John D John D BarrowBarrow
The Maths Of Whole The Maths Of Whole UniversesUniverses
A New Piece of GravityA New Piece of Gravity
- the ‘cosmological constant’ – does it exist?- the ‘cosmological constant’ – does it exist?
Gab + gab = Tab where = 8GN/c4 Geometry = mass-energy
Einstein’s Static UniverseEinstein’s Static Universe
(1917)(1917)
De Sitter’s Accelerating UniverseDe Sitter’s Accelerating Universe
Always expandingAlways expandingexponential curve R = exp[texponential curve R = exp[t/3]/3]
No matter – only No matter – only It has no beginning and no endIt has no beginning and no end
(1917)(1917)Willem De SitterWillem De Sitter
1872-19341872-1934
Friedmann’s universesFriedmann’s universes
1922,19241888-1925
Einstein’s Static UniverseEinstein’s Static Universeis unstableis unstable
Georges LemaîtreGeorges Lemaître(1894-1966)(1894-1966)
Einstein: ‘the biggest blunder of my life’Einstein: ‘the biggest blunder of my life’
Lemaître's UniverseLemaître's Universe
The best description of The best description of the visible universethe visible universe todaytoday
1927
> 0> 0
Tolman’s Oscillating Tolman’s Oscillating UniverseUniverse
Richard Tolman and Einstein at Cal Tech
1932
Tolman includes theTolman includes the22ndnd Law of thermodynamics: Law of thermodynamics:
Oscillations growOscillations grow(1932)(1932)
JDB + Dabrowski include theJDB + Dabrowski include thecosmological constant:cosmological constant:Oscillations always end Oscillations always end
(1995)(1995)
Kasner’s Anisotropic UniverseKasner’s Anisotropic Universe
Edward KasnerEdward Kasner(1878-1955)(1878-1955)
19211921
Volume abc time
(a,b,c) = (t(a,b,c) = (tpp, t, tqq, t, trr)) p + q + r = 1 p + q + r = 1 andand p p22 + q + q22 + r + r22 = 1 = 1-1/3 -1/3 p p 0 0 q q 2/3 2/3 r r 1 1
How Kasner Gave Us GoogleHow Kasner Gave Us GoogleKasner’s 9-yr old nephew Milton Sirotta invented the names
Googol = 10Googol = 10100100
Googolplex = 10Googolplex = 10GoogolGoogol
In 1996 Larry Page and Sergey Brin called their new internet search engine ‘BackRub’.Sean Anderson suggested Googolplex.
Shortened it to Googol but Anderson mistyped Google.comwhen he searched to see if Googol.com was available.
It was, Brin liked it better, so Google.com was registered on 15 Sept 1997
The Googleplex HQThe Googleplex HQ
expansionexpansion shearshear rotationrotation
The Ten UniversesThe Ten UniversesOfOf
Bianchi and TaubBianchi and Taub
Shear DistortionRotation
Gravitational Waves
But
Everyone sees the same historyEveryone sees the same historySpatially homogeneousSpatially homogeneous
Ordinary Differential EquationsOrdinary Differential Equations
space
time
The End of a closed Universe will not be SimultaneousThe End of a closed Universe will not be Simultaneous
Misner’s Mixmaster UniverseMisner’s Mixmaster Universe
An infinite number of things happen in a finite time as t An infinite number of things happen in a finite time as t 0 0
1969
x 1/x - [1/x] : 0 < x < 1 continued fraction map
y = x sin(1/x)
Gauss’s Probability DistributionGauss’s Probability Distribution
xxn+1 n+1 = 1/x= 1/xnn – [1/x – [1/xnn]]
As n the probability of outcome x tends to p(x) = 1/[(1+x)ln2] : p(x) = 1/[(1+x)ln2] : 00
11 p(x)dx = 1 p(x)dx = 1Error is < (0.7)n after n iterations
p(x)
x
In aLetter to Laplace
30th Jan 1812‘a curious problem’
that had occupied him for 12 years
Distribution of the fractional
parts
xxn+1n+1 = 1/x = 1/xnn – [1/x – [1/xnn] = T(x] = T(xnn))
T(x)
x
n stepsn steps = = initialinitial exp[ht]: where h = exp[ht]: where h = 22/[6(ln2)/[6(ln2)22] ] 3.45 3.45
ldT/dxl = 1/x2 > 1
as 0 < x < 1
T(x) =1/x – kT(x) =1/x – k
(1-k)(1-k)-1-1<x<k<x<k-1-1
k integerk integer
Gödel's Rotating UniverseGödel's Rotating Universe
Allows time travelAllows time travelto occur !to occur !
1949
The Big Bang UniversesThe Big Bang Universes
The Evidence of a Hot Early HistoryThe Evidence of a Hot Early History
A temporary lambdaA temporary lambda
The Inflationary UniverseThe Inflationary Universe
1981
Even Smaller BeginningsEven Smaller Beginnings
Chaotic InflationChaotic Inflation
Geography is more complicated than we thoughtGeography is more complicated than we thought
Eternal InflationEternal Inflation
More than 10More than 1010107777
by-universes from our patch aloneby-universes from our patch alone
What is the What is the Probability of someProbability of some‘‘universe’ arising in universe’ arising in
the Multiverse?the Multiverse?N(A)/N(B) = /Or – maybe you could take the limit? - But….Or – maybe you could take the limit? - But….
Fraction of even numbers in 1,2,3,4,5,6,7,8,… Fraction of even numbers in 1,2,3,4,5,6,7,8,… 1/2 1/2
Fraction of even numbers in 1,3,2,5,7,4,9,11,… Fraction of even numbers in 1,3,2,5,7,4,9,11,… 1/3 1/3
Answer depends on how you count the universesAnswer depends on how you count the universes
The Probability Measure ProblemThe Probability Measure Problem
Any Beginning in a Quantum Universe?Any Beginning in a Quantum Universe?
Lemaître's universe describes our visible universe to high precisionLemaître's universe describes our visible universe to high precision
The Universe is Accelerating AgainThe Universe is Accelerating Again
> 0
Dark Energy Dominates the UniverseDark Energy Dominates the Universe
The Return of Einstein’s Lambda ?The Return of Einstein’s Lambda ?