Lecture 7: The Maths of Whole Universes
Transcript of Lecture 7: The Maths of Whole Universes
![Page 1: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/1.jpg)
Concepts in Theoretical Physics
Lecture 7: The Maths of Whole Universes
John D Barrow
‘I am very interested in the Universe. I am specialising in the
Universe and all that surrounds it’
Peter Cook
![Page 2: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/2.jpg)
General Relativity
Newtonian
Gravity
GR NG for weak gravity + slow motion
V c, F c4/4G
10 potentials, gab
10 field eqns, Gab = Tab
V , F
since
F 1/r2 as r 0
1 potential, , 1 field eqn 2 = 4G
But not all solutions have counterparts
in both theories
![Page 3: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/3.jpg)
Einstein’s 10 partial differential eqnsSpacetime geometry = matter
Gab Rab - ½Rgab = Tab + gab
Every solution is an entire universe
Every solution of GR is an entire Universe
![Page 4: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/4.jpg)
A New Piece of Gravity
- the ‘cosmological constant’ – does it exist?
(r) = Ar2 + Br-1
is the most general gravitational potential
with Newton’s spherical property
Force -GM/r2 + c2r/3
Newton Einstein
![Page 5: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/5.jpg)
Time-dependent solutions of Einstein’s eqns
r a(t)r expand the space isotropically and homogeneously
ds2 = c2dt2 – dr2 – r2(d2 + sin2d2)
ds2 = c2dt2 – a2(t){ dr2 + r2(d2 + sin2d2)}
Most generally: the 3 spaces of constant curvature (k = 0,>0,<0)
ds2 = c2dt2 – a2(t){dr2/(1-kr2) + r2(d2 + sin2d2)}
K > 0 K = 0 K < 0
![Page 6: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/6.jpg)
Einstein’s Equations for a uniform, isotropic
universe: given p(), solve for a(t)
3a´2/a2 = 8G - 3kc2/a2 + c2,
3a´´/a = -4G ( + 3p/c2) + c2
´ + 3a´/a ( + p/c2) = 0
r(t)
r(t) = a(t)r0 v = dr/dt = (a´/a)r
Energy/unit mass = v2/2 – GM/r = constant
dE = -pdV
dE/dt = d(Mc2)/dt – pdV/dt
Volume, V= 4r3/3
m
M
´ = d/dt
(k = 0, +1, or -1)
![Page 7: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/7.jpg)
Einstein’s Static Universe (1917)
Non-Euclidean finite spherical geometry of space required
3a’2/a2 = 8G - 3kc2/a2 + c2 , 3a’’/a = -4G + c2
so a’ = a’’ = 0 requires 4G/c2 = = k/a2 > 0
a(t)
UNSTABLE !
![Page 8: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/8.jpg)
De Sitter’s Accelerating Universe (1917)
Always expanding
exponential curve a = exp[ct/3]
No matter – only
It has no beginning and no end
Willem De Sitter (1872-1934)
a(t)
![Page 9: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/9.jpg)
Alexander Friedmann’s universes
(1922,1924)
Alexander Friedmann,1888-1925
a(t)
=0
![Page 10: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/10.jpg)
Georges Lemaître's Universe
The best description of
the visible universe today
1927
> 0
a(t)
Georges Lemaître, 1894-1966
![Page 11: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/11.jpg)
Einstein’s Static Universe
is unstable
Albert Einstein and Arthur S Eddington
at the Cambridge University
Observatories, on Madingley Rd
Eddington-Lemaître Universe (1930)
a
![Page 12: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/12.jpg)
Tolman’s Oscillating Universe
Richard Tolman and Einstein at Cal Tech
1932
![Page 13: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/13.jpg)
Tolman includes the
2nd Law of thermodynamics:
Oscillations grow
(1932)
JDB + Dabrowski include the
cosmological constant:
Oscillations always end
(1995)
Adding the ‘Second Law’ and > 0
![Page 14: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/14.jpg)
Edward Kasner’s Anisotropic Universe
Edward Kasner
(1878-1955)
1921
Volume abc t
(a,b,c) = (tp, tq, tr)
p + q + r = 1 and p2 + q2 + r2 = 1
-1/3 p 0 q 2/3 r 1ds2 = c2 dt2 – t2p dx2 – t2q dy2 – t2r dz2, and Tab = 0
Speed of light
![Page 15: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/15.jpg)
The Ten UniversesOf
Bianchi and TaubShear Distortion
RotationGravitational Waves
Anisotropic expansion space
time
The Group Theory of Universes
Everyone sees the same history
Spatially homogeneous
Ordinary Differential Equations
3-parameter invariance group
A. Taub,1951
![Page 16: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/16.jpg)
Misner’s Chaotic Mixmaster Universe (1969)
An infinite number of things happen in a finite time
Xn+1 = 1/Xn – [1/Xn ] for 0 < Xn < 1 continued fraction map see lecture 1
C. Misner
![Page 17: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/17.jpg)
Kurt Gödel's Rotating Universe (1949)
Allows time travelto occur
Doesn’t expand
K. Gödel (1906-78)
![Page 18: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/18.jpg)
Big Bang Universes
![Page 19: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/19.jpg)
![Page 20: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/20.jpg)
A temporary causes acceleration at t 10-35 sec
The Inflationary Universe
A. Guth, 1981
![Page 21: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/21.jpg)
Even Smaller Beginnings
Solving the ‘horizon problem’ and creating
density fluctuations that can seed galaxies
![Page 22: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/22.jpg)
Observational tests: background radiation
![Page 23: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/23.jpg)
Chaotic Inflation
Expansion only approaches isotropy and homogeneity locally
![Page 24: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/24.jpg)
Eternal Inflation
More than 101077 by-universes from our patch alone
![Page 25: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/25.jpg)
Lemaître's universe describes our visible universe to high precision
The Universe is Accelerating Again Now
= 1.19 ×10−52 m−2 > 0
WHY?
What is its
origin?
A cosmic quantum vacuum energy?
It is a fluid with p = -c2 eqn of state
![Page 26: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/26.jpg)
Any Beginning in a Quantum Universe?
Lorentzian and Euclidean
Gott; Hartle & Hawking
![Page 27: Lecture 7: The Maths of Whole Universes](https://reader031.fdocuments.net/reader031/viewer/2022020622/61eed26dc9098439412e7f2c/html5/thumbnails/27.jpg)
Further reading
J.D. Barrow, The Book of Universes, Bodley Head, (2011)
S. Weinberg, The First Three Minutes, Basic Books (1993)
A.H. Guth, The Inflationary Universe, 2nd edn.( Vintage 1998)
S.W. Hawking and R. Penrose, The Nature of Space and Time, Cambridge UP, (1996)
B. Ryden, Introduction to Cosmology, Cambridge UP, (2106)
E.V. Linder, First Principles in Cosmology, 2nd edn, Addison Wesley, (1997)