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Energy, angular momentum, superenergy and angular supermomentum in conformal frames

Mariusz P. D?browski? and Janusz Garecki? a Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland

arXiv:0712.1358v2 [hep-th] 14 Dec 2007

(Dated: February 2, 2008)

Abstract

We ?nd the rules of the conformal transformation for the energetic quantities such as the Einstein energy-momentum complex, the Bergmann-Thomson angular momentum complex, the superenergy tensor, and the angular supermomentum tensor of gravitation and matter. We show that the conformal transformation rules for the matter parts of both the Einstein complex and the Bergmann-Thomson complex are fairly simple, while the transformation rules for their gravitational parts are more complicated. We also ?nd that the transformational rules of the superenergy tensor of matter and the superenergy tensor of gravity are quite complicated except for the case of a pure gravity. In such a special case the superenergy density as well as the sum of the superenergy density and the matter energy density are invariants of the conformal transformation. Besides, in that case, a conformal invariant is also the Bel-Robinson tensor which is a part of the superenergy tensor. As for the angular supermomentum tensor of gravity - it emerges that its transformational rule even for a pure gravity is quite complicated but this is not the case for the angular supermomentum tensor of matter. Having investigated some technicalities of the conformal transformations, we also ?nd the conformal transformation rule for the curvature invariants and, in particular, for the Gauss-Bonnet invariant in a spacetime of arbitrary dimension.

PACS numbers: 98.80.-k;98.80.Jk;04.20.Cv;04.50.-h

? ?

Electronic address: mpdabfz@wmf.univ.szczecin.pl Electronic address: garecki@wmf.univ.szczecin.pl

1

I.

INTRODUCTION

It is widely known that there exists a problem of the energy-momentum of gravitational ?eld in general relativity. Since the gravitational ?eld may locally vanish, then one is always able to ?nd the frame in which the energy-momentum of the gravitational ?eld vanishes in this frame, while it may not vanish in the other frames. In fact, the physical objects which describe this situation are not tensors, and they are called the gravitational ?eld pseudotensors. They form the energy-momentum complexes which are the sums of the obvious energy-momentum tensors of matter and appropriate pseudotensors. The choice of the gravitational ?eld pseudotensor is not unique so that many di?erent de?nitions of the pseudotensors have been proposed. To our knowledge, the most frequently used are the energy-momentum complexes of Einstein [1], Landau-Lifshitz [2], M?ller [3], Papapetrou [4], Bergmann-Thomson [5], Weinberg [6], and Bak-Cangemi-Jackiw [7]. Among them only Landau-lifshitz, Weinberg, and Bak-Cangemi-Jackiw pseudotensors are symmetric. On the other hand, only the Einstein complex is canonical. Bearing in mind the formalism of ?eld theory, it is obvious that there also exist the angular momentum complexes. Among them the Bergmann-Thomson angular momentum complex [5] and the Landau-Lifshitz angular momentum complex [2] are most widely used. The arbitrariness of the choice of pseudotensors and the fact that they usually give different results for the same spacetime inspired some authors [8–10] to de?ne quantities which describe the generalized energy-momentum content of the gravitational ?eld in a tensorial way. These quantities are called gravitational superenergy tensors [8] and gravitational angular supermomentum tensors [9] or super(k) -energy tensors [10]. The canonical superenergy tensors and the canonical angular supermomentum tensors have successfully been calculated for plane, plane-fronted and cylindrical gravitational waves, Friedmann universes, Schwarzschild, and Kerr spacetimes [8, 9]. By use of the superenergy and angular supermomentum tensors one can also prove that a real gravitational wave with Riklm = 0 possesses and carries positive-de?nite superenergy and angular supermomentum [8, 9, 11]. In Refs. [12, 13], the properties of the Einstein and Bergmann-Thomson complexes as well as the superenergy and supermomentum tensors for G¨del spacetime were studied. It o was shown that the former were sensitive to a particular choice of coordinates while the 2

latter were coordinate-independent. Some interesting properties of superenergy and supermomentum were found. For example, the relation between the positivity of the superenergy and causality violation. In this paper we are going to discuss another interesting problem - the sensitivity of the complexes, superenergy and supermomentum to the conformal transformations of the metric tensor [14]. Conformal transformations are interesting characteristics of the scalartensor theories of gravity [15–18], including its conformally invariant version [19–22]. The point is that these theories can be represented in the two conformally related frames: the Jordan frame in which the scalar ?eld is non-minimally coupled to the metric tensor, and in the Einstein frame in which it is minimally coupled to the metric tensor. It is most striking that the scalar-tensor theories of gravity are the low-energy limits of the currently considered as the most fundamental uni?cation of interactions theory such as superstring theory [23–26]. It has been shown that some physical processes such as the universe in?ation and density perturbations look di?erent in conformally related frames [25, 26]. This is the main motivation why we ?nd interesting to investigate the problem of energy-momentum in the context of conformal transformations. Inspired by similar ideas, in Ref. [27] it was shown that Arnowitt-Deser-Misner (ADM) masses were invariant in di?erent conformal frames for asymptotically AdS and ?at spacetimes as long as the conformal factor goes to unity at in?nity. Our paper is organized as follows. In Section II we give basic review of the idea of the conformal transformations of the metric tensor and discuss the transformational properties of the geometric quantities such as, for example, the Gauss-Bonnet invariant in D?spacetime dimensions. In Section III we discuss the conformal transformation of the Einstein energymomentum complex while in the Section IV the conformal transformation of the BergmannThomson angular momentum complex. In Section V we discuss the conformal transformation of the superenergy tensors and in Section VI the conformal transformations of the angular supermomentum tensors. In Section VII we give our conclusions.

3

II.

BASIC PROPERTIES OF CONFORMAL TRANSFORMATIONS

Consider a spacetime (M, gab ), where M is a smooth n?dimensional manifold and gab is a Lorentzian metric on M. The following conformal transformation gab (x) = ?2 (x)gab (x) , ? (II.1)

where ? is a smooth, non-vanishing function of the spacetime point is a point-dependent rescaling of the metric and is called a conformal factor. It must be a twice-di?erentiable function of coordinates xk and lie in the range 0 < ? < ∞ (a, b, k, l = 0, 1, 2, . . . D). The conformal transformations shrink or stretch the distances between the two points described by the same coordinate system xa on the manifold M, but they preserve the angles between vectors (in particular null vectors which de?ne light cones) which leads to a conservation of the (global) causal structure of the manifold [14]. If we take ? = const. we deal with the so-called scale transformations [16]. In fact, conformal transformations are localized scale transformations ? = ?(x). On the other hand, the coordinate transformations xa → xa only change coordinates and ? do not change geometry so that they are entirely di?erent from conformal transformations [14]. This is crucial since conformal transformations lead to a di?erent physics [16]. Since this is usually related to a di?erent coupling of a physical ?eld to gravity, we will be talking about di?erent frames in which the physics is studied (see also Refs. [20, 21] for a slightly di?erent view). In D spacetime dimensions the determinant of the metric g = det gab transforms as √ ?? = ?D ?g . g (II.2)

It is obvious from (II.1) that the following relations for the inverse metrics and the spacetime intervals hold g ab = ??2 g ab , ? d?2 = ?2 ds2 . s Finally, the notion of conformal ?atness means that gab ??2 (x) = ηab , ? 4 (II.5) (II.3) (II.4)

where ηab is the ?at Minkowski metric. The application of (II.1) to the Christo?el connection coe?cients gives [14] 1 c c ? Γc = Γc + ga ?,b + gb ?,a ? gab g cd?,d , ab ab ? 1 c ? Γc = Γc ? g ?,b + gb ?,a ? gab g cd?,d , ? ?c ? ? ab ab ? a ?,a ? Γb = Γb + D ab ab ? ?,a ? . Γb = Γb ? D ab ab ? (II.6) (II.7)

The Riemann tensors, Ricci tensors, and Ricci scalars in the two related frames gab and gab transform as [36] ? 1 a a ? δ ?;bc ? δc ?;bd + gbc ?;a ? gbd ?;a (II.8) Rabcd = Rabcd + ;c ;d ? d 1 a 2 a a a + 2 [δc ?,b ?,d ? δd ?,b ?,c + gbd ?,a ?,c ? gbc ?,a ?,d ] + 2 [δd gbc ? δc gbd ] gef ?,e ?,f , ? ? 1 a a ? Rabcd = Rabcd ? δ ?? ? δc ?? + gbc ??? ? gbd ??? ? ;a ? ;a (II.9) ;bc ;bd ;c ;d ? d 1 a a ? ? ? + 2 [δd gbc ? δc gbd ] gef ?,e ?,f , ? 1 1 ? Rab = Rab + 2 [2(D ? 2)?,a ?,b ? (D ? 3)?,c ?,c gab ] ? [(D ? 2)?;ab + gab ?] (II.10) , ? ? ? 1 1 ? (II.11) (D ? 2)?? + gab ? , ? g Rab = Rab ? 2 (D ? 1)?ab ?,c ?,c + ;ab ? ? ? ?,a ?,b ? R = ??2 R ? 2(D ? 1) ? (D ? 1)(D ? 4)g ab ? ?2

?

,

(II.12) (II.13)

R = ?

2

? R + 2(D ? 1)

?,a ?,b ? ? D(D ? 1)?ab g ? ?2

,

and the appropriate d’Alambertian operators change under (II.1) as

?

φ = ??2 φ = ?2

?

?,a φ,b ? ?,a φ ? (D ? 2)?ab g φ,b ? φ + (D ? 2)g ab

, .

(II.14) (II.15)

In these formulas the d’Alembertian

?

taken with respect to the metric gab is di?erent from ?

which is taken with respect to a conformally rescaled metric gab . Same refers to the covariant derivatives ? and ; in (II.8)-(II.11). ; For the Einstein tensor we have D?2 D?2 ? [4?,a ?,b + (D ? 5)?,c ?,c gab ] ? [?;ab ? gab ?] , Gab = Gab + 2 2? ? ? D?2 D?2 ? Gab = Gab ? ?? ? gab ? , ? (D ? 1)?,e ?,e gab + ? ;ab 2 2? ? 5 (II.16) (II.17)

An important feature of the conformal transformations is that they preserve Weyl conformal curvature tensor (D ≥ 3) Cabcd = Rabcd + 2 2 ga[d Rc]b + gb[c Rd]a + Rga[c gd]b , D?2 (D ? 1)(D ? 2) (II.18)

which means that we have (note that one index is raised) ? C abcd = C abcd (II.19)

under (II.1). Using this property (II.19) and the rules (II.1)-(II.3) one can easily conclude that the Weyl Lagrangian [28] ? Lw = ?α √ ? ??C abcd Cabcd = ?α ?gC abcd Cabcd = Lw g? (II.20)

is an invariant of the conformal transformation (II.1). In further considerations it would be useful to know the conformal transformation rules for the widely applied curvature invariants which are given by ? R2 = ??4 R2 + 4(D ? 1)2 ??2 ( ?)2 + (D ? 1)2 (D ? 4)2 ??4 g ab ?,a ?,b g cd ?,c ?,d ? 4(D ? 1)R??1 ? ? 2R(D ? 1)(D ? 4)??2 g ab ?,a ?,b + 4(D ? 1)2 (D ? 4)??3 ?g ab ?,a ?,b , ? ? Rab Rab = ??4 Rab Rab ? 2??1 (D ? 2)Rab ?;ab + R ? + ??2 4(D ? 2)Rab ?,a ?,b ? 2(D ? 3)R?,e ?,e + (D ? 2)2 ?;ab ?;ab + (3D ? 4) ( ?)2 ? (D ? 2)2 ?;ab ?,a ?,b ? (D 2 ? 5D + 5) ??,e ?,e , (II.22) + ??4 (D ? 1)(D 2 ? 5D + 8) (?,a ?,a )2 (II.21)

? ? Rabcd Rabcd = ??4 Rabcd Rabcd ? 8??1 Rbc ?;bc + 4??2 ( ?)2 + (D ? 2)?;bc ?;bc ? R?,b ?,b + 4Rbc ?,b ?,c + 8??3 (D ? 3) ??,c ?,c ? 2(D ? 2)?;bc ?,b ?,c + 2??4 D(D ? 1) (?,a ?,a )2 . (II.23)

In fact out of these curvature invariants one forms the well-known Gauss-Bonnet term which is one of the Euler (or Lovelock) densities [29, 30]. Its conformal transformation (II.1)

6

reads as ? ? ? ? ? ? RGB ≡ Rabcd Rabcd ? 4Rab Rab + R2 = ??4 RGB + 4(D ? 3)??1 2Rab ?;ab ? R ? + 2(D ? 3)??2 2(D ? 2) ( ?)2 ? ?;ab ?;ab ? 8Rab ?,a ?,b ? (D ? 6)R?,a ?,a + 4(D ? 2)(D ? 3)??3 (D ? 5) ??,a ?,a + 4?;ab ?,a ?,b + (D ? 1)(D ? 2)(D ? 3)(D ? 8)??4 (?,a ?,a )2 The inverse transformation is given by ? ;ab ? ? RGB ≡ Rabcd Rabcd ? 4Rab Rab + R2 = ?4 RGB ? 4(D ? 3)??1 2Rab ?? ? R + 2(D ? 2)(D ? 3)??2 2

? ?

.

(II.24)

? (II.25)

?

2

;ab ? ;a ;a ? 2?? ?? ? R?? ?? ;ab ? ;a ;a ? ?? ?? ? D??1 ?? ?? ;a ;a 2

? (D ? 1)(D ? 2)(D ? 3)??3 4

.

So far we have considered only geometrical part. For the matter part we usually consider matter lagrangian of the form ? Sm = ????D dD xLm = g √ ?gdD xLm = S , (II.26)

where the conformal transformation (II.1) have been used [24]. Then, the energy-momentum tensor of matter in one conformal frame reads as δ 2 ? T ab = √ ?? δ?ab g g which under (II.1) transforms as ? T ab = ??D?2 T ab , ? T = ??D T . ? T ab = ??D T ab (II.28) (II.29) 2 ?gcd δ √ ????D Lm = ??D √ g ?gLm , ?g ??ab δgcd g (II.27)

For the matter in the form of the perfect ?uid with the four-velocity v a (va v b = ?1), the energy density ? and the pressure p T ab = (? + p)v a v b + pg ab , the conformal transformation gives ? T ab = (? + p)?a v b + pg ab , ? ? v ? ?? 7 (II.31) (II.30)

where 2 δ √ T ab = √ ?gLm , ?g δgab and va = ? 1 dxa dxa = = ??1 v a . d? s ? ds (II.33) (II.32)

Therefore, the relation between the pressure and the energy density in the conformally related frames reads as ? = ??D ? , ? p = ??D p . ? It is easy to note that the imposition of the conservation law in the ?rst frame T ab;b = 0 , gives in the conformally related frame ? ? ? T ab;b = ? T . ? ?

,a

(II.34) (II.35)

(II.36)

(II.37)

From (II.37) it appears obvious that the conformally transformed energy-momentum tensor ? is conserved only, if the trace of it vanishes (T = 0) [6, 16, 19, 22]. For example, in the case of barotropic ?uid with p = (γ ? 1)? γ = const., (II.38)

it is conserved only for the radiation-type ?uid p = [1/(D ? 1)]? . Similar considerations are also true if we ?rst impose the conservation law in the second frame ? T ab;b = 0 , ? which gives in the conformally related frame (no tildes) T ab;b = ?,a T . ? (II.40) (II.39)

Finally, it follows from (II.29) that that vanishing of the trace of the energy-momentum tensor in one frame necessarily requires its vanishing in the second frame, i.e., if T = 0 in ? one frame, then T = 0 in the second frame and vice versa. This means only the traceless type of matter ful?lls the requirement of energy conservation. 8

III.

EINSTEIN ENERGY-MOMENTUM COMPLEX IN CONFORMAL FRAMES

Following our earlier work [12, 13] we consider the canonical double-index energymomentum complex [1–3]

E Ki k

=

√

?g(Ti k +E ti k ),

(III.1)

where T ik is a symmetric energy-momentum tensor of matter which appears on the righthand side of the Einstein equations and g is the determinant of the metric tensor. In fact, its de?nition results from the rewritten Einstein equations [3] √ where

F Ui kl

?g(Ti k +E ti k ) = F Ui kl ,l ,

(III.2)

are Freud’s superpotentials and

E ti k

gia = ?F Ui lk = α √ [(?g)(g kag lb ? g la g kb)],b ?g

(III.3)

k = α δi g ms (Γl Γr ? Γr Γl ) mr sl ms rl

1 k 1 k + g ms [Γk ? (Γk g tp ? Γl g kt)gms ? (δs Γl + δm Γl )] tp tl ml sl ,i ms 2 2

(III.4)

is the Einstein’s gravitational energy-momentum pseudotensor (α = c4 /16πG). It is useful to apply the property of metricity g ms = g ms ? Γm g as ? Γs g ma = 0 , ;i ,i ia ia to (III.4) and reduce it to a simpler form, i.e.,

E ti k k = α δi g ms (Γl Γr ? Γr Γl ) + Γa (Γk g tp ? Γl g kt ) mr sl ms rl ia tp tl

(III.5)

+ Γl (Γm g ak + Γk g am ) ? Γk (Γm g as + Γs g am ) . ml ia ia ms ia ia

(III.6)

Now, we apply the conformal transformation (II.1) to the Einstein pseudotensor (III.6). We assume that we start with the Einstein pseudotensor in the conformal frame (with tildes) and express it in the conformal frame (no tildes) as below

E ti

? k = ??2

E

k tk + ??1 (D ? 2)[δi (Γl ?,r ? g ms Γr ?,r ) + ?,i (g tp Γk ? g kt Γl ) i rl ms tp tl

(III.7) .

k + (g ak Γm ?,m + Γk ?,a ) ? 2Γa ?,k ] + ??2 (D ? 1)(D ? 2)[δi ?,r ?,r ? 2?,i ?,k ] ia ia ia

9

Note that in D = 2 the rule of transformation is very simple:

E ti

? k = ??2 tk , which seem E i

to re?ect the fact of the conformal ?atness of all the two-dimensional manifolds. On the other hand, it is not so simple in the case of a ?at space where all the Christo?el connection coe?cients vanish. In a general case it is clear that the transformed Einstein pseudotensor would di?er from the starting one. This seems to be natural conclusion in the context of its sensitivity to a change of coordinates (this is why we call it ”pseudotensor”) despite we do not change those coordinates here at all (cf. Section II). From (II.2) and (III.7) it follows that √ ? (III.8) ??E ti k = ?D ?g E ti k g ? √ k = ?D?2 ?g tk + ??1 (D ? 2)[δi (Γl ?,r ? g ms Γr ?,r ) + ?,i (g tp Γk ? g ktΓl ) i rl ms tp tl

E

+ (g

ak

Γm ?,m ia

k + Γk ?,a ) ? 2Γa ?,k ] + ??2 (D ? 1)(D ? 2)[δi ?,r ?,r ? 2?,i ?,k ] ia ia

under conformal transformation (II.1). On the other hand, the transformation rule of the material part of the Einstein complex (III.2) reads as (cf. (II.2) and (II.28)) ??Tik = g? i.e., the quantity √ √ ?gTik , (III.9)

?gTik is an invariant of the conformal transformation (II.1). Besides, if

the initial metric is Minkowskian, then the transformational rule (III.7) simpli?es since the terms which contain Γi vanish. kl

IV.

BERGMANN-THOMSON ANGULAR MOMENTUM COMPLEX IN CON-

FORMAL FRAMES

The Bergmann-Thomson angular momentum complex

BT M ijk

= ?BT M jki = ?BT M jik √

(IV.10)

consists of the sum of the material part

mM ijk

=

?g(xi T jk ? xj T ik ),

(IV.11)

and the gravitational part

gM ijk

=

?g(xi BT tjk ? xj BT tik ) α (?g)(g kj g il ? g kig jl ) , + √ ?g ,l 10

√

(IV.12)

where √ ?g BT tjk := √ ?gg jiE ti k + g ij ,l F Ui

[kl]

(IV.13)

is the Bergmann-Thomson energy-momentum pseudotensor of the gravitational ?eld [5] (see also [31]). As a consequence of the local energy-momentum conservation law

k E Ki ,k

= 0,

(IV.14)

which immediately follows from (III.2), the angular momentum complex satis?es local conservation laws

BT M ijk ,k

= 0.

mM ijk

(IV.15) and the gravitational

Under the conformal transformation (II.1), the material part part g M ijk of the complex transform as

mM

? ijk = ??2 m M ijk ,

(IV.16)

gM

? ijk = g M ijk + ??1 2α√?g(xi g jl ? xj g il )P k l + ?,b [4α(xi g lj ,t ? xj g li ,t )gla U [kt]ab ? 2(xi F U j[kb] √ ? xj F U i[kb] ) + 4αU [ji]kb] + α??2 6 ?g(xi g jl ? xj g il )Ql k ? 8?,b ?,t [(?g) (xi g kj ? xj g ki)g tb ? (xi g tj ? xj g ti )g kb ] , (IV.17)

where

k Pi k := δi (Γl rl g rd?,d ? g ms Γl ms ?,l ) + ?,i (Γktp g tp

? Γl tl g kt) ? 2Γm g kp?,p + Γkia g as ?,s + Γm g ak ?,m , im ia and

k Qi k := δi g ms ?,m ?, s ? 2g kp?,i ?,p , 1 U [kt]ab := √ [(?g)(g kak tb ? g ta g kb )], ?g α [km] = √ gia [(?g)(g ka g mb ? g ma g kb)],b , F Ui ?g FU i[kb] il := gF Ul [kb]

(IV.18)

.

(IV.19)

11

As we can see from above formulas the conformal transformation rule (IV.16) for the matter part of the Bergmann-Thomson angular momentum complex is fairly simple. Let us also mention that the transformation (IV.16) holds in any dimension of spacetime, while the transformation (IV.17) holds only in D = 4 dimensions. Besides, if the initial metric is Minkowskian, then the transformational rule (IV.17) further simpli?es since the terms which contain Γi , gik,l and g ik,l vanish. In this case also g M ijk = 0. kl

V.

SUPERENERGY TENSORS IN CONFORMAL FRAMES

Following [8] one is able to introduce the canonical superenergy tensor. The de?nition of the superenergy tensor Sab (P ), which can be applied to an arbitrary gravitational as well as matter ?eld is [12]

(b) b ? (b) (b)

T(a)

S(a) where

(P ) = Sa (P ) := lim

(y) ? T(a)

?

(P ) d? , (V.1)

?→P

1/2 σ(P ; y)d?

T(a) T(a)

(b)

(y) := Ti k (y)ei (y)ek (y), (a)

(b)

(b)

(b)

(P ) := Ti k (P )ei (P )ek (P ) = Ta b (P ) (a)

(V.2)

are the tetrad components of a tensor or a pseudotensor ?eld Ti k (y) which describe an energy-momentum, y is the collection of normal coordinates NC(P) at a given point P, σ(P, y) is the world-function, ei (y), (a)

i dual, respectively, ei (P ) = δa , (a) (a)

ek (y) denote an orthonormal tetrad ?eld and its

(b)

(b)

b a ek (P ) = δk , ei (y)ei (y) = δa , and they are paralell (a)

propagated along geodesics through P. At P the tetrad and normal components of an object are equal. We apply this and omit tetrad brackets for indices of any quantity attached to the point P; for example, we write T ab (P ) instead of T (a)(b) (P ) and so on. Firstly, the symmetric superenergy tensor of matter is given by (from now on we will also interchangeably use ? to mark the covariant di?erentiation) [12]

b m Sa (P ) b = δ lm ?(l ?m) Ta .

(V.3)

l In terms of the four-velocity of an observer taken as comoving v l = δ0 (v l vl = ?1), a more

convenient covariant form of (V.3) is

m Sa b

(P ; v l ) = hlm ?(l ?m) Ta b , 12

(V.4)

where hlm ≡ 2v l v m + g lm = hml . Secondly, the canonical superenergy tensor of the gravitational ?eld reads as

g Sa b

(V.5)

(P ; v l ) = hlm Wa b lm ,

(V.6)

where Wa b lm = 2α b [B alm + P balm 9 1 b b ? δa Rijk m (Rijkl + Rikjl ) + 2δa R(l|g Rg|m) 2 ? 3Ra(l| Rb|m) + 2Rb(ag)(l| Rg|m) ],

(V.7)

and 1 b B balm := 2Rbik (l| Raik|m) ? δa Rijk l Rijkm , 2 is the Bel–Robinson tensor, while 1 b P balm := 2Rbik (l| Raki|m) ? δa Rijkl Rikjm . 2 In vacuum, the gravitational superenergy tensor (V.6) reduces to a simpler form:

g Sa b

(V.8)

(V.9)

(P ; v l ) =

1 b i(kp) 8α lm b(ik) h [C (l| Caik|m) ? δa C (l| Cikp|m) ] . 9 2

(V.10)

It is symmetric and the quadratic form g Sab (P ; v l)v a v b is positive-de?nite. It is suggested [8, 12] that the superenergy tensor g Sab (P ; v l ) should be taken as a quantity which may serve as the energy-momentum tensor for the gravitational ?eld. Its advantage is that it is a conserved quantity in vacuum. The disadvantage is that the superenergy tensors

g Sa b

(P ; v l ) and

m Sa

b

(P ; v l) have the dimension: [the dimension of the components of an

energy-momentum tensor (or pseudotensor)] ×m?2 . This means it is rather that their ?ux gives the appropriate energy-momentum tensors or pseudotensors. However, in some other approach one is able to introduce average relative energy and angular momentum tensors which have proper dimension [32]. In fact, these new tensors di?er from the superenergy and angular supermomentum tensors by a constant dimensional factor of (length)2 . Now, we consider the conformal transformations of the superenergy tensors. For the matter superenergy tensor one has ?k m Si ? ? ? ? = hlm ?(l ?m) Ti k 13 (V.11)

= ??6 m Sik + ??2 hlm 2 ??4

;(l|

Ti k |;m) ? (??4 );ml Tik

k + ??3 hlm 2D kt(l| Ti t | (??4 );m) ? D plm Tik (??4 );p ? D pli Tp (??4 );m

+ ??6 hlm (P ktm Tit ),l ? (P tmi Ttk ),l + ??7 hlm Γk (D ptm Tit ? D t mi Ttp ) lp

t k ? Γp (D ktp Tit ? D t ip Ttk ) ? Γp (D kmt Tp ? D t mp Ttk ) ? D plm Tik ;p + D klp Tip ;m ? D pli Tp ;m lm li t + ??8 hlm D klp (D ptm Tit ? D t mi Ttp ) ? D plm (D ktp Tit ? D t ip Ttk ) ? D pli (D kmt Tp ? D t mp Ttk ) ,

where P abc = P acb ≡ ??1 D abc , and

a a D abc ≡ δb ?,c + δc ?,b ? gbc g ad ?,d .

(V.12)

(V.13)

For the gravitational superenergy tensor one has

g Sa

?

b

= ??4 g Sab + ? + + + + + + + + ? + + +

2α lm i] [b k] [b [(g k[b? l ? δ l ?i]k ) + (g i[b ? l ? δ l ?k]i )]g[a[k ?i]m] h 9

b δa k[i j] [i (g ? l ? δ l ?j]k )(g[i[k ?j]m] + g[i[j ?k]m] ) 2 α ?1 lm ?1 b c ? h (? );mc (g bc ?al + δa ?cl ? 2δlb ?ac ? 2δa ?bl 9 4α ?2 lm b(ik) gla ?bc + δlc ?ba ) + ? h [2R l g[a[k ?i]m] 9 δb j] [i i] [b (g k[b? l ? δ l ?i]k )Ra(ik)m ? a (g k[i? l ? δ l ?j]k )Ri(kj)m 2 b ?1 gc 4δa (? );lg (??1);mc g + 6(??1 );al (??1 );mb 1 b b (R m ?al + δa Rgm ?gl ? 2δlb Rgm ?ag ? 2Ram ?bl + gla Rgm ?bg + Rlm ?ba )] 8 4α ?3 lm 4 b ?1 ? h [ δa (? );c(m Rcl + ((??1 );mb Ral + (??1 );al Rbm ) 3 3 1 ?1 c (? );mc (Rba l + Rbc al )] 3 α ?4 2 b ? (? );rs g rs [hlb ?al + hlg (gla ?bg ? δa ?gl 36 8α ?5 lm 3 b b glg ?ba )] ? ? h [ ((??1 );al δm + δa (??1 );ml 9 4 4α ?6 lm 2 b ? h (? );rs g rs δa Rlm gal (??1 );mb )](?2 );rs g rs ? 9 1 3 b [Ral δm + gal Rbm ] + Rbmal 4 4 3 b α ?8 lm 2 b ? h (? );rs g rs (?2 );tp g tp (δa glm + δm gal ) , (V.14) 9 2

where

a ?a = 4??1 (??1 );bc g ae ? 2(??1 );c (??1 );d g cd δb . b

(V.15)

14

Bearing in mind (II.19), we have for a pure gravitational ?eld that ? Cilmb = git C tlmb = ?2 Cilmb , ? ? ? C klma = ??4 C klma . and so 1 k? ? ? ? ? ? ? B kiab = C klm Cilmb + C klm Cilma ? δi C lmn Clmnb a a b 2 1 k = ??2 C klm Cilmb + C klm Cilma ? δi C lmn Clmnb a b a 2 (V.16) (V.17)

,

(V.18)

which means that the four times covariant form of the Bel-Robinson tensor (V.8) for the pure gravitational ?eld is an invariant of the conformal transformation, i.e., ? Bkiab = Bkiab , (V.19)

Similarly, for a pure gravity, from (V.9) and from the formulas given in Section II, one can easily obtain that

g Sa

? b (P ; v l ) = ??D g S b (P ; v l ), ? a

(V.20)

from which we have that ??g Sab (P ; v l ) = g ? ? i.e., the tensorial density √ √ ?g g Sab (P ; v l), (V.21)

?g g Sab (P ; v l ) is an invariant of the conformal transformation ??g Sab =g Sab = g ? √ ?g g Sab = 0 , (V.22)

(II.1). For a conformally ?at spacetime and for a pure gravitational ?eld we have that

g Sa

?b=

which is, for example, the case of the Friedman universes.

VI.

ANGULAR SUPERMOMENTUM TENSORS IN CONFORMAL FRAMES

In this Section we further extend the notion of superenergy onto the angular momentum which has been introduced in Ref. [9]. The canonical angular supermomentum tensors can be de?ned in analogy to the canonical superenergy tensors as S (a)(b)(c) (P ) = S abc (P ) := lim

?

[M (a)(b)(c) (y) ? M (a)(b)(c) (P )]d? 1/2 σ(P ; y)d?

?

?→P

,

(VI.23)

15

where M (a)(b)(c) (y) := M ikl (y)ei (y)ek (y)el (y),

a b M (a)(b)(c) (P ) := M ikl (P )ei (P )ek (P )el (P ) = M ikl (P )δi δk δlc (a) (b) (c) (a) (b) (c)

(VI.24)

= M abc (P )

(VI.25)

are the physical (or tetrad) components of the ?eld M ikl (y) = ?M kil (y) which describe

(b)

angular momentum densities. As in (V.1)-(V.2), ei (y), ek (y) denote orthonormal bases (a)

i such that ei (P ) = δa and its dual are parallel propagated along geodesics through P and ? (a)

is a su?ciently small ball with centre at P. As in Section V we apply the fact that at P the tetrad and normal components of an object are equal and so we again omit tetrad brackets for indices of any quantity attached to the point P. For matter as M ikl (y) we take

mM ikl

(y) =

√ ?g(y i T kl ? y k T il ),

(VI.26)

where T ik are the components of a symmetric energy-momentum tensor of matter and y i denote the normal coordinates. The formula (VI.26) gives the total angular momentum densities, orbital and spinorial because the dynamical energy-momentum tensor of matter T ik comes from the canonical energy-momentum tensor by using the Belinfante-Rosenfeld symmetrization procedure and, therefore, includes the spin of matter [5]. Note that the normal coordinates y i form the components of the local radius-vector y with respect to the origin P. In consequence, the components of the m M ikl (y) form a tensor density. For the gravitational ?eld we take the gravitational angular momentum pseudotensor of Bergmann and Thomson (IV.12) to construct

gM ikl

(y) =F U i[kl] (y) ?F U k[il] (y) +

[kl]

i k |g|(yBT tkl ? yBT til ),

(VI.27)

im where F U i[kl] := gF Um

are von Freud superpotentials (III.3).

In fact, the Bergmann-Thomson pseudotensor can be interpreted as the sum of the spinorial part S ikl :=F U i[kl] ?F U k[il] and the orbital part O ikl := √

i k ?g(yBT tkl ? yBT til )

(VI.28)

(VI.29)

16

of the gravitational angular momentum densities. Substitution of (VI.26) and (VI.27) (expanded up to third order) into (VI.23) gives the canonical angular supermomentum tensors for matter and gravitation, respectively [9],

mS abc

(P ; v l ) = 2[hap ?p T bc ? hbp ?p T ac ],

(VI.30)

gS

abc

(P ; v l ) = αhpt [(g ac g br ? g bc g ar )?(t Rpr) + 2g ar ?(t R(b p r) ? 2g br ?(t R(ap 2 a + g bc (?r Rr(t ap) ? ?(p Rt) ) 3 2 b ? g ac (?r Rr(t bp) ? ?(p Rt) )]. 3

c) c) r)

(VI.31)

Both these tensors are antisymmetric in the ?rst two indices S abc = ?S bac . In vacuum, the gravitational canonical angular supermomentum tensor (VI.31) simpli?es to

gS abc

(P ; v l ) = 2αhpt [g ar ?(p R √

(b c) t r)

? g br ?(p R

(a c) t r) ].

(VI.32)

Note that the orbital part O ikl =

supermomentum tensor g S abc (P ; v l ) and m S abc (P ; v l ) of gravitation and matter do not require the introduction of the notion of a radius vector.

Only the spinorial part S ikl =F U i[kl] ?F U k[il] contributes. Also, the canonical angular

i k ?g(yBT tkl ? yBT til ) gives no contribution to g S abc (P ; v l).

After some algebra, one may show that there are the following transformational rules for the angular supermomentum tensors of matter m S ikl (P ; v a ) and for pure gravitation (which is composed of the Weyl tensor only) g S ikl (P ; v a) under conformal transformation (II.1):

mS

?ikl (P ; v a ) = ??8 m S ikl (P ; v a) + 4??8 [hp[i(P k]pr T rl ? + P lpr T k]r )] ? 24??9 hp[i?,p T k]l ; (VI.33)

|s| k) b p)

gS

ikl

(P ; v a ) = ??6 g S ikl (P ; v a) + 2α??6 hpt {g ib[P ? + P

(k l) s (t|s| C b p) (l (t|s| C i) b)

(l

(t|s| C (l b)

? P s(tb C

(l k) |s| p)

? P s(tp C

k) s] (l |s| i) p)

? g kb[P

|s| i) b p)

+P

(i l) s (t|s| c b p)

? P s(tb C

(l i) b p) ]

? P s(tp C

(l

s ]} (l k) b p)

? 4α??7 htp [g ib ?,t C 17

? g kb?,t C

,

(VI.34)

with hlm and P abc given by (V.5) and (V.12) and C abcd being the components of the Weyl conformal curvature tensor (II.18). It is obvious that in a conformally ?at spacetime, one has

gS

?ikl =g S ikl = 0.

(VI.35)

VII.

CONCLUSION

In this paper we have analyzed the rules of the conformal transformations of the energetic and superenergetic quantities which were proposed in general relativity and can be applied to some extended theories of gravity in which physics is studied in di?erent conformal frames. In particular, we have found the rules of the conformal transformation for the energetic quantities such as the Einstein energy-momentum complex, the Bergmann-Thomson angular momentum complex, the superenergy tensor, and the angular supermomentum tensor of gravitation and matter. We have shown that the conformal transformation rules for the matter parts of both the Einstein complex and the Bergmann-Thomson complex are fairly simple (Eqs. (III.9) and (IV.16)), while the transformation rules for their gravitational parts are more complicated (Eqs. (III.7) and (IV.17)). We have also found that the transformational rules of the superenergy tensor of matter (V.11) and the superenergy tensor of gravity (V.14) are quite complicated, except for the case of a pure gravity (V.20). In such a special case the superenergy density as well as the sum of the matter energy density and the superenergy density are invariants of the conformal transformation, i.e. ? ?? Tik +g Sik = g ? √ ?g Tik +g Sik .

Besides, in that case (of a pure gravity), a conformal invariant is also the Bel-Robinson tensor ? Bkiab = Bkiab , which is a part of the superenergy tensor. As for the angular supermomentum tensor of gravity - it emerges that its transformational rule (VI.34), even for a pure gravity, is quite complicated. This, however, is not the case for the angular supermomentum tensor of matter (Eq. (VI.33)). 18

Some other remarks from our investigations are as follows. The conformal transformation rule of the Einstein pseudotensor (III.7) vastly simpli?es in D = 2 dimensional spacetime. This seem to re?ect the fact that all the two-dimensional manifolds are conformally ?at. On the other hand, both a pure gravity superenergy tensor and a pure gravity angular supermomentum tensor vanish in all conformally ?at spacetimes. Because the superenergetic quantities are constructed of some combinations of the geometric quantities, we have studied the rules of their transformations from one conformal frame to another. In particular, we have derived the rules of the conformal transformation for the curvature invariants R2 , Rab Rab , Rabcd Rabcd (Eqs. (II.21)-(II.23)) and the GaussBonnet invariant RGB = Rabcd Rabcd ? 4Rab Rab + R2 (Eqs. (II.24)-(II.25)) in an arbitrary spacetime dimension. All the rules we found would be applied to the discussion of the conformal transformation of energetic and superenergetic quantities in some special models of spacetime [33]. Especially, these rules should be very e?ective to calculate energetic and superenergetic quantities in conformally ?at spacetimes. Quite recently, the form of the energy-momentum complexes within the framework of extended gravity f (R) theories have been studied [34]. In fact, it is very common to use conformal frames (Jordan and Einstein) in presentation of these theories [35] so that our results of Sections III and IV can be applied to study the sensitivity of the energy-momentum complexes to conformal transformations in these more general theories of gravity. On the other hand, since the complexes are sensitive to coordinate transformations, then it would be much better to study the superenergy and the angular supermomentum (which are covariant) in f (R) theories of gravity with the stress onto the problem of their sensitivity to conformal transformations.

VIII.

ACKNOWLEDGMENTS

The authors acknowledge partial support of the Polish Ministry of Education and Science grant No 1 P03B 043 29 (years 2005-2007). We thank Adam Balcerzak for assistance in

19

deriving one of the formulas.

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21