Argues That Gravity Should Be Viewed Not As a Fundamental Force, but As an Emergent Thermodynamic Phenomenon Due to Some Yet Undetermined Microscopic Theory

Home , Entropic force, Entropic gravity, Erik Verlinde

ABSTRACT

CHALLENGING ENTROPIC GRAVITY

A recent paper by Erik Verlinde [1] argues that gravity should be viewed not as a fundamental force, but as an emergent thermodynamic phenomenon due to some yet undetermined microscopic theory. Using well-established thermodynamic principles along with the holographic principle, Verlinde proposes that gravity can be seen as an entropic force as opposed to a fundamental one. Using this idea, Verlinde claims to be able to derive Newton’s 2nd law, Newton’s law of universal gravitation, and Einstein gravity. We present a challenge to this reformulation of gravity. Our contention is that a detailed examination of Verlinde’s argument shows that such a theory cannot correctly give Newton’s laws or Einstein gravity. We find that the Poisson equation for Newtonian gravity is not uniquely determined using Verlinde’s theory. We are also able to show that Verlinde’s theory does not uniquely determine Einstein’s field equations. Specific calculations can also be done in the Reissner-Nordströmand Schwarzschild metrics to show that the entropy of these black holes do not agree with accepted values of these entropies. We will begin by introducing results by Hawking, Unruh, and Bekenstein that motivate the relationship between gravity and thermodynamics. We then summarize Verlinde’s proposal of gravity being a thermodynamic phenomenon. From this new theory, we can make calculations using the Schwarzschild and Reissner-Nordströmmetrics to determine if Verlinde’s theory is compatible with well-known calculations for the temperature and entropy of black holes with these metrics. Finally, we will present a detailed explanation as to why we believe Verlinde’s derivation of the Poisson equation and Einstein’s field equations is flawed.

Jonathan James Roveto December 2011

CHALLENGING ENTROPIC GRAVITY

Jonathan James Roveto

A thesis

submitted in partial fulﬁllment of the requirements for the degree of

Master of Science in Physics

in the College of Science and Mathematics

California State University, Fresno

December 2011 APPROVED For the Department of Physics

We, the undersigned, certify that the thesis of the following student meets the required standards of scholarship, format, and style of the university and the student’s graduate degree program for the awarding of the master’s degree.

Thesis Author

Gerardo Mu˜noz(Chair) Physics

Frederick Ringwald Physics

Douglas Singleton Physics

For the University Graduate Committee:

Dean, Division of Graduate Studies AUTHORIZATION FOR REPRODUCTION OF MASTER’S THESIS

I grant permission for the reproduction of this thesis in part or in its entirety without further authorization from me, on the condition that the person or agency requesting reproduction absorbs the cost and provides proper acknowledgment of authorship.

Permission to reproduce this thesis in part or in its entirety must be obtained from me.

Signature of thesis author: ACKNOWLEDGMENTS I am very grateful to my advisor, Dr. Gerardo Muñoz,for his guidance throughout my years at Fresno State. I am thankful for his assistance in developing and editing this thesis along with the countless conversations over the years that have allowed me to mature as a student. I would also like to thank Dr. Frederick Ringwald for the support and encouragement given to me and my fellow students as we progressed through our studies and research. I am also grateful for the notes he has provided on this thesis. I would also like to thank Dr. Douglas Singleton for organizing the theoretical physics group meetings which allowed me to share this topic with him and others and gain further insight into the matter. I would also like to thank all of my professors at Fresno State for making my collegiate career an enjoyable experience. I especially would like to thank my professors in the Department of Physics for guiding me through this field. I am also grateful to the Nancy Wright and Roger Key for their part in keeping this department well-run and a wonderful place to study at. Finally, physics is a demanding field, and I would like to thank my friends and fellow graduate students for making the difficult life of a physics student feel a lot more enjoyable. TABLE OF CONTENTS

Page LIST OF FIGURES ...... vi NOTATION ...... vii BLACK HOLES AND THERMODYNAMICS ...... 1 Hawking Radiation ...... 1 Unruh Eﬀect ...... 3 Black Hole Entropy ...... 10 GRAVITY AS AN ENTROPIC FORCE ...... 12 Overview of Verlinde Gravity ...... 12 ENTROPIC BLACK HOLES ...... 19 Metric Calculation ...... 19 Temperature and Equivalence...... 22 Forces and Entropy...... 23 CHALLENGING VERLINDE GRAVITY...... 25 Newtonian Gravity Is Not Entropic ...... 25 General Relativity Is Not Entropic ...... 26 Final Remarks ...... 29 REFERENCES ...... 31 LIST OF FIGURES

Page

1.1. Spacetime diagram of left and right moving Rindler observers. Ob- server B (OB) is capable of sending signals to the right (and left) moving observers. However, Observer A (OA) is incapable of sending signals to these observers...... 6 NOTATION A quick note on the notation used in this thesis is in order. We employ the Einstein summation convention using the Latin alphabet to denote spacetime

a a 1 2 3 4 indices. Thus, contraction between two 4-vectors, VaV with V = (V ,V ,V ,V ) is given by

4 4 a X m n X mn V Va = gmnV V = g VmVn m,n=1 m,n=1

mn where g and gmn are the metric and inverse metric respectively. We use the (+, +, +, −) signature for the metric. The ﬂat space metric is given by

ηab = diag(1, 1, 1, −1) where diag is understood to be the diagonal components of the metric in matrix form. This means our Minkowski metric is given by

2 a b 2 2 2 2 ds = gabdx dx = dx + dy + dz − dt .

The exterior metric for a spherically symmetric, non-rotating, uncharged body of mass M is given by the Schwarzschild metric

2GM −1 2GM ds2 = 1 − dr2 + r2dθ2 + r2 sin2(θ)dφ2 − 1 − dt2. r r

∂T b b Partial diﬀerentiation will sometimes be denoted ∂xa = T ,a. The covariant derivative of a rank 1 tensor, T b, is given by the following rules:

∂T b ∇ T b = + Γb T c = T b a ∂xa ac ;a ∂T ∇ T = b − Γc T = T . a b ∂xa ab c b;a viii

The Christoﬀel symbol is deﬁned as

1 1 Γ c = gcd(∂ g + ∂ g − ∂ g ) = gcd(g + g − g ). a b 2 b ad a bn n ab 2 ad,b bn,a ab,n

Complex conjugation of a scalar value, φ, is given by φ¯. In this thesis, we employ a modiﬁed version of natural units where we set c = kB =h ¯ = 1 but keep the gravitational constant, G, intact. This thesis is dedicated to my parents. Their unending love and support has brought me to where I am today. BLACK HOLES AND THERMODYNAMICS

Hawking Radiation We begin by giving a brief overview of various phenomena that will form the foundation for the discussion of gravity as a thermodynamic phenomenon. In 1974, Hawking [2] proposed that black holes are not necessarily black. We will give an overview of the calculation that leads to this. Following Stephani [3], the exterior Schwarzschild metric for a spherically symmetric gravitating mass in Eddington-Finkelstein coordinates is given by

2GM ds2 = 2drdv + r2dΩ2 − 1 − dv2 (1.1) r where the Eddington-Finkelstein coordinates, in terms of the Schwarzschild coordinates and tortoise coordinate r?, are given by the ingoing null coordinate1

r ? v = t + r + 2GM ln( 2GM − 1) = t + r . This choice of coordinate systems is suited for the motion of photons moving radially towards or away from a spherical mass. The singularity at r = 2GM is removed showing that this is, in fact, a coordinate singularity and not a physical one. The singularity at r = 0 remains as it is a true singularity. We look for solutions of the wave equation for the massless

a Klein-Gordon ﬁeld ∇ ∇aφ = 0. The solution is given as a Fourier expansion

Z ∞ † φ = dω(ˆgωaˆω + g¯ˆωaˆω) (1.2) 0 in terms of a plane wave expansion

1null coordinates allow us to explicitly diﬀerentiate between in-moving and out-moving massless particles 2

ˆ −iωv ˆ −iω(t+r?) gˆω = hω(r, θ, φ)e = hw(r, θ, φ)e (1.3)

† whereg ˆω represent a complete set of Fourier coeﬃcients,a ˆω(ˆaω) are the usual annihilation(creation) operators, and ω is the positive frequency modes with respect to v. The system is assumed to be in vacuum with no incoming particles. The retarded time is given by the outgoing null coordinate

r u = t − r? = v − 2r? = v − 2r − 4M ln − 1 (1.4) 2M which allows a partial wave expansion for outgoing waves given by

−iω0u −iω0(v−2r?) gω0 = hω0 (r, θ, φ)e = hω0 (r, θ, φ)e (1.5) where ω0 denotes positive frequencies with respect to the retarded time u. This system of partial waves is not a complete system. Some incoming waves will be absorbed by the black hole and will be unable to propogate as out-going waves. In terms of these frequencies, the Fourier expansion can be written with gω0 in terms of the complete systemg ˆω and g¯ˆω

Z ∞ ˆ −iωv ¯ˆ iωv gω0 = dv(¯αωω0 hωe − βωω0 hωe ). (1.6) 0

If βωω0 is non-zero, there exists outgoing particles. It can be shown [3] that if one looks at the event horizon and does an eikonal approximation, one will find that these Bogoliubov coefficients are non-zero and outgoing particles exist. This phenomenon is the Hawking effect. A black hole appears to emit particles despite 3 the exterior metric being taken as a vacuum. Hawking was able to show that these outgoing particles exist as a thermal spectrum. As seen by an observer at infinity, the temperature of this spectrum is given by the Hawking temperature

κ T = (1.7) 2π where κ is the surface gravity of the black hole. For a Schwarzschild black hole, this

¯hc3 is given by kBT = 8πMG where we have restored the units. Numerically, this is −8 M roughly 6 × 10 M K [4] where M is the mass of the Sun. Even a (relatively) small object such as the Earth would only radiate at roughly 0.02 K if compactiﬁed to a black hole. Compare this to the cosmic microwave background of just under 3 K.

Unruh Effect A related phenomenon, which will be important in our discussion of Verlinde’s theory of gravity, is the Unruh effect [5]. Unruh was able to show that an accelerated detector in a Minkowski vacuum would become excited. The excitation probability is described by how a detector would react if it were immersed in a thermal bath of particles. We will go over a derivation as it is given in [6] with a slight change to conform to our metric. Our goal is to show that an accelerated observer will find a non-zero expectation value of the number operator in the Rindler vacuum - that is, the apparent vacuum seen by a constantly accelerating observer. To do this, we must first calculate the trajectory of a uniformally accelerating observer. Conversion to lightcone coordinates will be necessary for both Rindler and inertial observers. We can then solve the wave equation for a massless scalar field for the Rindler observer in terms of creation and annihilation operators in the Minkowski vacuum. This will 4 lead us to determining a non-zero expectation value for the particle density seen by the Rindler observer being accelerated in a Minkowki vacuum. For simplicity, the 1 + 1 dimensional metric can be written as

2 a b 2 2 2 ds = ηabdx dx = −dτ = dx − dt . (1.8)

The velocity of the Rindler observer is given in terms of the proper time, τ, as

ua(τ) = x˙(τ), t˙(τ) (1.9) where the dot represents diﬀerentiation by τ. The normalization condition, given by

a b b (1.8), is ηabx˙ (τ)x ˙ (τ) = −1. Differentiating this we can find aau = 0. In the instantly comoving inertial frame, the observer is at rest and ua(τ) = (0, 1). Taking into account the orthogonality of the velocity and acceleration, the constant acceleration is given by aa(τ) = (a, 0). For an inertial observer, we can define lightcone coordinates that are given by

u ≡ t − x, v ≡ t + x (1.10)

In lightcone coordinates, the metric takes the form

ds2 = −dudv. (1.11)

The lightcone coordinates are invariant under a Lorentz transformation given by

1 u˜ = αu, v˜ = v (1.12) α 5 where α is constant. The trajectory can be described in these coordinates as xa(τ) = (v(τ), u(τ)). These coordinates satisfy

u˙(τ)v ˙(τ) = 1 (1.13)

u¨(τ)¨v(τ) = −a2.

Diﬀerentiation ofu ˙ in the ﬁrst equation of (1.13) and simple substitutions using the second equation leads to

v¨2 = a2 (1.14) v˙ which has solutions of the form

A v(τ) = eaτ + B (1.15) a and using (1.13), we ﬁnd

1 u(τ) = − e−aτ + C. (1.16) Aa

We can use use a Lorentz transformation along with a shift of our origin to write the trajectory as

1 1 u(τ) = − e−aτ , v(τ) = eaτ . (1.17) a a

From here it can easily be shown by referring to (1.10) that the trajectory is hyperbolic - that is: 6

1 1 t(τ) = sinh(aτ), x(τ) = cosh(aτ). (1.18) a a

This gives rise to lightlike horizons as seen in Figure 1.1.

Figure 1.1. Spacetime diagram of left and right moving Rindler observers. Observer B

(OB) is capable of sending signals to the right (and left) moving observers. However,

Observer A (OA) is incapable of sending signals to these observers.

A co-moving frame can be deﬁned for the Rindler observer with coordinates xa = (ζ1, ζ2) that can similarly be put into lightcone coordinatesu ˜ = ζ2 − ζ1 and v˜ = ζ1 + ζ2. The relationship between the two sets of lightcone coordinates is given by

1 1 u = − e−au˜, v = eav˜ (1.19) a a and the metric is given by

ds2 = −dudv = −ea(˜v−u˜)dud˜ v˜ = e2aζ1 (dζ1)2 − (dζ2)2 (1.20) 7

This metric is conformally flat and describes the Rindler spacetime. These coordinates are incomplete and are incapable of spanning the entire Minkowski spacetime. To see this, consider a hypersurface of constant time. The physical distance covered by considering an infinite range of the spacelike coordinate −∞ < ζ1 < 0, where ζ1 = 0 is the observer’s position, is given by a finite value:

0 Z 1 1 d = eaζ dζ1 = . (1.21) −∞ a

We now look at solutions to a massless Klein-Gordon ﬁeld in these backgrounds. The action, in lightcone coordinates, can be written as

Z Z S = 2 ∂uφ∂vφdudv = 2 ∂u˜φ∂v˜φdud˜ v˜ (1.22) where the conformal flatness of the Rindler spacetime has greatly simplified the action. The field equations are simply given as

∂u∂vφ = 0, ∂u˜∂v˜φ = 0. (1.23)

The solutions are trivial to determine and can be decomposed into left-moving and right-moving plane wave solutions. The standard quantization treatment [7] can be applied in the domain x > |t| (the quadrant a right-moving Rindler observer has access to) and the ﬁeld operator can be written in the usual mode expansions. For simplicity, we shall look at only right-moving solutions given by

Z ∞ dω 1 ˆ √ −iωu − iωu + φ = 1/2 e aˆω + e aˆω (1.24) 0 (2π) 2ω Z ∞ dΩ 1 h i √ −iΩ˜uˆ− iΩ˜uˆ+ = 1/2 e bΩ + e bΩ 0 (2π) 2Ω 8 where ω and Ω are the respective positive frequencies for the Minkowski and Rindler

± ˆ± spacetimes. The operatorsa ˆω and bΩ satisfy the standard commutation rules

− + 0 hˆ− ˆ+ i 0 aˆω , aˆω0 = δ (ω − ω ) , bΩ, bΩ0 = δ (Ω − Ω ) . (1.25)

1/2 ˆ The factor of (2π) is correct for a 1 + 1 spacetime. The operatorsa ˆω and bΩ are the annihilation operators for the Minkowski and Rindler vacuums respectivelyn - that is,

− aˆω |0M i = 0 (1.26) ˆ− bΩ |0Ri = 0.

At this point, we must relate the creation and annihilation operators between the diﬀerent vacua. The Bogolyubov transformations allow us to relate the operators in the Minkowski vacuum with operators in the Rindler vacuum. These transformations are given by

Z ∞ ˆ− − + bΩ = dω aΩωaˆω − βΩωaˆω (1.27) 0 Z ∞ ˆ+ + ¯ − bΩ = dω a¯Ωωaˆω − βΩωaˆω 0 and are normalized as

Z ∞ ¯ 0 dω(αΩωα¯Ω0ω − βΩωβΩ0ω) = δ (Ω − Ω ) . (1.28) 0

Inserting (1.27) into (1.24), one can infer 9

Z ∞ 0 1 iωu dΩ −iΩ0u˜ ¯ iΩ0u˜ √ e = √ αΩ0ωe − βΩ0ωe . (1.29) 0 ω 0 Ω

One can then multiply this equation by e±iΩ˜u, integrate along −∞ < u˜ < ∞, and use standard Fourier integral identities to ﬁnd that the relationship between the Bogolyubov coeﬃcients is

2 2πΩ 2 |αΩω| = e a |βΩω| . (1.30)

The last step is to simply determine the particle density as seen by the Rindler observer in the Minkowski vacuum. We ﬁnd

This is the expectation value for seeing particles with frequency Ω by the Rindler observer in the Minkowski vacuum. Taking (1.28) with Ω0 = Ω, we ﬁnd

Z ∞ Z ∞ 2 2 2πΩ 2 2 dω(|αΩω| − |βΩω| ) = dω(e a |βΩω| − |βΩω| ) (1.32) 0 0 Z ∞ 2πΩ 2 = dω e a − 1 |βΩω| 0 = δ(0).

Using this relationship, one ﬁnds 10

∞ Z h 2πΩ i−1 ˆ 2 hNΩi = dω|βΩω| = e a − 1 δ(0) (1.33) 0 where the divergent delta function is due to integrating over the infinite spatial volume. In a finite box of volume V , the delta function would be replaced by δ(0) = V and one would find the average particle density of particles with frequency Ω is given by

ˆ −1 hNΩi h 2πΩ i n = = e a − 1 . (1.34) Ω V

This is a Bose-Einstein distribution with temperature given by the Unruh temperature

a T = . (1.35) Unruh 2π

This is the so-called Unruh eﬀect. An accelerated detector will respond as being in a thermal bath with a temperature given by (1.35) despite being in a Minkowski vacuum.

Black Hole Entropy The next piece of the puzzle was formalized by Bekenstein [8] [9]. Not only do black holes radiate with a thermal spectrum that can be characterized with a temperature, they also have an associated entropy. It was shown that a quantity proportional to the area of a black hole plus the entropy of any system exterior to a black hole would always increase. In general, processes involving black holes would tend to increase the area of the black hole. It turns out the entropy of a black hole system obeys the inequality: 11

A S ≥ (1.36) BH 4G where A is the area of the black hole horizon. For a Schwarzschild black hole in vacuum, this becomes an equality. Bekenstein generalized these thermodynamic properties and formulated laws for black hole thermodynamics. The fact that the entropy of a black hole can be given in terms of its area is strange to say the least. One would expect, such as with any typical thermodynamic system, that the entropy would increase according to the volume of the system. This relationship between the area of a black hole and the entropy of a gravitating system is the foundation for Verlinde’s theory of gravity. GRAVITY AS AN ENTROPIC FORCE

Overview of Verlinde Gravity Traditionally, spacetime is seen as a fundamental entity in our universe and gravity is the phenomenon describing how energy causes spacetime to curve. Verlinde proposes that this idea of spacetime as a fundamental quantity is incorrect. He believes that spacetime and gravity are a consequence due solely to the thermodynamic properties of a microscopic theory such as String theory, akin to how the ideal gas law is a consequence of the kinetic theory of gases. As we have already shown, there are strong indicators that there exists a relationship between gravity and thermodynamics. Verlinde takes this a step further to claim that spacetime, as we know it, is an emergent phenomenon and that Einstein’s equations are simply an equation of state used to describe spacetime in some unknown thermodynamic limit. The upshot of this is that there would no longer be a pressing need to quantize gravity if this theory is true as gravity would no longer be seen as a fundamental force of nature. Not only does Verlinde propose that gravity is thermodynamic, he proposes that any N-dimensional spacetime can be encoded on an (N-1)-dimensional boundary using the holographic principle. A hologram is capable of encoding a 3-dimensional ﬁgure using a 2-dimensional surface. The information is stored on a surface and is projected as a 3-dimensional object using various optical techniques. Verlinde proposes a similar mechanism exists for gravity. This is motivated by Bekenstein’s conclusion that the entropy of a black hole appears to be stored on its event horizon. Verlinde proposes that any given N-dimensional volume of spacetime can be encoded on an (N-1)-dimensional holographic screen. For a 3-dimensional volume of space in the Newtonian case, Verlinde claims that 2-dimensional 13 holographic screens at the boundary of this space can encode the details of the system bounded by the screen. Verlinde postulates that the equipotential surfaces of Newtonian gravity play the role of holographic screens. In the relativistic case, surfaces of constant redshift are the holographic surfaces. Given a holographic screen enclosing a gravitating mass, the space within contains a given amount of entropy according to Verlinde. One can then place a particle outside of this holographic screen. If the particle were to cross the holographic surface, the entropy contained within that surface would increase. Therefore, it is entropically favorable for the particle to move across that boundary. It is this entroptically favorable conﬁguration that causes the particle to cross the screen, not any fundamental force. This is the basis of Verlinde’s proposal that gravity is an entropic force. We will now give the details of Verlinde’s proposed theory. Motivated by an argument by Bekenstein, Verlinde claims a particle approaching a holographic screen causes the entropy on the screen to change by

1 ∆S = 2π when it comes within a Compton wavelength, ∆x = m . Verlinde now assumes the change in the entropy is linear in the mass of the particle and is given by

∆S = 2πm∆x (2.1)

The analogy of osmosis across a semi-permeable membrane is used to explain how a particle would have a reason to cross this holographic screen. If there is an entropically favorable configuration available by moving towards the screen, the particle would do so. Verlinde reasonably claims such a force is not fundamental; it is an effective force due to the microscopic theory involved. According to the 1st law of thermodynamics (assuming no change in internal energy of this heat bath), this 14 effective force is given by

F ∆x = T ∆S (2.2)

If we believe Verlinde, the volume enclosed by the holographic screen has a temperature as does any thermodynamic system in thermal equilibrium. Verlinde uses the Unruh temperature, given by (1.35) in conjunction with the change in entropy (2.1), to infer Newton’s 2nd Law

F = ma. (2.3)

The Unruh eﬀect is typically thought of as a detector seeing a temperature due to the detector being accelerated through space. Verlinde claims that, in fact, it is not the acceleration that gives rise to a temperature, but a temperature giving rise to an acceleration! Newton’s laws, in this theory, are simply the consequence of thermodynamics. Verlinde now attempts to derive Newton’s law of universal gravitation along the same lines. He postulates that the number of bits on the holographic screen is given by A N = (2.4) G where A is the area of the holographic screen. Momentarily restoring the units and

q G¯h noting the Planck length, lP = c3 , we ﬁnd c3A A N = = 2 . (2.5) Gh¯ lP

These bits of information apparently are each stored within 1 Planck area. This screen encloses a mass, M, with energy

E = M. (2.6) 15

Verlinde now assumes the energy distributed by these bits on the holographic screen follows equipartition. In other words, 1 E = NT. (2.7) 2

Finally, using (2.1), (2.2), and (2.4 - 2.7), along with the fact that the area of the holographic screen is 4πR2, one infers GMm F = , (2.8) R2

Newton’s law of gravity. Verlinde then attempts to derive the Poisson equation for Newtonian gravity.

GM Using the fact that a = −∇Φ, one can write ∆Φ = − R2 ∆x. Using (2.1) and (2.4) along with the fact that A = 4πR2, one writes the entropy change in terms of Φ, the Newtonian potential that Verlinde claims simply keeps track of the information of the system, as ∆S ∆Φ = − . (2.9) N 2

We now assume a static matter distribution given by ρ(r) enclosed by a holographic screen. Verlinde writes the temperature in terms of the gradient of the potential perpendicular to the holographic screen enclosing the static mass distribution. This temperature is ∇Φ T = . (2.10) 2π

The density of these bits on the holographic surface, ∂V , is given by 1 dN = dA. (2.11) G

Equipartition tells us that 16

1 Z E = T dN (2.12) 2 ∂V where the integration is done over the degrees of freedom on the holographic screen. Using the fact that E = M and using (2.10) - (2.12), Verlinde shows that 1 Z M = ∇Φ · dA. (2.13) 4πG ∂V

Now, using Gauss’ law and the integral form of the left hand side over the volume enclosed, Verlinde claims that

∇2Φ = 4πGρ, (2.14) the Poisson equation for gravity, holds true. This treatment of gravity can now be extended to a relativistic setting. Following Wald [10], one can show the acceleration of a particle in a static, curved spacetime can be written as

b −2φ a b a = e ξ ∇aξ (2.15) where ξb is the timelike Killing vector and 1 φ = ln(−ξaξ ) (2.16) 2 a is the natural generalization of the Newtonian potential. The redshift as seen from an observer at inﬁnity is given by eφ. The Killing vector satisﬁes the equation

∇aξb + ∇bξa = 0 [10] and is the generator of isometries in the spacetime. Movements along the Killing vector do not change the metric. Static metrics admit timelike Killing vectors that allow time translations along it that leave the metric unchanged. It can be shown that the acceleration is given by: 17

ab = −∇bφ. (2.17)

The same procedure as in the non-relativistic case is used by Verlinde. He writes the temperature measured by an observer at inﬁnity as eφ T = N b∇ φ (2.18) 2π b where the appropriate redshift factor is in place and N b is a spacelike unit outward normal perpendicular to the holographic screen and to ξa. The entropy gradient is rewritten

∇aS = −2πmNa. (2.19)

The negative sign signiﬁes that entropy decreases as a particle moves along the normal and away from the gravitating mass. Using the same equipartition argument as before, Verlinde shows 1 Z M = eφ∇φ · dA (2.20) 4πG ∂Σ where ∂Σ is the 2-dimensional surface of constant redshift bounding a spacelike volume Σ. This is the correct form for the Komar mass in a static spacetime. Verlinde now points out that the right hand side can be written as Z 1 a b c d M = dx ∧ dx abcd∇ ξ . (2.21) 4πG ∂Σ

Stokes’ theorem gives Z 1 a b M = Rabn ξ dΣ (2.22) 4πG Σ where na is a timelike normal to the surface ∂Σ. Verlinde ﬁnally substitutes the left hand side using the deﬁnition by Wald of the Komar mass in terms of the 18 energy-momentum tensor to give Z Z 1 a b 1 a b 2 (Tab − T gab)n ξ dV = Rabn ξ dV. (2.23) Σ 2 4πG Σ

The claim is that this derivation is suﬃcient to show that Einstein’s equations follow from treating gravity as an emergent thermodynamic phenomenon. ENTROPIC BLACK HOLES

Metric Calculation To see if this view of gravity as an entropic force has any merit to it, we will examine the Schwarzschild and Reissner-Nordströmmetrics. The Schwarzschild metric is the simplest, spherically symmetric solution to Einstein’s field equations. It is static and stationary, meaning its metric is time-independent and allows for a time-like Killing vector. The Reissner-Nordströmmetric allows for charged objects with spherically symmetric electric fields. For convenience, we shall first derive the Reissner-Nordströmsolution. Einstein’s field equations (EFEs) are 1 R − Rg = 8πGT (3.1) mn 2 mn mn

m where Rmn is the Ricci tensor, gmn is the metric tensor, R = Rm is the Ricci scalar, and Tmn is the energy-momentum tensor. We want to find the exterior solution for the Reissner-Nordströmmetric; therefore we only need to be concerned about the electric field’s contribution to

mn Tmn. Due to the spherical symmetry, our Maxwell field strength tensor, F , has components F tr = −F rt = E(r) with all other components being 0. In free space, the field strength tensor is divergenceless - that is [3] 1 √ ∇ F mn = √ ∂ ( −gF mn) = 0 (3.2) n −g n where g = det(gab) and we have used the fact that the field strength tensor is anti-symmetric. Since our source is spherically symmetric, we will use the ansatz

2 a b 2λ(r) 2 2 2 2 2 2 2ν(r) 2 2 ds = gabdx dx = e dr + r dθ + r sin (θ)dφ − e c dt (3.3) with Birkhoﬀ’s theorem [3] as our motivation. Our metric is therefore: 20

2λ(r) 2 2 2 2ν(r) gab = diag(e , r , r sin (θ), −e ). (3.4)

The only non-trivial part of (3.2) is the radial derivative. This gives

λ(r)+ν(r) 2 ∂r(e r sin(θ)E(r)) = 0 which leads us to ke−(λ(r)+ν(r)) E(r) = (3.5) r2 where k is a constant. Now that we have our ﬁeld strength tensor, we can ﬁnd Tmn. In vacuum, the energy-momentum tensor is given by [3] 1 T = F gabF − g F F ab (3.6) mn am bn 4 mn ab

The Ricci scalar can be determined by contracting the indices of the EFEs with each other. We ﬁnd

m −R = 8πGTm (3.7)

a where we have used the fact that ga = 4. The Maxwell energy-momentum tensor is traceless with respect to the metric, therefore the Ricci scalar vanishes. Thus, (3.1) reads

Rmn = 8πGTmn. (3.8)

The four non-zero terms are −4πGk2e2λ(r) 4πGk2 R = R = rr r4 θθ r4 4πGk2sin2(θ) 4πGk2e2ν(r) R = R = φφ r2 tt r4 21

The Ricci tensor is deﬁned as

c c c d d c Rab = ∂cΓa b − ∂bΓc a + Γc dΓa b − Γb cΓa d (3.9)

c and the Christoﬀel symbols, Γa b, are deﬁned as 1 Γ c = gcd(g + g − g ). (3.10) a b 2 ad,b bd,a ab,d

Combining (3.3),(3.9), and (3.10), we have 16 diﬀerential equations to solve. We

find that our only important component is the Rθθ component. Rθθ gives us 4πGk2 1 − e−2λ(r)(1 − 2λ0r) = (3.11) r2 where the prime represents differentiation by r. A quick use of the product rule allows us to integrate this equation and find 4πGk2 re−2λ(r) = r + + D (3.12) r where D is a constant. We identify this constant with the Schwarzschild radius,

RS = 2GM. Through working with the Ricci tensor components, we also would have found that λ(r) = −ν(r) + constant. We can absorb this constant into our deﬁnition of time, but more importantly we ﬁnd that R 4πGk2 e−2λ(r) = 1 − S + = e2ν(r). (3.13) r r2

2 2 For simplicity, we deﬁne RQ = 4πGk . Our solution is given by the metric dr2 R R2 ds2 = + r2dθ2 + r2sin2(θ)dφ2 − 1 − S + Q dt2. (3.14) R2 2 RS Q r r 1 − r + r2

Letting RQ = 0 reduces this down to the Schwarzschild metric. 22

Temperature and Equivalence We now examine the temperature an observer at inﬁnity will see due to a black hole. Consider a particle undergoing a radial acceleration. First we examine the Schwarzschild solution. We deﬁne a unit-normal in the radial direction 1 ! R 2 N a = 1 − S , 0, 0, 0 (3.15) r

a Such that N Na = 1. The Schwarzschild solution admits a time-like Killing vector ξa = (0, 0, 0, 1). Using this along with (2.16), (2.18), and (3.15), we ﬁnd the temperature is R T = S (3.16) 4πr2

The question now is when is the use of (2.18) justiﬁed. Equivalence [10] tells us that

r an observer can be considered an Unruh observer when, for example, the R trt component of the Riemann curvature tensor is small. Intuitively, it is only near the event horizon that an observer at rest would feel like an accelerated observer in a flat spacetime and where one could use equivalence. We find that this component is R (R − r) Rr = S S (3.17) trt r4 which means that equivalence holds only near the horizon. At r = RS, we find (3.16) reduces to 1 T = (3.18) S 8πGM which is the Hawking temperature for a Schwarzschild black hole. Next we run through the same steps for the Reissner-Nordströmmetric. We define the radial normal component in a similar manner. Using (3.14), we define our unit normal as 23

1 ! R R2 2 N a = 1 − S + Q , 0, 0, 0 (3.19) r r2 and use the same procedure to ﬁnd the temperature for the Reissner-Nordstr¨om black hole 1 R 2R2 T = S − Q . (3.20) RN 4π r2 r3

We ﬁnd equivalence holds at the two horizons given by 1 q r = R ± R2 − 4R2 (3.21) ± 2 S S Q which is in agreement with other calculations for the temperature of a Reissner-Nordstr¨omblack hole [11] when one uses the external horizon

1 q 2 2 r+ = 2 RS + RS − 4RQ . This temperature is ! r 2 2 4RQ 4RQ 1 + 1 − 2 − 2 RS RS T = . (3.22) RN !3 r 2 4RQ πRS 1 + 1 − 2 RS

An interesting point to note is that for an extremal black hole (RS = 2RQ), the temperature vanishes.

Forces and Entropy Finally, we examine the forces and entropy given by this model. For Schwarzschild, we use (2.2), (2.18), and (2.19) to ﬁnd the force seen by an observer at inﬁnity to be

a 1 1 mRS 1 GMm F = (F F ) 2 = = (3.23) S a q 2 q 2 RS 2r 2GM r 1 − r 1 − r 24 which is the correct force required to hold a particle near the event horizon as seen from an observer at infinity. Similarly for a Reissner-Nordströmblack hole, we find   2 m RS 2RQ 1 FRN = −   . (3.24) 2 3 q R2 2 r r Rs Q 1 − r + r2

Asymptotically we should see the charged object tend towards a Coulomb ﬁeld, so

2 Q2 2 Q2G we identify k = 4π which allows us to identify RQ as 4π where Q is the object’s charge. Thus we ﬁnd the force is given by   GMm mQ2G 1 F = − (3.25) RN 2 3 q  r 4πr 2GM Q2G 1 − r + 4πr2

We see that as a particle approaches the horizon as a black hole tends to extremity, a repulsive gravitational force contributes. By examining (2.19) and noting the entropy gradient is proportional to the normal vector, we can calculate the entropy associated with each black hole. For the Schwarzschild black hole, we ﬁnd at the horizon Z dr SS ∝ lim q ∝ RS ln(RS) (3.26) r→RS RS 1 − r which is in disagreement with the expected entropy for a Schwarzschild black hole,

2 πRS S = 4 . Upon doing the same calculation for the Reissner-Nordstr¨omblack hole and taking the limit as the outer event horizon, we ﬁnd Z dr 2 2 SRN ∝ lim ∝ RS ln(R − 4R ). (3.27) q 2 S Q r→RRN R RS Q 1 − r + r2

This is in disagreement with other calculations that have been done for the entropy of a Reissner-Nordstr¨omblack hole. CHALLENGING VERLINDE GRAVITY

Newtonian Gravity Is Not Entropic At this point, we would like to present ambiguities that arise in Verlinde’s treatment of gravity for both the Newtonian case and relativistic extension. Eq.(2.13) is the basis for our challenge to this theory. This tells us that the mass can be found by integrating the potential gradient over these equipotential surfaces. One can then use Gauss’ law to show 1 Z M = ∇2φdV. (4.1) 4πG V

The left hand side, of course, can be written as Z M = ρdV (4.2) V where the integration is over the volume enclosed speciﬁcally by the equipotential surface. Verlinde then claims that the Poisson equation follows from equating the integrands. However, since Gauss’ law implies closed surfaces, any modiﬁcation to the Poisson equation given by

∇2φ + u · ∇f(φ) = 4πGρ (4.3) where u is divergence-free and f(φ) is any arbitrary function of φ, will allow you to equate (4.1) and (4.2) at the integral levels. The function f(φ) is any arbitrary function of φ. This can be shown by integrating (4.3) as 26

Z Z (∇2φ + u · ∇f(φ))dV = (∇2φ + ∇ · (uf(φ))dV (4.4) V V Z Z = ∇2φdV + f(φ)u · dA V ∂V Z Z = ∇2φdV + f(φ) u · dA V ∂V Z = ∇2φdV V

where we have used the fact that the surface integral is done over equipotential surfaces and any divergence-free vector integrated over a closed surface in this way will not contribute. On these surfaces, f(φ) = constant. Verlinde’s use of equipotential surfaces and this property of closed surfaces tells us that while you can equate (4.1) and (4.2) at the integral levels, one can not justify the claim that the Poisson equation, at the integrand level, holds.

General Relativity Is Not Entropic Our challenge to the relativistic derivation runs along the same lines. Going into further detail, (2.20) can be written as Z 1 φ b M = e ∇bφN dA (4.5) 4πG ∂Σ where N b is a spacelike unit vector normal to the two-dimensional surface ∂Σ and parallel to ∇bφ. It can be shown that

−2φ b ∇aφ = e ξ ∇bξa. (4.6)

Thus (4.5) becomes Z −φ b a 4πGM = e ξ ∇bN dA. (4.7) ∂Σ 27

We now use the Killing equation ∇aξb + ∇bξa = 0 to write (4.7) as Z 1 −φ [b a] 4πGM = e ∇[bξa]ξ N dA. (4.8) 4 ∂Σ

We can now use Stokes’ theorem Z Z 1 ba ab BbadS = ∇b(B )dΣa (4.9) 2 ∂Σ Σ

ba where Bba is antisymmetric, dS is an antisymmetric surface element, and Σ is the 3-dimensional open volume bounded by the 2-dimensional surface ∂Σ. We identify 1 B = ∇ ξ , dSba = e−φξ[bN a]dA. (4.10) ba 2 [b a]

ab Computing ∇b B gives us 1 ∇ (Bab) = ∇ (∇[aξb]) (4.11) b 2 b b a = −∇b∇ ξ

a b = Rb ξ

c c m where we have used the Killing equation and [∇a, ∇b]A = R mabA . The volume element dΣa can be written nadΣ where na is normal to this spacelike volume Σ. We now have Z b a Rabξ n = 8πGM. (4.12) Σ

Assuming the Komar mass is justiﬁed, we ﬁnd Eq. (2.23) follows. This can be written as Z b a Θabξ n dΣ = 0 (4.13) Σ

1 a where Θab = Rab − 2 gab − 8πGTab. Verlinde claims that one can vary the normal, n , arbitrarily to determine all components of the Einstein equations at the integrand 28 level. However, because you contract Θab with the Killing vector, any vector orthogonal to ξb will vanish. In other words, one can claim the ﬁeld equations are given by 1 R − g R + f(R,T )∇ U(φ)∇ V (φ) = 8πGT (4.14) ab 2 ab a b ab where U(φ) and V (φ) are functions of the holographic surfaces. At the level of the

b integration, you would ﬁnd yourself back at (4.12) (recall ξ ∇bφ=0). Furthermore,

α the two-dimensional surface ∂Σ admits two vectors, ea , parallel to the surface and 2 b P α β orthogonal to ξ . One could add terms such as Cαβea eb , where α,β=1 C = C(R, φ, T, etc), without changing the equations at the integral level since

α a ea ξ = 0. Finally, one may wonder why the cosmological constant fails to appear. This is in contrast to a glaring issue that arises by introducing the Komar mass at (2.23). Verlinde claims that the particular combination of the energy-momentum tensor required to write M in (2.22) as a volume integral involving Tab is determined by coupling the conservation laws associated with both sides of this equation at the integrand level. This is in reference to the fact that

ab ∇bT = 0. (4.15)

However, Lovelock [16] proved that in N = 4 dimensions, any tensor of the form

ab ab A = A (gmn; gmn,r; gmn,rs) that is divergence-free in the second index (that is

ab A ;b = 0) can be written as

Aab = αGab + βgab (4.16)

ab ab 1 ab where α and β are constant and G = R − 2 g R is the Einstein tensor. In other words, Verlinde’s argument at this point is equivalent to imposing that the Einstein 29

ﬁeld equations hold, cosmological constant included. Furthermore, if the cosmological constant must be included, then the Komar mass is no longer valid as it only holds for asymptotically ﬂat spacetimes, which is not possible with Λ 6= 0. This is quite a Catch 22.

Final Remarks We have presented what we feel is a substantial challenge to Verlinde’s theory of entropic gravity. One might ask why can one combine the entropy-area law, a relativistic concept, with the Unruh observer, a quantum field theory concept, and arrive at a classical law, Newtonian gravity. It has been shown [13] that one need not invoke the Unruh observer nor the Holographic principle in the first place to derive a boundary theory of Newtonian gravity. This is probably a good thing considering others [14] have shown that non-negligible changes to F = ma can be found if one considers an Unruh observer undergoing centripetal acceleration. The derivation of the Einstein equations is a bit more concerning. One might ask on what grounds can we justify modifying (4.12) with functions involving φ. The potential, φ, be it in the equipotential surfaces of Newtonian gravity or the function describing the constant redshift surfaces of relativistic gravity, are abstract concepts. In this theory, these surfaces are promoted to actual physical objects - bookkeepers of “information”. They seem to be no less fundamental to this theory than the Riemann or energy-momentum tensor. If this can be reconciled, one might also ask “why Einstein?” This derivation relied on the entropy-area law of Bekenstein. However, Visser [15] has shown, using Euclideanization techniques, that entropy-area giving Einstein is not an if and only if statement. The entropy-area relationship can be shown to come about in (Riemann)2 gravity as well as Einstein gravity. The issue with the cosmological constant seems to be the most troubling in 30 terms of attempting to derive Einstein gravity from a wholly thermodynamic approach. Jacobson [12] was able to derive the full Einstein equations, cosmological constant included, using thermodynamic principles. If Verlinde wants to derive a theory of gravity from thermodynamics and holography, at no point is it fair to make assumptions that hold only if Einstein gravity is true in the first place, which is what we believe is happening by use of the Komar mass and tensor conservation laws. There are clear indications that there is a relationship between gravity and thermodynamics as shown by Bekenstein, Hawking, and others. Einstein’s equations may very well be the thermodynamic limit of some unknown phenomenon. However, we feel there are too many issues with Verlinde’s theory to believe that this is a valid reformulation of gravity. REFERENCES [1] E. Verlinde, On the origin of gravity and the laws of Newton, J. of High Energy Phys. JHEP04(2011)029

[2] S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 31 (1973) 161

[3] H. Stephani, Relativity 3rd. Ed., Cambridge University Press, Cambridge U.K. (2004)

[4] R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, University of Chicago Press, Chicago (1994)

[5] W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870

[6] V.F. Mukhanov, S. Winitzki, Introduction to Quantum Eﬀects in Gravity, Cambridge University Press, Cambridge U.K. (2007)

[7] M. Maggiore, A Modern Introduction to Quantum Field Theory, Oxford University Press, New York (2005)

[8] J.D. Bekenstein, Black Holes and Entropy, Phys. Rev. D 7 (1973) 2333

[9] J.D. Bekenstein, Generalized second law of thermodynamics in black-hole physics, Phys. Rev. D 9 (1974) 3292

[10] R.M. Wald, General Relativity, University of Chicago Press, Chicago (1984)

[11] L. Susskind, J. Lindesay, An Introduction to Black Holes, Information, and the String Theory Revolution, World Scientiﬁc Publishing Co., Singapore (2005)

[12] T. Jacobson, Thermodynamics of Spacetime: The Einstein Equation of State, Phys. Rev. Let. 75 (1995) 1260

[13] S. Hossenfelder, Comments on Comments on Comments on Verlinde’s paper “On the Origin of Gravity and the Laws of Newton”, arXiv:1003.1015v1 [gr-qc]

[14] M. Duncan, R. Myrzakulov, D. Singleton, Entropic derivation of F=ma for circular motion, Phys. Lett. B 703 (2011) 516

[15] M. Visser, Dirty black holes: Entropy versus area, Phys. Rev. D 48 (1993) 583

[16] D. Lovelock, The Four-Dimensionality of Space and the Einstein Tensor, J. Math. Phys. 13 (1972) 874 California State University, Fresno

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