PHY306 1

A brief history of A brief history of cosmologycosmology

Basic concepts spatial extent

finite (with edges) finite (unbounded) infinite

our location Earth at centre Sun at centre solar system near

centre solar system far from

centre no centre

past and future both finite

(creation, future destruction)

both infinite(no beginning, no end)

finite past, infinite future

dynamics static expanding cyclic

PHY306 2

Early ideas: astronomyEarly ideas: astronomy

Clearly understood concepts in Greek and Hellenistic astronomy shape and size of the Earth (Eratosthenes, BC 276-197) size and distance of the Moon (Aristarchos, BC 310-230) Sun is much larger than Earth (Aristarchos)

exact value was wrong by a large factor: method sound in principle, impossible in practice!

Ideas raised but not generally accepted Earth rotates on its axis (Heraclides, BC 387-312) Sun-centred solar system (Aristarchos)

PHY306 3

Early ideas: cosmologyEarly ideas: cosmology

Aristotle/Ptolemy Earth-centred, finite,

eternal, static

Aristarchos/Copernicus Sun-centred, finite,

eternal, static

At this time, little observational evidence for Sun-centred system!

PHY306 4

RenaissanceRenaissance

Birth of modern science scientific method

Galileo

better observations Tycho, Galileo

development of mathematical analysis

Kepler, Galileo, Newton

Newtonian cosmology

PHY306 5

Newtonian CosmologyNewtonian Cosmology

Newton’s Philosophiae Naturalis Principia Mathematica, 1687 Newtonian gravity, F = GMm/r2, and second law, F = ma Approximate size of solar system (Cassini, 1672)

from parallax of Mars

Finite speed of light (Ole Rømer, 1676)

from timing of Jupiter’smoons

No distances to stars No galaxies

PHY306 6

Newtonian CosmologyNewtonian Cosmology

Newton assumed a static universe Problem: unstable unless completely homogeneous

Consider mass m on edge of sphereof mass M and radius r

mass outside sphere does notcontribute (if sphericallysymmetric)

mass inside behaves like central point mass

if there exists an overdense region,everything will fall into it

2r

GMm

PHY306 7

Olbers’ ParadoxOlbers’ Paradox Named for Wilhelm Olbers, but known to Kepler

and Halley Consider spherical shell of radius r and thickness dr Number of stars in this shell is 4πr2n dr, where n is

number density of stars Light from each star is L/4πr2, therefore light from shell

is nL dr, independent of r therefore, in infinite universe, night sky should be

infinitely bright (or at least as bright as typical stellar surface – stars themselves block light from behind them)

Why is the sky dark at night?

PHY306 8

Resolution(s)Resolution(s) Light is absorbed by

intervening dust suggested by Olbers

doesn’t work: dust will heat up over time until it reaches the same temperature as the stars that illuminate it

(I’m not sure 17th century astronomers would have realised this)

Universe has finite size suggested by Kepler

this works (integral is truncated at finite r)

but now Newtonian universe will definitely collapse

Universe has finite age equivalent to finite size if

speed of light finite light from stars more

than ct distant has not had time to reach us

(currently accepted explanation)

Universe is expanding effective temperature of

distant starlight is redshifted down

this effect not known until 19th century

(does work, but does not dominate (for stars) in current models)

Olbers + Newton could have led to Olbers + Newton could have led to prediction of expanding/contracting prediction of expanding/contracting universeuniverse

PHY306 9

Further developmentsFurther developments

James Bradley, 1728: aberration proves that the Earth orbits the Sun

also allowed Bradley to calculate the speed of light to an accuracy of better than 1%

Friedrich Bessel, 1838: parallax distances of nearby stars

a discovery whose time had come: 3 good measurements in the same year by 3 independent people, after 2000 years of searching!

Michelson and Morley, 1887: no aether drift the speed of light does not depend on the Earth’s motion

PHY306 10

State of Play ~1900State of Play ~1900 We know

speed of light distance to nearby stars the Earth is at least

several million years old Our toolkit includes

Newtonian mechanics Newtonian gravity Maxwell’s

electromagnetism

We don’t know galaxies exist the universe is

expanding the Earth is several

billion years old We are worried about

conflict between geology and physics regarding age of Earth

about to be resolved lack of aether drift

PHY306 11

RelativityRelativity

Principle of relativity not a new idea!

Basic concepts of special relativity …an idea whose time had come…

Basic concepts of general relativity a genuinely new idea

Implications for cosmology

PHY306 12

RelativityRelativity

“If the Earth moves,why don’t we get leftbehind?”

Relativity of motion(Galileo) velocities are measured relative to given frame moving observer only sees velocity difference no absolute state of rest (cf. Newton’s first law) uniformly moving observer equivalent to static

PHY306 13

RelativityRelativity Principle of relativity

physical laws hold for all observers in inertial frames

inertial frame = one in rest or uniform motion

consider observer B moving at vx relative to A

xB = xA – vxt yB = yA; zB = zA; tB = tA

VB = dxB/dtB = VA – vx aB = dVB/dtB = aA

Using this Newton’s laws of motion

OK, same acceleration Newton’s law of gravity

OK, same acceleration Maxwell’s equations of

electromagnetism c = 1/√μ0ε0 – not frame

dependent but c = speed of light –

frame dependent problem!

PHY306 14

Michelson-Morley Michelson-Morley experimentexperiment

interferometer measures phase shift between two arms if motion of Earth

affects value of c, expect time-dependent shift

no significant shift found

PHY306 15

Basics of special Basics of special relativityrelativity

Assume speed of light constant in all inertial frames “Einstein clock” in which light

reflects from parallel mirrors time between clicks tA = 2d/c time between clicks tB = 2dB/c

but dB = √(d2 + ¼v2tB2)

so tA2 = tB

2(1 – β2) where β = v/c moving clock seen to tick more

slowly, by factor γ = (1 – β2)−1/2

note: if we sit on clock B, we see clock A tick more slowly

d

vt

stationary clock A

moving clock B

dB

PHY306 16

Basics of special Basics of special relativityrelativity

Lorentz transformation xB = γ(xA – βctA); yB = yA; zB = zA; ctB = γ(ctA – βxA)

mixes up space and time coordinates spacetime time dilation: moving clocks tick more slowly Lorentz contraction: moving object appears shorter all inertial observers see same speed of light c

spacetime interval ds2 = c2dt2 – dx2 – dy2 – dz2 same for all inertial observers

same for energy and momentum: EB = γ(EA – βcpxA); cpxB = γ(cpxA – βEA); cpyB = cpyA; cpzB = cpzA;

interval here is invariant mass m2c4 = E2 – c2p2

PHY306 17

The light coneThe light cone

For any observer, spacetime is divided into: the observer’s past: ds2 > 0, t < 0

these events can influence observer

the observer’s future: ds2 > 0, t > 0 observer can influence

these events

the light cone: ds2 = 0 path of light to/from

observer

“elsewhere”: ds2 < 0 no causal contact

PHY306 18

Basics of general Basics of general relativityrelativity

astronaut in freefall astronaut in inertial frame

frame falling freely in a gravitational field “looks like” inertial frame

PHY306 19

Basics of general Basics of general relativityrelativity

astronaut under gravity astronaut in accelerating frame

gravity looks like acceleration (gravity appears to be a “kinematic force”)

PHY306 20

Basics of general Basics of general relativityrelativity

(Weak) Principle of Equivalence gravitational acceleration same for all bodies

as with kinematic forces such as centrifugal force

gravitational mass inertial mass experimentally verified to high accuracy

gravitational field locally indistinguishable from acceleration

light bends in gravitational field but light takes shortest possible path

between two points (Fermat) spacetime must be curved by gravity

PHY306 21

Light bent by gravityLight bent by gravity

First test of general relativity, 1919 Sir Arthur Eddington photographs stars near Sun

during total eclipse, Sobral, Brazil results appear to support Einstein (but large error bars!)

photos from National Maritime Museum, Greenwich

PHY306 22

Light bent by gravityLight bent by gravity

member of lensing cluster

lensed galaxy

PHY306 23

ConclusionsConclusions

If we assume physical laws same for all inertial observers

i.e. speed of light same for all inertial observers

gravity behaves like a kinematic (or fictitious) force i.e. gravitational mass = inertial mass

then we conclude absolute space and time replaced by observer-

dependent spacetime light trajectories are bent in gravitational field gravitational field creates a curved spacetime

PHY306 24

Curved spacetime and Curved spacetime and implications for implications for

cosmologycosmologyGeneral Relativity implies spacetime is

curved in the presence of matter since universe contains matter, might expect

overall curvature (as well as local “gravity wells”) how does this affect measurements of large-scale

distances? what are the implications for cosmology?

PHY306 25

Curved spacetimeCurved spacetime

Two-dimensional curved space: surface of sphere distance between (r,θ) and (r+dr,θ+dθ) given by ds2 = dr2 + R2sin2(r/R)dθ2

r = distance from poleθ = angle from meridianR = radius of sphere

positive curvature

“Saddle” (negative curvature) ds2 = dr2 + R2sinh2(r/R)dθ2

(2D surface of constant negative curvature can’t really be constructed in 3D space)

Nick Strobel’s Astronomy Notes

PHY306 26

3D curved spacetime3D curved spacetime

Robertson-Walker metric ds2 = −c2dt2 + a2(t)[dx2/(1 – kx2/R2) + x2(dθ2+sin2θ dφ2)]

note sign change from our previous definition of ds2! a(t) is an overall scale factor allowing for expansion

or contraction (a(t0) ≡ 1) x is called a comoving coordinate (unchanged by

overall expansion or contraction) k defines sign of curvature (k = ±1 or 0),

R is radius of curvature path of photon has ds2 = 0, as before

PHY306 27

Implications for Implications for cosmologycosmology

comoving proper distance (dt = 0) between origin and object at coordinate x:

for k = +1 this gives r = R sin−1(x/R), i.e. r ≤ 2πR finite but unbounded universe, cf. sphere

for k = −1 we get r = R sinh−1(x/R), and for k = 0, r = x infinite universe, cf. saddle

for x << R all values of k give r ≈ x any spacetime looks flat on small enough scales

this is independent of a it’s a comoving distance

x

Rkx

xr

0221

d

PHY306 28

Implications for Implications for cosmologycosmology

cosmological redshift: change variables in RW metric from x to r: ds2 = −c2dt2 + a2(t)[dr2 + x(r)2dΩ2] for light ds = 0, so c2dt2 = a2(t)dr2, i.e. c dt/a(t) = dr

(assuming beam directed radially) suppose wave crest emitted at time te and observed at to

rrta

tc

rt

t

0

d)(

do

e

rrta

tc

rct

ct

0

d)(

doo

ee

first wave crest next wave crest

PHY306 29

Implications for Implications for cosmologycosmology

Then

but if λ << c/H0, a(t) is almost constant over this integral, so we can write

i.e.

ct

t

ct

tta

tc

ta

tc

oo

o

ee

e)(

d

)(

d

ct

t

ct

t

tta

ct

ta

c oo

o

ee

e

d)(

d)( oe

)()( o

o

e

e

tata

PHY306 30

Implications for Implications for cosmologycosmology

So expanding universe produces redshift z, where

Note: z can have any value from 0 to ∞ z is a measure of te

often interpret z using relativistic Doppler shift formula

)(

1

)(

)(1

ee

o

tata

taz

vv

c

cz1

but note that this is misleading: the object is not changing its local coordinates

PHY306 31

Implications for Implications for cosmologycosmology

Conclusions in general relativity universe can be infinite

(if k = −1 or 0) or finite but unbounded (if k = +1) universe can expand or contract (if overall scale

factor a(t) is not constant) if universe expands or contracts, radiation

emitted by a comoving source will appear redshifted or blueshifted respectively