1. Non-Commutative Einstein Equations and Seiberg–Witten Map Paolo Aschieri,Elisabetta Di Grezia, Giampiero Esposito, INFN, Turin and Naples. Friedmann Seminar, 30 May 2011, Rio de Janeiro

2. Plan (arXiv:1103.3348v1 [hep-th]) 1. Motivations 2. Non-commutative action and non-commutative Einstein equations for pure gravity 3. Expansion to first order of non-commutative Einstein equations 4. Non-commutative Einstein equations under the first- order Seiberg–Witten 5. Non-commutative Einstein equations under the first- order Seiberg–Witten map in the Schwarzschild background 6. Key open problems and future directions

3. 1. Motivations Why Non-Commutative Geometry? Classical Mechanics Quantum Mechanics It is not possible to know at the same time position and velocity. Observables become non-commutative (Planck constant) Classical Gravity Quantum Gravity Impossibility to test the structure of spacetime at infinitesimal distances; structure like a lattice or a non- commutative spacetime (Planck length). On this spacetime one can consistently formulate a gravity theory (Non-commutative D = 4 gravity coupled to fermions, P. Aschieri and L. Castellani JHEP06 (2009), 086)

4. Why Non-Commutative Gravity? Recent attempts to make sense of the non-commutative description of some classical theories of gravity by using the Seiberg--Witten map motivate our attempt of finding a non-commutative solution corresponding to an exact classical solution of pure gravity in a mathematically consistent formalism. Thus, the gauge approach to gravitation has been used to obtain non- commutative fields through the Seiberg--Witten map as in Ref. [0] Can we find a rigorous non-commutative counterpart of well-known classical solutions such as Schwarzschild, Kerr,…? We have tried to exploit the Seiberg--Witten map to answer this question.

5. 2. Non-commutative action and field equations for pure gravity (P. Aschieri, L. Castellani JHEP 2009) Classical pure Gravity action functional By replacing exterior products by deformed exterior products, one gets Non-commutative pure Gravity action functional

6. (3) Curvature 2-form Minkowski metric (2) Connection 1-form (1) Tetrad 1-form expanded on the Dirac basis of  matrices

7. where the classical spin-connection is From (2), (3) one has (4) R is the curvature on the Dirac basis; r and r are 2-forms ~

8. Explicit formulae for the components of R (4’)

9. Non-commutative Einstein equations V c - field equation V c - field equation ~ (5) (6)

10. The associated torsion Variation of the action with respect to the spin-connection yields the condition (T) (T) Torsion constraint

11. 3. Expansion to first order of Non- Commutative Einstein equations From the wedge-product of forms defined in Ref. [1] (P. Aschieri, L. Castellani), for any 1-form α 1, β 1 and any 2-form γ 2, one can write ( θ ρσ being antisymmetric and independent of position) (a)

12. Charge-conjugation conditions imply that (c) (b)

13. By virtue of (a), (b) and (c), Eq. (5) reduces to(5) and Eq. (6) reduces to(6) where (5’) (6’)

14. Equation (5’) to first order is satisfied, since V c is the classical tetrad, and the term(5’) for any solution of the vacuum Einstein equations by virtue of (a), (b) properties(a), (b)

15. 4. Non-commutative Einstein equations under the first-order Seiberg-Witten map There exists a map, the Seiberg-Witten map in Ref. [2], which relates non-commutative degrees of freedom to their commutative counterparts: to first order in the non- commutativity θ ρσ, it reads as The zeroth- and first-order terms in Eq. (7) are found to be (7) (8)

16. for the spin-connection one finds to first order in θ ρσ (9)

17. By substituting Eqs. (8) and (9) into Eq. (6’) we re-express Eq. (6’) in the form (9’)

18. 5. Non-commutative Einstein equations under the first-order Seiberg–Witten map in the Schwarzschild background Basis 1-forms of Schwarzschild (10) Using the identity

19. and Eq. (4’), (9), the left-hand side of Eq. (6’) takes the form where each K ϲ μνλ can be written as the sum of 6 terms, having defined

20. We obtain eventually

21. The field configurations in Eq. (8), obtained by applying the Seiberg–Witten map to the classical tetrad of Eq. (10), are not solutions of non- commutative Einstein equations.Eq. (8), (10), In order to search for solutions to non-commutative Einstein equations that, in the commutative limit, become Schwarzschild, we have therefore to revert to Eq. (9’), where no use of the Seiberg– Witten map is made, and look for solutions of Eq. (9’).(9’), However, the torsion constraint and non- commutative field equations are then found to be incompatible.

22. 6. Key open problems and future directions In conclusion, the calculation performed seems to show that there is a mismatch between: (I) using the Seiberg–Witten map in the non-commutative action in order to express all non-commutative fields in terms of the commutative tetrad and spin-connection, and then varying the action (that in general will be a higher derivative action) with respect to the classical fields only. (II) obtaining the non-commutative field equations by varying the action with respect to all non-commutative fields, and then trying to solve these equations by expressing the non-commutative fields in terms of the commutative ones via Seiberg–Witten map.

23. This mismatch could be due to the fact that in case (II), in order to obtain the field equations we have to vary also with respect to the extra fields V, ω, ω The field equations are not satisfied by considering the field configurations V, ω, ω obtained by the Seiberg–Witten map with V a and ω ab the classical black-hole tetrad and spin-connection. We also notice that we have chosen the non-commutativity directions ∂ and ∂ ∂r ∂θ not to be Killing vector fields for our classical black hole solution. ~~ ~~

24. We are studying how non-commutative gravity can be cast in Hamiltonian form, and how this formalism can be used so as to understand which type of constraint is the T-constraint (T) (T) We are studying if there are solutions of Eq. (6’) for a general form of non-commutative tetrad.(6’) The fact that the only solution of the first order Eq. is the classical solution confirms that the action has an expansion in even powers of theta. It then becomes necessary to evaluate the second-order Seiberg--Witten map.

25. References [0]. A.H. Chamseddine, Phys. Lett. B 504, 33 (2001). A.H. Chamseddine, Commun. Math. Phys. 218, 283 (2001). A.H. Chamseddine, “Invariant Actions for Non- commutative Gravity”, hep-th/0202137. [1]. P. Aschieri and L. Castellani, JHEP 0906, 086 (2009). [2]. Y.G. Miao and S.J. Zhang, Phys. Rev. D82, 084017 (2010).