INVESTIGATI ON OF VANI SHING OF A HORIZON FOR BIANCHI TYPE IX 
(THE MIXMJ\.S TER) UNIVERSE 
by 
D. M. Chitre 
D;i.,ss e rta tion subm:ltt.e d to the. Faculty of the. Gradu a t e. School 
of the. Unive.rs:lty of Maryland :ln part:la l fulfUlme.nt 
of the. r e.q_uire.me.nts for the. degr ee. of 
Doctor of Philosophy-
1972 
c. i 
Cop J E f SL 
This dissertati0n was held by the 
Gradu~te School in ~n attempt to 
coptact the author until August, 
1976. 
This 1,ra,c; not microfilmed. 
APPROVAL SHEET 
Title of Thesis: INVESTIGATION OF VANISHING OF A HORIZON FOR BIANCHI 
TYPE IX (THE MIXMASTER) UNIVERSE 
'( 
<?? 
Name of Candidate: D~ M. Chitre 
Do ctor of Philosophy 1972 
Thesis and Abstract Approved: C/2u:,l~ 
Signature of Supervising Professor 
Charles W. Misner 
Professor of Physics 
Department of Physics and Astronomy 
Date Approved: 
ABSTRACT 
TITLE OF THESIS: INVESTIGATION OF VANISHING OF A HORIZON FOR BIANCHI 
TYPE IX (THE MIXMASTER) UNIVERSE 
D. M. CHITRE 
THESIS DIRECTED BY: PROFESSOR CHARLES W. MISNER 
In this dissertation, the generic, non-rotating, homogeneous closed model 
universe ( th e "Mixmaster Universe", Bianchi Type IX) is studied to gain some 
insight into how the broad-scale homogeneity of the universe may have been 
p roduced at very early times. We begin our discus, sion by sketching the develop-
ment of relativistic cosmology until the last decade. In the second chapter 
we discuss particle horizons in the Robertson-Walker models . These standard 
models of the universe possess particle horizons. Thus, only a finite part of sucha 
universe could have been causally connected; while the isotropy of 2.7?K 
micro~ave radiation implies the universe to be homogeneous on a much larger 
scale than the size of the horizon . The third chapter discusses in detail the 
evolution of the Mixmaster Universe near the singularity using the Hamiltonian 
t echniques developed by Misner for these models . At a fixed time (or volume) 
e poch r2 , a Mixmaster Universe is specified by initial conditions' B+, B- ( shape 
0
a nisotropy ) and P+ , p_ (expansion rate anisotropy ). In the fourth chapter we 
d erive the equations for rays of high-frequency sound waves and light wav s. 
Wh en these equations are applied in the Mixmaster Universe, .we find that for 
certain subsets of initial conditions, some of these sound rays and light rays 
wou l d circumnavigate the corresponding universes in certain directions. Our 
results for light rays parallel t hose of Doroshkevich and Novikov, however we 
u se ent i rely different methods (Hamiltonian methods) for treating the Einstein 
equations . 
In the last chapter the evolution of the Mixmaster Universe is shown 
equivalent to a geodes ic flow within a bounded region of the Lobatchews ky plane. 
The boundary s ha pe makes this flow Ergodic. The ergodicity is proved by 
invoking a certain group of conformal transformations, G, which makes this 
flow of broken geodesics on the Lobatchewsky plane, D, into a continuous one 
on D/G, The Einstein equations in this problem lead to a natural measure on 
:in~tial conditions related to B+, P+? The measure of the circumnavigation sets 
depends upon the epoch and it goes to zero as the volume of the universe shrinks 
to zero. Finally, we compute the probability for circumnavigation along any 
one axis of the universe, It turns out to be roughly 1% for an empty universe 
and it decreas es to 0.02% for realistic models containing radiation and matter 
in them . 
ii 
ACKNOWLEDGEMENTS 
I wish to express my deep gratitude to m
y advisor, Professor 
 the course 
Charles Misner, for the constant guidanc
e he gave during
of the present work and the great patien
ce he showed during the 
Writing up of the work. I would like to
 thank Dr . V. Moncrief, 
ful discussions. I 
Professor Y. Nutku and Dr. M. Ryan for m
any use
proofreading and making 
Would also like to thank Dr. V. Moncrief
 for 
the necessary corrections on this work. 
nder, Mrs. J. Alexander, 
Finally; I would like to thank Mrs. B. A
lexa
and Miss J. Moffat for typing the thesis. 
 was supported by the University of Mary
land's 
This research
Center for Theoretical Physics, and by N
ASA Grant NGR-21-002-010. 
iii 
TABLE OF CONTENTS 
CHAPTER PAGE 
I . FORMULATION OF COSMOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 
II . PARTICLE HORIZONS . .......................... . ......... . .. . . 8 
III . THE NON-ROTATING BI/\.NCIU TYPE IX MODEL. ..... . ....... . ...... 17 
IV. PROPAGATION OF LIGHT AND HIGH-FREQUENCY SOUND WAVES . . ..... 27 
V. THE ERGODIC BEHAVIOR AND THE HORIZON PROBLEM ........... . .. . 32 
A. The Evo lu t ion of the Mixmas te r Universe ...... . ....... 32 
B. The Prob ability for t he vanishing oi a Horizon ...... . 40 
APPENDIX A. High Frequency Sound Waves t o El imin at e a Horizon 
i n the Mixmas t er Universe ........................ 54 
1. In troduction ....... . ...............................? 56 
2. The prop agation of Hi gh-Frequency Sound Waves ........ 62 
3. The Removal of Hor izons .......................... . ... 65 
4. u = 00 So lu tions of Einstein Equa tions ................ 70 
REFERENCES TO APPENDIX A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5 
APPENDIX B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6 
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 
iv 
LIST OF FIGURES 
Page 
Figure 
1. . ... 46 Description of particle horizon s ..... . ..............
.......
2.  47 The last scattering of microwave radiation .... . . .... .
.... . .....
3. ??..???? 48 The equipotentials of V(l3+, B_ ) for l arge " B" ? ???.????
4. movin g out in An idealization of equipo t ential walls which are s in 
B:t/3- plane while they are given l,y stationary circul
ar arc
....... 49 
the Lob atchewshy plane ........?. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?. 
? ? ? .?
5. e transformation which takes the equipotential wall A is th
represented by a circular arc into a diame t er ....?.
........... 50 
-1 
6. The effect of the transformation A RA is to map th
e reflected 
geodesic into a continued part of the incident traje
ctory ...... 51 
-1 
7. ated app li ca tion of the transformation S = A RA m
ake s a 
A repe
typica l solution of the Mixmaster Universe evolve a
long a 
eodesic ........... ?????????????????????????????? .. 5
2 
continuous g
8. ounded by arcs AB, BC, CP and PA represents th_e norma
l 
The area b
fundament a l region of the group, F ..................
...??..... 53 
CHAPTER I. 
Formulation of Cosmology 
The cosmological problem within the framework of general relativity 
consisted, for a long time, in finding~ simple model of the physical 
universe which is a solution to Einstein' s equations. The simplification 
was involved in idealizing the crudely observed symmetries of homogeneity 
and isotropy of the universe and extrapolating them to all times in the 
past. The investigation of world-models without any postulate of hom6 -
geneity was first done by Raychaudhuri (1955) who obtained an equation 
which showed how the expansion of th e universe was influenced by shear 
and rotation. Heckmann and SchUking (1959) classified severa l world-
? 
mode ls by s tudying the four-vector u of flow of incoherent matter. 
They a l so found homogeneous but anisotropic solutions for universes 
fill ed with pressurcless fluid . Analytic so lutions were also found for 
stress-energy tensors, corresponding to a pressureless fluid with mag-
netic field, or to a radiation fluid with p =I1 p by taking a specialized 
form for the anisotropy. Misner (1968) studied a general class of homo-
geneous, anisotropic solutions to the Einstein's equations in which the 
pressure anisotropies were included , and proved that a considerable num -
ber of those solutions would evo lve into the present day universe. B~ 
linskii and Khalatnikov (1969), Thorne (1967), Doroshekevich, Ze ldovich, 
and Novikov (1967) analysed cosmological models with pronounced aniso-
tropy at an early stage of cosmological expansion. This new way of 
studying cosmology where the mod e l s considered look ed very different 
1 
2 
in the distant past, exploited the richness of Einstein's equations and 
gave better insight into many problems -- the problem of particle hori-
zons being one of them. Before we turn to the study of horizons in this 
new light, let us sketch briefly the development of relativistic cosmo-
logy until the last decade and the status of hori zons in those cosmo-
logies. 
Building upon the observational work of earlier astronomers (Curtis, 
Shapley, Slipher) and greatly extending the investigation himself, Hubble 
(1924) in the early twenties proved the extragalactic nature of the Andro-
meda Nebula, discovered the expansion of the universe in the late twenties 
(192 9), and showed in 1936 that the large scale di_st:r;Lbution, of galaxi_es was 
homogeneous and isotropic to the telescope limit. These three crucial 
discoveries set the stage for cosmology for years to come. It was im-
mediately noticed that the models speculated by Friedman (1924) fitted 
very well with the above observations. Friedman was able to find solu-
(1. 1) 
With k =+land -l; the corresponding space metric s describing a three -
sphere and a three-hyperbo loid respectively. The Einstein equations, 
G?v - Ag?v = BTIG T for the above metrics reduce to 
7 ?v 
(1. 2) 
and 
k 8'ITG A
~ 1  
= -P (1. 3) 
+ 
2 2 
rir + R2 C " 
c R C 
respectively the energy 
Where A is cosmological constant and 
p, p are 
the 
3 
density and pressure as observed by t he comoving ob s ervers, 
Robertson (1929) extended this class of so lu tions by inc l uding the 
case where t he 3-space geome t ry is Euclidean. The me tr i c for tha t case is 
ds 2 (1.4) 
whic h has the same Eins t ein equat i ons as the t wo Friedman cases but wi th 
k=O. Robertson a nd la t er on Walke r (1936 ) s howed tha t t hese three cases 
( k= O, ?1) a r e t he only so lut ions satisfying the "Cosmo l og i cal Pr i ncip l e ." 
The co smol ogical princip l e which was f ir s t explicitly stated by Milne (1931), 
d emand s that the univers e be homogeneous and i s otr op i c . The homogenei t y of 
the unive r se means tha t through each ev ent in the univer se, ther e passes a 
spa c e like hyper s urfa c e who se events a r e phys i cally indis t i nguis habl e f r om each 
other. Whil e the i sotropy means tha t a comoving obser ver c annot dis tingu ish 
one of his space dir ec tions from the other s . Thus, if one r es tric t s t o 
Robert s on-Wa l ker mod el s fo r the univer se , theo r e tical cosmol ogy r edu c es t o 
the s tudy of equa t i ons (1.2) and (1.3), Through the equa tions of motion fo r 
the fluid, T?v?v = 0 we could obta in p and p in terms of R(t) and a n appro-
' 
pria t e equa tion of s t a t e would give a r e lations hip be t ween p and p. (Note , 
however tha t a solution to equa tion (1?2) is als o a solution t o equa tion 
(1.3) by virtue of one of the Bianchii identities provid ed it is not a s t a tic 
solution . Einste in in the pr e-Hubble days wa s trying to find a stat ic , c lo sed 
. 
mod e l f or the univ er se . With k=l, R=O, no r ealis tic flu i d can sa tis fy 
equation (1.3) unles s A#O. Thu s the need to introduce the co smol og i ca l 
cons t a nt A in the Eins t e in ' s equa tions per sis t ed till Hubble observed the gen-
eral rec e s s ion of the ga l ax i es . The variou s pa r ame t ers k, A, the ener gy 
d ensi ty, p , the s i ze of the univ er se now R a ppearing i n the theoretical 
0 
solution a r e dir ec tly or ind i r ectly connec t ed with the observed as tronomical 
Pa r ame t er s like the av er age dens ity now, the "Hubbl e constant, 
11 H = (R/ R) , and 
0 0 
.. 
4 
" 
the d cc e J e r c1 t j on pn r ;rn1e t er , q ="" (~) As the observations for these 
o R0 
a s tr or10rn i c,1 l p .1 r :1111 e Lc' r s get better, the ran ge of values for the theoretical 
Paramet ers gc t;; s m;.1? 11 e r . However , the observations are still not precise 
enou gh t o c} j s ti n~u ?i ~; 11 li e twe en variou s Robertson-Walker models (kO, ?1) or 
to Pr e d ic t t he v .:1.l. u e o f A. (S ee Sandage (1970) for the r evi ew of the measure-
me nt s o f t he ll ub bl e constant and the decelera tion parameter. ) 
Th e Ro l)(' r t: so n- hl :1 H :er mod e l s have a s ingula r ity at a finite time in the 
Pa s t wh e n til e e n e r gy dens ity a nd the curvature become infinite. To avoid the 
singula r j t y Boncl .i. nncl Cold (1948) proposed a tempora lly- invar iant or steady-
s t a t e mod e l un i v e r se . It is based on the so callecl perfect cosmologi cal principle , 
l.,e , tli e universe pr e s ents the same large scale view a t a ll times . The p os tulates 
of J 
? 10 mogene i ty a nd e qu iva l ence of all world-points complet e ly determine the 
metric ? I nd e ed, the 1n e: tric clearly must b e of the Rob ert son-Walker type and 
furt J7 ermor e , the e x pnn s ion factor in the Robertson-Walker metric must be such 
th
a t th e Hu bble c ons t o nt i s independent of the epoch, i.e . 
. 
1l = R/R = cons tan t 
i . e . R = e11t 
The 
cons tanc y of c urva ture of the space-like sections t constant requires 
b ,o and the metric b e c omes 
d s 2 (1. 5) 
'l'h e 
me tric i o f t l1e cl e Sitter type and i s a solution to the Einst e in equations, 
for 
Va nj_ ?h in ? e ne r gy-dens ity a nd A>O. To rectify the s itua tion, Hoyle (1949) modi-
f iec1 r . . 
~J.ns t cJ_n ' s e qu ,1 Lj on s to rea d 
(1 , 6) 
5 
where C?v is the matter creation term, Till a few years ago, observations were 
going on two fronts to decide between steady---state model and the "big-bang" 
Robertson-Walker models as a correct description of the universe. In the 
steady-state theory all matter must be continuously accelerated, whereas 
galaxies in Friedman models must decelerate. The deviation from a linear 
expansion law between distance and red-shift would provide the necessary in-
formation on the change of motion with time. A second approach has been t o 
look for evolutionary diff erences in galaxies in the great clusters. The 
steady-state mod el requires that galaxies must be form ed during all epochs. 
On the other hand, in Robertson-Walker models, most galaxies are formed at a 
unique epoch. Thus -integral properties such as color, and mass to light ratio 
for an adequate sample of galaxies would show dispersion in the steady state 
theory owing to the age difference of their stellar content. Although, these 
observations did not provide very clear cut answers, they favored evolving 
Robertson-Walker models . The discovery by Penzias and Wilson (1965) of the 
cosmic microwave background changed the situation considerably . There was no 
sa t isfactory way in which the steady-state theory could explain the 2. 7?K 
black-body spectrum of t he background radiation while the presence of relic 
thermal radiation is t o be expected if we trace the expansion of the universe 
back to a highly contracted, hot phase of the universe . Curiously, long before 
t he discovery of Penzias and Wilson and its explanation by Dicke etc, (1965 ) , Gamow 
and his colleagues (1949) had predicted a primeval fireball and had even predicted 
its present temperature somewhat larger than 2 . 7?K, The isotropy measurements 
of the black-body radiation gave an :unpetus to the study of anisotropic 
cosmologies which was already started by Heckmann and Schucking (1959) and Zel ' dovich 
(1965). The precise measurement (0.2%) ~ee Partridge (1969)] in its i so tropy put 
much more stringent limits on the anisotropy of cosmological models than those 
derived from the isotropy of the Hubble red-shifts . Zel ' dovich (1956) and 
6 
Thorne (1966) studi ed the anisotropic cosmologies in view of the possibility 
of a primordial magnetic field. Hawking and Tayler (196 6) a nd Thorne (1967) 
investigated the influence of anisotropy on the primordial he lium production. A 
number of people studied exact so lutions for anisotropic cosmologies. Misner 
(l968) studied a l arge class of homogeneous, but anisotropic universes and 
found that jus t abov e 10 10 ?K, neutrino viscosity was v ery efficient in reducing 
the anisotropy. Thi s led Misner (1967) to formulat e the principle of ''Chaotic 
Cosmology." The philosophy of chaotic cosmology was based on the existence of 
singularity proof of Hawking and Penrose (1970). Assuming that the singularity 
demanded by Einstein's equations is of the Friedman type, i.e. a singularity 
involving infinite densities a finite proper time in the past, the idea of 
chaot i? c cosmology is to start with arbitrary 1? n1? t1? a1  cond  1? t1? ons near the singu-
larity and try to prove that they would develop into the present day universe. 
Furthermore, one could try to formulate chaos in terms of maximum symmetry 
breaking near the singularity, or that the univers e goes through almost all 
irreg The first part of chaotic cou smology 1 ar stages near the singularity. 
Which consists of proving the cosmological principle has a great esthetic 
appeal while the second part could give rise to important consequences. 
I n this thesis, we will try to characterize the "chaotic" behavior of the 
Bianchi Type ( h ous
X   ag ne in se or ti rc o, p ico  m mo og den els) cosmologies near the 1  , 
Singularity. The Hamiltonian methods applied to these models by Misner (1969) 
describe the evolution of these universes (called the Mixmas_ter Universe ) 
i n terms of a particle motion in a potential well. The evolution is given com-
Pletely by specifying the initial conditions B+, B- (shape anisotropy) and 
p we will show in the limit of infinite +? p_ (expansion rate anisotropy). 
P
0 tenti'al that the evolution of the Mixmaster Universe is equivalent 
approximation 
to a geodesic flow within a bounded region of the Lobatchewsky plane. It 
7 
will b e s h.own that th e -bound ~r r s.h a J? e :ls such tha t I c\S the unive r se go es towards 
the s ingula rity , tf1 e collisions with the e ctuipot entia l wall s bound i n g th e r eg ion ma k e 
the g e odes i c f low e r godic . Thu s the cha o s of the Mixmaster Univ e r se will b e 
c 1i arac t e r i z e d by th e e r god ic n a tur e of the geod esic f l ow, o r mor e s p ec ifica lly, it 
me ans tha t i f we s t a rt with a we ll-d e fine d s t a t e of the unive r se ( g ive n b y certa in 
v a lue o f B?; P? at c erta in ~ ), as it evolves towa rd s the s ingular ity, it goes 
0 
throu g h a lmos t a ll poss ible a niso t r opic s tages. We will s tudy the c onsequ ence 
o f ergodic ity on the problem of hori zons . For certa in subse t s of init ia l c ondition s , 
s ome null- geod es i c s proceed to circumnaviga t e the corres ponding unive rs e . We 
will see tha t the Eins t e in equa tions l e ad to a n atura l measur e on the initia l 
c ondition s ( B?, P?) in this problem. Howev er, the measur e of the circumnaviga?-
tion s e ts d e p ends upon the epoch, so tha t the e r godic ity do es not gua r ant ee that 
a typica l so lutio n will evo lve throu gh_ s u ch a s et. Re ly i ng on the mix tur e property 
of the er godic ity, we will compute the proba bility for circumnavigation along 
any axis. It will turn out to be v e ry s ma ll but finit e , 
8 
CHAPTER II. 
Particle Horizons 
Hori zons are frontiers between observable and unobservab l e events. 
The first systematic s tudy of visual horizon s was done by Rindler (1956). 
He defined two type s of horizons. The first, called an event-horizon, 
for an observer A is a surface in space-time which divides all events 
into two non-empty classes: those that are observable by A and those 
that are forever unob servab le by A. The de Sitt' er universe can be 
seen to pos sess an event -horizon. The other type of hori zon is called 
a particl e horizon for a given comoving observer A at some time t . 
0 
It is a surface in the space-like hypersurface t = t which divides 
0 
all comoving observers into two non-empty classes: those that have 
a lready been observabl e by A at time t and those that have not. Pen -
o 
rose (1968) defines particle horizons differently. In hi s definition, 
the particle hori zon of an observer y (see Fig. 1) separates events I 
from which a particle with world line y can be observed, from events II 
from which the particle cannot be observed. Note that p and Q will form 
particle horizon in Rindler's definition. For spatially homogeneous 
universes where the three -space sections are symmetric, th e intersection 
of Penrose's particle horizon with the surface t = constant would give 
th e Rindler particle hori zon. We will restrict ourselves exclusively 
to the Rindler's definition of particle horizon . As an example, consi-
der Robertson - Walker model in the ea_rly times when th e universe was 
r a diation-d omi nat ed. Th e r adiation ener gy-dens ity i s giv en by 
P Rt+ = p R 1+ r ro o (2 . 1) 
wh ere p
r o is the ener gy- dens ity now and Ro  i s the present r adius of the 
unive r se . I gnor i ng the curvature t erm ahd cosmologi ca l cons t ant t erm 
i n comparison with the r adiation term in equa tion ( 2 .1) we obt a in 
2 p R t+ 
_!_ (dR) = 81rG ro o 
R2 dt 3c2 Rt+ 
or, i n tegr ating , 
~ = [ 32TTG p ll/ 1+ ~ 
R ro sec (2. 2) 
0 3 2 C 
The structure of the particle horizon at time t for the observer at r = 0 
will be given by s tudying the prop aga tion of null lines during the t ime 
interva l o tot: 
d*r  X (r) (1 + r 2 ) -
x (r ) = r fork= 0, - 1w  hile x (r) = 2 t an r fork= +l . Thus t he radius 
of the particl e hori zon at time tis given by 
X(r) C = 2 ?- (2 . 3) sec 
R -32T-TGP  l l/1+ 
o [ 3c2 ro 
1 3 
Taking p ro = 5 x 10- ergs/ cm
3 correspondin g to th e bl ack - body r adi ation 
28
temperature 2 . 7?K and R = 10 cm, we obta in th e coordinat e size for th e 
0 
hori zon at time t to be 
x (r) = 10- 8 lt/1 s ec 
10 
- 3 0 3 
For the present matter-density pm= 10 gm/cm, the universe remains 
radiation dominated until T ~ 3,000?K which corresponds to the age of 
r 
the uni. verse t -- 1013 sec. Thus, during the radiation dominated phase, 
the light can travel the coordinate di stance of 3 x 10- 2 which is a 
small fraction of the range of X going fiorn Oto 2TI for the closed uni -
verse . To get a better idea, let us compute the mass of the baryons 
in a typica l horizon. Consider the k = 0 case, for which the proper 
radius of the horizon at time twill be given by 
rH = R(t)r, 
where r is the maximum di s t anc e that light could have trave l ed. Substi-
tuting for R(t) from (2.2) and r from (2 . 3) we obtain 
= 2ct (2. 4) 
4
Writing P = aT , equations (2.2) and (2.1) would give the following 
r 
relationship be tween temperature T and the age of the universe , t: 
(2. 5) 
Thu s the total baryon mass in a horizon will be given by 
M = 4n r 3 
H 3 Pm H (2.6) 
where P , the matter density is given by 
m 
p R3 = p R 3 
m moo 
(2. 7) 
or 
pm = pmo [~J 
11 
where pm o is the matter density now and To  is the present radiation tern-
perature . Sub s tituting (2.4) and (2.7) into (2.5) and using (2.5) we 
get 
[-~-J 3 8c  3 T\ ? -[=32 =-1r_,~,---) (2. 8) 3 / 2 I 
3c7a 
-3 0 3 
Taking Pmo = 10 gm/cm, T = 2.7?K and T = 3,000?K, we compute the mass 
0 
of the horizon 
while the mass of the observable universe is roughly 10 51+ gm. Thus we 
find that in these standard evolutionary models of the universe, only a 
small fraction of the universe could have had communication between its 
various parts. As a result, no physical mechanism would make the pro -
perties of the two parts of the universe uniform when these two parts 
of the universe lie outside of each other ' s horizon. But the universe 
is observed to be homogeneous on a much larger scale than the size of 
the horizon. The 2. 7?K microwave relic radiation gives the deepest 
look into the universe. The microwave photons give us direct informa-
tion about the nature of the universe at the time they last interacted 
with matter. The main interaction with matter is the Thomson scattering 
by free electrons. The radiation first ceases to interact with matter 
when the plasma recombines as the temperature of the universe drops to 
~ 4000?K (see Peebles, 1968). The photons now travel fre e ly until the 
12 
epoch when the intergalactic plasma is turned on. To find out when the 
presently measured microwave photons scattered last with the intergalac-
tic plasma, we consider the optical depth T (e-T is the attenuation fac-
tor for the photon intensity) obtained by Bahcall and Salpeter (1965) 
for Thomson scattering in the universe with Hubble constant H 1= (10 10 years)- ? 
0 
Taking the deceleration parameter q = I1  and a dense intergalactic plasma 
0 
with e -5le  ct -r 3o  n density n = 10 cm , we obtain the optical depth to 
e bo e  
unity for a distance corresponding to a red-shift of z = 7. Thus the 
isotropy (s ee Partridge, 1969) of the present day observed 2,7?K micro-
wave radiation implies that the universe ha s been expanding isotropically 
at least since z = 7. And the inhomogeneity in the microwave radiation 
would be washed out only over a hori zon size at z = 7. Let us compute 
the relevant horizon size fork= 0 Robertson-Walker model, the metric 
for which is 
ds 2 = - c 2d t 2 + R2  ( t ) [d x 2 + d y 2 + dz 2] 
2 2d 2 t) [ - -C t 2 = R 2( -   2 + dx + d y + dz ] . 
2 
R (t) 
Defining a new time vari ab le n by 
cdt 
dn = R(t) 
we obtain a conformally flat metric: 
= R2  (n) [ - dn 2 dx 2 dy 2 dz 2 'I- + + ] (2. 9) 
Wh en the uni verse is radiation dominated R( t) a: It in which case n a: It 
.. 
13 
giving us 
R Cn) a: n . (2.10) 
2/3 
When the univers le /3  is matter dominated, R(t) a: t giving us n a: t ; or 
R (n) n 2 a: ? (2. 11) 
Suppose that the microwave photons which were scattered last at n scatt. 
(corresponding to z = 7) by intergalactic plasma at P and Q (see Fig. 2) 
are just observable no,v at n b s . Then setting ds
2 
= O in equation (2.9) 
0 
we get 
OP = OQ = n obs. n scatt. (2 .12) 
vhile n seat t . is related to the epoch z = 7 by the red-shift formula 
R(nobs. ) 
1 + z = (2.13) 
R(nscat t.) 
Since the universe is matter dominated during this relevant era, we can 
use (2.11) to obt ain 
[ n obs. n = 8 (2.14) scatt. r 
Thu s we arc seeing photons of exact ly the same temperatur (up to O. 3%) 
which were scattered by plasma at P and Q at n = nscatt. But light pro-
pagation between regions around P and Q before the time of microwave pho-
ton scattering is possible only if 
PQ < n
s catt . (2.15) 
.... 
14 
Thus, the maximal angular separation (0 ) between the plas
m m a regions 
which could have had prior causal communication is given by 
sinem / 2 
1 1 
2 (IB - l) 
or, 
So in Robertson-Walker models of the universe , the inhomogenejties in 
the temperature of microwave radiation would persist over regions sepa-
rated by angle 0 > em at z = 7. While we observe the microwave radia-
tion to have exactly the same temperature over widely different angles 
in the sky as opposed to the expected variations over regions separated 
by angle? (= PO?Q) > 22?. 
Thus t he isotropic models of the universe have an unpleasant fea-
ture in terms of the existence of the particle horizons. Misner (1969a) 
first pointed out the possibility of absence of horizons in a more gene-
ral, anisotropic, Bianchi Type IX model of the universe. That the struc-
ture of horizons is quite different in anisotropic mode ls can be seen in 
Kasner solutions (Kasner, 1921) of Einstein ~quations, whose importance 
as models for the initial singularity was first demonstrated by Lifshit z 
and Khalatnikov (1963). These solutions correspond to the gravitational 
field in empty, homogeneous Euclidean space and the metric for these solu-
tions is given by 
(2.16) 
15 
where p , p
1 2 , p 3 are three arbitrary numbers satisfying the relations: 
P + p + p l = 1 = p 2 2 + p3   2 + pl  2 2 3 (2? .1 7) 
Therefore, only one of those numbers is independent . So a singl e real 
parameter u (first used by Land au and Lifshitz, 1962) can represent p 1 , 
- u u + 1 u(u + 1) (2.18) 
1 + u + u 2 1 + u + u 2 1 + u + u 2 
Consider a particular Kasner so lution with p1 = 1 and p = p = 0. Then 2 3 
th e metric is 
ds 2 = - dt 2 + t 2dx 2 + dy 2 + dz 2 (2.19) 
In terms of the new time variable n = 1nt, the metric reduces to 
Thus the distance 6x covered by the light signal propagating in the x-
direction is 
Assumin g that the metric has the form (2.19) for all t from t = 0, then 
n take s on a ll real values from n = - oo . Thus at any given time n, any 
finit e di stance along the x-axis can be covered by a light signal in the 
time available since the initial singularity. Thus the particle hori zons 
16 
are washed out a lon g the x-axis by causa l propagation. In the nex t 
chapter , we will describe the behavior of Bianchi Type IX mod e l as 
inves tigat ed by Mi sner (19 69b), Belinski i, Lifshitz, Kh a l atnikov (1970). 
They find out that thi s model close ly approximat es Kasner solutions dur-
ing certain periods of its evolution . Here we will inves tigat e tha t 
s ub se t of Type IX solutions which, during their evo lut ion, approximate 
those Kasner s olutions which open particle horizons in one spatia l direc -
tion. Thi s subset of solutions (called the "circumnavigation so lu-
tions")will be characterised by the initial conditions /3 , /3 (shape 
+ -
ani sotropy) and p ,p (expansion rate anisotropy). \Ve will see how 
+ -
the Einstein equa tions lead to a natural measure on the init ia l condi-
tions ( /3 , p) in this problem, enabling us to compute th e probability 
+ + 
for a typica l Bianchi Type IX solution to have no hori zon a long any one 
axis. 
17 
CHAPTER III 
The Non-rotating Bianchi Type IX Model 
The me tric for the generic, non-rotating, closed homogen e ous cos-
mological model of Type IX can be written as 
(3 .1) 
Here R and are functions of time t only. The matrix B . . is lJ 
diagonal and trace less and measures the anisotropy of the universe. The 
a i are three independent differential forms which remain invariant unde r 
i 
the Bianchi Type IX homoegenity group. Thus the a satisfy the following 
relation 
da. = s. 'k a . .,, ak (3. 2) 
1 lJ J 
and they can be repres ented as 
al = sinijJd0 - cosijJsin0dc/J 
02 = cosijJd0 + sinijJsin0dcp (3.3) 
03 = - (dijJ + cos0dcp) 
where ijJ0</J are the Euler angles with O .::_ 1/J < 4n, 0 < 0 < TT and 
0 .s._ <P < 2n. The numerical factor of 1/4 in Equation (3.1) is chosen so 
that when S .. = 0 the space part of the metric is just the standard metric 
1] 
for a 3-sphere of radius R. Thus for B = O, ' one obtains the closed 
Rob ertson-Walker cosmological model which is both homogeneous and isotropic. 
3 
We will measure the volume of the universe (p roportional to lg = R ) by 
8 
a parameter ~ defined as 
L_ ________________ _ 
18 
R = (2/3rr) l/ 2 -n e 
For the two i ndependent anisotropy parameters choose 
Thus the variables !J, S+ and S_ characterize the three-geometries. To 
study the dynamics of these universes, the most elegant method i s to 
obtain the Hamiltonian action functional which Misner,[1969] did by 
a dapting the Hamiltonian formalism developed by Arnowitt, Deser and Misner 
[1962]. The idea of the Hamiltonian formalism is to cast the variational 
principle for Einstein's equations which is ol = 0 with 
I = (16rr) -1 I (3. 4) 
into a canonical form. Arnowitt, Deser and Misner [1962] show that the 
variational principle can be put in the form 
I ik ? I= (16r 4r   )-l Tr g 
ik d X (3. 5) 
r ir k where not all and gik are independent as they satisfy the constraint 
equations: 
ik 
- Tr = 0 (3.6) 
ik 
- 2rr lk 0 (3. 7) 
The quantity 3R is the curvature scalar formed from the spatial metric 
gij, "I" indicates the covariant derivative using this metric, and spa tial 
19 
i. 
(Latin) indices are raised and lowered using g J and g . .. The equations 
lJ 
( 3 .6) and (3. 7) are nothing but the Einstein's initial value eq uations 
Go?= O, up to certain multiplicative factors. The action integra l (3.5) 
for our me tric (3.1) would be 
where 
1 -2Sl( 2(3)
g   .. =1_R2( 2 (3 ) 
lJ 4 e ij 6n e e ij 
Thu s ' d gij = -2giJ d + 2gik d kj 
So 
(3. 8) 
Since is traceless , we separate out the traceless part of n j by k 
defining 
(3. 9) 
Since (3.8) is almost in a canonical form, taking Sl as a time coordinate, 
k 
we see that (2n ) 1T k will be the Hamiltonian so we write 
k 
H = ( 2n) 1T k (3 .10) 
Thus the action integra l now reduces to 
P arame t e rizin g in the same way as (3 kj , name ly 
20 
we obtain 
I= I (p+ dB++ p_dB_ - HdQ) (3.11) 
Thus we have obtained the standard canonical form provided B?' p? and 
can be varied ind e pendently and H is expressed as a function of B? ' 
p? and n. To see this note that the constraint equation (3.7) is iden-
tically satisfied in any case and the only remaining constraint equation 
(3.6 reduces to 
3 
( E:._ )2 3R + 1 (1rkk)2 (!_ pi 1 i 9., + 
2 21r k 3 o kTI 9.,)x 8 
1 k l_ ok 
<2nP 1Tm ) i + 3 i 0 m = 
or 
lci >2 
3 1T 9., 0 
i k 1 2 k 1 6 
Substituting p k p + p 2)' 3R (1-V) i = 6 (p+ TI k 21r H' and R2 
2 -n 
and writing R = e we can solve the 31T above equation for H, to obtain 
(3 .12) 
48 28 
where V( B) =; trace ( e - 2e- + 1) 
1) + 1 
- !!:_ e - 2 B+cosh 21:fB (3.13) 
3 
Thus, with n as the choice for the independent, coordinate time variable, 
the s t ate uf the universe at any epoch n, is given by the field amplitudes 
B+ a nd B_ and their congugate momenta P+ and p_. And the evolution of the 
universe is given by the Hamilton ' s eq u ations : 
21 
3H 
an (3 .14) 
In terms of the n time , the beginning of the universe a ppears at 
n =+ co . Tlte prope r t i me of comoving obse r ve rs is rela t e d t o rl by 
(see Mi sner [1969b]) 
dt =f t- -1 -312 1T H e drl (3 .15) 
Thus the non-rotating Bianchi Type IX problem is reduced to a Hamiltonian 
problem corresponding to a particle moving in two dimensions in a time de-
pendent potential. The "anisotropy potential" V( /3) is positive definite 
with 
V : 8 ( 2/3   2+ + /3 ) near /3 = 0. (3 .16) 
The equipotentials near the origin of the /3+/3- plane are closed curves for 
V < 1. The potential walls rise steeply away from (3 = 0, with the equ i -
potentials forming equilateral triangles in the /3+/3- plane. [See Figure 1.) 
The potential has reflection symmetry in the /3+ - axis and one side of the 
equilateral tri angle cuts the negative /3+ - axis. The corners of the triangle 
a re not closed however, but have channels leading of f to infinity. These 
channels narrow exponentially and the deviation f rom the straight-sided 
traingular shape takes up a very small part of the total equipotential. For 
example, near the positvie /3+ - axis, for large~+' V( /3 ) has the form 
V(/3) : 1 + 16/3 2 e 413+ (3. 17) 
In directions opposite the three corners, the potential rises exponentially 
for large distances from /3 = 0. Along the negative /3+ - axis, for example 
V has the value 
22 
 1 -BS
+ 
V( S) "'3 for S << -1 (3 . 18) e + 
Thus the equipot ntial contours for V > 1 consist of three disjointed 
curves, each of which runs off to infinity at the channels . Wh en not 
running off to infinity, the contours are approxmiately straight lines. 
Thus, the evolution of the universe is given by the motion of the sys tem 
point B = ( B+, S_) as a function of the time coordinate n , moving in a 
time-dependent potential well . In the first app roximation near the singu-
larity for n + 00 , we can neglect the potential t erm e -
4n (V-l) in the 
Hamiltonian [Eqn. (3 . 12) ] to give 
H (3 .19) 
The Hamil ton ' s equa tions , then give p? , H as cons t an ts of motion and 
~ = ??. Thus the universe point moves with velocity S ' = dS/dn --
d s-2 l-l 
2 2
 1 12 { ( d B+/dn) + (d B /d s-2) } of unit magni tude in straight lines . The 
approximation us ed would fail when V( B) becomes sufficiently l arge . The 
limiting equipotential ( the potential "wall" ) is one which would make 
B' to go to zero . Using the Hamilton ' s equations 
(3. 20) 
0 n e can rewrite Equation ( 3.12 ) as 
S ' 2 + S ' 2 + H-2 41 e- n V, for V >> 1 (3. 21 ) + 
so tha t the condition S ' = 0 gives 
2 4n 
V( Swall) = H e 
23 
Sub s tituting the asymp t o t ic form for the po t ?nLia l for 8 << -1 from 
+ 
Eq u a ti on ( 3 .18), we obt ain 
(3 . 22) 
The de pe nd en ce of His given by the Hamilt on equ a tion 
dH 
ds-2 
or 
2 
- 4 ( 1-8 ' ) (3. 23 ) 
Thus , when the sys t em point 8 i s we ll inside the potenti a l walls, 
8 1 '.:-'. 1 and His nearly cons t an t. From Equ a t i on (3 . 22) we then conclude 
that the sides of the linri t i n g equipo t en t i a l t ri angle move outwa rd at 
v e l ocity (d ~ ~11) I = ?w hen the s ys t em p oin t i s moving with unit velocity 
~nside the triangle . If the s ystem poin t is movi~g i n a straight line 
making an an gl e 8 with 8+ - axi s , t hen d +/<l0 = cos 8 and d8_ / dr2 = sin8 . 
Subs t ituting S+ = ( cos8 ) ~ + cans t m1d S ( s inO) ~ + cans t. i n the me t r ic 
(3. 1) we ob t a i n 
__ dt2 + .l__ [e2 CS+ + /:fo _)-2~ 01 2 + e2 ( S- - /38_) -2 n a2 2 
6 1T 
+ -43 + - 2n 3
2
a ] e 
d 2 + 617 r [ ?le ~ (2cos0 + /3sin 0 - 2) 1 2 =-t a 
r2 ( 2cos 0 - l.3sin 8 - 2) 22 
+ a.2 e a + a 3 
where a. , a. , 
1 2
a. a re some cons tants . S
3 in cc 1-1 i s cons t ant when t he sys tern 
is mov i n g free l y i nside t he pot ent i a l wa lls , Equ a tion (3.15) can be i n-
t egrate d t o give 
-3 St 
t a. e 
24 
Thus, the metric reduces to 
2 
ds 2 = -dt 2 + 61r r [A t 2 Pl a1 2 
2
+ A
 
 t 2p
2 3
2 cr 2 + A3  t P3 a ] (3.24) 
1 2 
where A1 , A2 , A3 are some constants and 
1 
P1 = cose -
1 si.n  e + 1. 
3 - 13 3 
l 
P2 = cose + -
1 si. n e + .!. (3.25) 
3 13 3 
2 
P3 = 3 cos e + 13.  
These coefficients satisfy which are 
just the relations which Kasner coefficients satisfy. Belinski, Khalatnikov, 
Lifshitz [1970] call these metrics generalized Kasner solutions. Thus 
the universe has a Kasner epoch when the system point S is moving freely . 
This epoch, then can be parameterized by "u". From equation (2 .18) sub-
stituting for p3 in (3.25), we get 
-u-(-u-+'-l)~ = 2 cose + 1 
2 3 
l+u+u 3 
or 
u 2+ u-1/2) 
case 
dQ 
u 2 +u+l 
Similarly, we would get (3. 26) 
dS 
= /3(u+ 1 / 2 sine 
dQ 2 
u +u+l 
Thu s the evolution of the Bianchi Type IX universe can be described as a 
series of Kasner epochs - the system point S moving in a certain direction 
25 
characterized by a certain value of the parameter u, with unit velocity un-
till it comes close to one of the equipotential walls (which are moving more 
slowly) and feels the potential~ and would then bounce off the wall and would 
follow freely a different Kasner epoch until it reaches the next wall. 
The Kasner solution which does not have a hori.zon. for causa], propagation 
along one axis, say the z-axis, has indices pl = p2 =- 0 and p3 = 1 which. 
corresponds to u = 00 [see Equation ( 2.18) ] . u = 00 corresponds to 
df3+ = 1 and d(:3_ 
a~r arr- 0, i.e. the sys tern point is moving horizontally, parallel 
to the S+ -axis in the positive direction. Note that the inclined potential 
walls are also moving with unit-velocity in the positive S+ direction. Thus, 
this epoch would correspond to the sys tern point following one of th.e inclined 
potential walls. In the next chapter we will investigate how closely a 
Bianchi Type IX solution would have to approximate this particular Kasner 
epoch for it to lead to a vanishing horizon along one axis . The Kasner solu-
tions corresponding to horizon vanishing along th,e 0th.er tlvo axes are given by 
pl = p
3 =  O, P2 = 1 and p2 = p3 = 0, p1 = 1 respectively: those are 
parameterized by u = 0 and u = - 1 and correspond to the system point 
moving parallel to the other two corner axes and moving in the same directions 
as those corners. 
As the system point bounces back and forth between the equipotential 
walls, let us note the physical significance of various paths as they relate 
to V ( S}. Consider a particular path when the sys tern point is close to S+ -axis 
and following the corner (/B_I << 1, s+ ? oo ) This corresponds to a 
"p ancake " singularity - a relative compression of the 3-axis with the 
other two axis approximately equal. With S = 0, it corresponds to the Taub 
Vacuum solution with the l engths in the 1 and 2-axes remaining finite and 
I[) 
equal. When the system pojnt is ne ar the potential walls, say for example 
the vertical wall which corresponds t o /3+ ? - 00 and /3 < 13 /3+' it 
corresponds to a " cigar" singularity with stretching of the 3-axis 
relative to the other ; the othe r two sides of the triangular equipotentials 
correspond to preferential stretching of the other two axes . 
27 
CHAPTER IV 
Prop agation. of Light a nd High-frequency Sound Waves 
We want to investigate possible modes of communication in the Bianchi 
Typ e IX model. In the Appendix A, we obtain an equation for th e propagation 
of high- frequency sound waves . It is a geodesic- like eq u a tion for sound 
rays describing a wave-packet moving in the 1/J-direction . The equ a tion reads 
dl/1 = ~V e ri e 2S+ dt s ( 4.1) 
3p,_
where V ( 
 l/2 
--;:;--) is the sound v e lo
s 3? cs i ty. Setting V = 1 one would get t . s 
the law of propaga tion for light going in the iJ;-direction. Or, indepen-
dently, one could get it from the geodesic equ ation D+ V = 0 as done in 
V 
Appendix B. For the light going in the t -direct 2i  on, setting v = v3 = 0 
(the compone nts of the tangent vector to the geodesic in the othe r two 
principal directions) in Equation (B 1.6) and using the null condition 
0 2 
(V 0)   2 - (V) = O, o . ne obtains 
(4.2) 
We want to study the above equations for large u (asymptotically 
u = 00 ) epoch when the universe approximates a genera lized Kasner solution 
with p = p = 0, p
1 2 =  1 which is 3 t hat Kasner solution for which light 
prop agates round the universe in the 1/1-direction. The u = 00 h ave two 
distinct cases - the axial and the non- axial ones . In the axial case, 
the sys t e rn point is close to the S+ - axis and is runnin g towards the corn.er, 
while the corner is movin g outw ,r ds with unit v e locity. It is more convenient 
to desc rib e the motion of the system point in terms of S defined as 
0 
(4.3) 
,, 
28 
(3 is then a measure of the horizontal dis tance from the corners . 
0 
nega tive and it decreases as the universe feels the potenti a l in the channel. 
The solution to Einstein's equations for this epoch can be describ e d as 
2 2 f3o 
f3 = Z ( e ) o K [ See Appendix A, Eqn. ( 4. 8) . ] 
where K is a small constant of motion, and Z is a spherical Bessel 
0 
function. For large values of the argument , it reduces to 
K -(3 2
f3 e O 
2 f3o 
1T sin ( e K - 1r I 4) 
' . 
which corresponds to the adiabatic harmonic oscillator given by Misner (1969). 
Substituting this solution in the equations for sound rays, we find that 
high-frequency sound waves can go round the universe in the l/J-direction 
during seven cycles of f3_ ; where we have taken 1 V corresponding to 
s 13 
for radiation fluid. While the light ray would do the same thing 
during four cycles of f3 . The result corresponds to the one quoted by 
Doroshekevich and Novikov (1970) which reads 
N 
e 
where N = number of times light goes roun
e d  in the l/J-direction and N = m 
number of maxima for f3 . Note that f3 is related to Q in their 
notation: Q = e 4 /3(3_. Note also that the argumen t of the Bessel function 
for f3 is decreasing, since (3 is negative and decreasing . For small 
0 
argument , we can write f3 as 
f3 ax+ blogx where X = 
Thus as x decreases, f3 increases and soon the approximation for 
/ (3_ / << 1 would fail. At this stage we can conside r the system point to 
have come out of the channel. 
29 
Next conside r the more general case, the off-axial case with B > 1 
and u very large. Suppose that the sys tern has bounced b ack off the vertical 
wall ( at Q = Qb) and is going towards the inclined wall with large u. The 
equipotential wall is moving outward with velocity 1 / 2 while the system 
point has ve 1 oc.i ty 1 3 2 + Zu towards the wall. Thus the system point wil 1 
catch up with t he wall and will experience its potential for a long time 
before getting bounced off. In the Appendix A, we compute the changes in 
v arious, relevant quantities as the system point enters and leaves the 
region where it experiences the potential of the wall. When we study the 
e quation for <  propagation of sound rays during .this epoch , we find the 
estimate of how large u should have been when it first bounced off the 
vertical wall, sothat a horizon is washed out during the next collision with 
t he inclined wall. The relevant equation [Appendix A, Eqn. (3. 7) J reads as 
13 
u > 
2V 
s 
( B+) wall is ne Qg  ative; it goes like ( B+)wall "' - 2 + a where a is a 
constant. Thus , we find t hat at any epoch, St , t here exist small sectors 
around t he lines parallel t o B+-axis s u ch that when t he system point is 
r ui ling along t hese sectors at Q, a horizon is removed in the lji-direction . 
The angular extent of these sectors ( 0 = /3/u) depeads upon Q and it 
decreases to zero as goes to 00 Therefore , at each epoch Q, we obtain 
a s ubset out of the set of all possible initial conditions ( B+ ' B_, 0) 
s u ch that the solutions corresponding to t he initial data specified by 
t his subset don ' t have a hor:lzon :ln tlte y-d:lrection , The totality of all 
s u ch s ubsets will be called a circumna\?igation set at that epoch. In the 
next chapter , we will show how to assign a set theoretical measure on these 
initial conditions . 
30 
Let us now follow the s ystem point after the first bounce with the 
inclined wall. As shown in the Appendix A (first shown by Belinskii and 
d /3 
Khalatnikov (1969)) u changes from u. to uf = - u., i.e. d St - < O. 
l. l. 
So the system point follows a Kasner epoch of straight line motion in the 
S plane till it comes close to the second inclined wall. A similar analysis 
as done in the Appendix A would show tha t a collision with this inclined 
wall changes u to -u -2, or the value of the parameter u now becomes 
u-2 and the system point l eads back towards the first incline d wall. The 
process r e pea ts itself till u becomes les s than unity in which case it 
stops rattling b ack and forth between the incline d walls and heads towards 
the vertical Wbll and would start rattling back and forth in one of the 
other two corners. The above description can be describe d ve ry elegantly by 
the Lifshi t z-K.h a latnikov bounce l aw u + u - 1 [ 1969 J. lie re we take our 
fundamenta l interval for u to be 1 < u < 00 Oth,e.r values of u can be seen 
to represent the solutions corresponding to the values of u in the fund a-
mental interval by simple relabeling of the axes. The operators permuting 
the Kasner exponents P1, P2, p are : 3 
P12 u + - ( 1 + u) (pl ' P2' P3) + (p2' pl' P3 ) 
P23 u + 1 / u (pl' P2' P3) ? (pl' P3' p2) 
and 
P31 u ? - u /( 1+4) (pl' P2, P3 ) + (p3' P2, pl) 
Thus the bounce law - u. given above can be transformed into 
l. 
= u - 1 by the operation r 23 B r 12 p . The evolution of the Mixn1as t er i 31
universe, the n consists of one Kasner epoch r epresented by u (~ 1) re-
placed by another with u - 1. So it continues till the integral part of 
t he initial value u is exhausted, i.e. until u becomes less than one. 
31 
The transformation p : u -+ 1/u will put u back in the f und amental 23
interval and that will start another series of Kasner epochs. These suc-
cessive series are called eras and the length of an era is given by the 
number of Kasner epochs it contains (see Belinski etc. (1970)). If the whole 
0 0 0 0 
sequence begins by the numb e r u = k + x , where k is the integral part of 
0 1 2 3
u ' then the lengths k , k , k , ... of the successive eras are given by 
0 
the numbers appearing in the expan sion of x in an infinite continuous 
fraction 
0 = 1 X 
1 
Belinskii etc. (1970 do an algebraic study of the above equ ation and prove that 
with increasing numbers of eras , .the values of x approach a s t ationary 
dis tribution, i.e. if we start with a certain probabilistic dis tribut ion 
for X in the range (O,l) at some epoch r2 and follow the evolution 
0 
of the system points t owards the singularity, the values of x t ake on 
a sta tionary distribution asymp totica lly. 
In the nex t chapter , we will use geometric method s to investigate the 
statistica l properties of the Mixmaster univers e . We will prove that the 
evolution of the universe is e rgodic in a certain phase space related to 
S+ ' 6_ and u . Thus we prove certain statistical prope rties not only of 
the anisotropic expansion (whi ch is related to u or x) but also of the aniso-
tropy (6+' 6_) of the un iverse . 
32 
CHAPTER V. 
The Ergodic Behavior and the Horizon Problem 
A. The Evolution of the Mixmaster Universe. 
The evolution of the Mixmaster Universe is given completely by 
specifying its shape anisotropy "B" and the expansion rate aniso-
tropy P+' p at some epoch,~. As we follow its trajectory, the sys-
0 
tern point seems to wander about rattling back and forth between the 
equipotential walls and moving along various directions (direction 
specified by 0; tane = p+/p_). In this chapter, we will study the 
exact nature of this wandering motion. We will show that the evolu-
tion of the Mixmaster Universe is equivalent to a geod es ic flow within 
a bounded region of the Lobatchewsky plane. We will find a certain 
group of transformations, G, which make this flow of broken geodesics 
on the Lobatchewsky plane, D, into a continuous one on D/G. 
As we saw earlier, the variational principle for Einstein's 
equations 61 = 0 with 
can be cast into a canonical form to obtain the Hamiltonian 
H = [ p 2 + p 2 + e -4 ~ (V - 1 ) ) 1 / 2 (5.1) 
+ 
where p and p are the momenta conjugate to the field a
+ m plitude- s B and B + 
respectively. 
33 
Let us rewrit e the corresponding Hami lton's equations: 
dS 
dfl = p /H 
dp _,1[2 
-e dp -4fl + cm- -e av dfl = 211 = 2H ~ (5. 2) 
and 
_4[2 
2e (V - l) 
H 
Let us introduce a new independent variable, A, ,defined by 
dA dfl = (5. 3
H )  
The Hamilton's equations can be written now as 
dB -
~ = p_ , 
(5. 4) 
adrp-f-l- 4 fl = 2e (V - 1) 
where 
- 2 2 -4fl L 
- PQ = H = [ p+ + p + e (V - 1) ] 2 (5. 5) 
As Misner (1971) points out, one can see that equations (5.4) are just 
a new set of Hamilton's equations 
(5. 6) 
with A as the independent variable and 
K = ?[ - Pn 2 + p + 2 + p 2 + e -4 Q (V - 1) ] (5. 7) 
as the new Hamiltonian. Indices A, B, etc. are used to labe l coordinates 
in n  supersr +:  :cc , so g = fl, g = 8+ , g = 8 . The variational principle 
34 
can now be written as 
(5. 8) 
the Einstein's equations in addition giving th e constraint K = 0. Con-
sider now the behavior of the potential energy like term, R in the Hamil -
tonian: 
~ 1 
3 - 4n-8B+ for B e + ? - oo 
The condition tha t V be important is then given by 
e 
or asymptotically, the "potentia l wall" de f ined by Sw a w1 il1 l  be given by 
B = .!. n - .!. 1 n ( 3il ) 
wall 2 8 (5 . 9) 
For large n approximation, we can ignore the variations in the second 
term,? ln(3H 2 ) (which would go as ln n). Thus the equipotential in 
t he B-plane bounding the region in which the potential (space curvature) 
terms are significant is given by 
B 1 =
w  a -ll - n + 2 a  (5 .10) 
Consider the following set of transformations (see Misner, 1971) which 
would make the above wa lls stationary in the new coordinat es : 
35 
n - t 2a = e cosh; 
e t B+ = sinh; cos ? 
t (5.11) 
B = e sinhs sin? 
Th e equipotential walls now will be given by 
t anh s 1 = 2 sec ? (5.12) 
and t he other two are obtained by replacing?+ ,? + 2n in the above for-
3 
mul ae . 
Subs tituting the new canonica l coordinates into th_e. act:lon, the. 
variational principle reads now 
(5 .13) 
wher e 
pt = et(p +s i nhc;cos? + p sinhssin? + pncoshs) 
t 
Pc; = e (p +c oshc;cos? + p cosh;s in? + p sinhs ) 
p<P = et(- p +s inhssin? + p sinhsco sh?) 
In t erms of these new canonical var iabl es , the Hamiltonian, K, i s given 
by 
p 2 
~ 
2K -2t = (- p 2 + p 2 + -~~-) e + R(s , ?). (5.14) 
t s sinh2s 
36 
The factor e- 2t can be removed by defining a new variable d\ 1 = e- 2td\, 
to give the new Hamiltonian 
p 2 
2K' = p 2 + Pc 2 + ? + RC s. ?) e2 t . 
t s sinh 2 s 
In the asymptotic limit, t aking R(s, ?) to be zero inside th e equipo-
tential walls and+ oo outside , we find that the evolution of the Mix-
master Universe inside the equipotential walls is governed by th e sta-
tionary Hami ltonian, 
p 2 
~ = 21 ( - p 2 + p 2 + ~ ) t s sinh 2 s (5.15) 
Note that for a particle llamiltonian 
the motion of the particle is along the geodesics of the manifold pos-
sessing the metric g Thus, the evolution of the Mixmaster Un iverse 
].lV 
as projected on the two-dimensional spaces,? will be given by a geo-
desic flow on the Riemannian manifold with the metric 
inside the region bounded by the curves given by (5.12). By making a 
transformation sinhs = 2r/(l - r 2 ), we obtain the metric on the Lobat-
chewsky plane which is a unit disc with constant negative curvature. 
The metric for it can be written in the cartcs ian coordinates as 
(5. 16) 
37 
The geodesics on this manifold are circular arcs, orthogonal to the 
circler= 1. The equipotential walls are now given by the three cir-
cular arcs as shown in Fig. 4. These arcs bound the fundamental r egion 
in which the system point moves along geodesics (circular arcs) until 
it hits the wall and is then bounced back along another geodesic and 
so it goes on. The "bounce law" would be governed by the shape of the 
border . 
A? t ypical solution to Einstein's equations ,would be given by a 
me s h of broken ge~desics. We will show that the so lut ion can be re-
presented by a continuous geodesic curve under a suitable set of trans-
formations. Consider the following transformation which tak es one wall 
(given by a circular arc) into a vertical diameter as shown in Fig. 5: 
-+ z -t (2 - /3 ) A z (2.12) 
(2 - /3) z + 1 
It is a conformal transformation, thus preserving the metric. The bounce 
law at any point on the diameter AB is very simple as it is governed by 
two constants of motion, the Hamiltoni an 
( I x2 y2)2 H' = ------~-- (p 2 + p 2) 
4 y (2.18) X 
and the vertical component of the momentum p . Thu s dd
y y  = p /p has the X y X 
same value before and after the bounce. Hence, the transformation z-+ - z 
would transform t he reflected geodesic onto a continued part of the inci-
-1 
dent geodes ic. By applying A now we recover the fundamental region. 
Figure 6 illustrates how th e transforma -t 1 ion R1 = A RA applied to a con-
tinuous part of the incident geodesic out side the fundamental region gives 
38 
the proper r efl ec t ed geodesic inside the fundamental region. Similarly, 
one could construct transformations R and R for th e other two walls or 
2 22!._ . 3 
3 l 
th ey can be generated from R and S = e Figure 7 finally shows how 
a repeat ed app lication of thes e transformations would give a repre senta-
a 
tion of brok en mes h of geodesic lines by a continuous one. Thus the evo-
lution of our mode l i s now given by a continuous geodesic flow on D/G; 
where Dis th e unit di s c with the metric given by (5.16), and G is the 
group genera t ed by R , R and R or by R ands~ To get an ori entable 
l 2 3 l 
manifo ld, consider a subgroup Fin G defined by 
F = fa. , R .? . . ' R. }where n is even l l l n (5.17) l 2 
Thus , considering points which are congruent under Gas identical, we 
obtain a closed, orientable, two-dimensional Riemannian manifold of con -
stant negative curvature. We will denote it by MF (~ D/F). Thes e groups 
(called Fuchsian groups) have been extensively studied (see Hed lund, [1939] 
an d refer ences there in). Some of the important properties of th ese groups 
for the relevance of geodesic flow on MF are the following: 
Corresponding to any such group F there exists a norma l fund amental 
r egion R. Thi s is a simply connected region bound ed by arcs of hyperbolic 
lines (circul ar a rc s perpendicul ar to the circle lzl = 1) which are con-
gruent in pairs, s uch that no two interior p~ints are congruent and any 
point of Dis congruent to some point within or on the boundary of R. 
The region R for our group Fis ABCP in Figure 8. 
Th e group F given by (5.17) is a Fuchsian group of the firs t kind 
which i s characterised by the f act t hat th e fundamental r egion does not 
39 
contain an interva l of lzl = 1. The geodesics on MF are determined uni-
qu e ly by specifying a point (x, y) in the fundamental region and a direc-
tion 8 (0 < 8 < 2TT) The set of these elements [(x, y, 0)] forms a Hausdorff 
space , denoted by nF. Measure in nF is defined by the integral 
J J J--4d-xd~yd0- -
(l -x2 _ y (52  .) 12 8 ) 
To prove certa in results for the geodesic flow on MF' the property that 
plays a crucial role is that the area of nF is finite. Hopf (1936) and 
Hedlund (1939) hav_e proved that the geodesic flow on MF is ergodic in the 
metrically transi tive sense , i.e., if we start with any arbitrary mea-
surab l e se t M of nF (points of M de
0 n ote initial co0 nditions of the Mix -
master Universe at s ome time n ), the corresponding universes given
0   by 
the geodesics specified by M will have non- zero intersection with any
0   
measurable set M of nF. The ?,er important result (again proved by 
Hedlund (1939]) says that the geodesic flow on MF is a "mixture", i.e., 
any measurable set M of positive mea sure tends, with increasing or de-
creasing time , to occupy a definite fractional part of any other mea-
surable set~!*, and th e fraction is simply the fractional part of nF 
which M occupies. This can be written as 
lim m(~I ? M*) = (mM) (mM *) 
t-++oo t  mnF 
wh ere mM denotes the measure of M. The mixture property tells us that 
th e se t s in nF tend towards homogeneous distribution. Thu s we obtain the 
40 
result that in whatever range of anisotropy shape (specified by a certain 
set of x and y value s ) and anisotropic expansion rates (specified by a 
c ertain range of e) the universe might have started near the singularity , 
we will expect it to have a uniform distributi on over nF -- far away from 
th e singularity. This th en characterises the "chaos " of the universe . 
If we start with a well-defined s t a t e of the universe and let it evolve 
towards th e singularity, we find th at it goes through a lmost all possible 
anisotropic stages . 
B. Th e Probability for the Vanishin g of a Hori zon . 
As seen ear lier , the light can circumnavigate the universe in the 
tjJ -direction if 
26 
2 + (5 .19) 
3e u 
___H_  __ .:_ 4TT, for large u 
In the asymptotic limit approximations, 
wh ere the position of the equipotential wall is given by 
Thu s , the circumnavigation sets ar e given by small sectors of angl e 
60"'/3= (5.20) 
u 
41 
When converted into the new coordinates, t he angl e is given by 
t [l 
+ x2 + y2 - 2x ] 
:::: 1 - 2e 1 - x2 
(5.21) 
08 -e 
- y2 
2TT 
Th e geodesic f l ow is given by the Hami l t oni an 
2 2 2 
1-1' 2 (1 - X - y) [ 2 2 ] 
= 4 Px + Py 
21
p -! 'cos 8 2HWriting = ------ and p 
'sin 0 
, the invari ant volume 
X (l _ X2 y2) y 
element would be given by 
= 4 dxAdyAH'dH'.Ad8 
(1 - x2 - /) 2 
Since th e motion of th e system point i s ergodic, let us choos e a stationary 
microcanonical ensemb l e distribution in the phase-space x, y, px' Py? If N 
is the total number of solutions, then the prob ability distribution function 
will be given by 
P(x, y, 0, H') N 8 (H' - h') = 
8TTh' (Area) 
(5.22) 
wher e the area, A, is given by 
A ff dxdy = 
fundamental (1 - 2 X /)2 
region 
Th e probability fo r a typical solution to lie in the circumnavigation set 
at time twill then be given by 
4 f P(x, y, 8, H1 )dxdyd8dH'H' 
N 2 2 (2 5 . 23) 
(1 - X - y ) ? 
int egra t ed over the circumnavigation set. Note that these sets given by (5.21) 
42 
decrease exponentially as t (or 0) increases. So it is a fairly good 
estimate to compute the above probability at the minimum possible value 
of t . The t m.i n. will be calculated so that our approximations are valid 
until that epoch. It turns out to be very close to the turn-around (the 
maximum expansion) epoch for an empty uni verse. For a more reaU s tic 
model (i.e., radiation and matter containing universe) the t m.i n. epoch, 
corresponding to the radiation energy being as dominant as the aniso-
tropic energy, is much earlier and thus reduces , the probability for a 
horizon vani s hing considerably. 
To see the validity of approximations, note that H1 = pt is a con-
stant given by th e va lue of pt from (5.13). Thus 
H' = H(O - 2:X ) - !3 +P+ - !3 p 
Wh en the system point is colliding with the vertical wall, p + "' 0 and th e 
posi tjon of the wall is given by 
2 1 e -4n H = 3 e 
-BS+ 
Combining these two equations, we obtain 
21  n 1 1 (5.24) + 4 10 g rn - 2a) - 4 10 g ( /3 H I ) 
Writing a= ?l og(/3 H1 ) we see that taking 
is a good approximation provided 
I?l og(O - 2a) I < I-? n + al (5. 25 ) 
43 
or 
In - 1 2a I > 2 
We now substitute the probability distribution function from equation (5.22) 
into (5.23) and integrate over the circumnavigation set given by (5.21). 
t . 
Takin g the va lue oft at t m.i n. i.e., e min . 
1
= -2 , the probability for a 
typica l solution not to have any horizon in the i/J-direction is then obtained 
as 
-2 (-1 
+ x2 y2 
N dxdy ) 
( 1 + - 2x) 
p = ff e 2 1 - x2 - y2 
< 
4,r 2NA (1 - x2 - y2)2 
+ x2 y2 
Since th e factor (1 
+ - 2x) is always les s than unity in the fund a-
1 - x2 - y2 
menta l region, a lower limit on the probability is given by 
p = 1 1 ~ 1% 
Let us now consider the effect of observed radiation on the evolution of 
th e Mixmaster Univers e . The Hamiltonian would be modified to read 
2 3 8
p TIG +  - g 0 0 R + -~- g T 
C 
Th e radiation energy densi ty T00 is given by 
where (T00 ) is the energy density now and R ? is the present radius of the 
0 0 
universe . Th e radius of the universe is given by 
}_ R _r,i 
R = 2 o e 
44 
By taking the length factor R to be th e present radius of the universe, we 
0 
get Q to be equa l to 0.70 for the present epoch. 
ergs/cm 3 corresponding to the black-body radiation temperature 2.7?K and 
R = 10 28 cm., we obtain 
0 
Substituting th e expression 3 0s 0  for R and T in the Hamiltonian we obtain 
R 6 
J- r2=p2+p2_.!:___R6 e --6 D 6 (1 _ V) + , 8nG _E_ e-{) D Too 
+ 64 0 R2 c4 64 
R 6 
= p 2 + p 2 + I R 4 e -4 D (V _ 1) + 8nG _o__  e -2 D 5 x 10 - 1 3 ? + 8 0 4 4 
C 
1 
W -r 8i 8t +in  g V = 3 e = 
1 4D-8a . 
3 e and calling D = Da for the epoch when the 
radiation energy is as important as the anisotropy-energy, we get 
-2D 
1 -8 a = 8nG ~ R 2 e a x - 1 3 
8 e 4 4 o 10 
C 
= 
or 
_8a + 2Da 
e = 10-4 
i.e. D = 4a 2 log 10 
a 
= 4a - 4.6 
Thus, t bm is given yi  n. 
t min
e . = D - 2a = 2a 4a .6 
So the probabi lity for a horizon vanishing in the t -direction computed 
at Q = D will be given by 
a 
45 
1 
p -( 2a () 2 a - 4.6) ~ = e 
2 (5 . 26) 4TT
Not e th a t a crude va lue fo r the es tima t e of minimum va lue of a will be 
give n by a ssum i ng tha t the ani sotropy decays as B+ ~ = 2 + a and de-
ma nd i n g th a t th ere s hould be no ani sotropy left a t T = 3, 000?K, which 
corre s pond s to the radiu s R = 2 x 10 22 cm. and correspondi
l n gl y~~ 7, 
giving a . =
m  3in .5. . Subs tituting , t he minimum va lu e of a i n equation 
( 5 . 26 ) we obtai n the probability for the vani shing of a horizon to be 
0.0 2%. 
Thu s , we con c lud e tha t ther e is a very sma ll but finite probability 
for a hori zon vani shing (in one direction) before the photons get de -
coup l e d from th e matter. 
46 
\ 
' ' ' \ I 
\ 
I 
' \ , 
' L ' I ' \ 
\ I 
' I I 
' I 
TT 7T ' ' ' , I \  ' I, 
\ 
\ , / t=o 
p 
/ \ 
,I 
\ 
\ 
\ 
/ 
A ' / I ' ' ' ' \ 
\ 
FIGURE 1. Descrip t ion of particle hori zons. 
47 
I 
0 
f} obs . 
y 
' ' ' 
' ' 
' 
' 
' 
FIGURE 2 . The last scattering of microwa
ve radiation. 
--
FIGUR E 3 . Th e cquipotcntia l s - of V
? ( ? S?-?   . ,..  ?  S?  ? ) ?  - ? f--- o( rF  i lg au rr g?,t e \: "o Su "r . tcsf?orT' :-~1-1rsi+1, rr:--r-~?---- -"-'- 
\,,. .,.-. _ 
~ 
11-z.. + ' 13 t 
FIGURE 4 . An idealization of equipotential'walls which are moving out in 8 8 plane 
+ -
while they are given by stationary circular arcs in the Lobatchewshy plane. 
.-
.c,-
\0 
---, I . -1 ? 7 f I ,"  ,. ( ' \- ( 1 ! c l ?1--r --
A: t:? . c: + ci .-nJ -P 
('2-.-13)-i: + I 
A 
B 
FIGURE 5 . A is the t rans formation whi ch takes the equipotential wall 
represent ed by a circular arc into a diameter . 
., ~--? 
V, 
0 
h, 
fZ :z -~ +(1..-J}) ~'-  - ~ fl-, ~ z --i> 
(2-JJ)?: - I 
I 
/ 
/ 
/ ' 
/ 
/ 
....... . 
' ' , / ' '\ 
' 
?---????' . , ..,_.,. .. ,., __  ... ~ 
FIGURE 6. The effect of the transformation A- 1 RA is to map the reflected 
geodesic into a continued parf of the incident trajectory. 
V, 
~ 
FIGURE 7. A?repeated application of the transformations S = A 1 - RA make.,
s ao  l tu yt pi io cn a lo  f the Mixmaster Universe evolve along a continuous geodesic. 
V, 
N 
C 
FIGURE 8. The area bounded by arcs AB, BC, CP and PA r
th epe ren so erm nta sl  fundamental region of the group, F. 
V, 
w 
~ 
I -
54 
APPENDIX A 
High- frequency Sound Waves To Elimina t e A Hori zon 
In The Mi xmas t e r Universe 
55 
ABSTRACT 
From the linear wave equation for small amplitude sound 
waves in a curved spacetime, there is derived a geodesic-like 
differential equation for sound r ays. to describe the motion of 
wav e p ackets . These equ a tions are applied in the generic, non-
rotating, homogeneous closed model universe (the "Mixmaster 
Universe", Bianchi Type IX). As for light rays [described by 
Doroshkevich and Novikov (DN)], these sound rays can circum-
naviga te the universe n ear the singularity to remove par ticle 
horizons only for a small class of these models and in special 
d irections . Although these results parallel those of DN, different 
Hamiltonian me thods are used for treating the Einstein equations . 
56 
1. Introduction 
The pres e nt day unive rse can be described very well by the Robertson-
Walker cosmological models. The extrapolation of these models for the 
early time s of the universe gives rise to the problem of particle horizons. 1 
A p article horizon at a particular epoch bounds each finite part of the 
universe which could have been spanned by a causal signal during the time 
available since the initial singularity . Since the Robertson-Walker 
mode l s possess p article horizons , only a finite part of such universe 
could have been causally connected. Thus, we are faced with the observation 
of the microwave background radiation having precisely ( ::_ 0. 2%) the same 
temperature 2 in widely different directions even though the regions of 
pl asma which scattered the radiation last h ad no prior causal relationship. 
The Rober ts on- Walker models therefore are too simplified to describe the 
early phase of the unive rse. Here we would consider a more general model 
of the universe - the non-rotating Bianchi Type IX model. It has a very 
different singularity behavior , 3 , 4 but it could evolve into the closed 
Rob e rtson-Walke r model at the presen t epoch . Misner 3 first pointed out 
the possibility of mixing by light in these models. 
Doroshkevich and Novikov 5 quote the results of their investigation of 
the prop agation of light in the Mixmas t e r Universe. Doroshkevich, Lukash 
and Novikov 6 in a recent preprint apply thes e results for findin g the likeli-
hood of horizon vanishing and find it to be very low. Our result s are in 
substantial agreemen t with theirs . In a future paper, we will show how our 
formulation and tre a tment of the problem gives us a natur a l probabalistic 
estima t e for hori zon vanishing. Here we will derive the equations for rays 
57 
of hi gh-fre qu e ncy sound waves in these generic models and study their be-
h avior in a c e rt a in class of solutions to Einstein ' s equations. The 
Hamilt oni a n me thods which we us to obtain information about the relevant 
solutions to Einstein ' s equations are quite different from the ones employed 
by Be linski et c ? or Doroshkenich and Novikov. 5 Also we do not reject the 
application of our calculations to epochs where quantum effects could enter. 
We look forw a rd to calcula tions in which quantum effects might be included 
and would me aningfully modify the interpretation of these small perturbations. 
58 
The metric of the Bianchi type IX for an anisotropic non- rotating 
universe can be written as 
e -2St( e 2 (3 ) .. o.o . , (Al. l) 
1-J ]_ J 
where 
o coswd0 + sin~sinOd? 
2 
and o = -(dW + cos0d?) 
3 
satisfy do i = ~ E: ijk o j /\ ok and are differential forms on the three-sphere 
p a r ameterized by Euler angl es W0? with O 2- w 2_ 41r, 0 2- 0 2- 1f and O 2- ? < 21r . 
Th e quantities St a nd dep end only on time, with St determining 
the vo lume and S . . , a di agonal traceless 3 x 3 matrix: 
1-J 
governing the aniso tropy (sh a pe ). Note tha t for S .. = O, this metric is 
1-J 
one form for the positive curvature Robertson-Walker me tric. As two 
i n d epend ent shape parameter s choose 
and 
-1 
Th e variational principle for Einstein ' s equations 61 = 0 with I= ( l61r) 
f 1 12 R ( - g ) d\~ c a n be c as t into a canonical form to obtain 7 the Hamiltonian :. 
59 
(Al. 2) 
p and p are the momenta conjugate to the field amp litudes B+ 
+ a nd B_ -
r es pectiv ely , with Q as the choice for the independent (coordinate time ) 
v ariable. An equation giving Q as a function of the cosmic time t is 
d t = -{fn ( Al. 3 ) 
The " anis otropy potential" V( B+, 8_ ) arises due to the anisotropy of 
th e curv at u re of the three-dimensional space sections of the unive rse . The 
p otential walls r ise s t eeply away from B = 0 , with the equipotentials 
asympto t ically fo rmin g equila t e r a l tri an g l es in the. B+ B_ plane as 
s h own in Fig . 1. One of the t h ree equivalent sides of the t riangle is 
d escribed b y the asymptotic form 
B -+- - oo 
+ (Al. 4) 
which is valid in the sector IB_I < - If B+? The corners of this 
tr iangula r p otential are flared open; for instance if 13 00 +  -+- with 
Is I < < 1, one finds 
V( B) ~ 168 2 e 48+ + 1. (Al.5 ) 
Th e e v o l u tion of the universe is des cribe d by the motion of the system 
p oint B = ( B+' B_) as a function of t he time coordinate n. When B 
i s we ll a way from the potential wa lls, the unive rse point moves with 
.--
1 
dB dB 
velocity B' = ~~ 2 2 11 2 = { (d
1/) + Cd a -) } of unit magnitude in straight lines 
and it can be par ame terized as 
2 
u +u-1/2) 
2 (Al. 6) 
u +u+l 
d/3 ilJ(u+l/2 
d ~ 2 (Al. 7) 
u +u+l 
wh re the parameter u goes from - 00 to 00 ? The potential walls move 
dB 
outward with velocity (in the sense of w all) o
d nQ e -half. The system 
point 13 would thus move in one direction with unit velocity till it 
\ 
comes close to one of the walls and feels the potential and would then 
b ounce off the wall changing its direction. Furthermore, Belinskii and 
Khalatnikov 4 h ave shown that all solutions would come arbitrarily close 
to the value s u 1 = -2, -1, - 2, 0, 1, 00 after rattling back and forth 
b e tween the walls. These values of u correspond to the system point 
movin g parallel to the three corner axes . 
When the system point is well inside the walls, the potential V 
can be neglected . But V = 0 just gives the Einstein equations R 
?v 
= O for Bianchi type I. One finds, 3 then that these epochs parallel 
Kasne 1 r solutions using a= - j logt + constant as the independent variable; 
th e Kasner metric being given by 
where the exponents p , p and 
1 p2 a3 re connected by the following two 
r elations : 
1. 
61 
Thu s the S point shifts from one Kas ner-like mode l to another at each 
collis ion with a pote nti a l wa ll . For the Kasner solution with p = Pz = 0, 
1 
p = 1, th e r e exi s t no ho
3 ri zons for causal propagation in the z-direction . 9 
Similarly, the r e is a b s ence of horizons in the other two directions for 
Kas ner me tri c s with p 1 = p = 3 0 , Pz = 1 and p2 = p = 3 0 , p = 1 1 
r e sp e ctive ly. This mo tiviates us to study the epochs of Bianchi type IX 
mode l which a pprox ima t e the s e Kasner- solutions for a long period of time. 
Th e s e epochs can b e s een to be the ones when the system point is moving 
p ar a llel to one of the axe s of the equipotential ,triangle and is either 
runn ing towards a corner or following an inclined wall. When the system 
point is runnin g tow a rd s a corner on the S+ -axis, the parameter "u" 
d esi gn a ting the direction of the velocity is asymptotically oo and S+ 
is v e ry l a rge ; IS_I  << 1 giving the universe a pancake shaped anisotropy 
corresponding to a relative compression of the 3-axis ( \/!-axis) with the 
other two axis approxima tely equal. While near the inclined walls, say 
for S + 00 , the 1 anisotropy is cigar shaped with the stretching of the 
1 - axis relative to the others. So we expect the null geodesics in the 
~- d irection to go around the universe during the u = oo epochs. 
In the next section we will derive the equations for the propagation 
of h igh-freque ncy sound waves and in the following sections we will study 
their b e havior during the epochs when u is very large. It. will be seen 
that the re exist a set of initial conditions for which the special Kasner-
like b ehavior persists long enough for these sound waves to go round the 
un.ive rse in the \/!-dire ction . This possibility of communication either by 
soun.d - wa ves or light rays a long a certain direction during the evolution of 
a un ive r se will b e called the removal of horizon in that direction for that 
un iv e r s e . 
62 
2. The Propagation of Hl gh-Freq uency Sound Waves 
Le t E , p and u be the e n ergy de ns ity, press ure and th e four ve locity 
of the fluid, and l et 1E , p ' and~ ? b e the small amp litude , high-fre quency 
perturb ations on the above solution. The propaga tion of the disturb ance is 
gov rned by the energy equa tion : 
(A2 .1) 
and the Euler e quation : 
? . \) 
( p + . E)u ; u 
\) (A2.2) 
Sub stituting p = p + p', E = E + 1 E and u = u + u' in the equations (A2.l) 
and ( A2 . 2) and linearis ing we obtain 
0 (A2 . 3) 
and 
(A2 . 4) 
Di fferentia ting e quations (1<2. .4) with respec t to? and substituting for 
u , ? '. ? from equation ( A,2.3) we ge t, 
1 -?-v 
E '?v u u = F (A2 . 5 ) 
wh e r F i s a scalar function which contain s the high-fre quency per turba-
tions 1E , p ' and~ ? only up to the ir first derivatives . 
Writing p ' = Aei4>, where cp is a rapidly varyin g function and se tting 
the dominant terms in the equation (A2.5) e qua l to zero, we obtain 
63 
( g )l\! + -)1-\) ) ,i, A u u 'I' ' ~) , ( A2 . 6) 
)1 \) 
wh e r e vs = ?(*) s is the s o und ve locity. The eq ua tion ( A2 . 6 ) is a 
Ha m~lton-Jacobi eq ua ti on c orres ponding to 
1 ( pv -?-v ) 1 1 - ?-v 
H = 2 g + u u P?Pv - v 2 2 u u PJJPV (A 2. 7) 
s 
as a p a rti c l e Ha miltoni an. To obtain the corres pondin g Lagr a ngian, we 
s o l v e f o r p fro m one se t of Ha mi l ton ' s equa tions : 
)1 
( g ?v + -? - v ) 1 - ?- v u u p\) - --2 u u p\) (A 2. 8) 
V . 
s 
wh e r e x 11 = ( t , e , cp ,~) . 
a 
No ting tha t ~ = at- _for comovin g coordina t e s , we can i nve r t c_A2 .8) to obtain 
(A 2. 9) 
Thu s , we ge t the Lag rangi an L as follows : 
L - H 
d x 11 = ---d x- v 2 - - J 1 dx
11 dxv 2 
d A d >- [g? v + (1-v )u u - - - - [ g + (1-v )u u ] ( A2.10) s ? v 2 d >- dt- ?v s ? v 
11 
-1 -d x--d-xv [ g + (1-v 2 )u-  -u J 
2 d t- d t- (A 2 .11) )1\! s ].1 \) 
Th e propa go t i on of rays i s the n given by the Lagr an ge ' s e qua ti ons , 
(A2 .12) 
Co n s i de r a pos sib l e s e t of soluti ons wi th 8 = cons t ant and cp con s t ant. 
Th e n the Lagr a n ge ' s eq ua tion s r e duce t o 
(A 2 .13) 
I 
-=-~==============.'..= - ~ , ,. -, ,.., ,., - , -=-===-=="""'-=============== U4 
ag,lnl, d1l1 
_..:t:.L ( ~ ) 2 
ao 0 d>. (A2 .14) 
d 0 (A2.15) 
d >-
d ( dl/1 ) 2 ~ (A2.16) 
d >- d>. ell/I 
Since gl/il/J is a function of t only, eqn. ( A2. 14) is identically satisfied , 
while ( A2 .15 ) and (A2,16) reduce to 
d -2$1 [cas e e 0 (A2 .17) 
d >. 
a nd 
d 0 
d>. 
For 0 = cons tant ,( A2 .17) reduces to ( A2 18). So the Lagrange ' s eqns. now 
r e duce to eqn . (AZ.13) and eqn . ( A2 8) which can b e solved for ~~ and ~f 
Putting H = 0 in the equation (112.10) we obtain 
0 
or 
+ ( d t ) 2 ( 1-v 2 ) 0 (A2.19) 
d>. s 
for e = constant , <P = constant class of solut.ions. From eqn . (A219) we 
obtain 
V 
~ = s V (/12 . 20) 
dt ?g33 -? s 
65 
By p ut ting vs= 1, we ge t the l aw o f prop aga t i on for li ght going in th e 
~! - direc t ion . 
3 . The Re mov a l of Horizons 
Le t us n ow s tudy the b e h avior of the above high-fre que ncy sound waves 
durin g the u = 00 e poch s . Firs t cons ide r the a xial case whe n the system 
point i s v e r y c l ose to the /3 + -axis and is running towards the corner . This 
i s the c ase wh ich Bc lins kii 7 e tc c a ll the case of small oscilla t ions . The 
a p propri a t e s olution to Einstein ' s eq uations as derived i n t h e n ext section 
i s 
/3 ) [ see ( A4. 8) ] 
whe r e z i s a Besse l f unction of order ze ro. K is a constant and 13 is 
O 0 
d efin e d a s 
Th e v ariation of /3 is given b y 
0 
d /3 
0 K [ see (A4.3)] 
d Q H 
= - 2( 13 + Q) in e quati on ( A2 . 20), we r eexpress the 
0 
e quation of the sound-wav e propaga tion in the ~-direction as 
d ~ = /6n 3Q e e 213 0 V 
dt s 
Usin g equ a tion(Al . 3), t h e ch a n ge in ~ ca n b e given i n t ernIB of t h e 
v aria ble Q a s 
d ~ d t 2 = - V (PJ . l ) 
dt dQ s H 
66 
Hence , th e change in 1/J along the sound wave between the e pochs n and n
1 2 
is given by 
n2 n2 
t, 1/J I di? I 2 V e2Bo drl s H 
n l Q2 nl 
= I 2 e 2Bo dQ vs dB H dB 0 
0 
nl 
Sub s tituting e qua tion (A4.3 ) in equation (A32), we obtain 
n2 
2 2Bo 
61/J V s f e dB K 0 (A3,3) 
nl 
"2Bo 
1 
V t, (-e-) 
2 s K (A3 .4) 
2B0 
Therefore , a change of Sn/vs in eK would give a change of 4n in i?. 
So 
Since B Zo  (~ K ) which for small K goes roughly as 
Ze2Bo 
f ne~Bo cos ( K - n/4) or 
2 2Bo 
? sin ( eK - n/4) ne ~ Bo 
13 would go through four cycles as its argument changes by 8n. Thus setting 
v = 1, we see tha t the light ray would circumnavigate the universe in the 
s 
~-direc tion (i. e . 1/J going from Oto 4n) during four cyc l es of S . This 
corres ponds to the DN result of N 1 
e = ~4m.  For radiation filled universes, 
the ve locity v of the sound-wave propagation will be 1/1:f; as a r es ult 
s 
these waves would go round the universe in the 1/J-direction durin g seven 
cycles of B Similarly, when the sys t ern point is running towards the othe r 
67 
two co r n e r s , the causal and the high - fre que ncy sound wave influence would 
circumnavi ga t e in the other two principal directions . 
Ne x t cons ide r the off-axia l case with u very large and B > 1. The 
appr opri a t e s olut i on to Eins tein ' s equa tions as derived in the next 
section again gives 
dB 
0 - K/H 
dD [See ( A4 . ll)] 
whil tot a l c h a n ge in B during one bounce with the inclined potential wall 
0 
for large u is give n by 
66 = 1 
0 2u. [See (A4. l 7)] 
1 
where u. is th e v a lue of u before the bounce . The change in ljJ along the 
1 
hi gh- fre qu ency sound wave ray going in the ljl-direction is again given by 
eq uation (A3. 3 
2v 
= s 6\jJ 
K dS 0 
S o during one collision with the wall, the change in ljJ would be 
V 
s 
K 
(A 3.5) 
68 
wh er the s ubscrip t i denotes the value s of the variables before the 
c oll' sion . The v alue of the constant K can b e obtained in t erms of u . and 
1. 
Hi from equ a t ion (Al. 6): 
2 
u +u- 1 / 2 
2 
u +u+l 
and th e equ ation (A4.ll) 
The n 61/J i s given in terms of the initial values as 
2 
2 u . +u.+l 61/J = V 1. 1. ( e2Bo) . .!__ s 3 Hi 1. u. 1. 
for l arge u. 
1. (A3 . 6) 
As the s ystem point evolves , consider the epoch when the system point 
h a d its first collision with the in c lined wall for large u . So the system 
point h as jus t boun ced b ack off the vertical wall and is going towards 
the inclined wall a t say Q = Qb . The position of the pot ntial wall is 
th n given by 
e - 4Qb 1 e - 8 ( B+) 3 wa 
ll 
= 1 3 e 
- 12Qb e -8( (3 0 ) wa ll 
Su b s titutin g the expression for Hin the equation (A3.6 ) and droppin g 
t h s ubscripts, we get 
69 
(V l c2f3ou) /C-:?l_ e -6rlbe -4 13 0) 
s 3 If 
= l__ V e 6 CS+)wall 
[3 s (A3 . 7) 
The r ef ore , for all solutions for which at the b eginning of the series of 
collisions with the inclined wall, the value of u is such that 
13
u   > 
2V 
s 
then the high-frequency sound wave communication has an open channel in 
the 1/J-direction . Since CS+) wall is negative (it goes as: s+ ~ --n+  
2 
const ant), we find that there exist small sectors around the lines pa rallel 
to the (3+ -axis such that when the system point is running along these 
sectors at~ ' a hor izon is removed in the ~-direction during the next bounce 
w ? th the inclined potential wall. The angular extent of these sectors 
d epends upon n and it goes to zero as n goes to 00 ? 
One concludes, therefore, that at each epoch ' there exist certain 
subs e t s of initial conditions [ i3 +' i3 - ; u( n)], such that some rays of 
high-frequency sound waves and null-ge odesics will proceed to circum-
n a vigate the corresponding universe. It will b e shown in a future 
publ:Lcation that the universe point wanders about in a truly ergodic 
fashi on and that by finding a measure on initial conditions, one can com-
pute the probability for a typica l solution to have no horizon along 
on e axis . 
70 
4. u = 00 SOLUTIONS OF EINSTEIN EQUATIONS 
In th i s s ecti on we will derive the rele vant infonna tion about u .. oo 
50 lutions whi ch we use d in th e las t secti on. Firs t conside r the axial case 
,;.., 11 ? JL t h e "' y s t e:rn point i s ve ry c los e to one of th e corne r a xe s and is running 
c owa rd s the corner . For the con1er on th e B+ -axis, the asymptotic form 
{ th e potcnti.11 i s 
0
2 
V( B) ~ 16B 46e + + l; B+ ? 00 and Is l <<l. 
Tl1 c n th e 1-l anu.ltoni a n of the sys t ern is 
(A4.1) 
To get a time -inde p c ncl ent lfamiltoni an, substitute 
B+ ~ s0 +Qin the action integrand 
-HdQ to give 
s O t h e new JI a mi l tonian is 
(M.2) 
And t l e c orre sponding Har.1ilton I s equations give 
K 
= - . CA4. 3) 
H 
.d (3 clK p 
H 
71 
dp+ - aK 32(L 2 eliBo 
== -- -
d S1 as H (A4, 5) 0 
dp 48
= -1.!L = - 168 e o 
d lt as H (N;.6 ) 
a nd 
d K = -3K- = 0 
d St alt 
Equ a ti o n ( A4. 7) tells us tha t K is a constant while Eqns .( A 4 . 3) , (A4 .4) and 
(A4.6) can be ma nipulate d to give 
dB p 
= 
d (3 0 K 
dp dp d f3 . 166 e 4Bo 
- - = /-=0  
d /30 d S1 dlt K 
Hence I 
i 
I 
4130 I 
- 1 dp_ -1613 e
---= 
K d (3 1<2--
0 
or 
d2f3 16 e4Bo 
( ~ + (~-- ) 8 = 0 which hK' a. s 
0 
th e so l ution 
2 e 2So . 
f3 = z ( - -- ) 0 K (A4. 8) 
wh e re z0 i s a Be sse l function of orde r zero . Note fi?om Eqns. (A4, 1) and (M2) 
th a t K a nd H are s trictly positive . The n from Eqn . (A4.3) s is always de-
0 
cre a s in g , s o Eqn. (A4.8 ) is valid s tartin g from some initial value of s until 
0 
e0 dee r a scs t o the point where the argument of the Bes s e l function gets 
sma ll and S gets l arge contradicting th e ls_l <<l ass umption. 
72 
Nexl con s ide r th e off--axia l case ((3 > 1). Wh e n the sys t em point is 
almost p a rallel to the B+ -axis (large u ) and is following one of the in-
clin e d pote ntial walls , the a s ymptotic form of the pote~tial is 
V ( (3 ) I\, l_ e 4 ( B+ + /JS_) 
3 
Th en th e Hami lt oni an of th e system is 
Sub s t ituting S+ = f 0 + a in the action, we get the time-independent 
Hami l t onia n 
(A4.10) 
The I-L1 mi lton ' s equations give 
d S
~0
P+ 
=  oK --= K = 
dQ. 1 3p+ K++P+ - H (A 4.11) 
d /3 
-- = oK 
p 
-
d 0. ap
- = - H (A4.12) 
d p+ aK 2 e 4( Bo + h sJ 
d7i" - - -- :-: as - (A4.1 3) 0 3 H 
dp - 31( e4( So + /Js_) 
--= --= 
-2 (A'.4 .14) 
d Q 08 71 H 
and 
( A.4.15) 
Fr om Eq~s . ( AL.13 ) and (A414) we ge t 
dp 
dQ = 0 
73 
or 
p = cons tant = a, say. (A4.16) 
5._ib s titutine for P+ nnd p i .n Eq. (f.4.16) from Eqs(A 4.11) and(AL1 .12) we 
obtai n 
1'.lso fr o1:i Eq . (A4. 15) K = 11 - P+ = H(l ) i s a constan t. These two 
cons urn t s of mot ion enable us to find B~  , 13 
1 after the bounce in ter ms of 
th e ir v a lues be.fore . Le t ui and uf b e the values,of the p a rameter u, 
ch a r acterising th e v e lociti es of the sys t em point well b e fore and we ll after 
the b ounce . The n the constancy of K = H(l - 8~) and /3 P+ - p = H(/3 B~ - /3~) 
gives r espectively. 
2  
u. + 1 
1
u. - -2 U 2 + u 
l l ) = H (1 - f f 2 ) 
f 
, . 
and 
u~ - 1 u2 - 1 
]_ 
 f
 
H. ( ) = Hf( ) 
]_ 
u ~ -+ u. + 1 u2 1 
1 ]_ f 
+ uf + 
He nc e. , 
H. 2 
1. u + u + 1 i i 
and uf = - ui ; where n a nd Hf are the values of H before and afte r the 1 
b ounc e , r espec t ive l y . 
Du r in g the col l ision with the wa ll, 
i/JB ) 
1-l.2 = 2. P+ + p~ + e 
-4 rn-  e 4CS+ + 
3 
cJB+  dB- -411 1 L/.' + 13 B ) 
= ti2 ( cl,1- ) 2 + Ji 2 ( dll . ) 2 + e C ~'+ 3 
74 
dll 
S o th e c.q u.J Li 11 d rl aclnH  gives 
dH 
H d S1 
df\ 
Usin g thi s resul t ;m<l solvin~ for dQ ' dn in t erms o f 11 , K and o: , 
on e obL a ins 
o: ] 2- _l H dH 
H 2 dQ 
or 
b e n c 
dB dB 
0 _ o_ dl-l K / dH 
d lL dS? I d Q H d l2 
A loHe.  r . l imit en the ch ange i n 13 0 during th e c ollis i on can b e compute d as 
dS d2
Th e mi
1
n 3 imum val u 0 e of 
dH i s at tha t value of H, whe r e dH 'Jo_ va nis hes ; 
Th e r e f 13 ) 2o  r Kt  e , i ( 
0 
Bu t 
( H . - Hf ) = I K [(u~3   + u. + 1) - (u 2 ] _ l. l. f + uf + 1) J 
4 
= -K u. ; s ince uf = -'u . 3 ]._ l. 
ll ?nee , 
1 
Ii ( !3 ) 
0 2 u . ( A4 . 17) 
l. 
75 
REFE RENCES TO APPENDIX A 
1. W. Rindle r, Monthly Notice s Roy. As tron. Soc . 116 , 6 (195 6). 
2 . P a rtrid ge and Wilki nson , Phys . Re v. Le tte r s , 18, 557 (196 7). 
3 . C. W. Misner , Phy s . Re v . Le tte rs , ~. 1071 ( 1969 ). 
4. . Kh a l a t nikov an d Be lin s ki , Sovie t Physics , J ETP , ~. 911 ( 1969). 
5 . Do ros hk.ev i ch and Novik.av, Astron . Zh . !i!_, 94 8 (1970). 
6 . Dorosh k.evi ch , Luk.as h and Novik.av , Pre print (I1oscow)(1 9 71) . 
7. Be lins ki, Kh a l a tnikov and Lifshitz , Adv ances in Phys . 19, 5 25 ( 19 70). 
8 . C. W. Mis ner , Ph y s . Re v . 186 , 1319 ( 1969 ). 
9 . S . W. Hawking , Ad a ms Prize Esaay , Camb ridge Univ . (1966) . 
76 
APPENDIX B 
1. The Geo desic Equ a ti on s 
As covar i ant ? b c1"is vec t o r r; , we t ak e the orthonormal? tc trad: 
w0  c dt 
1 1 -n e 8w = - e 1 a1 
t'61r 
2 - 1 -e- n 
{iH. l) 
w = e 82 a2 
v'61r 
3 - 1 - e -n w = e83 a3 
Y'6TT 
The s e t o f b as i s vec tors du a l to th
e w? is 
+ 
e 
0 at 
= e -Bl[sin''' ~ - C~ S ~! a + ,,, a J ~ ae. sine ~ cos'/' cote ~ 
a sini/J a 
= n -s [ \~ rr  c e 2 cos~ae + -si.n-  e sinlji cote ~ J . 
Yr-6;1:r- e n e-S3 ::: 
The conn e ction for ms w? as determined uniquely by 0 = dg = w w \) ?v ?v+  v? 
- ul \) and de}' = A w are \) 
0 1 . f 1 
w ::: 1 w 0 
= ( -n + 13 )w 1
0 2 (-0. 2 w  ::: 2 w 0 ?- + 62 ) (l) 
'(Bl.3 ) 
0 ?3 . . 3 
w 3 = w 0 = ( -n + (3 3) w 
I :i 1. u ::i n n ::i , , , --. ~ ? ? - ? 78 
and 
]_ 
w - 2 = -/6-; 1\ 28 3) 
3 
28 1 + .e2B2 - e w 
2 == w 1 C C 2
2 3 & eS\e 28 2 283 - e 281) 
1 (BJ,. 4) 
w == - w 2 = + e 
w 
3 2 
3 1 f6; n( 2 13 3 2 !3 l - e 28 2) 
2 
w = - w 3 = C e + e w
 
l 2 
+ + d 
 v = v?c? = J5: 
? d eno t es th e d i f f eren t i ation. wi t h 
r espect ? to t. If
is a t a n gen t vec t or t o a geodes ic p a r ame tr
i ze d by A, then 
-+ 
D+ v = 0 
V 
or 
(Bl.5) 
conne ction forms , i.e. 
\.Jhere rl.1 are th e components of the \) p 
.,?? 
. . ? dv? o dv? 
riting -d~ = v dt 
Computing the va l ues of r? ?. :_ .?? fro m (Bl. .3) and (Bl. 
4) and w
VP . 
= o?v ? v , th e geodes ic e qu a tions (Bl. 5) reduce to 
 
O?O 1 2 ? 2 2 ? ? 
3 2 ? ?
+ (V ) (-Wf\) + (v ) (-11-1-(3 ) + (V ) ( -n+8 ) = Q
 
V V 2 3
0 ? 1 ..,_ 0 1 ( . ? )  2 v 3 
r,---6 -0( 282 
v v -, v v - n-1-6 1 + v YbTI e e 
. (Bl. 6)" 
0?2 0 2 ? ? 3 1 /6; -0 28 3 e281) 
V V + V V (-S?l+{) ) + V V 6 e (e - = 
0 
2 
an<l 0 ?3 0 3 ? ' 1 2  ~ e -s'c e 281 -e 28 2) V V + V V ( - S?l+/3 ) + V V = 0 3
79 
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