CC BY
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Becthhk PYflH, Cephs 0U3UKÛ. № 11. 2003. c. 55-75
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UDC 530.12:531.51
The Effective Energy-Momentum Tensor in Kaluza-Klein Gravity With Large Extra Dimensions and Off-Diagonal Metrics
J. Ponce de Leon
Laboratory of Theoretical Physics Department of Physics University of Puerto Rico P.O. Box 23343, San Juan, PR 00931, USA
We consider a version of Kaluza-Klein theory where the cylinder condition is not imposed. The metric is allowed to have explicit dependence on the "extra" coordinate(s). This is the usual scenario in brane-world and space-time-matter theories. We extend the usual discussion by considering five-dimensional metrics with off-diagonal terms. We replace the condition of cylindricity by the requirement that physics in four-dimensional space-time should remain invariant under changes of coordinates in the five-dimensional bulk. This invariance does not eliminate physical effects from the extra dimension but separates them from spurious geometrical ones. We use the appropriate splitting technique to construct the most general induced energy-momentum tensor, compatible with the required invariance. It generalizes all previous results in the literature. In addition, we find two four-vectors, and induced by off-diagonal metrics, that separately satisfy the usual equation of continuity in 4D. These vectors appear as source-terms in equations that closely resemble the ones of electromagnetism. These are Maxwell-like equations for an antisymmetric tensor F^ that generalizes the usual electromagnetic one. This generalization is not an assumption, but follows naturally from the dimensional reduction. Thus, if could be identified with the electromagnetic tensor, then the theory would predict the existence of classical magnetic charge and current. The splitting formalism used allows us to construct 4D physical quantities from five-dimensional ones, in a way that is independent on how we choose our space-time coordinates from those of the bulk.
Key words and phrases: Kaluza-Klein Theory; General Relativity.
1. Introduction
Most of the recent advances in theoretical physics deal with models of our universe in more than four dimensions. Theories of the Kaluza-Klein type in many dimensions are used in different branches of physics. Superstrings (10D) and supergravity (11D) are well known examples.
In gravitation and cosmology the study of multidimensional theories called "brane world" models as well as the space-time-matter (STM) theory in 5D, which seeks the unification of matter and geometry, have become quite popular. Although, these theories have different physical motivation for the introduction of a large extra dimension, they share the same goals and philosophy [1].
The crucial question in these theories can be split up in two parts. (1) How to extract the correct 4D interpretation from geometrical objects, like scalar dilatonic fields and antisymmetric forms, which appear in more than four dimensions? (2) How to predict observable effects from the extra dimensions? It seems to be obvious that the successful answer to the second question requires the correct answer to the first one.
1.1. Large Extra Dimension
This paper deals with the first part of this question. We consider a general version of Kaluza-Klein theory in 5JD where the extra dimension is not assumed to be com-pactified. From a mathematical viewpoint, this means that the 5-dimensional metric tensor is allowed to depend explicitly on the fifth coordinate. This is the common assumption made in brane-world theory and STM [1-6].
However, we extend the discussion to the case where the five-dimensional metrics possess non-vanishing off-diagonal terms. The concrete question that we ask here is the following: What is the more general expression for an energy-momentum tensor in four-dimensions, that can be constructed with the elements of the 5D theory (The discussion can be easily extended to any number of dimensions. In this case the question is about the energy-momentum tensor constructed in one lower dimension).
The appearance of the extra dimensions in 4D physical laws is usually avoided by imposing the "cylindricity condition", which essentially requires that all derivatives with respect to the extra coordinate(s) vanish. However, this condition is too restrictive. Indeed, this, condition can be replaced by a less stringent one. Namely the requirement that physics in 4D should be invariant under coordinate transformations in 5D (to which we will refer as gauge invariance). Although cylindricity always implies gauge invariance, this invariance does not mandatory require cylindricity.
Therefore, in this work we require gauge invariance of 4D physical quantities instead of cylindricity. In a recent work [7] we applied the same requirement to study the motion of test particles in Kaluza-Klein Gravity. Gauge invariance allowed us to clarify a number of aspects of the so called "fifth force". In particular, we provided a new definition for this force where it is gauge invariant and orthogonal to the four-velocity of the particle [8].
1.2. Previous Results
Restricted versions of the problem under consideration here are well known in the literature.
The first version is the "classical" Kaluza-Klein theory, where the metric in 5D is restricted by the cylinder condition. This means that the extra dimension is assumed to be compactified. In this case, the five-dimensional Einstein equations, in vacuum, may be reduced to ones that resemble the 4D equations of general relativity in the presence of an electromagnetic field [2-10]. Here, the dimensional reduction is quite simple; in practice it amounts to integrating over the extra variable in the action integral.
The second version [11] relaxes cylindricity, i.e., the extra dimension is not assumed to be compactified. As a consequence, a simple integration in the action is no longer possible and the dimensional reduction is much more involved than in the classical case. However, a great simplification is attained if one restricts the discussion to the case where 7^4 = 0.1 This is the usual working scenario in brane-world and STM theories [1]. This restriction amounts to describing a space-time free of electromagnetic field. The induced AD energy-momentum tensor presents reasonable physical properties. In the context of brane-world theory, it predicts local and nonlocal five-dimensional corrections to general relativity. In STM, it constitutes the basis for the geometrical interpretation of matter [12]. As an important illustration of this, we mention that the 5D equations, under the appropriate symmetry conditions, lead to Friedmann-Robertson-Walker cosmological models with perfect fluid source, and satisfying the barotropic equation of state p = rip [13].
Both restricted versions use the tacit assumption (commonly seen in the literature) that the first four coordinates correspond to the physical space-time, while the remaining one is the extra dimension. The obvious question is: given an arbitrary 5-dimensional metric, how do we know, a priori, which one is the "extra" coordinate? What if we choose our space-time coordinates in some another way, say some combination (or function) of the bulk coordinates for example?
Mn this paper denotes the five-dimensional metric
1.3. General Case: Off-diagonal Metrics
Our work is free of these assumptions. In this paper we construct the most general "induced" energy-momentum tensor compatible with gauge invariance, instead of cylindricity, and 7^4 ^ 0. We provide a method to obtain physical quantities from 5D that is independent on how we choose our space-time coordinates. The dimensional reduction for this general version generalizes all previous results in the literature. In addition, we find two conserved four-vectors, which appear as source-terms in equations that closely resemble those of electromagnetisrn. In this analogy, one of them corresponds to an effective electrical charge and current, while the other 4-vector would correspond an effective magnetic charge and current.
The paper is organized as follows. In Section 2, we consider the 5D line element and field equations and provide some details of our splitting technique in an arbitrary set of coordinates. In Section 3, we do the splitting of the field equations. They are expressed in terms of three quantities only. These are well defined, gauge invariant, 4D quantities constructed from the elements of the 5D theory. In Section 4, we introduce the physical metric and construct the energy-momentum tensor as measured by an observer in physical 4D space-time. In Sections 5 and 6, we introduce what we call the coordinate frame and show that our general approach contains all previous results in the literature. In Section 7, we discuss the fully general case. We will see that, as a consequence, of the anholonomic nature of the theory, there is a quantity that can be interpreted as magnetic charge and current. It is represented by a four-vector that satisfies the usual continuity equation. Finally, in Section 8, we present a summary of our conclusions.
2. General Framework
In this section we will show how to construct four-dimensional quantities from five-dimensional ones. The construction is independent on the specific coordinates used in the 5D manifold and the way we choose coordinates in AD. Then we introduce the local frame, which is the frame attached lo an observer in 4D.
We will consider a general five-dimensional manifold with coordinates (A = 0,1,2.3,4) and metric tensor -jab(^c)- The 5D interval is then given by
dS2 = lABdÇAdÇB. (1)
The field equations are taken to be the 5D Einstein equations,
Uab ~ abH = kTAB (k = const), (2)
where 7Zab is the Ricci tensor constructed in terms of the Christoffel symbols
£BC = ]pAD (7DB,C + lDC,B - 1BC.D) , (3)
% = "fABTZab is the five-dimensional scalar curvature, and TAB a five-dimensional energy-momentum tensor.
We have already mentioned that are two casés where the (4+1) separation of these equations is quite straightforward. In the first case the metric is general, except for the restriction that it does not depend explicitly on the extra coordinate. In the second case the metric is allowed to depend on the extra coordinate but it is restricted by the assumption 7/i4 = 0. In both cases, the assumption that the first four coordinates represent the physical space-time, allows to obtain the 4D part of (2) and (3) by simply putting A,B = 0,1,2,3.
However, this simple method does not work in the general case under consideration here. Indeed, the 4D quantities defined with this straightforward identification (that
A, B — 0,1,2.3 are the space-time components of (2) and (3)) are not invariant under transformations in 5D [7,8].
Thus, in the general case where 74^ ^ 0 and the metric is allowed to depend on the extra variable, another splitting method should be used in order to assure gauge invariance of physical quantities.
In this paper we use the "local" frame of reference, introduced in a previous work [7], to reduce the five-dimensional equations and construct the physical quantities in 4D that are invariant under transformations in 5D.
2.1. Splitting Technique
To facilitate the discussion and set the notation, we start with a brief summary of our splitting technique. For details we refer the reader to Ref. [7] We assume that the 5D manifold in (1) allows us to construct, appropriately (see bellow), a four-dimensional hypersurface that can be identified with our 4D space-time. In this hypersurface we introduce a set of four parameters (¿x = 0,1,2,3), which are functions of
(4)
The derivatives of these functions with respect to £A
eM - — (5)
A - W
behave as covariant vectors2 with respect to changes = in 5D, and as
contravariant vectors with respect to transformations xM — x^{xv) in 4D. At each point these vectors are tangent to the hypersurface. Therefore, in the region where they are linearly independent, they constitute a basis for the 4D hypersurface under consideration. We will interpret this, appropriately defined 4D manifold, as the physical space-time and as the coordinates in it.
We can now introduce the vector ■ijjA, orthogonal to space-time. This is completely determined by
= 0, (6)
■ 1ab^B = e,
where e is taken to be +1 or -1 depending on whether the extra dimension is timelike or spacelike, respectively. We will also need the set of vectors e^, defined as
= 0.
They allow us to express any infinitesimal displacement in 5D as
d£A = ¿fa dxM + eiPAdx(4), (8)
where dx^) = and dx(4) = ifjBd£B represent the displacements along the
corresponding basis vectors. From this, it follows that
ef^ef (9)
2The index in parentheses numbers the vector, while the other one indicates its coordinate in 5D.
Вестник РУДН, Серия Физика. № 11. 2003. с. 55-75__59
2.2. Local Frame
Thus, in the neighborhood of each point the observer is armed with five independent
vectors, ё^ and фв, that constitute its frame of reference. In order to simplify further calculations we introduce a more symmetrical notation. To this end we set
(10)
(П)
e>A ~ VA, ef4) = ei>A.
Now, Eqs. (6), (7) and (9) become
_ rB eae(c) — °c>
ec (a1) — °C ■ Finally, we define our "local" hD metric 5(a)(* as
9{A){* = (12)
which neatly separates the five-dimensional manifold in (4 + 1) parts, viz.,
km- = (V ■ C3)
where gis the metric (to which we will refer as the induced 4D metric3 ) of the four-dimensional space-time section, viz.,
= (14)
The advantage of the local frame is that it provides a (4+1) separation which is invariant under arbitrary changes of coordinates in 5D. Namely, putting (^4),(* = 0.1,2,3 we obtain the space-time part of the local metric. We will use this frame, in the next Section, to perform the separation of the five-dimensional equations (2). The 5D interval becomes
dS2 = g{A){MA)dx^. (15)
In addition, from (11), it follows that
Iab = e>A eB 9(C)(D)- (16)
We also notice
7** = efai?Q)9™Q\ (I7)
where
g(P){Q) = ^ 0) _ (lg)
Finally, we mention that basis indexes are lowered and raised with g(A)(*> while 5D coordinate indexes are lowered and raised with 7ab-
3Later, in Section 4.1 we will define the physical or observable 4D metric as being conformally related
to the induced one.
3. Field Equations in the Local Frame
In this Section we obtain the explicit form of the field equations in an arbitrary local frame. They neatly separate in three sets of 1, 4 and 10 equations. These equations are expressed solely in terms of well defined, gauge invariant, 4D quantities constructed from the 5D theory.
In the local frame, the 5D line element becomes
dS2 =gMMdxMdxM+E{dxW)2. (19)
The field equations (2), using (16), can be written as
= k (j[A){* - ■. (20)
where
= epA)eftTZpQ (21)
are the components of the five-dimensional Ricci tensor on the local frame. T(A){* ~ e^je^TpQ and T are, respectively, the basis components of the 5D energy-momentum tensor and its corresponding trace.
Now, since the local frame provides a clear separation between the space-time and the extra part of the metric, setting A —> ¡j, and B —> v in H{a){* we obtain the 4D part of this 5D tensor. This is one of the advantages of using the local frame.
3.1. Christoffel Symbols
The first step, in order to find the basis components H(a)(*' >s to obtain an appropriate expression for the "coordinate" Christoffel symbols (3) in terms of basis vectors. For this we substitute (16) into (3). After some manipulations, we get
r-D _dP)-(Q)~D (MR) ,y(R) \ /99x
AC ^A eC e(R) \K{P)(Q) + V(P)(Q)j > \ZZ>
where K^j^ are the Christoffel symbols constructed with the local metric, viz.,
= 2^(R)(S) ($(s)(p)KQ) + ff(s)(Q)i(P) ~ ff(P)(Q)i(s)j , (23)
and
WL = \vmQ) + fom^m) ~ \s$(Ly (24)
Here we have introduced the antisymmetric and symmetric quantities f{p)(Q) and
c(M)
(P)(Q)'
W _ ¿(M) (¿N _ ¿N \ «n
N \e(P)\{Q) (Q)l(-P)J ' ^o)
MM) _ 4M) / -TV , .JV \
In Eqs. (23)-(26) and in what follows, the symbol indicates directional derivatives. Specifically, for an arbitrary quantity <t> we have
d<t> .A dcj)
ip)(Q) ~ e
The antisymmetric nature of Jyp)(q) implies that -Tjp^q) — 0. Indeed, setting M —> A in (25) and using orthogonality conditions (11) in (25) we obtain
7t(a) _ ¿P ¡.W zP ¿W /oq\
We now need to remember that e^ = (dxfi/d^A). Using this, and since (id2/diPdiQ) = (d2/dtQdtP), it follows that
= 0- (29)
Consequently, only the ten remaining components -^(p^q) are different from zero, in
general. In a previous work [7], we have seen that six of them are associated with the electromagnetic field, while the other four with the so called "fifth" force. In what
follows the index "(4)" in will be suppressed.
3.2. Ricci Tensor
The next step consists in substituting (22) into the usual definition for the Ricci tensor Kab- At this point one should proceed with care because this tensor involves second derivatives of the metric, which in the local frame become directional derivatives, and these derivatives do not commute, in general.4 Therefore, in order to keep the symmetry of the Ricci tensor, we should symmetrize the corresponding derivatives, where appropriate.
The calculations are quite long and involved. However, the final result is rather simple. It is expressed in terms of three quantities only. Namely;
(i) the four-dimensional Ricci tensor R^ constructed with the Christoffel symbols obtained from the induced 4D metric, viz.,
^ = (^K") + 3<TH(M) _ 9 »»>(*)) = OO)
(ii) the derivatives of the induced AD metric along the fifth dimension, which we denote as
(31)
and
(iii) the antisymmetric tensor ^a)(* defined in (25). Thus, omitting intermediate calculations, we have
(4)(4) = - (%) + K*XP) + 4^(A)(P^(A)(P) - * (Aa)^A) + £<m)<4) = ¿(A) (fy - + +
= Ku - e (>ivi(4) - MnH + - \ + DHrM)
(32)
2
(p)
- T^IL)?[v) ~ -T(Al)(p)f („) •
Here = 7{{%{vy TUl) = (h)(4) = D is the °Perator °f covariant
differentiation calculated with the AD Christoffel symbols5 (30). Thus, D^g^v = 0.
4lt may be worth mentioning that, although with different purposes, anholonomic frames have recently been considered for the study off-diagonal metrics in Einstein and brane gravity [I4-16|
5The parenthesis in the index of the covariant derivative is kept in order to remind us that it involves the use of directional derivatives (27). Thus, D(P)V\ = V\\(p) ~
In (32) indexes are lowered and raised by the induced metric g^. # = g^^u, and ^ = -(1/2) (gitv)|(4), which follows from (31) and g^gv\ = 5$.
The above equations (32) are totally general. For their derivation we have made no assumptions; neither on the character of the metric nor imposed restrictions on the frame of basis vectors used. In addition, they are (by construction) invariant under general transformations of coordinates in 5D. We note the simplicity of these
equations. In particular, as we can see, the symmetric quantity defined in (26)
appears nowhere in the final result.
4. Interpretation in Four-dimensions
As we have already mentioned in the Introduction, one of the crucial questions in discussing possible observational manifestations of the extra dimension is: how to give the correct four-dimensional interpretation to the geometrical quantities that appear naturally in theories formulated in more than four dimensions?
In our five-dimensional case these quantities are: the antisymmetric tensor
the four-vector and the symmetric tensor We are going to interpret them in terms of an effective four-dimensional energy-momentum tensor.
4.1. Four-dimensional Physical Metric
Firstly, we have to decide about another fundamental question, namely: how to identify the physical or observable space-time metric (which we will denote as from the induced one?.
The simplest choice is to assume g^ — g^. However, this choice may not describe all possible physical scenarios.6 In addition, there remains the freedom of multiplying this metric by a conformal factor, namely,
g^u = e20 g^ (33)
where 0 depends on the dilatonic fields present in the theory. In compactified Kaluza-Klein theory the necessity and uniqueness of the conformal factor has been widely discussed in many papers [19-24].
In this work we will adopt (33), which for /3 = 0 reduces to the case g^ = Moreover, in order to keep the generality of the discussion, and for future references, we allow /3 to be an arbitrary function of the space-time coordinates and the extra variable.
Physical predictions, in particular the interpretation of the matter content of the space-time as well as the behavior of test particles, crucially depend on the choice of Q.
Mathematically the transition from the induced metric to the physical one is quite simple. It amounts the replacement in the field equations of "induced" quantities (i.e., calculated with ¿?M„) by "physical" ones (calculated with g^). Christoffel symbols are obtained by substituting (33) into (30)
K = np + (pw8$ + fatf - P^gxp) . (34)
Thus, the Ricci tensor of the physical space-time, is given by
= -fiv - {dfra») + + wmpm ~ 9»u h(3wp{x) + £(a)/3(a)] • (35)
6In particular, in the context of STM, the identification v = together with the restriction of independence of the fifth coordinate, lead to a radiation-like equation of state for the induced matter. Also, for static spherically symmetric solutions they lead to an effective energy-momentum tensor which is anisotropic in a number of cases [2,17,18]
Becthhk PyflH, Cepha &u3um. № 11. 2003. c. 55-75
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Here /3(a) = 0\(a)< ar]d D is the operator of covariant differentiation constructed with the given above.7 Therefore, D^g^ = 0. In Eq. (35) and in what follows, indexes are lowered and raised by the physical metric
4.2. Four-dimensional Energy-momentum Tensor
We now proceed to construct the energy-momentum tensor TM!, corresponding to the physical metric g^lu specified in (33). Following the usual procedure, we define the energy-momentum tensor in 4D through the Einstein equations in 4D, namely
Tfj,v — R^v - -g^R, (36)
where R = gai3Ra/3 is the scalar curvature of the physical space-time. We also introduce the notation
^ = + ^
Such that = e20%v\ $ = $ = V^g»", and - e'2/3^x.
Now, we substitute Eq. (35) into (32) and change the covariant derivatives from D to D. Then, using the field equations (20) and the above notation we obtain the scalar curvature of the physical space-time as follows
R = ee™ (*2 - fl^) + +
+ (38)
In a similar way, we obtain from Eq. (36) an effective energy momentum tensor that clearly separates into six parts, namely,
¿V - - T<i> + r£'> + t^ + + iff + (39)
where
lV,)=ee2i3
- Iiv
3V|(4) + + 2,5,(4)) (v2 + *xP*Xp + 2^,(4)
- fl V
"" 2
Tiln) = (DM0H + D^Pm) - 2 betaMPM - g^ (2D(p)f3^ + 0{p)^> ThV = - (^)^m + >
= kef^TAB.
(40)
Let us note several features of the above equations:
1) depends only on the derivatives along the fifth dimension; T^ depends only on the four-vector T^y, T^i1^ depends only on the conformal factor;
T(/v') depends
only on the antisymmetric tensor T^V is a kind of "interaction" between
the fields T^) and /3^y, finally T^1* represents the space-time projection of the five-dimensional energy-momentum tensor on a general local basis.
7The relation between D and D is as follows: £>(M)V„ = D{li)Vp - (Pi{M)Vv + H^V^ - P't(a)Vagliv).
2) The interpretation of matter content of the space-time is fixed solely by the choice of four space-time basis vectors e^. They in turn define from (6) and (10). Once this choice is made none of the contributions to TMt, can be altered by changing coordinates in 5D.
3) The factor e appears in front of and T^vv\ Therefore their effects on gravity in AD will depend on whether the extra dimension is spacelike or timelike. The contribution from the other components of the energy-momentum tensor is the same in both cases. This can also be seen from (38).
4) The special nature of T^/J and T^l^ (which we will see describe neutral matter and electromagnetic field, respectively) is also reflected by the multiplicative factor in front of them. While the contribution from neutral matter is enhanced by a factor e2/3, the one from the electromagnetic field is diminished in the same proportion. A similar effect occurs with the scalar curvature (38).
5) Notice the absence of constants coefficients in (39). The only "arbitrary" assumption we have made so far is in the choice of the norm of vector ip in (6). Therefore, one could ask whether it is possible or not to introduce some particular constants in (39) by changing this norm, say putting 7abtPaiPb = ¿N2, N = constant. The answer to this question is negative; the whole construction is invariant under this change.
For completeness we provide an alternative expression, which is useful in particular
applications. It is obtained from the field equations (20) and 7£(4)(4) in8 (32):
- (e (*|(4) + W») + k- (2i(<<> - . (41)
Finally, we stress the fact that in Eqs. (35)—(41) we have lowered and raised indexes with the help of the physical metric g^.
4.3. Maxwell-like Equations
Equations (38), (39) constitute eleven of the fifteen field equations in five dimensions. The other four are obtained from (20), (32) and (33), viz.,
D
(A)
where
„-2/3
2 M + »
-20
T,
(nX/O {F{P)+0(P))
+
+
£0{<r) (
P* = e ( V* - Sty
= * - ¿j*)
(42)
(43)
We will see that the four equations (42) can be reduced to the inhomogeneous Maxwell equations, for electrical and magnetic field, containing sources. While the homogeneous Maxwell equations, ie., those without sources can be obtained from the condition
•^(.4)(*!(C) + + F(C)(A)\{* =
— + F(*{c)F{A)(i) + F{C)(A)F{*(A) : (44)
»Here we use that - (fc/3) {it™ - fg).
which follows from the antisymmetric nature of T(A){*< and the commutativity relation d2<p d2cf>
d^d^A) ~ dxWdx^ ~ ^1{A){* ~ ~ (45)
for an arbitrary function <j>. Condition (44) requires the identical fulfillment of the relations
^M^W) = -^(aOIM - (46)
and
-^(AOHKA) + -^(A)^) + -^(AXm)!^) = + + (47)
It is clear that this identity under certain conditions should lead to the one for the electromagnetic tensor.
5. Coordinate Frame
The equations in Sec. 4 are rather general, they are useful for the interpretation of bD metrics in an arbitrary frame. The choice of any particular local frame does not impose restrictions on the geometry. However, it should be clear that, the physical interpretation of any given metric in 5D crucially depends on this choice. In fact, the analysis of a given metric in two different (local) frames, will result in two different interpretations for the material content of the space-time.
Specific results in the literature frequently assume that the first four coordinates correspond to the physical space-time, while the remaining one is the extra dimension, viz.,
x" = (48)
The space-time basis vectors are
e® = (1,0,0,0,0),
= (0,1,0,0,0), = (0,0,1,0,0), e{*] = (0,0,0,1,0).
From (6) and (10) we find
,a f----*
(49)
ef4)=e^ = ^0,0,0,0,-J, (50)
where we have set 744 = s4>2. The above set of vectors constitute what we call coordinate frame.
The associated basis vectors are given by (11). Denoting 7/i4 = we obtain
ef0) = (1,0,0,0, ef1} = (0,l,0,0,->li),
ef2) = (0,0,1,0,-A2), (51)
ef3) = (0,0,0,1,-^), e^ =e$(Ao,A1,A2,A3,l).
The 5D line takes the well known form
d<S2 = e2/Vda^dx" + e$2 (d£4 + A^dx")2, ■
g^ = = -y^ - e&A^Av.
We will keep the use of £4, in order to avoid any confusion with the "physical" displacement along the extra dimension.9 From (25), and using the above basis vectors, we obtain the specific form of .F(M)(,,) and in the coordinate frame. Namely,
f(Hkp) = (api(m) ~ AM(P)) '
*m - ■ (53)
In this frame the rules for derivation, for an arbitrary function <f>, are
<Pa
$ ' (54)
<l>\(p) = -
1(4) = e-
where a comma denotes partial differentiation. The above quantities are invariant under the set of "gauge" transformations
x» = x",
e=i4+f(x°,x\,x\x% (55)
that keeps the shape of the line element (52). As a matter of fact, all physical quantities in equations (38)-(47) are invariant under these transformations.10 In what follows we will use the quantity
F„p - (Ap^ - AMp)) = {DMAP - D(p)Ap) =
= (Ap<li - A^p) + [ApAp,4 - ApAp,4). (56)
Which, in the case of no dependence on the extra variable, reduces to the usual electromagnetic tensor, viz.,
Fpp = (Ap,p~Ap,p). (57)
Therefore we will refer to (56) as the generalized electromagnetic tensor. Thus, in what follows
= eQF^. (58)
Since the choice of the coordinate-frame seems to be the simplest one, for a given metric, the question may arise of whether a simplified version for the effective energy-momentum-tensor (39), (40) can be derived in a simpler way by employing the coordinate frame from the outset.
The answer to this question is negative, at least for the general case where cylin-dricity is not assumed and 74^ / 0. Indeed, the explicit coordinate-frame version of (40) is quite cumbersome.
9In this frame, space-time displacements are <±em = d£4, while the ones along the extra dimension are ds(4) = = ei>(d£4 + ApAx").
10Under transformation (55): 7^ = 7^ + si2 {Apftl/ + Auf,p + f,p,f,v) and Ap - (Ap + f^). The induced metric g^ as well as the physical one g^v remain invariant. The derivatives change as g^v^x ~ 9p.u,a but the Christoffel symbols and remain invariant. Also, = +#,4/^,
Ap,1, - AplV + Apt4f^ + f,p,v and Ap\(u) = Ap|(l/) + /iit)„. Therefore, the antisymmetric tensor F(p)(p) and the four-vector T^) remain invariant.
5.1. ^-dependence of induced four-dimensional fields
Since the bulk metric 7ab is allowed to be a function of the extra coordinate, the induced physical fields g<P and F\IV depend on the space-time coordinates and also on the fifth coordinate. However the ^-dependence of these fields is non dynamic. Besides, these fields are gauge invariant under (55).
The physical meaning of (55) is that £4 can take any value independently at each space-time point. It reflects the arbitrariness in the choice of origin for £4 at each space-time point. Therefore, without loss of generality we can set £4 = ^ = const, everywhere in space-time. Consequently, 4D quantities will depend on x°, x1, x2 and x3 only, but not on £4.
6. Particular Cases and Previous Results
In general, we cannot expect to obtain simplified versions for (39), (40) except in the case where we introduce some physical simplifying assumptions. The simplest versions are generated by (45). They correspond to the cases where the (directional) derivatives commute. They are:
(i) no dependence of the extra variable,
(ii) no electromagnetic field,
(iii) no dependence of the extra variable and no electromagnetic field.
6.1. Classical Kaluza-Klein Theory: No Dependence on the
Extra Coordinate and 7^4 ^ 0
In this case from (39) vanishes. The other components reduce to TIlv ^ = £~~2 {^vpFv ~ -^9)ivFxpFxp^j ,
T^V = + № + ,
where semicolon denotes covariant differentiation and <pa = </>jQ.
We now go back to our comment (5), at the end of Sec. 4.2. In order to "derive" the dimensional units in 4D from 5D, some authors put a multiplicative coefficient in front of the extra part of the line element (52). The change $ —> K<P leads to a
coefficient K2 in front of while the other components remain unchanged.
Equation (41) reduces to the wave equation for the scalar field,
p-20
+ 2$af3a = e-^-^FXpFxp - <Pe2/37^j, (60)
while equations (42) and (47) yield equations of "electromagnetic type"11
(V3i^x)A = (61)
A] = 0. (62)
nIn (61) we have used f^H4) = e^Ae^BTAB = ef4)TAB = T4(/i)/<*>. Basis indexes are
"(4)
lowered and raised with the induced metric
6.1.1. /3 = 0
In this case we recover well known results of classical Kaluza-Klein theory. Setting
<P2 - (4G/c4)$2 and k = (8ttG/c4), we obtain T{JJI] = 0, t/P = 0 and =
(47r/c)J", where J" = ec<PT4M. I n this case, our results are identical to those in Ref. [9]. In this reference the connection between T4B and the four-vector charge-current density J'1 is taken as a postulate. In our approach this connection is not a postulate but is a byproduct of the dimensional reduction.
6.1.2.
We recover other particular cases in the literature. See bellow in Sec. 6.3.
6.2. STM and Brane-world Theory: Explicit Dependence on the Extra Variable and 7^4 = 0
The condition 7^4 = 0 requires A^ = 0. The components of the energy-momentum
tensor are identical to (59), except for the fact that now TjhX^ = 0. In this case
TpJ is the combination of two parts.12 One of them, ^T^J, depends only on the
extra derivatives of the physical metric. The other part, ^T^J, contains the extra derivatives of ¡3.
In what follows an asterisk denotes partial differentiation with respect to the extra coordinate, and
The scalar field satisfies the equation
(63)
<p"
-*- T
-2$a0a
e20
<Z>
\P
+ 4 [0 + 0) -l^+I^j + ^J)
(64)
Using this expression we obtain,
tW 4- =
' flV
$
<P
+
ee
W
<2>2
¿V ~ 2tP^pfi + I - ^g^if2 - <p\Pfxp)
+
+
ee
2/3
$2
4 P w - g^ I 2(h2 + § /3 +2^ 0 - 0 -eK™
(65)
These four-dimensional quantities have to be evaluated at some £4 = const. In brane-world models, for convenience, £4 = 0 is usually chosen.
12 (ifi)rp{I)
»2/3
and
(/3)rC) ^
h ( 4^ - + j - 3<?Ml, ( *0 +2 (p)
6.2.1. 0 = 0
Here we recover the effective energy-momentum tensor in STM and brane-world models.
In the context of STM, it is straightforward to verify that our equations (with TZ^ = 0) become identical to those in Ref. [11]. From (42) it follows that the AD
tensor PiLU is a conserved quantity provided T^ = 0, viz., D\PX = 0 [25]. These equations form the basis of STM. From a four-dimensional point of view, the empty 5D equations look as the Einstein equations with (effective) matter. In the case where the bulk metric is independent of the extra dimension, equations (64) and (65) show that the effective AD energy-momentum tensor is traceless, -^¡j/)^ = ^ other
words, independence of the 5D metric from the extra coordinate implies a radiation-like equation of state. Thus the existence of other forms of AD matter crucially relies on the ^-dependence of the bulk metric. A detailed investigation shows that we can recover a number of equations of state commonly used in astrophysics and cosmology [13,17]. In the brane-world scenario, tpfllJ is proportional to the extrinsic curvature of the
brane, which in turn is related, through Israel's boundary conditions, to the energy-
*
momentum tensor of ordinary matter on the brane. The quantities ¡f^, that is the
*
second derivatives of the metric with respect to the extra coordinate, as well as <P are related to the nonlocal Weyl corrections from the free gravitational field in the bulk. With this interpretation, one can show that (64) and (65) look exactly as the equations for gravity in brane-world models (for details see Ref. [1]).
6.2.2. 0^0
Let us first consider the "canonical metric" [2]. This metric has e2S — (l/L)2, where I is the extra coordinate and L is a dimensional coefficient. The (physical)
space-time metric is allowed to depend on the extra variable. With this choice T^J1^ =
) = T^V = 0. In this case, Eq. (65) correctly reproduces (with £ = -1) the effective energy-momentum tensor as discussed in Ref. [2] (pages 157, 159).
Finally, we consider the so called "warped metrics". These are similar to the canonical metrics. The difference is that here $ — 1 and 0 is an arbitrary function of the extra variable, and the space-time metric is independent on it. In this case
= -teg^e20 ^¡3 +2 . (66)
If we set e20 = fi it reduces to the one obtained in Ref. [26], with e = -1.
6.3. No Dependence on the Extra Coordinate and 7^4 = 0
We now consider the case where there is no electromagnetic field and that the conformal factor e20 is some arbitrary function of the scalar field, say, e20 = F(<P). From (59) we obtain T^P = 0 and
^ + n'7/) + = (1 + 2*/) ^ ^ + (1 + XPU - 2<P2f 2) t
(1 + 2*/) + (1 + 3*/ + 2<P2U + <P2f2)
<PQ<Pa
, (67)
where / = (l/2F)(dF/d<2>) and U = d//d<Z>.
If we now choose F = <P2n, where n is an arbitrary real constant, then we obtain the same results as in Refs. [27], and [28] (See also Ref. [2], page 194).
7. General Case: No Cylindricity and 74m ^ 0
The conclusion from Sec. 6 is that our general equations (38)-(47) reproduce well known results in the literature. This allows us to infer that (38)-(46) are reliable and can be used in the general case for further investigations in gauge invariant Kaluza-Klein theory.
The general case, with no cylindricity and 74,, 0, presents two new important features. The first one is the anholonomic nature of the theory. In any local frame, including the coordinate one, the derivatives (45) do not commute. The second new feature concerns the electromagnetic tensor. In order to keep the gauge invariance of the fields we are forced to consider (56), and not (57) as the physical relevant quantity13 . We now proceed to show that, as a consequence of these new features, one can construct two conserved four-dimensional "currents".
7.1. Induced Effective "Magnetic Charge" and "Magnetic
Current"
As a direct consequence of these new features, equation (47) leads to ones that resemble Maxwell equations with magnetic charge and current. In order to show this, we substitute (58) into (47) and use (53). We obtain
+ + hn\(v) = ~ (P^Mx + F^Mp + . (68)
*
Here, and in what follows Mu — A„t4 ~ e^Ayj(4) —Aw Let us now introduce the quantities [29] (Latin indexes run over 123)
Fi = Foi
(69)
B ■■ — F -
and define the vector B dual14 to Bij
Bl = ~^s^kBjk, (70)
where A is the determinant of the metric of the three-dimensional space.15 Setting ¡i i, v j and A —» 0 in (68) and using above definitions16 we get
> 1 d(y/XB)
VXJ'- \/A dxW Jm' (71)
jm — M0B - (E x M).
where M = (Mi,M2,M3). We note that M0 and M behave, respectively, as three-dimensional scalar and vector quantities with respect to 3D spatial transformations x1 = xl(xj), and are invariant under gauge transformations (55). Now setting p —► 1, v -4 2, A ->• 3 in (68) we get
f>m = ~(B • M).
13Under gauge transformations F^ —► F^v + {Au,4fn — A^jf^), while remains invariant.
14Wit.h these definitions, the fields E and B are gauge invariant.
15The space-time metric can be put in the form ds2 = [y^oda;0 + [goi/sfgm)^}2 + [-(g0jg0k/goo) + gik]dxldxk. The metric of the three-dimensional space is A¿j = —gtj + {gojgok/doo)-
16We note that -g = g00\, By = -V\eijkBk and (a x b)' = (1 /2\/\)e^k(ajbk - akbj). In the local frame, we use the following definitions for curl and divergence, respectively: (v x a)1 — (l/2VX)e^k(akm - aiKfc)), and v • a = (l/V^VAa*)^).
We note the formal analogy between (71), (72) and the Maxwell equations in the electric field E and magnetic field B. The quantities pm and jm play the role of "magnetic charge" and "magnetic current" density (in the local frame), respectively. We will refer to them as induced or effective magnetic charge and current. They satisfy the following equation
1 d
^¿^(VA|5m) + v-jm = E-(vxM), (73)
where we have used (46), which now reads — (M^^ - M^^). This equation resembles the continuity equation, except for the appearance of the term on the right-hand-side. This term, is a direct consequence of the anholonomic nature of the theory.
In order to obtain the expression for the conserved 4D magnetic current we introduce the third-rank antisymmetric tensor
J^x = (F„vMx- F„v A^j + (pvxMp- FuX A^ + A,)j , (74)
and define the four-vector J^ dual to JXpa
3!\/~5
J m = (75)
From (68) it follows that
1
J^A - (76)
Now, by virtue of the commutativity of partial derivatives we obtain
1 d
= 0. (77)
This is the continuity equation and shows that the four-vector of "effective" magnetic charge and current (75) is a four-dimensional conserved quantity. In three-dimensional form the above equations read
V • B — Pm, (79)
where pm = ^/gmJ^, = ^/gôôJm and the curl and divergence are defined as usual.17
Let us note several features of
1) Under gauge transformations (55) it transforms as
- J> - ¿^zf h" h- (8°)
However, this non-invariance is not a problem at all because (76) as well as the physical fields E and B are, by their definition (69), invariant under such transformations.18
17(vxa)' = (1/2%/A)£^k(aktj-ajik), and v«= (l/v/A)(v/Aai),i. We note that vxa = vxa+ a xA and v ' a = V ■ a - (l/\/X)(\/Aai)Mi.
*
18We recall that under gauge transformation Fap^ = Faf3^+ Fap f^. is invariant). Therefore,
the transformed quantities J& and Fsatisfy sf^gJL = -(epA/J72)^,A which is Eq. (76). This assures that the conservation of J& is gauge invariant.
2) The factor e appears nowhere in the definition of This means that the induced magnetic current is the same regardless of whether the extra dimension is spacelike or timelike. Also, it is invariant under scale changes of the scalar field, viz., $ -> K0.
3) The no-conservation of the magnetic charge pm and current jm, defined in the local frame, versus the conservation of pm and jm defined above, is not surprising. The reason is that pm is not the total induced magnetic charge. The total charge19 (which is conserved) is given by pm and it is pm = -(B • M) - (v x M) • A.
The conclusion from the above discussion is that if the antisymmetric tensor FpU can be identified with the electromagnetic one, then the theory predicts the existence of (classical) magnetic charge and current. In the case of no dependence on the extra coordinate (79) reduces to the usual v ' B = 0, which allows the existence of static, point-like, magnetic monopoles with B = const(r/r3) [30,31].
7.2. Induced Effective "Electric Charge" and "Electric
Current"
We now turn to the study of (42). After some manipulations, it can be written in
as
1
\f—g dxp
(81)
where we used the same identification between Tab and the four-vector charge and current density as in (61) and Ref. [9]. (d/dxl) denotes usual partial derivative and Jemd represents the induced current of geometrical origin, namely,
47T
— 7M =
e,ind
y/=9
■Ap + 2 tfF^Mp
2<P2e20 [^{AW - + eD[a)P. (82)
Once again, by virtue of the commutativity of partial derivatives and the antisymmetric nature of FpV, the total current is a four-dimensional conserved quantity, viz.,
The factor in the left-hand-side of (81) can be interpreted in terms of classical vacuum polarization as in Ref. [9]. Alternatively, the charge and current associated with this polarization, which is -3($p/<P)F'ip, can be put in the right hand-side of (81) as a part of the induced source.
Thus, in the general case (no cylindricity and -y^ ^ 0), the dimensional reduction provides the set of Maxwell-like equations (76) (or (78), (79)) in three-dimensional form) and (81). The effective 4D sources, (75) and (82), satisfy the usual conservation equations (77) and (83). We recall that all these four-dimensional equations have to be evaluated at some = const.
A similar expression can be found for jm, namely,
jm = MqB - (E x M) + (e XA) + Aq
(V x M) + (m x A j j
8. Conclusions
The splitting procedure we have used here is not intended to solve the field equations. Rather, it is a method to construct the appropriate four-dimensional quantities that are invariant under coordinate transformations in 5D. It has a number of advantages:
(i) The whole discussion is independent of the specific coordinates used in the five-dimensional manifold and the way we choose coordinates in AD.
(ii) It gives a unified approach to a number of particular scenarios discussed in the literature.
(iii) It uncovers the intrinsic anholonomic nature of the theory where, from one hand, the metric is allowed to depend on the extra coordinates and, the other hand, gauge invariance is required.
(iv) It provides some new physical insight. In particular, it shows how to generalize the electromagnetic tensor in such a way that the corresponding physical fields be gauge invariant. It also allows us to speculate about the possible relation between the conserved quantity J^ and the existence of classical magnetic charge. Since it depends solely on the variations of the metric along the extra dimension, this would be a unique effect from the extra dimension.
(v) Since the discussion is independent of the number of dimensions, our results can be used in brane world theories to interpret physical quantities in one lower dimension.
The picture that comes out from our paper is the following. For any given five-dimensional metric, in arbitrary coordinates in 5D, we choose our space-time basis vectors as we wish. Then from (6) and (10) we find e^. This completes our local frame. Next we construct the induced AD metric as in (14) as well as the quantities Tp and
Next we proceed to calculate the energy-momentum tensor. If we choose ¡3 = 0 we go ahead with the calculation and find the different components of J\IV. However, we do not know a priori whether the resulting energy-momentum tensor will satisfy appropriate physical conditions. Here comes a practical motivation to introduce the conformal factor. Allowing 0^0 we add a degree of freedom that can be used to satisfy some physical requirement as isotropy in the pressures or some equation of state.
Thus, the physical conditions imposed on AD matter, like the energy conditions, are the ones that ultimately should determine f3 and, therefore, the physical metric.
The physical metric obtained in this way can then be used to evaluate possible effects from the extra dimension. In particular, we can proceed to evaluate the four-dimensional invariants T^J7^, epvXp T\p as well as the conserved four-vectors J^ and Jg. All these are well-defined AD quantities20 invariant under transformations in 5D. They should help21 in the interpretation and formulation of concrete predictions. For this we need to consider specific exact solutions.
The interpretation of J>'n and Jp as, respectively, the four-vectors of magnetic charge-current density and electrical charge-current density is not mandatory. However, they are conserved AD quantities regardless of their particular interpretation.
The signature of the extra dimension can, in principle, be negative (e = -1) or positive (e = +1). Which means that it can be spacelike or timelike, respectively. In the first case F^, could be interpreted as the (generalized) electromagnetic tensor. However, timelike extra dimensions are also discussed in the literature. In particular they can lead to the existence of multidimensional objects, called "T holes" [24].
20We note that ami J^ depend on the induced metric only. While J^JF^and Jalso
depend on the choice of /3.
21"In relativity, invariants are diamonds. Do not throw away diamonds!"
Neutral matter is mainly represented by Tffl. This tensor shares a common property with the one of the electromagnetic type Tjilv\ Namely, that both depend explicitly on the signature of the extra dimension. However, they have different behavior with respect to scale changes. Rescaling the scalar field $ —> K$ induces a change
(1 /K)2T$ and T#v) K2jfvv). The scaling of the metric e2^„
induces a change T$ e20T$ and e'^T^. The first feature seems
to indicate that the effects from the extra dimension become more prominent in the presence of fields of the electromagnetic type rather than in neutral matter. The second one means that neutral matter dominates over other types of matter at large distances.
In summary, in this work we have provided the most general expression for the energy-momentum tensor compatible with the requirement of gauge invariance, instead of cylindricity. We have considered the most general expression for the physical metric as being conformal to the induced one. The choice of this factor should be limited by physical conditions imposed on the 4D energy-momentum tensor. We have found two four-vectors that separately satisfy the equation of continuity in AD. If the antisymmetric tensor F^ can be identified with the electromagnetic one, then the theory predicts the existence of classical magnetic charge and current.
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УДК 530.12:531.51
Эффективный тензор энергии - импульса в гравитации Калуцы - Клейна для больших дополнительных измерений и
недиагональных метрик
X. Понсе де Леон
Лаборатория теоретической физики Кафедра физики Университет Пуэрто-Рико США, РЯ00931, Сан Хуан, п/я 23349
Мы рассматриваем вариант теории Калуцы - Клейна без цилиндрического условия. Метрика принимается явно зависящей от дополнительных координат. Это обычный сценарий для теорий мировых бран и материи в пространстве - времени. Мы расширяем возможности теории путем рассмотрения пятимерных метрик с недиагональными членами. Условие цилиндричности заменяется требованием инвариантности физики в четырехмерном мире относительно координатных замен в пятимерном мире. Такая инвариантность не исключает физических эффектов от влияния дополнительных измерений, а только отделяет их от числа геометрических эффектов. Мы используем подходящую технику расщепления для построения наиболее общего индуцированного тензора энергии - импульса, совместного с требуемой инвариантностью. Предлагаемая структура тензора обобщает все предыдущие результаты, известные из литературы. Дополнительно мы находим два 4-вектора и , индуцированных недиагональной метрикой, которые по отдельности удовлетворяют обычному четырехмерному уравнению непрерывности. Эти векторы возникают как источники в уравнениях, очень напоминающих уравнения электромагнетизма. Эти уравнения обобщают обычные электромагнитные уравнения и имеют вид уравнений Максвелла для антисимметричного тензора Указанное обобщение является не допущением, а естественным следствием размерной редукции. Поэтому, если бы можно было отождествить Рц„ с тензором электромагнитного поля, то теория предсказывала бы классические магнитный заряд и ток. Примененный формализм расщепления позволяет построить четырехмерные физические величины, исходя из соответствующих пятимерных величин, причем независимо от выбора пространственно-временных координат в исходном многообразии.
