Quantum corrections to the Schwarzschild metric and reparametrization transformations

0.11 MB·English

by  G. G. Kirilin

Quantum corrections to the Schwarzschild metric and reparametrization
transformations
G.G. Kirilin∗
Budker Institute of Nuclear Physics SB RAS, Novosibirsk, Russia
(Dated: February 7, 2008)
Quantum corrections to the Schwarzschild metric generated by loop diagrams have been con-
sidered by Bjerrum-Bohr, Donoghue, and Holstein (BHD) [Phys. Rev. D68, 084005 (2003)], and
Khriplovich and Kirilin (KK) [J. Exp. Theor. Phys. 98, 1063 (2004)]. Though thesame field vari-
ablesinacovariantgaugeareused,theresultsobtaineddifferfromoneanother. Thereasonisthat
the different sets of diagrams have been used. Here we will argue that the quantum corrections to
metricmustbeindependentofthechoiceoffieldvariables,i.e. mustbereparametrizationinvariant.
Using simple reparametrization transformation, we will show that the contribution considered by
BDH, is not invariant under it. Meanwhile the contribution of the complete set of the diagrams,
considered byKK, satisfies therequirement of the invariance.
6
0 PACSnumbers: 04.60.-m
0
2
GENERAL STRUCTURE OF Supplementing the action (1) with a gauge fixing part
n
REPARAMETRIZATION TRANSFORMATION
a
√ g 1 1
J S = d4x − hα h hµβ h|µ , (5)
5 In the series of papers Weinberg [1, 2, 3], Boulware f Z 2 (cid:18) µ|α− 2 |µ(cid:19)(cid:18) |β − 2 (cid:19)

and Deser [4] have shown that the massless particles of and a corresponding action of ghosts η , we find the ex-
1 µ
v helicity ±2 are described by the effective theory, satisfy- pansion of the action up to second order in fluctuations:
0 ing the equivalence principle. Boulware and Deser have
2 shown that the corresponding effective action coincides S¯ +S¯ +S +S (η)= d4x√ g + + , (6)
g m f gh
0 with the classical Einstein action [8] Z − L L L
(cid:0) (cid:1)
1
with
0
06 S¯g =Z d4x(cid:2)−√−g¯R¯(g¯)(cid:3). (1) = R+ 1 gµν∂νφ˜∂µφ˜ m2φ˜2 , (7)
/ L − 2(cid:16) − (cid:17)
c In general case, it is necessary to supplement the la- 1 1
-q grangian density with a number (may be infinite) of L=(cid:18)Rµν − 2δµνR− 2Tµν(cid:19)hµν +(cid:16)φ˜||λλ+m2φ˜(cid:17)φ, (8)
termsofhigherordersin∂ g¯ . The naturalpropertyof
r α µν 1
g any effective theory is the reparametrization invariance. = h Pαβhγδ|λ+h Xαβhγδ , (9)
v: Itimplies thata scatteringamplitude onmass shelldoes L −4(cid:16) αβ γδ |λ αβ γδ (cid:17)
1
i not depend on the choice of field variables. In general + h Wαβhγδ+熵 η |λ +Rλη ,
X 4 αβ γδ µ |λ µ λ
relativity one of natural parametrizations of the gravi- (cid:16) (cid:17)
r 1
a mtaettiorincaltefineslodr:hαg¯β is=thge d+ecofmp(ohs)i,tiwonheorfetfheiscoavnarairabnit- +φ(cid:18)Pγµδν∂µφ˜Dν +Pγµδνφ˜|µν − 2gγδm2φ˜(cid:19)hγδ.
µν µν µν
trary symmetric tensor function, the expansion of which
In expressions (7)-(9) indices are raised and lowered by
begins with a linear in h term. For example, to derive
αβ means of the tensor g , R and R are the Ricci tensor
µν µν
the counter lagrangian of the gravity, interacting with a
and Riemann curvature of the background field, respec-
massless scalar field, ’tHooft and Veltman [5] have used tively. We introduce also the notation h = hλ. Indices
λ
the trivial parametrization
following vertical lines denote covariant derivatives rela-
tive to the metric tensor g . Matrices, appearing in the
g¯ =g +h , (2) µν
µν µν µν
expressions (7)-(9), have the following form:
where gµν is the background field, hµν is the operator 1
field,characterizingquantumfluctuations. Theactionof Pγαδβ =δ(αγδδβ)− 2gαβgγδ, (10)
scalar field in external gravitational field has the form
1
Xαβ =Pαβ R ρ σ+δσ Rρ δρ R
√ g¯ γδ ρσ (cid:20) (γ δ) (δ(cid:18) γ)− 2 γ) (cid:19)(cid:21)
S¯ = d4x − g¯mn∂ φ¯∂ φ¯ m2φ¯2 . (3)
m Z 2 n m − +(αβ γδ), (11)
(cid:0) (cid:1) ↔
Similarly to (2), we decompose the field φ¯ Wγαδβ =Tσ(αPγσδβ)+T(σγPσαδβ)
1
φ¯=φ˜+φ. (4) + 2Pγαδβ(cid:16)m2φ˜2−T(cid:17). (12)
2
EXAMPLE OF REPARAMETRIZATION
TRANSFORMATION
As an example, we parametrize the gravitational field
in the following way
a b c
a
g¯ =g +h h hα. (18)
µν µν µν − 4 µα ν
FIG. 1: Thediagrams taken intoaccount in Ref.[6]
As stated above, the lagrangianquadratic in fluctuation
changesduetothelineartermsonly. Thereparametriza-
In the expressions (10)-(12) indices with brackets are to tion(18)isequivalenttothereplacementofthematrices
be symmetrized. T is the stress tensor of the scalar X and W in the lagrangian density (9) by the matrices
µν
field: X +a and W +a , respectively, there
X W
1 αβ =δ(αPβ) Rκλ, (19)
T =∂ φ˜∂ φ˜ g gρσ∂ φ˜∂ φ˜ m2φ˜2 . (13) Xγδ (γ δ),κλ
µν µ ν µν ρ σ
− 2 (cid:16) − (cid:17) αβ = 1δ(αTβ). (20)
Wγδ 2 (γ δ)
The first variationof the action (8) eventually supply us
with the equations of motion for the backgroundfields: Graviton propagator corrections are generated by the
counter lagrangian of pure gravity. The counter la-
1 1 grangianhasbeenderivedinRef.[5],weaimheretofind
R g R= T , (14)
µν µν µν
− 2 2 its transformationunder the reparametrizationtransfor-
gµνφ˜ +m2φ˜=0. (15) mation (18). Using the general formula for the counter
|µν
lagrangianderived in Ref.[5], we find:
In Ref.[4] it has been shown that, at fixed gauge, the
√ g 1
three graviton vertex is matched by the gravitational L(a) = − Sp 2a(X )
interaction with stress tensor of the classical free spin count. 8π2(d 4)4 { X
−
a
2 field up to four parameters, corresponding to the + R(P )+a2(P P ) . (21)
reparametrizationof the field hµν [9]: 3 X X X o
Intheexpression(21)thematricesP,X and shouldbe
g¯µν =gµν +hµν +a1hµλhλν readas 10 10matrices in relationto the nuXmber of the
×
+a2hµνh+a3gµνhαβhβα+a4gµνh2. (16) componentsofthe symmetrictensorhµν. Addingupthe
results of Ref.[5] and (21) yields the counter lagrangian
Loopcorrectionstothescatteringamplitudehavebeen for the case of pure gravity
studied in Refs.[6], [7]. Corrections concerned were pro-
√ g 7 a2
portional to ln|q2|, where q2 is the transfer momentum Lcount. = 8π2(d− 4)(cid:20)(cid:18)20 + 8 (cid:19)RmnRmn (22)
squared. Inparticular,itwasfoundthat,afteraveraging −
1 a 14
over the fluctuations, corrections to the Schwarzschild + + +a R2 . (23)
metric appeared: (cid:18)120 8 (cid:18) 3 (cid:19)(cid:19) (cid:21)
This lagrangian gives the following corrections to the
g =gcl +g , (17)
µν µν µν pure time component of the metric (diagram Fig.1a):
where gµclν is the classical Schwarzschild solution, gµν is g1a = 43 +a 14 +2a G2~m. (24)
thequantumcorrectiontoit. Quiteapparently,thelead- 00 −(cid:20)15 (cid:18) 3 (cid:19)(cid:21) πc5r3
ing corrections to metric must be independent of the
Using the additionalvertices (19), (20), it is easy to find
way of parametrization of the field h . Actually, be-
µν
the contributions of the diagrams depicted in Figs.1b,c:
ing quadratic in fluctuations, additional terms in the
parametrization(16) generate additionalstructures to L g1b = 26 +a 37 +2a G2~m, (25)
onlyduetoreplacementofthefieldhµν inthelagrangian 00 (cid:20) 3 (cid:18) 3 (cid:19)(cid:21) πc5r3
density , i.e., these structures vanish after taking into
accountLthe equations of motion (14). However, in per- g1c = 5 +5a G2~m. (26)
00 −(cid:18)3 (cid:19) πc5r3
turbation theory it happens only if all diagrams have
been taken into account. In Ref.[6] only certain part of Summing up the results (24)-(26), we get the following
the diagramshavebeenconsidered,namely,thegraviton contributions of the diagrams Fig.1:
propagator corrections and the corrections to one of the
vertices (Fig.1). As we will show, the contribution of g1a+1b+1c = 62 2a G2~m. (27)
these diagrams is not reparametrizationinvariant. 00 (cid:18)15 − 3 (cid:19) πc5r3
3
diagrams, because the integration momentum (flowing
throughthe ”legswithcrosses”inFig.2a)is ofthe order
ofq. Itfollowsthattheleadingclassicalcorrectiontothe
Minkowski metric
a b
Gm
gcl = 2 (30)
FIG. 2: Tree diagrams 00 − c2r
is of the same order as the field h ; consequently, it
µν
The a-independent part of Eq.(27) coincides with the serves no purpose to distinguish them. Since the correc-
result of Ref.[6]. From Eq.(27) one can see that this tion (29) is not the leading one, therefore it is possible
contributionisnotreparametrization invariant. Whereas to turn back to the initial variables (2) rather than (18),
the sum of the contributions of all the diagrams, listed i.e.
in Ref.[7], is reparametrizationinvariant for the obvious
reason stated above gharm =gcl a gcl 2 = 2G2m2 , (31)
00 00− 4 00 c4r2
107G2~m (cid:0) (cid:1)
gqu = . (28)
00 30 πc5r3 wheregharm is the secondordertermin the expansionof
00
Parametrization dependence on the contribution of the the Schwarzschildmetric in the harmonic coordinates.
diagrams Fig.1 (i.e., diagrams containing a single gravi- It should be repeated once again that the quantum
ton propagator attached to one of the particles) is the correction(28)istheleadingone,thereforethetrick(31)
direct consequence of the fact that, in general relativ- doesnotpermittoturnbacktotheformervariables,i.e,
ity, separation of these diagrams from other loop ones the correctionmust be invariantby itself. An important
is a matter of convention only, because they do not con- point is that a-dependent contributions to the potential
tain the pole in q2 [10]. Being unrelated to the renor- vanish in the sum of the diagrams Fig.2a and Fig.2b
malization of the amplitude with pole in q2, these dia- only, i.e., even on the level of classical gravity one can-
gramsshouldbeconsideredinlinewithotherones. Asit not introduce the physically meaningful ”one-particle-
has been shown, reparametrization transformations mix irreducible potential” (contrary to the section VIII of
thisdiagramswith,forexample,diagramproportionalto Ref.[6]).
Sp PWPW (seeEqs.(10),(12)). DuetoEq.(14)there I would like to thank I.B. Khriplovich for his helpful
{ }
isnodifferencebetweenthecontributionofthediagrams comments and discussions. The investigation was sup-
Fig.1 on mass shell and, for example, the diagram pro- ported by the Russian Foundation for Basic Research
portional to Sp PWPW . through Grant No. 05-02-16627-a.
{ }
ASIDE ON CLASSICAL CORRECTIONS
The correction (28) is the leading one in l2/r2, where ∗ [email protected]
p [1] S. Weinberg, Phys.Lett. 9, 357 (1964).
lp is the Plank length. From the standpoint of leading [2] S. Weinberg, Phys.Rev. B 135, 1049 (1964).
corrections,the parametrizations(2), (18) areindeed in- [3] S. Weinberg, Phys.Rev. B 138, 988 (1965).
distinguishable, because, after averaging over the quan- [4] D.G.BoulwareandS.Deser,Ann.Phys.86,193(1975).
tum fluctuation, the information about parametrization [5] G. ’t Hooft and M. Veltman, Ann. Inst. H. Poincare A
of these fluctuation is lost. Therein lies the main differ- 20, 69 (1974).
ence between the leading quantum corrections and non- [6] N. E. J. Bjerrum-Bohr, J. F. Donoghue, and B. R. Hol-
leading classical corrections of the order r2/r2, there r stein, Phys.Rev.D68, 084005 (2003), hep-th/0211071.
g g [7] I.B.KhriplovichandG.G.Kirilin,J.Exp.Theor.Phys.
is the Schwarzschild radius. Let us consider this aspect 98, 1063 (2004), gr-qc/0402018.
in detail. The diagramdepicted in Fig.1c contributes to [8] We put ~ = c = 16πG = 1, restoring the dimension in
the classical correction to the Minkowski metric the finalresults only.
G2m2 [9] There are linear changes of variables such as hµν →
gcl =(2+a) . (29) c1hµν + c2gµνh, but we leave them aside for the sake
00 c4r2 of simplicity.
[10] In contrast to QED or QCD these corrections lead to
However, this correction is actually induced by the tree
therenormalizationoftheoperatorsofhigherdimensions
diagram (Fig.2a). The decomposition on the back- than (1),for example(23),thustheybearnorelation to
groundfieldanditsfluctuations(2)hasnosenseforsuch the renormalization of G.