research article field equations and lagrangian for the ...research article field equations and...
TRANSCRIPT
-
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; [email protected]
Received 1 August 2014; Accepted 2 December 2014
Academic Editor: Shinji Tsujikawa
Copyright Β© 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 Thiryβs 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 βπ = 0β 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 Thiryβsexpression 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 π
ππis the 5D metric and π
π] is thestandard 4D metric. The sign convention for the 4D metricand curvature tensors is (β + +) 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 5Dmetricπππ:
ππ] = ππ] + ππ΄ππ΄], π5] = ππ΄], π55 = π (1)
from which the inverse metric follows:
ππ]= ππ], π
5]= βπ΄
], π
55
= π΄2
+
1
π
. (2)
The terms quadratic in π΄π were introduced by Klein [7].The metric (1) is a general decomposition, and the quantitiesπ΄π and π are general at this point. But π΄π 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 π΄πΌ with the electromagneticvector potential at the outset, so that a normalizing constantis introducedmultiplyingπ΄πΌ. 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,π5πππ= 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 οΏ½ΜοΏ½ππ, whose components can
be summarized in the 3 pieces that distinguish the fifthcoordinate: οΏ½ΜοΏ½
π], οΏ½ΜοΏ½π5, οΏ½ΜοΏ½55. To proceed to the 5D Ricci tensor,we evaluate the contracted 5D Riemann curvature tensor,given by the usual formula:
οΏ½ΜοΏ½ππ= οΏ½ΜοΏ½π
πππ= ππΞΜπ
ππβ ππΞΜπ
ππ+ ΞΜπ
ππΞΜπ
ππβ ΞΜπ
ππΞΜπ
ππ. (3)
Since οΏ½ΜοΏ½5555
= 0 identically,
οΏ½ΜοΏ½55= οΏ½ΜοΏ½π
5π5=οΏ½
οΏ½οΏ½ΜοΏ½5
555+ οΏ½ΜοΏ½
]5]5 = οΏ½ΜοΏ½
]5]5
=
1
4
π2
πΉπΌπ½
πΉπΌπ½+
1
4π
(π]π) (π]π) β
1
2
β»π,
(4)
where β» β‘ βπβπin terms of the 4D covariant derivative β
π
and where πΉπΌπ½
β‘ ππΌπ΄π½β ππ½π΄πΌ. 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 π β‘ π2, in which case a puredivergence is obtained:
οΏ½ΜοΏ½55= βπβ»π +
1
4
π4
πΉπΌπ½
πΉπΌπ½. (5)
Since οΏ½ΜοΏ½5πΌ55
= 0 identically,
οΏ½ΜοΏ½πΌ5= οΏ½ΜοΏ½π
πΌπ5=οΏ½
οΏ½οΏ½ΜοΏ½5
πΌ55+ οΏ½ΜοΏ½
]πΌ]5 = οΏ½ΜοΏ½
]πΌ]5
=
1
2
πππ½π
βππΉπΌπ½+
1
4
π΄πΌπ2
πΉπ]πΉπ] +
3
4
πΉπΌπ½ππ½
π
+
1
4π
π΄πΌ(π
]π) (π]π) β
1
2
π΄πΌβ] (π
]π)
=
1
2
πππ½π
βππΉπΌπ½+
3
4
πΉπΌπ½ππ½
π + π΄πΌοΏ½ΜοΏ½55.
(6)
For the spacetime part of the 5D Ricci tensor, xTensorobtains
οΏ½ΜοΏ½π] = οΏ½ΜοΏ½
5
π5] + οΏ½ΜοΏ½πΌ
ππΌ]
= π π] β
1
2π
β]πππ +1
4π2(πππ) (π]π)
β
1
2
π ππΌπ½
πΉππΌπΉ]π½ + π΄ππ΄]οΏ½ΜοΏ½55
+ π΄π(οΏ½ΜοΏ½]5 β π΄]οΏ½ΜοΏ½55) + π΄] (οΏ½ΜοΏ½π5 β π΄ποΏ½ΜοΏ½55)
= π π] β
1
2
π2
ππΌπ½
πΉππΌπΉ]π½ β
1
π
βπβ]π
+ π΄ππ΄]οΏ½ΜοΏ½55 + π΄π (οΏ½ΜοΏ½]5 β π΄]οΏ½ΜοΏ½55)
+ π΄] (οΏ½ΜοΏ½π5 β π΄ποΏ½ΜοΏ½55) ,
(7)
where π π] is the standard 4D Ricci tensor.
For the 5D scalar curvature, xTensor obtains
οΏ½ΜοΏ½ = ππ]οΏ½ΜοΏ½π] β 2π΄
π
οΏ½ΜοΏ½π5+ (π΄2
+
1
π
) οΏ½ΜοΏ½55
= π β
1
4
ππΉπΌπ½
πΉπΌπ½+
1
2π2(πππ) (ππ
π) β
1
π
β»π
= π β
1
4
π2
πΉπΌπ½
πΉπΌπ½β
2
π
β»π,
(8)
where π is the standard 4D scalar curvature.
4. 5D Einstein Tensor
Let us now assemble the 5D Einstein tensor πΊππ
β‘ οΏ½ΜοΏ½ππβ
ππποΏ½ΜοΏ½/2. 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
πΊ55= οΏ½ΜοΏ½55β
1
2
ποΏ½ΜοΏ½
=
3
8
π2
πΉπΌπ½πΉπΌπ½
β
1
2
ππ .
(9)
Remarkably, the equationwhichwould describeπ 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
πΊ5] = οΏ½ΜοΏ½5] β
1
2
ππ΄]οΏ½ΜοΏ½
= π΄]πΊ55 +1
2
πππΌπ½
βπ½πΉ]πΌ +
3
4
πΉ]πΌππΌ
π.
(10)
We notice that
πΊ5] β π΄]πΊ55 = οΏ½ΜοΏ½5] β π΄]οΏ½ΜοΏ½55. (11)
For the spacetime part of the Einstein tensor, we constructanalytically from the foregoing and verify with xTensor:
πΊπ] = οΏ½ΜοΏ½π] β
1
2
ππ]οΏ½ΜοΏ½
= π π] β
1
2
ππ]π + π΄ππ΄]πΊ55
+ π΄π(πΊ]5 β π΄]πΊ55) + π΄] (πΊπ5 β π΄ππΊ55)
β
1
2
π(ππΌπ½
πΉππΌπΉ]π½ β
1
4
ππ]πΉπΌπ½πΉ
πΌπ½
)
β
1
2π
(βππ]π β ππ]β»π)
+
1
4π2[(πππ) (π]π) β ππ] (ππΌπ) (π
πΌ
π)]
= πΊπ] β
1
π
(βππ]π β ππ]β»π)
β
1
2
π2
(ππΌπ½
πΉππΌπΉ]π½ β
1
4
ππ]πΉπΌπ½πΉ
πΌπ½
)
+ π΄ππ΄]πΊ55 + π΄π (πΊ]5 β π΄]πΊ55)
+ π΄] (πΊπ5 β π΄ππΊ55) .
(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 Thiryβs results. Kaluzaβs 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 οΏ½ΜοΏ½ππto 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 οΏ½ΜοΏ½
ππ= 0. But that assumption
appears to have been in much of the subsequent literature.WhileThiry does construct the 5DEinstein tensorπΊ
ππ, he
does so with the implicit assumption that οΏ½ΜοΏ½ππ
is zero and soobtains what are truly vacuum equations in both 4D and 5Dterms. Therefore, Thiry recovers expression (5) for οΏ½ΜοΏ½
55, but
divided by π2. Thiryβs expression for οΏ½ΜοΏ½5πΌ
is missing a factorof π and missing the term in οΏ½ΜοΏ½
55. His expression for οΏ½ΜοΏ½
π] issimilar to (7) but is missing all the terms in οΏ½ΜοΏ½
55and οΏ½ΜοΏ½
π5.
As for the 5D Ricci tensor expressions, the Thiry expres-sions for the 5D Einstein tensor also drop terms. Theexpression given for πΊ
5πΌis similar to (10) but is missing the
term inπΊ55and is divided by π2.The expression given forπΊ
π]
is similar to (12) but is missing all the terms in πΊ55and πΊ]5.
Interestingly, during his discussion of the 5D EinsteintensorπΊ
ππ,Thiry fails to write down an expression forπΊ
55. He
merely refers back to his expression for οΏ½ΜοΏ½55.This has set some
subsequent investigations on a wrong path insofar as it isassumed that the equation for the scalar field is given by οΏ½ΜοΏ½
55=
0 in (5). This mischaracterizes the dynamical nature of thescalar field in theπΊ
55equation as shown in (9). Instead, we see
that the scalar field is dependent on the 4D scalar curvatureπ and is not dynamically coupled to the electromagnetic field,as setting οΏ½ΜοΏ½
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 constructThiryβs expressionfor οΏ½ΜοΏ½ fromThiryβs expressions for οΏ½ΜοΏ½
π], οΏ½ΜοΏ½]5, οΏ½ΜοΏ½55 and themetricπππ. 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 forπΊ
55, as well as for the other
components of πΊππ, but was inexplicably missing terms from
οΏ½ΜοΏ½ππ.
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 π
ππ, π1/2 = π1/2π1/2, where π1/2
is the determinant of the 4D metric ππ]. We also invoke
the cylinder condition, so that the integral Ξπ¦ over thefifth coordinate commutes with the field variables. Let us,therefore, write the 5D Hilbert action and expand it:
π = β«π1/2
οΏ½ΜοΏ½π5
π₯
= Ξπ¦β«π1/2
π1/2
οΏ½ΜοΏ½π4
π₯
= Ξπ¦β«π1/2
[π1/2
π β
π3/2
4
πΉπΌπ½
πΉπΌπ½β β] (
π]π
π1/2
)]π4
π₯
= Ξπ¦β«π1/2
[ππ β
1
4
π3
πΉπΌπ½
πΉπΌπ½βοΏ½οΏ½οΏ½
οΏ½2β] (π
]π)] π4
π₯,
(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 ππΌπ½, andtherefore consider variation of the Lagrangian:
L = π1/2
[πππΌπ½
π πΌπ½β
1
4
π3
ππΌπ
ππ½]πΉπΌπ½πΉπ]] . (14)
Variation of L provides, respectively, the equations for ππΌπ½,π΄πΌ, and π. 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
π]ππ]π. 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 famousπ coefficient. This π 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 π. Based largely on the work of Will, it isoften repeated that scalar field theories must have π > 500.However, it is also said that general relativity is recoveredin the limit as π β β, but many counterexamples tothat proposition have since been discovered [12], so ourunderstanding of the π limits is suspect.
Yet we see that, for the Kaluza Lagrangian (13) or (14),the question of π 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 π 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 βforeign.β 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. π-family theories seem extravagant by comparison. Carroll [13]shows how the scalar degree of freedom captured in the ππ term of the Lagrangian is a basic feature of gravity. For whensuch a relation asππ appears in a Lagrangian, one can recovera standard Lagrangian inπ with a coordinate transformationon the metric so that ππ
π] β π
π]. But as Carroll shows, thisjust results in a new term in the Lagrangian that has a kineticform. So in some sense, a Lagrangian inππ without a separatescalar term is the minimal scalar-tensor theory.
Now we proceed to evaluation of the Lagrangian (14).Consider first the field equations for π:
πΏL
πΏπ
= π1/2
(π β
3
4
π2
πΉπΌπ½
πΉπΌπ½) = 0. (15)
This is identical with (9) for πΊ55
= 0. Note that it is notequivalent to the equation for π for οΏ½ΜοΏ½
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 π΄π. For them, let ususe the Euler-Lagrange equations:
πL
ππ΄π
β β] [πL
π (β]π΄π)] = 0. (16)
Since Carroll [13] shows the algebra, we can quote his result(1.167) that
π (πΉπΌπ½πΉπΌπ½
)
π (βππ΄])
= 4πΉπ]. (17)
-
Journal of Gravity 5
Since πL/ππ΄π= 0, the field equation for π΄π is
β] (π3
πΉπ]) = 0. (18)
This is identical with (10) for πΊπ5
= 0 when πΊ55
= 0.Unlike the scalar-field equation, it is coincidentally the sameas its vacuum counterpart in the Ricci tensor, the equation forοΏ½ΜοΏ½]5 = 0.
Consider now the field equations for the inverse metricππΌπ½, 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
πΏπ1/2
= β
1
2
π1/2
ππΌπ½πΏππΌπ½
, (19)
ππΌπ½
πΏπ πΌπ½
= β]β](ππΌπ½πΏππΌπ½
) β βπΌβπ½(πΏππΌπ½
) . (20)
As per usual, we shall also neglect perfect divergences of theform
β«π1/2
βπΌππΌ
π4
π₯ β 0. (21)
Varying the first term in the Lagrangian (14) with respectto the metric
πΏLπ = π (πΏπ
1/2
π + π1/2
πΏππΌπ½
π πΌπ½+ π1/2
ππΌπ½
πΏπ πΌπ½)
= ππ1/2
πΏππΌπ½
(β
1
2
ππΌπ½π + π
πΌπ½)
+ π1/2
π (β]β]ππΌπ½πΏππΌπ½
β βπΌβπ½πΏππΌπ½
)
= π1/2
πΏππΌπ½
(ππΊπΌπ½+ ππΌπ½β]β]π β βπΌβπ½π) .
(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
πΏLπΉ= β
1
4
π3
πΉπΌπ½πΉπ]
Γ (πΏπ1/2
ππΌπ
ππ½]+ π1/2
πΏππΌπ
ππ½]
+π1/2
ππΌπ
πΏππ½])
= β
1
2
π3
π1/2
πΏππΌπ½
(ππ]πΉπΌππΉπ½] β
1
4
ππΌπ½πΉπ]πΉπ]) .
(23)
Now put together the pieces (22) and (23) to obtain theequation for the metric:
ππΊπΌπ½
=
1
2
π3
(ππ]πΉπΌππΉπ½] β
1
4
ππΌπ½πΉπ]πΉπ])
+ (βπΌβπ½π β ππΌπ½β»π) ,
(24)
which is equivalent to the equation for πΊπΌπ½
= 0 (12) whenπΊππ= 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; π = 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):
ΞΜπ
ππ=
1
2
πππ
(πππππ+ πππππβ πππππ) . (A.1)
All the connections are conveniently written as the sixconnections that distinguish the fifth coordinate, ΞΜ5
55, ΞΜ55], ΞΜ5
πΌ],ΞΜ]55, ΞΜ]5πΌ, ΞΜπ½π]. 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
ΞΜ5
55=
1
2
π5](βπ]π55) =
1
2
π΄]π]π, (A.2)
ΞΜ5
5] =1
2
π5πΌ
(π]ππΌ5 β ππΌπ5]) +1
2
π55
π]π55
=
1
2
π΄πΌ
ππΉπΌ] +
1
2
π΄πΌ
π΄]ππΌπ +1
2π
π]π,
(A.3)
ΞΜ5
πΌ] =1
2
π5π½
(ππΌππ½] + π]ππ½πΌ β ππ½ππΌ])
+
1
2
π55
(ππΌπ5] + π]π5πΌ)
= βπ΄π½Ξπ½
πΌ] +1
2
π΄π½
π΄]ππΉπ½πΌ
+
1
2
π΄πΌπ΄π½
ππΉπ½] +
1
2
(ππΌπ΄] + π]π΄πΌ)
+
1
2π
(π΄]ππΌπ + π΄πΌπ]π) +1
2
π΄πΌπ΄π½
π΄]ππ½π,
(A.4)
ΞΜ]55=
1
2
π]πΌ(βππΌπ55) = β
1
2
π]πΌππΌπ, (A.5)
ΞΜ]5πΌ=
1
2
π]π(ππΌππ5β πππ5πΌ) +
1
2
π]5ππΌπ55
=
1
2
π]π(ππΉπΌπβ π΄πΌπππ) ,
(A.6)
-
6 Journal of Gravity
ΞΜπ½
π] =1
2
ππ½πΌ
(ππππΌ] + π]ππΌπ β ππΌππ])
+
1
2
ππ½5
(πππ5] + π]π5π)
= Ξπ½
π]
+
1
2
ππ½πΌ
(π΄]ππΉππΌ + π΄πππΉ]πΌ + π΄ππ΄]ππΌπ) .
(A.7)
There are a couple of convenient identities:
ΞΜ5
55+ π΄]ΞΜ
]55= 0,
ΞΜ]]5 + ΞΜ
5
55= ΞΜπ
π5= 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, βGeometryβthe equations of Kaluzaβs unifiedtheory,β Comptes Rendus de lβAcadeΜmie des Sciences, vol. 226, p.216, 1948.
[3] H. Goenner, βSome remarks on the genesis of scalar-tensortheories,β General Relativity and Gravitation, vol. 44, no. 8, pp.2077β2097, 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] βxTensor computer tensor algebra file,β http://www.konflu-
ence.org/.[7] O. Klein, βQuantentheorie und fuΜnfdimensionale Relativi-
taΜtstheorie,β Zeitschrift fuΜr Physik, vol. 37, no. 12, pp. 895β906,1926.
[8] J. M. Overduin and P. S.Wesson, βKaluza-Klein gravity,β PhysicsReport, vol. 283, no. 5-6, pp. 303β378, 1997.
[9] R. Coquereaux and G. Esposito-Farese, βThe theory ofKaluza-Klein-Jordan-Thiry revisted,βAnnales de lβInstitut HenriPoincare Physique Theorique, vol. 52, p. 113, 1990.
[10] C. Brans and R. H. Dicke, βMachβs principle and a relativistictheory of gravitation,β Physical Review, vol. 124, no. 3, pp. 925β935, 1961.
[11] C.Will,Theory and Experiment in Gravitational Physics, revisededition, Cambridge University Press, Cambridge, Mass, USA,1993.
[12] V. Faraoni, βIllusions of general relativity in Brans-Dicke grav-ity,β Physical Review D, vol. 59, no. 8, 1999.
[13] S. Carroll, Spacetime and Geometry, section 4.8, AddisonWesley, San Francisco, Calif, USA, 2004.
-
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Superconductivity
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Physics Research International
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Solid State PhysicsJournal of
βComputationalββMethodsβinβPhysics
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
ThermodynamicsJournal of