delta\mathcal{S}_{mat}^{\lambda}}{\delta\lambda}\bigg{|}_{\lambda=1}=0\
.$ (29)

  * •

To consider a different ansatz, perform a coordinate transformation of the
terms in the virial integral identity (29). See Appendix B as an example.

With the convenient metric ansatz, it was possible to compute the virial
identity for a Kerr BH with scalar hair model. The latter becomes much simpler
than the one obtained in Paper II due to the choice of metric parametrization.
Concerning the latter example, a decomposition of the identity in a sum of the
stress-energy tensor terms was also possible, supporting the proposition
established in Paper II. However, this time, some of the parameters that
multiply the stress-energy tensor terms are matter term-dependent, something
not observed previously.

At last, while we did not establish any new no-hair or no-go theorem with the
aid of the new metric parametrization in this paper, the fact that the latter
significantly simplifies the computation of the virial identities can seed new
theorems in axial symmetry – just like it was done in spherical symmetry
through the use of the $m-\sigma$ parametrization.

## Acknowledgements

We would like to thank Carlos A. R. Herdeiro and Eugen Radu for the many
valuable discussions. This work is supported by the Center for Research and
Development in Mathematics and Applications (CIDMA) through the Portuguese
Foundation for Science and Technology (FCT - Fundação para a Ciência e a
Tecnologia), references UIDB/04106/2020, UIDP/04106/2020. We acknowledge
support from the projects PTDC/FIS-OUT/28407/2017, CERN/FISPAR/0027/2019,
PTDC/FIS-AST/3041/2020 and CERN/FIS-PAR/0024/2021. This work has further been
supported by the European Union’s Horizon 2020 research and innovation (RISE)
programme H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740.

##  Appendix A Virial identities in axial symmetry

The most generic BH solution containing both a spin and an electric charge is
given by the Kerr-Newman family of solutions. In a generic coordinate system,
the action that describes such an electro-vacuum solution is

$\mathcal{S}=\mathcal{S}_{grav}+\mathcal{S}_{Maxwell}\ ,$ (30)

where $\mathcal{S}_{grav}$ is the gravitational action given by (8), and
$\mathcal{S}_{Maxwell}$ is the electromagnetic contribution given by (17). For
a generic metric ansatz, the resulting virial identity is

$\int_{r_{H}}^{+\infty}dr\int_{0}^{\pi}d\theta\Big{[}I_{R}+I_{M}\Big{]}=I_{GHY}\
.$ (31)

###  A.1 Boyer-Lindquist coordinates

The Kerr-Newman metric can be expressed in the familiar Boyer-Lindquist (BL)
coordinates. To compute the virial identity of a charged, spinning BH in BL
coordinates, we will follow [23]

$ds^{2}=-f_{0}dt^{2}+f_{1}dr^{2}+r^{2}\big{(}f_{3}d\theta^{2}+f_{4}d\varphi^{2}\big{)}-2rf_{2}\,dtd\varphi\
,$ (32)

where the $f_{i}$ functions, with $i\in\\{0,1,2,3,4\\}$, are all
$(r,\theta)$-dependent.444Observe that the present metric is almost completely
generic with the exception of the explicit radial dependence in the angular
components. The latter seems to be a requirement for the finiteness of the
virial identity contribution from the Ricci/GHY component. The $4$-vector
potential, $A_{\mu}$, ansatz

$A_{\mu}=V(dt-a\sin^{2}\theta\,d\varphi)\ ,\qquad\qquad{\rm with}\qquad
V(r,\theta)=\frac{r\,Q}{\Sigma}\ ,$ (33)

with $Q$ the electrical charge. The metric functions $f_{i}$ in BL coordinates
are given as

$\displaystyle f_{0}$
$\displaystyle=\frac{\Delta-a^{2}\sin^{2}\theta}{\Sigma}\ ,\qquad\qquad
f_{1}=\frac{\Sigma}{\Delta}\ ,\qquad\qquad
f_{2}=\frac{a}{r\Sigma}\Big{[}\big{(}r^{2}+a^{2}\big{)}-\Delta\Big{]}\sin^{2}\theta\
,$ $\displaystyle f_{3}$ $\displaystyle=\frac{\Sigma}{r^{2}}\ ,\qquad\qquad
f_{4}=\frac{\sin^{2}\theta}{r^{2}\Sigma}\Big{[}\big{(}a^{2}+r^{2}\big{)}^{2}-a^{2}\Delta\sin^{2}\theta\Big{]}\
,$ (34)

with $\Delta=r^{2}-2Mr+a^{2}+Q^{2}$ and $\Sigma=r^{2}+a^{2}\cos^{2}\theta$.

While the computation of the virial identity was done with the unknown metric
functions $f_{i}$ the resulting expressions for the gravitating virial
contribution are too complex to write. The resulting Einstein-Hilbert
($I_{R}$) and Maxwell ($I_{M}$) contributions in BL coordinates come as

$\displaystyle I_{R}$ $\displaystyle=\frac{r_{H}\sin\theta}{r^{3}\big{(}\cos
2\theta+1+2\frac{r^{2}}{a^{2}}\big{)}^{2}}\bigg{[}12\big{(}a^{2}+Q^{2}\big{)}-13Mr+39r^{2}+\Big{(}4(a^{2}+Q^{2})-3Mr+r^{2}\Big{)}\cos
4\theta$
$\displaystyle+\frac{28r^{2}}{a^{2}}\Big{(}\frac{4Q^{2}r^{2}}{7a^{2}}-Mr+Q^{2}+r^{2}\Big{)}+16\Big{(}a^{2}-Mr+Q^{2}+\frac{5}{2}r^{2}+\frac{r^{2}}{4a^{2}}\big{(}-9Mr+9Q^{2}+5r^{2}\big{)}\Big{)}\cos
2\theta\bigg{]}\ ,$ (35) $\displaystyle I_{M}$
$\displaystyle=\frac{Q^{2}}{4r\Delta\Sigma^{6}\sqrt{\Omega}}\Bigg{\\{}\Omega\sin^{2}\theta\Big{[}\Delta\big{(}r^{2}-a^{2}\cos^{2}\theta\big{)}\Omega\Big{(}a^{2}(3r-2r_{H})\cos^{2}\theta-3r^{2}(r-2r_{H})\Big{)}-a^{2}r\Big{(}4a^{2}r\cos^{2}\theta(3r-2r_{H})\Sigma^{2}$
$\displaystyle+\Delta\sin^{2}\theta\big{(}r^{2}-a^{2}\cos^{2}\theta\big{)}\big{(}\Delta-a^{2}\sin^{2}\theta\big{)}\big{(}a^{2}\cos^{2}\theta-r(r-4r_{H})\big{)}\Big{)}\Big{]}+4a^{2}r^{3}(a^{2}+r^{2})^{2}\Sigma^{2}\big{(}\Delta-a^{2}\sin^{2}\theta\big{)}\cos^{2}\theta\Bigg{\\}}\
,$ (36)

where we reintroduced the metric functions $\Sigma$ and $\Delta$ and added the
function $\Omega=\left(a^{2}+r^{2}\right)^{2}-a^{2}\Delta\sin^{2}\theta$ to
simplify the $I_{M}$ term. The GHY term, computed using the normal vector
$n^{\mu}\partial_{\mu}=\frac{1}{\sqrt{f_{1}}}\partial_{r}$ and
$K_{0}=\frac{2}{r\sqrt{f_{3}}}$, associated with these coordinates is

$I_{GHY}=4\frac{a^{2}+Q^{2}}{r_{H}}\ .$ (37)

###  A.2 Isotropic coordinates

Consider the Kerr-Newman solution described by an axially symmetric metric
gauge in isotropic coordinates

$ds^{2}=-\frac{f_{1}}{S}H^{2}dt^{2}+f_{1}\big{(}dr^{2}+r^{2}d\theta^{2}\big{)}+\frac{r^{2}}{f_{1}}\sin^{2}\theta\Big{(}Wdt-
Sd\varphi\Big{)}^{2}\ ,$ (38)

where the metric functions $(f_{0},f_{1},W,S)$ are $(r,\theta)$-dependent and
$H\equiv H(r)$. The $4$-vector potential comes as

$A_{\mu}=\frac{V}{f_{1}}\Big{[}dt-a\sin^{2}\theta\,d\varphi\Big{]}\ ,$ (39)

while for the metric and Maxwell functions we have

$\displaystyle S$
$\displaystyle=\frac{1}{r^{4}}\Big{[}\Big{(}a^{2}+\big{(}-Hr+M+2r\big{)}^{2}\Big{)}^{2}-a^{2}H^{2}r^{2}\sin^{2}\theta\Big{]}\
,\qquad\qquad
f_{1}=\frac{1}{r^{2}}\Big{[}a^{2}\cos^{2}\theta+\Big{(}M-\big{(}H-2\big{)}r\Big{)}^{2}\Big{]}\
,$ (40) $\displaystyle W$
$\displaystyle=\frac{a}{r^{4}}\Big{[}a^{2}+(M+2r)\Big{(}M-2(H-1)r\Big{)}\Big{]}\
,\qquad\qquad H=1+\frac{a^{2}-M^{2}+Q^{2}}{4r^{2}}\ ,\qquad\qquad
V=\frac{Q}{r^{2}}\Big{[}M-(H-2)r\Big{]}\ .$ (41)

with

$\displaystyle I_{R}=$
$\displaystyle\frac{\sin\theta}{2Hf_{1}^{2}S^{2}}\Bigg{\\{}-2S\Bigg{[}-H^{2}f_{1}\Big{(}r(r-2r_{H})f_{1}^{\prime}S^{\prime}+\hat{f}_{1}\hat{S}\Big{)}+Hf_{1}^{2}\Big{(}H\big{(}(r-r_{H})S^{\prime}+\cot\theta\hat{S}\big{)}-r(r-2r_{H})H^{\prime}S^{\prime}\Big{)}$
$\displaystyle+r^{3}WW^{\prime}S^{\prime}(3r-4r_{H})\sin^{2}\theta\Bigg{]}+S^{2}\Bigg{[}H\Bigg{(}2f_{1}\Big{(}H\big{(}-\hat{\hat{f}}_{1}+\hat{f}_{1}\cot\theta-r(r-2r_{H})f_{1}^{\prime\prime}\big{)}-r(r-2r_{H})H^{\prime}f_{1}\Big{)}$
$\displaystyle+H\Big{(}\hat{f}_{1}^{2}+r(r-2r_{H})f_{1}^{\prime
2}\Big{)}+f_{1}^{2}\Big{(}-4r(r-2r_{H})H^{\prime\prime}-8(r-r_{H})H^{\prime}\Big{)}\Bigg{)}+r^{3}\sin^{2}\theta(3r-4r_{H})W^{\prime
2}\Bigg{]}$ $\displaystyle-H^{2}f_{1}^{2}\Big{(}r(r-2r_{H})S^{\prime
2}+\hat{S}^{2}\Big{)}+rW^{2}\sin^{2}\theta\Big{(}r^{2}(3r-4r_{H})S^{\prime
2}+(3r-2r_{H})\hat{S}^{2}\Big{)}\Bigg{\\}}\ ,$ (42) $\displaystyle I_{M}=$
$\displaystyle-\frac{2\sin\theta}{r^{3}Hf_{1}^{5}S}\Bigg{\\{}-2r^{3}Vf_{1}\Bigg{[}r(r-2r_{H})V^{\prime}f_{1}^{\prime}\Big{(}S^{2}+a^{2}W^{2}\sin^{4}\theta\Big{)}+2a^{2}VW^{2}\sin^{3}\theta\cos\theta\hat{f}_{1}\Bigg{]}$
$\displaystyle+f_{1}^{2}\Bigg{[}a^{2}V^{2}\sin^{2}\theta\Big{(}H^{2}\big{(}r^{3}f_{1}^{\prime
2}+(r-2r_{H})\hat{f}_{1}^{2}\big{)}+4r^{3}W^{2}\cos^{2}\theta\Big{)}+r^{4}(r-2r_{H})V^{\prime
2}\Big{(}S^{2}+a^{2}W^{2}\sin^{4}\theta\Big{)}\Bigg{]}$
$\displaystyle-2a^{2}VH^{2}f_{1}^{3}\Big{(}r^{3}\sin\theta
V^{\prime}f_{1}^{\prime}+2V\cos\theta(r-2r_{H})\hat{f}_{1}\Big{)}\sin\theta+r^{3}V^{2}\Big{(}\hat{f}_{1}^{2}+r(r-2r_{H})f_{1}^{2}\Big{)}\big{(}S^{2}+a^{2}W^{2}\sin^{4}\theta\big{)}$
$\displaystyle+a^{2}H^{2}f_{1}^{4}\Big{(}r^{3}\sin^{2}\theta V^{\prime
2}+4V^{2}\cos^{2}\theta(r-2r_{H})\Big{)}\Bigg{\\}}\ ,$ (43) $\displaystyle
I_{GHY}$ $\displaystyle=4(M-2r_{H})\ .$ (44)

Where the GHY term was computed using the normal vector
$n^{\mu}\partial_{\mu}=\frac{1}{\sqrt{f_{1}}}\partial_{r}$ and
$K_{0}=\frac{2}{\sqrt{f_{1}}r}$.

For the Kerr-Newman BH in both BL and isotropic coordinates, we were unable to
obtain a close form for the Einstein-Hilbert contribution. However, due to the
relation (31) – that has been numerically confirmed with arbitrary precision
–, we know that the integral must give $I_{GHY}$.

##  Appendix B Numerical metric transformation

Let us consider the metric that we presented in Paper II, which is a widely
used metric ansatz for numerical computation

$ds^{2}=-e^{2G_{0}}H(\bar{r})dt^{2}+e^{2G_{1}}\left[\frac{d\bar{r}^{2}}{H(\bar{r})}+\bar{r}^{2}d\theta^{2}\right]+e^{2G_{2}}\bar{r}^{2}\sin^{2}\theta\Big{(}d\varphi-
G_{W}dt\Big{)}^{2}\ ,\quad\;\;\;H(\bar{r})\equiv 1-\frac{r_{H}}{\bar{r}}\ ,$
(45)

where the parameterizing functions $\mathcal{G}_{i}$ with $i={0,1,2,W}$ are
all $(\bar{r},\theta)-$dependent. The transformation between the convenient
metric gauge (10) and (45) comes as

$\displaystyle F_{2}=e^{G_{2}}\bar{r}\sin\theta\ ,\qquad
F_{1}=\frac{e^{G_{1}}}{H}\ ,\qquad F_{W}=G_{W}\ ,\qquad F_{0}=e^{G_{0}}\
,\qquad\bar{r}=r\ ,$ (46)

With the new radial coordinate $r\in[r_{H},\,+\infty)$. With this coordinate
transformation, the virial identity for Kerr black holes with scalar hair is
easily obtained from (28) as

$\int
d^{3}x\sqrt{-g}\bigg{\\{}2\big{(}T_{t}^{t}-T_{r}^{r}-T_{\theta}^{\theta}\big{)}+\big{(}T_{t}^{t}-T_{\varphi}^{\varphi}\big{)}\bigg{[}1-\frac{4\omega
e^{2G_{2}}\bar{r}^{2}\sin^{2}\theta(\omega-
mG_{W})}{m^{2}e^{2G_{0}}+e^{2G_{2}}\bar{r}^{2}\sin^{2}\theta\left(\omega^{2}-m^{2}G_{W}^{2}\right)}\bigg{]}\bigg{\\}}=0\
.$ (47)

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