missing

# Finding new routes for integrating
Multi-Agent Systems using Apache Camel††thanks: Supported by Petrobras project
AG-BR, IFSC and UFSC.

Cleber J. Amaral1,2, Sérgio P. Bernardes1, Mateus Conceição1
Jomi F. Hübner1, Luis P. A. Lampert1, Otávio A. Matoso1, Maicon R. Zatelli1

###### Abstract

In Multi-Agent Systems (MAS) there are two main models of interaction: among
agents, and between agents and the environment. Although there are studies
considering these models, there is no practical tool to afford the interaction
with external entities with both models. This paper presents a proposal for
such a tool based on the Apache Camel framework by designing two new
components, namely _camel-jason_ and _camel-artifact_. By means of these
components, an external entity is modelled according to its nature, i.e.,
whether it is autonomous or non-autonomous, interacting with the MAS
respectively as an agent or an artifact. It models coherently external
entities whereas Camel provides interoperability with several communication
protocols.

##  1 Introduction

MAS literature has plenty of research about agents’ interactions, i.e., agents
sending and receiving messages to and from other agents (A-A). Many approaches
model almost any entity as an agent and thus the interaction remains something
among agents. However there are new approaches that questioned the
_agentification_ method proposing an MAS where non-autonomous entities are
conceived as artifacts in the environment. In these approaches, the
development of an MAS considers the design of both agents and artifacts. The
environment is not simply what is outside the system (the exogenous
environment), but it is designed accordingly to the system purpose (the
endogenous environment) [Ricci et al. 2006, Omicini et al. 2008]. In this
sense, we have two models of interactions: agent-to-agent (A-A) and agent-to-
environment (A-E). In the former, an agent _communicates_ with another agent
using an Agent Communication Language (ACL) and in the latter, an agent
_perceives and acts_ upon artifacts in the environment.

When we consider the integration with other applications, those two models are
adopted by the current development platforms. On the one hand, we have
approaches that use ACL for that purpose and other applications are seen by
the agents as other agents (having a mental state, implied by the ACL
semantics). On the other hand, we have approaches where other applications are
seen as part of the environment and agents perceive and act on them. Some
platforms provide an A-A approach while others an A-E approach, but, as far as
we know, no platform provides both. The designer is forced to conceive some
other application either as an agent or as an artifact, despite the
application properties.

In this paper, we propose to apply the same argument as [Omicini et al. 2008]
for the integration of MAS and external applications: some external
applications are autonomous and should be modelled as agents while others are
non-autonomous and should be modelled as artifacts.

For instance, in the Industry 4.0 context, it is expected the interaction of
many entities such as an autonomous planner sending commands to a non-
autonomous machine, which signalises what was done. Later, the planner must
choose a supplier after an auction to hire a freight to take the product to
the destiny, which is usually an human. This short example gives an idea of
how comprehensive and challenging the integration can be. We can notice that
both models of integration are required: the _autonomous_ planner and the
human are better modelled as agents, performing A-A interactions, and the
_non-autonomous_ machine should be integrated as an artifact which when
communicating with and agent performs and A-E interaction. Following this
concept, we have developed two components for JaCaMo platform, for integration
among agents, and between agents and the environment. The referred components
are used to set communication routes for the MAS and external entities, using
the framework Apache Camel [Ibsen and Anstey 2010], a mediation tool to
provide interoperability with many technologies.

##  2 MAS integration approaches

MAS are being applied as a core technology for distributed systems that needs
cooperation and negotiation [Roloff et al. 2016]. The integration of MAS and
external entities, i.e. any entity which was not defined its totality within
the MAS itself, regards concerns such as compatibility with standards,
interoperability and portability. We have found two main forms of integration:
(i) among agents (A-A); and, (ii) between agents and the environment (A-E).

###  2.1 Integration among agents (A-A)

The communication among agents is usually done by speech-acts which considers
utterances as actions, usually intending to change the mental state of
recipients. The utterance can inform beliefs, desires and intentions of
rational agents that attempt to influence other agents. The Knowledge Query
and Manipulation Language (KQML) is the first speech-act based language
providing high level communication in the distributed artificial intelligence
applications [Vieira et al. 2007]. In fact, once speech acts became widely
accepted in MAS community, the integration among different agent’s platforms
was facilitated.

Currently FIPA-ACL, which is very similar to KQML, is the main standard for
agents communication. FIPA-ACL uses performatives to make explicit an agent’s
intention for each sent message, for instance, _inform_ is used to influence
the recipient to believe in something, and _request_ to influence the
recipient to add something as a goal [Vieira et al. 2007].

Another communication aspect is related to the expected sequence of messages.
Conversations among agents usually follow some patterns which are often
referred to as interaction protocols. Typical patterns such as negotiation,
auction, and task delegation are defined using FIPA standards [Bellifemine et
al. 2005]. In addiction, there are communication infrastructures that allow
agents to be distributed over a network. The challenge in A-A is the
integration between an agent, for instance, using FIPA-ACL, and another agent
using another language, for instance, an human. This situation leads to the
necessity of some tool to make both end-points compatible.

###  2.2 Integration between agents and the environment (A-E)

There are systems or parts of a system which are better seen as resources or
tools that can be used by agents to achieve their goals. These entities,
called artifacts, have no internal goals, they are not autonomous and neither
proactive, but they supply useful functionalities for agents. An analogy for
agents and artifacts is the interaction between humans, as autonomous
entities, and tools they exploit in their activities [Ricci et al. 2006]. For
instance, a blackboard shared by agents would be modelled as an artifact,
being predictable and deterministic, if not, it would perform undesirable
autonomous behaviour.

Artifacts are placed in workspaces which represent areas of the MAS
environment. Agents can perceive changes and act within the workspaces they
are occupying, and on artifacts they are watching, i.e., being aware of events
and signals due to interactions with non-autonomous entities held inside of
virtual boundaries. The environment can reflect the effects of agents’ actions
and other phenomena. It is being treated as a first-class programming
abstraction with similar importance of agents programming abstraction [Ricci
et al. 2006].

Besides modelling non-autonomous entities, the artifacts can also be used for
other purposes: (i) for agents coordination using shared artifacts such as
organisational boards or coordination marks; (ii) for indirect communication
among agents, for instance, by blackboard artifacts; (iii) for implementing
the user interface of a system; (iv) for controlling transactions over
environment elements through distribution and synchronisation facilities; and
(v) for integration between the MAS and external entities [Boissier et al.
2019].

Regarding the use of artifacts to integrate external entities, the integration
is done usually through specific Application Programming Interfaces (APIs).
The main concern with this approach regards to the high programming effort
when there are different protocols in scenarios of heterogeneous devices.

##  3 Integrating A-A and A-E using Camel

Back to our example in the Industry 4.0 context, Figure 1 shows a process that
begins with a packed product, in a production line, up to its delivery to the
customer. On the first step, there is an industrial device, a non-autonomous
entity, that communicates using an industrial protocol. Once the device
signalises the end of the production, the order is checked out on the
Enterprise Resource Planning (ERP) software, another non-autonomous entity.
The supplier should choose the best offer for a freight, which may be done by
accessing suppliers’ systems to then interact with the winner, an autonomous
entity. Later, the delivery should be tracked by a monitoring system, which is
non-autonomous. Finally, when the product is near the destination, a message
must keep the client, an autonomous entity, informed.

Figure 1: Finishing production and delivering the product in Industry 4.0
context

This scenario illustrates the requirements for the integration: heterogeneous
endpoints with different kinds of interactions due to the autonomous and non-
autonomous nature of entities. We propose the use of the framework Apache
Camel for both integration models, A-A and A-E. We have thus two components:
_camel-jason_ for integrating MAS’s internal agents with external entities
modeled as agents, and _camel-artifact_ for integrating the former agents with
external entities modeled as environmental artifacts.

###  3.1 Apache Camel

Apache Camel is a lightweight Java-based framework message routing and
mediation engine [Ibsen and Anstey 2010]. Camel achieves high-performance
processes handling multiple messages concurrently, and provides functions such
as routing, exception handling, and testing. It uses structured messages and
queues based on Enterprise Integration Patterns (EIP) [Hohpe and Woolf 2003],
preserving loose coupling among the resources. Camel works as a middleware
that can be incorporated into an application through the use of _components_.
Communication among Camel components is defined in so-called _routes_ , which
set and manage how messages will be exchanged, possibly following sets of
rules and using data manipulation.

Figure 2: MAS architecture using _camel-jason_ and _camel-artifact_
components.

Routes define a single endpoint for each entity with an unique address. The
defined endpoint may receive data, through a _producer_ and send data through
a _consumer_. Consumers are entities that admit data in specific formats and
encapsulates it in a camel exchange object, an item that any other producer
understands and is able to decode. Producers are entities that receive
encapsulated data from the consumer decoding it in its entity’s message
structure.

In our implemented components, Camel is being embedded in two slightly
different manners regarding the models of integration, A-A and A-E, as shown
in Figure 2. In the case of A-A, it works as a communication infrastructure
that is used when the recipient is not found locally. In the case of A-E, the
external device is usually modelled reflecting real operations and signals
that it generates, typically having their individuals routes. In both cases,
the components are able to define tuned integration, covering a range of end-
points features. Notice that, the complexity of each supported protocol is
processed in a Camel component, which works as a bridge to Camel routes. There
are more than two hundred components available on Camel’s website111Supported
Camel components are listed in http://camel.apache.org/components.html and
many others on the community’s repositories.

###  3.2 _camel-jason_ component

The _camel-jason_ component enables agents to communicate with external
entities through ACL, whilst fulfilling the need of understanding those
entities as agents when modeling the MAS. In our proposal, the external entity
has a kind of virtual counterpart inside the MAS, a _dummy agent_. This
counterpart is seen by the agents as an ordinary agent of the system. Doing
so, agents can directly communicate with external entities assuming that they
are other agents (as receivers and senders of ACL messages).

_Camel-jason_ component provides a communication flow that is illustrated in
Figure 3 where an agent interact with a service A (an external entity). Since
the agent sees the service as another agent, it uses ACL for the
communication. When the service wants to contact the agent, the camel-jason
component translates the message into ACL and the agent receives it as if it
comes from an agent.

Figure 3: Communication flow using _camel-jason_ component.

On the top communication flow showed in Figure 3, we have an agent sending a
message to an external autonomous entity: (i) the agent send an ACL message
addressed to a dummy agent, which is created by _camel-jason_ component,
referring to _Service A_ ; (ii) the message is consumed by _camel-jason_
component consumer; (iii) the message is exchanged to the other side of the
route, possibly being transformed; and (iv) the message is processed by
_Service A_ component producer which prepares a service A compliance message,
which will be sent to some network address to be effectively consumed by the
_Service A_.

In the other way around, on the bottom of Figure 3, we have: (i) _Service B_
sends some data through the network reaching _Service B_ component consumer by
its network address; (ii) the message is exchanged through Camel route,
possibly being transformed; (iii) the message is processed by _camel-jason_
component producer which generates an ACL message; and (iv) the receiver agent
effectively consumes the ACL message.

The component uses a simplistic method to define the communication routes, in
which for many cases no actual programming is required, only XML definitions.
The user should know how to fill camel endpoint parameters according to the
compatible endpoint of the application. In cases data transformation is
required, camel brings some tools for simple transformation as well as complex
ones, using embedded programming codes if needed.

###  3.3 _camel-artifact_ component

In order to sustain the A-E model, the CArtAgO infrastructure is used, and the
_camel-artifact_ component was developed. This component allows agents to
perceive and act upon artifacts that represent external entities inside the
MAS.

Notwithstanding, _camel-artifact_ also allows the definition of communication
routes between CArtAgO artifacts and external entities. Routes for the _camel-
artifact_ component are implemented using the Java language. The user should
be aware of regular camel routes and how to define endpoints and their
respective parameters.

Figure 4: Communication flow using _camel-artifact_ component.

On the top communication flow showed in Figure 4, we have an artifact
integrated with some service A, an external non-autonomous entity. The
interaction between them is as follows: (i) the artifact sends a message to to
_Service A_ , through specific methods provided by _camel-artifact_ ; (ii) the
message, in form of an operation request, is consumed by _camel-artifact_
component consumer which generates a camel standardised message to be sent to
the external entity; (iii) the message is exchanged to the other side of the
route, possibly being transformed; and (iv) the message is processed by
_Service A_ compatible component producer which prepares a compliance final
message, with the proper format and structure, which will be sent to some
network address to be effectively consumed by the _Service A_.

In the other way around, on the bottom of Figure 4, we have: (i) _Service B_
sends some data through the network reaching _Service B_ component consumer by
its network address; (ii) the message is exchanged through Camel route,
possibly being transformed; (iii) the message is processed by _camel-artifact_
producer which generates an artifact operation request; and (iv) the recipient
artifact effectively consumes the operation request executing the referred
method.

##  4 Illustrative application

For a better understanding of how the _camel-jason_ and _camel-artifact_
components can be used, we will resume the example of Industry 4.0, presented
in Figure 1, and will build an implementation of this system.

The Figure 5 shows the MAS fully designed, with agents, external entities and
the camel components used to implement the integration. These components are
represented in the middle layer as artifacts and dummy agents. This
hypothetical scenario implements an MAS to integrate the production and
distribution stages of a product. The whole course can be divided in five
stages: (i) a Programmable Logic Controller (PLC) finishes the product
manufacturing, (ii) the information about the product is uploaded to an
Enterprise Resource Planning (ERP) software, (iii) a research starts in order
to contract the best freight company, (iv) the hired company starts
transporting the product, providing its tracking information, and (v) warns
the client via chat when it is near the final destination. The MAS is designed
to unify those stages and to be responsible for managing each process.
Moreover, Camel components are used as middleware between the MAS and external
entities to integrate them.

Figure 5: An industrial process illustrating the integration of an MAS with a
manufacturing device, updating an enterprise management software, choosing a
supplier for delivering a tracked product to the customer.

One common question when designing the MAS is how many agents should be used.
This is not mandatory, but a natural thought is to divide the process into
sections and designate a single agent to be responsible for each part. In this
case, we will consider that PLC and ERP stages represent the production part
of the process, so one agent, named _production_agent_ , will be responsible
for managing these processes. Next, there is the hiring stage, that
comprehends searching and hiring the best delivery company, for which the
_distribution_agent_ will be designated. The final stage could be thought as
the delivery process, where the last agent, named _delivery_agent_ , will be
responsible for consulting the tracking information and sending the message to
the costumer.

Another part of the designing process is the identification of which model,
agent or artifact, is more suitable to represent each external entity. A
common way to decide is observing its nature, i.e. autonomous or non-
autonomous. Following this idea, it could be decided that the PLC and ERP
software would be modeled as artifacts, since they are non-autonomous
entities; and the customer as an agent, an autonomous entity. Another
possibility to decide is looking at which type of communication the agents
will perform with each external entity, i.e. via message exchanging or
perception-action. For example, the action of hiring the delivery company,
informing the company about some new contract via email, seems natural to be
modeled as an A-A interaction. For this situation, the best suitable
performative for the message is tell, since the agent is telling the supplier
about a new hired delivery. On the other hand, when the _distribution_agent_
is searching for the best delivery company, consulting their prices and
conditions, it seems more suitable for this information to be perceived, like
people do in a websearch. The same idea can be used when the _delivery_agent_
tracks the position of the product, the information again is perceived by the
agent, like looking at a screen. If message exchange was used in this case the
agent would be flooded with unnecessary messages.

In fact, both interpretations, i.e., perception-action vs message exchanging
and autonomous vs non-autonomous, could point to the same conclusion. This
statement can be tested in the integration between the _delivery_agent_ and
the customer. The chat is done by message exchanging and it is performed by
autonomous entities. Therefore, both interpretation reinforces the idea that
the costumer should be modeled as an agent.

With the external entities modeled as MAS elements we could use _camel-
artifact_ and _camel-jason_ to integrate the internal agents with the
artifacts and agents, respectively. It is worth commenting that the external
entities could be easily exchanged, for instance making the auction via email
instead of using a web service. The interaction via email suggests modelling
the participants as agents and use the _camel-jason_ component for the
integration.

Now the camel routes can be developed, depending on which type of technology
the external entities use. As explained before, Camel have more than two
hundred endpoints available. In this example, OPC-DA, Rest, email, MQTT and
Telegram end-points are being used to create the routes.

The code, in XML, for the route from the _delivery_agent_ to Telegram can be
seen in Listing 1. The from tag signalises the consumer part of the route, and
to signalises the producer. In this case, the consumer address is the name of
the dummy agent to which the message had been sent (in this example,
customer), and the producer address is the authorisation token followed by the
chatId option.

⬇

1<route>

2 <from uri="jason:DummyCustomerAgent"/>

3 <to uri="telegram:bots/sometoken?chatId=-364531"/>

4</route>

Listing 1: Example of _camel-jason_ route definition

In this example, the _delivery_agent_ also uses the _camel-artifact_ in order
to obtain the delivery’s position from a MQTT server. The route, in Java, is
shown in Listing 2. When the position is published on the topic of interest
(latLong) the route redirects it to the artifact by specifying its name
(TrackedArtifact) and the operation (giveDistance) as headers. Here, we are
assuming that the calculations will be done by the artifact, but they could be
done in the route through a transformation, before sending to the artifact.

⬇

1from( "mqtt : foo? host=tcp://broker & subscribeTopicName=latLong" )

2 .setHeader( "ArtifactName" , constant ( "TrackedArtifact" ) )

3 .setHeader( "OperationName" , constant ( "giveDistance" ) )

4.to( "artifact : cartago" );

Listing 2: Example of _camel-artifact_ route definition

##  5 Related research

In this section, we went over works that have addressed agent technology in an
integrating context. Maturana and Norrie [Maturana and Norrie 1996] have
proposed a mediation and coordination tool for MAS. They have used mediator
agents as manufacturing coordinators. Following similar idea, Olaru et al.
[Olaru et al. 2013] have developed an agent-based middleware, which creates a
sub-layer of application layer that allows agents to mediate context-aware
exchange of information among entities. We think an autonomous entity as
middleware may increase complexity and compromise performance. Instead of
creating some kind of hierarchy, our approach gives connectivity power to MAS
entities.

Leading industrial suppliers are also providing solutions using agents such as
the Agent Development Environment (ADE), designed by Rockwell Automation
[Tichý et al. 2012]. It provides connectivity with common shop floor devices
and supports the development of agents. The limitation we have seen regards
especially connectivity with all sorts of entities (e.g. IoT sensors and
mobile devices, ERP and other software etc), which in our case is provided by
Camel.

Other research address the combination of MAS and Service-Oriented
Architecture (SOA). One way to achieve this merge is based on the creation of
a proxy function to provide interoperability between MAS and SOA, as found in
[Nguyen and Kowalczyk 2005, Shafiq et al. 2005, Greenwood and Calisti 2004,
Fayçal et al. 2010]. Another way is by implementing services as agents as we
found in [Mendes et al. 2009, Tapia et al. 2009, Carrascosa et al. 2009,
Argente et al. 2011]. The approaches using SOA are more mature to be applied
in practice. The ones that _agentified_ the services have also the advantage
to use MAS background, i.e., using ACL messages they are able to use
interaction protocols. In these studies integration is usually done through
specific APIs and they lack differentiation over autonomous and non-
autonoumous entities, and interoperability with heterogeneous entities, both
aspects increase development complexity.

Vrba et al. [Vrba et al. 2014] propose a gateway for wrapping an MAS as a
service to be used as a loosely coupled software component into the Enterprise
Service Bus (ESB). This gateway transforms agent messages to ESB messages and
vice versa, enabling communication between agents and ESB services. This
solution is closely to ours, only lacking support to the A-E approach, where
interaction is based on agents perceiving and acting upon artifacts in the
environment. The indiscriminately _agentification_ may increase complexity and
affect performance.

Cranefield and Ranathunga [Cranefield and Ranathunga 2013] developed a _camel-
agent_ component for _Jason_ agents. It is very similar to our developed
_camel-jason_ component. Essentially, the difference is that we are embedding
Apache Camel since our component works as an infrastructure, being transparent
to the agents. In their work, Jason was actually embedded in an Apache Camel
project where agents were smoothly placed in containers.

##  6 Conclusion

In this paper, we introduced two Camel components aiming the integration of
MAS with external entities: _camel-jason_ and _camel-artifact_. The former
integrates agents with external entities modelled as agents. The latter
integrates agents and external entities modelled as artifacts. The decision of
which component to adopt for each entity depends on the characteristics of the
external entity and the MAS developer _can_ choose the most suitable
component. For instance, he/she is not obliged to “agentify” every external
entity, even those that do not have agent properties.

The two components introduced in this paper, along with the communication
infrastructure provided by Camel and its existing components, makes the
integration between MAS and different entities simpler. Issues related to
interoperability, routing, and data transformations are partially solved in
the camel routes. Another advantage of using such components is that the agent
program does not need to deal with integration issues. Agents continue to
interact only with another agents and artifacts.

Finally, this is an ongoing work. In a future step we intend to compare our
approach with related works and to evaluate other aspects of using the
developed Camel components to integrate MAS and external entities, such as the
impact on the performance, security, openness, scalability, among others.

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However, as the late merger and ringdown stage is coming, this method’s
inconsistence manifests. Most manifestly, the relative inspiral motion’s
frequency following from $\omega{\equiv}\dot{\phi}$ and
$\omega{\sim}p_{\phi}/\mu{a^{2}}$ exhibits opposite trends of variation as the
time passes by, see FIG.16 for illustrations. This means that this method
cannot be used to tell us if the two Schwarzschild black holes involved in the
binary merger process own inner structure and experience shape deformation or
not.

Figure 16: The relative motion orbit of general relativity two particle system
and the corresponding gravitational wave form following from the conventional
EOB method. This method yields manifestly opposite variation trends for the
system’s inspiral frequency defined through $\omega{\equiv}\dot{\phi}$ and
$\omega{\sim}p_{\phi}/\mu a^{2}$ as the system evolution develops to the very
late stage. Figure 17: The spacetime is dynamically partitioned into three
regions, red, green and their inter-connection region. Each region has its own
static geometry whose applicable region rotates synchronously with the merger
participants. For our purposes, we only need to know the metric functions’
form in the red and green region in the equatorial plane.

The conventional EOB method’s inconsistency arises from its decomposition of
the essentially time dependent radiating geometry into “a static background
gravitation + dissipative test particle”. This doing is not self-consistent
because a test particle’s motion in a fixed background spacetime is well known
to be conservative so no radiation caused dissipation is allowed at all. To
avoid this inconsistency, we come back to the original general relativity two-
body action

$S{=}{-}\sum_{i=}^{\scriptscriptstyle
1,2}\\!\\!\int\\!\\!{d\tau}\sqrt{-||g_{\mu\nu}(X_{i})\dot{X}^{\mu}_{i}\dot{X}^{\nu}_{i}||}{-}\\!\\!\int\\!{F}^{\mathrm{diss}}{d\tau}$
(IX.1)

and write the time dependent radiation geometry as the combination of three
independent static components111To determine the binary system’s inspiral
motion, what we need is only the space time metric on the motion plane instead
of the whole space. So exactly speaking, the metric expressions (IX.2)-(IX.3)
are valid only on the motion plane. with time-dependent application region,
see FIG.17 for intuitive pictures. The first two components’ application
region happens to cover and rotates synchronously with the two merger
participants respectively,

$g^{1}_{\mu\nu}{=}\mathrm{diag}\\{1{-}\frac{2GE_{2}}{a_{1}},(1{-}\frac{2GE_{2}}{a_{1}})^{-1},a_{1}^{2},a_{1}^{2}\sin^{2}\\!\theta_{1}\\}$
(IX.2)
$g^{2}_{\mu\nu}{=}\mathrm{diag}\\{1{-}\frac{2GE_{1}}{a_{2}},(1{-}\frac{2GE_{1}}{a_{2}})^{-1},a_{a}^{2},a_{2}^{2}\sin^{2}\\!\theta_{2}\\}$
(IX.3)
$X_{1}^{\mu}{=}\\{\tau,a_{1}(\tau),\frac{\pi}{2},\phi_{1}(\tau)\\},X_{2}^{\mu}{=}\\{\tau,a_{2}(\tau),\frac{\pi}{2},\phi_{2}(\tau)\\}$
(IX.4)

By this region wise metric, the action of the general relativistic two-body
system becomes

$\displaystyle
S{=}{-}M_{1}\\!\\!\int\\!\\!d\tau\\!\sqrt{1{-}\frac{2GE_{2}}{a_{1}}{-}(1{-}\frac{2GE_{2}}{a_{1}})^{-1}\dot{a}_{1}^{2}-a_{1}^{2}\dot{\phi}_{1}^{2}}\rule{5.0pt}{0.0pt}$
(IX.5)
$\displaystyle{-}M_{2}\\!\\!\int\\!\\!d\tau\\!\sqrt{1{-}\frac{2GE_{1}}{a_{2}}{-}(1{-}\frac{2GE_{1}}{a_{2}})^{-1}\dot{a}_{2}^{2}-a_{2}^{2}\dot{\phi}_{1}^{2}}\rule{5.0pt}{0.0pt}$
$\displaystyle+S_{\mathrm{diss}}\rule{5.0pt}{0.0pt}$

Since the orbit decay phenomena caused by the gravitational wave radiation is
always a slow process relative to the rotation of the system, we can always
neglect the two participants’ radial coordinate’s variation in a single
circular orbit motion period thus setting $\dot{a}_{1}=\dot{a}_{2}=0$. So for
the circular orbit motion, the variation principle for $a_{1}$ and $a_{2}$
will yield that

$\omega^{2}_{1}\equiv\dot{\phi}^{2}_{1}=\frac{GE_{2}}{a_{1}^{3}},\omega^{2}_{2}\equiv\dot{\phi}^{2}_{2}=\frac{GE_{1}}{a_{2}^{3}}$
(IX.6)

For the non-circular orbit motion, as long as we understand $a_{1}$ and
$a_{2}$ as the length of the semi-major axis of the two participants’
approximately periodic elliptic orbit, equation (IX.6) will also be valid. At
the same time, to assure the unshifting of the two merger participants’
central of mass position, we impose

$a_{1}=\frac{M_{2}}{M}a,a_{2}=\frac{M_{1}}{M}a,M=M_{1}{+}M_{2}$ (IX.7)

As results, the synchronicity of the two participants’ inspiral motion implies
that

$\omega_{1}=\omega_{2}\Rightarrow
E_{2}=\frac{M_{1}^{3}}{M^{2}},E_{1}=\frac{M_{2}^{3}}{M^{2}}$ (IX.8)

It can be easily verified that, by this choice of $E_{1}$ and $E_{2}$,
equations (IX.6) will lead to a general relativistic version of Keppler’s
third law for the reduced mass particle’s motion in the total mass particle’s
gravitational field.

Equations (IX.7) and (IX.8) transform the two-body problem (IX.5) into a
single body one

$\displaystyle
L{=}{-}M_{1}\big{[}1{-}\frac{2GM}{a}{-}(1{-}\frac{2GM}{a})^{-1}\frac{M_{2}^{2}}{M^{2}}\dot{a}^{2}{-}\frac{M_{2}^{2}}{M^{2}}{a}^{2}\omega^{2}\big{]}^{\frac{1}{2}}$
$\displaystyle{-}\\!M_{2}\big{[}1{-}\frac{2GM}{a}{-}(1{-}\frac{2GM}{a})^{-1}\frac{M_{1}^{2}}{M^{2}}\dot{a}^{2}{-}\frac{M_{1}^{2}}{M^{2}}{a}^{2}\omega^{2}\big{]}^{\frac{1}{2}}$
$\displaystyle{+}L_{\mathrm{diss}}\rule{28.45274pt}{0.0pt}$ (IX.9)

Although destined to be descriptions for the relative motion of the two
bodies, this lagrangian differs from that of the reduced mass particle in the
background Schwarzschild field of the total mass particle, even in the equal
mass and exactly circular orbit case, because $\frac{M_{2}^{2}}{M^{2}}$ or
$\frac{M_{1}^{2}}{M^{2}}$ is always less than $1$. The logic of the one body-
lisation method (IX.9) is to capture the non-linearity of general relativity
gravitations non-perturbationally through the non-perturbative central fixing
condition (IX.7) and the relative motions’ synchronicity condition (IX.8). In
contrasts, the logic of conventional one body-lisation method of eob1998 ;
eob2000 ; eob2009  is to capture the non-linearity of general relativity
gravitations through the perturbative match of the relative motion hamiltonian
with that of a test body’s motion in the background of a static and corrected
Schwarzschild-like field. Since our one body-lisation operation (IX.7)-(IX.8)
are exact and call no post-newtonian approximation as input, we name it as an
eXact One Body (XOB) method.

For those who can’t help from evaluating how far is our one body-lisation idea
to the conventional EOB method, the post newton expansion of the hamiltonian
following from the conservative part of (IX.9) is as follows

$H\equiv\frac{1}{\mu}\big{[}\dot{a}\frac{\delta
L}{\delta\dot{a}}+\omega\frac{\delta
L}{\delta\omega}-L],\mu{\equiv}\frac{M_{1}M_{2}}{M_{1}{+}M_{2}},M{\equiv}M_{1}{+}M_{2}$
(IX.10)
$H_{0}=\frac{1}{\nu}+\frac{p^{2}}{2}-\frac{1}{q},p\equiv\dot{a},q\equiv\frac{a}{GM},\nu{\equiv}\frac{\mu}{M}$
(IX.11)
$H_{2}=(1-3\nu)\big{[}\frac{3p^{4}}{8}+\frac{3p^{2}}{2q}-\frac{1}{2q^{2}}\big{]}$
(IX.12)
$H_{4}=(1-5\nu+5\nu^{2})\big{[}\frac{3p^{6}}{16}+\frac{21p^{4}}{8q}+\frac{15p^{2}}{4q^{2}}-\frac{1}{2q^{3}}\big{]}$
(IX.13)
$H_{6}{=}(1{-}7\nu{+}14\nu^{2}{-}7\nu^{3})\big{[}\frac{35p^{8}}{128}{+}\frac{55p^{6}}{16q}{+}\frac{189p^{4}}{16q^{2}}{+}\frac{35p^{2}}{4q^{3}}{-}\frac{5}{8q^{4}}\big{]}$
(IX.14)

Except $H_{0}$, all the higher order terms do not coincide with the input of
the conventional EOB method eob1998 ; eob2000 ; eob2009 . However, just as we
pointed out above and illustrated in FIG.16, including the dissipation term in
the hamiltonian of the conventional EOB method is not self-consistent. The
inspiral frequency $\omega\equiv\dot{\phi}$ and
$\omega{\sim}p_{\phi}/\mu{}a^{2}$ following from such a method yield
manifestly opposite variation trends at the late time of the merger process.
In contrast, adding dissipation term in our hamiltonian (IX.10) is self
consistent, because the frequency $\omega$ it involves plays the role of the
merger participants’ inspiral speed and the resultant spacetime’s rotational
speed simultaneously. So the whole system allows radiation caused dissipation
consistently.

For the above reason, we will not take the higher order post newton expansion
of (IX.12)-(IX.14) differs from the input of reference eob1998 ; eob2000 ;
eob2009  as a negative diagnosis for our idea’s rationality. Instead we will
take the direct gravitational wave form output of our method as the
justification for its validity. Taking the most simple quadrupole radiation
power as source of dissipation, we will get evolution equations for $a(t)$

$\frac{dH}{da}\frac{da}{dt}=F_{\mathrm{diss}}=-\frac{32}{5}G\mu^{2}a^{4}\omega^{6}$
(IX.15)

From this equation we can solve the function $a(t)$ and calculate the
gravitational wave through

$h_{ij}\propto G\mu a^{2}\omega^{3}\cos 2{\int\\!\omega}dt$ (IX.16)

We compare in FIG.18 the relative motion orbit and gravitational wave forms
following from the EOB method and our XOB method. From the figure, we can
easily see that the two orbits and wave forms highly agree with each other
till the inner most circular orbit (ISCO) epoch of EOB. However, however, XOB
exhibits much rational or self-consistent behaviour after that epoch. For
example by XOB, the late time inspiral speed defined through
$\omega=\dot{\phi}$ and $\omega\sim p_{\phi}/\mu
a^{2}$($p_{\phi}{\equiv}\frac{\delta L}{\delta\omega}$) are both growing
functions of time. While the variation trends of those predicted by EOB are
contrary with each other.

Figure 18: The relative motion orbit and gravitational wave form of an equal
mass BHB system following from the standard EOB method and our XOB method.
Both members of the binary system are not inner-structured and experience not
any shape deformation during the merger process. The magenta dotted line marks
the ISCO radius defined by the EOB method. The red dashed line is the horizon
radius of the total mass black hole $r_{h}=2GM_{tot}$.

Two very impressive features of the gravitational wave forms following from
XOB and displayed in FIG.18 are (i) the decay of the gravitational wave forms
is invisible from the time interval displayed and (ii) the asymptotic
frequency of the inspiral motion is logarithmic divergence. Both features
imply that the binary system has infinite energy to be carried away by the
gravitational wave radiation. This is provided by the infinite decreasible
space of the gravitational potential energy between the two point particle
type participating black holes. Newtonian mechanically
$\varepsilon_{a\rightarrow 0}{=}-\frac{GM_{1}M_{2}}{a}$. General
relativistically, $\varepsilon_{a\rightarrow
a_{f}}{\propto}\frac{-GM_{1}M_{2}^{2}/(Ma)}{\sqrt{1-3GM_{2}^{2}/(Ma)}}{+}(M_{1}{\leftrightarrow}M_{2})$,
where $a_{f}=\mathrm{max}\\{\frac{3GM_{1}^{2}}{M},\frac{3GM_{2}^{2}}{M}\\}$ is
the asymptotical separation the two black holes can get close to, see equation
(X.3) of the next section for references. In the case of $M_{1}=M_{2}$, this
asymptotic value $a_{f}=\frac{3GM}{4}$ is less than the mathematic horizon
radius $r_{h}=2GM$.

The reason for XOB method’ power of being able to trace the internal relative
motion of binary systems and the validity of the quadrupole formula (IX.16)
for the gravitational wave form even when the two participating black holes
get closer than their horizon radius allowing is, to each of the participating
black holes, the horizon is not physical reality at all. That is, as the
outside fixed position observer or probe of each other, none of them will see
the other’s horizon. They only see their inspiral partners as frozen stars of
mass profile $M(r)\propto\frac{r}{2G}$ approximately. Correspondingly, the
concept of mass central point is still meaningful just as it is in the
newtonian mechanics. This is a conceptual revolution in a sense. If one cannot
understand this properly, he/she will have difficulties to understand why the
outside fixed position observers can see the gravitational wave forms
reflecting the relative motion of the two black holes inside the horizon
defined by their total mass $r_{h}\equiv 2GM_{\mathrm{tot}}$. This will be
even more clear when we consider the non-singular inner structure and banana
shape deformation of the participating black holes. In such cases, XOB’s
ability to trace the system’s evolution inside the mathematic horizon will
bring us the full gravitational wave forms with well behaved late time quasi-
normal modes feature and decodable information about the inner structure of
the participating black holes.

In the BHB merger process described by the XOB method, the exit from innermost
stable circular orbit (ISCO) does not happen on the epoch defined by the EOB
method $a=5.718GM_{\mathrm{tot}}$. Because, by XOB even after that epoch, the
relative motion orbit of the system is stable and the system’s evolution after
the horizon entrance $r<2GM_{\mathrm{tot}}$ is still traceable analytically
until the system enters the ring down phase. However, for comparing and
referring convenience, we will preserve this concept and take
$5.718GM_{\mathrm{tot}}$ as $a_{\scriptscriptstyle\\!ISCO}$, and mark it out
in most of the relative motion orbit and gravitational wave form picture
below.

##  X The full gravitational wave forms

Consider the non-singular inner structure of black holes and their banana
shape deformation caused radiation active suppression, we have to write the
relative motion orbit decay equation as

$\frac{da}{dt}=-\frac{32}{5}G\mu^{2}a^{4}\omega^{6}\zeta^{2}\cdot\big{(}\frac{dH}{da}\big{)}^{-1},\zeta=\frac{\sin
4GMz/a}{4GMz/a}$ (X.1)

where $\zeta$ is the radiation active factor of the binary system and $z$
grows with time slowly, see equation (VIII.20) and comments there for their
definition and concrete expressions. Focusing on the circular orbit or
understanding $a$ only as the semi-major axis length of the quasi-periodic
elliptic orbit,

$H[a,\nu,\omega]{=}\frac{M_{2}(1-\frac{2GM_{1}^{2}}{Ma})}{\sqrt{1{-}\frac{2GM_{1}^{2}}{Ma}{-}\frac{M_{1}^{2}\omega^{2}a^{2}}{M^{2}}}}{+}\frac{M_{1}(1-\frac{2GM_{2}^{2}}{Ma})}{\sqrt{1{-}\frac{2GM_{2}^{2}}{Ma}{-}\frac{M_{2}^{2}\omega^{2}a^{2}}{M^{2}}}}$
(X.2)

During each single quasi periodic evolution period the rotational speed of the
system $\omega^{2}=\frac{GM}{a^{3}}$ will be approximately constant and the
hamiltonian $H$ will become the function of $a$ and $\nu$ exclusively.

$\displaystyle
H[a,\nu]^{\omega^{2}\approx}_{GM{\\!/\\!}a^{3}}{=}\frac{M_{2}(1{-}\frac{2GM_{1}^{2}}{Ma})}{\sqrt{1{-}\frac{3GM_{1}^{2}}{Ma}}}{+}\frac{M_{1}(1{-}\frac{2GM_{2}^{2}}{Ma})}{\sqrt{1{-}\frac{3GM_{2}^{2}}{Ma}}}$
(X.3)

Figure 19: The zero point of $dH[a,\nu]/da$ happens between
$\frac{3}{2}{<}a{<}6$. The lower bound happens on $\nu=1/4$, the upper bound
on $\nu=0$. For the $\nu=1/4$ system, as long as the asymptotic value of $z$
satisfies the condition $\frac{3\pi}{8}{<}z_{t\rightarrow\infty}$, the
$a^{0}_{H^{\prime}}{<}a^{0}_{\zeta}$ maximal zero point of $\zeta$ will lie
outside that of $dH/da$.

In the special case of $M_{1}=M_{2}$ or $\nu=1/4$

$\frac{dH}{da}|^{\nu=1/4}_{\omega^{2}\approx
GM/a^{3}}=\frac{GM^{2}(2{-}3GM/a)}{2a^{2}(4{-}3GM/a)^{3/2}}$ (X.4)

In the general case of $\nu\neq 1/4$, the function form of $\frac{dH}{da}$
will be more complicated. But its basic feature of containing a zero point
somewhere inside the region $\frac{3}{2}\leqslant\frac{GM}{a}\leqslant 6$ is
very robust, referring FIG.19 for intuitions. Denoting this value of $a$ as

$a^{0}_{H^{\prime}}\equiv a:\frac{dH[a,\nu]}{da}=0$ (X.5)

By equation (X.1), the zero value of $dH/da$ will exhibit as a divergent point
on $\frac{da}{dt}$. However such a divergence will not bring us disconnections
on the variation of $a(t)$. So it can be cured by hand in the numeric
integration of eq(X.1). For latter uses, we define another two extra special
values of $a$ through

$a^{\infty}_{H}\equiv a:H[a,\nu]=\infty$ (X.6)
$a^{0}_{\zeta}\equiv\max\\{a:\zeta(t)\equiv\frac{\sin 4GMz/a}{4GMz/a}=0\\}$
(X.7)

where $a^{\infty}_{H}$ is the point on which $H$ itself diverge and
$a^{0}_{\zeta}$ is the maximal zero point of $\zeta$. In the special case of
$\nu=\frac{1}{4}$, according to equation (X.3), we can easily see that
$a^{\infty}_{H}=\frac{3GM}{4}$. On the evolution line of $a(t)$, wether the
divergence of $\frac{dH}{da}$ will be implemented depends on the relative
relation between $a^{0}_{H^{\prime}}$ and $a^{0}_{\zeta}$, see FIG.19 for
references.

Figure 20: The relative motion orbit and gravitational wave form of binary
systems consisting of equal mass members following from our XOB method. The
participating black holes in the upper row have point like singular inner
structure thus is described by the standard Schwarzschild metric exactly.
Their shape deformation parameter $z{\equiv}0$. The participants in the downer
row have extend inner structures but fixed length of $4GM$. So the shape
deformation parameter $z{\equiv}1$. This inner structure will experience
bending but not elongation as the merger progresses. The magenta dotted line
marks the ISCO radius defined by the EOB method. The purple dashed line is the
mathematic horizon radius $r_{h}=2GM_{tot}$.

Since $\zeta{=}\frac{\sin 4GMz/a}{4GMz/a}$, the concrete value of
$a^{0}_{\zeta}$ depends on the form of $z(t)$ inevitably. The standard
Schwarzschild black hole is defined by the singularity and horizon. By the
language of newonian mechanics, its matter distribution is singular and
experiences no stretching or bending. By our definitions of the banana shape
deformation, such black holes have $z\equiv 0$ and the corresponding radiation
active factor $\zeta\equiv 1$. In the case the two members have equal mass,
equation (X.1) allows us to trace their merger evolution down to
$a(t)\xrightarrow{t{\rightarrow}\infty}a^{\infty}_{H}=\frac{3}{4}GM$ and get

$\omega\equiv\dot{\phi}\xrightarrow{t{\rightarrow}\infty}\big{[}\frac{GM}{(3GM/4)^{3}}\big{]}^{\frac{1}{2}}=\big{(}\frac{4}{3}\big{)}^{\frac{3}{2}}(GM)^{-1}$
(X.8)
$H\xrightarrow{t{\rightarrow}\infty}-\frac{2M/3}{\sqrt{0}}\rightarrow-\infty$
(X.9)

The former equals to half of the real part of the late time quasi-mormal modes
frequency $\omega^{\mathrm{re}}_{022}$. The latter has the explanation of the
final black hole’s total mass or energy. Its divergence and negativity is a
reflection of the irrationality of the point particle type inner structure
picture of black holes. The upper part of FIG.20 displays the full three stage
gravitational wave forms correspondingly. To make the wave forms readable, we
reduced the late time frequency of the gravitational wave by a factor of
$0.3$. The decay feature of late time gravitational wave forms is non-visible
in the time interval displayed. The logarithmic divergence feature of
$\omega^{\prime}\equiv p_{\phi}/\mu a^{3}$ is due to the point particle type
inner structure of the black holes involved.

Now let us consider the extended inner structure and non-singular matter
distribution of the black holes. As the first step, we let the black hole have
fixed length which equals to the diameter of the corresponding Schwarzschild
horizon so that $4GMz{=}4GM$, or $z{\equiv}1$ constantly. Although the length
is fixed, we let the extended black hole be bendable as the inspiral and
merger progresses, so that the radiation active factor varies with time in the
most simple but still non-trivial way $\zeta=\frac{\sin 4GM/a}{4GM/a}$, see
equation (VIII.20) for the derivation this expression. When this radiation
active factor comes in, the asymptotic value of $a(t)$ is no longer the
singular point $a^{\infty}_{H}$ of the hamiltonian, but $a^{0}_{\zeta}$. In
the case the two participating black holes have equal mass,
$a^{\infty}_{H}=\frac{3GM}{4}$, $a^{0}_{\zeta}=\frac{4GM}{\pi}$. As results

$\omega\equiv\dot{\phi}\xrightarrow{t{\rightarrow}\infty}\big{[}\frac{GM}{(4GM/\pi)^{3}}\big{]}^{\frac{1}{2}}=\big{(}\frac{\pi}{4}\big{)}^{\frac{3}{2}}(GM)^{-1}$
(X.10)
$H\xrightarrow{t{\rightarrow}\infty}\frac{M/2(4-\frac{\pi}{2})}{\sqrt{4-\frac{3\pi}{4}}}\rightarrow
0.9473M$ (X.11)

The former still corresponds to half of the real part of the late time quasi-
normal modes frequency $\omega^{\mathrm{re}}_{022}$. The latter equals to the
final black hole’s total mass or energy. Combining with the full process
gravitational wave form displayed in the lower part of FIG.20. We easily see
that, relative to those following from the exact horizon wrapped point
particle type inner structure picture, the predictions following from the
inner structure picture characterised by the extended fixed length and banana
shape bendable and wrapped only by the approximate or asymptotic horizon are
much more acceptable observationally.

In the practical BHB merger process, the degree of the participating black
holes’ banana shape deformation varies with time. Just as we pointed out in
section VIII, when such a banana shape deformation happens, the participating
black holes will be elongated along the inspiral direction. So during the
whole merger process, we expect their length varies from zero to some final
length value $a_{f}$. Substituting the value of this final length and the
corresponding $\omega_{f}=\big{(}\frac{GM}{a_{f}^{3}}\big{)}^{1/2}$ into the
hamiltonian (IX.10), we should get an energy less than the total mass of the
initial black holes

$H[a_{\\!f},\nu,\big{(}\frac{GM}{a_{\\!f}^{3}}\big{)}^{\frac{1}{2}}]<M$ (X.12)

This is because the gravitational wave radiation dissipates energies of the
system inevitably. From this equation, we can solve out the lower limit of
$a^{\mathrm{fnl}}_{\mathrm{lowlim}}$ and the corresponding frequency
$\omega_{f}$, which equals to half of the real part of the merged black hole’s
quasi-normal mode’s frequency

$\omega^{\mathrm{re}}_{022}\equiv
2\dot{\phi}<2\big{[}\frac{GM}{a^{\mathrm{fnl3}}_{\mathrm{lowlim}}}\big{]}^{\frac{1}{2}}$
(X.13)

We display in FIG.21 $a^{\mathrm{fnl}}_{\mathrm{lowlim}}$ and
$\omega^{022}_{\mathrm{uplim}}$’s dependence on the symmetric mass ratio $\nu$
explicitly. This is a prediction of XOB method which is independent of the
inner structure picture of black holes, which means that even the black holes
are considered the conventional point-like singular objects, the quasi-normal
mode’s frequency of their binary merger system would not go beyond this upper
limit.

Figure 21: To assure energy conservation, the radius of the relative motion
orbit has a lower limit exactly computable through equation (X.12).
Corresponding with this relative radius lower limit, the real part of the
quasi-normal mode’s frequency $\omega_{022}$ of the merged black holes have an
upper limit which equals to the double of the relative motion frequency.

In principle, by tracing up the matter distributions’ evolution under the
radiation back reaction force’s driving, it is possible to determine the
function form of $z(t)$ theoretically. However, such a work is very difficult
to accomplish in practice. So we will illustrate the observational feature of
the black hole deformability’s difference by comparing the gravitational wave
forms originating from two shape deformation functions of

$z_{1}(t)=\frac{\pi a_{f}}{4GM}{\cdot}\frac{a_{f}}{a(t)},z_{2}(t)=\frac{\pi
a_{f}}{4GM}{\cdot}\big{[}\frac{a_{f}}{a(t)}\big{]}^{2}$ (X.14)

The corresponding radiation active factors read

$\zeta_{1}=\frac{\sin[\pi a_{f}^{2}a^{-2}]}{\pi
a_{f}^{2}a^{-2}},\zeta_{2}=\frac{\sin[\pi a_{f}^{3}a^{-3}]}{\pi
a_{f}^{3}a^{-3}}$ (X.15)

The black holes characterised by $z_{1}(t)$ experience the banana shape
deformation more intensively at the early stages of the merger process, while
those characterised by $z_{2}(t)$ experience the shape deformation more
intensively at the late stages of the merger process.

Figure 22: The relative motion orbit and gravitational wave form of BHB
systems consisting of equal mass members with inner structure and experience
banana shape deformation. The banana shape deformation parameter in the upper
black holes varies with time by $z{\propto}a_{f}a(t)^{-1}$, the lower varies
with $z{\propto}a_{f}^{2}a(t)^{-2}$. The the gravitational wave amplitude in
the lower figure not only grows more suddenly, but also decays more suddenly
around the $a\approx 2GM$ region. The magenta dotted line marks the ISCO
radius defined by the conventional EOB method. The red dashed line is the
horizon radius of the total mass black hole $r_{h}=2GM_{tot}$.

FIG.22 displays the relative motion orbit and the corresponding gravitational
wave forms of these two types of black holes. From the figure, we can see that
the $z{\sim}a(t)^{-1}$ black holes experience banana shape deformation more
mildly and more intensively during the early stages of the inspiral motion, so
their relative motion orbit radius evolves to the asymptotic value $a^{0}_{f}$
in longer time durations. Numerically this is $t^{1}_{isco}=222$,
$t^{1}_{h}=302$ while $t^{2}_{isco}=220$, $t^{2}_{h}=293$. We will call this
type of black holes softer than those characterised by $z{\sim}a(t)^{-2}$. The
most general form of the banana shape deformation function can be written as

$z(t)=k_{0}+k_{1}{\cdot}\frac{a_{f}}{a(t)}+k_{2}{\cdot}\frac{a^{2}_{f}}{a(t)^{2}}+\cdots$
(X.16)

Except $a_{f}$, all other parameters $k_{i}$ in this function are
dimensionless. For these dimensionless parameters, we conjecture that they are
independent of the black hole sizes. By carefully measuring of the late time
gravitational wave form following from various BHB merger process, we have
chances to determine these parameters observationally. So wether the black
holes in the nature have inner structures and how hard such inner structures
are are a question answerable experimentally.

##  XI Conclusion

In one sentence, our conclusion can be stated that black holes in the nature
have microscopic states understandable in standard general relativity and
inner-structures testable from the gravitational wave forms produced in the
BHB merger events, which is calculable analytically with our exact one body
method. Expansions of this sentence are as follows.

We provided exact time dependent solutions family to the Einstein equation
sourced by dust fluids and showed how such solutions family forms the general
relativity basis for the microscopic state of black holes both from the
viewpoint of outside fixed position observers and from the viewpoint of inside
co-moving observers. By the canonic quantisation scheme, we proved that the
degeneration degree of such solutions family has right exponential-area law
feature as required by the Bekenstein-Hawking entropy formula except some
polynomial corrections.

A time dependent collapsing star solution indeed contains a future singularity
and an asymptotically realisable horizon. But such a singularity and horizon
would not become realities in any finite future to the outside fixed position
observer by their physic time definition. They will be realised in the finite
future only to the freely falling co-moving observers which use the proper
time of the collapsing matter themselves as physic times. In another word, the
horizon and singularity are physical realities only to the super investigators
who adopt the global viewpoint or have the ability to see the future of
infinitely far away future of the outside fixed position observers. To the
physical probe or observers such as the participants of BHB merger events or
freely falling observers co-moving with the matter contents of a collapsing
star, the horizon and singularity of their being observed objects are either
non-implementable in any finite future or although reachable in finite futures
but non-detectable experimentally. The belief that black holes in the real
world carry exact horizon and singularity makes two mistakes of (A) taking the
horizon and singularity of the global viewpoint holders as the horizon and
singularity of the local viewpoint holders and (B) taking the physical law’s
general coordinate invariance as the physical reality’s observer independence,
which is not the case at all.

Since either participant involved in a BHB’s merger event can only see their
merger partner’s regular inner structures characterised by an approximately
linear mass function and an asymptotic horizon, both of them will experience
banana shape deformation under the inhomogeneous back reaction arising from
the gravitational wave radiations. We introduced the concept of radiation
activity $\zeta$ to quantify this deformation caused suppression of the system
ability to emit gravitational waves and derived out that $\zeta=\frac{\sin
4GMz/a}{4GMz/a}$, where $z$ is the elongation times of the banana shape
deformed black holes relative to their shape deformation before diameter.
$\zeta$ will be multiplied on the quadrupole of the binary system when the
quadrupole formula is used to calculate the gravitational wave radiations of
the system.

We then proposed an exact one body-lisation method to transform the general
relativity two-body problem into a single body one. This method gives up the
static spherical symmetric background of EOB and the conservative part of the
system’s action calls no post newtonian approximation as inputs. It puts the
two merger participants in a three-partitied spacetime which rotates
synchronously with the merger participants. The metric of the spacetime region
that accommodate the merger participants can be determined non-perturbatively
by requiring that the two participant’s inspiral motion is synchronous and
central fixed. This method avoids the intrinsic inconsistence of the
conventional EOB and provides descriptions for the relative motion of BHB
systems throughout the full inspiral, merger and ringdown stages. Taking the
quadrupole radiation as the source of dissipation, and neglecting the inner
structure of black holes by looking them as point particles wrapped by exact
horizons, we calculated the relative motion orbits and the gravitational wave
forms following from the BHB merger process and get results which are highly
consistent with those following from the conventional EOB but behave more
rationally at the very late time stages.

When considering the extended inner structure of black holes, even the most
simple fixed length banana mode, the gravitational wave forms following from
our XOB method are almost the same as those following from the combination of
EOB+NR+BHPT methods and measured in the real observations. We derived out an
exact upper bound on the real part of the final black holes’ quasi-normal
modes $\omega^{\mathrm{re}022}_{\mathrm{uplim}}$ irrespective of the black
hole inner structure details as a concrete prediction of XOB. We compared
gravitational wave forms following from BHBs whose members are characterised
by different shape deformation susceptibilities and point out that the
differences between them may probably be measurable in the current and future
observations. So the question on what inner structures the physic black holes
own is answerable experimentally.

Two possible directions for future works would be, to find observational
evidences for the black hole inner structures we proposed here through other
channel such as black hole images and gravitational wave echos, and to revise
and apply our XOB method to the BHBs whose members are spinning and carry
nonzero angular momentum.

## Acknowledgements

We thank very much to Asta Heinnesen, Maarten van de Meent, Gregorio Carullo,
Rodrigo Panosso, Daniel D’Orazio, Vitor Cardoso and Zhengwen Liu for their
comments on the manuscript of this work and everyday communications. This work
is done in Niels Bohr International Acadmia. We thank very much to the Chinese
Scholarship Council for their sponsor of oversea visiting researches and the
warm hosts provided by Vitor Cardoso and Julie de Molade. This work is
supported by CSC202006545026, NSFC grant no. 11875082, , the Villum
Investigator program of VILLUM Foundation (grant no. VIL37766) and the DNRF
Chair program (grant no. DNRF162) by the Danish National Research Foundation.

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