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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 ?