CC BY
27
UDC 530.1; 539.1
On some feature and application of the Faddeev formulation of gravity
V. M. Khatsymovsky
Budker institute of Nuclear Physics of Siberian Branch Russian Academy of Sciences, Novosibirsk, 630090, Russia.
E-mail: khatsym@gmail.com
In the Faddeev formulation of gravity, the metric is regarded as composite field, bilinear ofd = 10 4-vector fields. A unique feature is that this formulation admits the discontinuous fields. On the discrete level, when spacetime is decomposed into the elementary 4-simplices, this means that the 4-simplices may not coincide on their common faces, that is, be independent.
We apply this to the particular problem of quantization of the surface regarded as that composed of virtually independent elementary pieces (2-simplices). We find the area spectrum being proportional to the Barbero-Immirzi parameter 7 in the Faddeev gravity and described as a sum of spectra of separate areas.
According to the known in the literature approach, we find that 7 exists ensuring Bekenstein-Hawking relation for the statistical black hole entropy for arbitrary d, in particular, 7 = 0.39... fa genuine d = 10.
Keywords: Faddeev gravity, piecewise flat spacetime, connection, area spectrum.
1 Introduction
Here we briefly describe Faddeev formulation of gravity [1]. Let the metric be composed of d =10 4-vector fields,
J Ld4x = J gXvg»pSx»,vp^gd4x
= j nAB (fXAfB» - fXAJ»B}X)^~gd4x, (6)
gX» f\ f»A.
(1)
The Greek indices ...= 1, 2, 3, 4, Latin capitals A, B, ...= 1, ..., d. Simple example: locally, 4D Rie-mannian space can be considered as a hyper-surface in the 10D Euclidean space. If f A(x) were its coordinates, then we would have
fA
dfA¡dxx
nAB = Sab — fAf\B is projector onto the directions orthogonal to the four 10-vectors fA (the "vertical" directions). Varying w. r. t. fA and projecting onto the vertical directions we find
b»
»AT[Xv] + b»^AT[v»] + b»vAT[
»
nAB fXB,
(2)
This gives
But now fA(x) are a priori independent variables.
Now, in addition to the Christoffel connection, we can write an alternative affine connection
rp\
1 [ »v]
0
[»A]
(7)
(8)
i\,»v = fxf»A,v, = gXp&
ùP,»v^
(3)
almost everywhere. Then is the Einstein’s one.
nx
»V
rx
X »V>
and the action
with torsion
2 The piecewise constant fields
TX, [»v] = Q\,»v — &\,v», T[»v] = gXPTP, [»v] and curvature, whose final expression is simple
Sx =ir -ir +ir ir
»,vp »p,V »V,p 1 GV »p
-Qx
G p »V
n
(4)
AB
•(fA,v f»B,p — fA,pf»B,v ), nAB = Sab — fAfXB. (5)
The action is
The action does not contain any of the squared derivatives (fAM)2 and therefore it is finite on the discontinuous fA and g\M. Consider a piecewise fiat (composed of the flat 4-simplices) manifold. It can approximate (in the sense of curvature averaged over regions) general smooth manifold. Allowing discontinuities of the (induced on the 3D faces) metric g\M means that the 4-simplices may not coincide on their common faces, that is, be independent. Also we can approximate general smooth manifold by independent
0
A
»
hypercubes (were these hypercubes coinciding on their 3D faces, the metric would be strongly restricted).
To evaluate the action for this system, we consider [2] the neighborhood of a triangle a2 where the 4-simplices a4,..., a4,..., a^ and their 3D faces a3 = ai n u4+1 (an = an n a4) are meeting, and fA are independent in these 4-simplices.
In h Cd4x = h (fX\fB,„ — fA,fB,x)nAB Vgd4xthe Eq- (fA,X fB v — fA,^fB a) is K ^-function with support on a2 and the Eq. nAB Jg is a: Heaviside 0-function in the neighborhood of a2 with discontinuities on a3, i = 1, ..., n. The product of S mid ^-functions is not well-defined. However, the action is unambiguous in the leading order over variation of fA from 4-simplex to 4-simplex. Vice versa, the action obtained correctly reproduces the continuum one in the continuum limit, when this variation of fA tends to zero.
for the Cartan-Weyl form of the Einstein general relativity, provides nonzero contribution to the Faddeev action (in contrast to the case of Einstein gravity).
4 Discrete connection representation for the Faddeev action
To get this representation, we take 1) discrete connection representation for the Einstein action (that is, for Regge calculus), now with SO(IO) connection, +
2) discrete form of the local gauge violating condition
a2
va2AB UAB
2A(a2)
RAB (П)
(13)
3 Connection representation of the Faddeev action
Write down Cartan-Weyl action for SO(IO),
S = J fïfBRAB Vgd4x
R
AB
x^
dx^AB — d^u>AB + {шхшц — w^wx)
AB
AB
wAB fXfB =
we get the Faddeev action.
We can generalize the above connection representation by adding the 1 /7-term to the action so that
S =
S
П
AB
(fA,xfB,^- fA,nfB,x)Vs
1
- - eX^PfxA,f»B,p
d4x.
(12)
Using vertical components of the Eqs. of motion we get the Einstein action. It is remarkable that parameter 7, the analog of the Barbero-Immirzi parameter
Here va2AB is (dual) bi-vector of the triangle a2, A(a2) is the module of va2AB (the area), (Q) is holon-omy of the discrete connection, the product of SO(IO)-matrices QAb over tetrahedra a3 sharing a2. Excluding Q via Eqs. of motion we get Einstein (Regge) action [2].
The form of 2) is straightforward,
(9)
Excluding wAB via Eqs. of motion yields the Einstein action [3]. Performing this operation with the following condition fulfilled,
tta3AB
0.
(14)
(10)
// ,X^vp \
[fA fB + 2YjgfvAfpBj rAB (w)Jgd4x
+j ^AB fA f Jgd4x, (ii)
where are Lagrange multipliers at the local gauge violating condition. Excluding wAB via Eqs. of motion yields the generalized Faddeev action,
Here a2 is the face of the one of two 4-simplices a4 containing a3 and in which va2AB is defined.
To summarize, the versions of the Faddeev formalism considered can be continuum or discrete. Besides that, these can include connection (be of the I order) or be purely in terms of the tetrad or edge vectors (be of the II order). We have the following correspondence between the versions of the Faddeev gravity,
I continuum
I order
t
continuum
II order
->• 111
->•
discrete
I order
I
discrete
II order
(15)
Above we have constructed the discrete connection representation considering logical transitions i) ^ ii) ^ iii) from genuine Faddeev formulation i), and then we can exclude connections via Eqs. of motion, iii) ^ iv). Then it turns out that we can reproduce the well-defined (leading over variation of fA from 4-simplex to 4-simplex) part of the Faddeev action obtained by direct evaluation on the piecewise fiat manifold, i) ^ iv). That is, the diagram (15) is commutative.
5 Discrete connection action on hypercubes
As mentioned in Section 2, possibility of discontinuous metrics allows to operate with the hypercube minisuperspace decomposition of spacetime instead of the more complex and inconvenient simplicial decomposition. Summation over hypercubes is equivalent to summation over vertices (sites). The action is
Sdiscr ____
8nG
EE
V(f X)2(f »)2 — (f xf »)2
sites X,»
f X f » j» f X
JAJB JAJB
V — det y fXf»\\
RAB (i)
V(fX)2 (f »)2 — (f Xf »)2
+
8-kGy
V (eX»VpfvAfpB)2 . r
y ^ y ^ —------------------arcsin [
sites X,»
eX»VpfvAfpB RAB (i)
2^/(eX»VP fvAfpB )2 ( )
+ E E Ax»v]QX
AB
sites X,»,v
( X» XV xv x»\
'Uajb JAJB)
(16)
[4], Rx»(i) = Q{ (TAT0»)(TTix )i» \v,... label co-
TX
xX
we assume Minkowsky metric signature (+, +, +, -) and iX G SO(d-l,l).
6 Continuous time limit
The manifold is viewed as constructed of a set of 3D leaves of similar (here cubic) structure labeled by a parameter t. The difference dt for the neighboring leaves is assumed infinitesimal, and different variables
dt
fAA = O(dt), io = 1 + O(dt),
To = 1 + dt dt + O((dt)2).
(17)
S
discr _
dt
16nG
EE
sites X
0X»V
f»AfVB
— det llgX»|| (fA fB — fA fB )
(i{Ù X)AB + .... (18)
work [5]. There expression for the operator of the surface area requires point splitting regularization, and in order to preserve gauge invariance, this expression should be modified by introducing the path ordered exponents of the connection field operator. Then evaluation of the area operator on certain set of states in the Hilbert space of states (loop states) gives a discrete set of values.
The surface area operators in terms of the discrete Ashtekar type variables were considered in Ref. [6].
Now in the discrete framework, we simply need to quantize elementary area, or the area of the 2-simplex (triangle). Area bivectors are canonically conjugate to the orthogonal connection matrices. This fact leads to quantization of the elementary area in qualitative analogy with the quantization of angular momentum that is canonically conjugate to an orthogonal matrix of rotation in three-dimensional space. Now we are faced with the (d — 2)-dimensional angular momentum whose spectrum is well-known (see, e.g., Ref. [7]). We find that the elementary area spectrum is proportional to the Barbero-Immirzi parameter 7 in the Faddeev gravity [4],
A = 8nlpYa(j), a(j) = j(j + d — 4), l2p = G,
with multiplicity
g(j ) =
(j + d — 5)!(2j + d — 4) j !(d — 4)! ■
(19)
(20)
In particular, we can find the kinetic term pq in the resulting Lagrangian [4],
8 Requirements from black hole physics
The spectrum of horizon area in the loop quantum gravity (different from the spectrum of the generic surface area in that theory) was first used to calculate the black hole entropy in Ref [8]. The requirement that it be (41p)-1 Abh where Abh is the horizon area (Bekenstein-Hawking relation) means an Eq. for 7. In order to take into account the so-called holographic bound principle for the entropy of any spherical nonrotating system including black hole [9-11], in Refs [12-15] the formula for the spectrum of the horizon area was chosen to coincide with the general formula for the spectrum of the surface area, and corresponding value of y found.
Now the elementary area spectrum obtained can be used to evaluate statistical black hole entropy [14]. This gives Eq. for 7 for the different d’s,
7 Area quantization
The quantization of the surface area was first discussed in the continuum general relativity theory, namely, using Ashtekar variables as early as in the
E g(j )e-2nYa(j) = 1.
Calculation for, e. g., genuine d =10 gives
1
1
Y
Y = 0.393487933.... (22)
It can be recast to the connection representation both in the continuum and discrete (piecewise flat minisuperspace) case.
If one considers global embedding into external space, Physically reasonable area spectrum is possible
it may require as much as d = 230 dimensions [16] and which is consistent with black hole physics.
then
Y = 0.359772297.... (23)
Acknowledgement
The dependence on d is rather weak. The author is grateful to Ya.V. Bazaikin, I. B.
Khriplovich and I.A. Taimanov for valuable discus-9 Conclusion sions. This research has been supported by the Min-
istry of Education and Science of the Russian Federa-Faddeev formulation allows for discontinuous met- tion, Russian Foundation for Basic Research through
rics. Thus, spacetime can be virtually composed of Grant No. 11-02-00792-a and Grant 14.740.11.0082 of
independent microblocks. federal program "personnel of innovational Russia".
References
[1] Faddeev L. D. 2011 Theor. Math. Phys. 166 279-290.
[2] Khatsymovsky V. M. 2012 Faddeev formulation of gravity in discrete form [arXiv:1201.0808 [gr-qc]].
[3] Khatsymovsky V. M. 2012 First order representation of the Faddeev formulation of gravity [arXiv:1201.0806 [gr-qc]].
[4] Khatsymovsky V. M. 2012 On area spectrum in the Faddeev gravity [arXiv:1206.5509 [gr-qc]].
[5] Ashtekar A., Rovelli C. and Smolin L. 1992 Phys. Rev. Lett. 69 237-240 [arXiv:hep-th/9203079],
[6] Loll R. 1997 Class. Quant. Grav. 14 1725-1741 [arXiv:gr-qc/9612068].
[7] Vilenkin N. Ya. 1968 Special Functions and the Theory of Group Representations Translations of Mathematical Monographs (Amer. Math. Soc., Providence, Rhode Island) 22.
[8] Ashtekar A., Baez J., Corichi A. and Krasnov K. 1998 Phys.Rev.Lett. 80 904-907 [arXiv:gr-qc/9710007].
[9] Bekenstein J. D. 1981 Phys. Rev. D 23 287-298.
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11] Susskind L. 1995 J. Math. Phys. 36 6377-6396 [arXiv:hep-th/9409089],
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qc/0112074],
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Received 01.10.2012
В. М. Хацимовский
НЕКОТОРОЕ СВОЙСТВО И ПРИМЕНЕНИЕ ГРАВИТАЦИИ В ФОРМУЛИРОВКЕ
ФАДДЕЕВА
В формулировке Фаддеева гравитации метрика рассматривается как композитное иоле, билинейное по d = 10 4-векторным полям. Уникальное свойство состоит в том, что эта формулировка допускает разрывные поля. На дискретном уровне, когда пространство-время разлагается на элементарные 4-симплексы, это означает, что 4-симплексы могут не совпадать на их общих гранях, то есть, быть независимыми.
Мы применяем это к частной проблеме квантования поверхности, рассматриваемой как поверхность, составленная из виртуально независимых элементарных площадок (2-симплексов). Мы находим, что спектр площади пропорционален параметру Барберо-Иммирци 7 в гравитации Фаддеева и описывается как сумма спектров отдельных площадок.
В соответствии с известным в литературе подходом, мы находим, что существует 7, гарантирующее соотношение Бекенштейна-Хокинга для статистической энтропии чёрной дыры для призвольного d, в частности, 7 = 0.39... для изначального d = 10.
Ключевые слова: гравитация Фаддеева, кусочно-плоское пространство-время, связность, спектр площади. Хацимовский Владимир Михайлович, доктор физико-математических наук.
Институт ядерной физики им. Г.И. Будкера Сибирского отделения Российской академии наук.
Просп. Лаврентьева, 11, Новосибирск, Россия, 630090.
E-mail: khatsym@gmail.com
