Reduction Mechanics of the Cosmological Constant

Reduction Mechanics of The Cosmological Constant

Martin Sandstr¨om Department of Theoretical Physics Uppsala University

Supervisor: Giuseppe Dibitetto

Abstract The quantum creation of membranes by a totally antisymmetric tensor and gravitational fields is considered in arbitrary space-time dimensions. The creation process is described by instanton tunneling. As membranes are created, the energy density associated with the antisymmetric field decreases, reducing the effective cosmological constant to a lower value. For a collection of parameters and initial conditions, the creation mechanism goes to a halt as soon as the cosmological constant is near zero. A brief exploration of a canonical gravity representation is also considered, where the system of ADM-composition in arbitrary space-time dimensions is introduced.

Sammanfattning I detta arbete ger vi en teoretisk förklaring hur den kosmologiska konstanten kan re- duceras med hjälp av en naturlig mekanism. Relaterat till denna naturliga mekanism, är möjligheten av att en skapelseprocess av membran existerar och det visar sig endast vara möjligt när den kosmologiska konstanten har ett positivt värde. Empiriska experiment visar att det observerbara universum vi lever i, har ett positivt värde p˚aden kosmologiska konstanten och det möjliggör en teorerisk förklaring för skapelsen av membran.

1 Contents

1 Introduction 3 1.1 Creation Mechanism ...... 3 1.2 Neutralization of The Cosmological Constant ...... 4

2 Dynamics 4 2.1 Action Functional ...... 5 2.2 Gravitational Equations of Motion ...... 6 2.3 The Gibbons-Hawking-York Term ...... 7 2.4 Membrane Equations of Motion ...... 9

3 Solving the Equations 10

4 Instanton Solutions 12 4.1 Decomposition of Space-time ...... 12 4.2 Time Gauge ...... 19 4.3 Complete Set of Conﬁguration Parameters ...... 23 4.4 Extremizing of The Action ...... 23

5 Neutralization of The Cosmological Constant 25 5.1 Mass and Charge Dependence ...... 25 5.2 Reduction of The Cosmological Constant ...... 27

6 Conclusion 29

7 Appendix 30

A Mathematical Deﬁnitions 30 A.1 Exterior Derivative ...... 30 A.2 Lie Derivative ...... 31 A.3 Pull-back ...... 31 A.4 Hodge Dual ...... 32 A.5 Covariant Derivative ...... 33 A.6 Curvature Tensor ...... 34 A.7 Forms ...... 35 A.8 Generalized Kronecker Delta ...... 37

2 1 Introduction

In general relativity the cosmological constant plays a roll of being imposed as a fixed parameter, to be determined via observations on the dynamical nature of space-time. It is when the introduction of a totally antisymmetric field, coupled via gravity, the cosmological constant gets shifted by an effective term proportional to the energy density of the antisymmetric field. Hence, the cosmological constant can be treated as a dynamical variable rather than being imposed as a fixed parameter. If a source is introduced for the antisymmetric field, the cosmological constant changes to being a dynamical quantity - it is no longer constant. To be specific, consider in D = d + 1 space-time dimensions a totally antisymmetric field with d indices coupled to its natural source, the d − 1 spatial dimensional membrane. Just as in the case that an electric field can create particle pairs, the antisymmetric tensor field may create membranes. Now, couple this system to gravity and include a cosmological constant λ represent- ing a collection of physical processes, such as the mechanism of spontaneous symmetry breaking. The resulting cosmological constant Λ, is now a combination of λ and a term proportional to the energy density associated with the antisymmetric field. We will show that the spontaneous creation of closed membranes, tends to reduce the cosmological constant Λ to a lower value. The mechanism of membrane creation may evolve continuously in this manner, but as soon as Λ ≤ 0 the process goes to a halt. This implies that the cosmological constant is neutralized by a natural mechanism.

1.1 Creation Mechanism We shall begin by presenting a qualitative description of the membrane creation mechanism, and how the presence of it, changes the cosmological constant. Most of the technical analysis is reserved for later sections, but it may be helpful to establish some mathematical notation and derive the equations of motion for the antisymmetric field. Thus let xµ = zµ(ξ) be the d-dimensional history of the membrane in D = d + 1 space-time dimensions as a function of a d µ ν its coordinates ξ = 0, ··· , d − 1, and let gab = gµν z ,az ,b be the induced metric. Also let Aµ1···µd denote the components of the antisymmetric tensor field, with the associated field strength F = dA. The action we will use is Z p 1 Z √ SE = m ddξ dg − dDx gF F µ1···µD (1.1.1) 2D! µ1···µD e Z 1 Z √ + ddξA zµ1 ··· zµd a1···ad + dDx g∂ (F µ1···µD A ) d! µ1···µd ,a1 ,ad d! µ1 µ2···µD + SE,grav(λ) where m is the mass per (d−1) unit volume, and e is the coupling constant between the membrane and the antisymmetric tensor field. Here, SE,grav(λ) is the gravitational action, including an explicit contribution upon λ which is later assumed to be negative. The total derivative in (1.1.1) is included to ensure that the action has well defined variations with respect to Aµ1···µd . The field strength tensor can in general be written as F µ1···µD = g−1/2Eµ1···µD for some scalar field E. With this notation, the equations of motion from (1.1.1) for the antisymmetric tensor field are Z (∂ E) εµ1···µD = −e ddξ δD(x − z(ξ))zµ2 ··· zµD εa1···ad . (1.1.2) µ1 ,a1 ,ad This equation shows that away from the source, the scalar field E on either side of the membrane is a constant, and the two values of E differ in magnitude of |e|. Substitution of (1.1.2) into (1.1.1) shows that away from the membrane, the antisymmetric tensor field contributes a positive

3 cosmological constant term, proportional to the energy density E2. So the membrane divides space-time into two regions of diﬀerent energy densities, and each will be a portion of de Sitter or anti-de Sitter space characterized by some value of the total cosmological constant Λ. If the initial space-time is in a metastable state, a ﬁrst order phase transition to a true state will occur, where the true state quickly expands as a ”bubble” within the metastable state. The energy density of the initial metastable state must be less than the true state, as demanded by the conservation of energy, since some of the energy is used for the creation of the membrane. The probability per space-time volume, that a bubble will form, is given by

−B/ P ∼ e ~,B = SE[instanton] − SE[background], (1.1.3) where SE[instanton] is the euclidean action evaluated at the instanton solution to the euclidean equations of motion, in which the solution is the interpolation between real classical motions of the system, and so provides a semi-classical path by which the system tunnels from one region to the other. Also SE[background] is the background which corresponds to the metastable state, together with the conﬁguration of the classical single bubble. Once the membrane is created, it expands rapidly and coalesces with other such membrane. This is the essential mechanism in how the cosmological constant is altered.

1.2 Neutralization of The Cosmological Constant Accordingly to the conservation of energy, the energy density of the metastable state must be strictly larger than the stable state. Suppose we label the metastable and stable state as outside and inside, respectively. It can then be shown that for the realization of the membrane creation process, the magnitude of the outside scalar field must obey the inequality 1 |E | > |e|, (1.2.1) o 2 and if Eo is large enough so that (1.2.1) is satisfied, then membrane creation will on the average reduce the magnitude of E, and by consequence reduce the cosmological constant, Λo > Λi. On the other hand, if instead Λi ≤ 0, then for some values of the parameters and initial field values Eo, Λo, the membrane cannot be produced at all. This may happen, even if Eo is relatively large, satisfying (1.2.1). The properties of membrane production provide a natural mechanism by which the cosmological constant may be reduced to a value near zero. Suppose the initial cosmological constant Λo is positive, with λ having a arbitrary negative value, and suppose Eo is large enough satisfying (1.2.1). Then membranes will be produced which generally lowers Λ, and this process will continue as long as Λ > 0. But as soon as the value of Λ falls at or below zero, for some parameters and fields E, Λ, the neutralization process cannot continue and the membrane creation process turns to a halt. We will show that the membrane creation is halted if the parameters and the fields Λo, Eo satisfy a general inequality. Satisfaction of this inequality requires, however, the mass density m of the membrane to be large, and the magnitude of the coupling constant |e| to be small. It turns out that the rate for membrane creation is decreased as either m is large or |e| is small, at least for those processes which reduces the cosmological constant. Then as the membrane creation stops, when Λ is small, also forces the rate of membrane creation to be small.

2 Dynamics

In arriving at the description of the dynamics involving membrane creation, and so how the cosmological constant is changed, one may follow the indispensable treatment of such dynamics

4 through the theory of action. All treatment occur in euclidean space where timelike variables are anti-clockwise Wick rotated, and by the reality of the energy density associated with the antisymmetric tensor field, the anti-symmetric field components must be rotated in the same amount, but clockwise. Let (U, z) be a chart on a manifold M, where dim(M) = d + 1. Let ξ ∈ M, then z : ξ → x is a 1-1 map of U onto some subset of Rd. Denote this mapping by zµ = xµ(ξ) where zµ, µ = 0, 1, ··· , d, specifies the d-dimensional history of a membrane in D = d + 1 space-time dimensions as a function of its coordinates ξ = 0, 1, ··· , d − 1, and let d µ ν gab = gµν z ,az ,b be the induced metric. Also let the d-form A denote the antisymmetric field tensor with field strength F = dA.

2.1 Action Functional The action used throughout is Z Z 1 Z Z SE = +m η + e φ∗A − F ∧ ?F + d (A ∧ ?F ) + SE,grav(λ), (2.1.1) M ∂M 2 M M where the mass of the membrane is m, e is the coupling between the antisymmetric ﬁeld and gravity, and κ is constant. The term in(2.1.1) which is a total exterior derivative, is included to counteract terms in the action functional and so ensure well deﬁned variations of (2.1.1) with respect to A. Here SE,grav(λ) is the euclidean gravitational action 1 Z 1 Z SE,grav(λ) = − (R − 2ωλ) + K , (2.1.2) 2κ M κ ∂M

ij where λ is the cosmological constant, R = R ∧ ei ∧ ej is the Ricci scalar, and ∂M denotes the boundary of M. All integration is taken with respect to the orientation of the manifold M. The boundary term in (2.1.2) is the Gibbons-Hawking-York term, included to counteract the Palatini identity obtained from varying the Einstein-Hilbert action. This matching gives also a definition of the extrinsic curvature K, where K := σK. The volume-forms as viewed in (2.1.1), √ √ are defined as η := ddξpdg, ω := dDx g, and σ := ddx h, where we have used a shorthand, e.g. ddx := dx0 ∧ dx1 ∧ · · · ∧ dxd−1. The standard action functional analysis involves the infinitesimal change in the fields, and the equations of motion are obtained from the principle of extremal action, that is the action is constant with respect to first order changes in the fields involved. In our case the fields involved are the antisymmetric field, gravity, and the membrane itself. Without further ado, the first order change in the action (2.1.1) with respect to the antisymmetric field is Z Z E ∗ d δAS = e δAφ A + (−1) δA(A ∧ d ? F ), (2.1.3) ∂M M where we have used the identity d(a ∧ b) = da ∧ b + (−1)ma ∧ db, for a m-form a, to shift the exterior derivative over to the dual field ?F . The pullback φ∗A can be expanded as, 1 ∗ µ1 µd a1···ad d φ A = Aµ ···µ z ··· z ε d ξ, (2.1.4) d! 1 d ,a1 ,ad where we have defined,

(Aφ)µ1···µd := Aµ1···µd , (2.1.5) ∂zµ := zµ . ∂ξa ,a

5 Suppose for now that d ? F is an (n − d)-form, then the exterior product of δAA with d ? F is 1 δ A ∧ d ? F = δ A (d ? F ) εµ1···µdµd+1···µn−d dnx, (2.1.6) A d!(n − d)! A µ1···µd µd+1···µn−d 1 = δ A (?d ? F )µ1···µd dnx, d! A µ1···µd where (A.4.1) has been used. The local chart dimensions have been kept at n in order to make sense of the calculation. At the end of the day, one may limit to intended dimensions, in our case n → D. Assuming that the ﬁelds A, F , vanishes at ∂M, using Stokes’ theorem Z Z 0 = d(A ∧ ?F ) = (F ∧ ?F − A ∧ ? ? d ? F ) . (2.1.7) M M where we have used the property ? ? d ? F = (−1)(D−1)(n−D+1)(d ? F ). By (A.4.4), (2.1.7) is the same as (F,F ) − (A, ?d ? F ) = 0, thus by (2.1.6), 1 δ A ∧ d ? F = (dδ A) F µ1···µD dDx (2.1.8) A d! A µ1···µD 1 = − ∂ E εν1···νD δ A dDx + ··· , d! ν1 A ν2···νD where the relation F µ1···µD = E|g|−1/2εµ1···µD has been used. Combining (2.1.8) and the variation of (2.1.4), (2.1.3) becomes Z 1 Z E D d D µ2 µD a1···ad µ1···µD δAS = d x δAAµ ···µ e d ξδ (x − z(ξ))z ··· z ε + ∂µ Eε (2.1.9) d! 2 D ,a1 ,ad 1 + surface terms. By the principle of extremal action, the equations of motion are simply Z (∂ E) εµ1···µD = −e ddξ δD(x − z(ξ))zµ2 ··· zµD εa1···ad . (2.1.10) µ1 ,a1 ,ad

2.2 Gravitational Equations of Motion In (2.1.1) the ﬁeld strength F and the induced metric determinant dg, have intrinsic dependence upon the metric tensor. Hence, the variation of (2.1.1) with respect to the metric tensor is Z Z E 1 E,grav δgS = m δg η − δg F ∧ ?F + δgS (λ), (2.2.1) M 2 M Z p 1 Z √ = m dDx ddξ δD(x − z(ξ))δ dg − dDx E2δ g + δ SE,grav(λ), g 2 g g where Z Z E,grav 1 1 δgS = − δ (R − 2Λ) + δK . (2.2.2) 2κ M κ ∂M

The variation of the Ricci scalar R can be resolved as ik j ik j δR ≡ δg R k ∧ ? (∂i ∧ ∂j) + g δ R k ∧ ? (∂i ∧ ∂j) (2.2.3) jk ik j = R δgjkω + g δ R k ∧ ? (∂i ∧ ∂j) ,

6 √ where we have used (A.4.4),(A.7.16), and deﬁned ω := dDx g. Using the explicit expression (A.7.12), one can deduce that ik j ik j j l g δ R k ∧ ?(∂i ∧ ∂j) = g δ dΓ k + Γ l ∧ Γ k ∧ ?(∂i ∧ ∂j) (2.2.4) 1 = gik(∇ δΓj − ∇ δΓj )ω − Rgikδg ω, i kj j ki 2 ik

√ where ω := dDx g. Hence, by substitution of (2.2.4) in (2.2.2), we ﬁnd Z E,grav 1 jk 1 jk jk δgS = − ω δgjk R − R g + λg (2.2.5) 2κ M 2 Z mk i i 1 + g ∇mδΓ ik − ∇iδΓ mk + δK κ ∂M Z Z 1 jk 1 jk jk 1 1 k = − ω δgjk R − R g + λg − σ nkV − δK , 2κ M 2 κ ∂M 2 where we have used√ Stokes theorem to change the integration over to the boundary. Here K := σδK := ddx hδK where K is the trace of the extrinsic curvature, and V k is the deﬁned by, k jk i ji k V ≡ g δΓ ij − g δΓ ij, (2.2.6) and nk are the components of a normal vector perpendicular to the surface of M.

2.3 The Gibbons-Hawking-York Term The surface integral in the gravitational part of the euclidean action (2.1.1), is included as a counter-term to ensure that the equations of motion have a nicer form. In such a way the system have a description of extrinsic curvature, defined through first order variations of the k connection 1-forms. The simplification of the scalar nkV , will therefore be the object of interest, and resolved to match the counter-term. One could take a prior definition of the extrinsic curvature, and see if the counterterm is matched exactly to it, but assume that no such definition d is performed. This is the way we follow now. Take the basis to be {∂i}i=0 where the local 0 D k coordinate system is (x , ··· , x ). Hence by definition, the scalar nkV is k jk i ji k nkV = nkg δΓ ij − nkg δΓ ij. (2.3.1)

Now deﬁne the transverse metric components, hij, given by

gij ≡ hij + ninj, (2.3.2) gij ≡ hij + ninj,

i where nin = 1. The metric components hij has the following properties, j hijn = 0, (2.3.3) ik k hijh = δj , ij hijh = d, thus the transverse metric has dimensions equal to d, and it annihilates the normal vector components. The transverse metric thus is intrinsic to the boundary ∂M. In order to obtain the k desired explicit result of nkV , one notices that i ij 0 = ∇k n ni = g (nj∇kni + ni∇knj) , (2.3.4)

7 i that is n ∇kni = 0. The variation of the connection 1-form components are 1 gjkδΓi − gijδΓk = gjkgimδ (g + g − g ) (2.3.5) ij ij 2 mi,j jm,i ij,m 1 − gmkgijδ (g + g − g ) , 2 mi,j jm,i ij,m jk im = g g (δgmi,j − δgji,m) , where the Christoffel symbols are defined by (A.5.8). Thus (2.3.1) is simplified to,

k j im nkV = n g (δgmi,j − δgji,m) (2.3.6) j im = n h (δgmi,j − δgji,m) .

The variation of the metric tensor components, δgij, vanishes by assumption everywhere on the boundary ∂M. Therefore it is also true that the tangential derivatives vanishes there, δgij,k = 0. ik This implies that the transverse metric is transverse to the tangential derivatives, h δgij,k = 0, and so (2.3.6) becomes

k j im nkV = n h (δgmi,j − δgjm,i − δgji,m) (2.3.7) im k = −2h nkδΓ im.

ik Using that δ h nm,i = 0 on ∂M, one ﬁnds

k im k im im k nkV = 2h δ ni,m − nkΓ im + 2nm,iδh + 2h Γ imδnk (2.3.8) im i m = 2 h − n n δ (∇mni) m = 2δ (∇mn ) .

So the boundary term in the variation (2.2.5) together with (2.3.8), gives Z √ Z √ 1 m 1 m − h (δ (∇mn ) − δK) = − h δ (∇mn − K) . (2.3.9) κ ∂M κ ∂M

i The vanishing of the boundary term in (2.2.5), implies that K := ∇in , and (2.2.1) becomes

Z d E D d D g δ g jk δgS = +m d x d ξ δgjk δ (x − z(ξ)) g (2.3.10) 2pdg δg √ 1 Z g 1 Z √ − dDx δg gjk E2 − dDx g δg Gjk + λ gjk 2 jk 2 2κ jk Z √ Z mpdg 1 1 = − dDx g δg ddξ √ δD(x − z(ξ))gjk + G + λ + κE2 gjk . jk 2 g 2κ jk 2

Keeping in mind that indices are all euclidean, the equations of motion with respect to variations of the metric tensor g, are

Z pdg G = −Λ g − κm ddξ √ δD(x − z(ξ))g , (2.3.11) µν µν g µν where µ = 0, 1, ··· ,D, and we have deﬁned the Einstein tensor components and an eﬀective 2 cosmological constant as, Gµν ≡ Rµν − Rgµν /2 and Λ ≡ λ + κE /2, respectively.

8 2.4 Membrane Equations of Motion To complete the equations of motion, derived from (2.1.1), we need to vary the euclidean action with respect to the membrane history, that is the zµ’s, keeping excluding terms at a constant. The terms which carries explicit factors of dg and zµ, are thus considered but also an implicit variation of the components of φ∗A are included. At a glance the variation of (2.1.1) with respect to the membrane history is m Z p δ SE = − ddξ dgdgabδ dg (2.4.1) z 2 ab e Z d µ1 µd a1···ad µ1 µd a1···ad + d ξ Aµ ···µ δ z, ··· z, ε + (δAµ ···µ ) z, ··· z, ε , d! 1 d a1 ad 1 d a1 ad

pd d ab d The term g g δ gab can be expanded as

pd d ab d pd d ab µ ν pd d ab α µ ν d ab µ ν g g δ gab = g g δ gµν z,az,b = g g gµν,αδz z,az,b + 2 g gµν z,aδz,b. (2.4.2) Using integration by parts, one can resolve (2.4.2) to be

pd d ab d pd d ab α µ ν pd d ab µ ν g g δ gab = g g gµν,αδz z,az,b + 2 g g gµν z,aδz ,b (2.4.3) pd d ab µ ν pd d ab µ ν α − 2gµν g g z,a ,bδz − 2 g g gµα,ν z,az,bδz . The last term in (2.4.3) have the symmetry that

1 d dgabg zµ zν = g + g gabzµ zν , (2.4.4) µα,ν ,a ,b 2 µα,ν να,µ ,a ,b thus (2.4.3) becomes

pd d ab d pd d ab α µ ν pd d ab β α g g δ gab = − g g gαµ,ν + gνα,µ − gµν,α δz z,az,b − 2gβα g g z,a ,bδz (2.4.5) + boundary terms.

In (2.4.5), the ﬁrst term can be written as the components of the connection 1-forms, as deﬁned in (A.5.8). Hence (2.4.5) becomes

pd β µ ν α pd d ab β α = −2gαβ g Γ µν z,az,bδz − 2gβα g g z,a ,bδz + boundary terms, (2.4.6) µ pd d ab β α = −2gαβz,a∇µ g g z,b δz + boundary terms,

The variation of the second term of (2.4.1) can be resolved through manipulations, as Z ddξ d × A δ z,µ1 z,µ2 ··· z,µd +δzν1 ∂ A z,µ1 ··· z,µd εa1···ad (2.4.7) µ1···µd a1 a2 ad ν1 µ1···µd a1 ad Z = ddξ δzν1 [∂ A − d∂ A ] z,µ1 ··· z,µd εa1···ad + boundary terms, ν1 µ1···µd µ1 ν1···µd a1 ad 1 Z = ddξ δzν1 ∂ A z,µ1 ··· z,µd εa1···ad + boundary terms, d! [ν1 µ1···µd] a1 ad Z √ := ddξ g δzν1 Eε z,µ1 ··· z,µd εa1···ad + boundary terms ν1µ1···µd a1 ad

9 where we have used the deﬁnition of the anti-symmetrization operation, the identiﬁcation √ Fν1µ1···µd = E g εν1µ1···µd . Bringing everything together we can solve the equations of motion for (2.4.1), as

eE √ µ pd d ab α αµ1 µ2 µD a1···ad z ∇µ g g z = − g g z, ··· z, εµ µ ···µ ε . (2.4.8) ,a ,b md! a1 ad 1 2 D Bringing together (2.1.10), (2.3.11), (2.4.8), we arrive at the complete set of equations, Z (∂ E) εµ1···µD = −e ddξ δD(x − z(ξ))zµ2 ··· zµD εa1···ad , (2.4.9a) µ1 ,a1 ,ad Z pdg G = −Λ g − κm ddξ √ δD(x − z(ξ))z dgabz , (2.4.9b) µν µν g µ,a ν,b eE √ µ pd d ab α αµ1 µ2 µD a1···ad z ∇µ g g z = − g g z ··· z εµ µ ···µ ε . (2.4.9c) ,a ,b md! ,a1 ,ad 1 2 D The first equation in (2.4.9) relates the rate of change of the electric field E, and if the system does not satisfy the classical membrane history, E remain constant. Any crossing, between the two regions as separated, results in a discontinuous jump from one space-time to another thus any contribution to the change in E stems via the membrane. The second describes the spacetimes on either side of the membrane, where either side follow the equations of general relativity, with the exception that the cosmological constant is now dependent on E. Also in this situation, a discontinuity occurs from any crossing between the two regions. The third equation describes in some sense, the acceleration of the the membrane history zµ where the rate of change of the µ z,a’s is proportional to the charge coupling e, the electric field E, the membrane mass density µ m, the metric tensor, and the z,a’s themselves.

3 Solving the Equations

Now that the complete set of equations of motion (2.4.9) are in hand, the solutions of (2.4.9) are of our interest. As previously mentioned, two equations have singular terms which needs to be considered, and we leave this to the end of the section. When the membrane history is not the classical path the singular terms vanishes, and by taking the trace of the gravitational equations of motion in (2.4.9), leads to

2D 2D 1 R = Λ = λ + κE2 . (3.0.1) D − 2 D − 2 2

This equation remains true for two values of the electric field E, and we can make sense of this because of the separated regions of corresponding space-times, by the membrane as represented by the singularities in (2.4.9). Suppose we label these two regions by ”inside” and ”outside”, so that Eo, Ei, is the electric field on the outside and inside respectively. By this, the scalar curvature R is defined by values on the separate regions. Denote these curvatures by Ri,o, then the ”cosmological constants” are 1 Λ = λ + κE2, (3.0.2) o 2 o 1 Λ = Λ − κ E2 − E2 , i o 2 o i

10 where we have eliminated Ri from Λi, in terms of Λo. In analogy with how the step function is defined at a discontinuity, the value of the electric field at the membrane history, Eon, may be defined analogous as the average of the inside and outside electric fields, 1 E := (E + E ). (3.0.3) on 2 o i

To find Ei in terms of Eo we need to resolve the difference Ei − Eo, and this is related to the analysis on what occurs when crossing the inside and outside regions. The first equation of (2.4.9), can be contracted by the permutation symbol to yield e Z d D µ2 µD a1···ad ∂βE = − d ξ δ (x − z(ξ)) z ··· z εβµ ···µ ε , (3.0.4) d! ,a1 ,ad 2 D where the property (A.8.4) has been used. Let us define the unit normal vector components as 1 µ2 µD a1···ad −nµ ≡ z ··· z µ µ ···µ , (3.0.5) 1 d! ,a1 ,ad 1 2 D

1/2 a1···ad d −1/2 a1···ad where αµ2···µD := g εαµ2···µD , := g ε are the covariant permutation tensor components, = ±1. Then by using (3.0.5) in (3.0.4), one ﬁnds the equation

Z pdg ∂ E = +e ddξ √ δD(x − z(ξ)) n , (3.0.6) β g β and on the world volume it may be written as Z pdg nβ∂ E = e ddξ √ δD(x − z(ξ)). (3.0.7) β g Integrating (3.0.7), reveals that the diﬀerence in the outside and inside electric ﬁelds, are related by

Eo − Ei = e. (3.0.8) The constant has the ambiguity of being within a sign, just for the matter of what we would call ”inside” or ”outside” is of choice. As one will encounter later on, the initial configuration of the tunneling process, dictates a choice of what will be the inside and the outside. As an example; suppose the initial electric field is Eo, then since the energy density is proportional to the square of the electric field, energy conservation states that, 1 E2 − E2 = (E + E )(E − E ) = E e > 0, (3.0.9) o i o i o i 2 on where we have used (3.0.3), (3.0.8). By the assumption that the charge coupling is strictly positive, e > 0, one must have = +1. If the inside and outside are exchanged, so does the sign of (3.0.9) and consequently = −1. With the relation (3.0.8), the electric field on the worldline (3.0.3) takes the form 1 E = E − e, (3.0.10) on o 2 where we have eliminated the inside electric field Ei, in (3.0.3). The inside electric field can also be eliminated in (3.0.2), to yield Λi in terms of the outside electric field, as e2 Λ = Λ − κ eE − . (3.0.11) i o o 2

This concludes the conﬁguration parameters Λi,o, and the electric ﬁelds Ei,o and Eon.

11 4 Instanton Solutions

Some of the parameters to describe the system, has now been found, including the dynamical expressions of Λ, the electric ﬁelds in terms of and the charge coupling e. There are still a few parameters missing to the complete picture, as we will now resolve. Some of the possible freedoms have been left out in the calculations in the previous section, such as the inclusion of the extrinsic curvature. This is of no surprise, since we have left out the third equation of (2.4.9). Now, this equation depends upon a product of velocities, and one should extract the normal vector components from these terms. Without further ado, deﬁne the pullback of the extrinsic curvature as µ ν Kab = z,az,b∇µnν , (4.0.1)

d ab µ where its trace is deﬁned as K := g Kab. The velocities of the trajectories z are orthonormal µ to the unit normal components, that is z,anµ = 0. Hence, one can compose the expression

ab pd d µ g g z,anµ = 0, (4.0.2)

α By simply covariantly differentiating (4.0.2) by the operator z,b∇α, one resolves (4.0.2) to be the same as ab ab α pd d µ pd d α µ pd z,b∇α g g z,a nµ = − g g z,bz,a∇αnµ = − gKon, (4.0.3) where we have defined Kon as the trace of the extrinsic curvature, defined on the worldline. Now, comparing (4.0.3) with the third equation of (2.4.9), this relation follows:

eE K = − on . (4.0.4) on m so the trace of the extrinsic curvature, on the worldline, is proportional to the charge coupling e, the membrane mass m, the electric field. In the next section we shall follow an analogous analysis for the electric fields, to find the relations between the extrinsic curvatures on the outside and inside, denoted by Ko and Ki respectively. We will begin with a decomposition of space-time and further on define a Hamiltonian in deriving the equations of motion for the extrinsic curvatures.

4.1 Decomposition of Space-time The dynamics of the metric tensor components, can be understood through the gravitational equations of motion in (2.4.9), and in general it is true that the theory will remain invariant under general coordinate transformations. As we will now follow, the space-time will be separated in a particular way such that time is be singled out, separated from d spatial dimensions. This procedure may be a necessary step in recasting the gravitational equations of motion to canonical form. The treatment in this section will begin with a mathematical exercise, defining the separation in time and foliations, or sheets. Then we will resolve what quantities are intrinsic to the foliations at constant time, to follow up by defining an action principle in canonical variables. Given a manifold M equipped with an euclidean metric. The foliation of M into Σ × R, can be defined as,

∀t ∈ R, Σt ≡ {p ∈ M, t(p) = t} , (4.1.1) Σt ∩ Σt0 = ∅,

12 where t is smooth such that the gradient of t, between the sheets deﬁned by (4.1.1), is non-zero. d ν Take as the basis {∂µ}µ=0 of TpM associated with the coordinates z , that is we have ∂ ∂ ∂ ≡ = , (4.1.2) t ∂t ∂z0 ∂ ∂ ≡ , i ∈ [1, ··· ,D]. i ∂zi

0 i i 0 i For any i = 1, ··· , d the vector ∂i is tangent to t = c , z = c , where c , c , are constants such that ci 6= cj, i 6= j, holds, that is

∂i ∈ TpM(Σt), i = 1, ··· ,D. (4.1.3)

µ d ∗ µ µ The dual basis {dx }µ=0 in Tp M, satisﬁes the completion relation (dx , ∂ν ) = δν , where (·, ·) is the inner product operation, in particular (dt, ∂t) = 1. Generally, the vector ∂t can be written ∗ as a linear expression, involving the terms of a normal 1-form n ≡ αdt, dt ∈ Tp M(Σt), and a vector β ∈ TpM(Σt). That is we can write

∂t ≡ αn + β, (4.1.4) where β is called the shift, and α is called the lapse. Since (n, n) = 1 and (dt, ∂t) = 1, the inner product of the 1-form dt and the vector β, is

(dt, β) = (dt, ∂t) − α(dt, n) = 1 − (n, n) = 0. (4.1.5)

i Thus (n, β) = 0. Now, β = β ∂i so from (4.1.4)

t i i 1 · ∂t − αn ∂t − αn ∂i − β ∂i = 0, (4.1.6) where 1 ∈ R. Equating the vector components in (4.1.6), gives nt = α−1, (4.1.7) ni = βiα−1.

Since the normal 1-form can be expanded as n = αdt, the components are simply nt = α, ni = 0, and so the contravariant, and covariant normal vector components are given, respectively, by

1 β1 βD nµ = , , ··· , , (4.1.8) α α α

nµ = (α, 0, ··· , 0) .

The metric tensor components can now be expressed in terms of lapse and shift, as

2 µ ν i j 2 i gtt = g(∂t, ∂t) = g (αn + β, αn + β) = α gµν n n + gijβ β = α + β βi, (4.1.9) j gti = g(αn + β, ∂i) = gijβ = βi,

gij ≡ hij, and so the d + 1-decomposed metric is

2 µ ν 2 i j 2 j i i j ds = gµν dz dz = (α + hijβ β )dt + 2hijβ dtdz + hijdz dz , (4.1.10) 2 2 i i j j = α dt + hij dz + β dt dz + β dt .

13 This is the metric we shall now focus onward. By deﬁning the pullback of dzi, at constant t, as

i i ∂z a dz = a dξ , a = 1, ··· , d, (4.1.11) ∂ξ t the metric can be written in terms of the history parameters, ξ’s, as

2 2 2 a a b b ds = α dt + hab (dξ + β dt) dξ + β dt , (4.1.12) where i j ∂z ∂z i j hab = gij a b := gijEaEb , (4.1.13) ∂ξ t ∂ξ t is the pullback of the metric tensor components. The inverse metric tensor components gµν , can be found by noticing that 1 = nt = gttn = gttα, (4.1.14) α t βb = nb = gbtn , α t a ba ta ba a −2 δc = gcbg + gctg = hcbg + βcβ α , which in turn reveals,

gtt = α−2, (4.1.15) gat = α−2βa, gab = hab + α−2βaβb, where we have used (2.3.2). Using Cramer’s rule, one can relate the d + 1 determinant g = det(gαβ), to the pullback determinant, h = det(hab). The result is √ √ g = α h, (4.1.16) that is the g determinant is proportional to the lapse and the determinant h. The intrinsic features of the foliations Σt are of particular interest, since the resulting relations between the curvature scalar R and the extrinsic curvature scalar K, yields a nice form. By (2.3.2), the attempt of lowering the unit normal with hµν , only nulliﬁes it, but the operation of hµν on any 1-form ω ∈ Σt, is a mapping back to Σt. We thus have,

µ ν h ν n = 0, (4.1.17) µ h ν ωµ = ων ,

for ω ∈ Σt and n ⊥ Σt. We may therefore deﬁne the projection of an arbitrary (m,n)-tensor, to be

0 0 0 0 0 0 i1i2···im i1 i2 im j j j i i ···i PT := h 0 h 0 ··· h 0 h 1 h 2 ··· h n T 1 2 m 0 0 0 , (4.1.18) j1j2···jn i1 i2 im j1 j2 jn j1j2···jn where P is deﬁned as the operation of projection. The deﬁnition of (4.1.18) enables the projection of arbitrary tensor components. The interest in our hands, is how we may relate the curvature

14 scalar R, to the extrinsic curvature scalar K. Naturally, since R is contractions of the Riemann τ tensor components, the projection of R αµν on Σt, has the formal expression, ¯τ α τ Rαµν v = P (∇µ∇ν − ∇ν ∇µv ) , (4.1.19)

τ where the bar denotes dependence on the transverse metric, and v ∈ Σt, v nτ = 0. The expansion of the terms on the right hand side of (4.1.19), are by deﬁnition

τ µ0 ν0 τ α τ 0 β P (∇µ∇ν v ) = h µh ν h τ 0 ∇µ0 (h ν0 h β∇αv ) (4.1.20) µ0 τ τ 0 β µ0 ν0 α τ = h µh τ 0 (∇µ0 h β)∇ν v + h µh ν (∇µ0 h ν0 )∇αv µ0 α τ β + h µh ν h β∇µ0 ∇αv .

α α Using the covariant derivative of the projection operators, ∇µ(h β) = −∇µ(n nβ), one can ﬁnd the following relations

µ0 ν0 τ α β τ h µh ν h β(∇µ0 h ν0 )∇αv = KKµν v , (4.1.21) µ0 α τ τ 0 β τ λ h µh ν h τ 0 (∇µ0 h β)∇αv = KνλKµ v ,

ν0 β β where we have used that h ν h ν0 = h ν . Hence, by combining the analysis of (4.1.21), with (4.1.19), yields the projection of the Riemann tensor components as

¯τ α µ0 α τ β τ τ β Rαµν v = h µh ν h β(∇µ0 ∇α − ∇α∇µ0 )v + (KνβKµ − KµβKν )v (4.1.22) µ0 ν0 τ β α0 α τ τ α = h µh ν h βR α0µ0ν0 h αv + (KναKµ − KµαKν )v .

Thus we conclude that the projection of the Riemann tensor components are

¯τ τ α0 µ0 ν0 τ 0 τ τ Rαµν = h τ 0 h αh µh ν R α0µ0ν0 + Kµ Kνα − Kν Kµα. (4.1.23) In (4.1.19) we see a direct relation between the Riemann tensor components, and the extrinsic curvature, that is the Riemann tensor components are separated in two parts, intrinsic to the foliations Σt. Taking the trace, τ = µ of (4.1.23), reveals the relation

µ0τ 0 α0 ν0 µ µ R¯αν = h h αh ν Rτ 0α0µ0ν0 + Kµ Kνα − Kν Kµα. (4.1.24)

The intrinsic Ricci curvature scalar, is the contraction of (4.1.24), hence

αν µ0τ 0 α0ν0 2 αβ R¯ = h R¯αν = h h Rτ 0α0µ0ν0 + K − K Kαβ (4.1.25) α β 2 αβ = R − 2n n Rαβ + K − K Kαβ.

We may ﬁnd other expressions by considering diﬀerent contractions with the Riemann tensor components. Suppose the following projection is realized:

α µ0 ν0 α0 β α µ0 ν0 α0 h α0 h µh ν R βµ0ν0 n = h α0 h µh ν [∇µ0 , ∇ν0 ]n . (4.1.26)

By similar analysis, one can ﬁnd

α µ0 ν0 α0 α µ0 ν0 α0 α h α0 h µh ν ∇µ0 ∇ν0 n = h α0 h µh ν ∇µ0 Kµ0 = ∇¯ µKν . (4.1.27)

Hence (4.1.26) is β α µ0 ν0 α0 α α n h α0 h µh ν R βµ0ν0 = ∇¯ µKν − ∇¯ ν Kµ . (4.1.28)

15 A contraction, ν = α, in (4.1.28) reveals

β ν0 µ0 α0 α n h α0 h µR βµ0ν0 = ∇¯ µK − ∇¯ αK µ, (4.1.29) where we have used the deﬁning properties of the Riemann tensor components. The simpliﬁcation of (4.1.25), may be found by noticing that

αβ β α KαβK = (∇αn )(∇βn ) (4.1.30) β α β α β α = ∇α(n ∇βn ) − n [∇α, ∇β]n − n ∇β∇αn β α α β α β 2 = ∇α(n ∇βn − n ∇βn ) − n n Rαβ + K ,

α β and by the result of (4.1.30), we can eliminate the contraction n n Rαβ in (4.1.25), to ﬁnd

β α α β 2 αβ R − 2Λ = R¯ − 2Λ + 2∇α(n ∇βn − n ∇βn ) + K − KαβK , (4.1.31) where we have subtracted terms proportional to the cosmological constant, on both sides of (4.1.25) and rearranged terms. (4.1.31) is the relation we have sought for, which shows a relation of the Ricci curvature scalar R to the extrinsic curvature scalar K. The left hand side of (4.1.31) is just the integrand of the gravitational action, ignoring the Gibbons-Hawking-York term. Thus if we integrate both sides of (4.1.31) over space-time, we ﬁnd

1 Z √ S = − dDx g(R¯ − 2Λ + K2 − K Kab) (4.1.32) ΛEH 2κ ab 1 Z √ − ddx hn (nβ∇ nα − nα∇ nβ) κ α β β

:= SGC − SGHY , where we have deﬁned the Gauss-Codazzi action, SGC , as, 1 Z √ S ≡ − dDx g(R¯ − 2Λ + K2 − K Kab), (4.1.33) GC 2κ ab

SΛEH is the Einstein-Hilbert action with cosmological constant, and SGHY is just the Gibbons- Hawking-York term. If we examine (4.1.32) we see that SΛEH + SGHY = SGC . This implies that the equation of motion for SΛEH is the same as for SGC . Using (4.1.16) in (4.1.33), gives the action 1 Z √ S [h , α, βa] = − dDx h α R¯ − 2Λ + K2 − K Kab , (4.1.34) ADM ab 2κ ab where we have explicitly denoted this form of the action, as the ADM-action, and denoted the extrinsic curvature in spatial indices. It is a functional of the induced metric tensor components, the shift components, and the lapse. We may now change to an appropriate coordinate system. Let us pick the spatial coordinates as the foliations Σt themselves, which we will call ADM- t b b coordinates. This gives the simpliﬁcations Ea = 0, Ea = δa. The extrinsic curvature (4.0.1), in this coordinate system, becomes 1 K = −Γt n = (∇¯ β + ∇¯ β − h˙ ) (4.1.35) ab ab t 2α a b b a ab 1 = − £ h , 2 n ab

16 ˙ µ where we have defined hab := ∂thab, n = n ∂µ, and recognized the Lie-derivative of the transverse metric with respect to the unit normal. (4.1.34). One can notice that, 1 K Kab − K2 = K K hachbd + hbchad − 2habhcd , (4.1.36) ab 2 ab cd where we have invoked the symmetry of the extrinsic curvature, to symmetrize the right hand side of (4.1.36). The set of factors of the transverse metric components in (4.1.36), can be defined as 1 Gabcd := hachbd + hadhbc − 2habhcd , (4.1.37) 2 which is known as the deWitt metric tensor components. If Sab are fully symmetric tensor components, then abcd ab ab G Scd = S − Sh , (4.1.38) cd where the inverse operation where S := h Scd. Using (4.1.36) in (4.1.34), yields 1 Z √ S [h , α, βa] = dDx h α R¯ − 2Λ − GabcdK K . (4.1.39) ADM ab κ ab cd To reach the structure of the Hamiltonian, we need to define a Lagrangian density of the theory (4.1.39), hence √ ˙ a abcd LADM (hab, hab, α, β ) := α h R¯ − 2Λ − G KabKcd , (4.1.40) where we regard the Lagrangian density as a function of the lapse, shift, the transverse metric and its time derivative. The shift and the lapse are cyclic coordinates, and therefore the corresponding momenta vanishes. Thus, the only non-zero momenta is the one for the transverse metric. We define the momenta conjugate to hab, as ∂L Πab := ADM , (4.1.41) ˙ ∂hab so by definition the given momenta from (4.1.40), is √ √ ab abcd ab ab Π = − hG Kcd = − h K − h K (4.1.42)

ab ab ab By inverting Π in terms of K , defining habΠ ≡ Π, we find the extrinsic curvature and the time rate of change of hab, to be 1 Π Kab = √ Πab − hab , (4.1.43) h d − 1 ˙ 2α Π hab = ∇¯ aβb + ∇¯ bβa − √ Πab − hab . h d − 1 Now, If we define the ADM-Hamiltonian density as the following: a ab ab ˙ a ab HADM (α, β , hab, Π ) = Π hab − LADM (α, β , Π , hab), (4.1.44) where the expression in (4.1.44) is analogous to the Legendre transformation, where in our ab case the coordinates are the transverse metric components hab, and the momenta is Π . The Hamiltonian is the spatial integration over the Hamiltonian density, Z d HADM := d xHADM , (4.1.45)

17 ˙ where if we substitute for hab, the momenta, we ﬁnd the ADM-Hamiltonian density to be

√ 1 Π2 H = 2Πab∇¯ β − α h R¯ − 2Λ + Π Πab − . (4.1.46) ADM a b h ab d − 1

In (4.1.46), the independence ofα ˙ and β˙a, reveals the Hamiltonian constraints:

b ab H = 2∇¯ aΠ = 0, (4.1.47) √ 1 Π2 H0 = h R¯ − 2Λ + Π Πab − = 0. h ab d − 1

The ADM-Hamiltonian equations of motion are found by vary (4.1.46), with respect to the ﬁelds except the lapse and the shift since the coeﬃcients returns the constraints. Thus,

ab ab 0 δHADM = 2δΠ ∇¯ aβb + 2Π δ ∇¯ aβb − αδH (4.1.48) ab ¯ ab ¯ ab ¯c 0 = 2δΠ ∇aβb + 2Π ∇aδβb − 2Π βcδΓab − αδH , where the variations with respect to the shift and the lapse, are simply ignored, since they just generates the solutions (4.1.47). Further analysis of (4.1.48) shows that 1 Πabβ δΓ¯c := Πabβd ∇¯ δh + ∇¯ δh − ∇¯ δh (4.1.49) c ab 2 b da d ab a bd 1 1 = ∇¯ Πabβdδh − ∇¯ Πabβd δh , 2 d ab 2 d ab where the definition of the variation of the Christoffel symbols, have been used. What comes to the variation of the Hamiltonian density H0, it is dependent upon the transverse fields and the corresponding momenta. Varying H0, one finds

1√ 1 Π2 √ δH0 = hhcdδh R¯ − 2Λ − Π Πab − + hδR¯ (4.1.50) 2 cd h ab d − 1 2 1 ab Π + √ δ ΠabΠ − . h d − 1 We can further expand:

Π2 ΠacΠbd δ Π Πab − = δ h h ΠcdΠab − (4.1.51) ab d − 1 ac bd d − 1 ΠΠab Π = 2δh ΠacΠ b − + 2δΠab Π − h , ab c d − 1 ab ab d − 1

b ab where we have deﬁned Πc := hcaΠ . The variation of the Ricci curvature R¯ is,

¯ ¯ ab ab ¯ ¯ ab ab ¯ ¯c ¯ ¯c δR = Rabδh + h δRab = Rabδh + h ∇c δΓab − ∇b δΓac (4.1.52) ab ab 2 a b = R¯abδh + h ∇¯ − ∇¯ ∇¯ δhab,

18 2 ab where ∇¯ := h ∇¯ a∇¯ b. Bringing together (4.1.49), (4.1.51), (4.1.52), in (4.1.50), gives ab 2α Π δHADM = δπ ∇¯ aβb + ∇¯ bβa − √ Πab − hab (4.1.53) h d − 1 √ 1 + δh 2Πbc∇¯ βa − ∇¯ Πabβd + hα R¯ab − hab R¯ − 2Λ ab c d 2 ab ab 2 √ 2α ac b ΠΠ h cd Π ab 2 a b − √ Π Π c − − ΠcdΠ − + h h ∇¯ − ∇¯ ∇¯ α h d − 1 4 d − 1 1 √ + 2∇¯ Πbdβaδh − Πabβdδh + Πd δβa − h hab∇¯ 2 − ∇¯ a∇¯ b (αδh ). d ab 2 ab a ab

ab ˙ The coeﬃcient of δΠ in (4.1.53), is just the deﬁnition of hab in (4.1.43), which are the equations of motion for the transverse metric. The principle of extremal action, reveals that the equations of motion from varying SADM , are ˙ 2α Π hab := ∇¯ aβb + ∇¯ bβa − √ Πab − hab , (4.1.54) h d − 1 √ 1 Π˙ ab := ∇¯ Πabβc − Πbc∇¯ βa − Πac∇¯ βb − hα R¯ab − hab R¯ − 2Λ c c c 2 ab ab 2 √ 2α ac b ΠΠ h cd Π ab 2 a b + √ Π Π c − − ΠcdΠ − − h h ∇¯ − ∇¯ ∇¯ α, h d − 1 4 d − 1 µ ab H (hab, Π ) = 0, where Hµ := (H0, Hb) are the Hamiltonian constriants. The set of equations (4.1.54) are equivalent to the outside/inside gravitational set of equations (2.4.9), that is the second set of equations of motion in (2.4.9), have been recasted into terms of canonical theory.

4.2 Time Gauge In the previous subsection, we constructed a systematic analysis and separated space-time in spatial foliations, separated by a temporal vector. What one can now do, is to pick the explicit forms of the lapse and the shift. As this is just a re-definition of the distances between the sheets of t = constant, the space-time remains complete in sense of freedom of choice for α and βa. a One of these choices are called the time gauge, where if x (ξ) are vectors in Σt, the vector ∂t is a a defined to be perpendicular to the velocities x ,b ∈ TM(σt). This enforces α = 1 and β = 0, ab ab that is ∂t = dt. Also, using the definition of the trace of the momenta, Π − Πh /d = 0, the equations of motion attains the simpler form,

˙ 2 Π hab := √ hab, (4.2.1) h d(d − 1) " # √ 1 hab d − 4 Π 2 Π˙ ab := − h R¯ab − hab R¯ − 2Λ + 2 h d − 1 2d µ ab H (hab, Π ) = 0, and the variation of the Hamiltonian density (4.1.53) becomes √ ¯ ab ¯c ac ¯b δHADM = h∇c h δΓab − h δΓab . (4.2.2)

19 In the time gauge, the variation of the ADM-action, becomes 1 Z h √ i δS = − dDx ∂ (Πabδh ) + h∇¯ habδΓ¯c − hacδΓ¯b (4.2.3) ADM 2κ t ab c ab ab

= G(t2) − G(t1), where 1 Z G(t) := − ddxΠabδh , (4.2.4) 2κ ab are the generators of spatial translations. In terms of the trace of the momentum, we ﬁnd the Hamiltonian constraint H0 to be the same as

Π2 = hd(d − 1) R¯ − 2Λ . (4.2.5)

The explicit form of the trace of the momentum, is what we need to complete the instanton solutions the metric (4.1.12). At constant scalar curvature t2 +ρ ¯2 = k−1, ADM coordinates and time gauge, the metric becomes

2 2 a b 2 2 2 ds = dt + habdx dx = dt + dρ¯ +ρ ¯ dΩd−1, (4.2.6) where the assumption of the space to be of maximal symmetry, has been made. The dΩd−1 is the d − 1 spherical angular terms, where the components are deﬁned through

hρ¯ρ¯ = 1, (4.2.7) 2 hφ1φ1 =ρ ¯ , 2 2 2 2 hφiφi =ρ ¯ sin φ1 sin φ2 ··· sin φi−1, 2 ≤ i ≤ d − 1. The Ricci scalar R in D dimensions, determine the form of k, and has the simple form, R = kD(D − 1), (4.2.8) and Einstein’s equations in D = d + 1 space-time are G + DΛ = 0. Therefore, we√ can identify k by comparing Einstein’s equations with (4.2.8). For Λ > 0, a coordinate change Λ0ρ¯ = sin ψ, transforms (4.2.6) into, 0 2 2 2 Λ ds := dψ + sin ψdΩd−1, (4.2.9) where we have deﬁned Λ0 := 2Λ/d(d − 1). The transverse metric is now

0 −1 hψψ = (Λ ) , (4.2.10) 0 −1 2 hφ1φ1 = (Λ ) sin ψ, 0 −1 2 2 2 2 hφiφi = (Λ ) sin ψ sin φ1 sin φ2 ··· sin φi−1, thus the metric, is subject to an overall scale transformation. If Λ < 0, exchange the trigonometric functions by hyperbolic ones. Next is to find the Ricci scalar R¯, given the metric (4.2.9), and in the ADM-system with Gaussian coordinates; the first partial derivatives of the transverse metric is zero. Hence, one can ignore the product of Christoffel symbols as appearing in the Riemann tensor components. What is left for the Ricci scalar is the simple expression

¯ ab ¯ ¯c ¯c R = h Rab = ∂cΓab − ∂bΓac (4.2.11) √ 1 ab cd ab 1 = h ∂c h hab,d − h ∂a √ ∂b h . 2 h

20 Due to the spherical symmetry of our system, there are simpliﬁcations to be made. Suppose 0 ¯ we perform a rotation (ψ, φ1, ··· , φd−1) → (ψ , φ10 , ··· , φ(d−1)0 ). Then the component Rψψ transforms as 2 d−1 2 ∂ψ X ∂φj R¯ 0 0 = R¯ + R¯ = R¯ , (4.2.12) ψ ψ ∂ψ0 ψψ ∂ψ0 φj φj ψψ j=1 as it should be invariant under such coordinate transformations. The form dl02 in (4.2.9), transforms as

02 2 2 dl = dψ + sin ψdΩd−1 (4.2.13)   2 2 d−1 i−1 2 ∂ψ ∂φ1 X Y ∂φi + dψ02 + sin2 ψ dψ02 + sin2 φ + ··· ∂ψ0  ∂ψ0 ∂ψ0 j i=2 j=1 and as dl02 is an invariant with respect to the transformations in (4.2.13), the constraint on the rotations are   2 2 d−1 i−1 2 ∂ψ ∂φ1 X Y ∂φi + sin2 ψ + sin2 φ = 1. (4.2.14) ∂ψ0  ∂ψ0 ∂ψ0 j i=2 j=1

Substitution of (4.2.14) in (4.2.12), and rearranging terms, gives,

∂φ 2 0 = 1 R¯ − sin2 ψR¯ (4.2.15) ∂ψ0 φ1φ1 ψψ    d−1 2 i−1 X ∂φi Y + R¯ − R¯ sin2 ψ sin2 φ .  ∂ψ0  φiφi ψψ j j=1 j=1

As the general transformations are non-zero, we conclude that the coeﬃcients must be zero, and hence ¯ ¯ 2 Rφ1φ1 − Rψψ sin ψ = 0, (4.2.16) ¯ 2 2 Rφ2φ2 − Rψψ sin ψ sin φ1 = 0, ¯ ¯ 2 2 2 Rφ3φ3 − Rψψ sin ψ sin φ1 sin φ2 = 0, . .

Contraction of (4.2.16) with the inverse of (4.2.10), excluding Λ0, reveals that R¯φi − R¯ = 0, φi ψψ for i = 1, 2, ··· , d − 1. This implies the simple relation R¯ − dR¯ψψ = 0, that is to calculate the Ricci scalar R¯ we need only to find the explicit expression of the component R¯ψψ. From (4.2.11) we find √ 0 1 d(d − 1) R¯ = dR¯ψψ = −dΛ ∂ψ √ ∂ψ h = . (4.2.17) h ρ2 Substitution of (4.2.17) in (4.2.5), and passing the limit of ρ → ρ¯, gives the final expression for the trace of the momenta as √ 2Λ 1/2 Π = −σ d(d − 1) h ρ¯−2 − i,o , (4.2.18) i,o i,o d(d − 1)

21 where σi,o := ±1 are defined such that, for example, σi = +1(σi = −1) if the unit normal points in increasing (decreasing) area of the d-spheres concentric with the membrane in the inside region.ρ ¯ is the proper radius of the membrane, as measured in the inside and outside regions. The integration of the gravitational equations of motion in (2.4.9), over a small proper distance just encompassing the history, picks up the singularity at the history. The difference of the trace of the momenta, as defined in the outside and inside regions, are defined as √ Πo − Πi = hκmd. (4.2.19)

The average of Π, Πon, is found from (3.0.10) and (4.1.43), as √ 2e hE Π + Π = − on , (4.2.20) o i m(d − 1) thus by combining (4.2.20) with (4.2.19), the Π’s in terms of the electric ﬁeld, the mass of the membrane, the charge coupling, is 1 eE h−1/2Π = κmd − on (d − 1), (4.2.21) o 2 m 1 eE h−1/2Π = κmd + on (d − 1). i 2 m

So by (4.2.21) and (4.2.18), the signs of σi,o is found to be

1 1 κm2 σ = sign eE − e2 − , (4.2.22) o d o 2 2(d − 1) 1 1 κm2 σ = sign eE − e2 + . i d o 2 2(d − 1)

Now, to ﬁnd the proper radius of the membrane, one can pick either the outside or the inside Π as deﬁned in (4.2.21), and compare it with (4.2.18). Then by simple calculations one reveals that 2Λ 1 1 κm2 2 ρ¯−2 = o + eE − e2 − . (4.2.23) d(d − 1) m2 o 2 2(d − 1) This concludes the instanton solutions for the theory (2.1.1).

22 4.3 Complete Set of Conﬁguration Parameters The instanton solutions from the euclidean action (2.1.1), is deﬁned by the group of equations, 1 Λ = λ + κE 2, (4.3.1a) o 2 o 1 Λ = Λ − κ eE − e2 , (4.3.1b) i o o 2

Ei = Eo − e, (4.3.1c) 1 E = E − e, (4.3.1d) on o 2 1 1 κm2 σ = sign eE − e2 − , (4.3.1e) o d o 2 2(d − 1) 1 1 κm2 σ = sign eE − e2 + , (4.3.1f) i d o 2 2(d − 1) ( )−1/2 2Λ 1 1 1 κm2 2 ρ¯ = o + eE − e2 − , (4.3.1g) d(d − 1) m2 d o 2 2(d − 1) which relates the different parameters that defines the configurations of the systems. A nice feature of the collection (4.3.1) is that the inside can be recast in terms of the outside, so the set of independent equations in (4.3.1) is actually 5, rather than 7. As the proper radius is a non-negative quantity, we find a first restriction

1 1 κm2 2 2m2Λ eE − e2 − ≥ − o , (4.3.2) d o 2 2(d − 1) d(d − 1) which is automatically satisfied if Λo ≥ 0. How many types of configurations exists for the instanton? For a given region, σ = ±1, Λ > 0, Λ ≤ 0. This returns 16 configurations. But these configurations are not independent of each other. Suppose σo = +1, then the relation between the inside and outside is Λo < Λi. That means that for this particular choice of σo, it is not possible for having Λi > 0 and Λo < 0. Now, let σi = −1, then it is not possible to have Λi < 0 and Λo > 0. Due to the fact that σi ≥ σo, realizes that σi = −1 and σo = −1, in the same configuration, is also not possible. This exhaust the redundant configurations, reducing the 16 to 10. The possible solutions may be arranged in groups, labeling the instantons accordingly to types:

• Type 1: All instantons, excluding the conﬁgurations Λi ≤ 0, σi = −1, or Λo ≤ 0, σo = −1,

• Type 2: Λi,o ≤ 0, σo = −1, but excluding Type 3,

• Type 3: Λo ≤ 0, σo = −1, with Λi ≤ 0. σi = −1 Each of these types will be investigated, clarifying what conﬁgurations are possible for membrane creation to take place.

4.4 Extremizing of The Action If the equations of motion for the antisymmetric ﬁeld, are satisﬁed, (2.1.1) reduces to Z √ 1 Z √ 1 Z √ Π SE = m ddx h − dDx h(R − 2Λ) + ddx h , (4.4.1) 2κ κ d − 1

23 (Λ,σ) outside + + + - - + -- (Λ, + + σ ) - + + - - - inside

Figure 1: Indicated by green color; the possible conﬁguration values for Λ and σ. For example the set +− means Λ ≥ 0, σ = −1. where time gauge, and ADM coordinates, are implied. Suppose we are residing in either the inside region or the outside region. Then, from the gravitational equations of motion in (2.4.9), R = 2DΛ/(d − 2). Substitution of this into (4.4.1) gives the classical euclidean action as Z √ 2 Z √ 1 Z √ Π SE = m ddx h − dDx hΛ(x) + ddx h (4.4.2) κ(d − 1) κ d − 1 2Λ 1 := mA (¯ρ) − V (¯ρ, σ, Λ) + A (¯ρ)Π(¯ρ, σ, Λ), d κ(d − 1) D κ(d − 1) d where we have deﬁned Z √ 2πD/2 A (¯ρ) := ddx h = ρ¯d, (4.4.3) d Γ(D/2) Z D √ Vd(¯ρ) := d x g.

Suppose Λo, σo, is the false vacua, that is let it deﬁne the background from which the tunneling occur. Then as the net phase contribution to the probability is B = SE[Instanton] − SE[Background], we ﬁnd B as

2Λ dσ 2Λ 1/2 B = mA (¯ρ) − V (¯ρ, σ , Λ ) − i A (¯ρ) ρ¯−2 − i − (i ↔ o), (4.4.4) d κ(d − 1) D i i κ d d(d − 1) where we have used the deﬁnition for the trace of the momenta. The task now is to maximize B with respect to the proper radiusρ ¯. Suppose that the background and instanton space-time, is de-Sitter: Λi,o > 0, and σi = +1. Then, by diﬀerentiating with respect to the proper radius, one can resolve, √ ρκ¯ ∂B dΠ d2(d − 1) h dρ¯Λ0 ∂V i√ D − = κmd + − 2 − − (i ↔ o). (4.4.5) Ad(¯ρ) ∂ρ¯ (d − 1) h ρ¯ Πi Ad(¯ρ) ∂ρ¯

Using the metric (4.2.9), valid for de-Sitter space-time, together with σi = +1, we can solve for the inside volume and ﬁnd

Z √ A (¯ρ) Z b(¯ρ) V (¯ρ, σ , Λ ) = dDx g = |Λ0 |−D/2 d d(cos ψ) sind−1 ψ , (4.4.6) D i i i d inside ρ¯ 1

24 0 2 1/2 where b(¯ρ) := σi(1 − Λiρ¯ ) . Diﬀerentiation of (4.4.6) with respect toρ ¯, gives Z b(¯ρ) ! Z b(¯ρ) ∂VD,i 0 −D/2 Ad(¯ρ) d−1 ∂ d−1 = |Λi| d sign d(cos ψ) sin ψ d(cos ψ) sin ψ (4.4.7) ∂ρ¯ ρ¯ 1 ∂ρ¯ 1 √ d(d − 1) h A (¯ρ) = d . Πi ρ¯ Using the results from (4.4.7) in (4.4.5), the extremal of B with respect toρ ¯, reveals, ρκ¯ ∂B Π Π 0 = − = κmd + √i − √o . (4.4.8) Ad(¯ρ) ∂ρ¯ h h But (4.4.8) is just the same as (4.2.19), conﬁrming the equality of the results.

5 Neutralization of The Cosmological Constant

By the result of (4.4.6), the volume function VD is proportional to the magnitude of the integral over ψ. The magnitude itself is a function of b(¯ρ, Λ, σ), that is the parameter values governs the character of the volume function. Looking at (4.4.7), the change of volume with respect to proper radius is ∂ −1/2 V (¯ρ, σ, Λ) ∼ −σ 1 − Λ0ρ¯2 ρ¯d−1. (5.0.1) ∂ρ¯ D For Type 2 instantons, the instanton volume remains finite up to the proper radius. The background volume attains the values Λo ≤ 0, σo = −1. Thus for largeρ ¯, integrating (5.0.1) with respect toρ ¯ results in A (¯ρ) V (¯ρ, −1, Λ ≤ 0) ∼ +|Λ0 |−1/2 d + V , (5.0.2) D o o d c where ∂ρ¯Vc = 0. So the volume diverges at largeρ ¯. This means that, as the instanton remains bounded asρ ¯ → ∞ the background blows up, B → ∞, and so P ∼ e−B/~ → 0. For Type 3 instantons, both the instanton and background action, diverges at large proper radius, thus one must look at the difference in the actions. One may find that d − 1 B ∼ Λ (¯ρ) |Λ0 |1/2 − |Λ0 |1/2 . (5.0.3) κ d o i

As σi = −1, (4.3.1e) and (4.3.1f) shows that |Λo| > |Λi|, hence (5.0.3) has the property B → ∞, and therefore P → 0. In conclusion, the only conﬁgurations possible for membrane creation to take place are the Type 1 instantons, that is those green squares of coordinates in (4.3), excluding conﬁgurations of inside or outside values of (−−).

5.1 Mass and Charge Dependence So far, the treatment of the theory has governed the viability of the instanton parameters to imply a ﬁnite phase contribution to the probability of a membrane to be created. No major assumption on the mass m of the membrane itself nor the charge coupling e, has been made. We concluded that the phase B remains bounded for Type 1 instantons, so the treatment of B1 := B(Type 1) is of interest. Without further ado, by diﬀerentiating equation (4.4.4) with respect to the membrane mass m, produces ∂B 1 = A (¯ρ), (5.1.1) ∂m d

25 that is rate of change of B1 with respect to the proper radius, is proportional to the surface area Ad of the membrane. This implies that B is monotonically growing with increasing mass and so the probability of a membrane to be created, decreases with increasing m. By diﬀerentiating the euclidean action (2.1.1), with respect to the magnitude of the charge coupling e, one ﬁnds

∂S ∂S Z Z Z E = E sign(e) = A = dA = F, (5.1.2) ∂e ∂|e| ∂M M M where F = dA is the field strength. The integral over ∂M covers the instanton on the inside, and the membrane itself. The inclusion of the surface term in (2.1.1) patches the inside and outside, as the inside can not be covered with A on a single patch. By assumption, this surface term is away from the instanton, residing in the outside region at some fixed E = Eo as a boundary condition. Therefore, patch 1 covers the inside, the membrane and a finite portion of the outside. Patch 2 covers the outside region of the instanton. By consequence, the patches must cover the inside, but not necessarily the outside.

O M I

Figure 2: Integration over the potentials: O := outside, I := inside, M := membrane. All regions to the right of the red line is patch 1, and the region to the left of the green line, is patch 2. Integrations are with respect to the distance between the solid lines, where the distance tends to zero. The arrows extend such that the solid lines covers the respective integration regions.

26 From (3.0.5) the diﬀerence in the given surface orientation to the induced orientation, is −, thus the integration in (5.1.2) is Z √ ∂B1 D = −sign(e) d x hE = −sign(e)(Eo − e)VD(¯ρ, σi, Λi), (5.1.3) ∂|e| inside

= −[ sign(eEo)|Eo| − |e|]VD(¯ρ, σi, Λi). where we have used (4.3.1). Now, suppose the magnitude of the outside electric ﬁeld Eo, is small compared to the magnitude of the charge coupling e, that is |Eo| < |e|. From the set of equations 1 (4.3.1) with σo = +1, one ﬁnds that Eo − 2 e > 0 hence, E |e| |E |sign(eE ) − |e| = e o − |e| > − , (5.1.4) o o |e| 2 showing that ∂B1/∂|e| > 0. Thus B1 increases with increasing |e|, that is, the probability for membrane creation, decreases with increasing |e|, where |Eo| < |e|. If on the other hand, |Eo| > e, then for = +sign(eEo), σi = +1, we have ∂B1/∂|e| > 0 and hence the probability increases with increasing |e|. For = −sign(eEo) the probability will decrease with increasing |e|. For the reduction process of Λ, we will direct a particular interest in the case = +sign(eEo), |Eo| |e|, where the probability increases with increasing magnitude of the charge coupling.

5.2 Reduction of The Cosmological Constant

Given a particular set of variables, , e, m, and Eo; suppose that Λi > 0. As previously found, σo = +1 forces the inequality Λi < Λo. Similarly, for σo = −1, Λi > Λo. Now, assume that |Eo| is small, that is, 2|Eo| < |e|. From (4.3.1) one can resolve,

1 sign(Λ − Λ ) ∼ sign κe2 > 0, (5.2.1) o i 2 thus whenever 2|Eo| < |e|, the membrane production always increases the cosmological constant. But if 2|Eo| > |e|, then sign(Λo − Λi) = sign(eEo), (5.2.2) so the cosmological constant increases when sign(eEo) = −1, and decreases when sign(eEo) = +1. Which of these values of sign(eEo), characterizing the membrane creation processes, is more likely to occur? Define the difference in B evaluated at = +1 and = −1, respectively, as D(m) := B( = +sign(eEo)) − B( = −sign(eEo)). (5.2.3) We will show that the function D(m) is always negative, that is the phase contribution to the probability, is larger for = −1, and consequently increases the probability itself. Differentiating (5.2.3) with respect to the membrane mass m, gives the expression

( !)−d/2 ∂D(m) 2π(d+1)/2 2Λ 1 1 e2 κm2 2 = o + |eE | − − (5.2.4) ∂m Γ((d + 1)/2) d(d − 1) m2 d o 2 2(d − 1) ( !)−d/2 2π(d+1)/2 2Λ 1 1 e2 κm2 2 − o + |eE | + + , Γ((d + 1)/2) d(d − 1) m2 d o 2 2(d − 1)

27 where we have used (5.1.1), (4.4.3), and (4.3.1g). This expression can not be zero, for any m, thus D(m) is a monotonic function of the mass m. In the limit of large membrane mass, m → ∞, we ﬁnd from (4.3.1), with σo = −1, σi = +1, andρ ¯ → 0, the asymptotic of (5.2.3) to be, D(m) ∼ m−1 → 0, (5.2.5) as m → ∞. In the massless limit, m → 0, the sign of σi and σo, can be found from (4.3.1) to be equivalent, and equal to, 1 e2 σ = σ = sign eE − . (5.2.6) i o d o 2

The sign of σi and σo is determined purely from the sign of the factor eEo. Thus σi = σo = sign(eEo). Asρ ¯ tends to zero, m → 0, we ﬁnd that 2Λ 2Λ B() ∼ − i V (0, sign(eE ), Λ ) + i V (0, sign(eE ), Λ ). (5.2.7) κ(d − 1) D o i κ(d − 1) D o o

For σi = +1, this expression vanishes, but for σi = −1, the volume functions in (5.2.7) are the volume of de-Sitter space, which is proportional to (Λ)−D/2. Then, since (5.2.2) holds,

h −d/2 −d/2i B( = −sign(eEo)) ∼ − (Λi) − (Λo) > 0. (5.2.8)

Hence, since B vanishes for sign(eEo) = +1, one can resolve, D(m) < 0, m → ∞, (5.2.9) and consequently the probability of a membrane to be created, is more likely to occur when −sign(eEo) = +1, that is −sign(Λo − Λi) = +1. But this is the same as Λi < Λo, thus the cosmological constant is reduced in the membrane creation. This concludes the proposition that D(m) < 0. As we know the specific configuration of which membrane creation process is more probable, the care of asking for a simple algorithm to reduce the cosmological constant, is concerned. Thus we want to find the range of certain configuration values, especially Λ and E. The following shows how these values changes with successive created membranes, and we suppose σo = +1. Suppose that Λo ≤ 0. By the instanton solutions (4.3.1), the proper radius can only be non-negative. We find s e2 kdm2 −2Λ |eE | ≥ + + m o , (5.2.10) o 2 2(d − 1) d(d − 1) where we have solved the equation forρ ¯ in (4.3.1), in terms of the magnitude of the initial electric field, |Eo|. The sign of the second equation in (4.3.1), indicates as previously that we must have Λi < Λo, that is membrane creation always reduces the cosmological constant. Suppose now 1 2 that the initial electric field Eo, is large enough, so that Λo = λ + 2 κEo > 0, also let λ < 0, and assume that |e| is very small, compared to |Eo|. From (4.3.1), the consequence of the creation of a membrane, the outside cosmological constant and magnitude of the electric field, changes as,

Λo → Λi ≈ Λo − κ|eEo|, (5.2.11)

|Eo| → |Ei| ≈ |Eo| − |e|, thus the magnitude of the outside cosmological constant, gets reduced by a factor proportional to the magnitude of eEo, and the magnitude of the outside electric ﬁeld, is reduced by the

28 magnitude of the charge coupling. This process can be repeated in recursion, and we can build up the reduction process by successive creation of membranes. That is, we have a continuous process, governed by the general rules:

Λ → Λ − κ|eE|, (5.2.12) |E| → |E| − |e|, subject to the restriction that the outside electric field is large enough, so there is energy to dispense in the membrane creation process. The reduction continuous freely, even to a point where Λo ≤ 0, and at that point (5.2.10) is satisfied. The inequality (5.2.10) will certainly not hold when the inequality ≤ is changed to >, that is the evolution of E and Λ, comes to a halt when (5.2.10) no longer holds. Suppose that E = Ef , and Λ = Λf , are the values of the electric field and the cosmological constant, respectively, when such a halt occur. Then by reversing the inequality in (5.2.10), and taking the magnitude |E| to be large compared to |e|, we obtain at the halt: kdm2 |eE | < . (5.2.13) f 2(d − 1)

From the general rule (5.2.12), we ﬁnd that Λf can take values in the range,

− k|eEf | < Λf ≤ 0. (5.2.14)

6 Conclusion

The mechanism of neutralizing the cosmological constant through membrane creation, indeed seems to be a possible candidate for a natural process, but the conditions on parameters and fields, are of very specific nature. What comes from experimental data, indicates the current space-time to be de-Sitter, and this sheds hope given that the most probable process of neutralization starts from de-Sitter space-time. In the main paper given by Brown and Teitelboim[1], the analysis is thorough but lack some clarification and motivation in the derivations. Indeed, the inclusion on the definitions of distributions on the manifold, is a subject left out in this paper, but the main objective with this thesis was to understand how the mechanics can be realized, using intuitive standard methods as in Brown and Teitelboim[1]. The book by Castillo[4] contains many of the theoretical tools which lies in my interest, but has been left out due to time investment.

29 7 Appendix A Mathematical Deﬁnitions

Let M be a orientable, differentiable manifold, and denote the set of p-forms on M by Λk(M). Suppose a ∈ Λp(M), and b ∈ Λq(M). We define the exterior product of the p-form a with the q-form b, as (p + q)! a ∧ b ≡ A(a ⊗ b), (A.0.1) p!q! where A is defined as the antisymmetrization operator, given by, 1 X AT (X , ··· X ) ≡ (sign σ)T (X , ··· X ). (A.0.2) 1 k k! σ(1) σ(k) σ∈Sk

Here, T is an arbitrary tensor field, Xi are vector fields on M, Sk is the group of all permutations of the numbers (1, 2, ··· , k), and the sign is −1 if this permutation is odd, and +1 if the permutation i even. Now, a covariant tensor tx of rank d + 1, at the point x, is defined through the mapping tx : Tx(M) × · · · × Tx(M) → R, where Tx(M) is the tangent space at p ∈ M. Thus given a coordinate chart on M,(x0, ··· , xD), ∂ ∂ ∂ i1 i2 id+1 tx(X1, X2 ··· , Xd+1) = tx dx (X1) , dx (X2) ··· , dx (Xd+1) (A.0.3) ∂xi1 ∂xi2 ∂xid+1

i1 i2 id+1 = tx ∂i1 , ∂i2 , ··· , ∂id+1 dx ⊗ dx ⊗ · · · ⊗ dx (X1, X2 ··· , Xd+1), which shows that i1 i2 id+1 tx = (tx)i1,i2,··· ,id+1 dx ⊗ dx ⊗ · · · ⊗ dx . (A.0.4) The anti-symmetrization operator A, as deﬁned in (A.0.2), when operated from the left on a tensor product of forms, gives the exterior product as deﬁned in (A.0.1). Since (A.0.4) is a tensor product of 1-forms, and the components (tx)i1,i2,··· ,id+1 are totally skew-symmetric in all their indices and tx = A(tx). We have

tx = A(tx) (A.0.5)

i1 i2 id+1 = (tx)i1,i2,··· ,id+1 A dx ⊗ dx ⊗ · · · ⊗ dx 1 = (t ) dxi1 ∧ dxi2 ∧ · · · ∧ dxid+1 . (d + 1)! x i1,i2,··· ,id+1

A.1 Exterior Derivative ∞ Let g ∈ C (M); the diﬀerential of g at the point y ∈ M, dgy, is deﬁned by

dgy(Xy) ≡ Xy[g], Xy ∈ Ty(M). (A.1.1)

Let gx ∈ Tx(M) × · · · × Tx(M), be a contravariant tensor of rank (k, 0), and take the chart 0 D (x , ··· , x ). The diﬀerential of gx becomes,

dgx = Xx [g] (A.1.2) ∂ = dxi g dxj1 ⊗ dxj2 ⊗ · · · ⊗ dxjk ∂xi j1j2···jk ∂ = g dxi ⊗ dxj1 ⊗ dxj2 ⊗ · · · ⊗ dxjk . ∂xi j1j2···jk

30 Anti-symmetrization of dgx, Adgx, returns Adgx = dgx, thus,

dgx = A(dgx) (A.1.3)

i j1 j2 jk = A ∂i gj1j2···jk dx ⊗ dx ⊗ dx ⊗ · · · ⊗ dx 1 X h i = (sign σ)∂ g dxσ(i) ⊗ dxσ(j1) ⊗ dxσ(j2) ⊗ · · · ⊗ dxσ(jk) (k + 1)! σ(i) σ(j1),σ(j2),··· ,σ(jk) σ∈Sk

j0 j1 j2 jk = ∂[j0 gj1···jk]dx ∧ dx ∧ dx ∧ · · · ∧ dx , where we have deﬁned, 1 ∂ g ≡ ε j0j1···jk ∂ g , (A.1.4) [i0 i1···ik] (k + 1)! i0i1···ik j0 j1···jk ··· where ε ··· is the Levi-Civita symbol.

A.2 Lie Derivative Let φ be a one-parameter group of transformations on M with X being the infinitesimal generator −1 ∗ with respect to φ, and let t be a tensor field of type (0,k) on M. If the limit limh→0 h (φh(t)−t) ∞ exists, it is called the lie derivative with respect to X and is denoted by £X t. Let f ∈ C (M), ψ∗ : M → N be a differentiable map, and define the pullback of df as ψ∗df = d(ψ∗f). Then the Lie derivative on df with respect to the infinitesimal generator X is ∗ ∗ φh(df) − df dφhf − df £X df ≡ lim = lim = d(£X f). (A.2.1) h→0 h h→0 h The lie derivative on f with respect to X is, ∗ φhf(x) − f(x) f(φ(h, x)) − f(x) dφ £X f(x) = lim = lim ≡ (h, x) [f] (A.2.2) h→0 h h→0 h dh h=0 = Xx[f] = (Xf)(x), thus £X f = Xf. For a (0, k)-tensor t and a function f, we define £X (ft) ≡ f£X t + (£X f)t. Now, the Lie derivative on the (0, k)-tensor t with respect to X, in some local chart, is

i1 i2 ik £X t = £X ti1i2···ik dx ⊗ dx ⊗ · · · ⊗ dx (A.2.3)

i1 i2 ik = (Xti1i2···ik )dx ⊗ dx ⊗ · · · ⊗ dx

i1 i2 ik i1 i2 ik + ti1i2···ik [d(£x ) ⊗ dx ⊗ · · · ⊗ dx + ··· + dx ⊗ dx ⊗ · · · ⊗ d(£x )]

A.3 Pull-back ¯ Let ψ : M → N be a differentiable map. This induces a differentiable mapping ψ : Tp(M) → ¯ Tq(N), defined by ψ(X) ≡ ψ∗(X), for X ∈ Tp(M). Then, let t be a covariant tensor of rank (0, k), on N. The pullback on the tensor t, at p ∈ M, is defined as ∗ 1 k 1 k (ψ t) X , ··· ,X ≡ tψ ψ∗X , ··· , ψ∗X , (A.3.1) i i 0 k where X ∈ Tp(M), p ∈ M. We see that ψ∗X ∈ Tq(N), thus if (x , ··· , x ) is a chart in M, 0 l 1 (y , ··· , y ) is a chart in N, then we can expand the vector ψ∗X as ∂ ∂ ∂ ∂ 1 i 1 1 i j1 1 i ψ∗X = dy (X ) ≡ X [y ] = dx (X ) [y ] , (A.3.2) ∂yi ∂yi ∂xj1 ∂yi

31 thus the pullback of t, becomes

∗ 1 k 1 k (ψ t) X , ··· ,X = tψ ψ∗X , ··· , ψ∗X (A.3.3) ∂yi1 ∂ ∂yik ∂ j1 1 jk k = tψ dx (X ) , ··· , dx (X ) , ∂xj1 ∂yi1 ∂xjk ∂yik ∂ ∂ ∂yi1 ∂yik j1 jk 1 k = tψ , ··· , ··· dx ⊗ · · · ⊗ dx (X , ··· ,X ), ∂yi1 ∂yik ∂xj1 ∂xjk and we conclude that,

∂ ∂ ∂yi1 ∂yik ∗ j1 jk ψ t = tψ , ··· , ··· dx ⊗ · · · ⊗ dx . (A.3.4) ∂yi1 ∂yik ∂xj1 ∂xjk

A.4 Hodge Dual The Hodge dual is defined as the mapping ? :Λk(M) → Λn−k(M), where if ω is a p-form on M, ω ∈ Λp(M). Let α ∈ Λk(M), and β ∈ Λn−k(M), then since α ∧ β is defined, there exists a function g ∈ C∞(M), such that α ∧ β = fη. By definition:

i1 jk ik+1 jn α ∧ β = αi1,··· ,ik βik+1,··· ,in dx ∧ · · · ∧ dx ∧ dx ∧ · · · ∧ dx , (A.4.1)

i1···ik jk+1···jn 1 2 n = αi1,··· ,ik βik+1,··· ,in ε ε dx ∧ dx ∧ · · · ∧ dx , We define g to be the fully symmetric covariant tensor of rank (0, 2), on M, that is 0 D g(X,Y ) = g(Y,X), and g(X,X) ≥ 0, X,Y ∈ Tp(M). Take the coordinate chart (x , ··· , x ), i D defined at p ∈ M. Then the basis of Tp(M) are {∂/∂x }i=0. We define the contraction c of the vector field X, X ∈ Tp(M), with the metric, Xcg, X ∈ Tp(M), as the operation in which Xcg is a covariant tensor of rank (0, 1). If X = Xi∂/∂xi, locally, we find, ∂ ∂ ∂ Xcg = Xi g , dxk ⊗ dxl (A.4.2) ∂xi ∂xk ∂xl i k l i l = 2X gkl δi dx = 2X gildx .

The determinant of gij is defined here, to never be zero. If ω is a co-vector field, locally given by i i ω = ωidx , then there exist only one vector field, X, such that α = (1/2)Xcg, that is, αl = X gil. jl l gij has an inverse, defined by gijg = δi. Thus the metric tensor component gij is thought of a operator that changes co-vectors into vectors, and vice versa. Thus by definition,

i1jk+1 ikjn ij jk+1···jn −1 jk+1···jn εi1···ik g ··· g = det(g )ε = (det(gij)) ε . (A.4.3) This gives for (A.4.1): q i1···ik l1jk+1 lkjn α ∧ β = αi1···ik βik+1···in |det(gij)|ε εl1···lk g ··· g η (A.4.4) ≡ fη.

The Hodge dual, ?, is also deﬁned as the inner product,

j1ik+1 jkin f = (?α, β) ≡ k!(?α)j1···jk βik+1···in g ··· g , (A.4.5) thus comparing (A.4.4) and (A.4.5), we ﬁnd 1 q (?α) = αi1···ik |det(g )|ε . (A.4.6) jk+1···jn k! ij i1···ikjk+1···jn

32 A.5 Covariant Derivative Let M be a diﬀerentiable manifold. A connection on M assign to each vector ﬁeld X, X ∈ X(M) ∞ an operator ∇X from X(M) into itself, such that for all X,Y,Z ∈ X(M), a, b ∈ R, f ∈ C (M),

∇X (aY + bZ) = a∇X Y + b∇X Z, (A.5.1)

∇X (fY ) = f∇X Y (Xf)Y,

∇aX+bY Z = a∇X Z + b∇Y Z,

∇fX Y = f∇X Y.

1 n i j Take a chart (x , ··· , x ) of M, and let Y = Y ∂i, X = X ∂j then

j i j i ∇X Y = ∇X ∂i Y ∂j ≡ X ∇∂i (Y ∂j) (A.5.2) ∂Y j ∂Y k = Xi ∂ + Y j∇ ∂ = Xi + Y j(∇ )k ∂ . ∂xi j ∂i j ∂xi ∂i j k

3 k k The n number of diﬀerentiable functions, (∇∂i )j ≡ Γ ij characterizes the connection in this coordinate system. Let M be a Riemannian Manifold, there exists an unique connection, with i vanishing torsion, 0 = T (X,Y ) ≡ ∇X Y −∇Y X −[X,Y ], and such that ∇X g = 0. Let X = X ∂i, i j g = gijdx ⊗ dx , then by deﬁnition,

i j k 0 = ∇X g = X ∇∂i gjkdx ⊗ dx (A.5.3) i j k j k j k = X ∇∂i gjkdx ⊗ dx + gjk∇∂i dx ⊗ dx + gjkdx ⊗ ∇∂i dx .

k One can see that, Y = Y ∂k,

j j j ∇∂i dx (Y ) = ∇∂i (dx Y ∂k) − dx ∇∂i Y (A.5.4) j j k m k = ∂iY − dx ∂iY + Y Γ mi ∂k m j j m = −Y Γ mi = −Γ midx (Y ),

j j m thus, ∇∂i dx = −Γ midx , and we can resolve (A.5.3) as,

i j k j m k k j m 0 = ∇X g = X ∇∂i gjkdx ⊗ dx − gjkΓ midx ⊗ dx − gjkΓ midx ⊗ dx (A.5.5) i j j m k = X ∇∂i gmk − gjkΓ mi − gmjΓ ki dx ⊗ dx , hence we ﬁnd the relation,

j j ∇∂i gmk − gjkΓ mi − gmjΓ ki = 0. (A.5.6)

To resolve the connection coeﬃcients, to the metric coeﬃcients, we write,

∂igmk + ∂kgim−∂mgki = ∇∂i gmk + ∇∂k gim − ∇∂kmgki (A.5.7) j j j j j j = gjkΓ mi + gmjΓ ki + gjiΓ km + gijΓ mk − gjmΓ ik − gkjΓ im j = 2gijΓ km.

So we conclude that the Riemann connection, with no torsion, under the condition ∇X g = 0, is 1 Γi = gim (g + g − g ) , (A.5.8) jk 2 mj,k km,j jk,m

33 where we have used the notation ∂igjk ≡ gjk,i. The transformation property of the Christoﬀel symbols, in a local coordinate system xν , are given by,

i0 j k i0 k i0 ∂x ∂x ∂x i ∂x ∂x Γ 0 0 = Γ + , (A.5.9) j σ ∂xi ∂xj0 ∂xσ0 jk ∂xk ∂xj0 ∂xk0 thus the Christoffel symbols are not tensors. By considering another Christoffel symbol, Γ, with respect to the same local coordinate system as for Γ, defining a different connection, the difference of Γ and Γ,

i0 i0 i0 δΓ j0k0 ≡ Γ j0k0 − Γ j0k0 (A.5.10) 0 0 0 ∂xi ∂xj ∂xk ∂xi ∂xk ∂xi ∂xk = Γi − Γi + − ∂xi ∂xj0 ∂xk0 jk jk ∂xk ∂xj0 ∂xk0 ∂xk ∂xj0 ∂xk0 0 ∂xi ∂xj ∂xk = δΓi , ∂xi ∂xj0 ∂xk0 jk we see that the diﬀerence transform as a tensor.

A.6 Curvature Tensor The curvature tensor R, of the connection ∇, is the mapping R : X(M) × X(M) → X(M), given by R(X,Y ) ≡ ∇X ∇Y − ∇Y ∇X − ∇[X,Y ], (A.6.1) where [X,Y ] ≡ XY − YX is the Lie bracket. R is a rank (1, 3) tensor, it is not a tensor in the sense of that it does not map to a real number, but it is equivalent to the operation R(X,Y )Z, X,Y,Z ∈ X(M). Take function f ∈ C∞(M), then the equation, R(X,Y )(fZ) = fR(X,Y )Z, (A.6.2)

1 D i j k holds. Let With (x , ··· , x ) be a chart. We ﬁnd for X = X ∂i, Y = Y ∂j, and Z = Z ∂k:

R(X,Y )Z = ∇X (∇Y Z) − (X ↔ Y ) (A.6.3) j k j k = ∇X Y (∂jZ )∂k + Y Z ∇∂j ∂k) − (X ↔ Y ) i j k j k j k = X (∂iY )(∂jZ )∂k + Y (∂i∂jZ )∂k + Y (∂jZ )∇∂i ∂k j k j k j k + (∂iY )Z ∇∂j ∂k + Y (∂iZ )∇∂j ∂k + Y Z ∇∂i ∇∂j ∂k − (X ↔ Y ) j j k = (X(Y ) − Y (X )](∂jZ )∂k i i k l + (X Y − Y X)(Z )Γ ik∂l j j k l + (X(Y ) − Y (X ))Z Γ jk∂l j j k l + (Y X − X Y )(Z )Γ jk∂l i j k l l m l m l + X Y Z ∂iΓ jk − ∂jΓ ik + Γ jkΓ im − Γ ikΓ jm ∂l. We see that by a change of dummy index, two lines cancel, and by regrouping, we ﬁnd

j j l l k i j k l R(X,Y )Z = (X(Y ) − Y (X ))(∂jZ + Γ jkZ )∂l + X Y Z Rkji∂l j j l i j k l = ∂l(X(Y ∇j) − Y (X ∇j))Z + X Y Z Rkij∂l, where we have deﬁned a covariant derivative for vector components,

l l l k ∇jZ ≡ ∂jZ + Γ jkZ , (A.6.4)

34 and the Riemann tensor components,

l l l m l m l Rijk ≡ Γ ik,j − Γ ij,k + Γ kiΓ jm − Γ jiΓ km. (A.6.5) Contraction of l and j, in (A.6.5), deﬁnes the Ricci tensor components as,

jm l j l j Rik ≡ g glmR ijk = δl R ijk = R ijk, (A.6.6)

ik i and the contraction of the Ricci tensor, yields the Ricci scalar, deﬁned by R ≡ g Rik = R i

A.7 Forms i Let ei be a local basis, for the vector ﬁelds. If ∇ is a connection on M, the connection forms Γj, 2 with respect to the basis ei, are the n 1-forms deﬁned by

i i Γ j(X) ≡ θ (∇X ej), (A.7.1)

i i 3 i i for X ∈ X(M), and θ ej = δj. The n Christoffel symbols, Γ jk, are thus defined by Γ jk ≡ i Γ j(ek). We define the torsion 2-forms as, 1 T i(X,Y ) ≡ θi(T (X,Y )). (A.7.2) 2

j k Let X = X ej, Y = X ek, then by deﬁnition, 1 T i(X,Y ) = θi (∇ Y − ∇ X − [X,Y ]) (A.7.3) 2 X Y 1 = θi ∇ (Y ke ) − ∇ (Xje ) − XjY k[e , e ] 2 X k Y j j k 1 = θi((∇ Y k)e + Y k∇ e − (∇ Xj)e − Xj∇ e ) 2 X k X k Y j Y j 1 1 = θi((∇ Y k)e − (∇ Xj)e ) − (Y k(∇ θi)e − Xj(∇ θi)e ). 2 X k Y j 2 X k Y j Since,

i i i i Γ j(X) = θ (∇X ej) = ∇X (θ ej) − (∇X θ )ej (A.7.4) i i i = ∇X δj − (∇X θ )ej = −(∇X θ )ej i k i k i k =⇒ −Γ j(X)θ = (∇X θ )ejθ = (∇X θ )δj ,

i i j thus we ﬁnd ∇X θ = −Γ j(X)θ , and so, 1 1 T i(X,Y ) = θi((∇ Y k)e − (∇ Xj)e ) + (Γi (X)θj(Y ) − Γi (Y )θj(X)) (A.7.5) 2 X k Y j 2 j j The exterior derivative of a k-form α on M, is deﬁned as,

k+1 X i+1 (k + 1)dα(X1, ··· ,Xk+1) ≡ (−1) Xi(α(X1, ··· , Xˆi, ··· Xk+1)) (A.7.6) i=1 X i+j + (−1) α([Xi,Xj],X1 ··· , Xˆi, ··· , Xˆj, ··· ,Xk+1), i

35 where the vector ﬁelds denoted by ˆ, should be omitted. Hence, form a 1-form θi, X,Y in our basis, 2dθi(X,Y ) = Xθi(Y ) − Y θi(X) − θi[X,Y ] = Xθi(Y ) − Y θi(X), (A.7.7) and the torsion 2-form yields, 1 1 T i(X,Y ) = (Xθi(Y ) − Y θi(X)) + (Γi ⊗ θj − θj ⊗ Γi )(X,Y ) (A.7.8) 2 2 j j i i j = (dθ + Γ j ∧ θ )(X,Y ), that is, i i i j T = dθ + Γ j ∧ θ . (A.7.9) Similarly, deﬁne the curvature 2-forms

i 1 i R j(X,Y ) ≡ θ (R(X,Y )ej), (A.7.10) 2 where R(X,Y ) is defined as (A.6.1), hence, i i 2R j(X,Y ) = θ (∇X ∇Y − ∇Y ∇X − ∇[X,Y ])ej, (A.7.11) i k k k = θ ∇X (Γ j(Y )ek) − ∇Y (Γ j(X)ek) − Γ j([X,Y ]), i i k i i i i = ∇X Γ j(Y ) − (∇X θ )Γ j(Y )ek − ∇Y Γ j(X) + (∇Y θ )Γ j(X)ek − Γ j([X,Y ]) i i i i l k i l k = ∇X Γ j(Y ) − ∇Y Γ j(X) − Γ j([X,Y ]) + Γ l(X)θ Γ j(Y ) − Γ l(Y )θ Γ j(X) ek i i l = 2 dΓ j + Γ l ∧ Γ j (X,Y ). Thus we find that, i i i l R j = dΓ j + Γ l ∧ Γ j. (A.7.12) By taking the differential of (A.7.8), and using (A.7.12), we find, i i j i j i j dT = d(Γ j ∧ θ ) = dΓ j ∧ θ − Γ j ∧ dθ (A.7.13) i j i l l i j = R j ∧ θ − Γ l ∧ T − dθ − Γ j ∧ dθ . If the torsion 2-form vanishes, one can resolve that i j 0 = R j ∧ θ , (A.7.14) and from (A.7.9), we find i j i k j 0 = Γ j ∧ θ = Γ jkθ ∧ θ , (A.7.15) i i hence the connection is symmetric in its lower indices, Γ jk = Γ kj. We can also write the curvature tensor, as the exterior product,

i 1 i l k R j = R jlkθ ∧ θ . (A.7.16) 2 Taking the differential of the curvature 2-form, using (A.7.12), gives i i l i l dR j = 0 + dΓ l ∧ Γ j − Γ l ∧ dΓ j (A.7.17) i i k l i l l k = R l − Γ k ∧ Γ l ∧ Γ j − Γ l ∧ R j − Γ k ∧ Γ j i l i l = R l ∧ Γ j − Γ l ∧ R j. Substituting (A.7.16) in (A.7.14), we find i k l j 0 = R kjlθ ∧ θ ∧ θ , (A.7.18) i which is the same as the first Bianchi identities, R [kjl] = 0. We can write the equal constraint, i i i 0 = R kjl + R jlk + R lkj. (A.7.19)

36 A.8 Generalized Kronecker Delta By deﬁning the generalized Kronecker delta, X δµ1···µD ≡ sign(σ)δµ1 ··· δµD , (A.8.1) ν1···νD νσ(1) νσ(D) σ∈SD where SD is the group of all the permutations of the numbers (0, 1, ··· ,D), one can determine a relation between (A.8.1) and the Levi-Civita symbol, as

εµ1···µD ε = δµ1···µD , (A.8.2) ν1···νD ν1···νD so by contraction, X εµ1µ2···µD ε = δν1 δν2 ··· δνD sign(σ)δµ1 ··· δµD . (A.8.3) βµ2···µD β µ2 µD νσ(1) νσ(D) σ∈SD

One can express the right hand side of (A.8.3) to ﬁnd that

µ1µ2···µD µ1 ε εβµ2···µD = d!δβ . (A.8.4)

References

[1] J. David Brown and Claudio Teitelboim. Neutralization of The Cosmological Constant by Membrane Creation. Nuclear Physics, B297:787–836, 1988. [2] T. R. P. Carames and E. R. Bezerra de Mello. Spherically symmetric vacuum solutions of modiﬁed gravity theory in higher dimensions. arXiv, gr-qc/0901.0814v2, 2009.

[3] Sidney Coleman and Frank De Luccia. Gravitational eﬀects on and of vacuum decay. Phys. Rev. D, 21:3305–3315, Jun 1980. [4] Gerardo F. Torres del Castillo. Diﬀerentiable Manifolds: A Theoretical Physics Approach. Springer, 2010.

[5] Mirta Iriondo, Edward Malec, and Niall O´ Murchadha. The constant mean curvature slices of asymptotically ﬂat spherical spacetimes. Phys.Rev. D54 (1996), pages 4792–4798, 1995. [6] Cliﬀord V. Johnson. D-Branes. Cambridge University Press, 2003. [7] Kate Marvel and Neil Turok. Horizons and Tunneling in the Euclidean False Vacuum. arXiv, hep-th/0712.2719, 2007. [8] C. W. Misner R. Arnowitt, S. Deser. The Dynamics of General Relativity. arXiv, gr- qc/0405109v1, 2004. [9] Robert M. Wald. General Relativity. The University of Chicago Press, 1984.