1

Black Hole Thermodynamics

Revant Kasichainula∗ † , Rahul Janga∗ , Yao Lu∗ , Kevin Weng∗ , Anuj Chaudhri∗ ‡
∗
Department of Physics, BASIS Independent Fremont,
39706 Mission Blvd, Fremont, CA 94539, USA

Abstract—The study of thermodynamic quanti- a certain boundary. According to Einstein’s the-

ties, including entropy, temperature, and energy, ory of general relativity, black holes form when
within the framework of black hole dynamics has massive stars collapse under their own gravity at
led to the development of groundbreaking laws
the end of their life cycles. The core of the star
specific to black holes. These laws encompass the
zeroth law (which equates the temperatures of collapses into an infinitely dense point known as
two black holes in thermal equilibrium), the first a singularity, surrounded by an event horizon [1].
law (which relates changes in mass, entropy, and Black holes come in different sizes, from stellar-
energy), the second law (stating that the entropy mass black holes, which are several times more
of an isolated black hole never decreases), and the massive than the Sun, to supermassive black holes
third law (addressing the impossibility of forming
found at the centers of galaxies, which can have
black holes with vanishing surface gravity). The
information paradox arises from the clash between masses equivalent to millions or even billions of
quantum mechanics and general relativity when Suns.
applied to black holes. Quantum mechanics upholds Despite the name, black holes are not empty
information conservation, while classical general voids; they have mass, spin, and can also have
relativity suggests information loss when a particle electric charge. They interact with their surround-
falls into a black hole. This conflict challenges the
ings in various ways, such as attracting nearby
fundamental principle that information about the
initial state of a system can determine its future matter, forming accretion disks, and sometimes
evolution. Proposed solutions include the holo- emitting powerful jets of particles. Black holes
graphic principle and Hawking radiation. Hawking are crucial for our understanding of fundamental
radiation proposes that black holes emit thermal physics, including gravity, spacetime, and the
radiation due to quantum effects near their event behavior of matter under extreme conditions [1].
horizons. This radiation leads to a gradual loss of
They remain a subject of intense study and fas-
mass and energy by black holes over time, poten-
tially carrying away information in the process. cination among scientists and astronomers.
The ongoing conflict has led to development of
competing theories of quantum gravity, where the
solutions to the above paradoxes lie.
B. Types of Black Holes
Index Terms—Black Holes, General Relativity, The three main properties of a black hole are
Quantum Mechanics, Hawking Radiation mass, spin, and charge. The mass varies by the
different types of black holes. Since black holes
often have stars or gas orbiting around them,
I. Understanding Black Holes we are able to measure the mass of the black
A. What are Black Holes? hole. Black holes with greater mass can result in
stronger gravity [5].
Black holes are defined as regions in space
The most common black holes are called
where an enormous amount of mass is compressed
Stellar-Mass Black Holes, which range from three
into a tiny volume. They are regions in space
to ten solar masses (the size of our Sun) [6]. These
where gravity is so strong that nothing, not even
black holes form from the collapse of stars, and
light, can escape from them once it has passed
the star’s mass is often ten times greater than
† revantkasichainula@gmail.com the Sun. Intermediate Black Holes have masses
‡ anuj.chaudhri@gmail.com ranging from 100 to 10,000 solar masses [6]. They
are rare and typically form from the collisions Most of the stars live in pairs or other com-
between multiple stellar-mass black holes. Super- plicated systems. Our Sun makes an exception
massive black holes are the largest, with masses here because only one fifth of the star lives on its
from 100,000 to billions of solar masses, and own without a companion. The heavier star will
are formed in the early universe [6]. Almost all eventually evolve more rapidly and will run out of
galaxies, including our own, have a supermassive fuel sooner. When that happens, the heavier star
black hole at their center. will explode into a supernova and create a black
hole. As the star turns into a supernova, it will
C. Calculating the Schwarzschild Radius grow up in size and lower its surface temperature
[6]. Material from the star’s surface will slowly
Scientists use the Schwarzschild radius to clas- flow into the black hole. Two objects orbit each
sify different types of black holes. According to other, causing the flow of the material to form a
Equation 2, the Schwarzschild radius is the dis- non-linear path. At this stage, we can observe the
tance between the event horizon and the singu- black hole and its accretion disk.
larity point, where G is the gravitational constant
and c is the speed of light [4]. The Schwarzschild
radius can be derived from the conservation of II. Properties of Black Holes
energy equation 12 mv 2 = GM R
m
. By substituting A. No-Hair Theorem
and canceling the like terms, we will get R = 2GM
V2
. The No-hair theorem claims that the black
Substituting v with the speed of light we will get holes in stable condition after formation have 3
back the Schwarzschild Radius. main independent properties. These properties
include mass, electric charge, angular momentum.
2GM Once matter crosses the event horizon and enters
rs = (2)
c 2 the black hole, it effectively disappears from the
The spin of the black holes is caused by the observable universe, leaving no direct trace of its
rotation which creates a drag force toward the properties behind [7].
space around known as the frame-dragging. Ac- The No-hair theorem puts forth this claim and
cording to Einstein’s theory of general relativ- the name came into popularity because essentially
ity, the faster the spin rate, the radius of the “black holes have no hair” and can be described
accretion disk is smaller which also results in by a few parameters: mass, charge, and angu-
higher surface temperature [4]. The charge of a lar momentum, with no additional distinguishing
black hole always remains zero because the strong features. With two black holes that have the
tendency for charges would quickly be neutralized same three parameters, this conjecture says they
by attracting oppositely charged particles from are equal and indistinguishable, no matter the
the surrounding space [4]. difference in what each black hole is made of [7].
The three properties of the black holes that
are visible outside of the event horizon include
D. Evolution of Black Holes mass, angular momentum, and charge. The total
Since supermassive black holes are substan- mass inside a sphere containing the black hole is
tially more massive than other black holes, they found using the gravitational analog of Gauss’s
share the similar structure as stellar mass black law. Gauss’s law in electrostatics states that the
holes. In the early universe, stars were created electric flux through a closed surface is directly
from abundant dense clouds of hydrogen and proportional to the electric charge enclosed by
helium. They were heavy with masses of hundreds that surface. For Gauss’s law of gravitation, the
of solar masses. Then they burned the fuel and gravitational flux through any closed surface is
turned into first black holes. As the universe proportional to the enclosed mass [7].
expands, larger black holes merge with smaller Angular momentum is measured far away by
ones to enormous masses. They become local recognizing the gravitational dragging made by
centers of gravity and form whole galaxies around the Electromagnetic field. Gravitational dragging
them [1]. or the lens thirring effect tells us that massive

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objects not only curve spacetime due to their tracking the evolution of the Von Neumann en-
mass, but also drag spacetime around them as tropy of the radiation emitted by the black hole
they rotate. Charges are easily noticeable because as it evaporates. Initially, the entropy increases
charged objects repel other charges [11]. due to thermal effects, but eventually reaches a
maximum and decreases back to zero, indicating
the recovery of information encoded in the initial
B. Black Hole Information Paradox quantum state of the black hole. This resolves the
Black holes are a dissipative system. When paradox by demonstrating that no information is
an object falls into a black hole, all informa- permanently lost during the evaporation process
tion including shape and distribution of charge [9].
is evenly distributed along the horizon of the Holography is presented as another solution
black hole and lost to outside observers. Black through the holographic principle which says that
holes can form through the gravitational collapse information is encoded on the event horizon in
of massive stars and eventually reach the point a holographic manner. Another solution is the
where nothing can escape including light. firewall hypothesis concerns the existence of a
Steven Hawking proposed that black holes emit “firewall” near the event horizon altering tradi-
thermal radiation due to quantum effects near the tional understanding of the black hole structure.
event horizon. Hawking also showed that the ther- The information is carried out by quantum cor-
mal radiation emitted by black holes is random relations among all particles radiated from the
and holds no information about what went inside black hole. Radiated particles break these cor-
black hole. Hawking radiation, in simple terms relations with their infalling partners and these
however, is when a black hole slowly loses mass correlations between the emitted particles con-
and energy over time due to emission of particles tain information about anything that entered the
[30]. black hole. The energy released creates a firewall
A problem that becomes increasingly clear around the black hole [9]. Another theory to this
when understanding black holes is the black hole black hole paradox is remnants where the black
information paradox. The paradox concerns the hole does not completely evaporate and leaves
information loss associated with the black hole behind stable remnants carrying lost information.
and the violation of unitarity in quantum me-
chanics. With the evaporation of a black hole, D. Black Hole Metrics
the radiation emitted contains only information Static black holes, also called Schwarzschild
about total mass, charge, and angular momen- black holes, have mass but neither electric charge
tum. These properties can correspond to differentnor angular momentum. The Schwarzschild black
initial states and essentially meaning that the hole is described by the Schwarzschild metric,
initial state information is then lost [8]. which governs the spacetime geometry around a
In quantum mechanics the information encoded spherically symmetric, non-rotating mass [12].
in a system’s wave function at any point of time The Birkhoff theorem states that the
can be used to determine its state at other pointSchwarzschild metric is the only vacuum
solution, regardless of energy sources, that
time. With the loss of the initial state, this fun-
damental part of quantum mechanics is violated exhibits spherical symmetry. It essentially asserts
leading to the black hole information paradox. that there is no observable difference in the
gravitational field of a Schwarzschild black hole
and that of any other spherical object with
C. Solutions to the Black Hole Information Para- the same mass when observed from a distance.
dox Gravitational fields from a distance from a
One solution to the black hole paradox is the black hole behave similarly to those of any
Page curve which suggests that information is other spherical mass of equal magnitude. The
not permanently lost but encoded in the evolving gravitational effects of Schwarzschild black holes
entropy of Hawking radiation. The Page curve are equivalent to those of a massive star or planet
addresses the black hole information paradox by of the same mass.

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 The Reissner-Nordström metric provides a gen- Schwarzschild radius is defined as 2GM/c2 . Black
eral solution beyond the Schwarzschild metric, holes with a non-zero spin and electric charge will
incorporating properties such as electric charge. have a smaller radius.
This metric describes non-rotating charged black
holes that possess mass and electric charge but III. Structure of Black Holes
no angular momentum [11]. The Kerr metric de- A. Event Horizon
scribes rotating black holes that are non-charged The event horizon of a black hole is a boundary
and possess angular momentum as well as mass. in spacetime that marks the point beyond which
This rotation leads to unique gravitational ef- matter and light can only move inward toward
fects such as frame dragging, wherein spacetime the mass of the black hole, with no possibility of
around the black hole is dragged along with its escape.
rotation. The Kerr-Newman metric is the most Black holes, as discussed before, are a dissipa-
general stationary black hole solution that in- tive system, and with the presence of an event
corporates charge and angular momentum, rep- horizon - a region from which nothing, not even
resenting a rotating charged black hole [12]. light, can escape. Events occurring within the
Extremal black holes arise when black holes event horizon cannot be observed outside the
with the minimum possible mass are compatible black hole because information cannot escape,
with their charge and angular momentum. When making it impossible for external observers to
this minimum possible mass is exceeded, solutions detect events inside the black hole. Understand-
have naked singularities—gravitational singular- ing the properties and implications of the event
ities without an event horizon. A singularity is horizon is crucial for unraveling the mysteries of
a condition where gravity is predicted to be so black holes and their impact on the cosmos.
intense that spacetime breaks down. However, The event horizon poses significant challenges
the cosmic censorship conjecture asserts that all to observation and measurement, as events oc-
singularities that arise in general relativity are curring within it are effectively concealed from
hidden behind an event horizon. external observers. Gravitational time dilation, a
With the strong electromagnetic force, black consequence of general relativity, causes clocks
holes formed from the collapse of stars are ex- near a black hole to tick more slowly compared to
pected to retain the neutral charge of a star. distant observers, leading to a slowing of time for
However, things become more complicated when objects approaching the event horizon. Gravita-
considering angular momentum (J). The spin of tional redshift further complicates observations,
a near-extreme Kerr black hole is calculated to as light emitted from objects near the event hori-
be below this near-maximum value: GM 2 /c. The zon becomes increasingly redshifted, appearing
absolute value of J must be less than or equal dimmer and redder to external observers.
to this value. This is the Kerr bound, which The distinction between observations made by
places constraints on the angular momentum of a local observers falling into a black hole and those
rotating black hole relative to its mass. Dividing made by external observers provides insight into
both sides by GM 2 and considering that |J| is the nature of the event horizon. Locally, observers
always non-negative, we can remove the abso- experience no significant effects as they cross the
lute value. Rearranging the inequality gives us event horizon, per the equivalence principle of
0 ≤ J/GM 2 ≤ 1/c. relativity. However, external observers perceive
This inequality sets a limit on the maximum objects near the event horizon as frozen in time,
angular momentum and represents a direct rela- due to gravitational time dilation and redshift
tionship between the mass of a black hole and the effects.
amount of angular momentum it can have while At equilibrium, the event horizon of a black
remaining stable according to general relativity hole exhibits a spherical topology. However, for
[13]. The size of a black hole is determined by rotating black holes (Kerr black holes), the event
the radius of the event horizon (Schwarzschild horizon becomes oblate rather than perfectly
radius), which will be described in the following spherical due to the distortion of spacetime in-
sections, and is proportional to the mass. The duced by rotation [13].

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B. Singularity Light that is emitted tangentially to the photon
sphere (i.e., parallel to its surface) can escape and
At the center of a black hole, general relativity
travel outward into space. This light does not fall
tells us that there exists a point where space-
into the black hole but instead continues on its
time curvature becomes infinite. That point is
trajectory away from the black hole. Therefore,
the gravitational singularity, and for non-rotating
light emitted from the photon sphere at the right
static black holes, the singularity is a single point
angle can escape and be observed by distant
with zero volume. For rotating Kerr black holes,
observers, contributing to the observable light
the singularity is smeared out to form a ring lying
emitted by the black hole system [16].
in the plane of rotation [14].
However, light that crosses the photon sphere
The singularity contains all the mass of the
on an inbound trajectory, moving towards the
black hole solution and therefore has infinite den-
black hole, behaves differently. Due to the strong
sity. All that enters the event horizon must suffer
gravitational pull of the black hole, this light is
the fate of the singularity and will ultimately
ultimately captured by the black hole and cannot
reach the singularity and be crushed to infinite
escape. When light crosses the photon sphere
density, adding to the mass of the black hole.
on an inbound trajectory, it moves closer and
Spaghettification occurs when the gravitational
closer to the event horizon, eventually reaching
forces become so extreme that the observer is
a point where it cannot escape the gravitational
stretched and torn apart.
pull of the black hole, and it is pulled into the
Solutions to avoid the singularity are all theo- black hole’s event horizon. Once inside the event
retical, and one such theory is seen in charged horizon, the light is unable to escape from the
(Reissner-Nordström) or rotating (Kerr) black black hole, and it becomes part of the black hole’s
holes where entering the singularity would re- mass [16].
sult in exiting the black hole into a different
spacetime, essentially acting as a wormhole. This
would be a hypothetical structure that provides D. Ergosphere
shortcuts through spacetime. The ergosphere is a region surrounding a rotat-
Singularities, as discussed before, have in- ing black hole where objects cannot remain sta-
finitely large density and curvature. The break- tionary due to the effect of frame-dragging caused
down of general relativity occurs at such points by the black hole’s rotation. Frame-Dragging, as
because the quantum effects are so strong due mentioned before, is when rotating mass drags
to immense gravitational fields. Fields such as spacetime around it, causing nearby objects to
quantum gravity are still in development and be dragged along with the rotation [17]. Near the
hope to resolve the extreme conditions associated event horizon of a rotating black hole, this effect
with singularities [15]. becomes so strong that objects would have to
move faster than the speed of light in the opposite
direction to remain stationary. The ergosphere is
C. Photon Sphere
bounded by the black hole’s event horizon and
The photon sphere is a spherical boundary the ergosurface, which coincides with the event
around a black hole where photons can travel horizon at the poles but extends to a greater
on circular orbits at a certain radius. For non- distance around the equator [17].
rotating static black holes, this spherical bound- While objects and radiation can escape from
ary around a black hole where photons can the ergosphere, they cannot remain stationary
travel on circular orbits is at a radius 1.5 times within it and are forced to rotate along with the
the Schwarzschild radius (event horizon radius). black hole’s spin. The Penrose process, occurring
These photon orbits are dynamically unstable, near the ergosphere, allows particles to split, with
and any small perturbation, like the presence of one part falling into the black hole and the other
infalling matter, can cause the photon to either escaping with greater energy, extracting energy
escape the black hole or spiral inward towards the from the black hole’s rotation and can lead to
event horizon [16]. the slowing down of the black hole. Additionally,

5
the Blandford-Znajek process which is a varia- In order to initiate the derivations of the four
tion of the Penrose process, facilitated by strong laws of classical black hole thermodynamics we
magnetic fields near the black hole, is crucial for must first explore some key processes that will
the generation of the luminosity and relativistic enlighten us on the relations of energy within
jets observed in quasars and active galactic nuclei, their boundaries.
involving the extraction of energy from the black
hole’s rotation through magnetic interactions [17].
A. Key Processes
1) The Penrose Process: The Penrose Process
E. Accretion Disks is a mechanism by which energy can be extracted
Accretion disks form when a massive object, from a rotating black hole. The process involves
such as a black hole, exerts gravitational at- exploiting the rotational energy of a rotating
traction on nearby matter, pulling it into orbit black hole by extracting some of it and allowing
around itself. This nearby matter can include an object (such as a particle or photon) to escape
stars, gas clouds, or other sources that can be with more energy than it initially had [18]. A
drawn into orbit around the black hole. The disk, particle is split near the ergosphere into two parts;
typically flat and disk-shaped, is composed of gas, one of the resulting particles falls into the black
dust, and other materials that are heated to high hole, adding to its mass and angular momentum,
temperatures [15]. while the other one gains energy from the rotation
The light from the accretion disk is distorted and escapes to infinity [18]. The rotating black
due to gravitational effects - gravitational lensing. hole loses some of its energy, and the escaping
The gravitational field of the black hole warps the particle is released with more energy than the
fabric of spacetime around it, causing light from original, unsplit particle. The Penrose Process is
the accretion disk to follow distorted paths as it dependent on the conservation of energy within
travels towards an observer. This phenomenon, a black hole and is limited by the lifetime of the
known as gravitational lensing, alters the appar- black hole’s rotation.
ent shape and appearance of the accretion disk 2) Area Theorem: Formulated by Steven
when viewed from different angles. Light coming Hawking, the Area Theorem [19] states that the
from regions above and below the black hole’s total surface area of the event horizon of a black
equatorial plane follows different paths due to hole, or a collection of black holes, cannot de-
gravitational lensing, leading to the formation of crease over time. Mathematically, it is expressed
"humps" or distortions in the observed image of as dAdt
≥ 0 (where A is the total area of the
the accretion disk. event horizon, and t is time). The Area Theorem
These distortions change in size and shape as [19] is a statement about the evolution of black
the observer’s viewing angle changes, resulting hole event horizons. It states that the total area
in a dynamic and complex appearance of the of event horizons in a closed system can never
accretion disk [15]. decrease; it either remains constant or increases.
The Area Theorem is a crucial concept in the con-
text of classical black hole thermodynamics and is
IV. Classical Black Hole
closely related to the laws of black hole mechanics
Thermodynamics
formulated by physicist Jacob Bekenstein and
From the celestial enigmas of black holes, a later extended by Stephen Hawking. According
set of fundamental principles known as the Four to Bekenstein’s proposal [20], black holes possess
Classical Laws of Black Hole Thermodynamics entropy, which is proportional to the surface area
emerges, shedding light on the intricate interplay of their event horizon. The entropy of a black
between gravity, thermodynamics, and quantum hole [20] is defined as one-fourth of the area of its
mechanics within the cosmic arena. In this ex- event horizon in Planck units. This idea was fur-
ploration, we will unravel the basis for each of ther supported and refined by Stephen Hawking,
the four laws and delve into the implications they who showed that black holes can emit thermal
hold for our understanding of the universe. radiation, now known as Hawking radiation.

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 3) Cosmic Censorship: The cosmic censorship C. First Steps
conjecture [21], also formulated by Roger Penrose, With the understanding of these key processes,
suggests that certain types of singularities, specif- we can look at clues that will help us decipher
ically those arising from gravitational collapse the 4 classical laws of black hole thermodynamics.
(such as those within black holes), are always hid- First we must take into consideration that a clas-
den from external observers by an event horizon. sical black hole cannot emit anything, it would
In other words, the singularities are “clothed” or be futile to associate a non-zero temperature.
“censored” by the presence of an event horizon. Furthermore, Bekenstein’s proposal [20] noted
Two forms of the cosmic censorship conjecture are that the area of a black hole’s event horizon
posited to help understand the masking of black could be considered as an analog of entropy. This
hole singularities. analogy suggested a deeper connection between
a) Weak Cosmic Censorship: Weak Cosmic the behavior of black holes and the laws of
Censorship [22] describes how generic singular- thermodynamics and is the foundation for the
ities formed through gravitational collapse are laws. The development of the laws of black hole
always hidden within black holes, making them thermodynamics was motivated by the desire to
unobservable from the outside. establish a consistent framework that adheres to
b) Strong Cosmic Censorship: Strong Cos- the principles of thermodynamics. Thus, there has
mic Censorship [23] on the other hand posits to be some sort of relation between the change
that not only are singularities hidden within black in mass (dM ) of a black hole and the change in
holes, but they are also spacelike and cannot be horizon area (dA) and we want to find it [21]. Our
observed by any observer, even if they were able first clue comes from the Penrose Process which
to travel inside the black hole. shows a conservation of energy (or mass equiva-
lently) and angular momentum for an uncharged
rotating black hole as:

B. Censorship Importance
dM ≈ ΩdJ
The Cosmic Censorship Conjecture is valuable
in helping to maintain predictability and reg-
Including the angular velocity and electric po-
ularity in the evolution of spacetime. Without
tential of the horizon, the Penrose Process for a
cosmic censorship, it would be possible for singu-
rotating charged black hole results in two foun-
larities to exist without being hidden behind an
dational equations [21]:
event horizon. These "naked singularities" could
be visible and interact with the external universe.
The presence of naked singularities could lead to dA = 0
unpredictable and chaotic behaviors in the sur- dM = ΩdJ + ΦdQ
rounding spacetime [21]. If singularities were vis-
ible, it could imply that the laws of physics break
down in those regions. Predicting the evolution of Where:
spacetime near a naked singularity would become Ω = angular velocity
extremely challenging or even impossible, as the dJ = change in angular momentum of the black
singularities would no longer be hidden from hole
external observers. The censorship conjecture is Φ = electric potential of the horizon
further connected to certain energy conditions dQ = change in electric charge of the black
in general relativity [21]. If naked singularities hole
were allowed, it might imply violations of these
energy conditions in regions where singularities The expression represents change in mass of
are exposed. This could have repercussions for a rotating, charged black hole in reversible
our understanding of the physical limits and con- processes. From Einstein’s mass energy
straints in extreme gravitational environments. equivalence E = mc2 , we know that this

7
should have a connection with the change in
energy of the system and ultimately with the
dQ = 0
first law of thermodynamics.

The first law of thermodynamics is given by: Comparing these equations with the Penrose
process equations for a rotating charged black
hole,
dU = dQ + dW

dM = ΩdJ + ΦdQ
where changes in internal energy = heat added
to the system + work done on the system.

The most general form of the work done in an dA = 0

open system can be written as,
shows a striking similarity between them. For
an uncharged black hole, these reduce to,
dW = −pdV + ψde
dM = ΩdJ
where −pdV is the work done due to changes
in volume, and ψde is the electrical work in the
system due to changes in electric charge. Here ψ which has an analog in the first law as,
is the electric potential (actually electrochemical
potential) of the system. Plugging this back into
the first law of thermodynamics gives us the dU = −pdV
Gibbs equation, which is:
The Penrose process describes that changes in
dU = dQ − pdV + ψde mass of the system causes the black hole to rotate
faster thus increasing its angular momemtum. For
a classical thermodynamic system, this is equiv-
For reversible processes, the heat exchanged by alent to the idea of changes in internal energy
the system is related to the change in entropy as, accompanied by equivalent changes in volume.
The angular velocity plays the part of the ther-
modynamic pressure.
dQ = T dS In order to have a complete theory of Classical
Black Hole Thermodynamics, one should have a
relationship between dM and dA, something that
For a reversible adiabatic process in an other- plays the role of the “heat flow” term that is
wise open system, where there is no heat exchange represented in dQ = T dS.
with the surroundings (we are assuming changes
in charge are happening due to changes in particle D. The Missing Link: Surface Gravity
number and in this way the electric potential is It turns out that the missing term that will help
like an electrochemical potential), we can write us further the analogy between classical black
the first law of thermodynamics as, hole thermodynamics and thermodynamics and
relate dM and dA is given by:
dU = −pdV + ψde
dA
κ
8πG
augmented with the equation,

8
 where: κ = surface gravity E. Zeroth Law
Though it is defined in a rather technical way, it The surface gravity κ is defined locally on the
is essentially the gravitational acceleration at thehorizon of a black hole but remains constant over
event horizon of the black hole in the reference the entire horizon of a stationary black hole [24].
frame of a distant observer. This is given (in This constancy is akin to the zeroth law of regular
geometrical units) by: thermodynamics, where temperature, also defined
locally, is uniform within a system in thermal
equilibrium.
1 1
κ= = In the analogy between black hole thermody-
4M 2rs namics and regular thermodynamics, the black
hole temperature takes the form of surface grav-
ity. The constancy of surface gravity is derived by
where: rs = Schwarzschild radius assuming that the horizon is a Killing Horizon,
The most technical definition can be defined as- meaning that the Killing field is tangent to the
suming the event horizon is a killing horizon [20] null generators of the horizon [21]. Additionally,
(null horizon generators are orbits of the killing this constancy holds true when the black hole is
field) where κ would then be the magnitude of the either static or axisymmetric and exhibits "t-φ"
gradient of the norm of the horizon generating reflection symmetry [21].
Killing field χa = ξ a + Ωψ a , evaluated at the
horizon [21]. F. First Law
In the context of a rotating charged black hole,
the augmented equation:
κ2 = −(∇a |χ|)(∇a |χ|)
κdA
dM = + ΩdJ + ΦdQ
8πG
However conceptually, it can be represented in
simpler terms. Surface gravity is the strength of
the gravitational field at the surface of an object. provides a comprehensive depiction of the intri-
It is different from gravity which is the force cate relationships between the changes in mass,
of attraction between objects with mass. The angular momentum, and electric charge. The
formula for surface gravity is derived from the term 8πG
κdA
is associated with the change in the
more general law of gravity, considering a specific black hole’s horizon area, reflecting the thermo-
location at the surface of the massive object. But dynamic significance of the surface gravity κ, akin
for a black hole, things become more complex to temperature in regular thermodynamics [18].
due to the nature of its event horizon. The event The angular velocity Ω term in the equation
horizon is not a solid surface like the physical correlates with the rotation of the black hole,
surface of a planet, but rather a boundary beyond emphasizing the role of rotation in the ther-
which nothing, not even light, can escape. Surface modynamics of these astrophysical objects. This
gravity of a black hole, unlike for other massive equation essentially encapsulates the interplay of
objects, is inversely proportional to its mass. As mass, energy, and entropy within the dynamic
the mass of the black hole increases, the surface environment of black holes.
gravity decreases. When stationary matter is present outside the
black hole, the inclusion of additional terms re-
lated to matter reflects the impact of external
2GM influences on the black hole’s thermodynamic
rs =
c2 properties. In this scenario, the surface gravity
c4 assumes a role analogous to temperature [21],
gevent horizon =
4GM providing a thermodynamic perspective on the in-
teraction between the black hole and its external

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environment. This parallels the first law of regular reduced to zero in a finite number of steps [21].
thermodynamics, where conservation of energy is This assertion aligns with the analogous third law
a fundamental principle. of regular thermodynamics, which dictates that
absolute zero temperature cannot be reached in a
G. Using First Law to Find Relation Between dM finite number of operations. The Third Law in the
and dA context of black hole thermodynamics suggests
that achieving a state where the surface gravity
Consider a quasistatic process where a bit of is entirely eradicated requires an infinite sequence
mass is added to a black hole that is non-rotating of steps. This concept reinforces the uniqueness
and neutral: and complexity of extreme gravitational environ-
Mass change = flux of conserved energy current ments, contributing to our understanding of the
Tab ξ a through the horizon: ∆M = Tab ξ a κλdA
R
limitations imposed by the laws of thermodynam-
where: ics on black holes.
• dA = cross-sectional area element
• λ = affine parameter along the horizon gen-
I. The Grand Analogy
erators
• Killing vector ξ is given along the horizon
a Putting the 4 principle thermodynamics laws
by ξ = κλκ
a a and the 4 classical black hole thermodynamics
laws that we just derived together will illuminate
the full analogy. For the zeroth law of thermody-
Using Field Equations:
namics, we see that temperature is uniform within
a system in thermal equilibrium which is analo-
κ
 Z
∆M = a b
Rab κ κ λdλdA gous to the black hole’s constant surface gravity
8πG ! over the horizon. Extending to the first law, we
κ dρ see a conservation of energy, easily represented
 Z
∆M = dλdA
8πG dλ through the equation, dU = T dS − pdV + ψde,
κ whilst its black hole analog states dM = 8πG κdA
+
 Z
∆M = ρdλdA ΩdJ + ΦdQ showing an interplay between various
 8πG 
κ factors influencing the black hole’s mass. Simi-
∆M = ∆A (definition of ρ) larly, the second law of thermodynamics states
8πG
that entropy is always increasing and the second
law of classical black hole thermodynamics, pro-
Thus, we have found an equation that directly posed by Hawking through Area Theorem, states
relates dM and dA. that horizon area is always increasing. And fi-
nally, the third law of thermodynamics states that
H. Second and Third Laws temperature can never reach a value of absolute
zero, while its black hole analog similarly states
The Second Law, also known as the Area Theo-
that the surface gravity of the horizon cannot be
rem, establishes that, under the assumptions of a
reduced to zero in a finite number of steps.
positive energy condition and Cosmic Censorship,
the horizon area of a black hole is constrained to
never decrease [19]. This law is deeply rooted in J. Classical Faults
the concept that the total entropy of an isolated Despite the desperate pursuit of an analogy
system, including black holes, should always fol- that mirrors that of thermodynamics, the model
low an increasing trend. The area theorem, there- has several glaring flaws. Penrose and Hawking
fore, accentuates the irreversible nature of certain employed multiple theorems to demonstrate that
processes involving black holes and underscores energy flows both into and out of a black hole,
their adherence to fundamental thermodynamic portraying the black hole as an intermediary in
principles. the exchange of energy [19].
Moving to the Third Law, it posits that the Addressing three major problems with the
surface gravity of a black hole’s horizon cannot be analogy, the first is the disappearance of the

10
temperature of a black hole. Secondly, while en- which appear as virtual particles [35]. These par-
tropy is mathematically dimensionless, the area ticles spontaneously pop up without influence
of the black hole’s horizon is quantified in terms from external forces. This seemingly violates the
of length squared (L2). Lastly, there is a seeming law of conservation of energy, as according to
contradiction between the universal tendency for classical mechanics, particles cannot be created
total entropy to increase and the independent on a whim — they must come from somewhere.
increase of the horizon area of a black hole [21].
These quandaries find resolution through the
B. Conservation of Energy
incorporation of Quantum Field Theory (QFT).
The application of QFT in the context of black How can particles spontaneously exist without
hole thermodynamics helps reconcile the apparent violating energy conservation? In classical sys-
paradoxes and provides a more comprehensive tems, energy is a function of specific quantities
understanding of the intricate interplay between (momentum, velocity, fields, etc.). For instance,
quantum mechanics and gravity in extreme grav- kinetic energy depends on the momentum of the
itational environments. object, and potential energy depends on an ob-
ject’s relative location. The total energy — the
sum of these parts — should stay constant.
V. Hawking Radiation
In quantum systems, energy is now a possible
A. Introduction measurement outcome, meaning that measuring
Hawking radiation is the theoretical thermal the energy of specific particles may now result in a
black-body radiation of black holes [30]. This is variety of values [25]. The quantum counterparts
a counter-intuitive concept because black holes to the classical observables like mass and velocity
absorb all radiation (light) that crosses their event now don’t have very well-defined traits. The wave
horizon — how can they radiate away anything function — or quantum state — is now the
at all? There are many explanations for Hawk- combination of many of these possible measure-
ing radiation, with the most widely-accepted one ment outcomes. For example, the wave function
being Hawking’s own explanation in his book "A of a spinning particle can be represented as a
Brief History of Time". Hawking described a neg- weighted combination of the 2-state possibilities
ative energy flux across the event horizon of the (the particle can spin up or spin down).
black hole, which balances the outgoing positive Some factors of quantum systems can still
energy flux (the actual radiation emitted by the be classified into eigenstates, which is when a
black hole). Hawking used complementary pairs particular observable has a definite value. The
of virtual particles as an accompanying picture Heisenberg uncertainty principle states that po-
to his explanation. To understand his point, we sition and momentum cannot both be exactly
must first understand how virtual particles come defined, which means that a state cannot be a
to exist in our universe. superposition of both momentum and position
Virtual particles are theoretical, transient, and eigenstates. Energy also has eigenstates, and if
only exhibit some of the characteristics of real the system happens to be in an energy eigenstate,
particles. Their existence is limited by the un- then keeping track of energy is trivial. However,
certainty principle. For example, virtual photons most systems are dynamic, and thus, keeping
have the quantum numbers of a real photon but track of energy becomes much harder. These
do not obey the energy-momentum rule [35]: dynamic systems now cycle through many, many
different energy eigenstates [25].
Since we cannot feasibly keep track of so many
m2 c4 = E 2 − p2 c2 possible measurements of energy, we assign an av-
erage energy E to a quantum state. Our measure-
ments may fluctuate around this central average
A quantum vacuum, which is made up of value, but as long as our average energy stays
underlying quantum fields according to quan- the same (as long as we obey the Schrödinger
tum field theory, will have quantum fluctuations, equation), then energy is indeed conserved [25].

11
 However, there is a scenario in which our sys- particles will be drawn into the black hole, while
tem appears to not obey the Schrödinger equation its partner will escape and become a real particle.
— when our system is being measured. Wave Thus, this real particle must need some coun-
functions collapse when under observation, and terpart to balance the total energy — so some
this collapse process is unpredictable [25]. After energy is taken away from the black hole [31]. In
we do the measurement, the energy of the system a sense, the escaping real particle (to an external
will have a definite energy eigenstate, which will observer) has borrowed its energy from a black
not be the same as our average energy E. hole.
In quantum mechanical systems, energy This explanation has some problems. In reality,
doesn’t appear to be conserved as seen by there are no particles, only fields — and the
observers — this doesn’t mean it isn’t conserved, outgoing radiation wavelength with the particle
but just that observers are bad at keeping track scenario would be of all wavelengths, when Hawk-
of energy. ing himself calculated that Hawking radiation is
related to the size of the black hole.
C. Virtual Particles A revised model considers waves coming from
infinity which become affected by the event hori-
Now that we know no laws are being bro-
zon. The event horizon causes fluctuations in the
ken, let’s return to our discussion around vir-
wave, which, as they exit the influence of the
tual particles. We have already established that
black hole, begin to look like real particles to
particles may pop into existence in the midst of
a distant future observer. Hawking showed that
a quantum vacuum. These particles appear as
the waves leaving the black hole had more energy
particle-antiparticle pairs, where one particle has
than those entering, and the scattered waves have
a positive energy, and another has a negative
wavelengths similar to the size of the black hole’s
energy. These annihilate shortly after, restoring
event horizon [32]. However, it turns out that this
the total energy to 0. Virtual particles are perhaps
explanation is still incomplete.
best understood as a result of the mathematical
calculations which make up quantum mechanics
[36]. They appear in intermediate states of a E. A Better Explanation
calculation or system — they never appear in Hawking radiation was built upon math, and
asymptotic states. without a solid understanding of quantum field
One version of Heisenberg’s uncertainty princi- theory, no complete explanation of Hawking ra-
ple states that the products of the uncertainties diation can be given. However, this explanation
in energy of a particle and time over which this comes closest to the truth.
measurement is taken must be smaller than half A quantum vacuum has a zero-point energy,
of the reduced Planck’s constant: which is the energy of the so-called “empty space”
classical mechanics would have us believe in. This
∆E · ∆t ≥
~ energy includes the energy of non-interacting par-
2 ticles in their ground state. The zero-point energy
of space changes relative to the curvature of
space. Hawking’s calculations dealt with finding
However, this means that if this product is less
the difference between zero-point energies from
than the constant value, then virtual particles can
the curved space around black holes and flat
exist (just over an extremely small time interval).
spaces far away (at infinity). Hawking used the
Bogoliubov transformations to reconcile quantum
D. Hawking’s Explanation fields with gravity in lieu of a theory of quantum
Now we reconcile all these ideas together to gravity [32].
understand Hawking’s own explanation for Hawk- The difference in the zero-point energies be-
ing radiation. Hawking proposed a scenario where tween curved space (the area outside the event
virtual particle pairs appear right outside of the horizon) and flat space is what makes up Hawking
event horizon of a black hole. One of the virtual radiation. Radiation emanates from the curved

12
space surrounding the event horizon, and when This means that particles are an observer-
we write down corresponding quantum field equa- dependent concept which can start to change
tions (of the curved and flat spaces), we find an definitions when we add accelerating observers
asymmetry. By going further into detail, we will and curved space to the picture [34]. It may be
see how this discrepancy leads to actual radiation better to think of the Unruh effect as stating that
emanating away from the black hole. the very concept of a particle changes between
different observers — not that they are being
F. The Unruh Effect “created”.
Hawking radiation is built off of the Unruh
effect and the equivalence principle. To see why, G. Final Explanation
we will break down how the Unruh effect relates Now, how does this relate to everything else
to black holes, and the discrepancy between zero- we discussed? We just need one more topic to tie
point energies of curved vs. flat spaces. everything together. Einstein’s equivalence prin-
The Unruh effect states that accelerating ob- ciple, simply put, states that inertial and grav-
servers in a vacuum will perceive particles [28]. itational mass are the same thing. The heavily
An inertial observer in the same region of space curved space near a black hole means that there
would not detect any particles. Because this ac- is a strong gravitational field in that region [33].
celerating observer detects particles, they would Simply staying stationary right outside of the
also detect a temperature — meaning that if one event horizon requires an observer to accelerate
waved around a thermometer in empty space, (or else they would fall in). This means that an
they would measure a non-zero temperature (as observer outside of an event horizon experiences
the thermometer would be accelerating). The Un- the Unruh effect, and detects particles. To a
ruh temperature is given by: faraway external observer, these particles seem
real. This is the source of Hawking radiation.
a Simply put, black holes exist in a perpetual
TU = state of collapse, which is the source of Hawking
2πkB
radiation.
Some more conclusions we can draw is that
This means that larger accelerations corre- anything with mass would have its own “Hawking
spond with more particles being detected [28]. radiation”, however miniscule — all it needs to
How does the Unruh effect work? A quantum do is to curve space and exert a gravitational
vacuum is not nothingness — it is the lowest field. We also know that since smaller black holes
possible energy state of the underlying quantum curve space more (stronger tidal forces), they will
fields which make up our universe. The energy have a greater effect of Hawking radiation. This
states of these fields are defined by the Hamil- is why the radiation of black holes is inversely
tonian based on local coordinates (the Hamilto- proportional to their mass (as the larger a black
nian is an operator which represents the total hole, the bigger its surface area, and the smaller
energy of a system). Special relativity states that its effect on the curvature of space). This is
an accelerating observer and an inertial observer counterintuitive, but also tells us that smaller
who both lie in the same reference frame must black holes evaporate faster than larger ones do.
have different time coordinates (they experience
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