Kaluza-Klein Theory with Torsion

Kameshwar C. Wali

Cos PA, 18-20 November, 2009, Melbourne, Australia • INTRODUCTORY REMARKS

• BASIC FORMALISM

• A SET OF MINIMAL CONSTRAINTS ON TOR- SION

• COEFFICIENTS

• RICCI , RICCI SCALAR

• EINSTEIN EQUTIONS

• SOME RESULTS

• CONCLUDING REMARKS

1 • WORK DONE IN COLLABORATION WITH KARTHIK H. SHANKAR.

• WE CONSIDER A VARIANT OF 5D K-K THE- ORY, WITHIN THE FRAMEWORK OF EINSTEIN- CARTAN FORMALISM,BUT GENERALIZABLE TO ARBITRARY THAT INCLUDES TOR- SION.

• 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 AND SPIN OF MATTER FIELDS TO THE GEMETRICAL TOR- SION.

• METRIC AND TORSION TWO INDEPENDENT GEOMETRICAL CHARACTERISTICS OF SPACE- TIME.

• THIS GEOMETRICAL PROPERTY OF SPACE- TIME, TORSION, ARISES IN THE DEFINITION OF OPERATOR OF VECTOR AND TENSOR FIELDS, FORMING AN ANTI-SYMMETRIC PART OF THE AFFINE CON- NECTION COEFFICIENTS (ACCs).

2 • METRIC COMPATIBILITY LEADS TO THE DE- TERMINATION OF ACCs IN TERMS OF MET- RIC COMPONENTS AND THEIR DERIVATIVES, KNOWN AS , WHICH ARE SYMMETRIC.

• 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

• 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 • BASIC FORMALISM; NOTATIONS:

• LET i, j, k... and A, B, ..DENOTE COORDINATE AND INERTIAL FRAME INDICES, RESPECTIVELY.

i i • LET ei = ∂i AND θ = dx BE THE 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 • THE VIELBINES ei , eA SATISFY THE ORTHONOR- MALITY CONDITIONS,

A j j A i A ei eA = δi , ei eB = δB THE METRIC,

A B ij i j AB gij = ei ej ηAB, g = eAeBη ,

WHERE ηAB IS THE MINKOWSKIAN METRIC IN THE INERTIAL .

4 • COVARIANT DERIVATIVE OPERATOR AND COEFFICIENTS:

˜k C ∇ei ej = Γijek, ∇eA eB = ωABeC,

˜i C WHERE Γjk AND ωAB 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 ωBC = eB(∇ei eC)ej THE ANTI-SYMMETRIC PART OF THE AFFINE CONNECTION IS THE ,

i ˜i ˜i Tjk = Γjk − Γkj 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 • FURTHERMORE, WITH THE STANDARD AS- SUMPTION OF METRIC COMPATIBILITY, NAMELY,

∇ei gjk = 0, WE OBTAIN,

i i i Γ˜jk = Γˆjk + Kjk

WHERE

i im Γˆjk = (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 .

6 • 5D SPACE-TIME METRIC:

CONSIDER A FOLIATION OF THE 5D SPACE- TIME IN TERMS OF 4D HYPERSURFACES WITH xµ, (µ, ν, ..) COORDINATE INDICES ON THE HY- PERSURFACES, AND x5 THE 5D COORDINATE.

µ .b a µ a µ e.aeµ = δb , e.ae.ν = δν

.a .b µν µ ν ab gµν = eµ eν ηab, g = e.ae.bη ,

µ a WHERE e.a, e.µ ARE THE VIERBINES WITH (a, b) AS THE TETRAD INDICES ON THE HYPER- SURFACE.

• EXTENSION TO 5D DEFINE THE 5D VIELBEINES TO BE

ei = (eµ , eµ , e5 , e5 ), e.A = (e.a, e.a, e.5˙ , e.5˙ ) A .a .5˙ .a .5˙ i µ 5 µ 5

7 • EXTENDING THE ORTHONORMALITY RELA- TIONS, WE ARE LED TO THE UNIQUE CHOICE OF VIELBEINES,

eµ = 0, e5 = −eµ A e5 = Φ−1 .5˙ .a .a µ 5˙

.a .5˙ .5˙ e5 = 0, e = AµΦ, e5 = Φ

• THE METRIC IN THE 5D SPACE-TIME

.A .B ij i j AB gij = ei ej ηAB, g = e.Ae.Bη

2 2 2 gµν = gµν + AµAνΦ , gµ5 = AµΦ , g55 = Φ

µν µν µ5 µ 55 λ −2 g = g , g = −A , g = A Aλ + Φ

8 • A SET OF CONDITIONS ON THE TORSION:

• CONDITION 1

a TBC = 0 µ THIS IMPLIES WITH THE GIVEN METRIC, Tij = 0;THE ONLY NON-VANISHING COMPONENTS 5 OF TORSION ARE Tij . • CONDITION 2

5˙ A ω.BC = ω.B5˙ = 0, • WHICH IMPLY, ˜µ ˜5 .5˙ 5 Γ.i5 = 0, Γ.i5 = −e5 ∂ie.5˙ • THE ABOVE TWO CONDITIONS ARE SUFFI- CIENT TO DETERMINE ALL THE COMPONENTS i OF TORSION AND HENCE CONTORSION Kjk:

µ 5 T.ij = T.55 = 0

5 5 Tµν = −Tνµ = Fµν + Sµν

5 5 −2 σ T.µ5 = −T.5µ = −Φ gµσΓ.55, WHERE −1 Fµν = ∂µAν−∂νAµ,Sµν = AµJν−AνJµ; Jµ = Φ ∂µΦ

9 • THE IMPOSED CONDITIONS NOT ONLY DE- TERMINE THE TORSION IN TERMS OF THE METRIC, BUT MOST IMPORTANTLY, REQUIRE

∂5gµν = 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!!

• NOW, FROM THE GENERAL RELATION,

˜i ˆi i Γjk = Γjk + Kjk,

WE FIND, DUE TO REMARKABLE CANCELA- TIONS,THE RESULTING AFFINE CONNECTION COEFFICIENTS,

λ λ λ Γ˜55 = Γ˜ν5 = Γ˜5ν = 0, 5 Γ˜µν = ∇µAν + JµAν,

5 Γ˜5µ = ∂5Aµ + J5Aµ,

5 5 λ λ Γ˜µ5 = Jµ, Γ˜55 = J5, Γ˜µν = γµν,

λ WHERE γµν IS DERIVED FROM 4D SPACE-TIME λ λ λ WITH METRIC gµν; NOTE γµν 6= Γˆµν

10 • SUBSTITUTING THE ABOVE AFFINE CONNEC- TION COEFFICIENTS IN THE RICCI TENSOR, ˆ ˜j ˜j ˜j ˜m ˜j ˜m Rik = ∂kΓ.ji − ∂jΓ.ki + Γ.kmΓ.ji − Γ.jmΓ.ki

WE FIND,

R˜µν = Rµν, R˜µ5 = R˜5µ = R˜55 = 0,

WHERE Rµν 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.

• 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 • DISCUSSION OF SOME RESULTS:

• EQUATIONS

..5 5 µ ν ¡ 5 5 ¢ µ 5 5 ¡ 5¢2 x + Γ˜· µνx˙ x˙ + Γ˜· µ5 + Γ˜· 5µ x˙ x˙ + Γ˜· 55 x˙ = 0,

..λ λ µ ν x + Γ· µνx˙ 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:

ν ν ν ν Rµ − (1/2)Rδµ + Hµ = Σµ,

α α 5 −A Rµα − A Hµα = Σµ,

5 −1/2R = Σ5, Where

Hµν = ∇(µBν) − (∇ · B)gµν + J(µBν) − (J · B)gµν 5 and Bµ = T· µ5 is a vector in the 4D torsion free space- time.

Conservation of 4D stress tensor requires ν ∇νHµ = 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) fields do not de- pend upon the fifth coordinate x5, we are led to

.. (i) Φ˙ = 0, Jt = 0, or (ii) Φ =a ˙(t), Jt = a/a.˙

Case(i) yields, Hµν = 0 and the usual Friedman equation along with matter conservation, 8π (a/a˙ )2 = ρ. 3 case(ii) yields non-vanishing Hµν and a modified equa- tion ¡.. ¢ 8π (a/a˙ )2 + a/a = ρ, 3

3 Matter conservation implies ρa = ρo 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 • CONCLUDING REMARKS:

• TORSION IS AN INTEGRAL, GEOMETRICAL PROP- ERTY OF SPACE-TIME,

• THE MODEL WE HAVE EXPLORED HAS A VERY GENERAL MATHEMATICAL RESULT PERTAIN- ING TO THE USUAL METRIC AFFINE CON- NECTION COEFFICIENTS AND TORSION,

• 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- AND COSMOLOGY,

• WE HAVE EXPLORED QUALITATIVELY SOME OF THE CONSEQUENCES. NEEDS MUCH FUR- THER WORK.

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