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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
(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
4. Non-commutative Einstein equations under the firstorder Seiberg–Witten
5. Non-commutative Einstein equations under the firstorder Seiberg–Witten map in the Schwarzschild
6. Key open problems and future directions
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 noncommutative 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)
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 noncommutative 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,
We have tried to exploit the Seiberg--Witten map to
answer this question.
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
Minkowski metric
(1) Tetrad 1-form expanded on the
Dirac basis of g-matrices
(2) Connection 1-form
(3) Curvature 2-form
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
Explicit formulae for the components of R
Non-commutative Einstein equations
Vc - field equation
Vc - field equation
The associated torsion
Variation of the action with respect to the spin-connection
yields the condition
(T) Torsion constraint
3. Expansion to first order of NonCommutative Einstein equations
From the wedge-product of forms defined in Ref. [1] (P. Aschieri,
L. Castellani), for any 1-form α1, β 1and any 2-form γ2, one can
write (θρσ being antisymmetric and independent of position)
Charge-conjugation conditions imply that
By virtue of (a), (b) and (c), Eq. (5) reduces to
and Eq. (6) reduces to
Equation (5’) to first order is satisfied, since Vc is the classical tetrad,
and the term
for any solution of the vacuum Einstein
by virtue of (a), (b) properties
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 noncommutativity θρσ, it reads as
The zeroth- and first-order terms in Eq. (7) are found to be
for the spin-connection
one finds to first order in θρσ
By substituting Eqs. (8) and (9) into Eq. (6’) we re-express Eq. (6’) in the form
5. Non-commutative Einstein equations
under the first-order Seiberg–Witten map
in the Schwarzschild background
Basis 1-forms
Using the identity
of Schwarzschild
and Eq. (4’), (9), the left-hand side of Eq. (6’) takes the form
where each Kϲμνλ can be written as the sum of 6 terms, having defined
We obtain eventually
The field configurations in Eq. (8), obtained by
applying the Seiberg–Witten map to the classical
tetrad of Eq. (10), are not solutions of noncommutative Einstein equations.
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.
However, the torsion constraint and noncommutative field equations are then found to be
6. Key open problems and future
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.
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
not satisfied by considering the field
configurations V ,ω, ω obtained by the
Seiberg–Witten map with Va and ωab the classical black-hole tetrad
and spin-connection.
We also notice that we have chosen the non-commutativity
directions ∂ and ∂
not to be Killing vector fields for our classical black hole solution.
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
We are studying if there are solutions of Eq. (6’) for a
general form of non-commutative tetrad.
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.
[0]. A.H. Chamseddine, Phys. Lett. B 504, 33 (2001).
A.H. Chamseddine, Commun. Math. Phys. 218, 283
A.H. Chamseddine, “Invariant Actions for Noncommutative Gravity”, hep-th/0202137.
[1]. P. Aschieri and L. Castellani, JHEP 0906, 086
[2]. Y.G. Miao and S.J. Zhang, Phys. Rev. D82, 084017