Penta-dimensional theory of gravitation and electromagnetism

Transcription

1 Théore pentadmensonnelle de la gravtaton et de l électromagnétsme, Sém. Janet. Méc. anal. et méc. céleste 2 ( ), Penta-dmensonal theory of gravtaton and electromagnetsm By Yves THIRY Translated by D. H. Delphench 1. Axomatc presentaton of the theory. The basc element of the untary theory s comprsed of a fve-dmensonal dfferentable manfold V 5 of class C 2, such that the second dervatves of the coordnate changes on the ntersecton of two admssble coordnate domans are pece-wse C 2 (so V 5 has class [C 2, pece-wse C 4 ]). A Remannan metrc dσ 2 of normal hyperbolc type everywhere s defned on that manfold, and ts local expresson n an admssble coordnate system s: dσ 2 = αβ dx α dx β (α, β, any Greek ndex = 0, 1, 2, 3, 4). The fundamental tensor, whose components αβ are called the potentals for the coordnate system n queston, and whch have class C 1 wth frst dervatves that are pece-wse C 2, s sad to determne the elementary untary phenomenon;.e., the moton of a charged materal partcle. The followng set of hypotheses consttutes the cylndrcty postulate for V 5 : The Remannan manfold V admts a connected one-parameter group of global sometres of V 5 wth trajectores z that are orented to have dσ 2 < 0 and leave no pont of V 5 nvarant. The famly of those trajectores satsfes the followng hypotheses: They are homeomorphc to a crcle T 1. One can fnd a dfferentable manfold V 4 that has class [C 2, pece-wse C 4 ], lke V 5, such that there exsts a dfferentable homeomorphsm of class [C 2, pece-wse C 4 ] from V 5 onto the topologcal product V 4 T 1, under whch the trajectores z map to the crcle factors. V 4, whch can be dentfed wth the space whose elements are the trajectores z, s the quotent manfold of V 5 by the equvalence relaton that s defned by the sometry group. There wll then exst local coordnates x α n V 5 that are sad to be adapted to the group, whch wll always be used n what follows and are such that: 1. The x are an arbtrary local coordnate system on V 4. The manfolds x 0 = const. are manfolds that are defned globally on V 5 and homeomorphc to V 4. The

2 Thry Penta-dmensonal theory of gravtaton and electromagnetsm. 2 homeomorphsm of V 5 onto V 4 T 1 can be assumed to map each x 0 = const. onto the manfolds that are homeomorphc to V 4 n V 4 T The αβ that relate to adapted coordnates are ndependent of x 0. The vector that s the nfntesmal generator of the sometry group has contravarant components: = 0 (, any Latn ndex = 1, 2, 3, 4) 0 = 1 and ts square s 00, whch s negatve. One sets: = Those coordnates are defned up to a change of coordnates: j x = ψ ( x ), 0 0 j x = x + ψ ( x ), n whch ψ represents the restrcton of an arbtrary functon ψ (z) that s defned on V 4 to a local chart n V 4, and one calls ths transformaton a change of gauge. In an adapted frame [whch s a frame that has a pont x of V 5 for ts orgn, s orthonormal, and ts frst vector e 0 s a unt vector that s tangent at x to the trajectory z (x) that passes through x], the metrc on V 5 s expressed wth the Pfaff forms ω α : n whch: We set: wth dσ 2 = (ω 0 ) 2 + (ω 4 ) 2 (ω 1 ) 2 (ω 2 ) 2 (ω 3 ) 2 ω 0 = dx dx. ds 2 = [(ω 4 ) 2 (ω 1 ) 2 (ω 2 ) 2 (ω 3 ) 2 ] = g j dx dx j, 0 0 g j = j 00 j. If one consders the vector ϕ λ on V 5 that s defned by: 0 β ϕ λ = λ 00, n whch β s a constant,

3 Thry Penta-dmensonal theory of gravtaton and electromagnetsm. 3 and ts rotaton F λµ then one can ntrnscally defne the followng objects on the manfold V 4 : A scalar, A metrc of normal hyperbolc type: ds 2 = g j dx dx j, whch s conformal to the quotent metrc (ω 4 ) 2 (ω 1 ) 2 (ω 2 ) 2 (ω 3 ) 2 that s nduced by dσ 2 on V 4, and whch we smply call the conformal metrc. An antsymmetrc tensor F j and on each secton W 4 that s canoncally homeomorphc to V 4, one has a vector feld ϕ such that: g j = j + β 2 3 ϕ ϕ j, F j = ϕ j j ϕ. That beng the case, the manfold V 4, when endowed wth the conformal metrc ds 2 can be nterpreted as the space-tme of general relatvty, whle the g j are the gravtatonal potentals. The non-canoncal recprocal mage on V 4 of a vector ϕ that s assocated wth a secton W 4 wll be nterpreted as the electromagnetc potental vector. More smply, one says that ϕ s the electromagnetc potental vector, so the tensor F j can be nterpreted as the electromagnetc feld. The hypotheses that were made on V 5 wll nduce the metrc ds 2 on V 4, as well as nducng the classcal hypotheses of the axomatcs of the provsonal theory of electromagnetsm on the electromagnetc feld tensor. 2. The provsonal theory of electromagnetsm. Before commencng wth our elaboraton of the equatons of the theory, we shall present a rapd crtque of the provsonal theory of electromagnetsm, snce that study and the ambton to unfy the gravtatonal feld and the electromagnetc feld (at least n the broad sense) s the bass for the geness of the penta-dmensonal theory that s presented here. In the provsonal theory of electromagnetsm, the gravtatonal feld that s defned by the geometrc structure that s adopted for the unverse wll take on a satsfactory explanaton. By contrast, the electromagnetc feld s smply supermposed over that geometrc structure by ts ntroducton nto the rght-hand sde of the Ensten equaton wth the ad of an energy-mpulse tensor that s defned by startng wth some concepts of specal relatvty. In order to wrte down the equatons of the provsonal theory, let us now ntroduce some nductons that are dstnct from the feld, and let F j represent the tensor of magnetc nducton (B) and electrc feld (E), whle H j represents the tensor of the magnetc feld (H) and the electrc nducton (D). Upon assumng the followng relatons between the felds and the nductons:

4 Thry Penta-dmensonal theory of gravtaton and electromagnetsm. 4 H τ = B. D = ε E, n whch τ s the magnetc permeablty and ε s the delectrc strength, wth ε τ = 1, then one wll have the relaton: H j = ε F j between the tensors H j and F j. Under those condtons, we wrte out the energy-mpulse tensor of the pure electromagnetc feld schema n the form: τ j = 1 4 g j H kl F kl 1 2 (H k F jk + H jk F k ). The equatons of the provsonal theory are composed of: The Ensten equatons: S j = λ T j, The frst group of Maxwell equatons: j H j = J, The second group of Maxwell equatons: 1 η jkl 2 j F kl = 0, whch s equvalent to sayng that there locally exsts a potental vector ϕ (whose global exstence we shall assume) such that: F j = ϕ j j ϕ. In the pure electromagnetc feld schema, T j reduces to τ j, and J = 0. In the electromagnetc feld-matter schema, one takes: T j = ρ u u j + τ j, and one assumes the Lorentz transport equaton: J = µ u, n whch µ s the pure electrc charge densty. One can then show that k = µ / ρ remans constant along a streamlne, as we consder only the homogeneous schema, for whch k s an absolute constant. For a gven feld that corresponds to a pure electromagnetc schema, f one consders a small charged test mass wth charge / mass that s equal to k and one represents ts nteror feld by a pure matter schema then t wll result from a study of the matchng

5 Thry Penta-dmensonal theory of gravtaton and electromagnetsm. 5 condtons on ts world-tube, whch s generated by streamlnes of ts nteror feld, and passng to the lmt that the trajectory of that mass wll satsfy the dfferental system: u u j = k F j u. We shall now state a result that s more general then the one that we shall appeal to, but whch can play a role n the nterpretaton of the penta-dmensonal theory: For any moton of a homogeneous-charged perfect flud, the streamlnes wll be tmelke lnes that realze an extremum for the ntegral: x1 x0 F ds + k ϕ for varatons wth fxed extremtes, n whch ϕ s the vector potental form, and F s the ndex of the flud;.e.: exp p dp, p0 ρ + p when one assumes an equaton of state ρ = ϕ (p), wth k = µ / ρ *, ρ * = (ρ + p) / F. Upon usng that result for the charged pure matter schema, whch one can pass to by settng: p = 0, F = 1, ρ * = ρ, and upon returnng to the test mass, one wll see that the trajectores of that charged partcle are the tme-lke lnes that realze the extremum of the ntegral: x1 j 1/ 2 ( gjx x ) k x ɺɺ + ϕ xɺ du, 0 xɺ = dx du. Those trajectores are then geodescs of the Fnsler manfold that admts the metrc: ( L x, xɺ ) = ( g xɺɺ x ) + k ϕ xɺ. j j 1/ 2 3. The geodescs of V 5 and ther projectons onto V 4. Now return to the Remannan manfold V 5 that was envsoned n paragraph 1. Set: L 2 = αβ and suppose that One can then establsh the followng result: α xɺ xɺ β

6 Thry Penta-dmensonal theory of gravtaton and electromagnetsm. 6 The extremals on V 5 of the ntegral ntegral 0 ɺ L = h project onto V 4 along extremals of the ntegral: x1 x0 L du that correspond to the value h of the frst x1 j L( x, xɺ, h) du, x0 n whch h has the same value and L s gven by: L = 1 + h / j gj x x + hβ ϕ x ɺɺ ɺ, n whch g j and ϕ have the meanngs that they were gven n paragraph 1. That expresson for L s vald for the geodescs of V 5 that make L 2 postve. In order to make L 2 negatve, whch must lkewse play a role n the nterpretaton of the theory, one sets h = h 1 and consders the functon: L 1 = h1 / 1 j gj xɺɺ x + h1 β ϕ xɺ nstead of L. The functons L have the type of the square root of a quadratc form that s augmented by a lnear form of the knd that defned the trajectores of the charged partcle of the provsonal theory n the precedng paragraph. The penta-dmensonal untary theory was presented axomatcally, and ts propertes were presented as thngs that are mplct from those axoms. However, the geness of the theory must naturally follow along a dfferent path: A problem n the calculus of varatons lead to a descent process from a functon L to a functon L and an ascent process from L to L. In the case where 00 0, when the descent process s appled to a Remannan metrc L, t wll lead to a functon L that has the type of a square root of a quadratc form plus a lnear form. Hstorcally, one developed a cylndrcal manfold V 5 by the nverse ascent process, whch starts wth the functon that defnes the trajectores of a charged partcle n the provsonal theory. Return to the functon L and try to relate t to the provsonal theory. One wll then be led to set: (3.1) k = β h, or for L 1, k = β h1 1 + h / h / 1 1 and to study the varatons of x 0 by specfyng 0 ɺ L = h, whch gves:

7 Thry Penta-dmensonal theory of gravtaton and electromagnetsm. 7 (3.2) dx 0 k = 3 β ds β ϕ. One asserts that f one supposes that s constant then one wll recover the trajectores of the charged partcle n the space-tme V 4 of general relatvty precsely as projectons of the geodescs of V 5 along whch x 0 vary accordng to formula (3.2). One wll be n the presence of the Kaluza-Klen penta-dmensonal theory, whch adds nothng new to the provsonal theory of electromagnetsm, but smply represents t s a dfferent formalsm, whch s nonetheless useful and nterestng, and especally for the treatment of the global theorems n the provsonal theory. When one rejects the hypothess that = const., that wll lead to the pentadmensonal theory whose axoms I have presented. Consder the geodescs n V 5 along whch x 0 vares accordng to formula (3.2), n whch h s a well-defned constant, and consder the projecton of such a geodesc. It wll result from the feld equatons and the matchng condtons that wll be adopted that the projecton defnes the spato-temporal trajectory of a charged partcle. That projecton s extremal n V 4 of the ntegral that s assocated wth the functon: 1 k ds + ϕ, n whch k has the value that was gven by (3.1). If h s constant, but vares, then that functon wll dffer from the functon ds + k ϕ of the provsonal theory by more than just a multplcatve constant. The varatons of wll then be regarded as weak, and wll be regarded as close to 1, whch permts one to ntroduce the constant β. 4. The equatons of the theory. We take the feld equatons of the pentadmensonal theory to be the formal generalzatons of the feld equatons of general relatvty. In the external untary case, whch corresponds to the pure electromagnetc feld schema of general relatvty, the equatons are wrtten: S αβ = 0, n whch S αβ = R αβ 1 2 g αβ R s the Ensten tensor on V 5. They wll be characterzed by the followng varatonal prncple: They defne the extremum of the ntegral: αβ R αβ dx dx dx dx dx C

8 Thry Penta-dmensonal theory of gravtaton and electromagnetsm. 8 whch s taken over a fve-dmensonal dfferentable C for varatons of the 15 potentals and ther frst dervatves that are zero on the boundary of C. That comes down to specfyng the equatons S αβ = 0 solely n terms of V 4 when t s endowed wth the conformal metrc. The calculaton of S 0 nvolves the covarant dervatve, not of the electromagnetc feld tensor F j that was defned n paragraph 1, but of 3 F j. One wll then be led to ntroduce some nductons that are dstnct from the felds by settng: H j = 3 F j, and to nterpret 3 as the delectrc constant. In S j, the tensor τ j s constructed n the form that was ndcated n the provsonal theory and only the frst dervatves of ntervene. The equatons of the external untary case n local adapted coordnates on V 4, when t s endowed wth the conformal metrc: (4.1) wth: 3 Sj + 2 Kj χτ j = 0, j jh = 0, j 2σ + 2 1σ + χ HjF = 0, K j = 1 g 2 j 1 σ σ j σ, σ = log. χ = β 2 / 2 s nterpreted as the gravtatonal constant. If one gves the value of 1 to and suppresses the ffth equaton then one wll recover the equatons of the provsonal theory of electromagnetsm when they are adapted to the scope of Klen s theory. If s varable then they wll generalze the equatons of the pure electromagnetc feld schema of general relatvty, and the varatons of the ffteenth feld varable (vz., delectrc strength) wll be governed by a ffteenth equaton. In the nternal untary case, the equatons of the penta-dmensonal theory are wrtten: S αβ = Θ αβ. At best, Θ αβ s chosen to represent a charged matter dstrbuton. If one uses the formal generalzaton of a perfect flud schema Θ αβ = r v α v β ϖ αβ then the ntroducton of the ndex of the flud (whch wll undoubtedly be nterestng for the physcal nterpretaton) wll possbly permt one to narrow down the choce of trajectores of the provsonal theory that are extremals of the ntegral: F ds + k ϕ.

9 Thry Penta-dmensonal theory of gravtaton and electromagnetsm. 9 We shall use only the formal generalzaton of the pure matter schema Θ αβ = r v α v β, whch s a partcular case of the precedng one. r s a postve scalar, n whch v α are the covarant components of a vector whose square s + 1 or 1 and whose trajectores are call penta-dmensonal streamlnes. It results from the conservaton equatons that n V 5 those streamlnes wll be geodescs of that Remannan manfold, and one can just as well use geodescs that are orented to have dσ 2 < 0 as ones that are orented to have dσ 2 > 0. The penta-dmensonal streamlnes (orented to have dσ 2 > 0) that satsfy the ntegral ɺ L = 0, whch translates nto v 0 = h, project onto V 4 along tme-lke streamlnes such 0 that along those lnes: and they are extremals of: 1 + h / 2 dσ = ds 2, z1 1 + h / ds + β h ϕ. z0 If u s the unt velocty vector wth the conformal metrc ds 2 on V 4 then one wll have: v = 1 + h / along a streamlne. The frst fourteen equatons of the nternal untary case are then wrtten: u S j K j χ τ j = 1 + h / r u u j, j H j = 2h 1 + h / r β u, and one wll then be led to set: χ ρ = 1 + h / r, whch wll ndeed gve back: µ = 2h 1 + h / r β µ ρ = k = β h. 1 + h /, The ffteenth equaton can be put nto the form:

10 Thry Penta-dmensonal theory of gravtaton and electromagnetsm σ σ + χ H j F j = 2 r 1 h By defnton, we can wrte the equatons of nternal untary case n the followng form, whch generalzes the case of the electromagnetc feld-matter schema of general relatvty: S + K χτ = χ ρ u u j 3 2 j j j j jh µ u, = 2 1 h + 2 j 3 2σ + 2 1σ + χ HjF = χ ρ. 2 h 1+ 2 When one consders the equatons that correspond to the geodescs of V 5 that are orented to have dσ 2 < 0, t wll suffce to set h 1 = h n the precedng formulas. Remarks. The geodescs of V 5 that are orented to have dσ 2 > 0 can account for the trajectores n V 4 only for partcles that have a very small absolute value of µ / ρ. For the other ones (and that wll be case of the electron and photon), one wll then have to appeal to the geodescs that are orented to have dσ 2 < 0. One can gve varous other forms to the ffteenth equaton that are dvergence formulas and are useful for the treatment of global problems., 5. Study of the penta-dmensonal theory. 1. The Cauchy problem and the geodesc prncple. In the exteror untary case, consder the followng problem statement, whch relates to V 5 : One s gven the potentals αβ and ther frst partal dervatves on a hypersurface Σ n V 5 that s generated by the trajectores of the sometry group. Determne the values of those potentals outsde of Σ when one supposes that they satsfy the equatons S αβ = 0. Ths problem s the translaton to V 5 of the followng one: One s gven the gravtatonal potentals g j, a potental vector ϕ, and a scalar on a hypersurface S n V 4, along wth ther frst dervatves. Determne the values of those quanttes outsde of S when one supposes that they satsfy equatons (4.1) and Σ s the hypersurface n V 5 = V 4 T 1 that projects onto the ponts of S. The problem wll then admt one and only soluton that s cylndrcal under some hypotheses that must be specfed (e.g., analytcty or smple dfferentablty of the gvens). It wll then result that the characterstc manfolds S c of the equatons of the theory n V 4 are always manfolds that are tangent to the elementary cones n V 4. The characterstc manfolds Σ c n V 5 are

11 Thry Penta-dmensonal theory of gravtaton and electromagnetsm. 11 generated by the trajectores that project onto an S c and are, n turn, tangent to the elementary cones of V 5. Those manfolds are the wave surfaces of a untary feld such that crossng them can produce dscontnutes n the second dervatves of the potentals. The study of the Cauchy problem n the nteror untary case shows that the crcumstances that generalze the ones that are encountered n V 4 n general relatvty for the pure matter case to V 5 correspond to the ones that are encountered n V 4 for the case of electromagnetc feld-pure matter. The study of the prolongaton from the nteror to the exteror and the matchng condtons then lead to the geodesc prncple, and as was sad, that wll permt one to nterpret the geodescs n V 5 that are orented to have dσ 2 > 0 or dσ 2 < 0 as the pentadmensonal trajectores of charges partcles n the untary feld. 2. Global study of untary felds. That study s based upon the search for the hypotheses under whch the followng propostons wll be vald: The ntroducton of the matter dstrbuton (charged or not) nto a gven exteror untary feld can be acheved only n domans where that feld s not regular. An everywhere-regular exteror untary feld s trval. The crcumstances that one fnds wll nduce crcumstances n V 4 that determne whether the smlar propostons n V 4 are true for the varous schemas. The global study n Klen s theory and thus, n the provsonal theory leads to less satsfactory results. 3. Equatons of moton. The fve conservaton denttes of the tensor Θ αβ, namely: are equvalent to: D α (Θ α β) = 0 (D α : covarant dervatve operator for the Remannan connecton on V 5 ) The ntegral of the geodescs, whch one can wrte v 0 = h. A contnuty equaton that one can translate nto the statement that: n dx dx dx, n = 4 r v remans constant along a streamlne. Three equatons: v α D α v A = 0 (A = 1, 2, 3), whch gve the equatons of moton n the form:

12 Thry Penta-dmensonal theory of gravtaton and electromagnetsm. 12 d dt 4 Θ C A dv = 1 2 C Θ αβ α αβ dv (where the ntegrals are taken over some body C) that s used n general relatvty. The conservaton denttes can then be nterpreted as beng the penta-dmensonal equatons of moton of a test body. Those equatons are ntmately lnked wth the sothermal condtons. One can defne sothermal coordnates n V 5 by a formal generalzaton of the sothermal coordnates that are used n general relatvty. The sothermal condtons n V : 5 Φ ρ 1 αβ α ( ) = 0 translate n a V 4 that s endowed wth the conformal metrc nto the sothermal condtons: j ( ) j g g = 0, and by fxng the gauge transformaton upon whch the electromagnetc potental depends, the penta-dmensonal sothermal coordnates wll be found to be compatble wth the cylndrcal character of V 5. The search for an approxmate untary soluton leads one to consder some equatons of moton that are approxmate of order p, and one has the followng result: The equatons of moton that are approxmate of order p mply some sothermal condtons that are approxmate of order p that must be verfed. The calculaton n the frst approxmaton shows that t s only when one nterprets thngs n a V 4 that s endowed wth the conformal metrc that the potentals that relate to an uncharged schema wll be dentfed wth the ones n the pure matter case of general relatvty. The ssues that were just presented ndeed seem to mpose the nterpretaton that was gven here upon the theory. Bblography 1. F. HENNEQUIN, Étude mathématque des approxmatons en relatvté générale et en théore untare de Jordan-Thry, Bull. scent. Comm. Trav. hst. et scent. 1 (1956), Part 2: Mathématques, Gauther-Vllars, Pars, 1957; pp (Thess Sc. math., Pars, 1956). 2. P. JORDAN, Schwerkraft und Weltall, 2 nd ed., Fredr. Weweg and Son, Braunschweg, (De Wssenschaft, 107). 3. O. KLEIN, Quantentheore und fünfdmensonale Relatvtätstheore, Zet. Phys. 37 (1926),

13 Thry Penta-dmensonal theory of gravtaton and electromagnetsm A. LICHNEROWICZ, Théores relatvstes de la gravtaton et de l électromagnétsme, Masson, Pars, Y. THIRY, Étude mathématque des équatons d une théore untare à qunze varables du champ, J. Math. pures et appl. 30 (1951), (Thess, Sc. math. Pars, 1950).