General asymptotic solutions of the Einstein equations and phase
  transitions in quantum gravity
General asymptoti
 solutions
of the Einstein equations and
phase transitions in quantum gravity
Dmitry Podolsky
Helsinki Institute of Physi
s, University of Helsinki,
Gustaf Hällströmin katu 2, FIN00014, Helsinki, Finland
Email: dmitry.podolsky�helsinki.�
Abstra
t
We dis
uss generi
 properties of 
lassi
al and quantum theories of grav-
ity with a s
alar �eld whi
h are revealed at the vi
inity of the 
osmolog-
i
al singularity. When the potential of the s
alar �eld is exponential and
unbounded from below, the general solution of the Einstein equations
has quasi-isotropi
 asymptoti
s near the singularity instead of the usual
anisotropi
 Belinskii - Khalatnikov - Lifshitz (BKL) asymptoti
s. De-
pending on the strength of s
alar �eld potential, there exist two phases
of quantum gravity with s
alar �eld: one with essentially anisotropi
 be-
havior of �eld 
orrelation fun
tions near the 
osmologi
al singularity, and
another with quasi-isotropi
 behavior. The �phase transition� between the
two phases is interpreted as the 
ondensation of gravitons.
On leave from Landau Institute for Theoreti
al Physi
s, 119940, Mos
ow, Russia.
http://arxiv.org/abs/0704.0354v2
One pessimisti
 quotation from the golden era of �nding exa
t solutions of
the Einstein equations whi
h re�e
ted the relations between parti
le theorists
and experts in GR belongs to Ri
hard Feynman. Taking part in the Interna-
tional Conferen
e on Relativisti
 Theories of Gravitation at Warsaw, he was
writing to his wife [1℄: �I am not getting anything out of the meeting. I am
learning nothing. ... I get into arguments outside the formal sessions (say,
at laun
h) whenever anyone asks me a question or starts to tell me about his
�work�. The �work� is always: (1) 
ompletely un-understandable, (2) vague and
inde�nite, (3) something 
orre
t that is obvious and self-evident but worked out
by a long and di�
ult analysis, and presented as an important dis
overy, or (4)
a 
laim based on the stupidity of the author that some obvious and 
orre
t fa
t,
a

epted and 
he
ked for years, is in fa
t false ... (5) an attempt to do something
probably impossible but 
ertainly of no utility whi
h, it is �nally revealed in the
end, fails ... or (6) just plan wrong ... Remind me not to 
ome to any more
gravity 
onferen
es!� Certainly, I am well aware of that the work presented in
this essay 
ould belong to the 
lass (3) or (5) in the Feynman's 
lassi�
ation
(hopefully, not to the 
lass (6)!), but I will follow Feynman's own words [1℄:
�We all do it for the fun of it� trying to �nd my fun in identifying some links
whi
h 
onne
t the part of the 
ommon lore on general relativity named �Exa
t
solutions of the Einstein equations� to the problem of the GR quantization.
Of 
ourse, Feynman's interest was in the quantization of GR by applying the
path integral approa
h working so well in QED. Solutions of the Einstein equa-
tions de�ne saddle points of the a
tion
2 S = Sgravity + Smatter of the quantum
gravity with matter. However, the 
ontributions of these saddle points into the
partition fun
tion of the theory and �u
tuations near them
Dφmatter exp
(Sgravity + Smatter)
typi
ally have zero measure. In other words, the probability for an almost any
exa
t solution to des
ribe the observable features of the Universe or some parts
of it, to appear somehow from the quantum foam realized near the singularity
is in�nitely small, and the Feynman's anger is absolutely understandable.
Well, almost absolutely... Of 
ourse, there are several 
lasses of solutions
whi
h will be important for the quantum part of the story, too, and one 
an
without mu
h thinking immediately identify some:
1. Attra
tors : among them are Minkowski spa
etime, de Sitter (at least in
the sense of eternal in�ation [2℄) and anti de Sitter spa
etimes (a set of
AdS domains is mostly probably the global attra
tor of GR realized as
low-energy approximation of string theory [3℄); bla
k holes (S
hwarzs
hild,
Kerr, Reissner-Nordström, Kerr-Newman solutions), et
.
From now on, by the quantum theory of gravity we mean e�e
tive QFT of spin 2 �elds
[4℄ (plus matter �elds) � the one whi
h parti
les with energies E ≪ MP test. In this limit,
the e�e
ts of the non-renormalizability may be negle
ted. Although we dis
uss below the
situation whi
h is realized near the 
osmologi
al singularity, we limit the dis
ussion to time
s
ales t ≫ tP .
2. General solutions of the Einstein equations. As usual [5℄, a solution of the
Einstein equations is regarded as general if it 
ontains su�
ient number
of arbitrary fun
tions of 
oordinates. In the 
ase of Ri

i-�at spa
etimes,
this number is 4, and is equal to 8 in the presen
e of hydrodynami
 matter.
While any non-attra
tor type solution of the Einstein equations de�nes the
saddle point for the path integral (1) whi
h does have a vanishing 
ontribution
into the overall partition fun
tion, eventually it well settle down towards an
attra
tor solution due to the e�e
t of 
lassi
al perturbations and/or quantum
�u
tuations. The 
ontribution of attra
tor type saddle points into the partition
fun
tion (1) is therefore signi�
ant. However, the key word here is �eventually�.
For any non-attra
tor solution it takes a time tcoll before the solution rea
hes
its attra
tor asymptoti
s.
Let us 
onstru
t some initial state |Ψ(t = ti)〉 of quantum matter �elds in
a 
urved spa
etime and gravitons. The amplitude 〈Ψ(tf )|Ψ(ti)〉 is then de�ned
by the path integral (1) 
al
ulated on the 
losed S
hwinger-Keldysh 
ontour
from t = ti to t = tf and ba
k. Then, if tf ≪ tcoll, the 
orresponding attra
tor
saddle point does not give any noti
eable 
ontribution into the amplitude.
it is ne
essary to know the evolution of the quantum state |Ψ(t)〉 at time s
ales
t ≪ tcoll, we are for
ed to pay mu
h more attention to the type of saddle

orresponding to general solutions of the Einstein equations.
Certainly, the Einstein equations are hard to solve, and it is possible to �nd
something like their general solution only in physi
ally simpli�ed situations.
As was �rst shown by Belinskii, Khalatnikov and Lifshitz [6℄, asymptoti
ally,
the general solutions of the Einstein equations near the 
osmologi
al singularity
have the very same form for an almost arbitrary 
hoi
e of the matter 
ontent.
This asymptoti
s in the syn
hronous frame
is given by Kasner-like solution
ds2 = dt2 − γαβ(t,x)dxαdxβ , (2)
γαβ(t, x) = t
lαlβ + t
mαmβ + t
nαnβ . (3)
Both Kasner exponents p1, p2, p3 and Kasner axis ve
tors lα, mα and nα are
arbitrary fun
tions of spa
e 
oordinates. The Einstein equations provide two

onstraints on the Kasner exponents
p1 + p2 + p3 = 1, (4)
p21 + p
2 + p
3 = 1, (5)
as well as three other 
onstraints on arbitrary fun
tions of spa
e 
oordinates
present in (3). Taking into a

ount that the 
hoi
e of syn
hronous gauge
g00 = 1, g0α = 0 (6)
Of 
ourse, the time s
ale tcoll itself is a fun
tional of the initial state |Ψ(t = ti)〉.
Often, it is impossible to 
hoose the globally syn
hronous frame of referen
e due to the
limitations set by the 
asuality. However, everywhere in the text we dis
uss the physi
s in a
given 
asual pat
h.
leaves the freedom to make three-dimensional spa
e 
oordinate transformations,
one 
an easily see that the total number of arbitrary 
oordinate fun
tions in the
Kasner-like solution (2),(3) is equal to 4 as it should be expe
ted for a general
solution of Einstein equations 
orresponding to an empty spa
etime.
In the presen
e of the hydrodynami
 matter Kasner solution (2),(3) de-
s
ribes asymptoti
 behavior of metri
s near the singularity,
sin
e 
omponents
of energy-momentum tensor Tik grow slower at t → 0 then the 
omponents of
the Ri

i tensor.
Higher order 
orre
tions to the Kasner solution (2),(3), i.e.,
higher order terms in the expansion of γαβ(t, x) over powers of t play the role
of perturbations whi
h give rise to the time dependen
e of Kasner exponents pi
as well as Kasner axis ve
tors lα, mα and nα and to well-known BKL 
haoti
behavior. Therefore, the BKL solution is simultaneously a universal attra
-
tor for all solutions of the Einstein equations possessing a spa
elike singularity.
It means that no other saddle points 
ontribute into the amplitude (1) in the
vi
inity of the 
osmologi
al singularity.
In this essay, it will be �rst of all shown that in the presen
e of a s
alar �eld
with potential V (φ) whi
h is exponential and unbounded from below, the general
asymptoti
 solution of the Einstein equations is di�erent from the BKL solution
and is quasi-isotropi
 [8℄ (while the BKL solution is essentially anisotropi
). In
parti
ular, we will 
hoose potential of the form
V (φ) = −|V0|ch (λφ) . (7)
S
alar �eld potentials of this form appear in problems related to gauged super-
gravity models [10℄ and the ekpyroti
 s
enario [11℄. The 
osmologi
al singularity
realized in su
h theory is of the Anti de Sitter Big Crun
h type. The physi
s
in its vi
inity it is interesting by itself and even more so sin
e this type of
singularity seems to be realized quite often on the string theory lands
ape [3℄.
As in the 
ase dis
ussed in [6℄, it is 
onvenient to perform all 
al
ulations
in the syn
hronous frame of referen
e where g00 = 1, g0α = 0, gαβ = −γαβ ,
α, β = 1 . . . 3, i.e., the spa
etime interval has the form
ds2 = dt2 − γαβ(t, x)dxαdxβ . (8)
Near the hypersurfa
e t = 0 whi
h 
orresponds to the singularity, the spatial
metri
 
omponents behave as
γαβ(t,x) = aαβ(x)t
2q + cαβ(x)t
d + bαβ(x)t
(i,j)
(x)tfij . (9)
With the same pre
ision, one has in the vi
inity of singularity
φ(t,x) = ψ(x) + φ0(x)log(t) + φ1(x)t
f1 + φ2(x)t
f2 + · · · , (10)
Whi
h 
orresponds everywhere below to the spa
elike hypersurfa
e t = 0.
If there is a s
alar �eld in the matter 
ontent [7℄, BKL solution (3) remains general solution
of the Einstein equations with 
hanged Kasner 
onstraints (4),(5).
The quasi-isotropi
 solution for su
h potentials was �rst found at the ba
kground level in
[9℄, where it was also shown that it is the attra
tor. The goal we pursue in this essay is to
prove that the quasi-isotropi
 solution is also general and to understand how its instability
develops with the 
hange of the form of the potential.
with dots 
orresponding to higher order terms of φ(t,x) expansion in powers
of t. From the Einstein equations one �nds8 that the leading exponents in the
expansions (9) and (10) are de�ned by the expressions
, n = 2, d = 1− q, (11)
ψ(x) = Const, φ0(x) =
, f1 = 1− 3q, f2 = 2− q, (12)
cαα(x) = 2λφ1(x), c
α;β(x) =
1− 2q
1− 3q
φ0φ1,α(x), (13)
P̃ βα (x) + (1 − q)(qbγγ(x)δβα + (1 + q)bβα(x)) =
e−ψλφ2(x), (14)
− (1− q)bαα(x) =
(1− q)φ0φ2(x)−
−ψλφ2(x), (15)
where P̃ βα (x) is the 3-dimensional Ri

i tensor 
onstru
ted from 
omponents of
the tensor aβα(x) as 
omponents of metri
 tensor. Higher order terms in the
expansions (9) and (10) 
an be self
onsistently 
al
ulated by using the Einstein
equations and the orthogonality 
ondition
β = δ
α. (16)
One 
an immediately �nd from Eq. (16) that the higher order exponents in the
metri
 (9) are de�ned by
fij = i+ 2j − (3i+ 2j − 2)q, (17)
where i, j ∈ N. The n term in the metri
 expansion 
orresponds to i = 0, j = 1
and d term � to i = 1, j = 0. It is easy to see that there is no other exponents
in the expansion (9).
Let us examine the formulae (11)-(15) more 
losely and 
al
ulate the number
of arbitrary fun
tions present in this solution. First of all, one 
an immediately
see that the tensor aβα(x) is not 
onstrained by the Einstein equations. It has 6

omponents, and 3 of them 
an be made to be equal to 0 by a three-dimensional

oordinate transformation (the remnant gauge freedom of the syn
hronous gauge
(6)). Sin
e this tensor is used for lowering and rising the indi
es and represents
the leading term in the expansion (9), we will identify the term aαβt
ba
kground 
ontribution to γαβ(t, x). Furthermore, we see from Eqs. (14),(15)
that bαβ 
an be re
onstru
ted from the known tensor aαβ .
The tensor cαβ 
ontains three more arbitrary fun
tions of 
oordinates. In-
deed, it 
an be represented in the form
cβα(x) =
α + Y
;α + Y
Y γγ δ
α + c
(TT)β
α . (18)
Due to the limitations of spa
e we are unable to present the full derivation of the solution
here. It will be given in the forth
oming publi
ation [12℄.
The indi
es of all matri
es are lowered and raised by the tensor aαβ , for example, b
From Eq. (13) one 
an see that its tra
e part de�nes the value of φ2(x) 
on-
tributing to Eq. (10) and therefore provides one arbitrary fun
tion. Then, three

omponents of the ve
tor 
ontribution Yα(x) are �xed, and transverse tra
eless
part c
(TT)β
α (x) provides remaining two arbitrary fun
tions. We also note that
the cαβ term 
an be regarded as the leading term perturbation to the ba
kground

ontribution into γαβ . In parti
ular, it 
ontains the 
ontribution of s
alar per-
turbations (related to the tra
e of the tensor cβα) and tensor perturbations or
gravitons (related to the transverse tra
eless part of the tensor cβα).
The total number of arbitrary fun
tions in the solution (9),(10) is therefore
6, as one may expe
t for the general solution of the Einstein equations with a
s
alar �eld. By analysis similar to [6℄, one may show [9, 12℄ that the 
ontri-
butions of other matter �elds into the overall energy-momentum tensor grow
slower at t → 0 than the 
ontribution of the s
alar �eld. We 
on
lude that
the solution (9), (10) is the general asymptoti
 solution of Einstein equations
(with arbitrary matter 
ontent) near the 
osmologi
al singularity. Similarly to
the BKL solution, the quasi-isotropi
 solution is the universal attra
tor for all
solutions of the Einstein equations with s
alar �eld having the potential (7)
and arbitrary additional matter 
ontent whi
h possess the time-like singularity.
Again, under 
onsidered 
onditions, no other saddle points 
ontribute into the
amplitude (1) in the vi
inity of the Big Crun
h singularity.
It is instru
tive to understand how exa
tly the transition from the quasi-
isotropi
 regime (9),(10) near the singularity to the BKL anisotropi
 regime
(3) happens. This transition 
an be a
hieved by 
hanging the value of λwhile
keeping V0 �xed (or vise versa).
By 
onstru
tion, 2q < d = 1− q, i.e., the exponent aαβt2q in the expansion
(9) is leading. With the in
rease of q, the value of d de
reases and when q rea
hes
the 
riti
al value qc = 1/3, the 
ontributions aαβ(x)t
and cαβ(x)t
into the
expansion of the metri
 (9) be
ome of the same order. Similarly, one 
an 
he
k
that the values of higher order exponents (17) de
rease with the in
rease of q.
In parti
ular, all exponents with di�erent i's and similar j's be
ome of the same
order of magnitude at qc = 1/3. At q > qc = 1/3 the general asymptoti
 solution
of the Einstein equations near the singularity is given by Eq. (3) instead of Eq.
In fa
t, what we have just found is relevant for the quantum part of the
story, too, and in a sense is analogous to the spontaneous symmetry breaking
phenomenon in QFTs. Indeed, let us take the theory with a s
alar �eld
(Φ2 − v)2, (19)
set Φ(x, 0) = 0 as an initial 
ondition and 
ontinuously 
hange the value of
the parameter v. At v > 0 the solution Φ(t, x) = 0 of the 
lassi
al equations
of motion is perturbatively stable and 
orresponds to the true va
uum of the
theory at the quantum level. At ν < 0 the same solution be
omes 
lassi
ally
unstable, and Φ(t, x) rea
hes the �true� va
uum value Φ = ±
v during the time
t ∼ 1
log 1
(with the VEV of the operator Φ̂ having similar behavior at the
quantum level). Similar situation is realized in our 
ase.
At q < qc = 1/3 the quasi-isotropi
 solution (9),(10) is the general solution of
the Einstein equations; it is perturbatively stable by 
onstru
tion (without any
limitations on the weakness of the perturbations). At q > qc the quasi-isotropi
solution be
omes perturbatively unstable (perturbations de�ned by cαβ and
higher order terms grow faster than the ba
kground term aαβ at t→ 0).
Vise versa, at q > qc = 1/3 the BKL anisotropi
 solution if the Einstein
equations is general in the vi
inity of the 
osmologi
al singularity. It is stable
by 
onstru
tion with respe
t to arbitrary perturbations and the stability is lost
at q < qc.
This analysis remains valid for the quantum situation
sin
e the 
anoni-

al phase spa
e is in one-to-one 
orresponden
e with the spa
e of solutions of

lassi
al �eld equations [13℄, and both quasi-isotropi
 and BKL solutions are (a)
general and (b) universal attra
tors for other solutions of the Einstein equations
in the vi
inity of the time-like singularity.
The transition from the regime realized at q < 1/3 to the regime q > 1/3
probably 
orresponds in the quantum level to the 
ondensation of gravitational
perturbations. Indeed, one 
an interpret the higher order 
ontributions in the
expansion (9) as terms 
orresponding to the intera
tion between gravitational
degrees of freedom as well as higher order nonlinearities in the ba
kground. Our

on
lusion is based on the fa
t that at q = qc the spe
trum of the exponents
in the expansion (9) be
omes in�nitely dense. It is also possible to show that
the point of the �phase transition� qc = 1/3 
orresponds at the 
lassi
al level
to the situation when the 
hoi
e of globally syn
hronous frame of referen
e is
impossible near the singularity [12℄.
Let us summarize what have been found in the present essay. We have shown
that in the presen
e of the s
alar �eld with exponential potential unbounded
from below, the general asymptoti
 solution of the Einstein equations near
the 
osmologi
al singularity has quasi-isotropi
 behavior instead of anisotropi
found by [6℄. We have argued that at the quantum level there should exist a
phase transition between the quasi-isotropi
 and anisotropi
 phases, governed
by the strength of the s
alar �eld potential and interpreted this phase transition
as the 
ondensation of gravitational perturbations.
A
knowledgements
I am thankful to A.A. Starobinsky and D. Wesley for the dis
ussions and to
K. Enqvist for making helpful 
omments. While 
ondu
ting this work, I was
supported by Marie Curie Resear
h training network HPRN-CT-2006-035863.
One important 
omment regarding the quantization should be made. The quantum theory
of the s
alar �eld with the potential (7) is ta
hyoni
ally unstable and has neither well-de�ned
asymptoti
 |out〉 states, nor 〈out|in〉 S-matrix. However, the S
hwinger-Keldysh 〈in|in〉 S-
matrix is de�ned, and it is possible to make sense of the 
orresponding time-dependent theory
[12℄.
Referen
es
[1℄ R. Feynman, as told R. Leighton, �What do you 
are what other people
think?�, W.W. Norton, New-York, 1988.
[2℄ A. Linde, �Parti
le physi
s and in�ationary 
osmology�, Harwood, Switzer-
land, 1990 [hep-th/0503203℄; A. Linde, Phys. Lett. 175B, 395 (1983).
[3℄ A. Ceresole, G. Dall'Agata, A. Giryavets, R. Kallosh and A. Linde, Phys.
Rev. D 74 (2006) 086010 [hep-th/0605086℄; A. Linde, JCAP 0701 (2007)
022 [hep-th/0611043℄; T. Clifton, A. Linde, N. Sivanandam, JHEP 0702
(2007) 024 [hep-th/0701083℄.
[4℄ C. Burgess, in �Towards quantum gravity�, ed. D. Oriti, Cambridge Uni-
versity Press, 2006 [gr-q
/0606108℄; C. Burgess, hep-th/0701053.
[5℄ L.D. Landau and E.M. Lifshitz, �The 
lassi
al theory of �elds�, Pergamon
Press, 1979.
[6℄ V.A. Belinskii, E.M. Lifshitz and I.M. Khalatnikov, Adv. Phys. 19, 525
(1970); V.A. Belinskii, E.M. Lifshitz and I.M. Khalatnikov, Adv. Phys.
31, 639 (1982); B. Berger, D. Gar�nkle, J. Isenberg, V. Mon
rief, and M.
Weaver, Mod. Phys. Lett. A 13, 1565 (1998).
[7℄ V.A. Belinskii and I.M. Khalatnikov, Sov.Phys. JETP 36, 591 (1973);
L. Andersson, A. Rendall, Commun. Math. Phys. 218, 479 (2001)
[gr-q
/0001047℄.
[8℄ E.M. Lifshitz, I.M. Khalatnikov, ZhETF 39, 149 (1960) (in russian);
E.M. Lifshitz, I.M. Khalatnikov, Sov. Phys. Uspekhi 6, 495 (1964).
[9℄ J.K. Eri
kson, D. Wesley, P. Steinhardt, N. Turok, Phys. Rev. D 69 (2004)
063514 [hep-th/0312009℄.
[10℄ C.M. Hull, Class. Quant. Grav. 2, 343 (1985); R. Kallosh, A.
Linde, S. Prokushkin, M. Shmakova, Phys. Rev. D 65, 105016 (2002)
[hep-th/0110089℄; R. Kallosh, A. Linde, S. Prokushkin, M. Shmakova,
Phys. Rev. D 66, 123503 (2002) [hep-th/0208156℄.
[11℄ J. Khoury, B.A. Ovrut, P.J. Steinhardt, N. Turok, Phys. Rev. D 64, 123522
(2001) [hep-th/0103239℄; J. Khoury, B.A. Ovrut, N. Seiberg, P.J. Stein-
hardt, N. Turok, Phys. Rev. D 65, 086007 (2002) [hep-th/0108187℄; E.
Bu
hbinder, J. Khoury, B. Ovrut, hep-th/0702154.
[12℄ D. Podolsky, A. Starobinsky, in preparation.
[13℄ Č. Crn
ović and E. Witten, in Three hundred year of gravitation, eds. S.W.
Hawking and W. Israel, Cambridge University Press, 1987, p. 676; G.J.
Zu
kerman, in Mathemati
al aspe
ts of string theory, San-Diego 1986, Ed.
S.-T. Tau, Worlds S
ienti�
,1987, p.259.
http://arxiv.org/abs/hep-th/0503203
http://arxiv.org/abs/hep-th/0605086
http://arxiv.org/abs/hep-th/0611043
http://arxiv.org/abs/hep-th/0701083
http://arxiv.org/abs/gr-qc/0606108
http://arxiv.org/abs/hep-th/0701053
http://arxiv.org/abs/gr-qc/0001047
http://arxiv.org/abs/hep-th/0312009
http://arxiv.org/abs/hep-th/0110089
http://arxiv.org/abs/hep-th/0208156
http://arxiv.org/abs/hep-th/0103239
http://arxiv.org/abs/hep-th/0108187
http://arxiv.org/abs/hep-th/0702154
