PHY306 1 A brief history of cosmology A brief history of cosmology 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
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) - PowerPoint PPT Presentation
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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
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
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