Гравитационное действие Фаддеева на кусочно-постоянных фундаментальных векторных полях Текст научной статьи по специальности «Математика»

V. M. Khatsymovsky. Faddeev gravity action on the piecewise constant fundamental vector fields

UDC 530.1; 539.1

FADDEEV GRAVITY ACTION ON THE PIECEWISE CONSTANT FUNDAMENTAL VECTOR FIELDS

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 of d = 10 4-vector fields. We derive the minisuperspace (discrete) Faddeev action by evaluating the Faddeev action on the spacetime composed of the (flat) 4-simplices with constant 4-vector fields. This is an analog of the Regge action obtained by evaluating the Hilbert-Einstein action on the spacetime composed of the flat 4-simplices. One of the new features of this formulation is that the simplices are not required to coincide on their common faces. Also an analog of the Barbero-Immirzi parameter y can be introduced in this formalism.

Keywords: Einstein gravity, Regge calculus, composite metric, Faddeev gravity, discrete gravity.

1 Introduction

its curvature is

Regge calculus fl] is a minisuperspace formulation of gravity, that is, exact general relativity (GR) on a family of the metric fields (piecewise flat ones) sufficiently large to approximate any metric with any accuracy. Spacetime is taken as the set of the flat 4d tetrahedra. Let an be n-dimensional tetrahedron or n-simplex. Then the Einstein action is a sum over triangles a2,

1

2 I RVod4x = £ A

fA fpA.

fA = dfA/dx

XMV

fXf^A,i

(UA,V = dvUA), rxuv = gXprp»v, (4)

K:

MvP

_ pX _ pX I pX p7 _ pX pa

r MP,V r MV,P + r 7Vr MP r 7pr MV

= nAB (fA,v fMB,P — fA,pfMB,v ).

Here,

ÏÏab = Sab — fA fXB

(5)

(6)

is a projector. Note that it makes usual and covariant derivatives equivalent,

(1)

Here, Aa2 is area of the triangle a2, aa2 is the defect angle on the triangle a2. Regge calculus is expected to be the baze for quantizing GR [2-4] difficult to implement by the other methods because of the formal nonrenormalizability of GR.

Let us consider some another minisuperspace theory based on the Faddeev formulation of gravity [5]. In the Faddeev gravity, the metric is composed of d =10 4-vector fields,

nAB fxB,, = nAB VfxB.

Then we can write out the Faddeev action,

S = f gxv g^'Kx^vpVgd4 x ex^vpKxMvpd4

(fx

(7)

n

AB

--CXmvp fXA,MfvB,

Y

d4x.

(8)

(2)

Here, \,p,...= \,2, 3, 4; A,B,...= 1, ..., d. Simple example: locally, 4d Riemannian space can be considered cUS cl hyper-surface in the lOd Euclidean space. If f A(x) were its coordinates, then we would have

Here we have generalized it by adding the P-odd 1 /j-term where 7 is an analog [6] of the Barbero-Immirzi parameter [7-10] in the usual Einstein gravity (in the connection representation).

The variation of the action reads

SS

1

, = fAi Km — -SmK --— K 2VsSfXi Jm V X 2 X - - •

Xv pa

(3)

In the Faddeev gravity, however, fA(x) are independent variables. We can introduce an alternative connection

+ n

+

AB

v M v M v M

J B,v tXm + f B,mtvX + J B,XT mv

2yVS

(gXa Skt — SXT 9K7 )fB pTM

.

Here, the nAB-part of the field eqs SS = 0 turns out to be a linear homogeneous system for torsion

= fA (fA,V - f^J with a nondegenerate 24 x 24

X

2 a„ 2.

7

x

X

matrix leading to = 0. This gives f = rAv, the Cristoffel. Then = the Riemann for which

= o, and the /Apart of the field eqs turns out to be the Einstein eqs.

2 The piecewise constant fields

In GR, taking the metric as

ds2 = gnn(dxn)2 + dxadx^, (10)

where n is any of 1, 2, 3, 4, and a, ^,7,... = n, we can separate out the derivatives over xQ squared in the action as follows,

J =

(gaY g^ - g^ gjti g7,ing"nVgd4X + ... . (11)

If the metric is taken to be piecewise constant (with gv = const inside the 4-simplices), this implies continuity condition for the induced on the faces metric,

gaß |xn=xn + 0 — gaß |xn=xn-0 — 0 V

S :

n

AB

(/A

--eAMVp/AA,MfvB„

Y

d4x

<r2 at xi — 0,x2 — 0 (triangle in x3, x4-plane); 2) is a full derivative

Qab — d// - /A^/B ) so that

J Qabdx1dx2 — £(/Ad/B - /Ad/B),

(15)

(16)

where the contour C encircles the triangle a2. This gives

2Qab = ¿(x1)^2(a4)/2 (a4+1) i=i

-/1 (^4+i )/B (^4)1 +(A ^ B),

where the 4-simplices a4, i = 1 ^ n surround the triangle a2. The action is

(17)

VB (a2) ]T {[/A (a4)/B (a4+i)

i=i ^

-/A(a4+i)/B(a,4)] yg^

(12)

-^ [/4A(a,4)/3B(a,+i) - /4A(a4+i)/3B(a,4)] • (Ax3i Ax4i - Ax3i Ax4i) ,

(18)

if some 3-face is described locally by X — Xq . That is, the simplices should coincide on their common faces.

In the Faddeev gravity, the fields /A can be taken to be piecewise constant. The action

(13)

does not contain the square of any derivative and does not require continuity condition on /A(x) and thus on the metric = /a/ma- The simplices are not required to coincide on their common faces, and we can take /A(x) = const independently in these 4-simplices.

where a2 is formed by the edges a3,a4 with the 4-vectors Ax A i, AxA 1.

4 An invariant form of the action

We can rewrite this action in some invariant variables. This invariance is that one w. r. t. the choice of the coordinates

gned to the vertices. Choose some two edges a1, a1 in addition to the above a^a] spanning a2 so that a^a1, a^a] span some a4o D a2. The difference between xx of the endpoints of a^ is

Ax A i. Invariant edge variables /A or /3 are defined by ' ^

3 The discrete Faddeev action

/A — £ /A"Axil, /A — /AAxMi.

(19)

To evaluate the discrete action [11], we divide IR4 as the set of points x = (x1, x2,x3,x4) by the hypersurfaces a^xA+6 = 0 (mathematical hyperplanes) into polytopes, in particular, 4-simplices. are independent in these 4-simplices.

Then we note that the expression

QAB — /Al,A/iM,M - /A,A/B,p

Then we can write the contribution of a2 to the action,

1nAB (a2)]T

(14)

/A1 (a4)/B2 (a4+i) -/A1 (a4+i)/B1 (a,4)] yde^a"! (20)

/4iA(a4)/41B (ai+i) -/4lA(a4+i)/41B ,

1) is zero if /A depends only on one coordinate: contribution of the 3-simplices vanishes, and =

const • ¿(xi)^(x2) in a neighborhood of the 2-simplex

where nA — ¿A - /Ai fB\ ga1a" — /<Ai fa"a (metric edge components).

V. M. Khatsymovsky. Faddeev gravity action on the piecewise constant fundamental vector ßelds

5 Restoring the continuum action from the discrete minisuperspace one

We can obtain the continuum Faddeev action from the found discrete minisuperspace one. This reverse transition is intended to show that the information encoded in the minisuperspace formulation is sufficient to reproduce the essential degrees of freedom of the continuum theory. This procedure is analogous to the work [12] for the usual GR where the Regge action has been shown to tend to the Hilbert-Einstein one if simplicial decomposition of the given smooth manifold is made finer and finer. To do this, we choose the fields to approximate some fixed smooth fA(x) arbitrarily closely by making decomposition of spacetime into the 4-simplices finer and finer (the coordinate steps along the edges AxAi ^ 0 ). Then we choose the discrete variable fA(a4) to be fA(x) at x = xCT4, some central point in a4. The procedure is most easily illustrated by the case n = 4:

£ [fA(a4)fB (a4+i) - fA(*4+i)fB(*4)]

i=i

= A„fAAvfB - AvfAAf (21)

where Auf = f (a4) - f (a?) aid Avf = f K4) - f (a24) in the continuum limit become derivatives. This gives us the basic structure (some two-dimensional Jacobian) which being summed over the different contravariant components fA. and variations between the different pairs of simplices Af = f (a4) - f (a4') just leads to fA AfB ß - fA AfB ß t^16 Faddeev action.

The 1/7-part of the action is considered completely analogously.

6 Possibility of the cubic decomposition of spacetime

A new interesting property of the Faddeev formulation as compared to the Regge calculus is the

possibility to decompose the curved spacetime into the

IR4

rectangular parallelepipeds or cuboids

jx|xA + nAeA < xA

< xA + (nA + 1)eA,A = 1, 2, 3, 4}. (22)

Then /A(x) is assumed to be constant in each of these cuboids. Above derivation of the minisuperspace

a4

simplices but cuboids. In Regge calculus, the continuity of metric on the faces leads to that the spacetime composed of flat cuboids is flat as well. In the Faddeev gravity, continuity of /A (x) is not required, and the flat cuboids can be used for modeling the curved spacetime. The cuboid action looks just as the naively discretized action or the continuum one (8) in which the derivatives are substituted by their finite difference counterparts. It is much simpler than the simplicial one and, at the same time, it is a minisuperspace action.

7 Conclusion

To summarize, we have constructed some discrete formalism analogous to the Regge calculus, but based not on the usual GR, but on its Faddeev form. Some new features of this formalism are the following.

1) The invariant physical variables are edge 10-vectors /^i (a4) (or /^i (a4)), independent for the different 4-simplices a4 containing the edge a1.

2) The 4-simplices do not necessarily coincide on their common faces.

3) 1/7-term is present which is an analogue of the so called Barbero-Immirzi parity odd term in the connection representation of the GR action.

An advantage of the discrete Faddeev formalism is the possibility to use cubic decomposition instead of the simplicial one for modeling the curved spacetime due to possibility of the field /A discontinuities in the Faddeev gravity. The cuboid action looks simple as the naive discretization of the continuum one.

Also this possibility of the field /A discontinuities allows one to evaluate the area spectrum of any surface as the sum of spectra of independent triangles. In blackhole physics, reasonable physical arguments require a discrete area spectrum, but up to now, only one gravity theory (LQG) allows to obtain it [13-15]. We hope that the discrete Faddeev gravity approach is also able to handle this task.

Acknowledgement

This research was supported by the Ministry of Education and Science of the Russian Federation.

References

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Received 02.11.20Ц

В. M. Хацимовский

ГРАВИТАЦИОННОЕ ДЕЙСТВИЕ ФАДДЕЕВ А НА КУСОЧНО-ПОСТОЯННЫХ ФУНДАМЕНТАЛЬНЫХ ВЕКТОРНЫХ ПОЛЯХ

В формулировке Фаддеева гравитации метрика считается композитным полем, билинейным по 1 = 10 4-векторным полям. Мы выводим (дискретное) действие Фаддеева в конфигурационном минисуперпространстве путем вычисления действия Фаддеева на пространстве-времени, составленном из (плоских) 4-симплексов с постоянными 4-векторными полями. Это аналог действия Редже, полученного вычислением действия Гильберта-Эйнштейна на пространстве-времени, составленном из плоских 4-симплексов. Одна из новых черт этой формулировки состоит в том, что симплексы не обязаны совпадать на общих гранях. Также можно ввести в этом формализме аналог параметра Барберо-Иммирци 7.

Ключевые слова: эйнштейновская гравитация, исчисление Редже, композитная метрика, гравитация Фаддеева, дискретная гравитация.

Хацимовский В. М., доктор физико-математических наук. Институт ядерной физики им. Вудкера СО РАН.

Пр. Лаврентьева, 11, 630090 Новосибирск, Россия. E-mail: khatsym@gmail.com