Volume elements of spacetime and a quartet of scalar fields

Frank Gronwald^{1}, Uwe Muench^{1}, Alfredo Macías^{2}, Friedrich W. Hehl^{1}

1 Institute for Theoretical Physics, University of Cologne, D-50923 Köln, Germany 2 Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, P.O. Box 55-534, 09340 México D.F., México (received ...)

Starting with a ‘bare’ 4-dimensional differential manifold as a model of spacetime, we discuss the options one has for defining a volume element which can be used for physical theories. We show that one has to prescribe a scalar density σ. Whereas conventionally \sqrt{|det g_{ij}|} is used for that purpose, with g_{ij} as the components of the metric, we point out other possibilities, namely σ as a ‘dilaton’ field or as a derived quantity from either a linear connection or a quartet of scalar fields, as suggested by Guendelman and Kaganovich. file volume8.tex, rev., 1998-06-15

PACS numbers: 04.50.+h, 02.30.Cj, 04.20.Fy, 12.10.-g

I. INTRODUCTION

A fundamental premise is that gravity is intimately intertwined with the geometry of spacetime. This geometry is locally characterized by two independent concepts: The concept of a linear connection (parallel transport) and the concept of a metric (length and angle measurements).

Within a gauge approach to gravity the existence of the linear connection can be quite satisfactorily explained by the principle of local gauge invariance [1,2]. In contrast to this, it is not clear how to derive the metric from some fundamental principle. Usually the existence of the metric is simply assumed, sometimes in disguise of a local gauge group which contains an orthogonal subgroup. Therefore it is natural to ask whether the metric itself is a fundamental quantity, a derived quantity, or a quantity which can be substituted by some more fundamental field. To investigate this question is the main motivation for this article.

In physics it is of fundamental importance to integrate objects on spacetime. This requires the definition of a volume element. We will point out in Sec. II that this definition can be done without reference to any metric. Basically, a volume element can be defined on any differentiable manifold as the determinant of a parallelepiped defined in terms of n vectors, if n is the dimension of the manifold. Then no absolute volume measure exists. However, proportions of different volumes can be determined. Such a volume element is an (odd) density of weight −1. In order to define an integral, we then need an additional scalar density of weight +1.

Usual physical fields are no densities. Therefore the common practice is to take the components of the metric and to build a density according to \sqrt{|detg_{ij}|}. But there exist alternatives which open the gate to alternative theories of gravitation. In this light we will analyze, in Sec. III, one possibility, namely the quartet of scalar fields, as proposed in [3].

II. INTEGRATION ON SPACETIME AND THE VOLUME ELEMENT σ AS SCALAR DENSITY

We model spacetime as a 4-dimensional differentiable manifold, which is assumed to be paracompact, Hausdorff, and connected. In order to be able to formulate physical laws on such a spacetime, we have to come up with suitably defined integrals. If we want, for example, to specify a scalar action functional W of a physical system,

W = ∫ L = ∫ ϵ̃ L̂ ,  (2.1)

then, taking the integral in its conventional (Lebesgue) meaning, the Lagrangian L has to be an odd 4-form in order to make the integral (2.1) a scalar. Incidentally, a p-form ω = \frac{1}{p!} ω_{i_{1}...i_{p}} dx^{i_{1}} ∧ … ∧ dx^{i_{p}} is called even if it is invariant under a diffeomorphism x^{i} → x′^{i}(x^{j}) with det(∂x^{j}/∂x′^{i}) < 0. It is called odd if it changes sign under such a diffeomorphism [4,5,1].

Now, any odd 4-form can be split into a product of a 0-form (or scalar) L̂ and another odd 4-form ϵ̃. Let us define a trivial physical system by L̂ = 1. Then the integral measures the volume of the corresponding piece of spacetime,

Vol = ∫ ϵ̃ .  (2.2)

For that reason ϵ̃ is called a volume form or, more colloquially, a volume element of spacetime. This quantity can be split again into two pieces.

As the first piece we have the Levi-Civita ϵ in mind. Its components ϵ_{ijkl} are, by definition, numerically invariant under diffeomorphisms. We have

ϵ := \frac{1}{4!} ϵ_{ijkl} dx^{i} ∧ dx^{j} ∧ dx^{k} ∧ dx^{l} ,  (2.3)
with ϵ_{0123} = 1 = invariant .

Consequently ϵ transforms as an odd 4-form density of weight −1 (see, e.g., [1, Appendix A], for details),

ϵ′ = \frac{1}{J} ϵ = \frac{sgnJ}{|J|} ϵ ,  (2.4)

where J = det(∂x^{j}/∂x′^{i}) is the determinant of the Jacobian matrix of the diffeomorphism x^{i} → x′^{i}(x^{j}). We are denoting densities by boldface letters.

If we take the interior product ⌋ of an arbitrary frame e_{α} with the Levi-Civita ϵ 4-form density, then we find a 3-form ϵ_{α}; if we contract again, we find a 2-form ϵ_{αβ}, etc.:

ϵ_{α} := e_{α}⌋ϵ = \frac{1}{3!} ϵ_{αβγδ} ϑ^{β} ∧ ϑ^{γ} ∧ ϑ^{δ} ,  (2.5a)
ϵ_{αβ} := e_{β}⌋ϵ_{α} = \frac{1}{2!} ϵ_{αβγδ} ϑ^{γ} ∧ ϑ^{δ} ,  (2.5b)
ϵ_{αβγ} := e_{γ}⌋ϵ_{αβ} = \frac{1}{1!} ϵ_{αβγδ} ϑ^{δ} ,  (2.5c)
ϵ_{αβγδ} = e_{δ}⌋ϵ_{αβγ} = e_{δ}⌋e_{γ}⌋e_{β}⌋e_{α}⌋ϵ .  (2.5d)

Here, the coframe ϑ^{β} is dual to the frame e_{α}, that is, e_{α}⌋ϑ^{β} = δ_{α}^{β}. The (ϵ , ϵ_{α} , ϵ_{αβ} , ϵ_{αβγ} , ϵ_{αβγδ}) represent a basis for the odd form densities of weight −1. It is called ϵ-basis and can be used to define a metric independent duality operation. Instead of lowering the rank of the ϵ’s, we can also increase their rank by exterior multiplication with the coframe ϑ^{μ}:

ϑ^{μ} ∧ ϵ_{α} = +δ_{α}^{μ} ϵ ,  (2.6a)
ϑ^{μ} ∧ ϵ_{αβ} = −δ_{α}^{μ} ϵ_{β} + δ_{β}^{μ} ϵ_{α} ,  (2.6b)
ϑ^{μ} ∧ ϵ_{αβγ} = +δ_{α}^{μ} ϵ_{βγ} − δ_{β}^{μ} ϵ_{αγ} + δ_{γ}^{μ} ϵ_{αβ} ,  (2.6c)
ϑ^{μ} ∧ ϵ_{αβγδ} = −δ_{α}^{μ} ϵ_{βγδ} + δ_{β}^{μ} ϵ_{αγδ} − δ_{γ}^{μ} ϵ_{αβδ} + δ_{δ}^{μ} ϵ_{αβγ} .

(2.6d)

For the ϵ̃ (which is an odd 4-form density of weight 0), formulae analogous to (2.5),(2.6) are valid. We have just to add twiddles to the ϵ’s.

Since ϵ is an odd density of weight −1, we can split the volume element ϵ̃, if we postulate the existence of an even scalar density σ of weight +1, that is,

σ′ = |J| σ .  (2.7)

For our purpose here^{1}, we postulated an even scalar density, since the ϵ̃ in (2.1) and the Levi-Civita ϵ in (2.3) are both odd. Then eventually, equation (2.1) can be rewritten as

W = ∫ \underbrace{L}_{odd 4-f.} = ∫ \underbrace{ϵ̃}_{odd 4-f.} \underbrace{L̂}_{scalar}

= ∫ \underbrace{ϵ}_{odd 4-f. density, weight −1} \underbrace{σ}_{even scalar density, weight +1}  \underbrace{L̂}_{scalar} .  (2.8)

Conventionally, the square root of the modulus of the metric determinant is chosen as the scalar density σ:

0σ := \sqrt{|det g_{ij}|} .  (2.9)

As soon as a metric g = g_{ij} dx^{i} ⊗dx^{j} is given—the gravitational potential of general relativity—we can define_{0}σ.

Alternatively, we can promote the scalar density to a new fundamental field of nature, compare also the model developed in [1, Sec.6]. The value of such a density _{1}σ can be viewed as a scale factor of the volume element, see also [6,7]. Thus, from a physical point of view, it is interesting to investigate the role of _{1}σ as a scaling parameter which realizes a scale transformation on a physical system.

The possibility to take a metric independent scalar density_{1}σ in place of \sqrt{|det g_{ij}|} in order to build a proper volume element gives new opportunities to define a concept of scale invariance. In this context, the field _{1}σ is known as a dilaton field, which becomes non-trivial in the quantum theory after the conformal (scale) invariance is broken.

In a pure connection ansatz, we prescribe a linear connection Γ_{α}^{β} = Γ_{iα}^{β}dx^{i} (but no metric!). Define, as usual, the curvature-2-form by

R_{α}^{β} = dΓ_{α}^{β} − Γ_{α}^{γ} ∧ Γ_{γ}^{β}  (2.10)

and the Ricci-1-form by

Ric_{α} := e_{β}⌋R_{α}^{β} = Ric_{iα} dx^{i} .  (2.11)

Then

2σ := \sqrt{|det Ric_{ij}|} ,  (2.12)

with Ric_{ij} = Ric_{iα}e_{j}^{α}, is a viable scalar density, as first suggested by Eddington [8, Sec. 92], compare also Schrödinger [9] and the more recent work of Kijowski and collaborators, see [10]. Likewise, the corresponding quantity based on the symmetric part of the Ricci tensor,

2′σ := \sqrt{|det Ric_{(ij)}|} ,  (2.13)

also qualifies as a volume measure. Note that _{2}σ and 2′σ may have singular points in such a theory as soon as the Ricci tensor or its symmetric part vanish. There seems to exist no criterion around which would prefer, say, _{2}σ, as compared to _{2′}σ. Lately, such theories have been abandoned.

In contrast, the non-linear electrodynamics of Born-Infeld [11] with its structurally similar Lagrangian (f = maximal field strength)

3σ = \sqrt{| det (g_{ij} + F_{ij}/f) |} − \sqrt{| det g_{ij}|} ,  (2.14)

has been in the focus of renewed interest, see [12].

1In reference [1, Eq.(A.1.33)] we took an odd scalar density instead, which we also denoted by the same letter σ.

III. THE QUARTET THEORY

A further possibility is the introduction of a quartet of scalar fields. First, in close analogy to the components ϵ_{ijkl} of the Levi-Civita ϵ, we can define the totally antisymmetric tensor density ϵ^{ijkl} of weight +1. We put its numerically invariant component ϵ^{0123} = −1. Then we define [3]

4σ := −ϵ^{ijkl} (∂_{i}φ^{(0)}) (∂_{j}φ^{(1)}) (∂_{k}φ^{(2)}) (∂_{l}φ^{(3)})
= −\frac{1}{4!} ϵ^{ijkl} ϵ_{ABCD} (∂_{i}φ^{A}) (∂_{j}φ^{B}) (∂_{k}φ^{C}) (∂_{l}φ^{D}) ,

(3.1)

where A, …, D are indices of interior space. This definition yields, for the volume 4-form

\overline{η} := _{4}σ ϵ ,  (3.2)

the following relations:

\overline{η} = dφ^{(0)} ∧ dφ^{(1)} ∧ dφ^{(2)} ∧ dφ^{(3)}
= \frac{1}{4!} ϵ_{ABCD} dφ^{A} ∧ dφ^{B} ∧ dφ^{C} ∧ dφ^{D} .  (3.3)

If we introduce the abbreviation

∂_{A} := \frac{∂}{∂φ^{A}} ,  (3.4)

then the duality of dφ^{A} and ∂_{B} can be expressed as follows:

dφ^{A} [∂_{B}] = δ_{B}^{A} .  (3.5)

In analogy to the set of equation (2.5), we define the 3-form and the 2-form

\overline{η}_{A} := ∂_{A}⌋\overline{η} , \overline{η}_{AB} := ∂_{B}⌋\overline{η}_{A} , etc.  (3.6)

Explicitly they read

\overline{η}_{A} = \frac{1}{3!}ϵ_{ABCD}dφ^{B} ∧ dφ^{C} ∧ dφ^{D} ,  (3.7a)
\overline{η}_{AB} = \frac{1}{2!}ϵ_{ABCD}dφ^{C} ∧ dφ^{D} , etc.  (3.7b)

In analogy to (2.6) we have

dφ^{N} ∧ \overline{η}_{A} = +δ_{A}^{N} \overline{η} , dφ^{N} ∧ \overline{η}_{AB} = −δ_{A}^{N} \overline{η}_{B} + δ_{B}^{N} \overline{η}_{A} , (3.8)

and so on. We contract (3.8) and find

\overline{η} = \frac{1}{4} dφ^{N} ∧ \overline{η}_{N} , \overline{η}_{A} = \frac{1}{3} dφ^{N} ∧ \overline{η}_{AN} , etc. (3.9)

We differentiate (3.9):

d \overline{η} = −\frac{1}{4} dφ^{N} ∧ d \overline{η}_{N} , d \overline{η}_{A} = −\frac{1}{3} dφ^{N} ∧ d \overline{η}_{AN} , etc. (3.10)

Now, \overline{η}, as a 4-form, is closed:

d \overline{η} = 0 .  (3.11)

Provided dφ^{A} ≠ 0, we find successively,

d \overline{η}_{A} = 0 , d \overline{η}_{AB} = 0 , etc.  (3.12)

Using this information, we can partially integrate (3.9) and can prove that all these forms are not only closed, but also exact:

\overline{η} = d [\frac{1}{4} φ^{N} ∧ \overline{η}_{N}] , \overline{η}_{A} = d [\frac{1}{3} φ^{N} ∧ \overline{η}_{AN}] , etc. (3.13)

Using (3.3) and (3.7), we find

\frac{∂ \overline{η}}{∂ dφ^{A}} = \overline{η}_{A} , \frac{∂ \overline{η}_{A}}{∂ dφ^{B}} = \overline{η}_{AB} , etc.  (3.14)

or, because of (3.13):

d \frac{∂ \overline{η}}{∂ dφ^{A}} = 0 , d \frac{∂ \overline{η}_{A}}{∂ dφ^{B}} = 0 , etc.  (3.15)

Since the corresponding “forces” vanish too, as can be seen from (3.3) and (3.7),

\frac{∂ \overline{η}}{∂ φ^{A}} = 0 , \frac{∂ \overline{η}_{A}}{∂ φ^{B}} = 0 , etc.,  (3.16)

we find an analogous result for the variational derivatives:

\frac{δ \overline{η}}{δ φ^{A}} = 0 , \frac{δ \overline{η}_{A}}{δ φ^{B}} = 0 , etc.  (3.17)

Similarly, we have

\frac{δ\overline{η}}{δϑ^{α}} = 0 , \frac{δ\overline{η}_{A}}{δϑ^{α}} = 0 , etc.  (3.18)

and

\frac{δ\overline{η}}{δg_{αβ}} = 0 , \frac{δ\overline{η}_{A}}{δg_{αβ}} = 0 , etc.  (3.19)

That the volume element is an exact form is the distinguishing feature of this quartet ansatz. Under these circumstances, the volume (2.2) can be expressed, via Stokes’ theorem, as a 3-dimensional surface integral which doesn’t contribute to the variation of the action functional.

Using Eq. (2.8) and the volume element (3.3) and denoting the gravitational Lagrangian by V = V (g_{αβ}, ϑ^{α}, Q_{αβ}, T^{α}, R_{α}^{β}) and the matter Lagrangian by L_{m} = L_{m}(g_{αβ}, ϑ^{α}, Γ_{α}^{β}, Ψ, dΨ), the action W reads W = ∫ (V + L_{m}) = ∫ \overline{η} \underbrace{(V̂ + L̂_{m})}_{scalar} = ∫ d χ (V̂ + L̂_{m}) ,

(3.20)

see Guendelman and Kaganovich [3]. Note that the 3-form χ, according to (3.13), explicitly reads χ := φ^{N} ∧ \overline{η}_{N}/4. If we add a constant λ to the scalar Lagrangian, we find

∫ d χ (V̂ + L̂_{m} + λ) = W + λ ∫ d χ .  (3.21)

Since the 3-dimensional hypersurface integral ∫_{∂ Vol} χ doesn’t contribute to the variation, the scalar Lagrangian is invariant under the addition of a constant.

Variation with respect to φ^{A} yields the corresponding field equations

\frac{∂ (V + L_{m})}{∂φ^{A}} − d \frac{∂ (V + L_{m})}{∂ dφ^{A}} = 0 .  (3.22)

Suppose, see [3], that V̂ and L̂_{m} do not depend on the quartet field at all,

\frac{∂ V̂}{∂ φ^{A}} = 0 , \frac{∂ V̂}{∂ dφ^{A}} = 0 , \frac{∂ L̂_{m}}{∂ φ^{A}} = 0 , \frac{∂ L̂_{m}}{∂ dφ^{A}} = 0 , (3.23)

then the field equations for the quartet field read

(V̂ + L̂_{m}) \frac{∂ \overline{η}}{∂φ^{A}} − d[(V̂ + L̂_{m}) \frac{∂ \overline{η}}{∂ dφ^{A}} ] = 0 . (3.24)

The first term vanishes, since \overline{η} does not depend on φ^{A} explicitly, see (3.16). Then the Leibniz rule yields

d [( V̂ + L̂_{m}) \frac{∂ \overline{η}}{∂ dφ^{A}} ]  (3.25)
= \frac{∂ \overline{η}}{∂ dφ^{A}} d (V̂ + L̂_{m}) + (V̂ + L̂_{m}) \underbrace{d\frac{∂ \overline{η}}{∂ dφ^{A}}}_{=^{(3.15)}0} = 0 . (3.26)

Provided φ^{A} ≠ 0 and dφ^{A} ≠ 0, we can conclude that

d (V̂ + L̂_{m}) = 0 ,  i.e.,  V̂ + L̂_{m} = const . (3.27)

The gravitational field equations following from δg_{αβ} and δΓ_{α}^{β} are not disturbed by the existence of φ^{A}. Hence the usual metric-affine formalism applies in its conventional form (see [1], for recent developments cf. [13–16]), but the field equation (3.27) for the scalar field quartet φ^{A} has to be appended. Perhaps surprisingly, it is only one equation since, in addition to (3.18) and (3.19), we trivially have

\frac{δ\overline{η}}{δΓ_{α}^{β}} = 0 , \frac{δ\overline{η}_{A}}{δΓ_{α}^{β}} = 0 , etc.  (3.28)

IV. CONCLUSION

We transparently displayed the necessary structures for building up a volume element and pointed out several physical alternatives to the usual metric volume element, giving rise to different gravity theories. Within the framework of metric–affine gravity we can reproduce the essential features of the Guendelman-Kaganovich theory without the necessity to specify the gravitational firstorder Lagrangian other than by the property (3.23).

ACKNOWLEDGMENTS

We would like to thank Eduardo Guendelman and Alex Kaganovich for useful comments and remarks. This research was supported by CONACyT, grants No. 3544–E9311, and by the joint German–Mexican project DLR(BMBF)-CONACyT MXI 6.B0A.6A and E130– 2924.

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