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Kaluza-Klein Theory with Torsion

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Kaluza-Klein Theory with Torsion Kaluza-Klein Theory with Torsion Kameshwar C. Wali Cos PA, 18-20 November, 2009, Melbourne, Australia ² INTRODUCTORY REMARKS ² BASIC FORMALISM ² A SET OF MINIMAL CONSTRAINTS ON TOR- SION ² CONNECTION COEFFICIENTS ² RICCI TENSOR, RICCI SCALAR ² EINSTEIN EQUTIONS ² SOME RESULTS ² CONCLUDING REMARKS 1 ² WORK DONE IN COLLABORATION WITH KARTHIK H. SHANKAR. ² WE CONSIDER A VARIANT OF 5D K-K THE- ORY, WITHIN THE FRAMEWORK OF EINSTEIN- CARTAN FORMALISM,BUT GENERALIZABLE TO ARBITRARY DIMENSIONS THAT INCLUDES TOR- SION. ² TORSION, A GEOMETRICAL PROPERTY OF SPACE-TIME, REPRESENTS SPIN DEGREES OF FREEDOM. IN ANALOGY TO EINSTEIN EQUA- TIONS WHERE THE ENERGY-MOMENTUM OF MATTER FIELDS IS COUPLED TO THE GE- OMETRICAL RIEMANNIAN METRIC, WE CAN COUPLE ANGULAR MOMENTUM AND SPIN OF MATTER FIELDS TO THE GEMETRICAL TOR- SION. ² METRIC AND TORSION TWO INDEPENDENT GEOMETRICAL CHARACTERISTICS OF SPACE- TIME. ² THIS GEOMETRICAL PROPERTY OF SPACE- TIME, TORSION, ARISES IN THE DEFINITION OF COVARIANT DERIVATIVE OPERATOR OF VECTOR AND TENSOR FIELDS, FORMING AN ANTI-SYMMETRIC PART OF THE AFFINE CON- NECTION COEFFICIENTS (ACCs). 2 ² METRIC COMPATIBILITY LEADS TO THE DE- TERMINATION OF ACCs IN TERMS OF MET- RIC COMPONENTS AND THEIR DERIVATIVES, KNOWN AS CHRISTOFFEL SYMBOLS, WHICH ARE SYMMETRIC. ² WITH THE CHOICE OF A SET OF MINIMAL CONDITIONS ON TORSION,WE DETERMINE THE NON-VANISHING COMPONENTS OF TOR- SION IN TERMS OF METRIC COMPONENTS OF SPACE-TIME ² THE RESULTING ACCs ARE COMPLETELY DE- TERMINED FROM THE ASSUMED 5D METRIC, LEADING TO SOME REMARKABLE MODIFI- CATIONS OF THE CONVENTIONAL K-K THE- ORY 3 ² BASIC FORMALISM; NOTATIONS: ² LET i; j; k::: and A; B; ::DENOTE COORDINATE AND INERTIAL FRAME INDICES, RESPECTIVELY. i i ² LET ei = @i AND θ = dx BE THE BASIS OF THE TANGENT AND DUAL SPACES AT EACH POINT OF SPACE-TIME. LET THE CORRESPOND- i A ING INERTIAL BASIS BE eA = eAei AND θ = A i ei θ . A i ² THE VIELBINES ei ; eA SATISFY THE ORTHONOR- MALITY CONDITIONS, A j j A i A ei eA = ±i ; ei eB = ±B THE METRIC, A B ij i j AB gij = ei ej ´AB; g = eAeB´ ; WHERE ´AB IS THE MINKOWSKIAN METRIC IN THE INERTIAL COORDINATE SYSTEM. 4 ² COVARIANT DERIVATIVE OPERATOR AND AFFINE CONNECTION COEFFICIENTS: ~k C rei ej = ¡ijek; reA eB = !ABeC; ~i C WHERE ¡jk AND !AB ARE THE AFFINE AND THE RICCI ROTATION COEFFICIENTS, RESPEC- TIVELY. FROM THE TRANSFORMATION LAWS BETWEEN THE COORDINATE AND INERTIAL FRAMES, WE HAVE, A i j A !BC = eB(rei eC)ej THE ANTI-SYMMETRIC PART OF THE AFFINE CONNECTION IS THE TORSION TENSOR, i ~i ~i Tjk = ¡jk ¡ ¡kj AGAIN, FROM THE TRANSFORMATION LAW BETWEEN THE COORDINATE AND THE IN- ERTIAL FRAMES, WE HAVE THE RELATION, i B C i A Tjk = ej ek eATBC 5 ² FURTHERMORE, WITH THE STANDARD AS- SUMPTION OF METRIC COMPATIBILITY, NAMELY, rei gjk = 0, WE OBTAIN, i i i ¡~jk = ¡^jk + Kjk WHERE i im ¡^jk = (1=2)g [@jgkm + @kgjm ¡ @mgjk]; ARE THE USUAL CHRISTOFFEL COEFFICIENTS (SYMBOLS) THAT ARE SYMMETRIC, AND i i :i :i Kjk = (1=2)[T:jk + Tj:k + Tk:j]; IS KNOWN AS THE CONTORSION TENSOR. 6 ² 5D SPACE-TIME METRIC: CONSIDER A FOLIATION OF THE 5D SPACE- TIME IN TERMS OF 4D HYPERSURFACES WITH x¹; (¹; º; ::) COORDINATE INDICES ON THE HY- PERSURFACES, AND x5 THE 5D COORDINATE. ¹ :b a ¹ a ¹ e:ae¹ = ±b ; e:ae:º = ±º :a :b ¹º ¹ º ab g¹º = e¹ eº ´ab; g = e:ae:b´ ; ¹ a WHERE e:a, e:¹ ARE THE VIERBINES WITH (a; b) AS THE TETRAD INDICES ON THE HYPER- SURFACE. ² EXTENSION TO 5D DEFINE THE 5D VIELBEINES TO BE ei = (e¹ ; e¹ ; e5 ; e5 ); e:A = (e:a; e:a; e:5_ ; e:5_ ) A :a :5_ :a :5_ i ¹ 5 ¹ 5 7 ² EXTENDING THE ORTHONORMALITY RELA- TIONS, WE ARE LED TO THE UNIQUE CHOICE OF VIELBEINES, e¹ = 0; e5 = ¡e¹ A e5 = ©¡1 :5_ :a :a ¹ 5_ :a :5_ :5_ e5 = 0; e = A¹©; e5 = © ² THE METRIC IN THE 5D SPACE-TIME :A :B ij i j AB gij = ei ej ´AB; g = e:Ae:B´ 2 2 2 g¹º = g¹º + A¹Aº© ; g¹5 = A¹© ; g55 = © ¹º ¹º ¹5 ¹ 55 ¸ ¡2 g = g ; g = ¡A ; g = A A¸ + © 8 ² A SET OF CONDITIONS ON THE TORSION: ² CONDITION 1 a TBC = 0 ¹ THIS IMPLIES WITH THE GIVEN METRIC, Tij = 0;THE ONLY NON-VANISHING COMPONENTS 5 OF TORSION ARE Tij . ² CONDITION 2 5_ A !:BC = !:B5_ = 0; ² WHICH IMPLY, ~¹ ~5 :5_ 5 ¡:i5 = 0; ¡:i5 = ¡e5 @ie:5_ ² THE ABOVE TWO CONDITIONS ARE SUFFI- CIENT TO DETERMINE ALL THE COMPONENTS i OF TORSION AND HENCE CONTORSION Kjk: ¹ 5 T:ij = T:55 = 0 5 5 T¹º = ¡Tº¹ = F¹º + S¹º 5 5 ¡2 σ T:¹5 = ¡T:5¹ = ¡© gµσ¡:55; WHERE ¡1 F¹º = @¹Aº¡@ºA¹;S¹º = A¹Jº¡AºJ¹; J¹ = © @¹© 9 ² THE IMPOSED CONDITIONS NOT ONLY DE- TERMINE THE TORSION IN TERMS OF THE METRIC, BUT MOST IMPORTANTLY, REQUIRE @5g¹º = 0 ALL THE HYPERSURFACES IN THE FOLI- ATING FAMILY HAVE THE SAME 4D MET- RIC! ALTERNATELY, ONE MIGHT SAY, GRAV- ITY IS CONFINED TO 4D!! ² NOW, FROM THE GENERAL RELATION, ~i ^i i ¡jk = ¡jk + Kjk; WE FIND, DUE TO REMARKABLE CANCELA- TIONS,THE RESULTING AFFINE CONNECTION COEFFICIENTS, ¸ ¸ ¸ ¡~55 = ¡~º5 = ¡~5º = 0; 5 ¡~¹º = r¹Aº + J¹Aº; 5 ¡~5¹ = @5A¹ + J5A¹; 5 5 ¸ ¸ ¡~¹5 = J¹; ¡~55 = J5; ¡~¹º = γ¹º; ¸ WHERE γ¹º IS DERIVED FROM 4D SPACE-TIME ¸ ¸ ¸ WITH METRIC g¹º; NOTE γ¹º 6= ¡^¹º 10 ² SUBSTITUTING THE ABOVE AFFINE CONNEC- TION COEFFICIENTS IN THE RICCI TENSOR, ^ ~j ~j ~j ~m ~j ~m Rik = @k¡:ji ¡ @j¡:ki + ¡:km¡:ji ¡ ¡:jm¡:ki WE FIND, R~¹º = R¹º; R~¹5 = R~5¹ = R~55 = 0; WHERE R¹º REPRESENTS THE RICCI TENSOR ON THE TORSION-FREE 4D SPACE-TIME. NOTE ALSO RICCI SCALAR IN 5D R~= R, THE 4D TORSION- FREE RICCI SCALAR. ² REMARK:THE RESULTS ARE NOT SPECIFIC TO 4 AND 5 DIMENSIONS. IT IS STRAIGHT FORWARD TO EXTEND THEM TO ARBITRARY D AND D+1 DIMENSIONS. 11 ² DISCUSSION OF SOME RESULTS: ² GEODESIC EQUATIONS ::5 5 ¹ º ¡ 5 5 ¢ ¹ 5 5 ¡ 5¢2 x + ¡~¢ ¹ºx_ x_ + ¡~¢ ¹5 + ¡~¢ 5¹ x_ x_ + ¡~¢ 55 x_ = 0; ::¸ ¸ ¹ º x + ¡¢ ¹ºx_ x_ = 0: THE GEODESIC EQUATIONS ALONG HYPER- SURFACE ARE EXACTLY THE SAME AS THE GEODESIC EQUATIONS IN THE TORSION FREE 4D SPACE-TIME. THIS IS IN CONTRAST WITH THE CONVENTIONAL KALUZA-KLEIN THEORY.THE CONTRIBUTIONS OF VECTOR AND SCALAR FIELDS ARE COMPLETELY NULLIFIED BY THE CONTRIBUTIONS FROM TORSION TO THE 4D GEODESIC EQUATIONS. 12 EINSTEIN'S EQUATIONS: TO OBTAIN THE FIELD EQUATIONS, ONE NEEDS TO VARY THE ACTION WITH RESPECT TO INDE- PENDENT DYNAMICAL VARIABLES. IN EINSTEIN- CARTAN THEORY, THE INDEPENDENT VARIABLES ARE THE METRIC AND THE TORSION. IMPOSING THE GEOMETRICAL CONSTRAINTS USING CAR- TAN STRUCTURE EQUATIONS, WE HAVE EXPRESSED TORSION IN TERMS OF 5D METRIC. HENCE,TAKING THE LAGRANGIAN DENSITY TO BE THE RICCI SCALAR, VARYING THE ACTION WITH RESPECT TO 5D METRIC, WE OBTAIN THE FOLLOWING MODIFIED EINSTEIN EQUATIONS: º º º º R¹ ¡ (1=2)R±¹ + H¹ = §¹; ® ® 5 ¡A R¹® ¡ A H¹® = §¹; 5 ¡1=2R = §5; Where H¹º = r(¹Bº) ¡ (r ¢ B)g¹º + J(¹Bº) ¡ (J ¢ B)g¹º 5 and B¹ = T¢ ¹5 is a vector in the 4D torsion free space- time. Conservation of 4D stress tensor requires º rºH¹ = 0: 13 ROBERTSON-WALKER COSMOLOGY To illustrate the implications of the above equations, consider spatially flat homogeneous and isotropic uni- verse wit the metric, ¡ ¢ ds2 = ¡dt2 + a2(t) dx2 + dy2 + dz2 With the usual assumptions regarding homogeneity and isotropy, and assuming (for simplicity) ¯elds do not de- pend upon the ¯fth coordinate x5, we are led to :: (i) ©_ = 0; Jt = 0; or (ii) © =a _(t); Jt = a=a:_ Case(i) yields, H¹º = 0 and the usual Friedman equation along with matter conservation, 8¼ (a=a_ )2 = ½: 3 case(ii) yields non-vanishing H¹º and a modi¯ed equa- tion ¡:: ¢ 8¼ (a=a_ )2 + a=a = ½; 3 3 Matter conservation implies ½a = ½o is a constant in a matter dominated universe. 14 The dashed-dot curve denotes a(t), the dotted curve :: denotesa _(t), and the solid curve denotes a(t). The ¡1 time axis is in units of Ho . Initial conditions: a = 1,a _ = Ho and qo = ¡0:5, :: 2 where a = ¡qoHo . 15 ² CONCLUDING REMARKS: ² TORSION IS AN INTEGRAL, GEOMETRICAL PROP- ERTY OF SPACE-TIME, ² THE MODEL WE HAVE EXPLORED HAS A VERY GENERAL MATHEMATICAL RESULT PERTAIN- ING TO THE USUAL METRIC AFFINE CON- NECTION COEFFICIENTS AND TORSION, ² IN HIGHER DIMENSIONAL THEORIES, IT PRO- VIDES AN ALTERNATIVE WAY TO CONFINE GRAVITY, YET MODIFYING IT IN A SIGNIF- ICANT WAY TO BE RELEVANT TO ASTRO- PHYSICS AND COSMOLOGY, ² WE HAVE EXPLORED QUALITATIVELY SOME OF THE CONSEQUENCES. NEEDS MUCH FUR- THER WORK. 16.
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