11/2/03 Chris Pearson : Fundamental Cosmology 4: …cpp/teaching/cosmology/documents/...11/2/03 Chris Pearson : Fundamental Cosmology 4: General Relativistic Cosmology ISAS -2003 3

11/2/03 Chris Pearson : Fundamental Cosmology 4: General Relativistic Cosmology ISAS -2003

1

GENERAL RELATIVITY

“Matter tells space how to curve. Space tells matter how to move.”

John Archibald Wheeler

““Matter tells space how to curve.Matter tells space how to curve. Space tells matter how to move. Space tells matter how to move.””

John Archibald WheelerJohn Archibald Wheeler

Fundamental Cosmology: 4.General Relativistic CosmologyFundamental Cosmology: 4.Fundamental Cosmology: 4.General Relativistic CosmologyGeneral Relativistic Cosmology


2

GENERAL RELATIVITY

4.1: Introduction to General Relativity4.1: Introduction to General Relativity

目的

Suggested Reading on General Relativity• Introduction to Cosmology - Narlikar 1993• Cosmology - The origin and Evolution of Cosmic StructureColes & Lucchin 1995• Principles of Modern Cosmology - Peebles 1993

• Introduce the Tools of General Relativity• Become familiar with the Tensor environment• Look at how we can represent curved space times• How is the geometry represented ?• How is the matter represented ?• Derive Einstein’s famous equation• Show how we arrive at the Friedmann Equations via General Relativity• Examine the Geometry of the Universe

• このセミナーの後で皆さんは私のことが大嫌いになるかな~~~~


3

GENERAL RELATIVITY

4.1: Introduction to General Relativity4.1: Introduction to General Relativity

Newton v EinsteinNewton: • Mass tells Gravity how to make a Force• Force tells mass how to accelerate

Einstein: • Mass tells space how to curve• Space tells mass how to move

Newton: • Flat Euclidean Space• Universal Frame of reference

Einstein: • Space can be curved• Its all relative anyway!!


4

GENERAL RELATIVITY

4.1: Introduction to General Relativity4.1: Introduction to General RelativityThe Principle of Equivalence

†

mg = miRecall: Equivalence Principle

†

r a

A more general interpretation led Einstein to his theory of General Relativity

PRINCIPLE OF EQUIVALENCE:An observer cannot distinguish between a local gravitational field and an equivalent uniform acceleration

†

g =r a

• Imagine 2 light beams in 2 boxes• the first box being accelerated upwards• the second in freefall in gravitational field

• For a box under acceleration •Beam is bent downwards as box moves up

• For box in freefall ‹ photon path is bent down!!

Implications of Principle of Equivalence

Fermat’s Principle:• Light travels the shortest distance between 2 points• Euclid = straight line• Under gravity - not straight line• ‹ Space is not Euclidean !


5

GENERAL RELATIVITY

4.2: Geometry, Metrics & Tensors4.2: Geometry, Metrics & Tensors• The Interval (line element) in Euclidean Space:

ds

dx

dy

†

ds2 = dx 2 + dy 22-D

†

ds2 = dx 2 + dy 2 + dz23-D

†

ds2 = dr2 + r2dq 22-D

†

ds2 = dr2 + r2dq 2 + r2Sin2qdf 23-D

x

y

z

rsinq

rsinqdf

df

dq

dr

f

q

r

rdq dr

dS

rdq

dr

dS

ds

(dx2+dy2)1/2

dz


6

GENERAL RELATIVITY

4.2: Geometry, Metrics & Tensors4.2: Geometry, Metrics & Tensors• The Interval in Special Relativity (The Minkowski Metric) :

• The Laws of Physics are the same for all inertial Observers (frames of constant velocity)• The speed of light, c, is a constant for all inertial Observers

‡ Events are characterized by 4 co-ordinates (t,x,y,z)‡ Length Contraction, Time Dilation, Mass increase‡ Space and Time are linked

P The notion of SPACE-TIMESPACE-TIME

ds

(dx2+dy2 +dz2)1/2

dt

†

ds2 = c 2dt 2 - (dx 2 + dy 2 + dz2)

†

ds2 = c 2dt 2 - dr2 + r2dq 2 + r2Sin2qdf 2[ ]


7

GENERAL RELATIVITY

4.2: Geometry, Metrics & Tensors4.2: Geometry, Metrics & Tensors• The Interval in Special Relativity (The Minkowski Metric) :

x

y

t

dS2 =0

dS2 =0

dS2<0

dS2<0

Causally Connected Eventsin Minkowski Spacetime

Causally Connected Eventsin Minkowski Spacetime

dS2>0

dS2>0

†

ds2 = c 2dt 2 - (dx 2 + dy 2 + dz2)

o Area within light cone: dS2>0Events that can affect observer in past,present,future.This is a timelike interval.Observer can be present at 2 events by selecting anappropriate speed.

o Equation of a light ray, dS2=0, Trace out light cone from Observer in Minkowski S-TSpreading into the futureCollapsing from the past

†

dxdt

= ±c

o Area outside light cone: dS2<0Events that are causally disconnected from observer.This is a spacelike interval.These events have no effect on observer.


8

GENERAL RELATIVITY

4.2: Geometry, Metrics & Tensors4.2: Geometry, Metrics & Tensors• The Interval in General Relativity :

o In General

The interval is given by:

†

ds2 = giji, j= 0

n

Â dx idx j

• gij is the metric tensor (Riemannian Tensor) : • Tells us how to calculate the distance between 2 points in any given spacetime

• Components of gij • Multiplicative factors of differential displacements (dxi)• Generalized Pythagorean Theorem


9

GENERAL RELATIVITY

4.2: Geometry, Metrics & Tensors4.2: Geometry, Metrics & Tensors• The Interval in General Relativity :

†

ds2 = giji, j= 0

n

Â dx idx j

Euclidean Metric

†

gi, j = di, j

†

di, j = 0 i ≠ j( )

†

gi, j =

1 0 00 1 00 0 1

È

Î

Í Í Í

˘

˚

˙ ˙ ˙

Minkowski Metric

†

gi, j =

-1 0 0 00 1 0 00 0 1 00 0 0 1

È

Î

Í Í Í Í

˘

˚

˙ ˙ ˙ ˙

†

dx1 = xdx 2 = ydx 3 = z

†

dx 0 = t dx1 = xdx 2 = y dx 3 = z

†

ds2 = dx 2 + dy 2 + dz2

†

(n = 2) 3 Dimensions

†

ds2 = cdt 2 - (dx 2 + dy 2 + dz2)

†

(n = 3) 4 Dimensions


10

GENERAL RELATIVITY

4.2: Geometry, Metrics & Tensors4.2: Geometry, Metrics & Tensors• A little bit about Tensors - 1

†

ds2 = giji, j= 0

n

Â dx idx j

o Consider a matrix M=(mij)

o Tensor ~ arbitray number of indicies (rank)• Scalar = zeroth rank Tensor• Vector = 1st rank Tensor• Matrix = 2nd rank Tensor : matrix is a tensor of type (1,1), i.e. mi

j

o Tensors can be categorized depending on certain transformation rules:• Covariant Tensor: indices are low• Contravariant Tensor: indices are high• Tensor can be mixed rank (made from covariant and contravariant indices)• Euclidean 3D space: Covariant = Contravariant = Cartesian Tensors• 注意 mi

jk≠ mijk


11

GENERAL RELATIVITY

4.2: Geometry, Metrics & Tensors4.2: Geometry, Metrics & Tensors• A little bit about Tensors - 2o Covariant Tensor Transformation:

Consider Scalar quantity f. Scalar · invariant under coordinate transformations

†

f(x i ) = f(x' i )e.g. f(x1,x2,x3) = f'(x'1 ,x '2 ,x'3 )

So for,

†

Ai =∂f∂xi

†

A'i =∂'f∂x'i

†

A j =∂f∂x j

etc……

†

A'i =∂'f∂x'i

=∂x j

∂x'i∂f∂x j

=∂x j

∂x'iA j

†

A'i = ai, j A j

†

ai, j =∂x j

∂x'i

†

A'ij =∂xk

∂x'i∂xl

∂x' j

Akl

†

—f =∂f∂x1

ˆ x 1 +∂f∂x2

ˆ x 2 +∂f∂x3

ˆ x 3Taking the gradient

†

Ai =∂f∂xi

this normal has components

†

A'i =∂f '∂x'i

=∂f∂x'i

Which should not change under coordinate transformation, so

†

∂f'∂x'i

=∂f∂x j

∂x j

∂x'iAlso have the relation

Generalize for 2nd rank Tensors, Covariant 2nd rank tensors are animals that transform as :-


12

GENERAL RELATIVITY

4.2: Geometry, Metrics & Tensors4.2: Geometry, Metrics & Tensors• A little bit about Tensors - 3o Contravariant Tensor Transformation:

Consider a curve in space, parameterized by some value, r

†

x i(r)Coordinates of any point along the curve will be given by:

†

Ai =dx i

drTangent to the curve at any point is given by:

The tangent to a curve should not change under coordinate transformation, so

†

A'i = dx 'i

dr

†

dx'i

dr=

∂x'i

∂x j

dx j

drAlso have the relation

So for,

†

Ai =dx i

dr

†

A'i = dx 'i

dr

†

A j =dx j

dretc……

†

A'i = ai, j A j

†

ai, j =∂x'i

∂x j

†

A'i = dx 'i

dr=

∂x'i

∂x j

dx j

dr=

∂x'i

∂x j

A j

†

A'ij =∂x'i∂xk

∂x' j

∂xl

AklGeneralize for 2nd rank Tensors, Contravariant 2nd rank tensors are animals that transform as :-


13

GENERAL RELATIVITY

4.2: Geometry, Metrics & Tensors4.2: Geometry, Metrics & Tensors• More about Tensors - 4o Index Gymnastics

o Raising and Lowering indices:

Transform covariant to contravariant form by multiplication by metric tensor, gi,j

†

gij Ai = A j

†

gij Ai = A j

o Einstein Summation:

Repeated indices (in sub and superscript) are implicitly summed over

†

aii

Â ai = aiai

Aikk

Â Bk = AikBk

Aiki,kÂ BiBk = AikB

iBk

Einstein c.1916: "I have made a great discovery in mathematics; I have suppressed the summation sign everytime that the summation must be made over an index which occurs twice..."

†

ds2 = giji, j= 0

n

Â dx idx jThus, the metric May be written as

†

ds2 = gijdx idx j


14

GENERAL RELATIVITY

Minkowski Spacetime (Special Relativity)

†

gij AiA j < 0

gij AiA j > 0

gij AiA j = 0

spacelike

timelike

null

4.2: Geometry, Metrics & Tensors4.2: Geometry, Metrics & Tensors• More about Tensors - 5o Tensor Manipulation

o Tensor Contraction: Set unlike indices equal and sum according to the Einstein summation convention.Contraction reduces the tensor rank by 2.For a second-rank tensor this is equivalent to the Scalar Product

†

AiB j

i = jAiB j = A0B0 + A1B1 + A2B2 + A3B3


15

GENERAL RELATIVITY

4.2: Geometry, Metrics & Tensors4.2: Geometry, Metrics & Tensors• More about Tensors - 6o Tensor Calculus

o The Covariant Derivative of a Tensor:

Derivatives of a Scalar transform as a vector

†

Ai =∂f∂xi

How about derivative of a Tensor ?? Does transform as a Tensor?

†

∂Ai

∂x j

†

∂A'i∂x'm

=∂x j

∂x'i∂xn

∂x'm∂A j

∂xn +∂ 2x j

∂x'm ∂x'iA j

†

A'i =∂x j

∂x'iA j

Lets see…..Take our definition of Covariant Tensor Problem

Implies transformation coefficients vary with position

Correction Factor:Correction Factor:

†

Gijk

such that for ,

†

dAi = Gijk Akdx j

†

∂Ai

∂x j

Redefine the covariant derivative of a tensor as;

†

Ai; j ≡∂Ai

∂x j - Gijk Ak ≡ Ai, j - Gij

k Ak

• Christoffel Symbols• Defines parallel vectors at neighbouring points• Parallelism Property: affine connection of S-T• Christoffel Symbols = fn(spacetime)

, normal derivative

; covariant derivative


16

GENERAL RELATIVITY

4.2: Geometry, Metrics & Tensors4.2: Geometry, Metrics & Tensors• Riemann Geometry

General Relativity formulated in the non-Euclidean Riemann Geometry

†

Gkli = Glk

i

†

gik;l = 0‡ Simplifications

Covariant Differentiation of Metric Tensor

†

gik;l ≡ gik,l - Gilpgpk - Gkl

pgip 1

‡ 40 linear equations - ONE Unique Solution for Christoffel symbol

†

Gkli =

gim

2gmk,l + gml,k - gkl,m( )1

†

ds = constantpathÚSPECIAL RELATIVITY

†

ds2 = c 2dt 2 - (dx 2 + dy 2 + dz2) Shortest distance between 2 points is a straight line.

†

d ds = 0pathÚ But lines are not straight

(because of the metric tensor)GENERAL RELATIVITY

†

ds2 = gijdx idx j

Gravity is a property of Spacetime, which may be curved

Path of a free particle is a geodesic where

†

d2x i

ds2 + Gkli dxk

dsdx l

ds= 0

higher rank fl extra Christoffel Symbol


17

GENERAL RELATIVITY

4.2: Geometry, Metrics & Tensors4.2: Geometry, Metrics & Tensors• The Riemann Tensor

†

dAi = Gikl Aldxk

Recall correction factor for the transportation of components of parallel vectors between neighbouring points

†

Ai;k ≡∂Ai

∂xk - Gikl Al ≡ Ai,k - Gik

l AlSuch that we needed to define the Covariant derivative

To transport a vector without the result being dependent on the path,

Require a vector Ai(xk) such that -

†

∂Ai

∂xk = Gikl Al 1

1

†

∂∂xn

†

∂ 2Ai

∂xn∂xk =∂

∂xn Gikl Al( ) =

∂Gikl

∂xn Al + Gikl ∂Al

∂xn =∂Gik

m

∂xn + Gikl Gln

mÊ

Ë Á

ˆ

¯ ˜ Am

Interchange differentiation w.r.t xn & xk and use Ai,nk=Ai,kn

Necessary condition for 1

†

0 =∂Gik

m

∂xn -∂Gin

m

∂xk + Gikl Gln

m - Ginl Glk

m ≡ Rim

kn

Rim

kn is the Riemann TensorCan show:

†

Ai;nk - Ai;kn ≡ Rim

kn Am Where RHS=LHS=Tensor

• Defines geometry of spacetime (=0 for flat spacetime).• Has 256 components but reduces to 20 due to symmetries.


18

GENERAL RELATIVITY

4.2: Geometry, Metrics & Tensors4.2: Geometry, Metrics & Tensors• The Einstein Tensor

• Contracting the Riemann Tensor gives the Ricci Tensor describing the curvature

†

Rkl = gimRiklm ≡ Rmklm

• Contracting the Ricci Tensor gives the Scalar Curvature (Ricci Scalar)

†

R = gklRkl

• Define the Einstein Tensor as

†

Gik = Rkl -12

gikR

• Lowering the second index of the Riemann Tensor, define

†

gijRklmj = Riklm

• Symmetry properties of

†

Riklm fi Riklm;n + Riknl;m + Rikmn;l = 0 Bianchi Identities

Multyplying by gimgkn and using above relations ‡

†

Rik -12

gikRÊ

Ë Á

ˆ

¯ ˜ ;k ≡ 0

or…… The Einstein Tensor has ZERO divergence

Rim

kn is the Riemann Tensor


19

GENERAL RELATIVITY

4.3: Einstein4.3: Einstein’’s Theory of Gravitys Theory of Gravity• The Energy Momentum Tensor

o Non relativistic particles (Dust)†

T ikÏ Ì Ó

Define Energy Momentum Tensor

†

T ik = T 00 = mNN

Vol

Â c 2 = nmc 2 = roc2ß In comoving rest frame of N dust particles of number density, n

†

T ik = ruiukß In any general Lorentz frame

†

T 00 Density of mass /unit vol.

†

T i0 Density of jth component of momentum/unit vol.

†

T ik k-component of flux of j component of momentum

ro= ro(x) - Proper density of the flow

ui= dxi/dt - 4 velocity of the flow

Looks like classical Kinetic energy

In general,

†

T = p ƒ u = mnu ƒ u = ru ƒ u

Dust approximationWorld lines almost parallelSpeeds non-relativistic

Density, r = mass/vol Momentum, p = energy/volConsider Units

†

T ik The flux of the i momentum component across a surface of constant xk

• Conservation of Energy and momentum - Tki;k=0


20

GENERAL RELATIVITY

Fluid approximationParticles colliding - PressureSpeeds nonrelativistic

Looks like gas law

4.3: Einstein4.3: Einstein’’s Theory of Gravitys Theory of Gravity• The Energy Momentum Tensoro Relativistic particles (inc. photons, neutrinos)

1/3 since random directions

Relativistic approximationParticles frequently collidingPressureSpeeds relativistic

Particle 4 Momentum,

†

pi =Ec

, r p Ê

Ë Á

ˆ

¯ ˜ E 2 = c 2 r p 2 + m2c 4 ª

r p 2c 2

†

T 00 = ENN

Vol

Â = e e - energy density

†

T11 = T 22 = T 22 =r p 2c 2

3EN

Vol

Â ª13

e

o Perfect Fluid

General Particle 4 Momentum,

†

r p 0 = gmc 2

r p a = gmr v (a =1,2,3)

†

T 00 = gmc 2

N

Vol

Â ª rc 2

†

T11 = T 22 = T 22 =gr p 2c 2

3N

Vol

Â ª P

In rest frame of particle

For any reference frame with fluid 4 velocity = u

†

T ik = (P + rc 2)uiuk - gikP Pressure is important


21

GENERAL RELATIVITY

4.3: Einstein4.3: Einstein’’s Theory of Gravitys Theory of Gravity• The Einstein Equation

†

T ik = (P + rc 2)uiuk - gikP

Energy Momentum Tensor (matter in the Universe)

†

Gik = Rkl -12

gikR

Einstein Tensor (Geometry of the Universe)

†

Gik = k T ik = Rkl -12

gikR

After trial and error !!Einstein proposed his famous equation

†

k=8pGc 4

•Classical limit•must reduce to Poisson’s eqn. for gravitational potential

†

—2f= -4pGr

†

Gik = Rkl -12

gikR =8pGc 4 T ik

ご結婚

おつかれ

おつかれ

Basically: Relativistic Poisson Equation


22

GENERAL RELATIVITY

4.3: Einstein4.3: Einstein’’s Theory of Gravitys Theory of Gravity• The Einstein Equation

“Matter tells space how to curve. Space tells matter how to move.”“Matter tells space how to curve. Space tells matter how to move.”

Fabric of the Spacetime continuum and the energy of the matter within it are interwoven

†

Gik = Rkl -12

gikR - Lgik =8pGc 4 T ik

†

Gik = Rkl -12

gikR =8pGc 4 T ik注意

Does not give a static solution !!

Einstein inserted the Cosmological Constant term L

“The Biggest Mistake of My Life ”


23

GENERAL RELATIVITY

4.3: Einstein4.3: Einstein’’s Theory of Gravitys Theory of Gravity• Solutions of the metric

†

ds2 = gi, ji, j= 0

3

Â dx idx j

• Cosmological Principle:q Isotropy · ga,b = da,b, only take a=b terms

q Homogeneity · g0,b = g0,a = 0 †

ds2 = ga ,ba,b =1

3

Â dxadx bjust spatial coords

The Metric:

†

-ds2 = g00(dx 0)2 + gab dxadx b = g00(dx 0)2 + dl2

†

dt = -g00dx 0

cfi ds2 = c 2dt 2 - dl2Proper time interval:

How about ?

†

ga,b dxadx b

embed our curved, isotropic 3D space in hypothetical 4-space.

‡ our 3-space is a hypersurface defined by in 4 space.

‡ length element becomes:

†

(x1)2 + (x 2)2 + (x 3)2 + (x 4 )2 = a2

†

dl2 = (dx1)2 + (dx 2)2 + (dx 3)2 + (dx 4 )2

†

dl2 = (dx1)2 + (dx 2)2 + (dx 3)2 +x1dx1 + x 2dx 2 + x 3dx 3

a2 - (x1)2 - (x 2)2 - (x 3)2

Constrain dl to lie in 3-Space(eliminate x4)

Jim Muth


24

GENERAL RELATIVITY

4.3: Einstein4.3: Einstein’’s Theory of Gravitys Theory of Gravity• Solutions of the metric

†

x1 = rsinq cosf

x 2 = rsinq sinf

x 3 = rcosq

Introduce spherical polar identities in 3 Space

†

dl2 = dr2 + r2 sin2 qdf 2 + r2dq 2 +r2dr2

a2 - r2 =dr2

1- r2 a2 + r2(dq 2 + sin2 qdf 2)

†

dS2 = c 2dt 2 - R2(t) dr2

1- kr2 + r2(dq 2 + sin2 qdf 2)Ê

Ë Á

ˆ

¯ ˜

I. Multiply spatial part by arbitrary function of time R(t) : wont affect isotropy and homogeneity because only a f(t)II. Absorb elemental length into R(t) ·r becomes dimensionlessIII. Re-write a-2 = k

• r, q, f are co-moving coordinates and don’t changed with time - They are SCALED by R(t)• t is the cosmological proper time or cosmic time - measured by observer who sees universe expanding around him•The co-ordinate, r, can be set such that k = -1, 0, +1

dS

The Robertson-Walker Metric


25

GENERAL RELATIVITY

4.3: Einstein4.3: Einstein’’s Theory of Gravitys Theory of Gravity• The Robertson-Walker Metric and the Geometry of the Universe

†

dS2 = c 2dt 2 - R2(t) dr2

1- kr2 + r2(dq 2 + sin2 qdf 2)Ê

Ë Á

ˆ

¯ ˜ The Robertson-Walker Metric

http

://m

ap.g

sfc.

nasa

.gov

/

k ~ defines the curvature of space time

k = -1 : Hyperbolic Space

k = +1 : Spherical Space

k = 0 : Flat Space


26

GENERAL RELATIVITY

4.3: Einstein4.3: Einstein’’s Theory of Gravitys Theory of Gravity• The Robertson-Walker Metric and the Geometry of the Universe

k = -1 : Hyperbolic Space

k = +1 : Spherical Space

k = 0 : Flat Space


27

GENERAL RELATIVITY

4.3: Einstein4.3: Einstein’’s Theory of Gravitys Theory of Gravity• The Friedmann Equations in General Relativistic Cosmology

Solve Einstein’s equation :

†

Gik = Rkl -12

gikR =8pGc 4 T ik

Need:• metric gik• energy tensor Tik

†

dS2 = c 2dt 2 - S2(t) dr2

1- kr2 + r2(dq 2 + sin2 qdf 2)Ê

Ë Á

ˆ

¯ ˜ Metric is given by R-W metric:

†

g00 =1 g11 = -S2

1- kr2 g22 = -S2r2 g33 = -S2r2 sin2 q

†

gik =1

gik

†

g00 =1 -g = -Srsinq

1- kr2

Non-zero gik

Non-zero Gikl

†

G011 = G02

2 = G033 =

S•

cS

G110 =

S S•

c(1- kr2)G22

0 =S S

•

r2

cG33

0 =S S

•

r2 sin2 qc

G111 =

kr1- kr2 G22

1 = -r(1- kr2) G331 = -r(1- kr2)sin2 q

G122 = G13

3 =1r

G332 = -sinq cosq G23

3 = cotq

Use S so don’t confuse with Tensors

†

x 0 = ct x1 = r x 2 = q x 3 = fCo-ordinates


28

GENERAL RELATIVITY


Non-zero components of Ricci Tensor Rik

†

R00 =

3c 2

S••

SR1

1 = R22 = R3

3 =1c 2

S••

S+

2S2•

+ 2kc 2

S2

Ê

Ë

Á Á

ˆ

¯

˜ ˜

The Ricci Scalar R

†

R =6c 2

S••

S+

S2•

+ kc 2

S2

Ê

Ë

Á Á

ˆ

¯

˜ ˜

Non-zero components of Einstein Tensor Gik

†

G00 = R0

0 -R2

= -3

c 2S2

•

+ kc 2

S2

Ê

Ë

Á Á

ˆ

¯

˜ ˜

G11 = G2

2 = G33 ≡ R1

1 -R2

= -1c 2 2 S

••

S+

S2•

+ kc 2

S2

Ê

Ë

Á Á

ˆ

¯

˜ ˜

Sub fl Einstein

†

S2•

+ kc 2

S2 =8pG3c 2 T0

0

2 S••

S+

S2•

+ kc 2

S2 =8pG3c 2 T1

1 =8pG3c 2 T2

2 =8pG3c 2 T3

3


29

GENERAL RELATIVITY


Non-zero components of Energy Tensor Tik

= e

=-P

†

2 S••

S+

S2•

+ kc 2

S2 =8pG3c 2 T1

1 =8pG3c 2 T2

2 =8pG3c 2 T3

3

S2•

+ kc 2

S2 =8pG3c 2 T0

0

1

2

Assume Dust:• P = 0• e = rc2

†

d(rS3)dS

= 0 fi r = roSo

SÊ

Ë Á

ˆ

¯ ˜

3

fi T00 = roc

2 So

SÊ

Ë Á

ˆ

¯ ˜

3

, T11 = 0

†

∂dt

2

†

d(eS3)dS

+ 3PS2 = 0

actually implied by Tki;k=0

3

1

2

3

†

2 S••

S+

S2•

+ kc 2

S2 = 0

S2•

+ kc 2

S2 =8pGro

3c 2So

SÊ

Ë Á

ˆ

¯ ˜

3


30

GENERAL RELATIVITY


Return S(t)=R(t) notation

†

2 S••

S+

S2•

+ kc 2

S2 = 0 S2•

+ kc 2

S2 =8pGro

3c 2So

SÊ

Ë Á

ˆ

¯ ˜

31 2

†

r = roRo

RÊ

Ë Á

ˆ

¯ ˜

3

3

2

†

R2•

=8pGr

3R2 - kc 2 +

LR2

3Ê

Ë Á

ˆ

¯ ˜

†

R••

= -4pGr

3R +

LR3

Ê

Ë Á

ˆ

¯ ˜ 3

Our old friends the Friedmann Equations

Finally …………


31

GENERAL RELATIVITY

4.4: Summary4.4: Summary• The Friedmann Equations in General Relativistic Cosmology

• We have come along way today!!!

†

R2•

=8pGr

3R2 - kc 2 +

LR2

3Ê

Ë Á

ˆ

¯ ˜

†

R••

= -4pGr

3R +

LR3

Ê

Ë Á

ˆ

¯ ˜

The Friedmann Equations describe the evolution of the Universe

†

Gik = Rkl -12

gikR =8pGc 4 T ik

Deriving the necessary components of The Einstein Field Equation• Spacetime and the Energy within it are symbiotic• The Einstein equation describes this relationship

†

dS2 = c 2dt 2 - R2(t) dr2

1- kr2 + r2(dq 2 + sin2 qdf 2)Ê

Ë Á

ˆ

¯ ˜

The Robertson-Walker Metric defines thegeometry of the Universe

THESE WILL BE OUR TOOLBOX FOR OUR COSMOLOGICAL STUDIES


32

GENERAL RELATIVITY

4.4: Summary4.4: Summary

Fundamental CosmologyFundamental Cosmology4. General Relativistic Cosmology4. General Relativistic Cosmology終終終

次：次：次：Fundamental CosmologyFundamental Cosmology

5. The Equation of state5. The Equation of state& the Evolution of the Universe& the Evolution of the Universe

Documents

11/2/03 Chris Pearson : Fundamental Cosmology 4: …cpp/teaching/cosmology/documents/...11/2/03 Chris Pearson : Fundamental Cosmology 4: General Relativistic Cosmology ISAS -2003 3