Research ArticleField Equations and Lagrangian for the Kaluza MetricEvaluated with Tensor Algebra Software

L. L. Williams

Konfluence Research Institute, 6009 Olympic, Manitou Springs, CO 80829, USA

Correspondence should be addressed to L. L. Williams; lance@konfluence.org

Received 1 August 2014; Accepted 2 December 2014

Academic Editor: Shinji Tsujikawa

Copyright © 2015 L. L. Williams.This is an open access article distributed under the Creative CommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper calculates the Kaluza field equations with the aid of a computer package for tensor algebra, xAct.The xAct file is providedwith this paper. We find that Thiry’s field equations are correct, but only under limited circumstances. The full five-dimensionalfield equations under the cylinder condition are provided here, and we see that most of the other references miss at least some termsfrom them. We go on to establish the remarkable Kaluza Lagrangian, and verify that the field equations calculated from it matchthose calculated with xAct, thereby demonstrating self-consistency of these results. Many of these results can be found scatteredthroughout the literature, and we provide some pointers for historical purposes. But our intent is to provide a definitive expositionof the field equations of the classical, five-dimensional metric ansatz of Kaluza, along with the computer algebra data file to verifythem, and then to recover the unique Lagrangian for the theory. In common terms, the Kaluza theory is an “𝜔 = 0” scalar fieldtheory, but with unique electrodynamic couplings.

1. Introduction

In 1921, Kaluza [1] hypothesized that electrodynamics andgeneral relativity could be unified in terms of general rela-tivity extended to five dimensions. The essence of the Kaluzahypothesis is a specific form for the 5-dimensional (5D)metric tensor; it is an ansatz of the theory. The completefield equations for the Kaluza metric were not discoveredfor another 25 years, generally attributed to Thiry [2], butrecently were shown to be somewhat independently discov-ered by 4 research groups in the 1940s and 1950s: Jordanand colleagues in Germany, Thiry and colleagues in France,Scherrer working alone in Switzerland, and Brans and Dickeat Princeton [3]. The first English-language translation of theEuropean paperswasThiry [2] in the book byApplequist et al.[4]. The Thiry expressions [2] for the field equations havedominated the literature subsequently for the 5D theory,and the Brans-Dicke equations have dominated the literaturesubsequently for general scalar-tensor theories. However, thecomponents of the 5-dimensional Ricci tensor provided byThiry do not self-consistently produce his expression for the5D Ricci scalar, so one suspects there is a problem with them.

Moreover, there are diverse expressions in the literaturefor the associated Kaluza Lagrangian density, and construc-tions of ever-more-elaborate scalar field Lagrangians [3] havesomewhat polluted the minimal scalar field Lagrangian thatarises naturally in the Kaluza metric. We would like to basethe Lagrangian on the 5D scalar curvature, but there is somequestion as to the veracity of the scalar curvature calculatedby Thiry and others. We shall go on to verify that Thiry’sexpression for the scalar curvature is indeed correct.

The field equations are difficult to calculate, and neitherThiry nor any of the other references show their calculations.It is difficult to make sense of the differing results andcontradictions in the literature.The results presented here areperhaps found in German or French papers from the 1940sand 1950s, as suggested by [3]. But this paper may be the firstcomplete treatment of the Kaluza hypothesis in English. Thispaper removes any ambiguity by calculating the Kaluza fieldequations with the aid of a computer software package fortensor algebra, xTensor in the xAct package [5]. The xTensorfile is provided online [6].

Hindawi Publishing CorporationJournal of GravityVolume 2015, Article ID 901870, 6 pageshttp://dx.doi.org/10.1155/2015/901870

2 Journal of Gravity

2. The Metric Ansatz

In the following, we use Greek indices for spacetime com-ponents and the index 5 to indicate the fifth dimension.Roman indices span all 5 dimensions. Tildes are written over5D quantities, so that 𝑔

𝑎𝑏is the 5D metric and 𝑔

𝜇] is thestandard 4D metric. The sign convention for the 4D metricand curvature tensors is (− + +) in the taxonomy of Misner,Thorne, andWheeler, also known as the Landau and Lifshitztimelike convention.

The Kaluza field equations follow from applying themachinery of general relativity to the ansatz for the 5Dmetric𝑔𝑎𝑏:

𝑔𝜇] = 𝑔𝜇] + 𝜓𝐴𝜇𝐴], 𝑔5] = 𝜓𝐴], 𝑔55 = 𝜓 (1)

from which the inverse metric follows:

𝑔𝜇]= 𝑔𝜇], 𝑔

5]= −𝐴

], 𝑔

55

= 𝐴2

+

1

𝜓

. (2)

The terms quadratic in 𝐴𝜇 were introduced by Klein [7].The metric (1) is a general decomposition, and the quantities𝐴𝜇 and 𝜓 are general at this point. But 𝐴𝜇 will be identified

as proportional to the electromagnetic vector potential. Sothe 4D metric and the electromagnetic vector potential areunified in a 5D metric. The scalar field enters because a5D metric will have 15 components. At this point we areonly naming components of the 5D metric. The typicalKaluza treatment identifies 𝐴𝛼 with the electromagneticvector potential at the outset, so that a normalizing constantis introducedmultiplying𝐴𝛼. At this point, we keep the fieldsunitless and introduce normalization constants at the end.

Along with the form (1), there is another ansatz to theKaluza theory: the cylinder condition, asKaluza coined it. It isthe assumption that no field depends on the fifth coordinate,𝜕5𝑔𝑎𝑏= 0. Wesson and colleagues have developed the theory

without the constraint of the cylinder condition [8], therebyintroducing a huge degree of complexity to the theory. Thecurvature tensors in this paper were calculated by brute forcefrom the ansatz (1) and its connections under the cylindercondition. The connections are described in the Appendix.We proceed directly to the curvature tensors. All values canbe verified in the accompanying xTensor file.

3. 5D Ricci Tensor

Consider the 5D Ricci tensor �̃�𝑎𝑏, whose components can

be summarized in the 3 pieces that distinguish the fifthcoordinate: �̃�

𝜇], �̃�𝜇5, �̃�55. To proceed to the 5D Ricci tensor,we evaluate the contracted 5D Riemann curvature tensor,given by the usual formula:

�̃�𝑎𝑏= �̃�𝑐

𝑎𝑐𝑏= 𝜕𝑐Γ̃𝑐

𝑎𝑏− 𝜕𝑏Γ̃𝑐

𝑐𝑎+ Γ̃𝑐

𝑐𝑑Γ̃𝑑

𝑎𝑏− Γ̃𝑐

𝑏𝑑Γ̃𝑑

𝑎𝑐. (3)

Since �̃�5555

= 0 identically,

�̃�55= �̃�𝑐

5𝑐5=�

��̃�5

555+ �̃�

]5]5 = �̃�

]5]5

=

1

4

𝜓2

𝐹𝛼𝛽

𝐹𝛼𝛽+

1

4𝜓

(𝜕]𝜓) (𝜕]𝜓) −

1

2

◻𝜓,

(4)

where ◻ ≡ ∇𝜇∇𝜇in terms of the 4D covariant derivative ∇

𝜇

and where 𝐹𝛼𝛽

≡ 𝜕𝛼𝐴𝛽− 𝜕𝛽𝐴𝛼. Equation (4) was calculated

by hand and used to verify the operation of xTensor at leastfor this component of the Ricci tensor.

It is customary to set 𝜓 ≡ 𝜙2, in which case a puredivergence is obtained:

�̃�55= −𝜙◻𝜙 +

1

4

𝜙4

𝐹𝛼𝛽

𝐹𝛼𝛽. (5)

Since �̃�5𝛼55

= 0 identically,

�̃�𝛼5= �̃�𝑐

𝛼𝑐5=�

��̃�5

𝛼55+ �̃�

]𝛼]5 = �̃�

]𝛼]5

=

1

2

𝜓𝑔𝛽𝜇

∇𝜇𝐹𝛼𝛽+

1

4

𝐴𝛼𝜓2

𝐹𝜇]𝐹𝜇] +

3

4

𝐹𝛼𝛽𝜕𝛽

𝜓

+

1

4𝜓

𝐴𝛼(𝜕

]𝜓) (𝜕]𝜓) −

1

2

𝐴𝛼∇] (𝜕

]𝜓)

=

1

2

𝜓𝑔𝛽𝜇

∇𝜇𝐹𝛼𝛽+

3

4

𝐹𝛼𝛽𝜕𝛽

𝜓 + 𝐴𝛼�̃�55.

(6)

For the spacetime part of the 5D Ricci tensor, xTensorobtains

�̃�𝜇] = �̃�

5

𝜇5] + �̃�𝛼

𝜇𝛼]

= 𝑅𝜇] −

1

2𝜓

∇]𝜕𝜇𝜓 +1

4𝜓2(𝜕𝜇𝜓) (𝜕]𝜓)

−

1

2

𝜓 𝑔𝛼𝛽

𝐹𝜇𝛼𝐹]𝛽 + 𝐴𝜇𝐴]�̃�55

+ 𝐴𝜇(�̃�]5 − 𝐴]�̃�55) + 𝐴] (�̃�𝜇5 − 𝐴𝜇�̃�55)

= 𝑅𝜇] −

1

2

𝜙2

𝑔𝛼𝛽

𝐹𝜇𝛼𝐹]𝛽 −

1

𝜙

∇𝜇∇]𝜙

+ 𝐴𝜇𝐴]�̃�55 + 𝐴𝜇 (�̃�]5 − 𝐴]�̃�55)

+ 𝐴] (�̃�𝜇5 − 𝐴𝜇�̃�55) ,

(7)

where 𝑅𝜇] is the standard 4D Ricci tensor.

For the 5D scalar curvature, xTensor obtains

�̃� = 𝑔𝜇]�̃�𝜇] − 2𝐴

𝜇

�̃�𝜇5+ (𝐴2

+

1

𝜓

) �̃�55

= 𝑅 −

1

4

𝜓𝐹𝛼𝛽

𝐹𝛼𝛽+

1

2𝜓2(𝜕𝜇𝜓) (𝜕𝜇

𝜓) −

1

𝜓

◻𝜓

= 𝑅 −

1

4

𝜙2

𝐹𝛼𝛽

𝐹𝛼𝛽−

2

𝜙

◻𝜙,

(8)

where 𝑅 is the standard 4D scalar curvature.

4. 5D Einstein Tensor

Let us now assemble the 5D Einstein tensor 𝐺𝑎𝑏

≡ �̃�𝑎𝑏−

𝑔𝑎𝑏�̃�/2. We have not yet discussed matter sources, but it is

the Einstein tensor that is equated to the stress-energy tensor.

Journal of Gravity 3

For the 55-component, xTensor obtains

𝐺55= �̃�55−

1

2

𝜓�̃�

=

3

8

𝜓2

𝐹𝛼𝛽𝐹𝛼𝛽

−

1

2

𝜓𝑅.

(9)

Remarkably, the equationwhichwould describe𝜓 in terms ofas-yet-unspecified sources is merely algebraic. We will returnto this point in the discussion.

For the mixed component of the Einstein tensor, xTensorobtains

𝐺5] = �̃�5] −

1

2

𝜓𝐴]�̃�

= 𝐴]𝐺55 +1

2

𝜓𝑔𝛼𝛽

∇𝛽𝐹]𝛼 +

3

4

𝐹]𝛼𝜕𝛼

𝜓.

(10)

We notice that

𝐺5] − 𝐴]𝐺55 = �̃�5] − 𝐴]�̃�55. (11)

For the spacetime part of the Einstein tensor, we constructanalytically from the foregoing and verify with xTensor:

𝐺𝜇] = �̃�𝜇] −

1

2

𝑔𝜇]�̃�

= 𝑅𝜇] −

1

2

𝑔𝜇]𝑅 + 𝐴𝜇𝐴]𝐺55

+ 𝐴𝜇(𝐺]5 − 𝐴]𝐺55) + 𝐴] (𝐺𝜇5 − 𝐴𝜇𝐺55)

−

1

2

𝜓(𝑔𝛼𝛽

𝐹𝜇𝛼𝐹]𝛽 −

1

4

𝑔𝜇]𝐹𝛼𝛽𝐹

𝛼𝛽

)

−

1

2𝜓

(∇𝜇𝜕]𝜓 − 𝑔𝜇]◻𝜓)

+

1

4𝜓2[(𝜕𝜇𝜓) (𝜕]𝜓) − 𝑔𝜇] (𝜕𝛼𝜓) (𝜕

𝛼

𝜓)]

= 𝐺𝜇] −

1

𝜙

(∇𝜇𝜕]𝜙 − 𝑔𝜇]◻𝜙)

−

1

2

𝜙2

(𝑔𝛼𝛽

𝐹𝜇𝛼𝐹]𝛽 −

1

4

𝑔𝜇]𝐹𝛼𝛽𝐹

𝛼𝛽

)

+ 𝐴𝜇𝐴]𝐺55 + 𝐴𝜇 (𝐺]5 − 𝐴]𝐺55)

+ 𝐴] (𝐺𝜇5 − 𝐴𝜇𝐺55) .

(12)

It is striking that (12) produces the exact form of the Maxwellstress-energy tensor.

5. Comparison with Previous Results

It is important and illustrative to understand the assumptionsand limitations in Thiry’s results. Kaluza’s original papercontained the result, indeed, the compelling aesthetic featureof the theory, that the gravitational equations with electro-magnetic sources could be understood as vacuum equations

in a higher-dimensional space. Einstein himself was alwaysaware that the curvature terms of the Einstein equations had adeep and profound beauty that was not reflected in the ratherad hoc stress-energy terms they were equated to, and it washis vision to discover how the matter sources might actuallyspring from the fields, uniting matter and fields.

So it is sufficient to set the 5D Ricci scalar �̃�𝑎𝑏to zero and

still recover general relativity with electromagnetic sources,and it is a profound result. But such an assumption is notcorrect for a broader theory which would describe matter.Kaluza originally treated matter sources in the 5D theory andso did not presuppose to set �̃�

𝑎𝑏= 0. But that assumption

appears to have been in much of the subsequent literature.WhileThiry does construct the 5DEinstein tensor𝐺

𝑎𝑏, he

does so with the implicit assumption that �̃�𝑎𝑏

is zero and soobtains what are truly vacuum equations in both 4D and 5Dterms. Therefore, Thiry recovers expression (5) for �̃�

55, but

divided by 𝜙2. Thiry’s expression for �̃�5𝛼

is missing a factorof 𝜙 and missing the term in �̃�

55. His expression for �̃�

𝜇] issimilar to (7) but is missing all the terms in �̃�

55and �̃�

𝜇5.

As for the 5D Ricci tensor expressions, the Thiry expres-sions for the 5D Einstein tensor also drop terms. Theexpression given for 𝐺

5𝛼is similar to (10) but is missing the

term in𝐺55and is divided by 𝜙2.The expression given for𝐺

𝜇]

is similar to (12) but is missing all the terms in 𝐺55and 𝐺]5.

Interestingly, during his discussion of the 5D Einsteintensor𝐺

𝑎𝑏,Thiry fails to write down an expression for𝐺

55. He

merely refers back to his expression for �̃�55.This has set some

subsequent investigations on a wrong path insofar as it isassumed that the equation for the scalar field is given by �̃�

55=

0 in (5). This mischaracterizes the dynamical nature of thescalar field in the𝐺

55equation as shown in (9). Instead, we see

that the scalar field is dependent on the 4D scalar curvature𝑅and is not dynamically coupled to the electromagnetic field,as setting �̃�

55= 0 would lead one to believe.

Another problem with the Thiry expressions for thevacuum equations is that they are not self-consistent. Becauseof the missing terms, one cannot constructThiry’s expressionfor �̃� fromThiry’s expressions for �̃�

𝜇], �̃�]5, �̃�55 and themetric𝑔𝑎𝑏. Finally, we note that the beautiful emergence of the

electromagnetic stress tensor does not occur in the vacuum4D theory.

Among more recent review articles, Overduin and Wes-son [8] use the Thiry forms of the field equations. Thereview article by Coquereaux and Esposito-Farese [9] didprovide the correct expression for𝐺

55, as well as for the other

components of 𝐺𝑎𝑏, but was inexplicably missing terms from

�̃�𝑎𝑏.

6. 5D Lagrangian

We now turn to the Lagrange density of the 5D theory. Thiswill provide an independent check on the field equationsjust derived and allow us to determine the Lagrangian thatcorresponds to the field equations. But it will also allow us to

4 Journal of Gravity

evaluate the 5D theory in the context of other scalar field the-ories. We will find that unnecessary parameterizations of thescalar field have polluted the 5D Lagrangian, which actuallyneeds no kinetic term, contrary to what one commonly seesin the literature, even going back to the work of Jordan in the1940s [3].

There is of course a freedom in the choice of Lagrangian.The foregoing development was based on the ansatz (1) forthe 5D metric, from which the field equations are obtainedthrough the 5D Einstein equations. But we can also seek thefield equations from a corresponding Lagrangian. Just as theEinstein equations were extended directly to five dimensions,we expect that the Hilbert action should also support suchan extension. Therefore, one should consider a Lagrangianformed from the 5D scalar curvature.

We can combine (8) with the result, verified by xTensor,that the 5D determinant of 𝑔

𝑎𝑏, 𝑔1/2 = 𝜓1/2𝑔1/2, where 𝑔1/2

is the determinant of the 4D metric 𝑔𝜇]. We also invoke

the cylinder condition, so that the integral Δ𝑦 over thefifth coordinate commutes with the field variables. Let us,therefore, write the 5D Hilbert action and expand it:

𝑆 = ∫𝑔1/2

�̃�𝑑5

𝑥

= Δ𝑦∫𝜓1/2

𝑔1/2

�̃�𝑑4

𝑥

= Δ𝑦∫𝑔1/2

[𝜓1/2

𝑅 −

𝜓3/2

4

𝐹𝛼𝛽

𝐹𝛼𝛽− ∇] (

𝜕]𝜓

𝜓1/2

)]𝑑4

𝑥

= Δ𝑦∫𝑔1/2

[𝜙𝑅 −

1

4

𝜙3

𝐹𝛼𝛽

𝐹𝛼𝛽−���

�2∇] (𝜕

]𝜙)] 𝑑4

𝑥,

(13)

wherewe have set the scalar term to zero because it is a perfectdivergence, and as customary the corresponding surfaceterms are presumed not to contribute at the boundary. Aswe shall see, the impact of the scalar field in the gravitationalequations comes from the term in the scalar curvature.

Let us consider the field equations implied by (13),expanding to explicitly show the inverse metric 𝑔𝛼𝛽, andtherefore consider variation of the Lagrangian:

L = 𝑔1/2

[𝜙𝑔𝛼𝛽

𝑅𝛼𝛽−

1

4

𝜙3

𝑔𝛼𝜇

𝑔𝛽]𝐹𝛼𝛽𝐹𝜇]] . (14)

Variation of L provides, respectively, the equations for 𝑔𝛼𝛽,𝐴𝛼, and 𝜙. This would appear to be the unique Lagrangian

implied by the Kaluza hypothesis extended to the Hilbertaction. Yet this form is rarely written (it is found in thereview [9]). Also, (14) would seem to imply that the signof the fifth dimension in the metric should be timelike sothat the proper relative sign between the scalar curvature andelectromagnetic invariant is obtained. Some references claimthat the sign of the fifth coordinate should be spacelike.

In their review article, Overduin and Wesson [8] repeatthe Thiry equations. They also write down a Lagrangian forthe 5D theory, but they do not recover (13). Instead, theyhave a kinetic-type term for the scalar field proportional to

𝜕]𝜙𝜕]𝜙. Indeed, scalar field theory does typically have such

a quadratic term, and such a term was quickly added to thegeneralized 5D theory as early as the 1940s [3]. Brans andDicke [10] adopted a variation of the conventional kineticterm in their Lagrangian, parameterized with the famous𝜔 coefficient. This 𝜔 family of scalar field extensions togeneral relativity was further generalized and constrainedexperimentally by Will [11]. So that now, for any scalarfield extension to general relativity, the question arises asto the value of 𝜔. Based largely on the work of Will, it isoften repeated that scalar field theories must have 𝜔 > 500.However, it is also said that general relativity is recoveredin the limit as 𝜔 → ∞, but many counterexamples tothat proposition have since been discovered [12], so ourunderstanding of the 𝜔 limits is suspect.

Yet we see that, for the Kaluza Lagrangian (13) or (14),the question of 𝜔 is entirely irrelevant. It can be introducedonly to be set later to zero, but that has limited utility. Thequestion arises even why Brans and Dicke felt compelledto introduce an 𝜔 term, given that it is not necessary fora scalar field stress-energy source in the Einstein equations(12). But it was quite the contrary. When Brans and Dickerecover the dynamical scalar field term as in (12), they callit “foreign.” They presumably expected a kinetic-type actionin their Lagrangian with an adjustable coupling.

It is an expression of the elegance of theKaluza hypothesisthat the scalar field has no tunable coupling parameter. 𝜔-family theories seem extravagant by comparison. Carroll [13]shows how the scalar degree of freedom captured in the 𝜙𝑅term of the Lagrangian is a basic feature of gravity. For whensuch a relation as𝜙𝑅 appears in a Lagrangian, one can recovera standard Lagrangian in𝑅 with a coordinate transformationon the metric so that 𝜙𝑔

𝜇] → 𝑔

𝜇]. But as Carroll shows, thisjust results in a new term in the Lagrangian that has a kineticform. So in some sense, a Lagrangian in𝜙𝑅without a separatescalar term is the minimal scalar-tensor theory.

Now we proceed to evaluation of the Lagrangian (14).Consider first the field equations for 𝜙:

𝛿L

𝛿𝜙

= 𝑔1/2

(𝑅 −

3

4

𝜙2

𝐹𝛼𝛽

𝐹𝛼𝛽) = 0. (15)

This is identical with (9) for 𝐺55

= 0. Note that it is notequivalent to the equation for 𝜙 for �̃�

55= 0 that is usually

used to characterize the scalar field. It seems to constrainthe scalar field to the ratio of the scalar curvature to theelectromagnetic scalar. The scalar curvature, in the presenceof matter, equals the trace of the stress-energy tensor.

Consider now the field equations for 𝐴𝜇. For them, let ususe the Euler-Lagrange equations:

𝜕L

𝜕𝐴𝜇

− ∇] [𝜕L

𝜕 (∇]𝐴𝜇)] = 0. (16)

Since Carroll [13] shows the algebra, we can quote his result(1.167) that

𝜕 (𝐹𝛼𝛽𝐹𝛼𝛽

)

𝜕 (∇𝜇𝐴])

= 4𝐹𝜇]. (17)

Journal of Gravity 5

Since 𝜕L/𝜕𝐴𝜇= 0, the field equation for 𝐴𝜇 is

∇] (𝜙3

𝐹𝜇]) = 0. (18)

This is identical with (10) for 𝐺𝜇5

= 0 when 𝐺55

= 0.Unlike the scalar-field equation, it is coincidentally the sameas its vacuum counterpart in the Ricci tensor, the equation for�̃�]5 = 0.

Consider now the field equations for the inverse metric𝑔𝛼𝛽, which is equivalent to finding the equations for the met-

ric. We construct the variation of the Lagrangian in pieces.Carroll has a nice analysis of the scalar field Lagrangian, andwe use a couple of his results (4.69 and 4.122) to expedite ourvariation ofL. They are

𝛿𝑔1/2

= −

1

2

𝑔1/2

𝑔𝛼𝛽𝛿𝑔𝛼𝛽

, (19)

𝑔𝛼𝛽

𝛿𝑅𝛼𝛽

= ∇]∇](𝑔𝛼𝛽𝛿𝑔𝛼𝛽

) − ∇𝛼∇𝛽(𝛿𝑔𝛼𝛽

) . (20)

As per usual, we shall also neglect perfect divergences of theform

∫𝑔1/2

∇𝛼𝑉𝛼

𝑑4

𝑥 → 0. (21)

Varying the first term in the Lagrangian (14) with respectto the metric

𝛿L𝑅= 𝜙 (𝛿𝑔

1/2

𝑅 + 𝑔1/2

𝛿𝑔𝛼𝛽

𝑅𝛼𝛽+ 𝑔1/2

𝑔𝛼𝛽

𝛿𝑅𝛼𝛽)

= 𝜙𝑔1/2

𝛿𝑔𝛼𝛽

(−

1

2

𝑔𝛼𝛽𝑅 + 𝑅

𝛼𝛽)

+ 𝑔1/2

𝜙 (∇]∇]𝑔𝛼𝛽𝛿𝑔𝛼𝛽

− ∇𝛼∇𝛽𝛿𝑔𝛼𝛽

)

= 𝑔1/2

𝛿𝑔𝛼𝛽

(𝜙𝐺𝛼𝛽+ 𝑔𝛼𝛽∇]∇]𝜙 − ∇𝛼∇𝛽𝜙) .

(22)

The third line is obtained through two successive integrationsby parts, neglecting the divergence terms that are understoodto be boundary terms as per (21). This is how the dynamicalscalar field arises from the minimal Lagrangian.

Varying the second term in the Lagrangian (14) withrespect to the metric

𝛿L𝐹= −

1

4

𝜙3

𝐹𝛼𝛽𝐹𝜇]

× (𝛿𝑔1/2

𝑔𝛼𝜇

𝑔𝛽]+ 𝑔1/2

𝛿𝑔𝛼𝜇

𝑔𝛽]

+𝑔1/2

𝑔𝛼𝜇

𝛿𝑔𝛽])

= −

1

2

𝜙3

𝑔1/2

𝛿𝑔𝛼𝛽

(𝑔𝜇]𝐹𝛼𝜇𝐹𝛽] −

1

4

𝑔𝛼𝛽𝐹𝜇]𝐹𝜇]) .

(23)

Now put together the pieces (22) and (23) to obtain theequation for the metric:

𝜙𝐺𝛼𝛽

=

1

2

𝜙3

(𝑔𝜇]𝐹𝛼𝜇𝐹𝛽] −

1

4

𝑔𝛼𝛽𝐹𝜇]𝐹𝜇])

+ (∇𝛼∇𝛽𝜙 − 𝑔𝛼𝛽◻𝜙) ,

(24)

which is equivalent to the equation for 𝐺𝛼𝛽

= 0 (12) when𝐺𝑎𝑏= 0.So this shows that the unique 5D Hilbert Lagrangian

(14) under the Kaluza hypothesis (1) produces the fieldequations calculated from the connections and curvaturetensors. Furthermore, there is no kinetic scalar field term inthe Lagrangian; 𝜔 = 0 in Brans-Dicke terms. The Kaluzahypothesis appears to imply a minimally coupled scalar fieldwith no free parameters.

Appendix

5D Connections

This appendix gathers the connections calculated from (1),on which the curvature tensors presented above are based.We start with the conventional definition for the affineconnection (torsion-free, metric-compatible):

Γ̃𝑎

𝑏𝑐=

1

2

𝑔𝑎𝑑

(𝜕𝑏𝑔𝑑𝑐+ 𝜕𝑐𝑔𝑑𝑏− 𝜕𝑑𝑔𝑏𝑐) . (A.1)

All the connections are conveniently written as the sixconnections that distinguish the fifth coordinate, Γ̃5

55, Γ̃55], Γ̃5

𝛼],Γ̃]55, Γ̃]5𝛼, Γ̃𝛽𝜇]. We employ the cylinder condition and discard

all derivatives with respect to the fifth coordinate. These arestraightforward to calculate by hand but are verified withxTensor. They are

Γ̃5

55=

1

2

𝑔5](−𝜕]𝑔55) =

1

2

𝐴]𝜕]𝜓, (A.2)

Γ̃5

5] =1

2

𝑔5𝛼

(𝜕]𝑔𝛼5 − 𝜕𝛼𝑔5]) +1

2

𝑔55

𝜕]𝑔55

=

1

2

𝐴𝛼

𝜓𝐹𝛼] +

1

2

𝐴𝛼

𝐴]𝜕𝛼𝜓 +1

2𝜓

𝜕]𝜓,

(A.3)

Γ̃5

𝛼] =1

2

𝑔5𝛽

(𝜕𝛼𝑔𝛽] + 𝜕]𝑔𝛽𝛼 − 𝜕𝛽𝑔𝛼])

+

1

2

𝑔55

(𝜕𝛼𝑔5] + 𝜕]𝑔5𝛼)

= −𝐴𝛽Γ𝛽

𝛼] +1

2

𝐴𝛽

𝐴]𝜓𝐹𝛽𝛼

+

1

2

𝐴𝛼𝐴𝛽

𝜓𝐹𝛽] +

1

2

(𝜕𝛼𝐴] + 𝜕]𝐴𝛼)

+

1

2𝜓

(𝐴]𝜕𝛼𝜓 + 𝐴𝛼𝜕]𝜓) +1

2

𝐴𝛼𝐴𝛽

𝐴]𝜕𝛽𝜓,

(A.4)

Γ̃]55=

1

2

𝑔]𝛼(−𝜕𝛼𝑔55) = −

1

2

𝑔]𝛼𝜕𝛼𝜓, (A.5)

Γ̃]5𝛼=

1

2

𝑔]𝜇(𝜕𝛼𝑔𝜇5− 𝜕𝜇𝑔5𝛼) +

1

2

𝑔]5𝜕𝛼𝑔55

=

1

2

𝑔]𝜇(𝜓𝐹𝛼𝜇− 𝐴𝛼𝜕𝜇𝜓) ,

(A.6)

6 Journal of Gravity

Γ̃𝛽

𝜇] =1

2

𝑔𝛽𝛼

(𝜕𝜇𝑔𝛼] + 𝜕]𝑔𝛼𝜇 − 𝜕𝛼𝑔𝜇])

+

1

2

𝑔𝛽5

(𝜕𝜇𝑔5] + 𝜕]𝑔5𝜇)

= Γ𝛽

𝜇]

+

1

2

𝑔𝛽𝛼

(𝐴]𝜓𝐹𝜇𝛼 + 𝐴𝜇𝜓𝐹]𝛼 + 𝐴𝜇𝐴]𝜕𝛼𝜓) .

(A.7)

There are a couple of convenient identities:

Γ̃5

55+ 𝐴]Γ̃

]55= 0,

Γ̃]]5 + Γ̃

5

55= Γ̃𝑎

𝑎5= 0.

(A.8)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

References

[1] T. Kaluza, Sitzungsberichte der K. Preussischen Akademie derWissenschaften zu Berlin, vol. 966, 1921.

[2] M. Y. Thiry, “Geometry—the equations of Kaluza’s unifiedtheory,” Comptes Rendus de l’Académie des Sciences, vol. 226, p.216, 1948.

[3] H. Goenner, “Some remarks on the genesis of scalar-tensortheories,” General Relativity and Gravitation, vol. 44, no. 8, pp.2077–2097, 2012.

[4] T. Applequist, A. Chodos, and P. G. O. Freund,Modern Kaluza-Klein Theories, Addison-Wesley, New York, NY, USA, 1987.

[5] xTensor, in the xAct package, http://www.xact.es/.[6] “xTensor computer tensor algebra file,” http://www.konflu-

ence.org/.[7] O. Klein, “Quantentheorie und fünfdimensionale Relativi-

tätstheorie,” Zeitschrift für Physik, vol. 37, no. 12, pp. 895–906,1926.

[8] J. M. Overduin and P. S.Wesson, “Kaluza-Klein gravity,” PhysicsReport, vol. 283, no. 5-6, pp. 303–378, 1997.

[9] R. Coquereaux and G. Esposito-Farese, “The theory ofKaluza-Klein-Jordan-Thiry revisted,”Annales de l’Institut HenriPoincare Physique Theorique, vol. 52, p. 113, 1990.

[10] C. Brans and R. H. Dicke, “Mach’s principle and a relativistictheory of gravitation,” Physical Review, vol. 124, no. 3, pp. 925–935, 1961.

[11] C.Will,Theory and Experiment in Gravitational Physics, revisededition, Cambridge University Press, Cambridge, Mass, USA,1993.

[12] V. Faraoni, “Illusions of general relativity in Brans-Dicke grav-ity,” Physical Review D, vol. 59, no. 8, 1999.

[13] S. Carroll, Spacetime and Geometry, section 4.8, AddisonWesley, San Francisco, Calif, USA, 2004.

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