Possible Enhancement of noncommutative EFFECTS IN gravity

Objective

Look for consequences of gravity on noncommutative (NC) space-time

In particular, gauge theory formulation due to Chamseddine Chamseddine

Closure requires introducing abelian gauge fields, which mix with gravityClosure requires introducing abelian gauge fields, which mix with gravity

and induce first order correctionsand induce first order corrections

If theory contains electromagnetism,If theory contains electromagnetism,

em fields induce NC corrections in metricem fields induce NC corrections in metric

gravity induces NC corrections in em fields gravity induces NC corrections in em fields

Get new bounds on NC scale from neutron star gravitational red shift Get new bounds on NC scale from neutron star gravitational red shift measurementsmeasurements

Outline

• A little background

• Noncommutative gravity

• Solutions and physical interpretation

• Possible applications to neutron stars, cosmology

• Conclusions

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 Idea goes back to Heisenberg, as short distance cut-off for QFTQFT.

NC geometry arises from string theory. Speculation (hope) NC scale << Planck scale Experimental bounds from atomic physics, collider

physics, astrophysics . How ‘bout gravity?.

Different proposals

Noncommutative gravity

May serve as effective theory for quantum gravity

Aschieri, Blohmann, Dimitrijevic, Meyer, Schupp,Wess Aschieri, Blohmann, Dimitrijevic, Meyer, Schupp,Wess Advantage: Action has have full diffeo symmetryAdvantage: Action has have full diffeo symmetry Disadvantage: complicatedDisadvantage: complicated

Chamseddine Chamseddine Disadvantage: Diffeo symmetry brokenDisadvantage: Diffeo symmetry broken Advantage: Possible to compute corrections toAdvantage: Possible to compute corrections to solutions of GR solutions of GR generalizes gauge theory formulation of GRgeneralizes gauge theory formulation of GR

………………....

gauge theory formulation of GRgauge 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

Simplistic approach to Noncommutativity

• Space-time coordinates NC operators Assume some choice gives Heisenberg algebra breaks diffeo symmetry• Star product realization

Gronewald-Moyal starGronewald-Moyal star

Prescription: Replace point-wise product by Prescription: Replace point-wise product by

Noncommutative SL(2,C) gauge theory?

Gauge Algebra doesn’t close

Chamseddine `02Chamseddine `02

Enlarge gauge group to GL(2,C)

Enlarge space of vierbeins

Gauge variations

GL(2,C) curvature

Noncommutative GL(2,C) gauge theory

Dynamics?

Fully consistent dynamics – nontrivial problem

Basic assumption: recover standard gravity in commutative limit when

More interesting possibility: theory contains electromagnetism in commutative limit

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

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 map NC GL(2,C) GL(2,C)

Gravity and abelian gauge fields get mixedGiven noncommutative solution:

Leading corrections to commutative solution: vierbeine:

connections:

assume metric invariant under GL(2,C) and reduces to standard expression when f=0

First order corrections:

Applications to black holes

Schwarzschild solution

SW map

Kerr-Newman solution

Introduce time-space noncommutativityIntroduce time-space noncommutativity

Noncommutative corrections for

Read off correction to standard gravitational redshift

Other corrections:

angular momentum contribution

charge contribution

Mansour Haghighat, A.S., e-Print: arXiv:1008.1598 [gr-qc]

Apply to neutron stars

Millisecond pulsars

EXO0748-676 redshift measured to precision of .1%

from solution

Bound should improve with precision measurement from

Magnetars

Previous formula gives

Flat expanding universe

• Invariant measure

• Vierbeine and spin connections

• Killing vectors

Any choice of satisfies previous condition

time-space noncommutativity

space-space noncommutativity

Uniform electric-like fields Uniform electric-like fields in flat expanding universe in flat expanding universe

time-space noncommutativity

space-space noncommutativity

incredibly tiny at current time!

radiation dominated era

t < trm

Inflation era

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?

ConcludingConcluding remarksremarks

Can get first order noncommutative effects in gravityCan get first order noncommutative effects in gravity

Abelian gauge fields generated from the space-time background Abelian gauge fields generated from the space-time background

Multipole fields around black holes, Multipole fields around black holes, Cosmic electric-like fields from Robertson -Walker metricCosmic electric-like fields from Robertson -Walker metric

Corrections to space-time metric generated by abelian gauge fieldsCorrections to space-time metric generated by abelian gauge fields

Corrections to Kerr-Newman metricCorrections to Kerr-Newman metric

QuestionQuestion??

FullyFully c consistent for NC GL(2,C) dynamics containing Einstein-Maxwell ? onsistent for NC GL(2,C) dynamics containing Einstein-Maxwell ?