New representation of the field equations looking for a new vacuum solution for QMAG

Transcription

1 New representation of the field equations looking for a new vacuum solution for QMAG Elvis Baraković University of Tuzla Department of Mathematics 19. rujna 2012.

2 Structure of presentation

3 Structure of presentation Metric affine gravity (MAG)

4 Structure of presentation Metric affine gravity (MAG) Quadratic metric affine gravity (QMAG)

5 Structure of presentation Metric affine gravity (MAG) Quadratic metric affine gravity (QMAG) Solutions of QMAG

6 Structure of presentation Metric affine gravity (MAG) Quadratic metric affine gravity (QMAG) Solutions of QMAG PP spaces

7 Structure of presentation Metric affine gravity (MAG) Quadratic metric affine gravity (QMAG) Solutions of QMAG PP spaces Main result

8 Structure of presentation Metric affine gravity (MAG) Quadratic metric affine gravity (QMAG) Solutions of QMAG PP spaces Main result Discussion

9 Metric affine gravity Alternative theory of gravity.

10 Metric affine gravity Alternative theory of gravity. Natural generalization of Einstein s GR, which is based on a spacetime with Riemannian matric g of Lorentzian signature.

11 Metric affine gravity Alternative theory of gravity. Natural generalization of Einstein s GR, which is based on a spacetime with Riemannian matric g of Lorentzian signature. We consider spacetime to be a connected real 4-manifold M equipped with Lorentzian metric g and an affine connection Γ. SPACETIME MAG={M, g, Γ}

12 Metric affine gravity MAG R 0 T 0, GR R 0 T = 0.

13 Metric affine gravity MAG R 0 T 0, GR R 0 T = 0. The 10 independent components of the symmetric metric tensor g µν and 64 connection coefficients Γ λ µν are unknowns of MAG.

14 Metric affine gravity In QMAG, we define our action as S := q(r) (1) where q(r) is a quadratic form on curvature R.

15 Metric affine gravity In QMAG, we define our action as S := q(r) (1) where q(r) is a quadratic form on curvature R. The quatratic form q(r) has 16 R 2 terms with 16 real coupling constants.

16 Metric affine gravity In QMAG, we define our action as S := q(r) (1) where q(r) is a quadratic form on curvature R. The quatratic form q(r) has 16 R 2 terms with 16 real coupling constants. Why we use quadratic form?

17 Quadratic metric affine gravity The system of Euler Lagrange equations: S g S Γ = 0, (2) = 0. (3)

18 Quadratic metric affine gravity The system of Euler Lagrange equations: S g S Γ = 0, (2) = 0. (3) Objective: To study the combined system of field equations (2) and (3) which is system of real nonlinear PDE with real unknowns.

19 Solutions of QMAG Special case of q(r) is q(r) := R κ µν λµν Rλ κ so we get Yang - Mills theory for the affine connection.

20 Solutions of QMAG Special case of q(r) is q(r) := R κ µν λµν Rλ κ so we get Yang - Mills theory for the affine connection. Riemannian and non Riemannain solutions.

21 Solutions of QMAG Special case of q(r) is q(r) := R κ µν λµν Rλ κ so we get Yang - Mills theory for the affine connection. Riemannian and non Riemannain solutions. Definition We call a spacetime {M, g, Γ} Riemannian if the connection is Levi Civita, i.e. Γ λ µν = { λ µν} and non Riemannian otherwise.

22 Solutions of QMAG Only after these variations we set the connection to be Levi Civita and consider Riemannian solutions of the field equations.

23 Solutions of QMAG Only after these variations we set the connection to be Levi Civita and consider Riemannian solutions of the field equations. D. Vassiliev proved that the following spacetimes

24 Solutions of QMAG Only after these variations we set the connection to be Levi Civita and consider Riemannian solutions of the field equations. D. Vassiliev proved that the following spacetimes Einstein spaces (Ric = Λg), pp-spaces with parallel Ricci curvature (pp metric+ Ric = 0), and Riemannian spacetimes which have zero scalar curvature and are locally a product of Einstein 2 manifolds (Levi Civita+R = 0), are solutions of the system (2),(3).

25 Solutions of QMAG Definition We call a spacetime {M, g, Γ} a pseudoinstanton if the connection is metric compatible and curvature is irreducible amd simple..

26 Solutions of QMAG Definition We call a spacetime {M, g, Γ} a pseudoinstanton if the connection is metric compatible and curvature is irreducible amd simple.. It is the case that there are only three types od pseudoinstantons:

27 Solutions of QMAG Definition We call a spacetime {M, g, Γ} a pseudoinstanton if the connection is metric compatible and curvature is irreducible amd simple.. It is the case that there are only three types od pseudoinstantons: scalar pseudoinstanton: R (1) 0, pseudoscalar pseudoinstanton: R (1) 0, Weyl pseudoinstanton: R (10) 0.

28 Solutions of QMAG Definition We call a spacetime {M, g, Γ} a pseudoinstanton if the connection is metric compatible and curvature is irreducible amd simple.. It is the case that there are only three types od pseudoinstantons: scalar pseudoinstanton: R (1) 0, pseudoscalar pseudoinstanton: R (1) 0, Weyl pseudoinstanton: R (10) 0. Theorem (Vassiliev) A pseudoinstanton is a solution of the field equations (2), (3).

29 Classical pp spaces PP spaces are well known spacetimes in GR (Brinkmann,Peres).

30 Classical pp spaces PP spaces are well known spacetimes in GR (Brinkmann,Peres). Meaning of the pp?

31 Classical pp spaces PP spaces are well known spacetimes in GR (Brinkmann,Peres). Meaning of the pp? A very simple formula for curvature: only trace free Ricci and Weyl.

32 Classical pp spaces PP spaces are well known spacetimes in GR (Brinkmann,Peres). Meaning of the pp? A very simple formula for curvature: only trace free Ricci and Weyl. Definition A pp wave is a Riemannian spacetime whose metric can be written locally in the form ds 2 = 2dx 0 dx 3 (dx 1 ) 2 (dx 2 ) 2 + f (x 1, x 2, x 3 )(dx 3 ) 2 (4) in some local coordinates (x 0, x 1, x 2, x 3 ).

33 Generalized pp-waves In a classical pp space we consider the polarized Maxwell equation da = ±ida. (5)

34 Generalized pp-waves In a classical pp space we consider the polarized Maxwell equation da = ±ida. (5) We seek plane wave solution of (5): A = h(ϕ)m + k(ϕ)l ϕ : M R, ϕ(x) := l dx. M

35 Generalized pp-waves In a classical pp space we consider the polarized Maxwell equation da = ±ida. (5) We seek plane wave solution of (5): A = h(ϕ)m + k(ϕ)l ϕ : M R, ϕ(x) := l dx. Definition A generalized pp-wave is a metric compatible spacetime with pp metric and torsion M T := 1 Re(A da). 2

36 New representation of the field equations We write down explicitly our field equations (2), (3) under following assumptions:

37 New representation of the field equations We write down explicitly our field equations (2), (3) under following assumptions: (i) our spacetime is metric compatible, (ii) curvature has symmetries R κλµν = R µνκλ, ε κλµν R κλµν = 0, (iii) scalar curvature is zero.

38 Main result The main result is

39 Main result The main result is Lemma Under the above assumptions (i) (iii), the field equations (2), (3) are

40 Main result The main result is Lemma Under the above assumptions (i) (iii), the field equations (2), (3) are 0 = d 1 W κλµν Ric κµ + d 3 ( Ric λκ Ric ν κ 1 4 g λν Ric κµ Ric κµ ) (6)

41 New representation of the field equations where d 1, d 3, d 6, d 7, b 10 are some real constants. 0 = d 6 λ Ric κµ d 7 κric λµ ( + d 6 Ricκ η ( d 7 Ric η λ (K µηλ K µλη ) g λµw ηζ κξ (K ξ ηζ K ξ ζη ) g µλric η ξ K ξ ηκ +g µλ Ricκ η K ξ ξη K ξ ξλ Ricκµ + 1 ) 2 g µλricκ ξ (K η ξη K η ηξ ) (Kµηκ Kµκη) gκµwκζ λξ (K ξ ηζ K ξ ζη ) gµκric η ξ K ξ ηλ +g κµric η λ K ξ ξη K ξ ξκ Ric λµ + 1 ) 2 gµκric ξ λ (K η ξη K η ηξ ) + b 10 (g µλ W ηζ κξ (K ξ ζη K ξ ηζ ) + gµκwηζ λξ (K ξ ηζ K ξ ζη ) +g µλ Ricκ ξ (K η ηξ K η ξη ) + gµκric ξ λ (K η ξη K η ηξ ) +g κµric η λ K ξ ξη g λµricκ η K ξ ξη + RicµκK η λη Ric µλk η κη + 2b 10 (W η µκξ (K ξ ηλ K ξ λη ) + Wη µλξ (K ξ κη K ξ ηκ ) ) K µξη W ηξ κλ K ξ ξη Wη µλκ ) (7)

42 Discussion Theorem Generalized pp spaces of parallel Ricci curvature are solutions of the system (6), (7).

43 Discussion Theorem Generalized pp spaces of parallel Ricci curvature are solutions of the system (6), (7). The proof is done by brute force.

44 Discussion Theorem Generalized pp spaces of parallel Ricci curvature are solutions of the system (6), (7). The proof is done by brute force. Singh: On axial vector torsion in vacuum quadratic Poincaré gauge field theory solutions of the vacuum field equations with purely axial torsion.

45 Discussion Theorem Generalized pp spaces of parallel Ricci curvature are solutions of the system (6), (7). The proof is done by brute force. Singh: On axial vector torsion in vacuum quadratic Poincaré gauge field theory solutions of the vacuum field equations with purely axial torsion. Conjecture There exits a purely axial torsion waves which are solution of the field equations (2), (3).

46 Discussion We are going to try to generalize pp waves as follows

47 Discussion We are going to try to generalize pp waves as follows Conjecture There exists a new class of spacetimes with pp metrics and purely axial torsion which are solution of the field equations (2), (3).

48 Discussion We are going to try to generalize pp waves as follows Conjecture There exists a new class of spacetimes with pp metrics and purely axial torsion which are solution of the field equations (2), (3). Expectations:

49 Discussion We are going to try to generalize pp waves as follows Conjecture There exists a new class of spacetimes with pp metrics and purely axial torsion which are solution of the field equations (2), (3). Expectations: to prove two conjectures above.

50 Discussion We are going to try to generalize pp waves as follows Conjecture There exists a new class of spacetimes with pp metrics and purely axial torsion which are solution of the field equations (2), (3). Expectations: to prove two conjectures above. to give a physical interpretation of the new solutions and compare them with existing Riemannian solutions.

51 Structure of presentation Metric affine gravity Solutions of QMAG PP spaces New representation of the field equations Discussion Thank you! Welcome to Tuzla! Elvis Baraković New representation of the field equations looking for a new vac