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Home , Special relativity

Variational Principle Approach to

Chakkrit Kaeonikhom

Submitted in partial fulfilment of the requirements for the award of the degree of

Bachelor of in B.S.(Physics)

Fundamental Physics & Research Unit The Tah Poe Academia Institute for & Cosmology Department of Physics, Faculty of Science Naresuan University

March 15, 2006 To Mae Variational Principle Approach to General Relativity

Chakkrit Kaeonikhom

The candidate has passed oral examination by members of examination panel. This report has been accepted by the panel as partially fulfilment of the course 261493 Independent Study.

...... Supervisor Dr. Burin Gumjudpai, BS MSc PhD AMInstP FRAS

...... Member Dr. Thiranee Khumlumlert, BSc(Hons) MSc PhD

...... Member Alongkorn Khudwilat, BS(Hons) MSc

I Acknowledgements

I would like to thank Burin Gumjudpai, who gave motivation to me to learn story that I have never known, thank for his inspiration that lead me to the elegance of Physics, thank for his explanation of difficult concepts and thank for his training of LATEX program. I also thank Daris Samart for willingness to spend his discussing to me and for help on some difficult calculations. Thanks Chanun Sricheewin and Alongkorn Khudwilat for some discussions and way out of some physics problems. I also thank Artit Hootem, Sarayut Pantian and all other members in the new founded Tah Poe Academia Institute for Theoretical Physics & Cosmology (formerly the Tah Poe of Theoretical Physics: TPTP). Thank for their encouragement that push me to do my works. Finally, thank for great kindness of my mother who has been teaching me and giving to me morale and .

II Title: Variational Principle Approach to General Relativity

Candidate: Mr.Chakkrit Kaeonikhom

Supervisor: Dr.Burin Gumjudpai

Degree: Bachelor of Science Programme in Physics

Academic Year: 2005

Abstract

General relativity theory is a theory for which Galilean relativity fails to explain. Variational principle is a method which is powerful in physics. All physical laws is believed that they can be derived from action using variational principle. Einstein’s field equation, which is essential law in general relativity, can also be derived using this method. In this report we show derivation of the Einstein’s field equation using this method. We also extend the gravitational action to include boundary terms and to obtain Israel junction condition on hypersurface. The method is powerful and is applied widely to braneworld gravitational theory.

III Contents

1 Introduction 1 1.1 Background ...... 1 1.2 Objectives ...... 1 1.3 Frameworks ...... 1 1.4 Expected Use ...... 2 1.5 Tools ...... 2 1.6 Procedure ...... 2 1.7 Outcome ...... 3

2 of classical and introduction to special relativity 4 2.1 Inertial reference frames ...... 4 2.2 Failure of ...... 4 2.3 Introduction to special relativity ...... 6

3 Introduction to general relativity 11 3.1 and curvature ...... 11 3.1.1 Transformations of scalars, vectors and ...... 11 3.1.2 ...... 13 3.1.3 Parallel transport ...... 15 3.1.4 Curvature tensor ...... 16 3.2 The ...... 18 3.3 Einstein’s law of ...... 19 3.3.1 The - tensor for perfect fluids ...... 19 3.3.2 Einstein’s field equation ...... 20

IV 4 Variational principle approach to general relativity 24 4.1 Lagrangian formulation for field equation ...... 24 4.1.1 The Einstein-Hilbert action ...... 24 4.1.2 Variation of the metrics ...... 26 4.1.3 The full field equations ...... 28 4.2 equation from variational principle ...... 29 4.3 equation with surface term ...... 30 4.3.1 The Gibbons-Hawking boundary term ...... 30 4.3.2 Israel junction condition ...... 33

5 Conclusion 38

A Proofs of identities 40

A.1 ∇cgab =0 ...... 40 ab A.2 ∇cg =0...... 40

A.3 Covariant derivative for scalar field, ∇aφ ...... 42 a a A.4 R bcd = −R bdc ...... 42 a a a A.5 R bcd + R dbc + R cdb =0...... 43 A.6 Bianchi identities ...... 43 b A.7 Conservation of : ∇ Gab = 0 ...... 44

B Detail calculation 45 B.1 Variance of electromagnetic wave equation under Galilean transformation 45 B.2 Poisson’s equation for Newtonian gravitational field ...... 46 a B.3 Variation of Cristoffel symbols : δΓbc ...... 47

V Chapter 1

Introduction

1.1 Background

Classical mechanics is useful to explain physical phenomena but it fails to explain gravity. General relativity (GR) is proposed to be a satisfactory theory for gravity. Studying GR leads to the Einstein’s field equation which can be derived in standard way. Our interest is to apply variation principle to derive the field equation in GR.

1.2 Objectives

To study general relativity and which is applied to derive, with vari- ational principle, Einstein’s field equation and Israel junction condition.

1.3 Frameworks

• To explain failure of Newtonian mechanics, special relativity.

• To study general relativity.

• To use variational principle for Einstein’s field equation and for Israel junction condition.

1 1.4 Expected Use

• To obtain Einstein’s field equation from variational method and to obtain junc- tion condition by including of surface term in action.

• A derived-in-detailed report for those who interest with thoroughly calculation from variational method in general relativity.

• Attaining understanding of concept of general relativity and having skills on tensor calculus.

1.5 Tools

• Text books in physics and .

• A high efficiency personal computer.

• Software e.g. LATEX, WinEdit and Photoshop

1.6 Procedure

• Studying special relativity.

• Studying tensor analysis and calculation skills.

• Studying concepts of general relativity.

• Studying Einstein’s field equation by evaluating .

• Studying Einstein’s field equation by variational principle.

• Including surface term in action and deriving junction condition.

• Making conclusion and preparing report and other presentation.

2 1.7 Outcome

• Understanding of basic ideas of , special relativity and gen- eral relativity.

• Attaining skills of tensor calculation.

• Understanding in detail of the variation method in general relativity.

3 Chapter 2

Failure of classical mechanics and introduction to special relativity

2.1 Inertial reference frames

Newton introduced his three laws of as of classical mechanics. These laws have successfully explained motion of most objects known to us. The Newton’s first law states that a body remains at rest or in uniform motion. This law introduces a called inertial frame. All proceeded dynamical laws base on this law. But what and where is it? To know or to measure of a particle, we need a frame of reference. For example when we measure of a car, a particular spot on ground is inertial frame. The ground is on the Earth and is not really inertial frame due to gravity. Furthermore, all in the possess gravity therefore nowhere is really locate on true inertial frame! However we will discuss about inertial frame again in chapter 3.

2.2 Failure of Galilean transformation

In Newtonian mechanics, the concept of time and are completely separable. Furthermore time is assumed to be absolute quantity and independent of any ob- servers. Consider inertial frames of reference moving with constant velocity to each other. In classical mechanics there is transformation law between two inertial frames.

4 v

vt (x, y, z, t), (x’, y’, z’, t’)

Figure 2.1: of two inertial frames

That is so called Galilean transformation:

x0 = x − vt y0 = y z0 = z t0 = t. (2.1) Newton’s laws are with respect to Galilean transformation

0 0 Fi = mx¨i = mx¨i = Fi (2.2) The principle of Galilean transformation yields that the velocity of in two dif- ferent inertial frames is measured with different values u0 = c − u and u = c. (2.3) However the Galilean transformation fails to explain electromagnetic wave equation; electromagnetic wave equation is not invariant under Galilean transformation. Con- sider electromagnetic wave equation: 1 ∂2φ ∇2φ = c2 ∂t2 or ∂2φ ∂2φ ∂2φ 1 ∂2φ + + − = 0. (2.4) ∂x2 ∂y2 ∂z2 c2 ∂t2

5 Figure 2.2: Illustration of four-dimensional , time axis is orthogonal to all three-dimensional spatial axes

Using chain rule and equation (2.1) to transform coordinate, the wave equation becomes c2 − v2 ∂2φ 2v ∂2φ ∂2φ ∂2φ 1 ∂2φ + + + − = 0. (2.5) c2 ∂x02 c2 ∂t0∂x0 ∂y02 ∂z02 c2 ∂t02 This equation contradicts to Einstein’s postulates in special relativity that physical laws should be the same in all inertial frames. Therefore we require new transforma- tion law, which we shall discuss in the next section.

2.3 Introduction to special relativity

In framework of Newtonian mechanics we consider only three-dimensional space and flat while in special relativity we consider space and time as one single entity called spacetime. We do not separate time from space and we have four of spacetime. Considering four-dimensional spacetime, time axis is orthogonal to all axes of space (x, y, z). The mathematical quantity which represents space and time is components of spacetime coordinates

xa = (x0, x1, x2, x3) = (ct, x, y, z) in Cartesian coordinate (2.6)

Special Relativity (SR) is a theory for physics in flat spacetime called Minkowski spacetime. If we are talking about spacetime, we must have events. Any events

6 Figure 2.3: Lightcone and trajectory of a particle in four-dimensional spacetime possess four coordinates describing where and when the particle is. Trajectories of particles in four-dimensional spacetime form a set of particles’ worldline. If we reduce to one dimensional space, we will have two spatial dimensions left. Plotting these two spatial coordinate axes versus time, we have as shown in Fig. 2.2 The electromagnetic wave equation introduces in which √ is c = 1/ µ0²0, where µ0 and ²0 are permeability and permittivity of free space respectively. The value c is independent to all reference frame. This statement is confirmed by Michelson-Morley experiment. Einstein’s principle of special relativity states that

• The law of physical phenomena are the same in all inertial reference frames.

• The velocity of light is the same in all inertial reference frames.

The spacetime interval or line-element in four dimensional spacetime is

ds2 = −c2dt2 + dx2 + dy2 + dz2. (2.7)

We can see that if ds2 = 0, this equation becomes equation of spherical wave of light with radius cdt. We can classify four-vectors into 3 classes namely spacelike, lightlike and timelike vectors. Any line elements are called

spacelike if they lie in region ds2 < 0 lightlike or null if they lie in region ds2 = 0 timelike if they lie in region ds2 > 0.

7 equation (2.7) can be expressed in general form as

X3 2 a a b a b ds = dx dxa = gabdx dx = gabdx dx . (2.8) a,b=0

We use Einstein’s summation convention. In this convention we do not need to use summation symbol in equation (2.8). If upper indies and lower indies of four-vectors are similar (repeated) in any terms, it implies sum over that indices. The symmetrical tensor gab is called . When considering flat Minkowski spacetime, we write ηab instead of gab,   −1 0 0 0  0 1 0 0  g = η =   . (2.9) ab ab  0 0 1 0  0 0 0 1

The can be computed in form,       −1 0 0 0 dx0 T dx0  0 1 0 0   dx1   dx1  ds2 =        0 0 1 0   dx2   dx2  0 0 0 1 dx3 dx3

= −(dx2)2 + (dx1)2 + (dx2)2 + (dx3)2 = −cdt2 + dx2 + dy2 + dz2. (2.10)

Coordinate transformation law in SR is Lorentz transformation, a or boosts in one direction between two inertial frames moving relative to each other with constant velocity. In relativity, time is not an absolute quantity. Time measured in each reference frame are not equal. Time measured in one inertial frame by observer on that frame is called . Since speed of light is the same in all inertial frames, then the transformation law between two frames is written as

cdt0 = γ(dt − vdx/c) dx0 = γ(dx − vdt) dy0 = dy dz0 = dz. (2.11)

8 This set of equations is Lorentz transformation. The symbol γ is , 1 γ = p . (2.12) 1 − v2/c2 Lorentz transformation can also be written in general form,

a a b dx = Λ bdx . (2.13)

a where Λ b is Lorentz matrix,   γ −βγ 0 0  −βγ γ 0 0  Λa =   . (2.14) b  0 0 1 0  0 0 0 1 p Here we define1 β ≡ v/c and γ ≡ 1/ 1 − β2. Therefore equation (2.13) is       cdt0 γ −βγ 0 0 cdt  dx0   −βγ γ 0 0   dx    =     . (2.15)  dy0   0 0 1 0   dy  dz0 0 0 0 1 dz

Using equation (2.10) and equation (2.11), we obtain

−c2dt2 + dr2 = −c2dt02 + dr02 or − c2dt2 + dx2 = −c2dt02 + dx02 (2.16) Therefore ds2 = ds02 (2.17)

where dr2 = dx2 + dy2 + dz2 and dr02 = dx02 + dy02 + dz02. Equation ( 2.16) is rotational transformation in (x, ct) space. Values of β and γ range in 0 ≤ β ≤ 1 and 1 ≤ γ ≤ ∞ respectively. If we introduce new parameter ω and write β and γ in term of this new variables as

β = tanh ω γ = cosh ω, (2.18) where ω in the of in this (x, ct) space, then Lorentz transformation becomes       ct0 cosh ω − sinh ω 0 0 ct  x0   − sinh ω cosh ω 0 0   x    =     . (2.19)  y0   0 0 1 0   y  z0 0 0 0 1 z

1β is speed parameter. For β = 1.

9 We can see that the Lorentz boost is actually rotation in (x,ct) space by angle ω. Lorentz transformation yields two important phenomena in SR. These are time dila- tion and contraction. The dilation of time measured in two inertial reference frames is described by

∆t = ∆t0γ (2.20) and of matter measured in two inertial reference frames is described by L L = 0 (2.21) γ where ∆t0 and L0 are proper time and respectively.

10 Chapter 3

Introduction to general relativity

3.1 Tensor and curvature

In last chapter, we consider physical phenomena based on flat space which is a spa- cial case of this chapter. In this chapter is is of interest. A physical quantity needed here is tensors which are geometrical objects. Tensor is invariant in all coordinate systems. Vectors and scalars are subsets of tensors indicated by rank (order) of tensors i.e. vector is tensor of rank 1, scalar is zeroth-rank tensor. Any tensors are defined on M which is n-dimensional generalized object that it locally looks like Euclidian space Rn.

3.1.1 Transformations of scalars, vectors and tensors

a a In general relativity, vectors are expressed in general form, X = X ea where X is a a component of vector and ea is vector of the component. Here Λ b is general transformation metric. Contravariant vector or tangent vector Xa transforms like ∂x0a X0a = Xb = Λa Xb (3.1) ∂xb b or can be written in chain rule, ∂x0a dx0a = dxb. ∂xb a We define , δ b as a quantity with values 0 or 1 under conditions, ½ 1 if a = b δa = b 0 if a 6= b.

11 Therefore ∂x0a ∂xa = = δa . (3.2) ∂x0b ∂xb b Another quantity is scalar φ which is invariant under transformation,

φ0(x0a) = φ(xa). (3.3)

Consider derivative of scalar field φ = φ(xa(x0)) with respect to x0a, using chain rule, we obtain ∂φ ∂xb ∂φ = . ∂x0a ∂x0a ∂xb

a Covariant vector or 1-form or dual vector Xa in the x -, trans- forms according to ∂xb X0 (xa) = X (xa) = Λb X (xa). (3.4) a ∂x0a b a b There is also a relation a b a Λ bΛ c = δ c. (3.5) For higher rank tensor transformation follows

0 0 0k 0k l1 ll 0 0 0 ∂x 1 ∂x k ∂x ∂x 0k k ...k k1k2...... k T 1 2 k 0 0 = ...... T k l1...... ll k k 0l0 0l0 l1...... ll ∂x 1 ∂x k ∂x 1 ∂x l 0 0 k k l1 l k1k2...... k = Λ 1 ...... Λ k Λ 0 ...... Λ l 0 T k . (3.6) k1 kk l1 ll l1...... ll

The metric tensor

Any symmetric covariant tensor field of rank 2 such as gab, defines a metric ten- sor. Metrics are used to define and length of vectors. The square of the infinitesimal distance or interval between two neighboring points (events) is defined by 2 a b ds = gabdx dx . (3.7) ab ab Inverse of gab is g . The metric g is given by

ac c gabg = δb . (3.8)

We can use the metric tensor to lower and raise tensorial indices,

...... b... T...a... = gabT...... (3.9) and

...a... ab ...... T...... = g T...b.... (3.10)

12 3.1.2 Covariant derivative

Consider a contravariant vector field Xa defined along path xa(u) on manifold at a point P where u is free parameter. There is another vector Xa(xa + δxa) defined at xa + δxa or point Q. Therefore

Xa(x + δx) = Xa(x) + δXa. (3.11)

We are going to define a tensorial derivative by introducing a vector at Q which is parallel to Xa. We can assume the parallel vector differs from Xa by a small amount ¯ denoted by δXa(x). Because covariant derivative is defined on curved geometry, therefore parallel vector are parallel only on flat geometry which is mapped from manifold. We will explain about it in next section. The difference vector between the parallel vector and the vector at point Q illus- trated in Fig. 3.1 is

[Xa(x) + δXa(x)] − [Xa(x) + δX¯ a(x)] = δXa(x) − δX¯ a. (3.12)

We define the covariant derivative of Xµ by the limiting process

n £ ¤o a 1 a a ¯ a ∇cX = lim X (x + δx) − X (x) + δX (x) δxc→0 δxc 1 £ ¤ = lim δXa(x) − δX¯ a . δxc→0 δxc δX¯ a(x) should be a linear function on manifold. We can write

¯ a a b c δX (x) = −Γbc(x)X (x)δx (3.13)

a where the Γbc(x) are functions of coordinates called Cristoffel symbols of the second a kind. In flat space Γbc(x) = 0. But in curved space it is impossible to make all the a Γbc(x) vanish over all space. The Cristoffel symbols of second kind are defined by µ ¶ 1 ∂g ∂g ∂g Γa = gad dc + bd − cb . (3.14) bc 2 ∂xb ∂xc ∂xd

Following from the last equation that the Cristoffel symbols are necessarily symmetric or we often called it that torsion-free, i.e.

a a Γbc = Γcb. (3.15)

13 XXaa X a X a X a Q X aaX xa ()u P

Figure 3.1: Vector along the path xa(u)

We now define derivative of a vector on curved space. Using equation (3.13) and chain rule, we have

· a ¸ a 1 ∂X (x) c a b c ∇cX = lim δx + Γbc(x)X (x)δx . δxc→0 δxc ∂xc

Therefore the covariant derivative is

a a a b ∇cX = ∂cX + ΓbcX . (3.16)

c The notation ∂c is introduced by ∂c ≡ ∂/∂x . We next define the covariant derivative of a scalar field φ to be the same as its ordinary derivative (prove of this identity is in Appendix A),

∇cφ = ∂cφ. (3.17)

Consider

a ∇cφ = ∇c(XaY ) a a = (∇cXa)Y + Xa(∇cY ) a a a b = (∇cXa)Y + Xa(∂cY + ΓbcY ) and

a a ∂cφ = (∂cXa)Y + Xa(∂cY ).

From equation (3.17), we equate both equations together:

a a a a a b (∂cXa)Y + Xa(∂cY ) = (∇cXa)Y + Xa(∂cY + ΓbcY ).

14 Renaming a to b and b to a for the last term in the right-hand side of the equation above because they are only dummy indices therefore,

a a b a (∇cXa)Y = (∂cXa)Y + (∂cXa − ΓacXb)Y .

Covariant derivative for covariant vector is then

b ∇cXa = ∂cXa − ΓacXb. (3.18)

The covariant derivative for tensor follows

a.... a.... a d.... d a.... ∇cTb.... = ∂cTb.... + ΓcdTb.... + .... − ΓcbTd.... − ..... (3.19)

The last important formula is covariant derivative of the metric tensor,

∇cgab = 0 (3.20) and ab ∇cg = 0. (3.21) The proves of these two identities are in Appendix A.

3.1.3 Parallel transport

The concept of parallel transport along a path is in flat space. A parallel vector transporting from a point to another point maintains its unchange in magnitude and direction. In curve space, components of a vector are expected to change under parallel transport in different way from the case of flat space. Consider the parallel transport along a curve xa(τ) with a tangent vector Xa = dxa/dτ where τ is a parameter along the curve. Beginning from the formula of the covariant derivative

a b X ∇aX = 0, (3.22) we get µ ¶ dxa dxb dxc ∂ + Γb = 0 dτ a dτ ac dτ a b a c dx ∂ dx b dx dx a + Γac = 0 dτ ∂xµ dτ¶ dτ dτ d dxb dxa dxc + Γb = 0. dτ dτ ac dτ dτ

15 Figure 3.2: Parallel transport of vectors on curved space

We rename a to b and b to a, then

d2xa dxb dxc + Γa = 0. (3.23) dτ 2 bc dτ dτ

This equation is known as the geodesic equation. The geodesic distance between any two points is shortest. The geodesic is a curve space generalization of straight line in flat space.

3.1.4 Curvature tensor

We now discuss the most important concepts of general relativity, That is concept the of or curved geometry which is described in tensorial form. We will introduce the Riemann tensor by considering parallel transport along an a infinitesimal loop illustrated in Fig. 3.3. The R bcd is defined by the commutator of covariant derivatives,

a b a R bcdX = (∇c∇d − ∇d∇c)X . (3.24)

16 a X aa a x x X' 1 a X' 2

xa xaax dxa

xaadx

Figure 3.3: Transporting Xa around an infinitesimal loop

Consider

a a a (∇c∇d − ∇d∇c)X = ∇c∇dX − ∇d∇cX a a b a a b = ∇c(∂dX + ΓdbX ) − ∇d(∂cX + ΓcbX ).

a ∇dX is a tensor type(1, 1). Using equation (3.19) we get

a a a b e a a b a e e b (∇c∇d − ∇d∇c)X = ∂c(∂dX + ΓdbX ) − Γcd(∂eX + ΓebX ) + Γce(∂dX + ΓdbX ) a a b e a a b a e e b −∂d(∂cX + ΓcbX ) + Γdc(∂eX + ΓebX ) − Γde(∂cX + ΓcbX ) a a b a b e a e a b = ∂c∂dX + Γdb∂cX + ∂cΓdbX − Γcd∂eX − ΓcdΓebX a e a e b a a b a b +Γce∂dX + ΓceΓdbX − ∂d∂cX − Γcb∂dX − ∂dΓcbX e a e a b a e a e b +Γdc∂eX + ΓdcΓebX − Γde∂cX − ΓdeΓcbX .

a e a e We rename e to b in the terms Γce∂dX and Γde∂cX . Assuming torsion-free condition of Cristoffel symbols and using equation (3.24), we have Riemann tensor expressed in terms of Cristoffel symbols:

a a a e a e a R bcd = ∂cΓbd − ∂dΓbc + ΓbdΓec − ΓbcΓed. (3.25)

a R bcd depends on the metric and the metric’s first and second derivatives. It is anti-symmetric on its last pair of indices,

a a R bcd = −R bdc. (3.26)

The last equation introduces identity:

a a a R bcd + R dbc + R cdb = 0. (3.27)

17 Lowering the first index with the metric, the lowered tensor is symmetric under interchanging of the first and last pair of indices. That is

Rabcd = Rcdab. (3.28)

The tensor is anti-symmetric on its first pair of indices as

Rabcd = −Rbacd. (3.29)

We can see that the lowered curvature tensor satisfies

Rabcd = −Rabdc = −Rbacd = Rcdab

and Rabcd + Radbc + Racdb = 0. (3.30)

The curvature tensor satisfies a set of differential identities called the Bianchi iden- tities:

∇aRbcde + ∇cRabde + ∇bRcade = 0. (3.31) We can use the curvature tensor to define Ricci tensor by the contraction,

c cd Rab = R acb = g Rdacb. (3.32)

Contraction of Ricci tensor then also defines curvature scalar or Ricci scalar R by ab R = g Rab. (3.33) These two tensors can be used to define Einstein tensor 1 G = R − g R, (3.34) ab ab 2 ab which is also symmetric. By using the equation (3.31), the Einstein tensor can be shown to satisfy the contracted Bianchi identities

b ∇ Gab = 0 (3.35)

3.2 The equivalence principle

In chapter 2, we introduced some concepts about inertial frames of reference. We will discuss about its nature in this chapter. An inertial frame is defined as one in which a free particle moves with constant velocity. However gravity is long-range and can not be screened out. Hence an

18 purely inertial frame is impossible to be found. We can only imagine about. According to Newtonian gravity, when gravity acts on a body, it acts on the gravitational , mg. The result of the force is an of the inertial mass, mi. When all bodies fall in vacuum with the same acceleration, the ratio of inertial mass and gravitational mass is independent of the size of bodies. Newton’s theory is in principle consistent with mi = mg and within high experimental accuracies mi = mg to 1 in 1,000. Therefore the equivalence of gravitational and inertial mass implies:

In a small laboratory falling freely in gravitational field, mechanical phenomena are the same as those observed in an inertial frame in the absence of gravitational field.

In 1907 Einstein generalized this conclusion by replacing the word mechanical phe- nomena with the laws of physics. The resulting statement is known as the principle of equivalence. The freely-falling frames introduce the local inertial frames which are important in relativity.

3.3 Einstein’s law of gravitation

In this section, we use Riemannian formalism to connect matter and metric that leads to a satisfied gravitational theory.

3.3.1 The energy-momentum tensor for perfect fluids

The energy-momentum tensor contains information about the total energy density measured by an arbitrary inertial observer. It is defined by the notation T ab. We start by considering the simplest kind of matter field, that is non-relativistic matter or dust. We can simply construct the energy-momentum tensor for dust by using four- velocity ua defined as ua = (c, 0, 0, 0) for and the proper density ρ:

T ab = ρuaub. (3.36)

Notice that T 00 is the energy density. In general relativity, the source of gravitational field can be regarded as perfect fluid. A perfect fluid in relativity is defined as a fluid that has no and no heat conduction. No heat conduction implies T 0i = T i0 = 0 in rest frame and energy can flow only if particles flow. No viscosity implies vanishing of force paralleled to the interface between particles. The should always be perpendicular to the

19 interface, i.e. the spatial component T ij should be zero unless i = j. As a result we write the energy-momentum tensor for perfect fluids in rest frame as   ρc2 0 0 0  0 p 0 0  T ab =   (3.37)  0 0 p 0  0 0 0 p where p is the pressure and ρ is the energy density. Form equation (3.37), it is easy to show that ³ p ´ T ab = ρ + uaub + pgab. (3.38) c2 We simply conclude from the above equation that the energy-momentum tensor is . The metric tensor gab here is for flat spacetime (we often write ηab instead of gab for flat spacetime). Notice that in the limit p → 0, a perfect fluid reduces to dust equation (3.36). We can easily show that the energy-momentum tensor conserved in flat spacetime:

ab ∂bT = 0. (3.39) Moreover, if we use non-flat metric, the is

ab ∇bT = 0. (3.40)

3.3.2 Einstein’s field equation

Einstein’s field equation told us that the metric is correspondent to geometry and geometry is the effect of an amount of matter which is expressed in energy-momentum tensor. Matters cause spacetime curvature. We shall use Remannian formalism to connect matter and metric. Since covariant divergence of the Einstein tensor Gab vanishes in equation (3.35), we therefore write 1 G = R − g R = κT . (3.41) ab ab 2 ab ab If there is gravity in regions of space there must be matter . The proportional constant κ is arbitrary. Contract the Einstein tensor by using the metric tensor, we obtain

ab ab G = g Gab = κg Tab 1 gabR − gabg R = κgabT ab 2 ab ab 1 R − δaR = κT. 2 a 20 a δa is 4 which is equal to its dimensions. Therefore equation (3.41) can be rewritten as ³ 1 ´ R = κ T − g T . (3.42) ab ab 2 ab We want to choose appropriate value of constant κ. Let us consider motion of particle which follows a geodesic equation

d2xa dxb dxc + Γa = 0. (3.43) dτ 2 bc dτ dτ

In Newtonian limit, the particles move slowly with respect to the speed of light. Four-velocity is ua ∼= u0, therefore the geodesic equation reduces to d2xa dx0 dx0 = −Γa = −c2Γa . (3.44) dτ 2 00 dτ dτ 00 Since the field is static in time, the time derivative of the metric vanishes. Therefore the Christoffel symbol is reduced to: 1 ³ ´ Γa = gad ∂ g + ∂ g − ∂ g 00 2 0 d0 0 0d d 00 1 = − gad∂ g . (3.45) 2 d 00 In the limit of weak gravitational field, we can decompose the metric into the Minkowski metric, the equation (2.9), plus a small perturbation,

gab = ηab + hab, (3.46) where hab ¿ 1 here. From definition of the inverse metric, we find that to the first order in h, 1 gab = gab −1 = (ηab + hab) ∼= ηab − hab.

Since the metric here is diagonal matrix. Our approximate is to only first order for small perturbation of inverse matric. Equation (3.45) becomes 1¡ ¢ Γa = ηad − had ∂ h 00 2 d 00 1 ∼= ηad∂ h . (3.47) 2 d 00 21 Substituting this equation into the geodesic equation ( 3.44),

d2xa c2 = ηad∂ h . (3.48) dτ 2 2 d 00

2 0 2 Using ∂0h00 = 0 and since d x /dτ is zero, we are left with spacelike components of the above equation. Therefore,

d2xi c2 = ηij∂ h . (3.49) dτ 2 2 j 00 Recall Newton’s equation of motion

d2xi = −∂iΦ (3.50) dτ 2 where Φ is Newtonian potential. Comparing both equations above, we obtain 2 h = Φ. (3.51) 00 c2 In non relativistic limit (dustlike), the energy-momentum tensor reduce to the equa- tion (3.36). We will work in fluid rest frame. Equation (3.38) gives

2 T00 = ρc (3.52) and

ab T = g Tab ∼ 00 = g T00 ∼ 00 = η T00 = −ρc2. (3.53)

We insert this into the 00 component of our gravitational field equation (3.42). We get ³ 1 ´ 1 R = κ ρc2 − ρc2 = κρc2. (3.54) 00 2 2 This is an equation relating derivative of the metric to the energy density. We shall i 0 expand R 0i0 in term of metric. Since R 000 = 0 then

i i i i b i b R00 = R 0i0 = ∂iΓ00 − ∂0Γi0 + ΓibΓ00 − Γ0bΓi0.

22 The second term is time derivative which vanishes for static field. The second order of Christoffel symbols (Γ)2 can be neglected since we only consider a small perturbation. From this we get · ¸ 1 ³ ´ R ∼= ∂ Γi = ∂ gib ∂ g + ∂ g − ∂ g 00 i 00 i 2 0 b0 0 0b b 00 1 1 = − ∂ gib∂ g − gib∂ ∂ g 2 i b 00 2 i b 00 1 1 ∼= − ∂ ηij∂ h − ηij∂ ∂ h 2 i j 00 2 i j 00 1 = − δij∂ ∂ h 2 i j 00 1 = − ∇2h . (3.55) 2 00 Substituting equation (3.51) into the above equation and using equation (3.54), we finally obtain 1 ∇2Φ = − κρc4. (3.56) 2 The connection to Newtonian theory appears when this equation is compared with Poisson’s equation for Newtonian theory of gravity. The theory of general relativity must be correspondent to Newton’s non-relativistic theory in limiting case of weak field. Poisson’s equation in a Newtonian gravitational field is

∇2Φ = 4πGρ (3.57) where G is classical gravitational constant. Comparing this with equation (3.56) gives the value of κ, 8πG κ = . (3.58) c4 From equation (3.41) then the full field equation is

1 8πG G = R − g R = T . (3.59) ab ab 2 ab c4 ab

23 Chapter 4

Variational principle approach to general relativity

4.1 Lagrangian formulation for field equation

4.1.1 The Einstein-Hilbert action

All fundamental physical equation of classical field including the Einstein’s field equa- tion can be derived from a variational principle. The condition required in order to get the field equation follows from Z δ Ld4x = 0. (4.1)

Of course the quantity above must be an invariant and must be constructed from the metric gab which is dynamical variable in GR. We shall not include function which is first the derivative of metric because it vanishes at a point P ∈ M. The Riemann tensor is of course made from second derivative set of the metrics, and the only independent scalar constructed from the metric is the Ricci scalar R. The well √ definition of Lagrangian density is L = −gR, therefore Z √ 4 SEH = −gR d x (4.2)

24 is known as the Einstein-Hilbert action. We derive field equation by variation of action in equation (4.2), Z √ 4 δSEH = δ −gR d x Z ³ ´ 4 √ ab = d xδ −gg Rab Z Z Z 4 √ ab 4 √ ab 4 √ = d x −gg δRab + d x −gRabδg + d xRδ −g.

Now we have three terms of variation that

δSEH = δSEH(1) + δSEH(2) + δSEH(3) (4.3)

The variation of first term is Z 4 √ ab δSEH(1) = d x −gg δRab. (4.4)

Considering the variation of Ricci tensor,

c c c c d c d Rab = R acb = ∂cΓab − ∂bΓac + ΓcdΓba − ΓbdΓac (4.5) δR = ∂ δΓc − ∂ δΓc + Γd δΓc + Γc δΓd − Γd δΓc − Γc δΓd ab ³c ab b ac ba cd cd ba ´ ac bd bd ac c c d d c d c = ∂cδΓab + ΓcdδΓba − ΓacδΓbd − ΓbcδΓad ³ ´ c c d d c d c − ∂bδΓac + ΓbdδΓac − ΓbaδΓcd − ΓbcδΓad and the covariant derivative formula:

c c c d d c d c ∇cδΓab = ∂cδΓab + ΓcdδΓba − ΓacδΓbd − ΓbcδΓad (4.6) and also

c c c d d c d c ∇bδΓac = ∂bδΓac + ΓbdδΓac − ΓbaδΓcd − ΓbcδΓad (4.7) we can conclude that

c c δRab = ∇cδΓab − ∇bδΓac. (4.8)

Therefore equation (4.4) becomes Z ³ ´ 4 √ ab c c δSEH(1) = d x −gg ∇cδΓab − ∇bδΓac Z · ³ ´ ³ ´ ¸ 4 √ ab c c ab ab c c ab = d x −g ∇c g δΓab − δΓab∇cg − ∇b g ∇bδΓac + δΓac∇bg .

25 Remembering that the covariant derivative of metric is zero. Therefore we get Z ³ ´ 4 √ ab c c δSEH(1) = d x −gg ∇cδΓab − ∇bδΓac Z · ³ ´ ³ ´¸ 4 √ ab c ab c = d x −g ∇c g δΓab − ∇b g δΓac Z h i 4 √ ab c ac b = d x −g ∇c g δΓab − g δΓab Z 4 √ c = d x −g ∇cJ where we introduce c ab c ac b J = g δΓab − g δΓab. (4.9) If J c is a vector field over a region M with boundary Σ. Stokes’s theorem for the vector field is Z p Z p 4 c 3 c d x |g| ∇cJ = d x |h|ncJ (4.10) M Σ where nc is normal unit vector on hypersurface Σ. The normal unit vector nc can a be normalized by nan = −1. The tensor hab is induced metric associated with hypersurface defined by

hab = gab + nanb. (4.11) Therefore the first term of action becomes Z p 3 c δSEH(1) = d x |h|ncJ = 0. (4.12) Σ This equation is an integral with respect to the volume element of the covariant di- vergence of a vector. Using Stokes’s theorem, this is equal to a boundary contribution at infinity which can be set to zero by vanishing of variation at infinity. Therefore this term contributes nothing to the total variation.

4.1.2 Variation of the metrics

Firstly we consider metric gab. Since the contravariant and covariant metrics are symmetric matrices then, ab b gcag = δc . (4.13) We now consider inverse of the metric: 1 1 gab = (Aab)T = Aba (4.14) g g

26 ba where g is determinant and A is the cofactor of the metric gab. Let us fix a, and expand the determinant g by the ath row. Then

ab g = gabA . (4.15)

If we perform partial differentiation on both sides with respect to gab, then ∂g = Aab. (4.16) ∂gab Let us consider variation of determinant g: ∂g δg = δgab ∂gab ab = A δgab ba = gg δgab. Remembering that gab is symmetric, we get

ab δg = gg δgab. (4.17) Using relation obtained above, we get √ 1 δ −g = − √ δg 2 −g 1 g = √ gabδg . (4.18) 2 −g ab ab We shall convert from δgab to δg by considering

d cd δδa = δ(gacg ) = 0 cd cd g δgac + gacδg = 0 cd cd g δgac = −gacδg .

Multiply both side of this equation by gbd we therefore have

cd cd gbdg δgac = −gbdgacδg c cd δbδgac = −gbdgacδg dc δgab = −gacgbdδg . (4.19) Substituting this equation in to equation (4.18) we obtain √ 1√ δ −g = − −g gabg g δgdc 2 ac bd 1√ = − −g δbg δgdc 2 c bd 1√ = − −g g δgdc. 2 cd

27 Renaming indices c to a and d to b, we get √ 1√ δ −g = − −g g δgab. (4.20) 2 ab The variation of Einstein-Hilbert action becomes Z Z √ 1 √ δS = d4x −gR δgab − d4xR −g g δgab EH ab 2 ab Z √ h 1 i = d4x −g R − g R δgab (4.21) ab 2 ab The functional derivative of the action satisfies Z X ³ δS ´ δS = δΦi d4x (4.22) δΦi i where {Φi} is a complete set of field varied. Stationary points are those for which δS/δΦi = 0. We now obtain Einstein’s equation in vacuum: 1 δS 1 √ EH = R − g R = 0. (4.23) −g δgab ab 2 ab

4.1.3 The full field equations

Previously we derived Einstein’s field equation in vacuum due to including only gravi- tational part of the action but not matter field part. To obtain the full field equations, we assume that there is other field presented beside the gravitational field. The action is then 1 S = S + S (4.24) 16πG EH M where SM is the action for matter. We normalize the gravitational action so that we get the right answer. Following the above equation we have 1 δS 1 ³ 1 ´ 1 δS √ = R − g R + √ M = 0. −g δgab 16πG ab 2 ab −g δgab We now define the energy-momentum tensor as 1 δS T = −2√ M . (4.25) ab −g δgab This allows us to recover the complete Einstein’s equation, 1 R − Rg = 8πGT . (4.26) ab 2 ab ab

28 4.2 Geodesic equation from variational principle

Consider motion of particle along a path xa(τ). We will perform a variation on this path between two points P and Q. The action is simply Z S = dτ. (4.27)

In order to perform the variation, it is useful to introduce an arbitrary auxiliary parameter s. Here ds is displacement on spacetime. We have ³ dxa dxb ´1/2 dτ = − g ds (4.28) ab ds ds We vary the path by using standard procedure: Z δS = δ dτ Z ³ dxa dxb ´1/2 = δ − g ds ab ds ds Z · ¸ 1 ³ dxa dxb ´−1/2 dxa dxb dδxa dxb = ds − g − δg − 2g . 2 ab ds ds ab ds ds ab ds ds Considering the last term, · ¸ dδxa dxb d dxb dg dxb d2xb −2g = − 2g δxa + 2 ab δxa + 2g δxa . ab ds ds ds ab ds ds ds ab ds2 The two points, P and Q are fixed. We can set first term of above equation to zero. Therefore we obtain Z · ¸ 1 ³ dxa dxb ´−1/2 dxa dxb dg dxb d2xb δS = ds − g − δg + 2 ab δxa + 2g δxa 2 ab ds ds ab ds ds ds ds ab ds2 Z · ¸ 1 ds2 dxa dxb dg dxb d2xb = dτ − δg + 2 ab δxa + 2g δxa 2 dτ 2 ab ds ds ds ds ab ds2 Z · ¸ 1 dxa dxb dg dxb d2xb = dτ − δg + 2 ab δxa + 2g δxa . 2 ab dτ dτ dτ dτ ab dτ 2 By using chain rule we get Z · ¸ 1 ∂g dxa dxb ∂g dxc dxb d2xb δS = dτ − ab δxc + 2 ab δxa + 2g δxa 2 ∂xc dτ dτ ∂xc dτ dτ ab dτ 2 Z · ¸ 1 dxa dxc dxc dxa dxc dxa d2xa = dτ − ∂ g δxb + ∂ g δxb + ∂ g δxb + 2g δxb 2 b ac dτ dτ c ba dτ dτ a bc dτ dτ ba dτ 2 Z · ¸ 1³ ´dxa dxc d2xa = dτδxb − ∂ g + ∂ g + ∂ g + g . 2 b ac c ba a bc dτ dτ ba dτ 2

29 We set the variation of action to zero, δS = 0 and multiply by gdb,

1 ³ ´dxa dxc d2xa gdb − ∂ g + ∂ g + ∂ g + δd = 0 2 b ac c ba a bc dτ dτ a dτ 2 We therefore recover geodesic equation:

d2xd dxa dxc + Γd = 0 (4.29) dτ 2 ac dτ dτ

4.3 Field equation with surface term

In section 4.1 we have derived field equation without boundary term which is set to be zero at infinity. In this section, we shall generalize field equation to general case which includes boundary term in action.

4.3.1 The Gibbons-Hawking boundary term

We begin by putting boundary term in the first part of the action in section 4.1 of Einstein-Hilbert action, Z µ ¶ Z √ 1 √ 1 ab 4 1 c 3 δSEH = −g Rab − gabR δg d x + −hncJ d x. (4.30) 16πG M 2 16πG Σ Considering vector J c in the last term of equation (4.9) we have

c ab c ac b J = g δΓab − g δΓab.

Using formula (prove in appendix B) 1 ³ ´ δΓc = gcd ∇ δg + ∇ δg − ∇ δg , (4.31) ab 2 a bd b ad d ab we get · ¸ 1 ³ ´ J c = gab gcd ∇ δg + ∇ δg − ∇ δg 2 a bd b ad d ab · ¸ 1 ³ ´ −gac gbd ∇ δg + ∇ δg − ∇ δg 2 a bd b ad d ab 1 1 1 = gabgcd∇ δg + gabgcd∇ δg − gabgcd∇ δg 2 a bd 2 b ad 2 d ab 1 1 1 − gacgbd∇ δg − gacgbd∇ δg + gacgbd∇ δg . 2 a bd 2 b ad 2 d ab 30 Interchanging dummy indices a and b in the second term, a and d in the fourth term, and a and d in the last term, we obtain result

c ab cd¡ ¢ J = g g ∇aδgbd − ∇dδgab . (4.32)

c Now our discussion is on hypersurface. Lowering index of J with metric gce,

ab¡ ¢ Jc = g ∇aδgbd − ∇dδgab . (4.33)

By using equation (4.11), we have

c c¡ ab a b¢¡ ¢ n Jc = n h − n n ∇aδgbd − ∇dδgab c ab c ab a b c a b c = n h ∇aδgbd − n h ∇dδgab − n n n ∇aδgbd + n n n ∇dδgab.

From boundary condition, the first term of this equation vanishes since it is projected on hypersurface which variation of metric and induced metric vanish at Σ such as

δgab = 0, (4.34) δhab = 0 (4.35) and we use definition nanbnc = 0. (4.36) Therefore we obtain c c ab n Jc = −n h ∇dδgab. (4.37)

a1...ak Now let us consider arbitrary tensor T b1...bl at P ∈ Σ. It is a tensor on the tangent space to Σ at P if

a1...ak a1 ak d1 dl c1...bk T b1...bl = h c1 ...h ck h b1 ...h bl T d1...dl (4.38)

b Defining ha ∇b as a projected covariant derivative on hypersurface by using notation

Da, it satisfies

a1...ak a1 ak e1 el f d1...dk DcT b1...bl = h d1 ...h dk h b1 ...h bl h c∇f T e1...ek . (4.39) The covariant derivative for induced metric on hypersurface automatically satisfies

d e f ¡ ¢ Dahbc = ha hb hc ∇d gaf + nenf = 0 (4.40)

b b since ∇dgef = 0 and habn = 0 (remembering n is perpendicular to hypersurface. Therefore its dot product with metric on hypersurface is zero). Next we introduce extrinsic curvature in the form

c Kab = ha ∇cnb (4.41)

31 and we can use contract it with induced metric,

ab ab c ab K = h Kab = h ha ∇cnb = h ∇anb. (4.42)

Variation of extrinsic curvature given by

¡ ab ¢ δK = δ h ∇anb ab ab ab c ab c = δh ∇anb + h ∂aδnb − h δΓabnc − h Γabδnc ab ab c ab c = h ∂aδnb − h Γabδnc − h δΓabnc ab ab c = h ∇aδnb − h δΓabnc bc a ab c = h h c∇aδnb − h δΓabnc ¡ ¢ = D hbcδn − habδΓc n ch b abi c ¡ bc ¢ bc ab c = Dc δ h nb − nbδh − h δΓabnc ab c = −h δΓabnc. (4.43) Using equation (4.31), h1 ¡ ¢i δK = −habn gcd ∇ δg + ∇ δg − ∇ δg c 2 a bd b ad d ab 1 = habnd∇ δg . (4.44) 2 d ab Notice that when substituting equation (4.37) to this equation, we obtain 1 δK = − ncJ . (4.45) 2 c Considering boundary term from Einstein-Hilbert action Z Z √ √ c 3 c 3 −hncJ d x = −hn Jcd x Σ Σ Z √ = −2 −hδKd3x ZΣ Z √ √ = −2δ −hKd3x + 2 δ −hKd3x Σ Σ and using equation (4.20), Z Z Z √ √ √ c 3 3 ab 3 −hncJ d x = −2δ −hKd x − −hKhabδh d x. Σ Σ Σ Since δhab vanishes at boundary. Therefore we get Z Z √ √ c 3 3 −hncJ d x = −2δ −hKd x. (4.46) Σ Σ

32 Now we have variation of Einstein-Hilbert action in the form Z µ ¶ Z √ 1 √ 1 ab 4 1 3 δSEH = −g Rab − gabR δg d x − δ −hKd x. (4.47) 16πG M 2 8πG Σ We generalize this equation by naming the first term as the variation of gravitational action, δSGravity, we finally get Z 1 √ δS = δS − δ −hKd3x EH Gravity 8πG Z Σ 1 √ δS = δS + δ −hKd3x Gravity EH 8πG Z Σ Z √ 1 √ 4 1 3 ∴ SGravity = −gRd x + −hKd x. (4.48) 16πG M 8πG Σ The last term is known as Gibbons-Hawking boundary term. In next section, we will vary this term using junction condition.

4.3.2 Israel junction condition

We begin from considering the action

S = S + S Gravity EH Z GH Z 1 √ 1 √ = −gRd4x + −hKd3x. 16πG M 8πG Σ Its variation is Z Z 1 √ 1 √ δS = −gG δgabd4x + −hn J ad3x Gravity 16πG ab 16πG a ZM Z Σ 1 √ 1 √ + δ −hKd3x + −hδKd3x. (4.49) 8πG Σ 8πG Σ In this section, we are not interested in boundary when performing variation of grav- itational action then there is no boundary condition. Variation of K is

¡ a b¢ δK = δ hb ∇an a b a b a b c a b c = δhb ∇an + hb ∂aδn − hb δΓacn + hb Γacδn a b a b c a b = δhb ∇an + hb δΓacn + hb ∇aδn . (4.50) Considering the first term of equation (4.50),

a b ¡ a a ¢ b δhb ∇an = δ δb + n nb ∇an ¡ a a¢ b = n δnb + nbδn ∇an .

33 Using identity c δnb = nbncδn , (4.51) we get

a b ¡ a c a¢ b δhb ∇an = n nbncδn + nbδn ∇an ¡ a c a c¢ b = n ncδn + δc δn nb∇an c a b = nbδn hc ∇an c b = nbδn Kc = 0. (4.52) Since extrinsic curvature is associated with hypersurface then its dot product with normal vector on Σ vanishes yielding equation (4.52). We continue to do variation of the second term of the equation (4.50) by using formula (4.31), we then have 1 ³ ´ h aδΓb nc = h ancgbd ∇ δg + ∇ δg − ∇ δg . b ac 2 b a cd c ad d ac Therefore variation of K is 1 ³ ´ δK = hadnc ∇ δg + ∇ δg − ∇ δg + h a∇ δnb. (4.53) 2 a cd c ad d ac b a √ Applying similar procedure to the equation (4.20) for −h, we obtain √ 1√ δ −h = − −h h δhab. (4.54) 2 ab Using the equation (4.32), finally the gravitational action is Z Z 1 √ 1 √ δS = −gG δgabd4x − −h h Kδhabd3x Gravity 16πG ab 16πG ab MZ Σ 1 √ ¡ ¢ + −hn gcbgad ∇ δg − ∇ δg d3x 16πG a c bd d cb ZΣ · ¸ √ ³ ´ 1 ad c a b 3 + −h h n ∇aδgcd + ∇cδgad − ∇dδgac + 2hb ∇aδn d x. 16πG Σ Considering cb ad d¡ cb c b¢ d cb nag g = n h + n n = n h . (4.55) Therefore Z Z 1 √ 1 √ δS = −gG δgabd4x − −h h Kδhabd3x Gravity 16πG ab 16πG ab MZ · Σ √ ³ 1 d cb d cb ad c + −h n h ∇cδgbd − n h ∇dδgcb + h n ∇aδgcd 16πG Σ ´ ¸ ad c ad c a b 3 +h n ∇cδgad − h n ∇dδgac + 2hb ∇aδn d x.

34 Interchanging dummy indices between d and c and rename b to a of the first and the second term in bracket of the third integral term results that the first term and the last term in bracket cancel out. So do the second and the forth term in bracket of the third integral term. Therefore Z Z 1 √ 1 √ δS = −gG δgabd4x − −h h Kδhabd3x Gravity 16πG ab 16πG ab MZ Σ √ ³ ´ 1 ad c a b 3 + −h h n ∇aδgcd + 2hb ∇aδn d x. (4.56) 16πG Σ Considering the first term in the third integral term

ad c ad c ad c h n ∇aδgcd = h ∇a(n δgcd) − h δgcd∇an .

Using equation (4.19), we get

ad c ad c ad ef c h n ∇aδgcd = h ∇a(n δgcd) + h gdegcf δg ∇an ad c ad ef c = h ∇a(n δgcd) + h (hde + ndne)gcf δg ∇an ad c a ef = h ∇a(n δgcd) + h eδg ∇anf ad c ef = h ∇a(n δgcd) + Kef δg .

Considering

δgef = δ(hef + nenf ) = δhef + nf δne + neδnf ef f e a e f b = δh + n n n δna + n n n δnb = δhef . (4.57)

Therefore ad c ad c ef h n ∇aδgcd = h ∇a(n δgcd) + Kef δh . (4.58) Substituting this equation in equation (4.56), we obtain Z Z 1 √ 1 √ δS = −gG δgabd4x − −h h Kδhabd3x Gravity 16πG ab 16πG ab MZ Σ 1 √ h ¡ ¢ i + −h had∇ ncδg + K δhef + 2h a∇ δnb d3x 16πG a cd ef b a Z Σ Z 1 √ 1 √ ¡ ¢ = −gG δgabd4x + −h K − h K δhabd3x 16πG ab 16πG ab ab MZ Σ √ h i 1 ad ¡ c ¢ a b 3 + −h h ∇a n δgcd + 2hb ∇aδn d x (4.59) 16πG Σ

35 Considering the two terms in bracket of the last integral term h i ad c a b ad ¡ c ¢ c a b h ∇a(n δgcd) + 2hb ∇aδn = h ∇a δ n gcd − gcdδn + 2hb ∇aδn ad a c a b = h ∇aδnd − h c∇aδn + 2hb ∇aδn ad a b = h ∇aδnd + hb ∇aδn bd a a b = g hb ∇aδnd + hb ∇aδn a ¡ bd b¢ = hb ∇a g δnd + δn ¡ ¢ = h a∇ gbdn n δnc + δb δnc b ah d c ci a ¡ b b ¢ c = hb ∇a n nc + δ c δn ¡ b c¢ = Db h cδn (4.60) and substituting the last equation into equation (4.59), we obtain Z Z 1 √ 1 √ ¡ ¢ δS = −gG δgabd4x + −h K − h K δhabd3x Gravity 16πG ab 16πG ab ab MZ Σ √ 1 ¡ b c¢ 3 + −hDb h cδn d x. (4.61) 16πG Σ Since the last term in the equation (4.61) is divergence term. It yields vanishing of this integral with boundary at infinity on Σ. Finally, the variation of gravity is Z 1 √ δS = −gG δgabd4x Gravity 16πG ab MZ √ 1 ¡ ¢ ab 3 + −h Kab − habK δh d x. (4.62) 16πG Σ The action for matter on hypersurface is Z 3 Smat = Lmatd x (4.63) Σ and its variation is Z √ ab δSmat = −htabδh (4.64) Σ where the energy-momentum tensor on hypersurface is

1 δSmat tab = 2√ . (4.65) −h δhab The total action is

SGravity = SEH + SGB + Smat. (4.66)

36 Since we include energy-momentum tensor on hypersurface, then Einstein tensor in the bulk vanishes.1 Therefore the total action gives the Israel junction condition

Kab − habK = −8πGtab. (4.67)

The energy-momentum tensor on hypersurface is not necessarily conserved because energy can flow from the hypersurface to the bulk. The idea has been recently widely applied to the research field of braneworld gravity and braneworld cosmology (see e.g. [8] for review of application of Israel junction condition to braneworld gravity.)

1The bulk is the region of one left beyond the hypersurface. The bulk and hypersurface together form total region of the manifold.

37 Chapter 5

Conclusion

Relativity begins with concept of inertial reference frame which is defined by Newton’s first law. Therefore relative frames in one direction is considered allowing us to do Galilean transformation for inertial frame moving not so fast. However under this transformation, the speed of light is no longer invariant and electromagnetic wave equations are neither invariant. Special relativity came along based on Einstein’s principle of spacial relativity. In this theory, there is a unification of time and space so called spacetime. It uses Lorentz transformation under which electromagnetic wave equations are invariant. Moreover it reduces to Galilean transformation in case of small velocity. Therefore SR implies no absolute inertial reference frame in the universe. Einstein introduced transformation law when including effects of gravitational field and also introduced the equivalence principle which suggest that purely inertial reference frame is not sensible. It introduces us local inertial frame in gravitational field. Therefore Newtonian mechanics based on inertial frame fails to explain gravity. SR based on flat space is extended to general relativity using concept a of curved space. Curved space is indicated by Riemann tensor, R bcd. This theory attempts to explain gravity with geometry. It tells us that mass is in fact curvature of geometry This fact is expressed in the Einstein’s field equation,

Gab = 8πGTab.

All physical laws are believed to obey principle of least action, δS = 0. The dynamical variable in GR is gab therefore the Lagrangian density in action must be a function of gab. We used Ricci scalar in term of second derivative of metric. We did not choose function of first derivative of metric since it vanishes at any point on manifold. With

38 the action Z √ 4 SEH = −gRd x we can derive Einstein’s field equation by neglecting boundary term (surface term). If including boundary term then, we must have dynamical variable on boundary or hypersurface, Σ such as hab which is an induced metric on the hypersurface. Bounding the manifold, we obtain extrinsic curvature K which is constructed from hab in the action on the hypersurface. That is Z √ 3 SGH = −hKd x. Σ This is called Gibbon-Howking boundary term. The result of this variation is Israel junction condition,

Kab − habK = −8πGtab which is shown in detail calculations in this report. This result can be applied widely to braneworld gravity.

39 Appendix A

Proofs of identities

A.1 ∇cgab = 0

The identity ∇cgab = 0 is proved by using covariant derivative to tensor type(0, 2):

d d ∇cgab = ∂cgab − Γcagbd − Γcbgad (A.1)

Using equation (3.14), we have

1 ³ ´ 1 ³ ´ ∇ g = ∂ g − g gde ∂ g + ∂ g − ∂ g − g gde ∂ g + ∂ g − ∂ g c ab c ab 2 bd c ae a ce e ac 2 ad c be b ce e bc 1 ³ ´ 1 ³ ´ = ∂ g − δb ∂ g + ∂ g − ∂ g − δa ∂ g + ∂ g − ∂ g c ab 2 e c ae a ce e ac 2 e c be b ce e bc 1³ ´ 1³ ´ = ∂ g − ∂ g + ∂ g − ∂ g − ∂ g + ∂ g − ∂ g c ab 2 c ab a cb b ac 2 c ba b ca a bc = 0. (A.2)

ab A.2 ∇cg = 0

ab The identity ∇cg = 0 is proved by using covariant derivative to tensor type(2, 0):

ab ab a bd b ad ∇cg = ∂cg + Γcdg + Γcdg (A.3)

40 Therefore 1 ³ ´ 1 ³ ´ ∇ gab = ∂ gab + gbdgae ∂ g + ∂ g − ∂ g + gadgbe ∂ g + ∂ g − ∂ g c c 2 c de d ce e cd 2 c de d ce e cd 1 1 1 = ∂ gab + gaegbd∂ g + gbdgae∂ g − gaegbd∂ g c 2 c de 2 d ce 2 e cd 1 be ad 1 ad be 1 be ad + g g ∂cgde + g g ∂dgce − g g ∂egcd 2 · 2 2 ¸ · ¸ 1 ³ ´ 1 ³ ´ = ∂ gab + gae ∂ g gbd − g ∂ gbd + gbd ∂ g gae − g ∂ gae c 2 c de de c 2 d ce ce d · ¸ · ¸ 1 ³ ´ 1 ³ ´ − gae ∂ g gbd − g ∂ gbd + gbe ∂ g gad − g ∂ gad 2 e cd cd e 2 c de de c · ¸ · ¸ 1 ³ ´ 1 ³ ´ + gad ∂ g gbe − g ∂ gbe − gbe ∂ g gad − g ∂ gad 2 d ce ce d 2 e cd cd e 1 ³ ´ 1 ³ ´ = ∂ gab + gae ∂ δb − g ∂ gbd + gbd ∂ δa − g ∂ gae c 2 c e de c 2 d c ce d 1 ³ ´ 1 ³ ´ − gae ∂ δb − g ∂ gbd + gbe ∂ δa − g ∂ gad 2 e c cd e 2 c e de c 1 ³ ´ 1 ³ ´ + gad ∂ δb − g ∂ gbe − gbe ∂ δa − g ∂ gad 2 d c ce d 2 e c cd e a δc are constant then their partial derivatives vanish. Therefore we get

1 1 1 1 ∇ gab = ∂ gab − gaeg ∂ gbd − gbdg ∂ gae + gaeg ∂ gbd − gbeg ∂ gad c c 2 de c 2 ce d 2 cd e 2 de c 1 1 − gadg ∂ gbe + gbeg ∂ gad 2 ce d 2 cd e 1 1 1 1 = ∂ gab − δa∂ gbd − gbdg ∂ gae + gaeg ∂ gbd − δb∂ gad c 2 d c 2 ce d 2 cd e 2 d c 1 1 − gadg ∂ gbe + gbeg ∂ gad 2 ce d 2 cd e 1 1 1 1 = ∂ gab − ∂ gba − gbdg ∂ gae + gaeg ∂ gbd − ∂ gab c 2 c 2 ce d 2 cd e 2 c 1 1 − gadg ∂ gbe + gbeg ∂ gad. 2 ce d 2 cd e Interchanging dummy indices d and e, we get 1 1 1 1 ∇ gab = − gbeg ∂ gad + gadg ∂ gbe − gadg ∂ gbe + gbeg ∂ gad c 2 cd e 2 ce d 2 ce d 2 cd e = 0. (A.4)

41 A.3 Covariant derivative for scalar field, ∇aφ

We will show that covariant derivative for scalar field is just ordinary derivative. Considering

¡ b¢ ∇aφ = ∇a XbX b b = Xb∇aX + X ∇aXb b b c = 2Xb∂aX + 2XbΓacX ¡ b¢ b c bd¡ ¢ = 2∂a XbX − 2X ∂aXb + XbX g ∂agdc + ∂cgad − ∂dgac b c d c d c d = 2∂aφ − 2X ∂aXb + X X ∂agdc + X X ∂cgad − X X ∂dgac.

Interchanging dummy indices c and d yields that the last and the forth term cancel out. Therefore we get

b c d ∇aφ = 2∂aφ − 2X ∂aXb + X X ∂agdc b c ¡ d¢ c d = 2∂aφ − 2X ∂aXb + X ∂a gdcX − X gdc∂aX b c d = 2∂aφ − 2X ∂aXb + X ∂aXc − Xd∂aX b d = 2∂aφ − X ∂aXb − Xd∂aX ¡ b¢ = 2∂aφ − ∂a XbX

= ∂aφ (A.5)

a a A.4 R bcd = −R bdc

a a Proving the identity R bcd = −R bdc is as follows

a a a e a e a R bcd = ∂cΓbd − ∂dΓbc + ΓbdΓec − ΓbcΓed ¡ a a e a e a ¢ = − − ∂cΓbd + ∂dΓbc − ΓbdΓec + ΓbcΓed ¡ a a e a e a ¢ = − ∂dΓbc − ∂cΓbd + ΓbcΓed − ΓbdΓec a a ∴ R bcd = −R bdc (A.6)

42 a a a A.5 R bcd + R dbc + R cdb = 0

a a a Proving the identity R bcd + R dbc + R cdb = 0 is as follows a a a e a e a 1.R bcd = ∂cΓbd − ∂dΓbc + ΓbdΓec − ΓbcΓed a a a e a e a 2.R dbc = ∂bΓdc − ∂cΓdb + ΓdcΓeb − ΓdbΓec a a a e a e a 3.R cdb = ∂dΓcb − ∂bΓcd + ΓcbΓed − ΓcdΓeb

a a a a a e a e a a ∴ R bcd + R dbc + R cdb = ∂cΓbd − ∂dΓbc + ΓbdΓec − ΓbcΓed + ∂bΓdc a e a e a a −∂cΓdb + ΓdcΓeb − ΓdbΓec + ∂dΓcb a e a e a −∂bΓcd + ΓcbΓed − ΓcdΓeb = 0 (A.7)

A.6 Bianchi identities

Considering covariant derivative of Riemann tensor evaluated in locally inertial coor- dinate:

∇aˆRˆbcˆdˆeˆ = ∂aˆRˆbcˆdˆeˆ (A.8) Hats on their indices represent locally inertial coordinate. Notice that for locally inertial coordinate, Cristoffel symbols which contain first-order derivatives of metrics must vanish at any points. But the second derivatives of metrics in Riemann tensor do not vanish. Therefore R = g Raˆ ˆbcˆdˆeˆ aˆˆb³ cˆdˆeˆ ´ aˆ aˆ = gaˆˆb ∂dˆΓcˆeˆ − ∂eˆΓdˆcˆ aˆfˆ¡ ¢ = gaˆˆbg ∂dˆ∂cˆgeˆfˆ + ∂dˆ∂eˆgcˆfˆ − ∂dˆ∂fˆgcˆeˆ − ∂eˆ∂dˆgcˆfˆ − ∂eˆ∂cˆgdˆfˆ + ∂eˆ∂fˆgdˆcˆ 1¡ ¢ ∴ R = ∂ ∂ g − ∂ ∂ g − ∂ ∂ g + ∂ ∂ g . (A.9) ˆbcˆdˆeˆ 2 dˆ cˆ eˆˆb dˆ ˆb cˆeˆ eˆ cˆ dˆˆb eˆ ˆb dˆcˆ We therefore obtain

∇aˆRˆbcˆdˆeˆ + ∇cˆRaˆˆbdˆeˆ + ∇ˆbRcˆaˆdˆeˆ 1¡ = ∂ ∂ ∂ g − ∂ ∂ ∂ g − ∂ ∂ ∂ g + ∂ ∂ ∂ g 2 aˆ dˆ cˆ eˆˆb aˆ dˆ ˆb cˆeˆ aˆ eˆ cˆ dˆˆb aˆ eˆ ˆb dˆcˆ +∂cˆ∂ ˆ∂ˆgeˆaˆ − ∂cˆ∂ ˆ∂aˆgˆ − ∂cˆ∂eˆ∂ˆg ˆ + ∂cˆ∂eˆ∂aˆg ˆˆ d b d beˆ b daˆ db¢ +∂ˆb∂dˆ∂aˆgeˆcˆ − ∂ˆb∂dˆ∂cˆgaˆeˆ − ∂ˆb∂eˆ∂aˆgdˆcˆ + ∂ˆb∂eˆ∂cˆgdˆaˆ = 0. (A.10)

43 b A.7 Conservation of Einstein tensor: ∇ Gab = 0

Considering Bianchi identities,

∇eRabcd + ∇dRabec + ∇cRabde = 0 ae bc¡ ¢ g g ∇eRabcd + ∇dRabec + ∇cRabde = 0 a b ∇ Rad − ∇dR + ∇ Rbd = 0.

Renaming indices b to a, we have

a 2∇ Rad − ∇dR = 0 1 ∇aR − ∇ R = 0 ad 2 d ¡ 1 ¢ ∇a R − g R = 0. ad 2 ad We then see that twice-contracted Bianchi identities is equivalent to

a ∇ Gad = 0. (A.11)

44 Appendix B

Detail calculation

B.1 Variance of electromagnetic wave equation un- der Galilean transformation

Consider electromagnetic wave equation

∂2φ ∂2φ ∂2φ 1 ∂2φ + + − = 0. (B.1) ∂x2 ∂y2 ∂z2 c2 ∂t2

Transforming t → t0,

∂φ ∂t0 ∂φ ∂x0 ∂φ ∂y0 ∂φ ∂z0 ∂φ = + + + . ∂t ∂t ∂t0 ∂t ∂x0 ∂t ∂y0 ∂t ∂z0

Using x0 = x − vt, y0 = y, z0 = z, t0 = t, we get ∂φ ∂φ ∂φ = 0 − v 0 ∂t ∂tµ ∂x¶ µ ¶ ∂2φ ∂ ∂φ ∂ ∂φ = − v ∂t2 ∂t ∂t0 ∂t ∂x0 µ ¶ ∂t0 ∂2φ ∂x0 ∂2φ ∂y0 ∂2φ ∂z0 ∂2φ = + + + ∂t ∂t02 ∂t ∂x0∂t0 ∂t ∂y0∂t0 ∂t ∂z0∂t0 µ ¶ ∂t0 ∂2φ ∂x0 ∂2φ ∂y0 ∂2φ ∂z0 ∂2φ −v + + + ∂t ∂t0∂x0 ∂t ∂x02 ∂t ∂y0∂t0 ∂t ∂z0∂t0 ∂2φ ∂2φ ∂2φ = − 2v + v2 . (B.2) ∂t02 ∂x0∂t0 ∂x02

45 Transforming x → x0 ∂φ ∂t0 ∂φ ∂x0 ∂φ ∂y0 ∂φ ∂z0 ∂φ = + + + ∂x ∂x ∂t0 ∂x ∂x0 ∂x ∂y0 ∂x ∂z0 ∂φ = ∂x0 ∂2φ ∂2φ = (B.3) ∂x2 ∂x02 The y-axis and z-axis do not change, we have ∂2φ ∂2φ = (B.4) ∂y2 ∂y02 and ∂2φ ∂2φ = . (B.5) ∂z2 ∂z02 Substituting all equations of transformation into equations (B.1), ∂2φ ∂2φ ∂2φ 1 ∂2φ 2v ∂2φ v2 ∂2φ + + − + − , ∂x02 ∂y02 ∂z02 c2 ∂t02 c2 ∂x0∂t0 c2 ∂x02 we finally have equation (2.5): c2 − v2 ∂2φ 2v ∂2φ ∂2φ ∂2φ 1 ∂2φ + + + − = 0 (B.6) c2 ∂x02 c2 ∂t0∂x0 ∂y02 ∂z02 c2 ∂t02

B.2 Poisson’s equation for Newtonian gravitational field

Considering a close surface S enclosing a mass M. The quantity g · ndS is defined as the gravitational flux passing through the surface element dS where n is the outward unit vector normal to S. The total gravitational flux through S is then given by Z Z er · n g · ndS = −GM 2 dS. (B.7) S S r 2 2 By definition (er · n/r dS) = cos θ dS/r is the element of solid angle dΩ subtended at M by the element surface dS. Therefore equation (B.7) becomes Z Z g · ndS = −GM dΩ = −4πGM. S Z = −4πG ρ(r)dV v

46 From application of divergence theorem, the left side of this equation can be rewritten Z Z g · ndS = ∇ · gdV. (B.8) S v Therefore we obtain ∇ · g = −4πGρ. Recall that g = −∇Φ. (B.9) We therefore have result as in equation (3.57), ∇2Φ = 4πGρ.

a B.3 Variation of Cristoffel symbols : δΓbc

From equation (3.14) we have 1 ¡ ¢ Γa = gad ∂ g + ∂ g − ∂ g bc 2 b dc c bd d bc 1 ¡ ¢ 1 ¡ ¢ δΓa = δgad ∂ g + ∂ g − ∂ g + gad ∂ δg + ∂ δg − ∂ δg . bc 2 b dc c bd d bc 2 b dc c bd d bc Considering variation of gad,

ad d f ad δg = δf δd δg = δ dg gef δgad f de h i d ef ¡ ad¢ ad = δf g δ gdeg − g δgde h i d ef a ad = δf g δ(δe ) − g δgde ed ad = −g g δgde, therefore 1 ³ ´ 1 ³ ´ δΓa = − gedgadδg ∂ g + ∂ g − ∂ g + gad ∂ δg + ∂ δg − ∂ δg . bc 2 de b dc c bd d bc 2 b dc c bd d bc 1 ¡ ´ = −gadΓe δg + gad ∂ δg + ∂ δg − ∂ δg bc ed 2 b dc c bd d bc 1 ³ ´ = gad ∂ δg + ∂ δg − ∂ δg − 2Γe δg 2 b dc c bd d bc bc ed ³ 1 ad e e e e = g ∂bδgdc − Γbcδged − Γbdδgec + ∂cδgbd − Γbcδged − Γdcδgeb 2 ´ e e −∂dδgbc + Γbdδgec + Γdcδgeb 1 ¡ ¢ ∴ δΓa = gad ∇ δg + ∇ δg − ∇ δg . (B.10) bc 2 b dc c bd d bc 47 Bibliography

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