c quantum theory 3%e general relativity gru adde …i. introduction ① cosmology is o object: 1-...
TRANSCRIPT
I.
Introduction
① Cosmology iso
1-object :
,
Universe
- method"
Physical cosmology"
.
mm
⇒
"
physics"
" C special E
-79,
quantum
Adderelativity theory
general ,
- OE -
- ome 3%erelativity
-
-→ parade creatimfannihilation
Gru-
- 849k¥ • ⇐¥Y!I
'
%GM I s Htt > = EH >
gut quantum<
! quaffing ,g{LQG ? mechanics 0×010342
String theory?oEot3h42
xquantumheld theory in curved space-time
"
need to describe the beginning of the Universe"
Inflation
Strong nuclear fro
) short -
range* four fundamental Interactions
{Weak nuclear force
electro -
magnetic force
) long-rangegravitational force
( Standard model of particle physics )
fermions Bosons"
Leptons fuel (1) WEH
Jian"If"Early universe
.
of ,( It
, -
. .
i 8)
Quarks (4) (5) Eb) y
The most important
interaction in
Cosmology
?Gravity
! !
Why ? mass always accumulates !
② Units : Natural unit Cparticle physics ) & Astronomical
unit ( Oom version )
C re 3×105 km/s = 3×10"
ants dyn = g.am/sz
K I 10-34
Is = 10-27ergs
{erg
= dyn.cm\KB
II.4×10-23
Jfk = 1.4×10't erglk
② - I.
natural unit ⇒ C .
- K - KB -
- I : Note : let1,6×0
- ' 'J
. 6×1512erg
.
= If Not'
erg
⇒ erg-
- 6.3×10" eV
Is = 3 X to"
cm = 1027ergt
= 1.6×10's
eV- '
= 1.6×1024 GeV"
{ ik = , !!, ?:t " " er'
=
s-xw.am/g=erg-er8-= 6.3×10 "eV I 6.3×10
"
GeVam -152
=
to"
erg-ywsye.ge= to
"
erg
Useful Conversion : Kc = I = 3×15"
erg.
an= 18.9×10-6 eV - an
I 200 MeV .fm-
femtometer C- 15km )
2.7K = 2.4×10-4 eV I 12 an"
x ( 0.08am )- I
= ( o . 8mm )"
Peak of Planck fu. n 343T @ K = 0.23mm → D= 1.4mm
run me
2. 4×10-4 eV ~
3-8×10"
5'
r
380GHz3193T @ w =1140GHz
⇒ f =180GHz
mm
② - 2. Planck unit
Gn = 6.7×10-8 omfg
15=6/7×10-8x ( txio"
GeV- '
13¥10"
GeV )/( 1.6×1024 GeV-
Y-
~ 13¥ × 10-8+39-23- 48
gey- z
~ b- X 10-39 GeV -2=1 i GeV = 7×10-20
⇒ Is = 1.6×1024 Get'
= 2×1043 ⇒ tpenfx 15445
Im = b- Xlo"
GeV"
= 7×10"
⇒ Ipe n I X to- 33
m
Ig = 6.3×1023 GeV = 4×104 ⇒ wipe n 2×10-5of\
, , , g. , , ,,µµ , , , ,,
⇒
q ,, my ,
In natural unit,
Mpl u to " GeV
side note : Relativist 's unit sets 8IGn=I ,instead of Gn .
" Reduced Planck mass
"
Mpe = ~ Mg ~ Zxiocevm
② -3 . Astronomical Unit
Mo . = 2x 1033g ,Ro. = 8×10
"
an → So.
=
= # x to'
{Lo .
= 4×1033 erg Is n I glans
6×1027- Me,
= 6×1027g ,
Rot = 6×108 an → So,
=
¢¥p= ¥ X 103
~ 6 glans
I All = too light second = too x 3×10 "
an = tfxlo "
an
Iparsec
= I"
parallax = tf = Itu-r 216000 AU = 3×10' '
an
"
1/80/3600It
( o 8000000
216000 All = 216000 x too light second = alyr = 3.26 Lynmm
to the center of the Galaxy = 8- tkpc = 2. b- x 1022cm
to M 31 ~ 0.78 Mpc~ 2. 3X 1024am{ tothe
edgeof the Universe ~ 14 Gpc - 4×1028 an
③ Review of General theory of relativity
key quantity of dynamics : Tat ) from ICE ) & J CE )
* Newtonian theory ofgravity
eat . of motion : I .
- ima =MDI
dt2
{gravitational held
equation for the
grau. potential :
otcxie)
( Poisson 'sequation ) 024C I , -4 = 4Th Gns CI , -4 & I = - m 0/0
Equivalently ,
motion takes to extrcmize the action :
tfS [ ICED =/ de Ll the ,
nice , ie ]
ti
S C Tues t faith ] - Scotch ]
= f dtflckthtfxth ,nicest foie , se ) - LC Its
,niceties )
=
-2,4¥,
+
ftp..fi/toC8x4=-.fE.ise..+¥C¥÷s÷ ) . ¥E¥
. ) toes ,
= I at El :# . ¥ sci -
- o
⇒ 3¥.
-
- ¥E⇒)⇒ mic -
-
- mat
L = T - U = tzmxz - my
* Einstein 's theory ofgravity
igravity
= Curvature of spacetime .
① Spacetime : ( I -13 ) dimension
go =L
KM = ( t, X. y ,
Z ) I nd = K
,K2
=L ,x
' = Z
curved spacetime is described by the metric tensor :
gyu
⇒ defines the inner product of any giventwo Vectors
.
e.
g . spacetime intervalof
Einstein Convention
do = ( dxn , dxu ) = % gyu dxndxu =
gyu dxmdxu
nowIll
Minkowski Htt.
no gravity )spacetime
:
des;:[If:!!!;' oh -
-
f'
!;)Inner product of energy
-
momentum 4 - vector i P' = C E, p )
p'
= ( pm, Po ) =
yµ pmpu
= -
E- 2+11512
= -
m'
("
rest"
mass )
C at the local inertial observer )
⇒ gyu pnpu = - m- C general observer )
. NoteJust like the "
rest"
mass , space-timescalar muse be the same for all observers
.
mum
o
"
spacetime tells mass how to move" C
geodesic
equation)
=
i geodesic minimizes the spacetimeinterval along the path
⇒ Action S = - mfdt ( de = -
dsa = - gyudxmdxo )
! SR limit
=- mf def
'
- mfdtC I - Kia ) = fdtfmtkm.li)
S -
-
- mfde = - mf.dk/-8mdIITYI-
⇐-
Leg . ) 3¥ =¥(f¥ ) with he - mf-gmdffdf.IT =- md¥
¥ L'lxncxl ) = - Guiche ⇒ 2L }÷,
= -
magmarinse
12L3¥ = -
mega. Shiv-
magnify= - m
.
( gain
-1quasi )
⇒ f¥= -Fi( game + grain )
n-
-It ) ( gait + senior ]
=ILg. to + Had'¥ ]2
: ¥t⇒=¥¥d¥,
= - E . #I s . "e¥I¥¥+zaed¥.de#+g..dIIetsmdII.]
I 2
= ¥ .= -
EsmaKI¥l=- Yasmin :¥I¥
-
=t⇒'
⇒
( s . "e¥Id¥+%e¥¥¥¥ + g.it#.-ismd:I3--smald¥1 I
zg.io#.=gm.aIEoiI-s...e.dfIdEI-q.pdoEIE
Here, only
"
a"
is the real index,
and all others are dummy .
28*917.
= ( % . a
- gaap -
gear ) III 1¥
④ tzgdm ( where ganga = SI )~
↳ inverse metric
g.97¥ = = - Yasir ( g + spa . .-
spa . HII IET
= - t ;i¥i¥÷
⇒ DIET t TIP III III -
- o i
geodesic equation
-
↳ Christoffel symbol.
" "
. Christoffel Symbol : Tae
't= Tag leg
of; covariant derivative of the
-
-
#
coordinatebasis
.
velocity vector i V
'= WEE = doffs of
⇒ acceleration : a'
= doY = dat ee + uao¥eE
UP Tag eg"
-
= Has EE -- u
'
Effie:
= ( II t Tofu ez
: geodesic equation ⇐ a o ( Parallel transportation)
mm
o Examples on surface
C X )FfO Newtonian vs
.
Relativistic Motion due to Gravity
"÷÷
Central - force,
two - body problem"
Spacetime geodesic"
.
O"
mass tellsspacetime
how to Curve"
.
gyu ⇒ e. am .
What determines gyu?
distribution of mass
Einstein Equation"
Einstein tensor
" "
energy- momentum tensor
"
Gyu =
STIGTru
i ) Einstein tensor
Gm = Ru - 42 gyu R
-
H IRico
,Scalar R -
-
gyu RN =gmRµ=RMµRicci tensor
Ryu = Rdr an
run
I
Riemann tensor ( Tra - Qq ) Ud = Rdppu UP
Ramu = IIe - III. t to .
- TETE
Ii ) Energy - momentum tensor
Tmo =µ
- component of momentum flow through he = Gust
. hypersurface .
~
Jpg III fcp , 8D C pkm )
Too n
energy density
Tot ~
energyflow i To - momentum density{
Ty - Stress tensor ( ~ minus )
( perfect fluid >
Tm= ( Stp ) Unh t Pgm ( un
-
-
gu w )
Note : Tru = Tvr C symmetric )
Continuity equation: Tn Tmo = o
⇐
8¥ + y ,Levi ) =o Tn SE
Ito ,csuiui , = o
I ( Hatspacetime )
⇐Or -11=0
( RTI = 2T 's t FukTT - EETI )
⇒ anti = dit + Tfa Tf -TITI= o
a-
Again ,
"
connection"
!
* Action for the Einsteinequation
SCgyu
] = SEH t Sm
= ¥qJd4x Fg R + fd4x Fg Lm
I Ts matter Lagrangian density
Ricci scalar
Fgu=¥a #
Egg::*= fttqf -
'
kissing Rap tf Ryu t Fg go SgRg÷ )me
-
→ surface term.
= I÷g[ Rm - Ya Rgm ]quo
→ o.
to = - EFF.
⇒ Rm - Yim R = 8 TG Tru
CH uses Jacobi formula : I eat = e*A forany
nxn matrix A
⇒ IAI = etrh A
⇒ SIGI = lgltrlg 's g )
⇒ ffg = Yzfggmsgru
= - Ifs gaggert Kit
"-
-4
Example of matter action ( scalar field : 0 thx ) )
Sm C ¢ ] = folk Ff L ( 0, 24 ; x )
= folk Fg ( -
'
kg"
On 4 a ¢ - V coli )
⇒ Ssn = forex Fg Sgm ( %gr f
Lay04 - Veon ) -
'
k 9404 ]-
=- Yasu
= forex Fg 8g " [ Ya 8in ('
k Tay pay tuco , ) - Ya 74 Or of ]mm
=- Ya Tm