Gravitational Law in Extra Dimensions

Theoretical Physics

SA104X Degree Project in Engineering Physics KTH Stockholm

Michael Bühlmann 890724-T296 [email protected]

Supervisor: Tommy Ohlsson

Department of Theoretical Physics Royal Institute of Technology KTH Stockholm, Sweden

May 21, 2013 Abstract

Some recent theories which try to amend shortcomings of current models in physics suggest the existence of additional dimensions. Such extra dimensions would modify the inverse square law of gravity. A short overview over gravitational theory is presented and some of the extensions to general relativity and models which use extra dimensions, so-called Kaluza–Klein theories are discussed. A derivation of the correction to Newton’s gravitational law due to extra dimensions is performed and yields an additional term of Yukawa-type. We determine its interaction range and strength and derive the gravitational constant in the extra dimensional space. Experiments which probe Newton’s inverse square law are presented and recent results and constraints on corrections of Yukawa-type are discussed. This report is part of my bachelor degree project in theoretical physics at KTH. Contents

1 Introduction 4 2 Development of Gravitational Theory 6 2.1 Newtonian Gravitation ...... 6 2.2 General Relativity ...... 7 2.3 Beyond General Relativity ...... 7

3 Extradimensional Theories 9 3.1 Kaluza–Klein Theories ...... 9 3.1.1 UED: Universal extra dimensions ...... 9 3.1.2 ADD Model ...... 10 3.1.3 RS Model: Warped extra dimensions ...... 10 3.2 String Theory ...... 10

4 Gravitational Law in Extra Dimensions 12 4.1 Derivation of the gravitational law in (4 + n) dimensions ...... 12 4.1.1 General form of the potential ...... 12 4.1.2 Limit cases ...... 13 4.1.3 Deviation from Newton’s Law ...... 13 4.1.4 Compactiﬁcation on spheres and warped extra dimensions ...... 14 4.2 Discussion ...... 15

5 Experimental Evidence 19 5.1 Types of experiments ...... 19 5.1.1 Astronomical measurements ...... 19 5.1.2 Eötvös-type experiments ...... 19 5.1.3 Cavendish-type experiments ...... 20 5.1.4 Casimir force measurement ...... 20 5.2 Constraints obtained by current experiments ...... 20

6 Summary 23 A Appendix 24

3 1. Introduction

The aim of physics is to model our world under all kind of different circumstances ranging from the smallest scales in particle physics up to the evolution of stars, galaxies and the universe as a whole. Such descriptions of our nature are obtained by doing observations and experiments and trying to fit these data to mathematical models. The models are verified against other experiments and depending on the result discarded, improved or verified again. During the last centuries more precise descriptions of nature, which are valid within a longer range of circumstances, have been obtained by this process. The precession of the perihelion of Mercury for example can correctly be described by general relativity but not by Newton’s gravitational law and the predictions of the Standard Model were experimentally confirmed with good precision. However there are some observations which can not be described by the Standard Model. By measuring rotation curves of galaxies it was shown that the mass in the galactic plane must be more than the material that could be seen [5]. This kind of matter is nowadays known as dark matter, but the Standard Model does not predict any kind of such matter. Also it is not known why gravity is such a weak force compared to the other fundamental interactions. This is also known as the hierarchy problem. Another problem within the Standard Model is related to the mass of neutrinos, namely that they are supposed to be massless, whereas observations of neutrino flavor oscillations indicate that this is not the case [11]. The Standard Model also does not include gravitational interaction. In order to overcome these shortcomings of the Standard Model physicists proposed many different extensions. Two of the most famous examples are supersymmetry models [19], where each particle has a heavy so-called superpartner, and grand unified theories which have the goal of describing all the interaction in the Standard Model as a manifestation of a single unified interaction [18]. Another approach which has been made is the introduction of additional spatial dimensions, so-called extra dimensions. Even though we only perceive three spatial and one time dimension, there could be more than those, provided that the extra dimensions are hidden from us by some mechanism. The introduction of extra dimensions can account for some of the shortcomings of the Standard Model, as we will see. As written above, physicists do experiments to verify their models. We will see that the introduction of extra spatial dimensions leads to deviations in the gravitational law. Hence one way of testing our world on extra dimensions is to measure the gravitational force with high precision. In this way constraints on the nature of the extra dimensions can be obtained and certain models can be discarded.

This report will give a short overview on extradimensional theories, their implications on gravity and the current experimental situation. The outline is as follows. We will start in chapter 2 with a short historical review on how our understanding of gravitation developed during the last centuries. We will also discuss why physicists are looking for theories beyond

4 1. INTRODUCTION general relativity and with what kind of attempts they have come up with. In chapter 3 we will focus on a few modern theories which use extra dimensions and point out some of the implications they bring along. We then move on to the main topic of this report, the derivation of the gravitational law in extradimensional spaces (chapter 4). We look at different limit cases and compare different kind of models and geometries for the additional dimensions. In chapter 5 we will discuss different kind of experiments which probe nature for deviations in the gravitational law and what constraints they set on the existence of extra dimensions. Finally in chapter 6 we will summarize the results of the previous chapters and draw our conclusions.

5 2. Development of Gravitational Theory

Modern research on gravitation began with Galileo Galilei (1564-1642) and his observations on falling objects. Galileo showed that all objects accelerate equally fast under gravitation, which stood in contradiction with the Aristotelian belief that heavy objects fall faster. However, a mathematical description of this phenomenon was only available about a century later thanks to Newton.

2.1. Newtonian Gravitation

The Philosophiae Naturalis Principia Mathematica by Isaac Newton [24] revolutionized not only the understanding of gravitation but physics and mathematics as a whole. The initial reason for Newton to write it was a question from the English astronomer and physicists Ed- mond Halley about whether an inverse square force directed towards the Sun could account for the elliptical orbits of the planets [15]. In order to answer this question he first had to develop mathematical tools (today known as calculus) and a new theory of mechanics, basing on the three newtonian laws. Using those ideas Newton showed in his principa that a body upon which an inverse square force is acting indeed moves on an ellipse and obeys the laws that Kepler had discovered by observation. Not only did Newton explain the orbits of the planets, but he also showed that the same force was responsible for the orbit of the Moon, the tides in the sea and the fall of an apple off a tree. The gravitational force is an universal force which affects everything that has mass. Experimental data revealed that the force between two point-like objects is proportional to their (gravitational) masses m1 and m2 and inverse proportional to the square of the relative distance r. Introducing a proportionality constant G, the gravitational constant, we can write the force as m m F⃗ = G 1 2 r,ˆ (2.1) r2 where rˆ points from the first object to the second. We can write the gravitational force as the gradient of a potential which makes cal- culations easier, since we are only dealing with a scalar field instead of a vector field. In the following sections we will denote the classical newtonian potential as V4 since it is the potential of the 4-dimensional space-time1. It reads G m m V (r) = − 4 1 2 , (2.2) 4 r

1Actually this potential is just an approximation for a ﬂat space-time with a weak gravitational ﬁeld, see section 2.2.

6 2. DEVELOPMENT OF GRAVITATIONAL THEORY

where G4 = G is the gravitational constant in the 4-dimensional space-time. In chapter 4 we are going to see that when we introduce more dimensions, the gravitational constant will also change. Many people asked themselves whether the gravitational force is a pure inverse square law and why this might be the case. In 1873 the French mathematician Joseph Louis François Bertrand proved using perturbation theory, that only the inverse square law and Hooke’s law (harmonic oscillator) produce stable and closed orbits of the planets [6] which agreed with astronomical observations. Furthermore several experiments have been done (see chapter 5) to probe Newton’s theory and for most of the circumstances it seems to hold rather precisely. However, the anomalous advance of Mercury’s perihelion could not be explained until Albert Einstein came up with his famous theory of general relativity.

2.2. General Relativity

In 1905 Albert Einstein published his theory on special relativity which greatly changed our understanding of space and time. It made the laws of electromagnetism valid in all inertial frames, i.e. it removed the need of an absolute reference frame in which electromagnetic waves travel through a so-called ”aether”. Two years later Einstein started thinking about how to incorporate gravity into the relativistic framework. In a thought-experiment he argued that free fall is inertial motion and that there is an equivalence of a gravitational field and the corresponding acceleration of the reference frame. This argument is called the equivalence principle. Until 1915 Einstein continuously developed his theory of general relativity and incorpo- rated new mathematical tools like Riemannian differential geometry developed by Bernhard Riemann in the late 19th century. In 1915 he published his general theory of relativity in the form in which it is used today [9]. In this theory gravitation itself is not a miraculous force by itself but an effect of the distortion of space-time by matter, affecting the inertial motion of other matter. The interaction of gravity with matter is described by the Einstein field equations, a set of ten partial differential equations. Einstein predicted that light would bend in the neighborhood of a massive object and time would run slower in a strong gravitational field, which both has been experimentally verified [15]. He also successfully calculated the advance of the perihelion of mercury, yet the theory only started to be widely used after 1960 when physicists began to understand the concept of black holes [15]. Einstein’s theory incorporates Newton’s gravitational law as a limit case of weak gravitational fields and large distances. The inverse square law appears naturally with no other exponent than 2 possible.

2.3. Beyond General Relativity

After the success of unifying special relativity with gravity physicists tried to unify other fundamental theories and forces. In 1921 Theodor Kaluza extended General Relativity to ﬁve dimensions in order to include electromagnetism in the same theoretical framework [12]. Continuing in this direction, Oscar Klein proposed a few years later that the fourth spatial dimension is curled up into a small, unobservable circle. In this so-called Kaluza–Klein (KK) theory, the gravitational curvature of the extra spatial dimension behaves as an additional force similar to electromagnetism. By separating the resulting equation both Einstein’s ﬁeld

7 2. DEVELOPMENT OF GRAVITATIONAL THEORY equations and Maxwell’s equation can be retrieved. These and other models of electromagnetism and gravity were pursued by Albert Einstein in his (unsuccessful) attempts at a classical unified field theory (see [12] for an overview of the different approaches considered by Einstein). In the sixties Sheldon Glashow, Abdus Salam and Steven Weinberg successfully developed a unified theory of electromagnetism and weak interaction, the so-called electroweak theory [14]. Since then the electroweak theory became a template for further attempts at unifying forces. Several proposals for so-called Grand Unified Theories have been made, although none is currently universally accepted. A problem for experimentally verifying such theories is that very high energies are involved which are beyond the reach of current accelerators. A main difficulty is the formulation of a quantum gravity theory. Currently, there is still no complete and consistent quantum theory of gravity, although several models have been suggested (for a historical overview see [27]). One candidate is string theory and M- theory (section 3.2), where point particles are exchanged with one dimensional strings in an 11-dimensional space. String theory promises to be a unified description of all particles and interactions, however the huge number of solutions, called string vacua, and the lack of possibilities to verify the theory, is criticised [29].

8 3. Extradimensional Theories

3.1. Kaluza–Klein Theories

Nowadays people use Kaluza–Klein theory (KK theory) to refer to any theory with extra spatial dimensions. In this section we will shortly discuss what kind of different Kaluza– Klein theories exist and which implications the existence of extra dimensions would have. A general feature of KK theories is the so-called compactification of the additional dimensions. The extra dimensional space is not infinite as the 4-dimensional space-time, but can be described as a compact manifold. For example we can wind up each of the extra dimensions on a circle with radius Ri, in which way we obtain a generalized torus. Or we can consider the extra dimensions to form a generalized sphere. One possibility for explaining why we cannot perceive the hypothetical extra dimensions is that the compactification scale is simply too small in order to directly notice them.

3.1.1. UED: Universal extra dimensions The universal extra dimensions (UED) model was proposed in 2001 by Thomas Appelquist, Hsin-Chia Cheng and Bogdan Dobrescu [2]. It describes a higher-dimensional version of the SM where all particles are allowed to propagate in the extra dimensions. Since particles have more freedoms to move they will also have additional contributions to their kinetic energy. However, we cannot observe the motion of the particles along the extra dimensions and thus we would interpret this kinetic energy as an additional part of its rest energy, i.e. its mass. In compact spaces a particle can only obtain discrete momenta. This implies that for a Standard Model (SM) particle, where the particle is assumed to be at rest in the extra dimensions, discrete versions with higher rest mass exist. These heavier particles are called KK particles and each SM particle has its own KK tower of heavier versions. For example if we consider n extra dimensions compactiﬁed on a torus each with radius R and a SM particle with rest mass m, the rest masses of its corresponding KK particles are [20]

l2 m2 = m2 + , (3.1) l R2 where l is called the KK number. This equation can be generalized to the case where we n have diﬀerent radii Ri. By using an integer vector l ∈ Z the mass of the KK particle can be written as

∑n l2 m2 = m2 + i . (3.2) l R2 i=1 i We will see this equation again in chapter 4.

9 3. EXTRADIMENSIONAL THEORIES

An experimental bound on the size of the compactification radius R is given by the fact that those KK states have not been detected in collider experiments in the TeV energy range. Their masses would thus have to be greater than a few TeV, which implies a strong constraint on the compactification radius, i.e R ≤ 1 ≈ 10−20m. It is very unlikely to Ecollider find experimental evidence in such tiny dimensions. Each particle is also assigned an additional conserved number, the KK parity which can be +1 (even KK number) or −1 (odd KK number) [21]. Hence all the SM particles have KK parity +1 (KK number is 0). The lightest KK particles (those with a KK number of 1) have a parity of -1 and are stable, because the only decay which is allowed by energy conservation would be a decay into SM particles. But SM particles have the opposite parity and thus the decay is forbidden. This property makes the lightest KK particles possible dark matter candidates, so-called Kaluza–Klein dark matter, and is a basic motivation for the UED model.

3.1.2. ADD Model The ADD model was first proposed in 1998 by Nima Arkani-Hamed, Savas Dimopoulos and Gia Dvali [3]. It assumes that the SM particles are confined to a 4-dimensional subspace, a so-called brane, which corresponds to the classical 4-dimensional space-time. A theory confining our world into a brane is called a braneworld theory. Since SM particles only live in the brane they cannot penetrate the space outside of it, the so called bulk. Hence there are also no KK particles as in the UED model and dimensions are allowed to be larger. The ADD model is also referred as the model with large extra dimensions. Gravity itself is not part of the SM and therefore can penetrate the whole space. Within this model the weakness of the gravitational force compared to the other fundamental forces could be explained by the fact that it is spread out in a larger space. In the ADD model, the n extra dimensions are usually compactified on the n-dimensional torus T n and for simplicity it is assumed that all radii have the same value [20].

3.1.3. RS Model: Warped extra dimensions In 1999 Lisa Randall and Raman Sundrum proposed a 5-dimensional model, the so-called RS model [25]. It is similar to the ADD model described above but in the RS model the additional dimension is not flat as in the ADD-model, but curved. This is why it is said to have warped extra dimensions. The extra dimension is compactified on a circle where the upper half is identified with the lower half. The particles are confined to the two points where the two halves meet, i.e. on each of these two boundary points stands a four dimensional world like the one we’re living in. By imposing the principles of general relativity on this setup, a natural explanation on the weakness of gravity, i.e. the hierarchy problem, arises.

3.2. String Theory

Another framework which uses extra dimensions is string theory (see e.g. [4]). However it is very different from the extra dimensional theories mentioned above. In string theory point particles are exchanged by one dimensional objects called strings. The different particles that we can observe are different quantum states of the strings. To be mathematically

10 3. EXTRADIMENSIONAL THEORIES consistent, additional dimensions have to be introduced. Those are usually compactified on a very small scale. Different versions of string theory have been developed, but nowadays the so-called M- theory is the one considered most, as it contains all the other theories as limiting cases. M-theory itself incorporates eleven dimensions and is a candidate for a theory of everything, as it is a self-contained mathematical model which describes all the fundamental interactions and different types of matter. As already mentioned in section 2.3, many physicists are criticising string theory as it lacks verifiability and predicts an infinite amount of solutions.

11 4. Gravitational Law in Extra Dimensions

4.1. Derivation of the gravitational law in (4 + n) dimensions

In this section we derive the corrections to the gravitational force due to compactified extra dimensions. We follow the derivation performed in [16], but with some more details and clarifications. Let (n + 4) be the dimension of our extended space time, where we have n extra dimensions denoted by yi, i = 1, 2, . . . , n. In this derivation we assume that the extra dimensions are compactified on a n-dimensional torus T n. A more generic solution an arbitrary n- dimensional compact manifold is described in [16] and some results are shortly mentioned in subsection 4.1.4. We will start with the examination of the gravitational potential in (4 + n) dimensions of a massive object in the Newtonian limit (flat space-time, (3+n) spacial and one time-like dimension) and then impose compactification on the result.

4.1.1. General form of the potential Let us assume that we have a massive point-like object with mass M at position r⃗ = (⃗x,y1, y2, . . . , yn), where ⃗x is the coordinate in the three base dimensions, and let us derive | | the gravitational∑ potential at the origin. We can write the distance to the object as r = r⃗ = 2 n 2 1/2 (⃗x + i=1 yi ) . We assume that the gravitational potential of the object must satisfy the (n + 3) dimensional Poisson equation

3+n △n+3Vn+4 = 4π Gn+4 ρ = 4π Gn+4M δ (r⃗) , (4.1) which is equal to zero as long as r⃗ is not zero. We assume that the resulting potential is spherical symmetric, hence all the derivatives along the non-radial directions become zero. The laplacian in (n + 3) dimensions then reads

∂2 N − 1 ∂ △ = + . (4.2) n+3 ∂r2 r ∂r By simply plugging in we can verify that G M V = − n+4 (4.3) n+4 |r⃗|n+1 is a solution to equation (4.1). However, the solution does not satisfy our boundary conditions. We want it to be periodic in yi, i.e. we want it to be independent under the shifts yi → yi + 2πRi. This can

12 4. GRAVITATIONAL LAW IN EXTRA DIMENSIONS be obtained by writing equation (4.3) as ∑ Gn+4M Vn+4,c = − ∑ , (4.4) 2 n − 2 (n + 1)/2 m∈Zn (⃗x + i=1(yi 2πRimi) ) where m is a vector in the n-dimensional lattice Zn. We use the index c for the potential to indicate that this is the potential in the compactiﬁed space.

4.1.2. Limit cases

We can shortly check the result by looking at very large Ri’s, i.e. at the limit Ri >> |⃗x|. In that case only the term with m = 0 survives, since for all the other terms the denominator goes to inﬁnity and we get equation (4.3) back. Let us look at the other case when the Ri’s are small compared to |⃗x|. We substitute zm = y − 2πRm, where Rm = (R1m1,R2m2,...,Rnmn). Note that the zm lie on a lattice and each cell of that lattice has the volume (2πR1)(2πR2) ... (2πRn) = ST n , which is the volume of the n-dimensional torus. Now we can rewrite the potential as ∑ ∫ Gn+4M 1 Gn+4M 1 n V = − · S n ≈ − d z, n+4,c (n + 1)/2 T (n + 1)/2 ST n 2 2 ST n Rn 2 2 m∈Z (x + zm) (x + z ) where we used that the volume of the torus is small compared to x. We can further calculate this integral by switching to spherical coordinates and using the surface of the (n-1)- n−1 n/2 n −1 dimensional unit sphere S given by SSn−1 = 2π Γ( /2) : ∫ (√ ) ∞ n−1 n G M S n−1 z G MS n−1 π Γ( ) 1 V ≈ − n+4 S dz = − n+4 S 2 · n+4,c (n + 1) 2 n+1 n 2 2 / n ST 0 (x + z ) ST 2 Γ( 2 ) x G MS n 1 = − n+4 S · (4.5) 2 ST n x Comparing equation (4.5) with the classical gravitational potential in three dimensions, we see that the relation between G4+n and the standard gravitational constant G4 is given by

2 ST n G4+n = · G4 . (4.6) SSn

4.1.3. Deviation from Newton’s Law To obtain a more accurate result we have to start again with equation (4.4). We use a special form of the poisson summation formula to simplify the n-dimensional sum. The general formula is given and proved in Appendix A. In our case x = yi and L = −2πRi. We can apply the formula for all n extra dimensions to get ∫ G M ∑ eiym˜ V = − n+4 e−iym˜ dny, n+4,c 2 2 (n + 1)/2 ST n Rn (x + y ) m∈Zn where we introduced m˜ := ( m1 ,..., mn ). Note that |m˜ |2 is exactly the additional mass R1 Rn term of the KK particle as occurring in equation (3.2).

13 4. GRAVITATIONAL LAW IN EXTRA DIMENSIONS

We can switch to spherical coordinates, with y denoting the radius and ϕ1, . . . , ϕn−1 denoting the angles, where ϕ1 is the angle between y and m˜ . The integral only depends on y and ϕ1. Thus we can perform the integral over all the other angles and get the surface of the (n-2)-dimensional unit sphere SSn−2 . Then the potential looks like ∫ ∫ ∑ ∞ n−1 π Gn+4MS n−2 − y | | − V = − S e iym˜ dy dϕ eiy m˜ cos ϕ1 sinn 2(ϕ ) . n+4,c 2 2 (n + 1)/2 1 1 ST n (x + y ) m∈Zn 0 | 0 {z } (I) Let us focus on the last integral. Using the formula 8.411 (7) from [13] we can write it as √ n−2 1 1 n n−1 Γ( + ) Γ( ) 2 2 πΓ( ) − n 2 2 2 2 2 +1 (I) = − J n−2 (y|m˜ |) = n J n − (y|m˜ |) y ( ) n 2 −1 2 1 y|m˜ | 2 2 2|m˜ | 2 2 n n 2 2 π 2 − n 2 +1 = n J n − (y|m˜ |) y , −1 2 1 SSn−2 |m˜ | 2 n where J n − (z) is the the Bessel function of order ( − 1). The potential becomes 2 1 2

∫ n n n ∑ −iym˜ ∞ 2 2 y 2 J n − (y|m˜ |) Gn+4Mπ 2 e 2 1 V = − n dy . n+4,c −1 2 2 (n + 1)/2 ST n |m˜ | 2 (x + y ) m∈Zn | 0 {z } (II) We can calculate the integral using formula 6.565 (3) from [13] √ n −1 −x|m˜ | |m˜ | 2 πe (II) = n n+1 2 2 Γ( 2 )x

(n+1)/2 SSn π Putting everything together and remembering that G4 = G4+n = G4+n (n + 1) we 2ST n Γ( /2)ST n obtain G M ∑ G M ∑ V = − 4 e−x|m˜ |e−iym˜ = − 4 e−x|m˜ |, (4.7) n+4,c x x m∈Zn m∈Zn where in the last step we set y = 0 for further simplifying the result. We can get a ﬁrst approximation by taking only the largest terms, i.e. the summands with |m| = 0 and |m| = 1. This corresponds to the 0th and 1st KK state (see section 3.1). We get ( ) G4M − x V ≈ − 1 + 2d · e Rm , (4.8) n+4,c x m where Rm is the largest compactiﬁcation radius and dm the corresponding degeneracy, i.e. how many compact dimensions with radius Rm exist.

4.1.4. Compactification on spheres and warped extra dimensions How compactification is done on any arbitrary compact manifold using representations of fundamental groups is described in [16]. If we compactify the n extra dimensions on a n-sphere with radius R instead of an n-torus we obtain a slightly different result: ( ) G M √ V = − 4 1 + (n + 1)e− nr/R . (4.9) sphere x

14 4. GRAVITATIONAL LAW IN EXTRA DIMENSIONS

Another type of model considers non-compact but warped extra dimensions (see RS model, subsection 3.1.2). In this case an approximation to the gravitational potential is given by [25]: ( ) G M 2 V = − 4 1 + , (4.10) warped x 3k2x2 where 1/k is the so-called warping scale.

4.2. Discussion

In this section we will shortly summarize the results in the previous section and visualize them in order to compare the diﬀerent approximations and see how the potential depends on the structure of the extra dimensions. We showed the following behavior of the gravitational potential for a point-like mass in a toroidal compactiﬁed potential:

• As x ≫ R, that is the compactiﬁcation radius is tiny small compared to the inter- acting distance, the potential is the same as the classical 4-dimensional potential in equation (2.2). The extra dimensions seem to be non-existent.

• As x ≪ R, that is the compactiﬁcation radius is very large compared to the inter- acting distance, the potential is the same as the potential in a space where the extra dimensions are not compactiﬁed, see equation (4.3)

• As x ≈ R we derived an approximation of the potential which has an additional exponential term that depends on the geometry of the extra dimensions and the com- pactiﬁcation scale.

For compactiﬁed extra dimensions the approximation of the potential of a point-like mass M can be written as G M ( ) V = − 4 1 + αex/λ . (4.11) x The additional exponential term in the potential is also called Yukawa potential after the theoretical physicist Hideki Yukawa (1907-1981). Here α is called the interaction strength of the force and λ the interaction length. For toroidal compactiﬁcation we obtained in the previous section

α = 2dm λ = Rm, (4.12) where Rm is the largest radius of the torus and dm its multiplicity. For spherical compacti- ﬁcation it can be shown that [16] √ n α = n + 1 λ = . (4.13) R Non-compactiﬁed but warped extra dimensions behave like 1/x3 for x ≪ 1/k.

15 4. GRAVITATIONAL LAW IN EXTRA DIMENSIONS

We also derived a connection between the 4-dimensional and (4+n)-dimensional gravitational constant for toroidal compactiﬁcation:

∏n 2 ST n G4+n = G4 ∝ G4 Ri (toroidal) (4.14) S n S i=1 ∏n G4+n = G4 · 2 Ri (spherical). (4.15) i=1 The derivation of the second equation for spherical compactification can be found in [16]. In the next step we will graphically compare the different limiting cases and our approximation with the exact result. For simplicity we choose n = 1 for our extra dimension and compactify it on a circle with radius R. To avoid to choose a specific R we divide our x R V (x) equations by R and express the distance in /R and the potential in V · /MG4 = /V4(R). The resulting potential is drawn in Figure 4.1. We can see that the exact solution (equation (4.7)) behaves as 1/x for x ≫ R (classical potential in a 4-dimensional space- time) and 1/x2 for x ≪ R (classical potential in a 5-dimensional space-time) as expected. Our approximation obtained in equation (4.8) describes the exact solution quite well in the region around x = R. This region is also where we would expect to see first deviations in experiments, thus it is justified to use this approximations for the evaluation of experimental data. In a next step we investigate how the potential changes if we change the number of extra dimensions and their geometry. We compare toroidal and spherical compactification both in two and three dimensions and contrast it with the potential of warped extra dimension(equation (4.10)). Figure 4.2 shows the different potentials. For toroidal compactification it is assumed that all extra dimensions have the same compactification radius and for the warped extra dimensions k is assumed to be 1/R. We see that for all the models the potential starts deviating from the classical potential in about the same region. More extra dimensions lead to a faster decrease as x gets smaller whereby the effect is larger for toroidal compactification than for spherical. Warped extra dimensions behave very similarly to compactified ones in the region where x ∼ R. Since possible deviations in experiments would be observed in this region, it would be very hard to determine the properties of the extra dimensions.

16 4. GRAVITATIONAL LAW IN EXTRA DIMENSIONS

. .Exact: (4.7) 101 . .Newton in 4d: (2.2) . .Newton in 5d: (4.2) . .Approximation: (4.8) ) R ( 4 V / V

100 . normalized potential:

10−1 . 100 101 normalized distance: x/R

Figure 4.1.: Comparison of diﬀerent approximations to the exact result in the region where x ∼ R for toroidal compactiﬁcation with n = 1. The approximation of equation (4.8) describes the behavior for x ≥ R quite well.

17 4. GRAVITATIONAL LAW IN EXTRA DIMENSIONS

103 . .Newton in 4d . .n = 2 toroidal compactification . .n = 3 toroidal compactification . .n = 2 spherical compactification ) 102 R . .

( n = 3 spherical compactiﬁcation 4

V . .

/ warped extradimensions V

101

100 . normalized potential:

−1 10 . 10−1 100 101 normalized distance: x/R

Figure 4.2.: Comparison of diﬀerent geometries of the extradimensional space. Compactiﬁed extra dimensions behave quite similarly, which makes it hard to experimentally determine the geometry of the extradimensional space. The potential for warped extra dimensions behaves as 1/x3 for x ≪ R.

18 5. Experimental Evidence

5.1. Types of experiments

There are several types of experiments which can be used to test the Newtonian gravitational law on diﬀerent interaction distances. In this section we shortly introduce the most important experiments, starting at largest scales.

5.1.1. Astronomical measurements The most obvious test of gravity that comes to our mind is the observation of planetary orbits. This kind of observations have been done for a long time with increasing precision, resulting in rigid constraints on hypothetical deviations from Newton’s inverse square law respectively general relativity. A very interesting experiment is the LLR (Lunar Laser Ranging) [1,10]. The LLR data consists of range measurements from telescopes on Earth to reﬂectors placed on the Moon by American astronauts and unmanned Soviet lander. The precision is in the range of a few centimetres, which allows a very exact measurement of the orbit of the Moon. This data can be compared to predictions by theory, where we also have to include general relativistic eﬀects and the non-spherical gravitational potential of the Earth [1].

5.1.2. Eötvös-type experiments In Eötvös-type experiments the difference between inertial and gravitational mass of an object is measured. This kind of experiments are done to verify the equivalence principle stated by General Relativity (see section 2.2). The existence of an additional force term which is not proportional to the objects mass would lead to a difference beween the gravitational mass and the inertial mass of the object [22]. Therefore the results of Eötvös-type experiments set certain constraints on hypothetical deviations from Newton’s gravitational law1. The classical setup of the Eötvös experiment are two different masses on the opposite side of a rod. The rod itself is hung up with a thin fibre. In the reference frame of the earth the forces acting on the system are the string tension, gravitation and the centrifugal force. The gravitational force is depending on the gravitational mass whereas the centrifugal force is caused by the rotating reference frame and thus depends on the inertial mass. Assuming that we are not doing the experiment at the equator or at the poles there will be a net torque on the system if the inertial mass is not proportional to the gravitational mass. Hence the rod will start to rotate and this rotation can be detected.

1Here I do not agree with this statement of one of my sources [22]. The Yukawa-type potential we derived in the previous chapter is in fact proportional to the gravitational masses, hence in my opinion this kind of deviation could not be detected by this experiment. A statement by V. M. Mostepanenko was not available before the deadline of this project

19 5. EXPERIMENTAL EVIDENCE

The ﬁrst experiment of this type was started in 1885 by the Hungarian physicist Loránd Eötvös (1848–1919) and has been greatly improved since then. The relative diﬀerence between the two masses is constraint to be less than 10−11 [26].

5.1.3. Cavendish-type experiments In these types of experiments the deviations of the gravitational force from Newton’s law are directly measured. The first experiment of this kind was performed by Henry Cavendish in 1798 [8]. It was the first measurement of the gravitational force and the gravitational constant in a laboratory. Cavendish used a torsion balance made of a rod with two spheres attached on both ends. This torsion balance was placed in between two other, heavier spheres which deflected the torsion balance until it was in equilibrium. By measuring the deflection Cavendish determined the density of the earth from which he further could calculate the gravitational constant. Nowadays similar experiments can be performed with much higher precision, using lasers to determine the deflection and a more complicated set-up. In this way the gravitational law can be verified over a large distance range and constraints on possible deviations can be set (see for example [28]).

5.1.4. Casimir force measurement The Casimir effect is a phenomenon occurring between uncharged metal plates placed a few micrometers apart in vacuum [17]. The effect was predicted by Casimir in 1948 [7] and can be described by quantum field theory in which all the fundamental fields, such as the electromagnetic field, are quantized at every point in space. At the most basic level, the field at each point in space is a simple harmonic oscillator. Since the quantized simple ℏω harmonic oscillator has a non-zero lowest possible energy E0 = 2 (the so-called zero-point energy), the vacuum itself contains energy. By placing the two metal plates, we constrain the allowed frequencies of the oscillator, resulting in a lower energy density within the plates. As the system tries to be in the state with the lowest energy, a force is pulling the two plates together. This force is known as the Casimir force and is the dominant background force at separations of a few micrometers and below [23]. By measuring the force acting between the two plates or other, more complex geometrical objects, and comparing them to theoretical values, we can set constraints on other force terms like the Yukawa term in the gravitational law.

5.2. Constraints obtained by current experiments

Results of recent experiments are presented in Figure 5.1 and Figure 5.2 where each line represents the constraints obtained by a single experiment. The regions of (λ,α) above the curves are ruled out, since the corresponding deviation of the force would have been larger than the error of the experiment2. Regions below the curves lie in the uncertainty region of the experiment and cannot be ruled out. Figure 5.1 is reproduced from [10], ﬁgure 2.13. The constraints on the far right are obtained by planet observations and the big bump is set by the LLR experiment (see sub-

2Usually the border is set to 2σ which yields a 95% conﬁcence level

20 5. EXPERIMENTAL EVIDENCE

102 101 100 . predicted for 1 extra dimension 10−1 −2 α 10 10−3 10−4 excluded by experiment 10−5 10−6 −7 interaction constant 10 10−8 permitted area 10−9 10−10 10−11 . 100 101 102 103 104 105 106 107 108 109 1010 1011 1012 1013 1014 1015 interaction range λ [m]

Figure 5.1.: Constraints on the interaction range λ and interaction constant α obtained by various large scales experiments and measurements. The dotted lines show the expected values for the ADD model with one extra dimension. They are in the ruled out area, meaning that this model is experimentally excluded. The constraint lines are reproduced from ﬁgure 2.13 in [10]. section 5.1.1). The constraints more on the left are large scale geophysical experiments and measurements. Figure 5.2 shows laboratory results. For references to the corresponding experiments see [22] for curves 1-5, 7 and 8 (same enumeration) and [23] for curve 6 (corresponds to curve 2 in Fig 1). The other, newer experiments discussed in [23] do not improve the constraints. Curves 1 and 2 show the best Eötvös-type experiments whereas 3 and 4 are obtained by Cavendish-type experiments. All the experiments which give insights on smaller interaction ranges (5-8) measure the Casimir force and compare it to theory. We see that for large interaction lengths λ the constraints on α are very strong. When we draw the expected values of α (equation (4.12) for 1 ≤ n ≤ 11) into the diagram we can already exclude interaction lengths above 1mm. But as λ decreases the constraints get weaker and allow the existence of extra dimensions. For the ADD model it is possible to get a rough estimate on the compactiﬁcation scale R depending on the number of extra dimensions by comparing Planck scales for the ADD model (see [20, 22]):

32 −17 R ≈ 10 n cm. (5.1) For one extra dimension we ﬁnd that R ≈ 1015cm and according to equation (4.12) λ = R and α = 2. By looking at Figure 5.1 we see that this possibility is excluded by the

21 5. EXPERIMENTAL EVIDENCE

1028 . .Eötvös experiments 8 . .Cavendish experiments 1023 . .Casimir force experiments

1018 α 7 6 1013 excluded by experiment 108 5

interaction constant 3 predicted by theory 10 4 3 . − 2 10 2 1 permitted area 3

10−7 . 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 103 104 interaction range λ [m]

Figure 5.2.: Constraints on the interaction range λ and interaction constant α obtained by various laboratory experiments. Lines 1 and 2 follow from Eötvös, lines 3 and 4 from Cavendish type experiments. Lines 5-8 are obtained from the measurements of the Casimir force. Values for α and λ in the region above the curves are ruled out by these experiments. Curves 1-5, 7 and 8 are reproduced from [22] (same enumeration) and curve 6 from [23] (corresponds to curve 2 in Fig 1). experiments and thus the existence of only one extra dimension is not possible at least with the models considered here. Considering n = 2 we obtain λ ≈ 1mm and α = 4. Those values are slightly above the curves in Figure 5.2, but because our λ is only a rough estimate this possibility cannot be ruled out. If however n = 3 we have λ ≈ 5nm and α = 6 (if we assume that all dimensions have the same compactiﬁcation scale). These numbers lie in the permitted area and hence models with three or more extra dimensions do not conﬂict with recent experiments. It is not possible to obtain similar constraints for the UED model, since the expected values for the interaction range λ are below 10−20m (see subsection 3.1.1). Experiments which measure the gravitational force on such tiny distances are still far away in the future if it will be possible at all.

22 6. Summary

We saw how our understanding of gravity evolved during the last centuries and why physicists are looking for theories beyond current models. The introduction of extra dimensions might account for phenomenons which we cannot describe with current models, like dark matter. Currently there are several models of extra dimensions under discussion, each with certain advantages. We derived the gravitational potential in an extradimensional space and then imposed toroidal compactification on it. We studied several limiting cases and did a first approximation which yielded an additional term to the Newtonian potential of Yukawa type. The resulting potential has the form G M ( ) V = − 4 1 + αex/λ , x with G4 the common gravitational constant, α the interaction strength and λ the interaction length. By our derivation of toroidal compactification we obtained α = 2dm and λ = Rm where Rm is the largest compactification radius and dm the amount of dimensions with compactification radius Rm. We also shortly looked at the parameters for other geometries of the extra dimensional space. Several experiments have been performed which probe Newton’s inverse square law on different length scales. We looked at some different types of experiments, namely those of Eötvös type, Cavendish type, measurement of the Casimir force and astronomical measurements and we saw how they constrain the interaction strength and range parameter of the Yukawa potential. Due to the constraints set by those experiments, the ADD model with only one additional dimension can be ruled out. Models with more than one extra dimensions however cannot be excluded by the current data. Coming experiments are expected to further strengthen the constraints, especially on very short interaction lengths.

23 A. Appendix

Poisson Summation Formula

In the derivation of the gravitational potential we used a speciﬁc form of the Poisson summation formula which I want to proof here.

Theorem. Assume f(x) ∈ S(R) is a Schwartz function. Then ∑ ∑ 1 2πikx f(x + L · n) = fˆ(k)e L (A.1) |L| m∈Z k∈Z where fˆ(k) is the Fourier transform of f(x). ∑ Proof. Deﬁne h(x) := n∈Z f(x + Ln). h(x) is a periodic function with period L, hence we can write it as its Fourier series. The Fourier coeﬃcients are ∫ L 1 − 2πimx h = h(x)e L dx m |L| 0 ∫ ∑ L 1 − 2πimx L = | | f(x + Ln)e dx y = x + Ln L ∈Z 0 n ∫ ∫ ∑ L(n+1 1 −2πim (y+Ln) 1 − 2πimy = f(y)e L dy = f(y)e L dy |L| |L| R n∈Z Ln 1 = fˆ(m) |L|

Thus we can write h(x) as ∑ ∑ 2πikx 1 2πikx h(x) = h e L = fˆ(k)e L k |L| k∈Z k∈Z

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