Simon Badger, Johannes Henn, Jan Plefka and Simone Zoia
Scattering Amplitudes in Quantum Field Theory
– Monograph –


# Survey on Security Attacks in Connected and Autonomous Vehicular Systems

S M Mostaq HossainIEEE Student Member
Dept. of Computer Science
Tennessee Tech University
Tennessee, USA
<EMAIL_ADDRESS>Shampa BanikIEEE Student Member
Dept. of Computer Science
Tennessee Tech University
Tennessee, USA
<EMAIL_ADDRESS>Trapa BanikIEEE Student Member
Dept. of ECE
Tennessee Tech University
Tennessee, USA
<EMAIL_ADDRESS>Ashfak Md ShibliIEEE Student Member
Dept. of Computer Science
Tennessee Tech University
Tennessee, USA
<EMAIL_ADDRESS>
###### Abstract

Connected and autonomous vehicles, also known as CAVs, are a general trend in
the evolution of the automotive industry that can be utilized to make
transportation safer, improve the number of mobility options available, user
costs will go down and new jobs will be created. However, as our society grows
more automated and networked, criminal actors will have additional
opportunities to conduct a variety of attacks, putting CAV security in danger.
By providing a brief review of the state of cyber-security in the CAVs
environment, this study aims to draw attention to the issues and concerns
associated with security. The first thing it does is categorize the multiple
cyber-security threats and weaknesses in the context of CAVs into three
groups: attacks on the vehicle’s network, attacks on the Internet at large,
and other attacks. This is done in accordance with the various communication
networks and targets under attack. Next, it considers the possibility of cyber
attacks to be an additional form of threat posed by the environment of CAVs.
After that, it details the most up-to-date defense tactics for securing CAVs
and analyzes how effective they are. In addition, it draws some conclusions
about the various cyber-security and safety requirements of CAVs that are now
available, which is beneficial for the use of CAVs in the real world. At the
end, we discussed some implications on Adversary Attacks on Autonomous
Vehicles. In conclusion, a number of difficulties and unsolved issues for
future research are analyzed and explored.

###### Index Terms:

Connected autonomous vehicles, vehicular network, cyber-attacks, vehicular
security, adversary attacks.

##  I Introduction

Improvements in automotive technology have accelerated during the past decade.
The traditional idea of safety in the automobile industry has been
revolutionized as a result of the increasing complexity of modern vehicle
systems, which coincides with a rapid rise in the use of electronic components
and wireless technologies. In addition, there has been a rise in the interest
in the development of vehicular networks (VN)[1] and connected autonomous
vehicles (CAV)[2], which has led to the emergence of new security issues and
vulnerabilities. In the meanwhile, the industry standards for in-vehicle and
vehicular communications do not adhere to the well-established computer
security regulations due to the hardware limits and changes in network
architecture. The market for CAVs is rapidly expanding around the world. The
forecast for 2050 puts it at $7 trillion [3]. In addition, numerous major
automakers and IT behemoths are working furiously to beat the clock and
deliver usable CAVs to the market. Several ambitious research initiatives have
been launched around the world due to the substantial business potential of
CAVs.

CAVs could revolutionize many industries, but they also present serious
privacy and safety concerns [4]. Specifically, the rising degree of
interconnectivity and mechanization raises the stakes for security threats. An
attacker might take control of a vehicle or its accessories. Information
sharing and wireless connectivity are primary drivers of successful cyber
attacks on automobiles. Consequently, some of the most pressing security and
privacy concerns for IVs are related to the confidentiality of user identity
and data, the integrity of data transmissions (both incoming and outgoing),
and the safety of the vehicle’s Electronic Control Units (ECUs) [5].

In this paper, we talk about how we think about the security of connected
vehicles based on recent threats and ways to protect against them. By giving
an overview of these threats, we show how important it is that connected
vehicles have good security, in general, to protect them from cyber-attacks.
We also show that secure systems would be broken even if one of the security
features wasn’t covered by any of the three categories. We think this paper is
helpful for making and using connected vehicle services, as well as
researching cyber security for connected vehicles.

This paper’s remaining sections are structured as follows. In Section II, we
present a literature review on the topic of CAV cyber-security. In Section
III, we present the implications of Adversary Attacks on Autonomous Vehicles,
and in Section IV we discuss the limitations. Section V is the final section
of this work.

Figure 1: The potential attacks in connected and autonomous vehicles.

##  II Research Overview

Autonomous vehicles, in contrast to conventional ones, must communicate with
external networks consisting of other vehicles and infrastructure. These
external networks present both a potential attack vector and a hacking
opportunity [6]. Autonomous vehicles rely on the processing of enormous data
sets to function. When dealing with image data, deep learning algorithms are
especially susceptible to adversarial attacks, and the likelihood of false
positives increases with the volume of data to be processed. Attackers can
take advantage of flaws and vulnerabilities in autonomous vehicles [7], and
the technology is still in its infancy in terms of both hardware and software,
making it challenging to make it dependable under all scenarios.

###  II-A Attacks on Vehicle-Based Ad-Hoc Networks (VANETs)

  1. 1.

Internal Network Attacks

    1. 1.1

key and Passcode attacks
One of the safeguards of linked cars is a passcode and keys. Passwords and
keys using IR-based technology may be cracked after multiple failed attempts.
The passcodes of linked vehicles can be broken by a brute-force attack [8].
Passengers’ personal information may be compromised by a disruption in
Bluetooth connectivity.

  2. 2.

External Network Attacks

    1. 2.1

V2X Attacks on Networks
There are numerous advantages to the idea of linked automobiles, and new
technology in vehicular networks makes it possible for vehicles to communicate
data with one another and with fixed infrastructure, but there is also a risk
that cybercriminals would exploit this to get access to the underlying network
and commit crimes [9]. Its capacity to link to a smartphone, the cloud, and
other devices is described by its V2X communication. Dedicated Short Range
Communication (DSRC) is a duplex communication protocol channel designed for
automotive use that operates at 5.9 GHz with a bandwidth of 75 Mhz, while IEEE
802.11 p and WAVE are commonly used for vehicle-to-everything (V2X)
communication. They are all susceptible to attack because of recognized flaws.
Overtaking, changing lanes, and exchanging information with oncoming cars all
rely on data exchanged over vehicle-to-vehicle (V2V) networks, which might be
vulnerable to attacks. An impersonation attack involves a hostile car
connecting to a host vehicle using a spoofed identifier, establishing contact
with the host vehicle, and then sending and receiving malicious and sensitive
data, which is then logged and stored[8]. Unencrypted and insecure protocols
are a fundamental flaw of V2V communication since they allow attackers to
listen in on conversations between vehicles and steal important information
like authentication keys, which can subsequently be used in authentication
attacks. Threats to Vehicle-to-Infrastructure (V2I) Communications: When a
vehicle establishes a connection with the base station, it opens up a two-way
line of communication. Through the car’s connection to vulnerable intelligent
traffic signs and cellular network nodes, attackers can get unauthorized
access to the network and ECUs of the vehicle.

    2. 2.2

Distributed Denial Of Services (DDOS)
In the case of Internet-connected vehicles, denial-of-service attacks are
among the most dangerous possibilities. system service is denied as a result
of several attack mechanisms, causing congestion and infrastructure damage
[10]. As a result, there is a risk that the vehicles would collide, which
might be fatal for the people inside.

TABLE I: Summary of various attacks on Connected and Autonomous Vehicular Systems Sl. | Types | Impacts | Attack methods | Features | Year [Ref.]
---|---|---|---|---|---
1 |  Remote sensor attacks |  Regard injected signals as legal signals |  Various attack vectors |  Cannot distinguish the legal input and deceived |  2022[23]
2 |  GPS spoofing attacks |  Manipulate the received GPS signals arbitrarily |  Broadcast fake GPS signals with high signal strength |  Ignore true GPS signals |  2021[24]
3 |  Location trailing attacks |  Discover behaviors and activities of the vehicle, the profile of the driver |  Relate pseudonymous position samples to specific vehicles |  A single pseudonym is not sufficient to protect vehicles’ privacy |  2022[25]
4 |  Close proximity vulnerabilities |  The system or communication mechanism may be attacked |  Bluetooth or key-less entry and ignition systems |  The attack may be primitive and happen occasionally |  2021[26]
5 |  CAN and SAE J1939 buses vulnerabilities |  Unauthorized nodes can eavesdrop on any conversation taking place on the bus |  The result of the system of arbitration |  An attacker can compromise a network by inserting a malicious frame |  2020[27]
6 |  Attacks using a flashing ECU software |  Disclose information, corrupt information, degrade hardware performance, trick a remote device |  Reverse engineering, code modification, fuzzing attacks |  Use unpatched vulnerabilities to gain access |  2021[28]
7 |  Business process integration scams |  Integrated business services attacks |  Gain the client-level access, system-level access, access to the other vehicle systems |  The adversary uses the client-level access to implement another attack |  2021[26]
8 |  DoS attacks |  Delay and interfere the receiver’s response |  Insert useless messages, create some issues on network nodes |  Dangerous and fatal |  2023[29]
9 |  Impersonation attacks |  Send phony messages, modify received messages, and discard important ones |  Fake or use another identity |  Multiple identities are spoofed |  2021[30]
10 |  Replay attacks |  Confound authorities, mislead traffic, endanger transportation |  Record and re-transmit the early valid packets later |  Occur in certain procedures for securing and exchanging keys |  2020[31]
11 |  Routing attacks |  Disturb the normal routing process, drop passing packets |  One compromised or cooperative node |  Exploit the drawback and vulnerability of routing protocols |  2020[32]
12 |  Data falsification attacks |  Mislead receivers’ reaction and result in fatalities |  Distribute or disseminate fabricated security alerts and other information |  An unnecessary warning can open the way ahead. Congestion in the event of a falsified accident message |  2022[33]
13 |  Eavesdropping attacks |  Leak actual identity and private key |  Overhear the wireless medium |  The victim is not conscious of the attacks |  2021[34]
14 |  Password and key attacks |  Crack password or key |  Multiple values are examined until the mechanism is compromised |  The attacks fall into three groups |  2021[26]
15 |  Infrastructure attacks |  Cut off the communication in the infrastructure |  Vulnerability related to message transfer |  Attacks are related to cloud service |  2020[35]
16 |  Conditional attacks |  Create unsafe environments and critical safety issues |  N/A |  Communicated data differ arbitrarily from reality. No deviations exceed the threshold |  2021[26]
17 |  Intrusions into AI defenses |  Modify training algorithm and training data |  Data poisoning, escape, theft, and inference attacks on models |  Take place throughout the data-gathering, data-training, and data-predicting stages |  2020[36]

###  II-B Attacks based on sensor vulnerabilities

  1. 1.

Global Positioning System (GPS)
Using GPS data with a high degree of accuracy allows one to both identify and
navigate the car. To get around the difficulties of accessing GPS data, the
number of satellites in the public domain was increased so that anyone with
internet access could simply acquire the data. Data transparency and open
access make it easier for hackers to manipulate information and provide false
directions or even take over the vehicle’s navigation system [11]. Because of
this, there were concerns raised about the passengers’ safety and security.
GPS spoofing and jamming are two terms that refer to the same activity, which
involves hackers transmitting data and signals that are not accurate to the
GPS system. As the GPS sensors are adjusted to pick up greater signals, the
unrealistic signal grows stronger and the vehicle’s location drifts further
and further from the target [12].

  2. 2.

Inertial Measurement Unit (IMU)
IMU stands for inertial measurement unit and is comprised of a gyroscope and
an accelerometer. Its purpose is to measure the vehicle’s speed, as well as
its acceleration and orientation. They also keep an eye on the ways in which
environmental dynamics, such as the gradient, are shifting. The data produced
by the sensors can be adjusted or left unchanged in order to determine the
road’s incline. This causes the car to travel more slowly on inclines, which
in turn slows the traffic behind it [13].

  3. 3.

Light Detection and Ranging (Lidar)
The technique of Light Detection and Ranging (LiDAR) can be utilized for the
purposes of localization, obstacle detection, and collision avoidance. The
system relies on the time it takes for the light to travel to and from the
vehicle to determine the distance to the object. In the event that the hacker
transmits a signal to the scanner at the same frequency, it is safe to presume
that the object has been located [14]. As a result, the autonomous vehicle
will either proceed more slowly or come to a complete halt.

  4. 4.

Monoscopic and Stereoscopic Cameras
Cameras can perform a wide variety of detecting tasks, including lane
detection, traffic sign recognition, headlight detection, obstacle
identification, and more. It is possible to disable the functionality of
cameras by shining high-beam torches or the headlights of the opposing vehicle
in their direction. It is possible that it will give rise to safety problems
such as false detection or the failure to identify the items. The camera’s
complementary metal oxide (CMOS) sensors can be damaged by extremely bright
illumination [15].

###  II-C Exploits and attacks against hardware

  1. 1.

Attack based on OBD port
Most cars manufactured after 2008 will include what is called an ”OBD” port,
which stands for ”onboard diagnostics [9]. The onboard diagnostics (OBD) port
is utilized to acquire vehicle diagnostic information. A vehicle’s performance
and any problems it may have are logged and shown. CAN bus is used for
communication between electronic control units. It’s a USB-like handheld
device that plugs into a port often found below the dashboard, across from the
driver’s seat. which can then be connected to a computer through a USB port or
wirelessly by Bluetooth [16]. A malicious actor could compromise a car’s
network by communicating with the vehicle’s ECUs through a PC connected to the
vehicle.

  2. 2.

Threat of Firmware Tampering to ECU
More than one hundred electronic control modules called engine control units
(ECUs) regulate the sensors and actuators of every vehicle subsystem [17].
Despite the fact that ECU code is propitiatory and therefore secure, attackers
have started to re-flash them with malicious custom firmware in recent
attacks. Since we suppose the attacker has actual access to the ECU, we label
this as a direct access attack. The external interface is used by the attacker
to upgrade the firmware of the ECU, changing the ECU’s behavior. Using
authentication for software updates and hashing techniques to ensure the
security of the ECU firmware code during its lifetime [17]; modifying the ECU
memory; tampering with security keys.

  3. 3.

Rogue updates
One source of Rogue upgrades is the latest firmware in connected vehicles.
These patches aren’t officially sanctioned by the manufacturer, and they’re
missing critical security and safety patches. Vehicle data is vulnerable to
serious cyber attacks that can reveal personal information. To the extent that
sufficient security flaws are provided, hackers can exploit them to plant
malware in the connected cars’ firmware and take command of them. Exploitation
may occur via: Physical Access or remote access

    1. 3.1

Physical Access
Due to the increasing direct integration of the physical layer with the ECUs,
the danger of cyber-attacks has increased in recent years [17]. All the
modules, including the sensors, controls, and communications, are vulnerable
to hacking. Overloading the vehicle’s physical layer or attacking individual
electrical modules are two examples of possible attacks.

    2. 3.2

Remote Access
Various networks, including WiFi, Bluetooth, 4G, etc., provide remote access.
Unintentional direct connection of ECUs to the CAN bus [17]. Because of its
internet connection, the firmware could be compromised by malicious code. The
automotive industry is uncertain about the nature of the dangers they face or
the best course of action to eliminate them. Even the automakers themselves
are unsure of the best course of action for recovery.

###  II-D Autonomous vehicle adversarial attacks

Probabilistic estimations provide the basis of Deep Learning’s ability to
perform categorization tasks. When the likelihood that a set of data is
associated with a particular class is high, we say that confidence in that
class is high [18]. Autonomous vehicles utilize cutting-edge sensing and
perception technology, and deep learning algorithms like Deep Neural Networks
(DNN) [19]. However, DNNs can be breached by specific malicious attacks,
making it possible to launch such attacks on autonomous cars. Adversarial
attacks are a form of cyberattack in which the target DNN is tricked or misled
into making potentially harmful judgments by slight modifications or additions
to the original data [20]. In [21], stop signs can be modified physically by
applying stickers to specific areas. It was misread by the software designed
to identify traffic signs.

###  II-E Miscellaneous Attacks

Many researchers and hackers were hard at work putting together a variety of
different approaches. The keyless entry system is vulnerable to spoofing,
malware attacks, and programs installed in the infotainment and navigation
systems. New varieties of attacks, such as those that target electric vehicles
by taking advantage of the electric charger, which connects the devices and
logs the data. The variety of OEMs and tier-1 suppliers in the automotive
industry increases the likelihood of a wide variety of vulnerabilities and
attacks. This is because of the many different types of cars available, each
with its own advantages and disadvantages, as well as the advent of autonomous
driving technology. This complicates efforts to ensure the safety and
visibility of several original equipment manufacturers. In Table I we have
tried to draw a brief picture of different types of attacks on connected
autonomous vehicle systems. In addition, digital twin technology [37] is about
to be adopted by the automotive sector. It’s a new frontier in the study of
autonomous, linked vehicles. Although the digital twin offers superior
security defense, there have been certain difficulties from an attack
standpoint. As a result, there has been a progressive shift in the scope of
the study.

##  III Research problem definition

The Adversary Attacks on Autonomous Vehicles could be Studied in Detail. With
the help of the concept of the author’s suggestion [22], the following three
implications could be the solution for that.

  1. 1.

First, it’s crucial to use a variety of defensive strategies in concert.
There is no silver bullet for protecting driving models from the five
different ways they might be attacked. While feature squeezing has a good
attack detection rate for all attacks, it has a high false positive rate of
40% for regular input pictures. It is recommended to use a combination of
these defensive measures to ward off enemies. It’s possible that automobile
sides aren’t the best place for detection techniques that require a lot of
processing power. Edge computing allows for a quicker response time, and the
addition of hostile detection middleware on edge nodes can increase the
security of autonomous vehicles [22].

  2. 2.

Second, more research is required to understand how various DNN topologies of
regression models affect their susceptibility.
VGG16 is safer than previous versions in both white-box and black-box attacks.
The complex model structure of VGG16 makes it more challenging to solve. It is
possible that more complicated models are more robust in terms of safety. Due
to computational constraints, autonomous cars can’t transport highly
complicated driving models; therefore, we need to create a model with just the
proper level of structure complexity and processing requirements. With the
help of edging computing, even complex driving models may be put into use.
Complex models can be abstracted and communicated to vehicles, roadside units,
and the cloud in order to reduce the strain on the vehicle’s resources[22].

  3. 3.

Third, Protecting driving model detail and researching adversarial example
transferability is critical.
The results of the white-box assault experiment (RQ1) reveal that driving
models are vulnerable to attacks and that attackers who have knowledge of
their attributes may be able to devise effective adversarial attacks. Hide
hyper-parameters and driving model neural network structure. Information
extraction can be used to attack deep learning algorithms, according to a
recent study. Exploring methods like preventing model extraction. The black
box attack experiment (RQ2) shows that hostile cases generated on one driving
model aren’t transferrable. This contradicts classification model attack
transferability findings. Driving models are regression models, hence
transferability differences between classification and regression models
should be studied. Different regression methods can learn V-B hyper-planes.
Test this hypothesis. Driving model transferability research will help
construct and defend against black-box attacks[22].

##  IV Limitations

The work [22] suggests that hackers may be able to infiltrate a self-driving
car’s system during an over-the-air software or firmware update. Malware could
intercept photos and produce hostile samples before perception. Hostile
instances cannot be seen by humans or detected by the car’s infotainment
system, unlike direct interference with vehicle components. In contrast to
adversarial attacks, basic attacks, such as replacing the driving scenery with
random pictures, are easy to identify. We tested three different CNN-based
driving models. Due to their reliance on sequence-to-sequence structures and
input driving videos, the top three driving models on Udacity’s leaderboard
were not selected. Untested CNN with RNN models exist. The Udacity dataset is
used only in our experiments, as before. More primary models and data sets are
required to further our findings. CleverHans and Foolbox are open-source
adversarial attack and defense software. Three classifier tools. Steering
angle regression models will require adapting these tools. Avoid feature
squeezing with optimization. This technique generates one valid adversarial
example on MNIST in 2030 seconds. We didn’t use it because it’s too slow for
real-time driving model attacks.

##  V Conclusion

In this paper, In this document, we’ve tried to include most of the most
common cyber attacks and exploits that can be used against a connected,
autonomous vehicle. This is the hottest area, and attacks in it are constantly
evolving in sophistication. Meanwhile, hackers are always coming up with new
ways to trick and hack the vehicle. However, the security component of
vehicles is not getting the attention that it deserves as the development of
autonomous vehicles continues to proceed at a rapid pace. At a time when many
nations are making efforts to have autonomous vehicles on the road as quickly
as feasible, this could pose a serious threat to the adoption and security of
autonomous vehicles. Scientists should stop squabbling and start working
together on projects that will finally give cyber security the attention it
deserves during the planning and development phases. Finally, we address
Adversary Attacks on Autonomous Vehicles and offer some consequences based on
our findings.

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# Cosmological horizons from classical double copy

Jun-Lang He1,2 Jia-Hui<EMAIL_ADDRESS>1Key Laboratory of
Atomic and Subatomic Structure and Quantum Control (MOE), Guangdong Basic
Research Center of Excellence for Structure and Fundamental Interactions of
Matter, Institute of Quantum Matter, South China Normal University, Guangzhou
510006, China  2Guangdong-Hong Kong Joint Laboratory of Quantum Matter,
Guangdong Provincial Key Laboratory of Nuclear Science, Southern Nuclear
Science Computing Center, South China Normal University, Guangzhou 510006,
China  3Key Laboratory of Atomic and Subatomic Structure and Quantum Control
(Ministry of Education), Guangdong Basic Research Center of Excellence for
Structure and Fundamental Interactions of Matter, School of Physics, South
China Normal University, Guangzhou 510006, China  4Guangdong Provincial Key
Laboratory of Quantum Engineering and Quantum Materials, Guangdong-Hong Kong
Joint Laboratory of Quantum Matter, South China Normal University, Guangzhou
510006, China

###### Abstract

The classical doubly copy provides relations between classical solutions in
gravitational theories and solutions in gauge theories. In this paper, we
consider the cosmological horizons in the gravity side from the perspective of
gauge theories in the classical double copy paradigm. We give several examples
to show how to locate the cosmological horizons by using only the single- and
zeroth-copy data on the base spacetime. It is found that the proposed double
copy procedure holds not only for cases with flat base spacetime but also for
certain case with curved base spacetime.

##  I Introduction

The relationships between gauge and gravity theories have been actively
explored for decades. A remarkable discovery is the perturbative double copy
[1, 2, 3], which relates scattering amplitudes of the gauge and gravity
theories. The success of the perturbative double copy has motivated
investigations into classical solutions. Surprisingly, it turns out that some
exact classical solutions of Einstein equations can be obtained from classical
solutions of Abelian Yang-Mills (Maxwell) equations and associated auxiliary
scalar fields, which is the classical doubly copy. Conventionally, the
gravitational solution is called the “double copy”, while the gauge solution
is called the “single copy” of the gravitational solution and the scalar is
termed the “zeroth copy”.

Kerr-Schild double copy is a notable framework of the classical double copy,
which was first proposed in [4]. Spacetime metric which admits Kerr-Schild
coordinates can be written as an obviously double copy form of a Maxwell gauge
field (accurately, an additional Klein-Gordon scalar field is also needed). A
typical example is that the Schwarzschild metric sourced by a point mass at
the origin is related to the gauge field sourced by a point charge at the
origin. After the pioneer work [4], the Kerr-Schild double copy forms of a
number of Kerr-Schild spacetime solutions have been studied in literature [5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. By the way, there is
another framework of classical double copy for classical solutions of gravity
and gauge fields which is called Weyl double copy that relates the field
strengths of the two sides [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35].

The main idea of the double copy paradigm is to understanding gravity theory
through gauge theory, i.e. quantities in gravity theory are expected to be
described by quantities in the corresponding gauge theory [36]. Kerr-Schild
double copy of classical solutions responds the idea from the aspect of the
construction of local fields. Black holes are important and interesting
classical solutions in the gravity side and horizons are typical structures of
black holes. In [12], the authors found an interesting connection between
black hole horizons and single copy gauge field: when the spacetime has
multiple horizons, the single-copy electric field changes signs between
horizon radii. Recently, a procedure for locating black hole horizons in Kerr-
Schild spacetimes using only single- and zeroth-copy data on flat base
spacetime is given in [37]. Motivated by these works, we here address the
problem of locating the cosmological horizons in the gravity side by using
just associated single- and zeroth-copy data on the base spacetime.

This paper is organised as follows. Section II is a brief review of the Kerr-
Schild double copy. In Section III, we introduce the local definition of
cosmological horizon of a spacetime metric which admits a Kerr-Schild form.
The quantities that play important roles in our discussion are expansions of
null geodesic congruences. In Section IV, we calculate several examples to
show how to use the single- and zeroth-copy data on base spacetime to
determine the corresponding cosmological horizons. The final section is
devoted to a summary.

##  II Kerr-Schild double copy

Here we give a brief review of the Kerr-Schild double copy, first introduced
in [4]. Considers a spacetime metric that admits a Kerr-Schild form with a
flat base metric, i.e.

$g_{\mu\nu}=\eta_{\mu\nu}+\Phi k_{\mu}k_{\nu},$ (1)

where $\eta_{\mu\nu}$ is the flat metric, $\Phi$ is a scalar field and
$k_{\mu}$ is a vector that is null and geodetic with respect to both
$\eta_{\mu\nu}$ and $g_{\mu\nu}$, i.e.

$g^{\mu\nu}k_{\mu}k_{\nu}=\eta^{\mu\nu}k_{\mu}k_{\nu}=0,\qquad
k^{\mu}\nabla_{\mu}k^{\nu}=k^{\mu}\partial_{\mu}k^{\nu}=0.$ (2)

The inverse of the spacetime metric is

$g^{\mu\nu}=\eta^{\mu\nu}-\Phi k^{\mu}k^{\nu}.$ (3)

The index on $k_{\mu}$ can be raised with either $g_{\mu\nu}$ or
$\eta_{\mu\nu}$, since
$g^{\mu\nu}k_{\mu}=\eta^{\mu\nu}k_{\mu}-\Phi(k^{\mu}k_{\mu})k^{\nu}=\eta^{\mu\nu}k_{\mu}.$

From the Kerr-Schild form metric (1), the “single copy” Maxwell field is
defined as

$A_{\mu}=\Phi k_{\mu}.$ (4)

When $g_{\mu\nu}$ is a solution to Einstein’s equations, the above Maxwell
field is guaranteed to satisfy the Maxwell equations. $\Phi$ is the “zeroth
copy” satisfying the Klein-Gordon equation on flat metric. The classical
double copy of Kerr-Schild spacetimes thus maps certain classical solutions of
Maxwell and Klein-Gordon equations to classical solutions of Einstein
equations,

$g_{\mu\nu}=\eta_{\mu\nu}+\frac{1}{\Phi}A_{\mu}A_{\nu}.$ (5)

It is also possible to extend the classical double copy to certain curved
spacetime [7, 38]. Consider the generalized Kerr-Schild metric

$g_{\mu\nu}=\overline{g}_{\mu\nu}+\Phi k_{\mu}k_{\nu},$ (6)

where the base metric $\overline{g}_{\mu\nu}$ now is curved. $k_{\mu}$ is
again null and geodetic with respect to both the full and base metrics

$g^{\mu\nu}k_{\mu}k_{\nu}=\overline{g}^{\mu\nu}k_{\mu}k_{\nu}=0,\qquad
k^{\mu}\nabla_{\mu}k^{\nu}=k^{\mu}\overline{\nabla}_{\mu}k^{\nu}=0.$ (7)

The single and zeroth copy of (6) are still defined by $A_{\mu}=\Phi k_{\mu}$
and $\Phi$ which satisfy the Maxwell and Klein-Gordon equations on the base
metric respectively.

##  III Local definition of cosmological horizons

A co-dimension 2 spacelike surface $\mathcal{S}$ in four dimensional spacetime
(with metric $g_{\mu\nu}$) always has two different future-directed null
normal vectors, which are denoted by $n^{\mu}$ and $l^{\mu}$. Taking the
normalization $n_{\mu}l^{\mu}=-1$, we can write the induced metric on
$\mathcal{S}$ as

$h_{\mu\nu}=g_{\mu\nu}+n_{\mu}l_{\nu}+l_{\mu}n_{\nu}.$ (8)

The expansion $\theta^{(V)}$ of $\mathcal{S}$ along a normal vector $V^{\mu}$
is defined as

$\theta^{(V)}=\mathcal{L}_{V}\ln\sqrt{\det h}=h^{\mu\nu}\nabla_{\mu}V_{\nu},$
(9)

where $\mathcal{L}_{V}$ denotes the Lie derivative along $V^{\mu}$.

Then, we introduce how to use the notion of expansion to define cosmological
horizons in a Kerr-Schild spacetime[39, 37, 40]. The method here is similar to
that proposed to locate event horizons in [37]. In flat spacetime, we choose a
hypersurface $\mathcal{H}$ with normal vector $\tilde{H}_{\mu}$. Then, we
foliate $\mathcal{H}$ into co-dimension 2 spacelike surfaces
$\mathcal{S}_{\lambda}$ parameterized by $\lambda$ such that Kerr-Schild
vector $k_{\mu}$ is one of its two null normals. We also assume that $k_{\mu}$
is outgoing. The other ingoing null normal vector $l_{\mu}^{(0)}$ of
$\mathcal{S}_{\lambda}$ can be determined as

$l_{\mu}^{(0)}=\frac{H_{\rho}H_{\sigma}\eta^{\mu\nu}}{2}k_{\mu}+H_{\mu},$ (10)

where $H_{\mu}=-\tilde{H}_{\mu}/k\cdot\tilde{H}$. The induced metric and its
inverse of each $\mathcal{S}_{\lambda}$, inherited from $\eta_{\mu\nu}$, are
given by

$\displaystyle\mathring{h}_{\mu\nu}=\eta_{\mu\nu}+k_{\mu}l_{\nu}^{(0)}+l_{\mu}^{(0)}k_{\nu},~{}\mathring{h}^{\mu\nu}=\eta^{\mu\nu}+k^{\mu}l_{\rho}^{(0)}\eta^{\nu\rho}+l_{\rho}^{(0)}\eta^{\rho\mu}k^{\nu},$
(11)

where we use a circular accent to denote quantities defined on flat spacetime.

Now we move to the curved Kerr-Schild spacetime with metric $g_{\mu\nu}$ in
the form of Eq.(1). Since the hyperspace $\mathcal{H}$ and its foliation
$\mathcal{S}_{\lambda}$ are described in terms of the differential forms
$H_{\mu}$ and $k_{\mu}$, they are in fact independent of the metric. Thus, the
same hypersurface $\mathcal{H}$ and its foliation $\mathcal{S}_{\lambda}$ are
used in the Kerr-Schild spacetime. Since $k_{\mu}$ is null with respect to
both $\eta_{\mu\nu}$ and $g_{\mu\nu}$, it is still a null normal of the co-
dimension 2 foliation $\mathcal{S}_{\lambda}$. The second null normal
$l_{\mu}$ can be determined uniquely

$l_{\mu}=l_{\mu}^{(0)}-\frac{\Phi}{2}k_{\mu}.$ (12)

Then, the induced metric on $\mathcal{S}_{\lambda}$ and its inverse are

$h_{\mu\nu}=g_{\mu\nu}+k_{\mu}l_{\nu}+l_{\mu}k_{\nu},~{}h^{\mu\nu}=g^{\mu\nu}+k^{\mu}l^{\nu}+l^{\mu}k^{\nu}.$
(13)

Now consider the expansion of $\mathcal{S}_{\lambda}$ along its null normals.
In flat spacetime, using the induced metric in Eq.(11), the expansions can be
calculated as

$\begin{split}\mathring{\theta}^{(k)}=\mathring{h}^{\mu\nu}\mathring{\nabla}_{\mu}k_{\nu},\qquad\mathring{\theta}^{(l^{(0)})}=\mathring{h}^{\mu\nu}\mathring{\nabla}_{\mu}l_{\nu}^{(0)}.\end{split}$
(14)

Similarly, in Kerr-Schild spacetime, the expansions of $\mathcal{S}_{\lambda}$
are

$\theta^{(k)}=h^{\mu\nu}\nabla_{\mu}k_{\nu},\qquad\theta^{(l)}=h^{\mu\nu}\nabla_{\mu}l_{\nu}.$
(15)

In the appendix of [37], the authors showed that for Kerr-Schild spacetimes
the expansions on the $\eta_{\mu\nu}$ and $g_{\mu\nu}$ backgrounds are related
via

$\theta^{(k)}=\mathring{\theta}^{(k)},\qquad\theta^{(l)}=\mathring{\theta}^{(l^{(0)})}+\frac{\Phi}{2}\mathring{\theta}^{(k)}.$
(16)

In flat spacetime, the two expansions are always of opposite sign. For the
assumption that $k_{\mu}$ is outgoing and $l^{(0)}_{\mu}$ is ingoing,
$\mathring{\theta}^{(k)}>0$ and $\mathring{\theta}^{(l^{(0)})}<0$. $k_{\mu}$
describes a diverging null geodesic congruence and $l_{\mu}^{(0)}$ describes a
converging null geodesic congruence. In curved spacetime, the situation is
different and a cosmological horizon is determined by the following criterion
[39]

$\displaystyle\theta^{(l)}=0,~{}\theta^{(k)}>0,~{}\mathcal{L}_{k}\theta^{(l)}>0.$
(17)

Due to the relations in Eq.(16), the above criterion can be rewritten as

$\displaystyle\mathring{\theta}^{(l^{(0)})}+\frac{\Phi}{2}\mathring{\theta}^{(k)}=0,~{}\mathring{\theta}^{(k)}>0,~{}\mathcal{L}_{k}(\mathring{\theta}^{(l^{(0)})}+\frac{\Phi}{2}\mathring{\theta}^{(k)})>0.$
(18)

Thus, we locate the cosmological horizon of a curved spacetime by using just
the corresponding single- and zeroth-copy quantities in base spacetime.

##  IV Examples

###  IV.1 De Sitter spacetime

In static coordinates the four-dimensional de Sitter metric takes the form

$ds^{2}=-(1-\frac{\Lambda r^{2}}{3})dt^{2}+(1-\frac{\Lambda
r^{2}}{3})^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta d\varphi^{2},$ (19)

where $\Lambda$ is the positive cosmological constant ($\Lambda>0$). Making a
coordinate transformation $d\tau=dt\pm\frac{\frac{\Lambda
r^{2}}{3}}{1-\frac{\Lambda r^{2}}{3}}dr$, the de Sitter metric can be written
in the Kerr-Schild form

$ds^{2}=-d\tau^{2}+dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\varphi^{2}+\frac{\Lambda r^{2}}{3}(d\tau\pm dr)^{2}.$ (20)

The first four term in the above equation constitute the line element of flat
metric $\eta_{\mu\nu}$ in spherical coordinates. The Kerr-Schild vector
$k_{\mu}$ and the zeroth copy $\Phi$ are

$k_{\mu}dx^{\mu}=-d\tau\mp dr,\qquad\Phi=\frac{\Lambda r^{2}}{3},$ (21)

from which the single copy gauge field constructed by $\Phi$ and $k_{\mu}$ is

$A_{\mu}dx^{\mu}=\Phi k_{\mu}dx^{\mu}=\frac{\Lambda r^{2}}{3}(-d\tau\mp dr),$
(22)

which describes a uniform charge density filling all space by replacing
cosmological constant $\lambda$ to charge density $\rho$ [5]. The Kerr-Schild
vector’s integral curves are future-directed and ingoing when taking a
negative sign in Eq.(21), and future-directed and outgoing when taking a
positive sign. Here we take the positive sign, i.e.
$k_{\mu}dx^{\mu}=-d\tau+dr$. We now look at the family of hypersurfaces
$\mathcal{H}_{(r)}$ at constant $r$ with normalized normal vector

$H_{\mu}dx^{\mu}=-dr.$ (23)

On flat spacetime, the hypersurfaces $\mathcal{H}_{(r)}$ are foliated by co-
dimension 2 spacelike surfaces $\mathcal{S}_{\lambda}$ normal to $k_{\mu}$.
Then, according to Eq.(10), the other null normal $l_{\mu}^{(0)}$ of
$\mathcal{S}_{\lambda}$ is

$\displaystyle
l_{\mu}^{(0)}dx^{\mu}=\frac{H_{\rho}H_{\sigma}\eta^{\rho\sigma}}{2}k_{\mu}dx^{\mu}+H_{\mu}dx^{\mu}=\frac{1}{2}(-d\tau-
dr).$ (24)

From the relation (12), the ingoing normal of $\mathcal{S}_{\lambda}$ on de
Sitter spacetime is

$\displaystyle
l_{\mu}dx^{\mu}=l_{\mu}^{(0)}dx^{\mu}-\frac{\Phi}{2}k_{\mu}dx^{\mu}=-\frac{1}{2}(1-\frac{\Lambda
r^{2}}{3})d\tau-\frac{1}{2}(1+\frac{\Lambda r^{2}}{r^{2}})dr.$ (25)

Then, the expansions of $\mathcal{S}_{\lambda}$ along the two null normals on
the flat spacetime are

$\mathring{\theta}^{(k)}=\frac{2}{r},\qquad\mathring{\theta}^{(l^{(0)})}=-\frac{1}{r},$
(26)

and on the de Sitter spacetime are

$\theta^{(k)}=\frac{2}{r},\qquad\theta^{(l)}=\frac{\Lambda r^{2}-3}{3r}.$ (27)

The results in Eq.(26) is consistent with the general conclusion that
$\mathring{\theta}^{(k)}$ and $\mathring{\theta}^{l^{(0)}}$ are of opposite
sign in flat spacetime. From Eq.(27), we see that in de Sitter spacetime
$\theta^{(k)}$ is always positive while $\theta^{(l)}=0$ on the marginal
surface $r=\sqrt{3/\Lambda}$. Then, we explicitly compute the Lie derivative
of the expansion $\theta^{(l)}$ at $r=\sqrt{3/\Lambda}$,

$\displaystyle\mathcal{L}_{k}\theta^{(l)}|_{r=\sqrt{3/\Lambda}}=k^{\mu}\partial_{\mu}\theta^{(l)}|_{r=\sqrt{3/\Lambda}}=\frac{2\Lambda}{3}>0.$
(28)

Thus, according to the criterion (17), the surface $r=\sqrt{3/\Lambda}$ is the
cosmological horizon. Using just the information of a gauge field sourced by a
uniform charge density filling all space on flat spacetime, we locate the
cosmological horizon of the corresponding Kerr-Schild spacetime.

The above discussion can be extended directly to $D$-dimensional de Sitter
spacetime given in [41], and in Kerr-Schild coordinates the line element is

$ds^{2}=-d\tau^{2}+dr^{2}+r^{2}d\Omega_{d-2}^{2}+\frac{r^{2}}{L^{2}}(d\tau\pm
dr)^{2},$ (29)

where $L^{2}=\frac{(D-1)(D-2)}{2\Lambda}(D>2)$ and $\Lambda$ is the
cosmological constant. The zeroth copy and Kerr-Schild vector are

$\Phi=\frac{r^{2}}{L^{2}},\qquad k_{\mu}dx^{\mu}=-d\tau\mp dr.$ (30)

Similarly, we take $k_{\mu}=(-1,1,0,...,0)$ as the outgoing null normal and
the normalized one-form is $H_{\mu}dx^{\mu}=-dr$. The other null normal
$l_{\mu}^{(0)}$ of the foliation $\mathcal{S}_{\lambda}$ is

$\begin{split}l_{\mu}^{(0)}dx^{\mu}=(\frac{H_{\rho}H_{\sigma}\eta^{\rho\sigma}}{2}k_{\mu}+H_{\mu})dx^{\mu}=\frac{1}{2}(-d\tau-
dr).\end{split}$ (31)

On the $D$-dimensional de Sitter spacetime, the corresponding null normal is

$l_{\mu}dx^{\mu}=\frac{1}{2}(-1+\frac{r^{2}}{L^{2}})d\tau-\frac{1}{2}(1+\frac{r^{2}}{L^{2}})dr.$
(32)

Then, the expansions of the foliation along the two null normals $k^{\mu}$ and
$l^{\mu}$ on Kerr-Schild spacetime can be obtained,

$\theta^{(k)}=\frac{D-2}{r},\qquad\theta^{(l)}=\frac{D-2}{2}(-\frac{1}{r}+\frac{r}{L^{2}}).$
(33)

We can see $\theta^{(l)}$ becomes zero when
$r=L=\sqrt{\frac{(D-1)(D-2)}{2\Lambda}}$. Then, we also check that

$\begin{split}\mathcal{L}_{k}\theta^{(l)}|_{r=L}=k^{\mu}\partial_{\mu}\theta^{(l)}|_{r=L}=\frac{D-2}{L^{2}}>0.\end{split}$
(34)

Thus, we locate the cosmological horizon at
$r=\sqrt{\frac{(D-1)(D-2)}{2\Lambda}}$.

###  IV.2 De Sitter black holes

The line element of a four-dimensional Schwarzschild-de Sitter black hole is

$ds^{2}=-(1-\frac{2M}{r}-\frac{\Lambda
r^{2}}{3})dt^{2}+(1-\frac{2M}{r}-\frac{\Lambda
r^{2}}{3})^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta d\varphi^{2},$ (35)

where $M$ is the mass parameter and $\Lambda$ is the positive cosmological
constant, and its Kerr-Schild form is

$\begin{split}ds^{2}=-d\tau^{2}+dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\varphi^{2}+(\frac{2M}{r}+\frac{\Lambda r^{2}}{3})(d\tau\pm
dr)^{2}\end{split}$ (36)

where the zeroth copy and Kerr-Schild vector are

$\Phi=\frac{2M}{r}+\frac{\Lambda r^{2}}{3},\qquad k_{\mu}dx^{\mu}=-d\tau\mp
dr.$ (37)

The corresponding single copy is defined as $A_{\mu}dx^{\mu}=\Phi
k_{\mu}dx^{\mu}=(\frac{2M}{r}+\frac{\Lambda r^{2}}{3})(-d\tau\mp dr)$, which
is the gauge field sourced by a point charge plus a uniform charge density
filling all space.

We choose $k_{\mu}dx^{\mu}=-d\tau+dr$ as the outgoing null normal. Now, we
consider the family of hypersurfaces $\mathcal{H}_{(r)}$ at constant $r$ with
normalized normal vector $H_{\mu}dx^{\mu}=-dr$. The other ingoing null normal
on flat spacetime is $l_{\mu}^{(0)}dx^{\mu}=\frac{1}{2}(-d\tau-dr)$. On Kerr-
Schild spacetime, the ingoing null normal is

$l_{\mu}dx^{\mu}=-\frac{1}{2}(1-\frac{2M}{r}-\frac{\Lambda
r^{2}}{3})d\tau+\frac{1}{2}(1+\frac{2M}{r}+\frac{\Lambda r^{2}}{3})dr.$ (38)

The expansions in flat spacetime are
$\mathring{\theta}^{(k)}=\frac{2}{r},\mathring{\theta}^{(l^{(0)})}=-\frac{1}{r},$
and in the Schwarzschild-de Sitter spacetime are

$\theta^{(k)}=\frac{2}{r},\qquad\theta^{(l)}=\frac{f(r)}{3r^{2}},$ (39)

where $f(r)=\Lambda r^{3}-3r+6M$.

The Schwarzschild-de Sitter black hole has an black hole event horizon and a
cosmological horizon when $0<M<1/3\sqrt{\Lambda}$ is satisfied. The locations
of these horizons correspond to the positive roots of
$1-\frac{2M}{r}-\frac{\Lambda r^{2}}{3}=0$, i.e. $f(r)=\Lambda r^{3}-3r+6M=0$.
Equation $f(r)=0$ has two positive roots $r_{1}$, $r_{2}$ and a negative root
$r_{0}$ which satisfy $r_{0}<r_{1}<r_{2}$ and $r_{0}=-(r_{1}+r_{2})$ [42].
Thus, $f(r)$ can be written as $f(r)=(r-r_{0})(r-r_{1})(r-r_{2})$. We compute
the Lie derivative of the expansion $\theta^{(l)}$ along $k^{\mu}$ at
$r=r_{2}$ and obtain

$\begin{split}\mathcal{L}_{k}\theta^{(l)}|_{r=r_{2}}=k^{\mu}\partial_{\mu}\theta^{(l)}|_{r=r_{2}}=\frac{r_{2}(r_{2}^{2}-r_{1}^{2})+r_{2}^{2}(r_{2}-r_{1})}{3r_{2}^{2}}>0.\end{split}$
(40)

Thus, the surface $r=r_{2}$ is the cosmological horizon. One can check that
the above derivative is negative at $r=r_{1}$.

Then, we consider the four-dimensional Reissner-Nordstrom-de Sitter (RNdS)
black hole with a Kerr-Schild form as follows

$\begin{split}ds^{2}=-d\tau^{2}+dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\varphi^{2}+(\frac{2M}{r}-\frac{Q^{2}}{r^{2}}+\frac{\Lambda
r^{2}}{3})(d\tau\pm dr)^{2},\end{split}$ (41)

where the zeroth copy and Kerr-Schild vector are

$\Phi=\frac{2M}{r}-\frac{Q^{2}}{r^{2}}+\frac{\Lambda r^{2}}{3},\qquad
k_{\mu}dx^{\mu}=-d\tau\mp dr.$ (42)

We also take $k_{\mu}dx^{\mu}=-d\tau+dr$ and consider the family of
hypersurfaces $\mathcal{H}_{(r)}$ at constant $r$ with normalized normal
vector $H_{\mu}dx^{\mu}=-dr$. The ingoing null normal in flat spacetime is
also $l_{\mu}^{(0)}dx^{\mu}=\frac{1}{2}(-d\tau-dr)$ and in Kerr-Schild
spacetime is

$l_{\mu}dx^{\mu}=-\frac{1}{2}(1-\frac{2M}{r}-\frac{\Lambda
r^{2}}{3})d\tau+\frac{1}{2}(1+\frac{2M}{r}+\frac{\Lambda r^{2}}{3})dr.$ (43)

The expansions in flat spacetime are
$\mathring{\theta}^{(k)}=\frac{2}{r},\mathring{\theta}^{(l^{(0)})}=-\frac{1}{r}$
and in the RNdS spacetime are

$\theta^{(k)}=\frac{2}{r},\qquad\theta^{(l)}=\frac{g(r)}{3r^{2}},$ (44)

where $g(r)=\Lambda r^{4}-3r^{2}+6Mr-3Q^{2}$. Equation $g(r)=0$ has three
positive roots $R_{1},R_{2},R_{3}$ and a negative root $R_{0}$, which satisfy
$R_{0}<R_{1}<R_{2}<R_{3}$ and $R_{0}=-(R_{1}+R_{2}+R_{3})$ [43]. We rewrite
the function $g(r)$ as $g(r)=(r-R_{0})(r-R_{1})(r-R_{2})(r-R3)$ and compute
the Lie derivative of the expansion $\theta^{(l)}$ along $k^{\mu}$ at
$r=R_{3}$,

$\begin{split}\mathcal{L}_{k}\theta^{(l)}|_{r=R_{3}}=k^{\mu}\partial_{\mu}\theta^{(l)}|_{r=R_{3}}=\frac{(R_{3}-R_{1})(R_{3}-R_{2})(R_{1}+R_{2}+2R_{3})}{3R_{3}^{2}}>0.\end{split}$
(45)

Thus, we locate the surface $r=R_{3}$ as the cosmological horizon.

For higher-dimensional Schwarzschild(RN)-de Sitter spacetime, their
cosmological horizons can be located similarly with the double copy method. In
addition, we point out that if we choose the ingoing Kerr-Schild vector as one
null normal of the foliation, we can also locate the event horizons of the de
Sitter black holes with the criterions proposed in [37, 39].

###  IV.3 Four-dimensional black plane

In this subsection we consider the case of four-dimensional black plane with
zero charge. The Kerr-Schild double copy about black plane spacetime was
studied in [7], where the background metric is a plane de Sitter spacetime.
Although the background spacetime is curved, every step of the procedure for
locating cosmological horizon still holds. Next, we locate the cosmological
horizon using single and zeroth copy data. The black plane metric with
vanishing charge is given by [44]

$ds^{2}=-(\alpha^{2}r^{2}-\frac{4\pi
M}{\alpha^{2}r})dt^{2}+(\alpha^{2}r^{2}-\frac{4\pi
M}{\alpha^{2}r})^{-1}dr^{2}+\alpha^{2}r^{2}(dx^{2}+dy^{2}),$ (46)

where $M$ is the ADM mass and $3\alpha^{2}=-\Lambda<0$. This solution is
asymptotically de Sitter not only in the transverse directions, but also in
the membrane directions and has a single cosmological horizon at
$r=(\frac{4\pi M}{\alpha^{4}})^{\frac{1}{3}}$. The vacuum solution ($M=0$)
corresponding to (46) is

$ds^{2}=-\alpha^{2}r^{2}dt^{2}+(\alpha^{2}r^{2})^{-1}dr^{2}+\alpha^{2}r^{2}(dx^{2}+dy^{2}),$
(47)

which is plane de Sitter spacetime. Regard this spacetime as a background
metric, the black plane metric in Kerr-Schild coordinates is

$\begin{split}ds^{2}=-\alpha^{2}r^{2}d\tau^{2}+(\alpha^{2}r^{2})^{-1}dr^{2}+\alpha^{2}r^{2}(dx^{2}+dy^{2})+(\alpha^{2}r^{2}\frac{4\pi
M}{\alpha^{4}r^{3}})(d\tau\pm\frac{1}{\alpha^{2}r^{2}}dr)^{2},\end{split}$
(48)

where the zeroth copy and Kerr-Schild vector are

$\Phi=\alpha^{2}r^{2}\frac{4\pi M}{\alpha^{4}r^{3}},\qquad
k_{\mu}dx^{\mu}=-d\tau\mp\frac{1}{\alpha^{2}r^{2}}dr.$ (49)

We take the outgoing Kerr-Schild vector
$k_{\mu}dx^{\mu}=-d\tau+\frac{1}{\alpha^{2}r^{2}}dr$ in plane de Sitter
spacetime. We also consider the family of hypersurfaces $\mathcal{H}_{(r)}$ at
constant r with normalized normal $H_{\mu}dx^{\mu}=-dr$. The ingoing null
normal in the plane de Sitter spacetime is
$l_{\mu}^{(0)}dx^{\mu}=-\frac{1}{2}\alpha^{2}r^{2}d\tau-\frac{1}{2}dr$. In the
Kerr-Schild spacetime, the ingoing normal is

$l_{\mu}dx^{\mu}=-(\frac{2\pi
M}{\alpha^{2}r}-\frac{\alpha^{2}r^{2}}{2})d\tau+(\frac{2\pi
M}{\alpha^{4}r^{3}}+\frac{1}{2})dr.$ (50)

The expansions in the plane de Sitter spacetime are
$\bar{\theta}^{(k)}=\frac{2}{r},\bar{\theta}^{(l^{(0)})}=-\alpha^{2}r.$ Please
note that $k_{\mu}$ and $l_{\mu}^{(0)}$ have the same sign due to the
background is plane de Sitter spacetime, which has a cosmological singularity
at $r=0$. The expansions in black plane spacetime are

$\theta^{(k)}=\frac{2}{r},\qquad\theta^{(l)}=\frac{4\pi
M-\alpha^{4}r^{3}}{\alpha^{2}r^{2}}.$ (51)

$\theta^{(l)}=0$ at $r=(\frac{4\pi M}{\alpha^{4}})^{\frac{1}{3}}$ . The Lie
derivative of the expansion $\theta^{(l)}$ at $r=(\frac{4\pi
M}{\alpha^{4}})^{\frac{1}{3}}$ is

$\begin{split}\mathcal{L}_{k}\theta^{(l)}|_{r=(\frac{4\pi
M}{\alpha^{4}})^{\frac{1}{3}}}=k^{\mu}\partial_{\mu}\theta^{(l)}|_{r=(\frac{4\pi
M}{\alpha^{4}})^{\frac{1}{3}}}=-3\alpha^{2}>0.\end{split}$ (52)

Thus, the surface $r=(\frac{4\pi M}{\alpha^{4}})^{\frac{1}{3}}$ is the
cosmological horizon.

##  V Summary

In this paper, we show through several examples how to locate the cosmological
horizons for stationary Kerr-Schild spacetime in the classical double copy
paradigm. Our work is a generalization of that on the locating of black hole
event horizons in [37]. It is found that the cosmological horizons of the de
Sitter spacetime, Schwarzschild(RN)-de Sitter black holes and black plane
spacetime indeed can be located by only using the corresponding single- and
zeroth-copy data.

When foliating the chosen hypersurfaces to co-dimension 2 spacelike surfaces
$\mathcal{S}_{\lambda}$, we choose an outgoing Kerr-Schild vector $k_{\mu}$ as
a null normal of the foliation, instead of an ingoing one in the locating of
black hole event horizons [37]. An independent ingoing null normal of
$\mathcal{S}_{\lambda}$ in the flat base metric can be determined in Eq.(10).
After computing the expansions of $\mathcal{S}_{\lambda}$ along the two null
normals, the location of cosmological horizon is determined by Eqs.(16)(17).

For the de Sitter spacetime and Schwarzschild(RN)-de Sitter black hole cases,
the base metrics are all flat. Last but maybe more interesting is the black
plane case, where the base metric is a curved plane de Sitter spacetime. Our
result shows that the method for locating cosmological horizon still holds in
this curved background case.

The discussed asymptotically de Sitter black holes in this paper are all
spherically symmetric. It will be interesting to consider the application of
the double copy method to the rotating or axially symmetric spacetime with
cosmological horizons.

## Acknowledgments

This work is supported in part by Guangdong Major Project of Basic and Applied
Basic Research (No.2020B0301030008).

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