The General Theory of Relativity, in the end, captured inertial motion and its close cousin of free-fall motion in a gravitational field, in the most

Transcription

1 Lab Note, Part : Gravitational and Inertial Mass, and Electromanetism as Geometry, in - Dimensional Spacetime Jay. Yablon (jyablon@nycap.rr.com), ebruary 6, 008 ***Note: This includes Lab Note, Part 3, dated ebruary 1, 008, beinnin on pae 17*** 1. Introduction It has been understood at least since Galileo s refutation of Aristotle which leend situates at the Leanin Tower of Pisa, that heavier masses and lihter masses similarly-disposed in a ravitational field will accelerate at the same rate and reach the round after identical times have elapsed. Physicists have come to describe this with the principle that the ravitational mass and the inertial mass of any material body are equivalent. As a material body becomes more massive and so more-susceptible to the pull of a ravitational field (back when ravitation was viewed as action at a distance), so too this increase in massiveness causes the material body in equal measure to resist the ravitational pull. By this equivalence, the result is a wash, and so with the nelect of any air resistance, all the bodies accelerate and fall at the same rate. (The other consequence of Galileo s escapade, is that it strenthened the role of experimental testin, in relation to the pure thouht upon which Aristotle had relied to make the obvious but untested and in fact false arument that heavy objects should fall faster. In this way, it spawned the essence of what we today know as the scientific method which remains a dynamic blend of thouht and creativity, with experience and cold, hard numbers derived from measurement of masses, lenths, and times.) Alon his path to developin the General Theory of elativity (GT), Albert Einstein made a brief stop in 1911 in an imainary elevator, to conduct a edanken in which he concluded that the physical experience of an observer fallin freely in a ravitational field before terminally hittin the round is no different from what was commonly thouht of as Newton s inertial motion in which a body in motion remained in motion unless acted upon by a force. (GT later showed that this was not quite true, the asterisk to this insiht arisin from the so-called tidal forces.) And, he concluded that the force one feels standin on the floor of an elevator in free fall to which a constant force is then applied, is no different from the force one feels when standin on the surface of the earth. 1

2 The General Theory of elativity, in the end, captured inertial motion and its close cousin of free-fall motion in a ravitational field, in the most eleant way, as simple eodesic motion in a curved eometry alon eodesic paths which coincide precisely with the paths one observes for bodies movin under ravitational influences. This was a triumph of the hihest order, as it placed ravitational theory on the completely-solid footin of iemannian eometry, and became the old standard aainst which all other physical theories are invariably measured, even to this day. ( Marble and wood is another oft-employed analoy.) However, the question of absolute acceleration, that is, of an acceleration which is not simply a eodesic phenomenon of unimpeded free fall throuh a swathe carved out by eometry, but rather one in which an observer actually feels a force which can be measured by a weiht scale in physical contact between the observer and that body which applies the force, is in fact not resolved by GT. To this day, it is hotly-debated whether or not there is such a thin as absolute acceleration. Surely, the forces we feel on our bodies in elevators and cars and standin on the round are real enouh, but the question is whether there is some way to understand these forces which are impediments to what would otherwise be our own eodesic free fall motion in spacetime under the influence of ravity and nothin more as eodesic forces in their own riht, simply of a different, supplemental, and perhaps more-subtle character than the eodesics of ravitation. That is the central question to be examined in this lab note. If we think of the ravitational mass of a material body more enerally as its interaction mass for the specific circumstance in which the interaction is ravitational, then the answer to the question whether the real forces we feel when our bodies are absolutely accelerated miht still be described in terms of eometric eodesics, may still lurk amidst Galileo s leendary escapade at Pisa, but with a twist. In this situation, the interaction mass of a material body is now inequivalent to its inertial mass, because that interaction is now electrical rather than ravitational. Here, Aristotle has his day, because electricallyheavier bodies do fall faster than electrically-lihter ones. How do electrical masses now come into play? When we fail to maintain our ravitational eodesic motion by failin to morph throuh the floor of the elevator, or when we fail to continue our ravitational free fall by not fallin unimpeded throuh the earth s surface, it is because we are stopped by the collective electrical repulsion between billions of electrons in our bodies and billions more in the elevator floor or the earth s round. It is because the

3 electrical interaction has now trumped the ravitational interaction and taken us off of our ravitational eodesics. Or, perhaps, if we can obtain a eometric insiht into electrodynamics, it is because we are now leavin the ravitational eodesic, and the atoms in our body are instead embarkin upon a different sort of eodesic path which now coincides with the path that has lon been observed as the Lorentz force motion of a chared mass in an electromanetic field. In liht of the quantum revolution of the 0 th century, one other consideration is in order. In this discussion, we are talkin not about quantum phenomenon, but about bulk phenomenon which lend themselves to completely classical description. In the same way that a bulk material body follows a eodesic path throuh ravitation, the question we raise is whether bulk electrical bodies, or lare numbers of electrons in an electric field, can also be understood, via their Lorentz force motion, to be followin eodesic paths made of marble no less fine than the marble with which General elativity directs the paths of material bodies throuh spacetime eometry in a manner that coincides precisely with what we observe and measure to be a ravitational path. We want to understand why we don t fall throuh the elevator or throuh the earth. Not only do we want to understand this in a way that avoids contradictin the eodesic principles of ravitation, we want to do so in a way that seamlessly extends these principles in a totally-self consistent way, into the electromanetic arena. ive-dimensional theories (or hiher), have frequently been a foundation upon which to try to mere classical ravitation with classical electrodynamics. Kaluza and Klein bean the trend, Einstein looked favorably on the effort, many others have followed, but to this day, there is as yet no theory which has been fully compellin in all aspects, and which at the same time, is motivated to a fifth dimension in a completely natural way, conservatively based on solid principles of observational physics which are already firmly-established.. Usin Dirac s Gamma- to Motivate a ifth, Timelike Dimension One of the most important connections in all of physics is iven by the Dirac relationship: ν ν ν { γ γ γ γ } η 1, (.1) whereby the Dirac γ matrices, 0,1,, 3, are defined so as to reproduce the Minkowski metric ν tensor dia( ) ( 1, 1, 1, 1) η under anticommutation. This relationship not only underlies 3

4 Dirac s equation, but also ensures that the Klein-Gordon equation applies to fermions as well as bosons. It is firmly established in all respects, and certainly must be rearded as one of those physical relationships which is made of marble over wood. Also made of marble, is the axial Dirac matrix first motivated by Weyl: γ γ γ, (.) γ iγ which is defined from matrix-multiplyin the other four Dirac matrices, and which has a wellestablished and riorously-observed physical meanin in relation to the left- and riht-chiral handedness of elementary fermions. We know that when the γ are sandwiched between Dirac spinors in the form j ψ γ ψ, the resultin current source density j (also thouht of as a probability and flux density) transforms as a four-vector in spacetime. We also know that j ψγ ψ is a pseudo-scalar. Thouh there are five such Dirac amma matrices, only four these are multiplicatively independent. We will now motivate a five-dimensional spacetime, based on (.1) and (.), in the followin way: The five-dimensional spacetime we employ will be one which is defined as a eometry in which j ( j, j ) j and j, taken toether, all transform toether as a five-vector, with 0,1,,3,. In this five-dimensional space, we employ uppercase Greek indexes. We maintain the lower case 0,1,, 3 for the usual -dimensional spacetime subspace. The metric tensor for such a five-dimensional eometry, must therefore be formed from the anticommutator of all five of the γ, similarly to (.1). That is, if j ( j, j ) transform as a five-vector, then we must define a five-dimensional, x Minkowski metric tensor accordin to: Ν η Ν Ν { γ γ γ γ } 1. (.3) is to Given the well-known anticommutation properties of the five Ν ( ) ( 1, 1, 1, 1, 1) γ, one can readily deduce that Ν dia η, and that η 0 for Ν. The usual Minkowski metric tensor ν Ν η is of course preserved in the 16 x components of η for which, Ν, ν 0,1,, 3. Importantly, because η 1, we find that this fifth dimension has a timelike, rather than a

5 spacelike sinature. Put succinctly: this five-dimensional eometry consists of two timelike and three spacelike dimensions. We next define infinitesimal coordinate intervals in the usual way, includin a fifth interval, that is, (,,,, ). Because γ is known as the axial matrix and because it is associated with a timelike metric sinature as noted just above, we shall refer to as the axial time coordinate, and will continue to refer to The x 0 x as the ordinary time coordinate. 1 3 x, x, x coordinates of course retain their role as ordinary space coordinates. Geometrically, in liht of the two timelike dimensions, it will often be very useful to reard time not as a time line but as a time plane. Thus, followin eynman, we miht not only think about worldlines which move forwards and backwards in time, but also which move sideways in time, and at various anles throuh the time plane. In fact, it is particularly helpful if one draws a vertical coordinate axis for ordinary time axis for axial time x orthoonal to a horizontal coordinate x, to represent the time plane. Then for material bodies at rest, 0, one may speak about the anle at which their worldlines move throuh this time plane. As we shall see, this may lead to a solely-eometric way to understand rest mass, electric chare, and electrical Lorentz orce motion, as eodesic motion throuh curved, non-euclidean eometry. The next step is to specify a metric interval d Τ for this five-dimensional spacetime. One miht reard this as a flat spacetime and so define η Ν Ν. However, if our objective is to understand the motions of electrical bodies on the basis of eodesic paths throuh a eometry, we must take one final step, and allow this five-dimensional eometry to be a curved, non-euclidean eometry just like that which is used in GT. Thus, we shall establish a metric tensor η κh just as in GT, and specify the weak-field limit accordin to Ν Ν Ν η, i.e., 0 Ν Ν h Ν. We further maintain the usual interval in the -dimensional spacetime subspace, usin d τ ν ν, and we thereby specify metric intervals in this - dimensional spacetime with axial time, accordin to: Ν Ν ν ν ν ν ν ν. (.)

6 (As an aside, for completeness, havin extended η Ν Ν to incorporate curvature and hence ravitation, we should also return to (.3), and redefine the Dirac Gamma matrices so as to incorporate these curvatures as well. Thus, we now define a new set of Dirac matrices Γ ( x ) from the contravariant Ν ( x ) Ν Ν Ν { Γ Γ Γ }, accordin to: 1. (.) These Γ, which are now fields rather than constant matrices, and which approach the usual in the weak-field limit, now implicitly include ravitational effects. When employed in Dirac s equation, these Γ lead to some very interestin ways to interpret the Schwiner manetic moments as indicative of ravitational effects near the Planck lenth which ive clues as to the true size of the elementary fermions, and these may also bear a relationship to the Γ γ Λ used in perturbation theory to represent non-diverent perturbative corrections. But these are topics for an entirely different paper. Let s return to the main thread of discussion by returnin to (.).) γ 3. A Possible Geometric Interpretation of est Mass In applyin (.), we extend all of the customary GT relationships from four to five dimensions. Thus, (thus the Ν and Ν Ν Ν is a symmetric tensor, inverses are specified by Ν δ Ν are used to lower and raise indexes), the covariant derivative of the metric tensor is defined by 0, the -D Christoffel connections are Ν; ( ) Γ 1, Τ Τ,, rank vector, hence Γ Γ Τ, and the covariant derivative of a first A is A A ; A,. Now, let s use alebraic manipulation to rewrite (.) in both of the followin forms: 1 ν ν and (3.1) ν ν 0 ν 1 ν. (3.) Usin a velocity four-vector u /, these can be combined to obtain: 6

7 ν 0 1 ν ( u u 1). (3.3) If we define a momentum four-vector p mu for a mass m in the usual way, and contrast (3.3) to the equation p m m ( u u 1) 0 p for the enery-momentum of an onshell mass m, we see that m ( u u 1) 0 d τ / in (3.3) plays a role identical to that of the mass m in. In the mass shell equation, m is of course introduced by hand, because the most one can deduce from the four-dimensional metric equation d τ ν ν is u u 1 0 ; then we need to multiply throuh by a mass m which we simply take out of thin air based on our empirical knowlede that masses exist in nature. In contrast, in (3.3), the d τ / multiplier of u u 1 0 arises totally out of the five-dimensional eometry, with nothin introduced by hand. d τ / is marble, and m is wood. We can capture this very-tellin correspondence, by writin: m. (3.) d Τ That is, in some way to be further determined, the rest mass of a material body appears to be proportional to the ratio of d τ to d Τ, and so may have a simple eometric foundation based on the trajectory of a worldline in the eodesics of this five-dimensional eometry. 0 x x time plane. Now, we are ready to examine the. The Geodesic Equation in ive Dimensions The five-dimensional calculation to follow is analoous to one way of derivin the eodesic equation in four dimensions, thouh we will in any event carry out the calculation in full a) to help the reader review the basic role of the metric equation as the first interal of the equation of motion b) to convince the reader that the -D eodesic equation is in fact correct, and c) to establish a careful discipline about workin in five dimensions, makin sure that all of our calculations are carried out in a -covariant rather than only a -covariant manner. 7

8 We return to the metric equation (.), and now rewrite this, aain via trivial alebraic rearranement, as: Ν 1 Ν. (.1) Takin the covariant derivative of each side, employin 0, renamin indexes, commutin, and usin Ν Ν, enables us to then write: Ν; Ν 0 Ν. (.) ; The covariant derivative ;,, which we substitute into (.). After some rearranin of terms, and assumin that 0, we can write: Ν Ν d 0. (.3) inally, we contract the and Ν indexes, do some index renamin, and obtain the eodesic equation in -Dimensions: Ν d x Τ 0. (.) This looks just like the four-dimensional equation, but for the indexes summin over all five dimensions rather than four, and the d Τ rather than d τ in the denominators. But, we can multiply throuh fully by /, and rewrite (.) as: d x Τ 0. (.) Now, let us work with (.) above. Equation (.) is a set of five independent equations. Let s separate it into the four spacetime equations represented by: d x Τ 0 (.6) 8

9 and the sinle axial-dimension equation: d x Τ 0. (.7) d x Equation (.6) is for, which is the observed acceleration of a worldline in the observed Τ four dimensions of spacetime. But, the five-dimensional summation in Γ in (.6) adds some new terms to the usual ravitational eodesic equation. Specifically, (.6) expands to: d x τ τ τ τ 0 (.8) which on account of Γ Γ Τ, we consolidate with some index renamin to: d x τ τ τ τ 0 (.9). Geodesic Motion of a Chared Mass in an Electromanetic ield Now, it is time to contrast (.9) above to the Lorentz orce Law when taken toether with the ravitational eodesic equation. This is, for example, set forth in equation (0.1) of Gravitation by Misner, Wheeler and Thorne: d x τ τ q m τ τ 0. (.1) At the outset, let us set aside the term Γ in (.9). We can do, this, for example, by settin Γ 0. We also move the / in front of the Γ τ, so that (.9) now becomes: d x τ τ Γ τ τ 0 (.) Now, we contrast (.1) directly with (.). The term τ Γ τ in (.) is the marble of τ q eometry, while the correspondin term τ is the wood of an empirically-derived m 9

10 relationship. However, comparin these terms, we can ive a totally eometric footin to the Lorentz force law if we note the proportionalities: q, and (.3) m τ Γ τ. (.) Combinin (3.), m /, with (.3), additionally yields: q (.) In fact, if we substitute the proportionalities in (.3) and (.) into (.1) as if they were equalities, then the eodesic equation (.), totally-based in eometry, is the Lorentz force law. Lorentz force motion now appears to be motion alon a eodesic, just like ravitational motion. Additionally, this eodesic motion appears to introduce an absolute acceleration, and hence force, because of the inequality of electrical and inertial mass, which via (.3) has its oriins in the eometric statement. This eometric statement, is now at the root of the force, and hence absolute acceleration, experienced by a chared mass in an electromanetic field. Yet, this force and absolute acceleration, still is a form of eodesic motion described by (.), and so rests fully upon the marble of eodesic motion throuh curved eometry. In fact, let s take this eometric understandin of inertial mass and electric chare even a 1 3 step further. Consider a material body viewed at rest, by settin 0, hence d τ 0. Then, accordin to (.): 00 Ν Τ d Ν. (.6) urther, take the weak field approximation where, 0. Now, (.6) becomes: η Ν Ν 0 ( ) ( ) d Τ. (.7) Draw 0 x and x as orthoonal axes, where 0 x is vertical and x is horizontal (sideways axial time). rom (.7), d Τ is clearly the hypotenuse. Define an anle Θ such that tan 0 Θ /. 10

11 Now, in this weak field 1 rest frame 3 0, we can write (3.), (.3), η Ν Ν and (.), respectively, as: m ( ) ( ) sin Θ. (.8) q m 0 cot Θ (.9) q cosθ (.10) 0 ( ) ( ) In this way, ravitational and inertial mass, as well as electric mass (chare), obtain a totally eometric interpretation in terms of the anle of a worldline throuh the time plane. eferrin especially to (.9), movement throuh ordinary time and inertial mass, movement throuh axial time 0 x contributes to ravitational x contributes to electrical mass, and real force and absolute acceleration arises from anular movement throuh the time plane in which a worldline projects both 0 x and x components. 6. Symmetric Gravitation and Antisymmetric Electrodynamics: Can they be Compatible? Now, let us turn back to the association τ Γ τ in (.). One of the fundamental difficulties which has been encountered by physicists attemptin to unify classical electrodynamics with GT, is that the former is an antisymmetric field theory while the latter is symmetric. How to combine oil and water in this way has perplexed physicists for over a century. So, the question arises, does the relationship τ Γ τ provide a path for a seamless and internally-consistent union of these two theories? As it stands, τ 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 tensor ν ν. However, now that we are in -D, we have to be careful that we are raisin indexes properly, usin the full Ν and not just its four-dimensional subset. To do this, we need to reconize that in -D, (.) should be eneralized to Τ Γ Τ, which means that 11

12 there are four additional, independent components in antisymmetric field strenth by requirin that Ν Τ (assumin we maintain an Ν ). We are not at this juncture concerned about what these new components miht be; the only reason for usin than Τ rather τ is to make sure we handle the raisin of the lower index properly in -dimensions. So, oin into -D, and usin 1 Τ ( ) Γ Τ,, Τ Τ, ( ) 1 Ν 1 ΤΝ ΤΝ 1 ΤΝ Τ Γ Τ Τ,, Τ Τ,, we rewrite (.) as:. (6.1) Althouh unconcerned for now about the extra components in customary path and reard Ν, we shall follow the Ν as a totally-antisymmetric tensor, thereby extendin this basic property of electrodynamics to these extra components, whatever they may be. That is, we continue to employ the condition Ν Combinin Ν ΤΝ Ν Ν. Ν Ν with (6.1) now lets us write: Τ Ν ( ) ( ) Τ,, Τ Τ, Τ,, Τ Τ,. (6.) This lets us express the antisymmetric field strenth relation Ν Ν completely in terms of certain relationships involvin first derivatives of the ravitational potential, as expressed via the metric tensor. Now, let us reduce this. rom (6.), we can rename indexes and use the symmetry of the metric tensor to write: Τ Ν ( ) ( ) Τ Ν Τ, Τ,, Τ Τ, Τ,, Τ, (6.3) which further reduces with some index chanes and the symmetry of the metric tensor to: ΤΝ 0. (6.), This is an alternative way of sayin that Ν Ν. We can further simplify this usin the inverse relationship ΤΝ δ Ν, which we can,, differentiate with respect to the th ΤΝ ΤΝ ΤΝ dimension to obtain ( ), 0 ΤΝ ΤΝ,,. This can then be used to reduce (6.) to the very simple:, i.e., Ν, 0. (6.) 1

13 Ν The above,, 0, is a purely eometric statement completely equivalent to Ν Ν. The symmetric field theory of ravitation is fully compatible with the antisymmetric field theory Ν of electrodynamics, so lon as we require that, Do Maxwell s Equations Become Components of Einstein s Gravitational ield Equation? We have shown the possibility that Lorentz force motion miht be described as simple eodesic motion in a five-dimensional spacetime with axial time, and that the inequality of electrical and inertial mass which causes one to feel a force and prevents one from fallin throuh the floor of Einstein s elevator or into the earth s core, may well emanate from the simple proportionality q m. But equations of motion are only one part of a complete field theory. The other part is a specification of how the sources of that theory influence the fields oriinatin from those sources. In a complete theory, the equations of motion then describe motion throuh the fields oriinatin from the sources. To complete the field theory which we have motivated thus far, one therefore would also need to also examine the Einstein equation in five dimensions: κt Ν Ν 1 δ Ν, (7.1) as well as the iemann Identity: 0, (7.) ΝΒ ΒΝ ΒΝ to see if amon their new axial (index ) components, one miht find the Maxwell equations ν ν j ;, and ν ; ν ; ; ν 0. In five dimensions, one would of course specify the iemann tensor in the usual way, albeit with an extra index. That is: ΒΝ Γ Β, Ν ΒΝ, ΒΝΓ Γ ΒΓ Ν. (7.3) Now, let s consider the component of this equation, that is: ΒΝ Γ Β, Ν ΒΝ, ΒΝΓ Γ ΒΓ Ν. (7.) The second term in the above, with explicit substitution of Γ ΒΝ is iven by: 13

14 Γ [ ( )] 0. (7.) 1 ΒΝ, Β, Ν Ν, Β ΒΝ,, Ν This is equal to zero, as a consequence of, 0, equation (6.), which is the same thin as Ν Ν, and because ordinary derivatives commute. Thus, by virtue of Ν a.k.a., 0, (7.) simplifies to: Ν Ν ΒΝ Γ Β, Ν ΒΝΓ Γ ΒΓ Ν, (7.6) consistin of only three terms. Now, we make use of Γ proportionality, in terms of the field strenth tensor, as such: 1 Τ Τ eneralized from (.), to rewrite (7.6) as a ΒΝ Β, Ν ΒΝ Γ Ν Β Β; Ν. (7.7) What is absolutely fascinatin, and of enormous eventual import, is that this expression for the mixed field strenth tensor. Β is identical to its ravitationally-covariant derivative Β ; Ν rom here, we can et to both of Maxwell s equations almost immediately. irst, let s contract (7.7) down to the icci tensor, and use the proportionality liberally to eliminate the minus sin, as such: Β Β Β ; j, (7.8) Β Β where j Β is the five-dimensional, covariant electric source current. Because ; 0, this can be raised into mixed form with some index renamin as: j ; ; ; ;. (7.9) Note that ; 0, and so 1 Ν ; ;, because Γ ; ; 0 by virtue of, 0, that is, Ν Ν. Now, we can return to Einstein s equation (7.1), for and Ν, and use (7.9) as well as δ 0, to write: 1 κt δ ; j, (7.10) This is the first of Maxwell s equations, for the field of an electric chare. We find, in particular, that j T is a four-vector situated alon the axial components of the enery momentum 1

15 tensor. The electric current, which is an electrical source density, is now also simply part and parcel of the eneralized ravitational source T Ν! What about Maxwell s manetic equation? Here, we lower all indexes in (7.7) and then use the symmetry ΝΒ Β; Ν ΒΝ ΝΒ to rewrite (7.7) as:. (7.11) Then, we turn to some more eometric marble, namely the iemann identity (7.). Takin the component of this identity, we write: 0, (7.1) ΝΒ ΒΝ ΒΝ Β; Ν ΒΝ; Ν; Β We may then consider the spacetime subset equations, to write: 0, (7.13) ναβ αβν βνα αβ ; β ; α α ; β Now, Maxwell s manetic equation also rests on eometric marble. Maxwell s electrodynamics in this manner, becomes fully unified with Einstein s ravitation. The equation for an electric source, j ;, is specified in eometry as: 1 κt δ. (7.1) The manetic equation 0 is specified in eometry as: αβ ; β ; α α ; β 0, (7.1) ναβ αβν βνα τ τ d x q inally, the eodesic equation 0 τ τ, which includes m the Lorentz force law, is specified in eometry as equation (.9): d x τ τ τ τ 0. (7.16) (We note, as an aside, that Ν Ν Ν a.k.a., 0 implies that Γ. 1, Other than to match the Lorentz force law term for term, see (.1) and (.), there is no apparent fundamental reason why we must have Γ 1 0. If Γ 0, the final term in (7.16), 1

16 is an extra term in the Lorentz force law, and usin q from (.9), this term is of the m, form 1 q 1 q q Γ, and so includes the couplin ratio m m m. If Γ 1, 0, then this term drops out entirely, and we revert to the exact comparison, Ν made between (.1) and (.). Additionally, if 0, and iven that, 0 so, 0,, this means taken toether that 0, so that constant throuhout the five-dimensional spacetime eometry. If we presume that the five-eometry is locally, asymptotically flat and therefore can always be transformed into eodesic coordinates at a sinle -dimensional event, then because 1 in eodesic coordinates (i.e., at a sinle event, ) it must also be 1 1 everywhere else. So the condition Γ 0 would require that 1, everywhere.) Irrespective of the ultimate disposition of, Γ 1,, all of the foreoin does appear to place Maxwell s electrodynamics onto the solid eometric footin of Einstein s ravitational theory. Even the inequivalence of electrical and inertial mass, and the real, measurable forces and the absolute accelerations which accompany this, are nevertheless the result of material bodies pursuin eodesic worldlines throuh a five-dimensional spacetime eometry. 16

17 Lab Note, Part 3: Gravitational and Electrodynamic Potentials, the Electro-Gravitational Laranian, and a Possible Approach to Quantum Gravitation Jay. Yablon (jyablon@nycap.rr.com), ebruary 1, The Electrodynamic Potential as the Axial Component of the Gravitational Potential Workin from the relationship Τ Γ Τ which eneralizes (.) to five dimensions, and reconizin that the field strenth tensor ( φ, A, A A ) A accordin to 1, A and the metric tensor 3 Ν ν ; ν ν ; ν is related to the four-vector potential A A, let us now examine the relationship between. This is important for several reasons, one of which is that these are both fields and so should be compatible in some manner at the same differential order, and not the least of which is that the vector potential A is necessary to establish the QED Laranian, and to thereby treat electromanetism quantum-mechanically. (See, e.., Witten, E., Duality, Spacetime and Quantum Mechanics, Physics Today, May 1997, p. 8.) 1 Startin with Τ Γ Τ, expandin the Christoffel connections ( ) Ν, makin use of, 0 which as shown in (6.) is Γ ΒΝ 1 Β, Ν Ν, Β ΒΝ, equivalent to Ν Ν, and usin the symmetry of the metric tensor, we may write: 1 ( ) ( ) 1 1 Τ Γ Τ,, Τ Τ,, Τ Τ,. (8.1) It is helpful to lower the indexes in field strenth tensor and connect this to the covariant potentials A, eneralized into -dimensions as A A, usin A ; A Τ Τ; ( ) ( ) ; Τ AΤ ; Τ, Τ Τ,, Τ Τ, The relationship ( ), Τ Τ, expresses clearly, the antisymmetry of, as such:. (8.) in terms of the remainin connection terms involvin the ravitational potential. Of particular interest, is that we may deduce from (8.), the proportionality A. (8.3) ; Τ, Τ (If one forms A ; Τ AΤ ; from (8.3) and then renames indexes and uses Ν Ν, one arrives back at (8.).) urther, we well know that A ; Τ AΤ ; A, Τ AΤ,, i.e., that the covariant 17

18 derivatives of the potentials cancel out so as to become ordinary derivatives when specifyin, i.e., that is invariant under the transformation A ; Τ A, Τ. Additionally, the Maxwell components (7.10) of the Einstein equation, are also invariant under A, because (7.10) ; Τ A, Τ also employs only the field strenth. Therefore, let is transform ; Τ A, Τ A in the above, then perform an ordinary interation and index renamin, to write: A. (8.) In the four spacetime dimensions, this means that the axial portion of the metric tensor is proportional to the vector potential, A, and that the field strenth tensor and the ravitational field equations κt Ν Ν 1 δ Ν are invariant under the transformation A ; Τ A, Τ used to arrive at (8.). We choose to set A ; Τ A, Τ, and can thereby employ the interated relationship (8.) in lieu of the differential equation (8.3), with no impact at all on the electromanetic field strenth or the ravitational field equations, which are invariant with respect to this choice. 9. Unification of the Gravitational and QED Laranians The Laranian density for a ravitational field in vacuo is ravitation, where is the metric tensor determinant and ν ν is the icci tensor. Let us now examine a Laranian based upon the -dimensional icci scalar, which we specify by:. (9.1) To start, we return to deduce the Β component of (7.6), namely: Ν Γ, Ν ΝΓ Γ Γ Ν, (9.) as well as, which is easily found by contractin the remainin free indexes in the above: Τ Τ Τ Τ Τ Γ, Τ ΤΓ Γ Γ. (9.3) Now, let us return to the discussion in the next-to-last pararaph of section 7, where we 1 considered, but reached no conclusions about, the question of whether Γ is, or is, 18

19 1 not, equal to zero. Let us now make the (inductive) hypothesis that Γ 0, and see what (deductive) results emere from this hypothesis. Ν irst, as already noted, taken toether with, 0, and because 1 in eodesic coordinates, this means that that, 1 constant everywhere in the -dimensional spacetime. Second, the eodesic equation (.9) now does reduce to (.), which, via (.3) and (.), can be made identically equivalent with the Lorentz force law (.1). Third, from (8.), A axial component of the covariant vector potential must now be constant everywhere in spacetime, that is A 1, the constant. ourth, this means that the axial components of the field strenth tensor must all become zero. To see this, we simply A ; Τ AΤ ; A, Τ AΤ, take: A A A A 0. (9.) Τ ; Τ Τ;, Τ Τ,, Τ Τ, The first term, 0, by virtue of the hypothesis just made. The latter term, 0,, Τ Τ, Ν because this is just a component of, 0, i.e., Ν Ν. Another way of statin (9.), is that only the ordinary spacetime components Ν are nonzero. Earlier, we eneralized ν to ν of the field strenth tensor Ν. Now, we find that all of these added components are, 1 zero. ifth, and finally, Γ 0 directly simplifies (9.) to: Ν Γ ΝΓ, (9.) and (9.3) to: Γ Γ. (9.6) Τ Τ Τ Τ Now, it will be helpful to start with the mixed icci tensor covariant form, in a -covariant manner, as such: Ν Ν Ν Ν, and lower this into Ν. (9.7) rom this, is it easily found, makin use of 1, that the component equation:. (9.8) 19

20 We then rearrane this into and insert the result into (9.1), thus arrivin at:. (9.9) inally, we make use of the spacetime components of: (7.9) written as j ; (8.) written as Τ A ; (9.6), written as Γ ΤΓ ; and the oft-employed Γ Τ Τ, to rewrite (9.9) as: Γ Τ 1 Τ Τ Τ a b A j, (9.10) 1 where we have absorbed the proportionality into the unknown constants a, b. However, the term Τ Τ Τ τ τ, with the final step taken by virtue of 0 Τ from (9.). Now, choosin a b 1, (9.10) finally reduces to: 1 τ 1 1 τ A j QED ( ) ravitation QED. (9.11) 1 τ Lo and behold: the QED Laranian density ( A j ) is QED automatically added to the four-dimensional icci scalar as part of the five-dimensional icci τ scalar. More to the point: is a seamlessly-interated electro-ravitational Laranian density, in vacuo. Choosin the constant factors a b 1, even the factor of ¼ and the neative sin of the QED Laranian density are all automatically introduced. Employed in the Euler- Larane equation, QED can be used in the usual manner to obtain Maxwell s equation ν j ν ;. But of even reater interest, is that we now brin QED into the mix, directly from ravitational theory in five dimensions, which raises a possible approach to quantum ravitation. 10. A Tentative Path Toward Quantum Gravitation Maintainin to be the four-dimensional metric determinant of ν, the electroravitational Laranian density, in vacuo, is now specified by: 1 τ ( A j ). (10.1) τ Thouh derived from a -dimensional spacetime with axial time, and wholly-founded upon eometrodynamics in five dimensions, all that remains in (10.1) are objects specified in ordinary 0

21 -dimensional spacetime. The Einstein-Hilbert action now includin QED is then specified in the usual way by interatin over the invariant -volume element dv d x : S c c 1 τ ( ν A ) d x ( τ A j ) Expandin, dv 16πG 16πG. (10.) A A, and interatin by parts in the usual way, then allows one to τ ; τ τ ; specify a path interal Z is ( A) iw ( J ) DAe e and associated transition amplitude ( J ) W for the QED portion of the above. But what is particularly intriuin, is that 1 QED, in five dimensions, is naturally added to the icci scalar, see (9.11). What Zee, A. in Quantum ield Theory in a Nutshell, Princeton (003), pp. 167 and 60 refers to as The Central Identity of Quantum ield Theory, is iven enerally by: 1ϕ K ϕ J ϕ V 1 ( ϕ ) V ( δ 1 /δ J ) J K J Dϕ e e e. (10.3) In (10.), we find that the icci curvature scalar, in this identity, plays the role of V. Given that the icci scalar, which is a classical ravitational scalar, is firmlyentrenched toether with QED in equation (9.11), iven that this oriinates strictly on the eometrodynamic basis of the -dimensional spacetime eometry with axial time, iven that appears to map neatly to V in the Gaussian identity (10.3), and iven that we know a reat deal about how QED is utilized in quantum field theory, all of the foreoin may provide a new pathway for understandin how to quantize ravitation. 11. The Geometric Maxwell Tensor Before concludin this lab note, it is also helpful to recast the Maxwell enery tensor 1 τ ( δ ν ) 1 T ν Maxwell π ν Γ into eometric form. Aain we start with Τ 1 Τ, which, in liht of (9.), 0 Τ τ, can be written without any information loss as the four-dimensional 1 Γ τ τ. irst, we return to (9.). Usin Γ 1 Τ Τ, this becomes: 1 Ν Γ ΝΓ Ν, (11.1) 1

22 and with 0, Τ 1 ν ν. (11.) The contraction of this, also evident via (9.6), becomes: 1 τ. (11.3) τ Employin (11.) and (11.3) then enables us to specify the Maxwell tensor, eometrically, as: 1 1 ( τ ν δ ν τ ) ν δ. (11.) 1 T ν T ν Maxwell π ν To maintain a balanced set of spacetime indexes, (11.) suests that the Maxwell tensor must actually be the axial Β and Ν ν component of larer, fourth-rank enery tensor T ΒΝ which has the same symmetries as the iemann tensor ΒΝ T ν Maxwell, and for which ν T. The Poyntin vector is then to be found residin on the T 0 0 k k components of the iemann tensor, k 1,, 3, and so too, acquires a totally eometric foundation. and These expanded tensors have 0 independent components. Particularly, ΒΝ ΝΒ yields a symmetric tensor of two antisymmetric x tensors. An ΒΝ antisymmetric x tensor has 10 independent components, yieldin a symmetric 10x10 tensor with independent components. However, identity (7.) imposes five constraints amon these independent components, one for each of the five indexes which is omitted from any particular equation based on (7.). The result is 0 independent components. Maxwell s tensor (11.) clearly contains 10 of these independent components, leavin the remainin 0 components for further exploration. ΝΒ