Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mso.anu.edu.au/~brian/A3002/lect3.pdf Äàòà èçìåíåíèÿ: Thu Jul 19 03:52:42 2007 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 01:07:09 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: ï ï ï ï ï ï ï ï

Final Parametric Solution
+=
1 " !0 r 1 (1 " cos) )= 2 ! 0 r0
3/ 2

A Quick Overview of Relativity
· S p e c i a l R e l a ti v i ty :
­ The manifestation of requiring the speed of light to be invariant in all inertial (nonaccelerating) reference frames ­ Leads to a more complicated view of the world where space and time have to be considered together ­ space-time

1 (1 % 1 " !0 * = & ) " sin ) # = 2 !0 '2 $

H 0t

M in k o w s k i S p a c e
· Special relativity defined by cartesian coordinates, xµ on a 4-dimensional manifold. · Events in special relativity are specified by its location in time and space ­ a fourvector ­ e.g. Vµ
x 0 % ct = t $ ! x1 % x ! #x 2 x %y ! 3 ! x %z "
µ

Minkowski Metric
· Metric tells you how to take the norm of a vector ­ e.g. the dot product. In Minkowski space, the dot product of two vectors is, where we use the summation convention (lower and upper indices are summed over all possible values) Minkowski Metric
)
µ(

& $ $ =$ $ $ %

'1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

# ! ! ! ! ! "

A # B " % µ$ Aµ B$ = ! A0 B 0 + A1 B1 + A2 B 2 + A3 B

3

ds 2 " $ µ# dx µ dx# = !dt 2 + dx 2 + dy 2 + dz

2

T im e
· N o te th a t fo r a p a r ti c l e w i th fi x e d coordinates, ds2 =-dt2 < 0 D e fi n e P r o p e r ti m e a s d ! 2 - d s 2 T h e p r o p e r ti m e e l a p s e d a l o n g a tr a j e c to r y th r o u g h s p a c e ti m e r e p r e s e n ts th e a c tu a l ti m e m e a s u r e d b y th e o b s e rv e r.

T e n s o rs
· General Relativity requires curved space · Use Tensors ­ which are a way of expressing information in a coordinate invariant way. If an equation is expressed as a tensor in one coordinate system, it will be valid in all systems. · Tensors are objects like vectors, or matrices, except they may have any range of indices and must transform in a coordinate invariant way.


The Metric Tensor
The metric tensor in GR is the foundation of the subject. It is the generalisation of the Minkowski metric. It describes spacetime in a possibly a non-flat, noncartesian case. e.g. Spherical coordinate flat case
x0 ' t & # x1 ' r sin ! cos " # %x 2 x ' r sin ! sin " # 3 # x ' r sin ! $
µ

An Example ­Maxwell's equations

g

µ)

& $ $ =$ $ $ %

'1 0 0 010 0 0 r2 0 0 0

# ! ! ! ! r 2 sin 2 ( ! "

0 0 0

The basic components of GR equations
· R i e m a n n c u r v a tu r e te n s o r ­ w h i c h i s a c o m p l i c a te d e x p r e s s i o n d e r i v e d fr o m t h e m e t r i c t e n s o r ­g i v e s c u r v a t u r e · G e o d e s i c s ­ t h e s h o r te s t s p a c e - ti m e d i s ta n c e b e tw e e n tw o p o i n ts . Im a g i n e a s e t o f p a th s i s p a r a m e te r i s e d b y a s i n g l e p a r a m e te r "

Einstein's Equation of motion
G
µ!

1 $ Rµ! # Rg 2

µ!

= 8"GTµ

!

Lefthand side describes curvature of space time Tµ% is called the stress-energy Tensor and contains a complete description of energy and momentum of all matter fields. Lots of information in the Tµ% and Gµ% - end up with very complicated linked highly non-linear equations.

· T e s t p a r ti c l e s m o v e a l o n g g e o d e s i c s

G R a n d C o s m o lo g y
· Robertson Walker Metric (independent of Einstein's Equations) Provides a completely descriptive metric for a homogenous and isotropic universe

Einstein's equation for a perfect fluid 8"G
· a is the scale factor ­ tracks a piece of the Universe. · p and & are pressure and density of matter (perfect fluid)
a 2 + kc 2 = a=$ 1 %a ' c2 & a k = a2 k = a2 3

#a

2

& dr 2 # ds 2 = (cdt ) 2 ' a (t ) 2 $ + r 2 (d) 2 + sin 2 )d( 2 )! 2 %1 ' kr " · r,#,$ are spherical coordinates · a(t) describes the size of a piece of space over time · k tells you the curvature (-1,0,1) -> (open, flat, closed)

4 "G (# + 3 p)a 3 divide 1st eq. (2 k 1 8"G by (ac)^2 # * + 2= 2 )a c3 move k/a^2 to 1 % a (2 ( 1 % 8"G # $' * * ' side, factor out c^2 & a) * c2 ' 3 & ) 2 $2 ( 1 % a ( % 8"G % a ( #' * $ 1* ' *' * c 2 & a) ' 3 & a) & )
factor out (adot/a)^2

a = H 0 and a k 1 % a( = '* a2 c 2 & a )

#
2

crit

as before

%# ( $ 1* ' & # crit )

!


Summary of Behaviour of Density parameter and Geometry
FLAT $"0 = 1 k = 0 "( t ) = 1( & & OPEN %"0 < 1 k = #1 "( t ) < 1)for all time &" > 1 k = +1 "( t ) > 1& CLOSED ' 0 *
" " "
c,0

Critical Density Value.
=
2 3H 0 8#G 2

c,0

% ( H0 = 9.2 x10$27 kg / m 3 ' * & 70 km / s / Mpc ) = 1.4 x10$7 M
sun

Contributions to Density
!

"0 =

$

# i,0 = # crit ,0

c,0

$

% ( H0 / pc 3 ' * & 70 km / s / Mpc )

2

"

i, 0

"0 = "

M ,0

+ "% ,0 + "& ,0 + "

' ,0

+ "?,

!
0

!

Contributions to '
· Within a Megaparsec (Mpc) There are two large galaxies - Andromeda and Milky Way - 1012solar masses of material - So density is roughly 10-6solar Masses per cubic parsec. · Radiation from Big Bang has approx 'rad=5x10-5 · Current Concordance Model of Cosmology has
­ ­ ­ ­ ('i=1.00 ' M= 0 . 2 7 ' ) = 0 .7 3 ' e v e r y th in g e

Solving Einstein's Equations
Robertson-Walker Metric
& dr 2 # ds 2 = (cdt ) 2 . a (t ) 2 $ + r 2 (d2 2 + sin 2 2d1 2 )! 2 %1 . kr " redefine as conformal time dt 0 = c/ d- = d2 2 + sin 2 2d1 2 a(t) & , dr 2 )# * ' ds 2 = (a (0 )) 2 $(d0 ) 2 . * + r 2 (d-) '! 2 + 1 . kr (" %

Friedman Equation
1 c2 " $ #
2 a% k 1 8(G = ) '+ a & a2 c 2 3

re-writing 1st G.R. eq

2 1 " da % 8(G 2 )a * k $ '= c 2 # dt & 3c 2

ls e

< 0 .0 1
!

re-arranging above and moving out a^2

"=c

#

dt a(t)

# da & 2 8)G 4 *a + ka % (= 3c 2 $ d" ' y, a a0 -0 =
2

2

Normalise equation to current epoch
& 1 da (= ' a0 d"
2 0

8)G 2 3H 0

d #a % d" $ a0

!

.d #a 0% / d" $ a0

2 &1 2 -0 H 0 # a & 4 4 # a & 2 *% ( a0 + k% ( a (3 a0 = * 0c 2 $ a0 ' '2 $ a0 '

. d # a &12 a 2- H 2 * 0 % (3 = 0 0 0 c2 *0 / d" $ a0 '2
chain rule for differentiation

# a &4 # a & % ( + k% ( $ a0 ' $ a0 '

2

1 c2

" da % 2 8(G 2 )a * k $ '= # dt & 3c 2
replace (1/c da/dt) with above expression. move a^2 over

Boundary condition
replacing dn/dt with relation above.

!

!

1 c2 1 c2

" a % 2 1 8(G k )* 2 $'= 2 # a& c 3 a 1 8(G k 2 )0 * (H 0 ) = 2 c3 a0

H0 =
2

a (now) a0

at current epoch

Which at current epoch gives

!


a

2 0

H 02 k 2 & 8)G( 0 = a0 $ $ 3c 2 ' a 2 c2 0 %

# ! ! "

previous equation with a0^2 mult through

replace in Omega_0 get rid of a0^2 in denominator

Radius of Curvature of the Universe