Noncommutative corrections to classical black holes

Archil Kobakhidze*

School of Physics, The University of Melbourne, Victoria 3010, Australia(Received 4 December 2007; published 11 February 2009)

We calculate leading long-distance noncommutative corrections to the classical Schwarzschild black

hole sourced by a massive noncommutative scalar field. The energy-momentum tensor is taken Oð‘4Þ inthe noncommutative parameter ‘ and is treated in the semiclassical (tree-level) approximation. These

noncommutative corrections dominate classical post-post-Newtonian corrections if ‘ > 1=MP. However,

they are still very small to be observable in present-day experiments.

DOI: 10.1103/PhysRevD.79.047701 PACS numbers: 11.10.Nx, 04.20.�q, 04.25.Nx, 04.70.Bw

I.—Recently, a number of different attempts [1–4] havebeen made to define theory of gravitation on space-timewith canonical noncommutativity given by the noncom-mutative algebra of coordinates:

½x̂�; x̂�� ¼ i‘2���; (1)

where ‘ is a length parameter of noncommutativity and��� is a constant antisymmetric matrix.1 The essentialdifference between these formulations of noncommutativegravity lies in the treatment of symmetries of generalrelativity in a noncommutative setting. The resulting ac-tions contain rather complicated noncommutative correc-tions to the Einstein-Hilbert Lagrangian of classicalgeneral relativity, and thus it is rather difficult to analyzecorresponding equations of motions. It has been observedin all these formulations (see also [6]) that the lowest ordernoncommutative corrections are Oð‘4Þ.2 Thus in the lead-ing approximation pure gravity action can be treatedundeformed.

Our aim in this paper is to calculate noncommutativelong-distance corrections to classical black holes.3 Wefollow the formalism of the effective field theory whichhas been applied to calculate commutative quantum cor-rections to classical black holes in [9]. According to thediscussion in the previous paragraph, the leading interac-tions of the linearized graviton field with matter are givenby the Lagrangian,

L int ¼Z

d4x1

2h��T

��NC; (2)

where T��NC is the noncommutative energy-momentum ten-

sor which can be expanded in a series of the noncommu-

tative parameter ‘2,

T��NC ¼ T��

0 þ T��1 þ T��

2 . . . (3)

The nth term in this expansion isOð‘2nÞ, that is, T��0 is the

usual commutative energy-momentum tensor. The gravi-ton field for a nearly static source can be solved as ([wework in the harmonic gauge, @�ðh�� � 1

2���hÞ ¼ 0]

h��ðxÞ ¼ �16�GN

Z d3 ~q

ð2�Þ3 ei ~q ~r 1

~q2

��T��NCðqÞ �

1

2���TNCðqÞ

�; (4)

where TNC ¼ ���T��NC, and T

��NCðqÞ ¼ hp2j :T��

NCðxÞ:jp1i,q� ¼ ðp2 � p1Þ�. Once the explicit form of energy-

momentum tensor is known, the matrix elements can becalculated perturbatively. In what follows we consider onlythe lowest semiclassical (tree-level) approximation in theperturbation theory.Here we would like to stress an important point con-

cerning the Lagrangian (2). This linearized Lagrangian isindependent on the particular model of canonical noncom-mutative gravity. The reason is that in all known noncom-mutative Hermitian theories of gravitation [1–4] thenoncommutative corrections necessarily contain highercurvature terms. They obviously drop out in the linearizedapproximation considered above. Thus, the results ob-tained below must be correct for any of the models ofnoncommutative gravity proposed [1–4].II.—Consider a massive scalar field which is a source of

the Schwarzschild black hole. The energy-momentum ten-sor reads

T��NCðxÞ ¼

1

2ð@��?@��þ @��?@��Þ

� 1

2���ð@��?@���m2�?�Þ

� T��0 ðxÞþ���m

2‘4

16����@�@�@�@�þ . . . ;

(5)

T��0 ¼ @��@��� 1

2���ðð@�Þ2 �m2�2Þ. The ? product

*archilk@unimelb.edu.au1Other approaches to noncommutative gravity can be found in

recent reviews [5]. See also references therein.2The same conclusion has been reached in [7] within the

theory of gravitation with Lie-algebra noncommutativity.3Noncommutative corrections Oð‘4Þ to classical black hole

metrics has been recently calculated in [8] within a specifictheory [1] of noncommutaive gravity, by solving truncatedequations of motion.

PHYSICAL REVIEW D 79, 047701 (2009)

1550-7998=2009=79(4)=047701(3) 047701-1 � 2009 The American Physical Society

in the first line is defined as aðxÞ ? bðxÞ ¼ aðxÞbðxÞ þP1n¼1

in‘2n

2nn! ��1�1 . . . ��n�n@�1

. . . @�na@�1

. . . @�nbðxÞ. In the

second line of the above equation we retain only Oð‘4Þterms with four derivatives at most. This is justified in ourcase since we are interested in low-momentum massiveparticles, m � j ~pj. The scalar field is quantized in a stan-dard way:

�ðxÞ ¼Z d3 ~p

ð2�Þ3=2 ffiffiffiffiffiffiffiffiffiffiffiffiffi2Eð ~pÞp ½að ~pÞe�ikx þ aþð ~pÞeikx�;

½að ~pÞ; aþð ~p0Þ� ¼ �3ð ~p� ~p0Þ; að ~pÞj0i ¼ 0;

j ~pi ¼ aþð ~pÞj0i:

(6)

Using (5) and (6), it is easy to calculate the followingmatrix element:

h ~p2j :T��2 ð0Þ:j ~p1i � ��� m

2‘4

16����

� hp2j :@�@�@�@�:jp1i

¼ ��� m2‘4

64ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEð ~p1ÞEð ~p2Þ

p ð���P�q�Þ2

� ��� m3‘4

16�0i�0jqiqj; (7)

where in the last step we approximate P� ¼ ðp1 þ p2Þ� �2m�0�, Eð ~p1Þ, Eð ~p2Þ � m. Using (7) we obtain noncom-muative corrections from (4):

�hNC00 ¼ GNm3‘4�0i�0j

8�2

Zd3 ~q

qiqj

~q2ei ~q� ~r

¼ GNm3‘4�0i�0j

4�

���ij

r3þ 3rirj

r5

�; (8)

�hNCkm ¼ ��km�hNC00 ; (9)

�hNC0k ¼ 0: (10)

Thus, the noncommutative Schwarzschild metric with theabove long-distance corrections looks like

g00 ¼�1� 2

GNm

rþ 2

G2Nm

2

r2� 2

G3Nm

3

r3þ . . .

�þ �hNC00 ;

(11)

gij ¼���ij

�1þ 2

GNm

r

��G2

Nm2

r2

��ij þ

rirj

r2

�

þ 2G3

Nm3

r3rirj

r2þ . . .

�þ �hNCij ; (12)

g0i ¼ 0: (13)

Recall that the metric is given in the harmonic gauge.The first term in square brackets in (8), entering the

expression for g00 (11), looks like a post-post-Newtonian

correction in commutative general relativity [for gij in

(12) the noncommutative correction is similar to a third-order post-Newtonian correction], except the fact thatthe strength of this noncommutative correction isdetermined by ðGN‘

4Þ, and not byG3N . Therefore, if 1=‘ �

MP, than the noncommutative correction dominates overthe post-post-Newtonian one of the classical theory by afactor ð‘MPÞ2. Despite this potential enhancement, non-commutative corrections are still small to be detectable inpresent-day experiments. Indeed, for solar system objects,the noncommutative correction is compatible with aclassical post-Newtonian correction only if ‘ > 10�8 cm(atomic size). Such a large noncommutative scale iscertainly excluded by particle physics observations.Interestingly, due to the second term in (8), also con-tributing to (11), g00 is not the scalar quantity anymore.This is due to the violation of Lorentz invariance in ca-nonical noncommutative space-times. It would be interest-ing to investigate observational effects these correctionsfurther.III.—In this paper we have calculated leading semiclas-

sical corrections to the classical Schwarzschild metric. Ourresults are independent of particular realization of gravita-tional theory on noncommutative space-time, since wehave considered only the corrections steaming from inter-actions of a linearized gravitational field with a noncom-mutative matter energy-momentum tensor.These corrections are second order in noncommutative

parameters, i.e. Oð‘4Þ, and resemble quantum correctionsto classical black hole geometries calculated in a commu-tative case in [9]. Noncommutative corrections are signifi-cantly larger than the classical post-post-Newtoniancorrections if the scale of noncommutativity is largerthan the Planck scale, ‘ � 1=MP. Nevertheless, they arestill very small to be observable in present-dayexperiments.Unfortunately, the formalism we have used in our cal-

culations is applicable only for large distances, r � GNm.Thus our results are not capable of probing whether thegeometry at the origin of a black hole is singular or it getssmeared, according to the naive expectations from thespace-time noncommutativity. On the other hand, it wouldbe interesting to go beyond the semiclassical approxima-tion and calculate radiative corrections. If the phenomenonof UV-IR mixing, which is typical for noncommutativequantum field theories, persists in this case, then one wouldexpect an interesting connection between large-distancecorrections and small-distance geometry. Finally, the re-sults presented here can be straightforwardly extended byconsidering other noncommutative fields, e.g. spinor andgauge fields, to obtain noncommutative corrections to Kerrand Reissner-Nordtrom black holes.

I am indebted to Masud Chaichian and Anca Tureanu fore-mail correspondence. This work was supported by theAustralian Research Council.

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