Kaluza-Klein Theory and Lorentz Force Geodesics. Jay R. Yablon * 910 Northumberland Drive Schenectady, New York,

Transcription

1 Kaluza-Klein Theory and Lorentz orce Geodesics ay. Yablon * 90 Northumberland Drive Schenectady, New York, Abstract: We examine a Kaluza-Klein-type theory of classical electrodynamics and ravitation in a five-dimensional iemannian eometry. Based solely on the condition that the electrodynamic Lorentz force law must describe eodesic motion in this fivedimensional eometry, it appears possible to place all of Maxwell s electrodynamics, the theory of electrodynamic potentials, and the QED action on a solid eometrodynamic footin. We make no choice as between the fifth dimension bein timelike or spacelike, but simply point out the impact in those places where this choice makes a difference. In the end, we deduce the Maxwell stress enery tensor, and in the process, learn that this fifth dimension must be spacelike. PACS: 0.0.h * jyablon@nycap.rr.com

2 . Introduction The possibility of employin a fifth spacetime dimension to unite classical ravitation and electrodynamics has intriued physicists for almost a century. [], [] Early theorists became perhaps overly-occupied with makin assumptions about the scale or topoloy of the extra coordinate dimension. [3] ollowin the path of Wesson and other current-day theorists [], we seek here to expose the main features of Kaluza- Klein theory irrespective of any particular model, and most importantly, to make the connection between Einstein s ravitation and Maxwell s electrodynamics which some have looked to -dimensional theories to provide, as clear and solid as possible, and as independent as possible of the detailed choice of model. Most fundamentally, we adopt the view of the above-noted theorists that matter and electrodynamic chare are induced in the observed four dimensions of spacetime, from a vacuum in five dimensions, and so, in keepin with the spirit of Wheeler s proram, [] are of completely eometrodynamic oriin. Particularly, we seek to show how classical electrodynamics emeres entirely from an Einstein-Hilbert Action of the eneral form S dv where is a suitably-defined icci curvature scalar, interated over a suitable multidimensional spacetime volume, and 8πG c is the constant from Einstein s equation T δ. The reader will observe that this omits any Laranian density Matter of matter, i.e., that it is not of the form S ( ) Matter dv and so is in the nature of action equation for the vacuum.[6] In different terms, we seek to induce the entirely of Maxwell's electrodynamics with sources, as well as the Maxwell stress-enery tensor, out of a ravitationally-based vacuum. The main line of development will be deduced, based on a sinle proposition: we shall require that the Lorentz force of electrodynamics, fully eodesic motion in the five-dimensional eometry. d x m q τ τ, must be represented as The foundation of this effort will be a five-dimensional iemannian eometry, without any chanes or enhancements, which merely extends the entire apparatus of ravitational theory into one more dimension. In five dimensions, we employ Ν with uppercase Greek

3 indexes, Ν 0,,,3, for the metric tensor, so with lowercase, 0,,, 3 is the ordinary metric tensor in the spacetime subspace. Inverses are defined in the usual manner accordin to Ν δ Ν and so and Ν raise and lower indexes in the customary manner, but must be applied over all five dimensions to achieve proper five-covariance. The covariant derivative of the metric tensor 0, as always. ; While most authors who still study Kaluza-Klein theories treat the fifth dimension as spacelike and a few have considered this to be timelike, e.., [7], [8], [9], we shall approach the fifth dimension as independently of this choice as possible. Where this choice does make a difference, we shall point this out. If we define η h in the usual manner with 6πG c, then for the weak-field limit η. If the fifth dimension is timelike, ( η ) (,,,, ) ; if it is spacelike, then dia( η ) (,,,, ) dia case, η 0 for Ν. Note that the constant in Einstein s equation. In either T δ is related to the foreoin, with fundamental constants restored, by 8 c c πg, with the overbar used to distinuish these two constants constant will appear frequently in the various equations herein.,. The At the end of section 0 see equations (0.) and (0.) infra, in the course of establishin the Maxwell stress-enery tensor, we will deduce that this fifth dimension must be spacelike.. Geodesic Motion in ive Dimensions, and the Lorentz orce We start by maintainin the usual interval in the -dimensional spacetime subspace, usin d τ, and define the five-space interval as: dτ Ν σ σ. (.) The above is independent of whether the weak field η, i.e., of whether the fifth ± dimension is timelike or spacelike, and is enerally model-independent. Like any metric equation, (.) can be alebraically-manipulated into: 3

4 Ν, (.) dτ dτ which is the first interal of the equation of motion. In five dimensions, we specify the Christoffel connections in the usual manner, that is, Τ ( ) Γ Τ Γ Τ. As noted, we employ ; 0 derivative Γ, Τ Τ, Τ, as usual, with the usual first rank covariant, hence A ; A, Γ A. We then take the covariant derivative of each side of (.) above, and after the usual reductions employed in four dimensions, and multiplyin the result throuh by dτ /, we arrive at a five-dimensional eodesic equation which bears an exact resemblance to the four-dimensional ravitational equation: d x Γ Τ Τ 0. (.3) The above contains five independent equations. We are interested for now in the four equations for which, which specify motion in ordinary spacetime: d x Γ Τ Τ 0. (.) This expands, usin the metric tensor symmetry d x Γ σ τ Γ σ σ Γ, to: Ν 0. (.) Now, let us contrast (.) above to the ravitational eodesic equation which includes the Lorentz force law, namely, equation (0.) of [0]: d x Γ σ τ q m σ σ 0. (.6) We now take a critical step: We require that the Lorentz force as expressed above, must be represented as nothin other than eodesic motion in the five-dimensional eometry. The first two terms in (.) and (.6) are identical, and they specify eodesic motion in an ordinary ravitational field absent any electrodynamic fields or sources. The absence of any mass or

5 chare in the first two terms captures the Galilean principle of equivalence, and further expresses Newtonian inertial motion in a ravitational field via the Christoffel connections Γ. If we require the Lorentz force to also be fashioned as eodesic motion throuh eometry, then we can do so by definin the third terms in (.) and (.6) to be equivalent to one another, and the fourth term in (.) to be zero. Therefore, we now define: σ σ q Γ σ σ, and (.7) m Γ 0. (.8) One miht wish to consider Γ 0, in which case Γ in (.) would become an additional term in the Lorentz force law, but in the absence of experimental evidence for any deviations from the Lorentz force law, we shall proceed on the basis of (.8). The relationships (.7) and (.8), ensure that Lorentz force motion is in fact, no more and no less than eodesic motion in five dimensions. All else throuh section 7 will be deduced from (.7) and (.8). 3. Placin the Lorentz orce on a Geometrodynamic ootin as Geodesic Motion Now, let us focus on equation (.7). We can divide out write the remainin terms as. σ from (.7), and then q Γ σ σ, (3.) c m where we have explicitly restored c. Now, we separate the proportionalities q m and Γ σ σ, and turn the proportionalities into equalities by restorin their dimensional and numeric constants, startin with the former proportionality. Irrespective of whether the fifth dimension is timelike or spacelike, we take to be iven in dimensions of time, so that is a dimensionless ratio. In the event that the fifth dimension is spacelike, one need merely divide throuh by c. In rationalized Heaviside-Lorentz units, the electric chare strenth q (for a unit chare such as the electron, muon and tauon) is

6 related to the dimensionless (runnin) couplin α q πc which approaches α / at low enery. The value of α is the same in all systems of units but the numerical value of q is different, so it is imperative that the exact expression for q m be based on α rather than q, and be independent of where the π factor appears. urther, to match dimensions with c the mass m needs to be multiplied by a factor of now define: G. Takin all of this into account, we cα q q. (3.) b Gm b πg m c m where b is a dimensionless, numeric constant of proportionality that we are free at this moment to choose at will, which we will carry throuhout the development, and which will ultimately be deduced to be b 8 when we obtain the Maxwell stress-enery tensor, see equations (0.) and (0.) infra. The equivalence between the first two terms is independent of the system of units but the terms containin q are in Heaviside-Lorentz units. Then, we substitute (3.) into (3.) to obtain: Γ σ σ. (3.3) The definitions (3.) and (3.3), toether with Γ 0 from (.8), when substituted into (.), turn the five-dimensional eodesic equation (.) into the Lorentz force law, and places this electrodynamic motion onto a totally-eometrodynamic footin. Of course, (3.3) is of further value, because it also relates the mixed field strenth tensor σ to the extra-dimensional connection components Γ σ, and this will lead to numerous other results. Althouh the Γ Τ are not themselves tensors in eneral, (3.3) does suest that that particular components transform in the same way as the mixed tensor suestion is formally validated by the result (6.), infra. σ Γ σ do, multiplied by a the constant factor. This The question of whether the foreoin are fair suppositions, now rests on the correctness and sensibility of the deductions to which they lead. 6

7 . Timelike versus Spacelike for the ifth Dimension, and a Possible Connection to Intrinsic Spin The results above are independent of whether the extra dimension is timelike or spacelike. In this section, we make a brief diression to examine each of these alternatives in a very basic way. This section can be safely skipped by the reader wishin to proceed straiht into the main line of development. 3 Transformin into an at rest frame, 0, the spacetime metric equation d τ reduces to d τ ± 0 00, and (3.) becomes: 0 00 q ±. (.) b πg m or a timelike fifth dimension, the physics ratio x may be drawn as a second axis orthoonal to q / m (which, by the way, results in the q / m material body in an electromanetic field actually feelin a Newtonian force in the sense of 0 x, and ma due to the inequivalence of electrical and inertial mass) measures the anle at which the material body moves throuh the or a spacelike fifth dimension, where one may wish to employ a compactified, hypercylindrical x, x 0 time plane. x φ (see [], iure ) and is a constant radius (distinuish from the icci scalar by context), dφ. Substitutin this into (3.), leavin in the ± ratio obtained in (.), and insertin c into the first term to maintain a dimensionless equation, then yields: dφ ± c b cα ± Gm b πg q m. (.) We see that here, the physics ratio q / m measures an anular frequency of fifth-dimensional rotation. Interestinly, this frequency runs inversely to the mass, and by classical principles, this means that the anular momentum is independent of the mass, i.e., constant. If one doubles the mass, one halves the tanential velocity, and if the radius stays constant, then so too does the anular momentum. Toether with the ± factor, one miht suspect that this constant anular momentum is, by virtue of its constancy independently of mass, related to intrinsic spin. In fact, 7

8 followin this line of thouht, one can arrive at an exact expression for the compactification radius, in the followin manner: Assume that x is spacelike, castin one s lot with the preponderance of those who study Kaluza-Klein theory. In (.), move the c away from the first term and move the m over to the first term. Then, multiply all terms by another. Everythin is now dimensioned as an anular momentum all to m v, which we have just ascertained is constant irrespective of mass. So, set this ± n, which for n, represents intrinsic spin. The result is as follows: dφ m ± b 3 c α ± G b c q ± n. (.3) πg Now, take the second and fourth terms, and solve for with n, to yield: b G b 3 α c α L P, (.) where L P 3 G c is the Planck lenth. This ives a definitive size for the compactification radius, and it is very close to the Planck lenth. (Keep in mind that we will eventually find in (0.) infra that b 8, so (.) will become / α.) What is of interest, is that α is a runnin couplin. At low probe eneries, where α / ,. 83 b LP. However, this is just the apparent radius relative to the low probe enery. If one were to probe to a reime where α becomes lare, say, of order unity, α then b LP is quite close to the Planck L P lenth of Wheeler s eometrodynamic vacuum foam. [0] at 3., [] * Since we have based the foreoin on a unit chare with spin ½, and since this is independent of the mass, the foreoin would appear to characterize the compactification radius for all of the chared leptons, and to provide a eometric foundation for intrinsic spin. This suests that for α or on the order of unity, the compactification radius of the fifth dimension may become * By way of review, the Planck mass, defined from the term atop Newton s law as a mass for which c, is thus M P c G GM P. In the eometrodynamic vacuum, the neative ravitational enery between Planck masses separated by the Planck lenth the Planck masses themselves. The Schwarzschild radius of a Plank mass 3 L P G c precisely counterbalances and cancels the positive enery of 3 GM / c G c L. S P P 8

9 synonymous with the Planck lenth itself, or the Schwarzschild radius of the vacuum, or somethin close to one of both of these. While (.) applies enerally for a compactified spacelike fifth dimension, before proceedin too far with this intrinsic spin interpretation (.3), however, it is worth notin that for a neutral body, q 0, such as the neutrino, we have d φ / 0, and so there is no fifthdimensional rotation. More enerally, any electrically-neutral body must be considered to be non-movin throuh the x dimension, 0. This would suest that the neutrino has no intrinsic spin, which is, of course, contradicted by empirical knowlede. So, (.3), while intriuin, does need to be studied further. Also, the intrinsic spin interpretation (.3) suests conversely, that any elementary scalar particle which has no intrinsic spin, must be electrically neutral. This is, in fact, true of the hypothesized His boson. [3] Despite the above puzzle reardin the neutrino, which we will return to in the hindsiht of the conclusion, the use of the term intrinsic to describe an inherent quantized anular momentum of elementary particles, covers up what is actually a deep inorance of what this really means. Why? or a material body to have an anular momentum, one must implicitly consider a radius with which that body circles about an oriin. At the same time, nobody believes that intrinsic spin represents an anular momentum about a radius in the three usual spatial dimensions. By associatin intrinsic spin with motion throuh a fourth, compactified, hyper-cylindrical spatial dimension, one simultaneously makes sense of intrinsic spin and of a compact fourth spatial dimension. The material body now has a spatial radius outside of the usual three spatial dimensions to ive meanin to its intrinsic spin, and the compactified fourth dimension now takes on meanin as somethin which is physically observed, via the phenomenon of intrinsic spin, and not merely a fictional idea that ives people pause about Klauza-Klein theories specifically, and dimensional compactification in eneral.. Symmetric Gravitation and Antisymmetric Electrodynamics Now, followin the brief diression in section, let us turn back to the association Γ σ σ in (3.3), which arises from the requirement that the Lorentz force be represented as eodesic motion in five dimensions. We know that is an antisymmetric tensor. By virtue of (3.3), this will place certain constraints on the related Christoffel connections 9

10 ( ) Γ Τ, Τ Τ, Τ,, and it is important to find out what these are. These constraints, in the next section, will provide the basis for placin Maxwell s equations onto a purely eometrodynamic footin. to irst, because we are workin in five dimensions, we will find it desirable to eneralize. We make no a priori supposition about the additional components in than to require that they be antisymmetric, Ν, other. Any other information about these new components is to be deduced, not imposed. Second, we eneralize (3.3) into the full five dimensions, thus: Γ. (.) By virtue of (.8), Γ 0, we may immediately deduce that: Γ b 0. (.) As it stands, is a mixed tensor, and it would be better to raise this into contravariant form where we can clearly examine the consequences of havin an antisymmetric field strenth Ν. Thus, let us now raise the lower index in (.), and at the same time equate this to the Christoffel connections, as such: ( ) Ν Ν Ν Γ,,,. (.3) Now, we use (.3) to write Ν completely in terms of the metric tensor and its first derivatives, as: Ν ( ) ( ) Ν Ν,,,,,,.(.) enamin indexes, and usin the symmetry of the metric tensor, this is readily reduced to:: ΤΝ 0. (.) Τ, This is an alternative, eometric way of sayin that Ν. We can further simplify this usin the inverse relationship Τ, Τ Τ, ΤΝ ΤΝ ΤΝ differentiate to obtain ( ), 0 ΤΝ Τ δ Ν, which we can ΤΝ ΤΝ, i.e.,,, Τ. This Τ 0

11 can then be used with to reduce (.) to the very simple expressions, for both the covariant and contravariant metric tensor:, 0 ; 0. (.6), This states that all components of the metric tensor are constant when differentiated with respect to the fifth dimension.,,, Γ from (.8), see also Now, we return to write out ( ) 0 ΤΝ ΤΝ (.). Combined with, 0 above and Τ,, Τ we further deduce that:, 0 ; 0 (.7), This means, quite importantly, that constant and constant, everywhere in the fivedimensional eometry. To fix these constant values, consider the weak-field limit dimension is timelike, dia ( ) (,,,, ) η. If the fifth η and. If it is spacelike (briefly explored reardin intrinsic spin in section ), then dia( ) (,,,, ). But, by (.7), if the above expressions for and they are true everywhere. Therefore: η and are true anywhere, then, or, (.8) respectively, for a timelike or spacelike fifth dimension. In either case, timelike or spacelike, Τ τ τ. The inverse δ then leads also to the null condition: Τ τ τ 0, (.9) τ τ which applies irrespective of the timelike versus spacelike choice. inally, usin (.) toether with (.6) and (.7), we may deduce: ( ) 0 Γ,,,. (.0)

12 Takin this toether with (.), Γ b 0, we have now deduced that all of the newlyintroduced fifth-dimensional components for the mixed field strenth tensor are zero, i.e., Γ 0. (.) The free index in 0 above can easily be lowered to also find that the covariant: 0. (.) But, since the ordinary spacetime components of are non-zero, one should take care to ensure that the contravariant tensor components 0 as well, that is, we want to make sure that the fixed index in (.) can properly be raised. One can employ (.) toether with the explicit components for Γ to write: ( ). (.3) Ν Ν Ν Γ,,, Expandin this to separate the from the components, and applyin (.6), (.7) and (.9) as needed, toether with Ν to eliminate the only term which (.6), (.7) and (.9) cannot directly eliminate, one can indeed deduce that in addition to (.) and (.): 0. (.) Now, the free index can be easily lowered, referrin also to (.), to find that: Γ Γ 0. (.) i.e., 0. So, we find that all of the newly-introduced fifth-dimensional components of the field strenth tensor, whether in raised, lowered, or either mixed form, are equal to zero. Equations (.), Γ 0, and (.), Γ Γ 0, taken toether, tell us that as well, the rule that any Christoffel connection with two or more fifth-dimension indexes, is also equal to zero. M Combinin (.) with 0 as well as 0, we may deduce two further relationships: Γ Γ 0 and Γ Γ Τ 0, (.6) Τ which are variations of the two or more fifth dimension index rule noted above.

13 It is also helpful as we shall soon see when we examine the iemann tensor, to make note of the fact that: Γ ( ) ( ) 0. (.7) Τ,,, Τ Τ, Τ,, Τ, Τ,, Τ,, This makes use of (.6) and the fact that ordinary derivatives commute. A further variation of (.7) employs (.) to also write, for the field strenth tensor: Γ, 0. (.8), i.e.,, 0. Lust like the metric tensor, all components of the field strenth tensor are constant when differentiated with respect to the fifth dimension. Aain, at bottom, every result in this section is a consequence of relationships (.) and (.), taken in combination with the antisymmetric field strenth the tools required to turn to the iemann tensor, and to Maxwell s equations. Ν. Now, we have 6. Maxwell s Equations as Pure Geometry We have shown how Lorentz force motion miht be described as simple eodesic motion in a five-dimensional Kaluza-Klein spacetime eometry. But equations of motion are only one part of a complete (classical) field theory. The other part is a specification of how the sources of that theory create the fields oriinatin from those sources. In a complete theory, the equations of motion then describe motion throuh the fields oriinatin from the sources. It is now time to place Maxwell s equations on a firm eometric footin. In five dimensions, we specify the iemann tensor in the usual way, albeit with an extra fifth-dimensional index. That is: Γ, Ν Γ Ν, Γ ΝΓ Γ Γ Ν. (6.) Now, let s consider the component of this equation, that is: Ν Γ, Ν Γ Ν, Γ ΝΓ Γ Γ Ν. (6.) By virtue of Γ Τ, 0, equation (.7), which is in turn a consequence of 0,, which is in turn a consequence of Ν, the second term zeros out, and (6.) becomes: Ν Γ, Ν Γ ΝΓ Γ Γ Ν. (6.3) 3

14 Substitutin (.), i.e., Γ into the above, and with some minor term rearranement, we immediately arrive at the very critical expression: (, Ν Γ Ν Γ Ν ) Ν Ν b ;. (6.) In particular, these three remainin terms of for the ravitationally-covariant derivative Ν turn out to be identical with the expression ; Ν of the mixed field strenth tensor, times the constant factor equations in the followin way:. This leads us immediately to a eometric foundation for Maxwell s As reards Maxwell s electric chare equation, we contract (6.) down to its icci tensor component and define a five-current with covariant -space index: ( Τ Τ Τ ), Γ Τ Γ ;. (6.) Now, we separate this into the two equations as such: ( σ σ τ τ σ σ τ ) σ, σ Γ τσ Γ ; σ, and (6.6) Τ Τ (, Γ Τ Γ Τ ) ;. (6.7) In (6.6), note that because 0 and Γ Τ 0 (see.), we can easily drop the, Τ indexes down to σ, τ. In (6.7), however, we leave ; as is because as we shall note in a moment, this term is not zero. In (6.6), we discern the four-covariant derivative σ σ σ τ τ σ ; σ, σ Γ τσ Γ σ τ, which is what allowed us to drop ; to σ ; σ. This means that σ σ is the observed ; electromanetic current source density, with covariant index. This is Maxwell s electric chare equation, on a eometric foundation. or the fifth-dimensional component in (6.7), we can use Τ 0 to eliminate the first two terms inside the parenthesis, but the third term is not zero. or the third term, we aain employ the substitution Γ from (.). Thus: 6 b ;. (6.8)

15 In the above, we have used Τ Τ Τ Τ Τ Τ. Note, that we raise and lower indexes while they are five-dimensional, then we reduce to lowercase Greek indexes via 0. σ Now, we bein to notice a sinificant result: Despite the ; ; σ ; term in (6.8) containin components of a mixed tensor which vanish in their own riht, namely 0, this term for is not equal to zero, and so, ; 0. ather, we find that the covariant σ derivative term ; ; σ ; 0 does not vanish even thouh 0, and in fact, leaves a very central term found in the QED free-field Laranian and in σ T Maxwell ( δ ), the Maxwell stress-enery QCD ( ree ) tensor in Heaviside-Lorentz units. One may think of ; 0 as bein ravitationally induced out of 0, solely as a non-linear ravitational effect, because in the absence of ravitation, covariant derivatives approach ordinary derivatives and so ;, 0. This induced term oriinates from the final term Γ Γ Ν of the σ iemann tensor, via the proression Τ Τ Γ ΝΓ Γ ΤΓ 6 b Τ, startin from (6.), and usin Γ from (.). So, the upshot of (6.8), is that the fifth component of the five-covariant current source density in a five-dimensional spacetime, ;, is not zero despite 0, is ravitationally-induced from the term Γ ΝΓ in the iemann tensor, and carries the scalar which is central to QED and the Maxwell stress-enery tensor and which, in the free-field Laranian density, represents the kinetic enery of a photon. Turnin now to Maxwell s manetic equation, we first lower the index in (6.),, and use to write: Ν Ν ( Ν ), Ν Γ Ν Γ ; Ν Ν Ν ;. (6.9) Maxwell s manetic equation then arises straiht from the -dimensional rendition of the first Bianchi identity: 0. (6.0) Ν Ν

16 Makin use of (6.9), the component of this is: ( ) ( ) 0 Ν Ν Ν b ; Ν Ν; Ν;, Ν Ν, Ν,, (6.) where we account for the well-known fact that in the cyclic combination of (6.) with antisymmetric tensors, the Christoffel terms in the covariant derivatives cancel identically, so the covariant derivatives becomes ordinary derivatives. In the immediately obtain Maxwell s manetic equation Ν α subset of this, we 0. (6.) α,, α α, In liht of our earlier havin found some new terms in Maxwell s electric chare equation arisin from the fifth dimension, see, e.., the equation in (6.8), one may ask whether there are any additional electrodynamic terms of interest in the (6.) above, in the circumstance where more than a sinle fifth-dimensional index is employed. Because, it is clear that with more than two fifth-dimensional indexes, e..,, (6.) will identically reduce to zero. But we should explore whether there is any additional electrodynamic information to be leaned when exactly two fifth-dimensional indexes are used in (6.). Thus, we may examine, say: ( ) ( ) 0 b ; ; ;,,,. (6.3) σ We learn from (6.8), especially ; σ 0, not to automatically eliminate a field σ σ strenth term such as when it appears in a covariant derivative, i.e., ; σ. However, the miration of covariant to ordinary derivatives in the cyclic combination of (6.) removes this complication. We know from (.) that 0, so their the ordinary derivatives of these in (6.3), by,,, will vanish as well. The remainin ( ), 0 virtue of (.6), 0, and (.8),, 0. Thus, (6.3) is identically equal to zero, not, only because of the Bianchi identity, but because of the inherent properties of the and developed in section. Thus, there is no additional electrodynamic information to be leaned from (6.3). 6

17 We have now placed each of Maxwell s equations on a solely eometric footin. Maxwell s source equation in covariant (lower index) form is specified by (6.6), namely, σ ; σ. The fifth component of this source equation, (6.8), contains the very central term QCD ( ree ), which is central to QED and to the Maxwell stressenery tensor. Maxwell s manetic equation is simply a fifth-dimensional component (6.) of the first Bianchi identity 0. And, the Lorentz force equation (.6), Ν Ν upon which the foreoin eometrization of Maxwell s equations is based, is merely the equation for four-space eodesic motion in the five-dimensional eometry, d x Γ Τ Τ 0, (.). With source equations producin fields and with material bodies in those fields movin over eodesics that are identical to and synonymous with the Lorentz force, Maxwell s classical electrodynamics with the Lorentz force law now rests on the firm eometrodynamic footin of a five-dimensional Kaluza-Klein eometry. Now, let s turn our efforts toward derivin the enery tensors and scalars associated with the foreoin. 7. Calculation of the ive-dimensional Curvature Scalar We bein discussion here by derivin the five-dimensional icci curvature scalar ( ) with. σ, where the ordinary four-dimensional curvature scalar σ. We ll start In (6.), we have already found. So, all we need do is raise the index usin ; ;, i.e., Τ ( Τ Τ, Γ Γ Τ ) ;, (7.) and then take the component. Above, we simply employ the definition of the covariant derivative of a second-rank contravariant tensor, particularly, of Now, we separate (7.) into: σ τσ ( σ τ τσ, σ Γ Γ τσ ) σ σ ; ;., and (7.) Τ Τ (, Γ Τ Γ Τ ) ;. (7.3) 7

18 In the former equation, (7.), we employ the same set of reductions used in (6.6), and we see that contains the contravariant current source density ( ρ, ( ), (), (3) ). In (7.3), the first two terms can be eliminated because Τ 0, so with suitable upper-to-lower-case reduction of Greek indexes also via Τ 0, we have: ; Γ τσ. (7.) While (.) tells us that Γ, this is the first time we have had to work with Γ τσ, and because Γ τσ Γ, this cannot be related directly to τσ. So, let s find out where the term comes in. Another way to arrive at (7.) from (6.) is to write: ( Τ Τ, Γ Τ Γ Τ ) ;, (7.) which merely entails usin the to raise the indexes in a five-covariant manner. This equation is identical to (7.), just in a different form. The M component is then: σ σ ( (, τ τ σ τ ) σ σ Γ τσ Γ σ τ Γ σ τ ) ;, (7.6) where we aain use suitable Τ 0 -based reductions, and have also expanded the final term Γ Τ Τ in (7.) into its spacetime and fifth-dimensional parts. Contrastin with (6.6), we see that Γ Γ σ σ τ τ σ, σ τσ σ τ is simply the lower index current density. And, in the remainin term, we may now employ the (.) substitution becomes: Γ Τ Τ. So, (7.6) now ( ) ( ), (7.7) usin Τ Τ Τ from (6.8), and Τ Τ Τ from followin (6.8). So, simply put, also contains the term, but it arises from the raisin of the index in, and so contains the term combination. It also helps to see directly as:, (7.8) 8

19 This expression (7.8) will play a central role in the section 0 derivation of the Maxwell tensor. eturnin to compare (7.) and (7.7), this also means that: Γ τσ. (7.9) So, now we have all the inredients needed to write out the five-dimensional curvature scalar, leavin as a remainin unknown still to be deduced. Usin (7.7), we simply write: ( ) ( ) 6 b. (7.0) The four-dimensional icci scalar if there is a way to deduce. σ σ is still an unknown in (7.). Now, let us see 8. The Einstein Hilbert Action, and Derivation of the Enery Tensor and the icci Tensor, from ive-dimensional Variation At this phase of development, we are at a juncture: Up until this point, all of the development has been based on a sinle supposition introduced just after (.6): the requirement that the Lorentz force must be represented as nothin other than eodesic motion in a fivedimensional eometry, as implemented throuh (.7) and (.8). Other than perhaps our imposin the requirement that Ν, every step taken since then has been fully deductive, with no other assumptions. We have even left open the question of whether the fifth dimension is timelike or spacelike, simply explorin the consequences in the alternative, as pertinent. This has enabled us to place Maxwell s equations, deductively, on a fully eometric footin, fully specify the fifth-dimensional components of the icci tensor, and obtain the σ five dimensional icci scalar (), but only up to the four-dimensional scalar σ, which still stands out as undetermined. Determinin, would ive us a window into turn into the remainin, and this in T components, amon which, one would expect to find the Maxwell stress enery tensor, which would be a final check on the validity of this entire path of development. So, we need to find. To deduce, we must now, finally, make a new supposition beyond that of Lorentz force eodesics, which we do as follows: 9

20 Some theorists, particularly those who have adopted the so-called Space-Time-Matter view [], seek the derivation of Einstein s equations out of a five-dimensional iemannian eometry without the introduction of explicit matter source terms. There are perhaps several ways to frame this objective: the one we shall choose here, as set forth in the introduction, will be to employ an Einstein-Hilbert action of the eneral form, which is to say, not usin an action S ( ) term Matter follows: S dv, omittin any source Matter dv. We do this is as Let us now posit that the action of the five-dimensional iemannian eometry that we have been explorin herein, is to be defined over the four-dimensional spacetime of our common physical experience, in the form: ( ) () dv ( ) S dv. (8.) This is a completely eometric definition of the action, without any explicit source term, of the eneral form ( ). S dv, but in which is replaced by the five-dimensional scalar Now, althouh there is no explicit source term in (8.), the component serves the role Matter dv, we see of an implicit source term, because if one contrasts (8.) with S ( ) that one can associate: S ( ) () dv ( ) dv ( Matter ) dv. (8.) Then, employin from (7.7), we have now effectively defined: Matter 8 ( ) 8 8 δ Ν 8. (8.3) eferrin to the old adae that δ is made of marble but T is made of wood, the definin of allows us to fashion a T or marble as well, because is a Matter completely eometric object. 0

21 Now, we can use variational principles to immediate calculate the enery tensor. Specifically, the variation of the -dimensional metric tensor determinant () is specified by δ () δ ( ) term Matter accordin to: (See [6]): T ( ) Matter Matter δ. The -dimensional enery tensor may be defined from the matter δ δ Matter. (8.) Then, we simply substitute the five-eometry-based Matter from (8.3) into the above, thus: ( δ Ν ) ( ) T, (8.) We note from (8.) that the four-dimensional enery tensor T b 8 is symmetric, T T with δ 0, but that the fifth-dimensional components T appear to be non-symmetric, because δ Ν δ Ν. Keep in mind,. The transposed (8.) is: Ν ( δ ) ( ) T, (8.6) Ν The mixed tensors formed from the above by raisin M, respectively, are: ( ) Ν ( ) b Ν δ δ b T Ν, and (8.7) ( ) Ν ( ) b δ b Ν Ν T. (8.8) This non-symmetry is further emphasized by the two different mixed tensors (8.7) and (8.8). There are two possibilities: either the fifth-dimensional components T T Ν Ν really are and ouht to be non-symmetric, or we will need to take steps to make this tensor symmetric. We defer this for the moment pendin a bit more development. irst, despite the non-symmetry, the five-dimensional trace enery from (8.7) and (8.8) turn out to be identical: T 3 ( ). (8.9)

22 3 Note that the factors arises because with an extra dimension, δ. If we now consider the Einstein equation in five dimensions as T Ν Ν δ Ν (), then this contracts down to T. Therefore, from (8.9) we deduce to for either (8.7) or (8.8): 3 ( ) () ( ), (8.0) inally, from the inverse Einstein equation, we deduce from (8.7) and (8.8) respectively, also 3 usin the common (8.9), that the factors cancel, all of the δ Ν terms cancel, and we are left with: 3 Ν T T b Ν 3 δ Ν δ () Ν, (8.) Ν T Ν 3 δ ΝT() b Ν, (8.) In retrospect, (8.) and (8.) could have been leaned directly from (8.7) and (8.8), which were written suestively for that very reason. However, it is useful to confirm that this works via the use of the inverse field equation, even with the extra dimension. Lowerin the upper indexes in the above, we obtain the respective covariant: b δ Ν, (8.3) Ν b δ Ν, (8.) which also in non-symmetric in the fifth-dimensional components Ν Ν, just like the enery tensor, contrast (8.). However, what we also deduce from either (8.3) or (8.) that the covariant curvature tensor in four spacetime dimensions is: δ 0, (8.0) δ that is: 0. The δ Ν which first made its appearance in (8.3) and (8.), is effectively a screen factor which shuts all four-dimensional components of the covariant icci tensor down to zero, and leaves a four-dimensional vacuum under the five-dimensional variation (8.). Althouh 0, this is not so for T, because by (8.) or (8.6) and (7.8):

23 ( b ) T T 8 8, (8.) Althouh non-zero, this tensor is symmetric in four dimensions. inally, we set out at the beinnin of this section to deduce the four-dimensional icci scalar. Combinin (7.7) with (8.8) yields ( ), i.e.: 0. (8.) More directly, this also comes from 0, (8.0). However, the ordinary, four-dimensional trace enery is not zero, but from (8.), is: ( ) T. (8.3) usin The derivation in this section made use of a five-dimensional variation, i.e., a variation δ. In section 0, we shall see how a four-dimensional variation δ leads to the Maxwell stress enery tensor. But first, we pause to examine the non-symmetry of the fifthdimensional component of the icci tensor,, and the enery tensor T T. Ν Ν Ν Ν 9. A Non-Symmetry icci Tensor for the ifth-dimensional Components? What are we to make of the fact that and Ν Ν T T Ν Ν in section 8 above? It is helpful to directly examine the definition of the iemann tensor (6.): Γ, Γ, Γ Γ Γ Γ, (9.) and the reverse-indexed: Γ, Γ, Γ Γ Γ Γ, (9.) The first and fourth terms are clearly identical, because Γ Γ. The third terms are also identical if one renames indexes. However, the second terms are not necessarily the same, Γ Γ, Γ Γ,, and specifically: ( ) ( ), and (9.3),,,,,,,,,,, ( ) ( ), (9.),,,,,,,,,,, 3

24 which, as a eneral rule, are not by identity, the same. If they are the same, it has to be because of particular constraints on the. But as a eneral rule, it is possible to entertain field equations, i.e., icci tensors and enery tensors which are not transposition symmetric, even when the. Γ Γ and This possibility has lon been known, and is the precise problem that Einstein pointed out in [], see his contrast of equations (a) and (b). It has also been noted that startin with a eneral (thouh still symmetric) connection allowed Eddinton and Einstein followin him in 93 to obtain a non-symmetric icci tensor, the antisymmetric part of which could then be taken as a representation of the (antisymmetric) electromanetic field tensor. [] Given the foreoin, as well as the fact that althouh non-symmetric in five dimensions, the fourdimensional enery tensor and icci tensor (8.) and (8.0) retain their T T and transposition symmetry, we shall accept the non-symmetric and Ν Ν T T Ν Ν as is, and not attempt to make these symmetric in the Ν indexes. That is, we shall take and Ν Ν T T Ν uncovered in the previous section as an indication that in nature, wherein Maxwell s Ν electric chare source equation is effectively represented alon those fifth-dimensional components, (see sections 6 and 7) the fifth-dimensional components of and T are nonsymmetric. Therefore, we return to (8.9), which we redefine in the opposite manner as before, reversin and Ν, as follows: b δ Ν (9.) We do this so as to be consistent with the results in section 6. Thus, from (9.) we find, just as in (6.6) and (6.7), respectively, that: δ (9.6) δ. (9.7) However: δ 0, (9.8)

25 which demonstrates explicitly the non-symmetric character of Ν Ν. The entire fifth column (, ) ( ), of the covariant icci tensor contains the Maxwell source chare current density term, while the fifth row we may summarize that transformin as part of a five-vector with a, except for, is zero. Combined with 0 from (8.0),., 0 As we shall now see, allowin the icci and enery tensors to stay non-symmetric will validate itself by leadin to the Maxwell stress-enery tensor, which we take to be a point of contact between theory and settled empirical observation. 0. Derivation of the Maxwell Stress-Enery Tensor, usin a our-dimensional Variation In section 8, we derived the enery tensor based on the variational calculation (8.), in five dimensions, i.e., by the variation δ. Let us repeat this same calculation, but in a slihtly different way. In section 8, we used (8.3) in the form of Matter b 8 b δ 8 Ν, because that ave us a contravariant δ Matter / δ Matter aainst which to obtain the five-dimensional variation. Let us instead, here, use the very last term in (8.3) as Matter, writin this as: τ ( ) ( δ ) 8 8 τ. (0.) It is important to observe that the term is only summed over four spacetime indexes. The fifth term,, see, e.., (6.8). or consistency with the non-symmetric (9.), we employ δ rather than δ. By virtue of this separation, in which we can only introduce as in section 8, we can only take a fourdimensional variation δ T Matter / ( ) and not δ Matter δ δ δ, which, in contrast to (8.), is now iven by: Matter Matter. (0.) Substitutin from (0.) then yields:

26 T τ ( δ ) ( b ) τ. (0.3) Now, the non-symmetry of sections 8 and 9 comes into play, and this will yield the Maxwell tensor. Because δ 0, the first term drops out and the above reduces to: T τ ( ) ( b ) τ. (0.) Note that this four-dimensional tensor is symmetric, and that we would arrive at an enery tensor which is identical if (0.3) contained a δ rather than δ. One aain, the screen factor δ 0 is at work. In mixed form, startin from (0.3), there are two enery tensors to be found. If we raise the index in (0.3), the first term becomes δ 0 and we obtain: T τ ( ) δ ( ) b τ. (0.) with this first term still screened out. However, if we transpose (0.3) and then raise the index, the first term becomes T and this term does not drop out, i.e., τ ( ) δ ( ) b τ. (0.6) So, there are two mixed tensors to consider, and this time, unlike in section 8, these each yield different four-dimensional enery tensors. Contrastin (0.) and (0.6), we see that δ 0 has effectively broken a symmetry that is apparent in (0.6), but hidden in (0.). At this time, we focus on (0.), because, as we shall now see, this is the Maxwell stress-enery tensor τ ( δ ) T τ, before reduction into this more-reconizable form. Purposely leavin constant factors separated, the trace equation of (0.) is then: ( ) T. (0.7) and so, via the inverse equation T δ T, from (0.) and (0.7): τ ( ) δ ( ) b 3. (0.8) τ 6

27 Note that here, traceable to the screened term, lost via δ 0, that one cannot simply lean from (0.) as we were able to for (8.9). It was necessary to use the full inverse field equation T δ T. Now, we take the trace of (0.8) to obtain: 3 ( ) T 6 b. (0.9) Interestinly, this does not look to be the same as the trace in (0.7), yet these are the same. This means that a further relationship must subsist, and if we look closely, (0.9) is the same as (0.7), multiplied by a factor of 3. If x 3x, then x 0 T 0, which is characteristic of Maxwell s tensor. So, settin (0.7) equal to (0.9), we obtain:, so this is an indication that the trace ( ) 6 ( ) b b 3 and we find after reducin, that: ( ), (0.0). (0.) Now, we return to the enery tensor (0.) and shift some terms to rewrite this as: τ ( ) ( ) τ δ δ T. (0.) Then, we substitute makin use of / c, to obtain: from (0.) into (0.), and do some further rearranin, includin 6 8 τ T ct τ δ. (0.3) b b If we now set c as well as: b 8 and, (0.) then (0.3) now reduces, rather fortuitously, to the Maxwell stress-enery tensor: τ ( δ ) T τ, (0.) in the Heaviside-Lorentz units that we have been employin from the outset. The factor b which we have employed all alon is now determined to be b 8. urther, because we have 7

28 deduced that we no loner need to straddle between a timelike and a spacelike fifth dimension: we have deduced that the fifth dimension must be spacelike. Also, despite the fivedimensional non-symmetry that we started with, the net result is still a symmetric tensor in fourdimensions. The stress-enery tensor is an important result, because this tensor is underpinned by extensive empirical evidence. We can then also derive the mixed icci tensor correspondin to the stress-enery (0.). We start with (0.8), substitute (0.), and reduce, to obtain: 6 b τ τ ( ) δ b ( ). (0.6) Clearly, this is also traceless, as it should be. urther use of / c with c, and b 8 and from (0.), then reduces to: τ ( δ ) τ, (0.7) which is summarized by the traceless field equation T, as expected. inally, in bein able to derive the traceless equation (0.) which amon many thins tells us that electromanetic enery propaates at the speed of liht, we have solved the essential riddle which concerned Einstein in [6], see equations () versus (a) and (3) therein, which was to find a compatibility between the field equation T δ which contains a nonzero scalar trace, and (0.) and (0.7) above which are scalar-free. More fundamentally, since (0.) was derived by riorously applyin the field equation T δ, we have demonstrated that Einstein s equation, which one ordinarily applies to trace matter which can be placed at rest, is also fully compatible with, and is indeed the foundation for, the enery tensor of traceless, luminous electromanetic radiation.. elation between the Electrodynamic Vector and Gravitational Tensor Potentials We now formally introduce the four-vector potential A ( φ, A, A A ) field strenth tensor accordin to ; ;,,, related to the, A A A A, where, as is well-known, the covariant derivatives become ordinary derivatives in the particular combination used to form. Introducin the vector potential A is desirable and indeed required, for as Witten points out, ([7] at p. 8) A is essential to the quantum mechanical treatment of electromanetism. 3 8

29 So far, we have restricted ourselves strictly to classical electrodynamics and classical ravitation, based on five-dimensional iemannian eometry. We now venture a tentative, introductory foray, from here, into the quantum world. Once aain, we start with (.), written out usin with 0 from (.6), as: Τ, ( ) ( ) Τ Γ Τ Τ,, Τ Τ,, Τ Τ,. (.) It is helpful to lower the indexes in field strenth tensor and connect this to the covariant vector potentials A, eneralized into -dimensions as A via Τ A ; Τ AΤ ; A, Τ AΤ, ( ) A A Τ ( ) ( ) ; Τ Τ; Τ, Τ Τ,, Τ Τ, The relationship b ( A A ) ( ) antisymmetry of Τ Τ ; Τ Τ;, Τ Τ, expresses clearly, the in terms of the non-zero connection terms ( ), Τ Τ,, as such:. (.) involvin the ravitational potential. Of particular interest, is that we may extract from (.), the relation: b A ; Τ, Τ h, Τ, (.3) usin also η h for the ravitational potential enery h. If one forms A from (.3) and then renames indexes and uses Ν, one arrives back at ; Τ AΤ ; (.). So (.3) is just the simplest form of equation (.). The reason we did not remove the covariant derivative via, is that in (.3), A Τ is considered Τ A ; Τ AΤ ; A, Τ AΤ, ; separated from A Τ;, and the covariant derivatives do not become ordinary unless and until one forms the combination. When the terms are separated as in (.3), Τ A ; Τ AΤ ; A, Τ AΤ, the covariant derivatives must be left intact. Γ Equation (.3) makes perfect sense classically: after all, the oft-employed of (.) is simply a first order differential equation between the vector potential A and the ravitational field h, and each is a dynamical field. Equation (.3) above merely states that differential equation explicitly. But quantum mechanically, (.3) raises questions, because we are talkin about a relationship between a spin- photon and a spin- raviton, and so need now to come to better terms with what (.3) implies quantum mechanically. 9

30 Equation (.3) is a first order differential equations which tells us up to a constant factor, that the covariant derivative of the electrodynamic potential A is equal to the ordinary derivative of the ravitational potential h. In the weak field limit / linear approximation, where covariant derivatives become approximately equal to ordinary derivatives, we have, Τ A ; Τ A, Τ, and so, interatin based on this linear approximation, we obtain: A. (.) Keep in mind, (.3) is exact and non-linear; (.) only applies to the weak-field, linear approximation A. ; Τ A, Τ Now, the reader will recall that the term and has shown up repeatedly throuhout many of the prior equations, oin all the way back to (7.7), and most recently, in the enery tensor (0.3) which later turned into the Maxwell stress-enery tensor, via the keystone relationship (0.) which enabled us to uncover the Maxwell tensor. Equation (.) is in lower-index (covariant) form. We now wish to obtain a suitable contravariant expression for various equations where index in (.) to write component, and use or which is akin to (.), so that this can be employed in the appear. To properly raise (.), let us first raise the free A ΝΤ. Next, we write Τ, then take the M δ δ Ν A Ν to obtain: Ν ΝΤ σ ΝΤ ΝΤ σ Ν Ν σ Ν Ν Τ σ Τ Τ δ σ δ δ σ A. (.) We now separate this out into: σ δ σ A A, and (.6) σ δ σ A A. (.7) The latter (.7) reduces to A. Since 6πG c and the Planck enery E P c G, and also usin b 8 from (0.), we may restate (.7) as: A. (.8) 8 π E P 30

31 Apparently, the fifth component of Ν A has a hue enery, on the scale of the Planck vacuum. It is a little trickier to reduce (7.6), because A reduces to A 0, which is another way of sayin that A <<< EP, i.e., that in the linear approximation, the spacetime part of the electromanetic vector potential, A, has an enery much less than the Planck enery. That is obvious, by definition. Let s instead make additional use of η h, and especially, η h h, and also from (0.) to rewrite (.6) as: ( h ba ). (.9) This yields a workable expression, and we find that the fifth component potential is added to A in this linear approximation. Now, we make further use of h of the ravitational h φ η φ which in the linear approximation has the ravitational field equation σ T σ φ with aue condition 0 φ. Quantummechanically, φ is of course representative of spin- ravitons, and we know that representative of a spin- photon. Also, φ φ is the scalar (spin-0) trace of φ. We A is eneralize to five dimensions, i.e., h φ η φ. Because, h φ 0. But h φ η φ φ, and so (.9) can be turned into: ( φ ba ). (.0) This suests that a spin- photon, toether with the decomposes into a vector potential, interpreted quantum mechanically as φ components of a ravitational wave, interpreted quantum mechanically as a spin- raviton. If (.0) tells us about the decomposition of in the linear approximation, and we already know that so h φ 0, what about the ordinary spacetime components of the contravariant? How do these decompose? Goin back to (.), we still start with ΝΤ Τ, but now we write: ΝΤ σ ΝΤ ΝΤ σ Ν Ν σ Ν Ν Τ σ Τ Τ δ σ δ δ σ A,(.) 3

32 where we aain use δ spacetime components are: Ν A Ν which we earlier used for (.). The four-dimensional ( φ ba ) A σ σ δ A A where we have also used ( φ ba ) from (.0) and. (.) with c. Proceedin forward, we then employ η h η ( φ η φ) to write: ( ) ( ) ( φ ba A η φ η φ ba ) A φ. (.3) estructurin a bit, and usin b 8 from (0.), we finally rewrite (.3) as: ( φ) φ φ A A A η. (.) Thus, in the linear approximation A ; Τ A, Τ, we find that decomposes into the Minkowski tensor η, a term φη which effectively multiplies the Minkowski tensor by the factor ( φ) which includes a spin-0 scalar φ, a ravitational wave φ with a quantum mechanical interpretation as a raviton, a A φ term which may be thouht of as containin both a photon affects liht propaation), and finally, the term A and the φ components of a raviton (remember, ravitation A A which contains two photons, and which is provocatively reminiscent of the [ G ], G terms which appear in the field strenth [ ] tensor G i[ G, G ] of Yan Mills theory, which in forms lanuae is simply dg ig. So, we have now obtained contravariant expression for namely (.0), ( φ ba ) where which is akin to (.),, so that this can be employed in the various equations appear. Alon the way, we have seen the contravariant decompose into spin- ravitons, spin- photons, and spin-0 scalars. Now, let s see how (.9) comes into play, particularly in those equations which contain the term. To sample (.0) further, let s for example, substitute this into the keystone relationship (0.), which we rewrite as 0 with. Because we 8 3