1. Possible Enhancement ofnoncommutative EFFECTS IN gravity Objective Look for consequences of gravity on noncommutative (NC) space-time In particular, gauge theory formulation due to Chamseddine Closure requires introducing abelian gauge fields, which mix with gravity and induce first order corrections If theory contains electromagnetism, em fields induce NC corrections in metric gravity induces NC corrections in em fields Get new bounds on NC scale from neutron star gravitational red shift measurements

2. Outline • A little background • Noncommutative gravity • Solutions and physical interpretation • Possible applications to neutron stars, cosmology • Conclusions

3. Motivation, Background Possible breakdown of Riemannian description of space-time at the Planck scale. New uncertainty relations derived for at the Planck scale, S. Doplicher, K. Fredenhagen, J.E. Roberts, Phys.Lett.B331:39-44,1994. Realized by replacing x,y,z,t by noncommuting operators . Idea goes back to Heisenberg, as short distance cut-off for QFT. NC geometry arises from string theory. Speculation (hope) NC scale << Planck scale Experimental bounds from atomic physics, collider physics, astrophysics . How ‘bout gravity? .

4. Noncommutative gravity May serve as effective theory for quantum gravity Different proposals Aschieri, Blohmann, Dimitrijevic, Meyer, Schupp,Wess Advantage: Action has have full diffeo symmetry Disadvantage: complicated Chamseddine Disadvantage: Diffeo symmetry broken Advantage: Possible to compute corrections to solutions of GR generalizes gauge theory formulation of GR ………..

5. gauge theory formulation of GR gauge group = SL(2,C) vierbein one forms: spin connection one forms: spinor notation: Metric invariant under SL(2,C) gauge transformations Utiyama ‘56 , Kibble ‘61

6. Simplistic approach to Noncommutativity Space-time coordinates NC operators Assume some choice gives Heisenberg algebra breaks diffeo symmetry Star product realization Gronewald-Moyal star Prescription: Replace point-wise product by

7. Noncommutative SL(2,C) gauge theory? Gauge Algebra doesn’t close

8. Chamseddine `02 Enlarge gauge group to GL(2,C) Enlarge space of vierbeins Gauge variations GL(2,C) curvature Noncommutative GL(2,C) gauge theory

9. Dynamics? Fully consistent dynamics – nontrivial problem Basic assumption: recover standard gravity in commutative limit when More interesting possibility: theory contains electromagnetism in commutative limit

10. Solutions Say S are solutions to commutative theory with isometry I in direction K . Choose such that only are nonzero. Then are Invariant under I Noncommutative field eqs = commutative field eqs S is also a solution to the noncommutative theory Example: For static solutions take

11. Physical interpretation? Approach 1 Try to make sense of noncommutative fields. postulate a NC metric – not unique, no guiding principle Approach 2 Map solution back to the commutative theory Seiberg-Witten mapNC GL(2,C) GL(2,C)

12. Gravity and abelian gauge fields get mixed Given noncommutative solution: Leading corrections to commutative solution: vierbeine: connections:

13. assume metric invariant under GL(2,C) and reduces to standard expression when f=0 First order corrections:

14. Applications to black holes Schwarzschild solution SW map

15. Kerr-Newman solution

16. Mansour Haghighat, A.S., e-Print: arXiv:1008.1598 [gr-qc] Introduce time-space noncommutativity Noncommutative corrections for Read off correction to standard gravitational redshift Other corrections: angular momentum contribution charge contribution

17. Apply to neutron stars Millisecond pulsars EXO0748-676 redshift measured to precision of .1% from solution

18. Bound should improve with precision measurement from Magnetars Previous formula gives

19. Flat expanding universe Invariant measure Vierbeine and spin connections Killing vectors Any choice ofsatisfies previous condition time-space noncommutativity space-space noncommutativity

20. Uniform electric-like fields in flat expanding universe time-space noncommutativity space-space noncommutativity incredibly tiny at current time! radiation dominated era t < trm

21. Inflation era

22. Some thoughts on the Field Action Require: Invariance under NC GL(2,C) Should contain Einstein-Hilbert action in commutative limit linear terms in GL(2,C) curvature Evaluate at f=0 quadratic term Stability?

23. Concludingremarks Can get first order noncommutative effects in gravity Abelian gauge fields generated from the space-time background Multipole fields around black holes, Cosmic electric-like fields from Robertson -Walker metric Corrections to space-time metric generated by abelian gauge fields Corrections to Kerr-Newman metric Question? Fully consistent for NC GL(2,C) dynamics containing Einstein-Maxwell ?