Feb 1, 2008 - John F. Donoghueb, and Barry R. Holsteinb,c a University of Copenhagen. The Niels ...... discussed by Duff[18]. While that discussion properly ...
arXiv:hep-th/0211071v1 8 Nov 2002
Quantum Corrections to the Schwarzschild and Kerr Metrics N. E. J. Bjerrum-Bohra John F. Donoghueb , and Barry R. Holsteinb,c a University of Copenhagen The Niels Bohr Institute Blegdamsvej 17, Copenhagen Ø DK-2100, Denmark b Department of Physics University of Massachusetts Amherst, MA 01003 c Institut f¨ ur Kernphysik Forschungszentrum J¨ ulich D-52425 J¨ ulich, Germany February 1, 2008
Abstract We examine the corrections to the lowest order gravitational interactions of massive particles arising from gravitational radiative corrections. We show how the masslessness of the graviton and the gravitational self interactions imply the presence of nonanalytic pieces p ∼ −q 2 , ∼ q 2 ln −q 2 , etc. in the form factors of the energy-momentum tensor and that these correspond to long range modifications of the metric tensor gµν of the form G2 m2 /r2 , G2 m¯h/r3 , etc. The former coincide with well known solutions from classical general relativity, while the latter represent new quantum mechanical effects, whose strength and form is necessitated by the low energy quantum nature of the general relativity. We use these results to define a running gravitational charge.
1
1
Introduction
In this paper we will discuss the long distance classical and quantum corrections to the Schwarszchild and Kerr metrics using the techniques of effective field theory. We will show how the nonanalytic radiative corrections to the energy-momentum tensor can be used to obtain the classical nonlinear terms in these metrics at long distance, and calculate the analogous nonanalytic quantum corrections. For the Schwarzschild metric we consider the case of a massive scalar particle. Here we clear up some numerical disagreements in related calculations that have emerged in the literature. We then present the Kerr results, using a massive fermion as a source, and show that the spin-independent quantum corrections are the same as those of the scalar particle. We also elucidate various theoretical issues and compare with other results in the literature. Effective field theory is ideally suited for discussing the quantum effects of general relativity at scales well below the Planck mass[1, 2]. While it is expected that the degrees of freedom and the interactions of gravity will be modified beyond the Planck scale, at low energies these ingredients are accurately described by general relativity. Effective field theory separates the known quantum effects of the low energy particles from the unknown physics at high energy. The latter effects are represented by the most general series of effective lagrangians consistent with the symmetry of general relativity. However the propagation of the low energy particles yields identifiable quantum effects that can be isolated by the techniques of effective field theory. The present study builds on two sets of recent work. One of these is the use of effective field theory to study quantum corrections to the gravitational potential[1, 3, 4, 5, 6, 7, 8]. While the basic principles of these studies are the same, there are some differences and/or disagreements. Since there is not a universal definition of the meaning of a potential, different authors use different definitions of potential in terms of Feynman diagrams, and hence obtain different answers. Even in a case where the same definition is used, different results have been obtained. We will provide some clarification of these disagreements. A related paper [8] provides a full and detailed calculation of the scattering potential of scalar particles. In the present paper we note that a subset of diagrams is more readily interpreted as a change in the metric, and we calculate these effects. The other precedent for the present paper is the calculation of the the leading quantum corrections to the Reissner-Nordstrom and Kerr-Newman
1
metrics using effective field theory[9]1 . These metrics involve charged particles, so that the quantum corrections involved photon loops, not graviton loops. However, this provided a particularly clear laboratory for the study of metric corrections. Interestingly, we saw that the classical nonlinearities in the metric can be calculated straightforwardly using Feynman diagram techniques. At the same time we saw that there was a clear identification of certain nonanalytic terms with long-distance quantum effects in the metric. We use the insights of that study to investigate the present problems, which involve graviton loops. In the present case the interpretation is not as clear, although the calculations are well defined. We will be using harmonic gauge throughout this paper. In this gauge, the Schwarzschild metric has the form[11, 13], 1− 1+
g00 =
Gm r Gm r
!
=1−2
Gm G2 m2 + 2 2 + ... r r
g0i = 0 gij
Gm 2 G2 m2 ) − = −δij (1 + r r2 = −δij
Gm G2 m2 1+2 + r r2
!
1+ 1− −
Gm r Gm r
!
ri rj r2
ri rj G2 m2 + ... r2 r2
(1)
The Kerr metric[14] refers to a particle with spin and, keeping only terms up to first order in the angular momentum, has the harmonic gauge form 1− 1+
g00 = g0i = gij
Gm r Gm r
!
+ ... = 1 − 2
Gm G2 m2 + 2 2 + ... r r !
2G 2G2 m ~ − (S × ~r)i + . . . r3 r4
2G ~ × ~r)i + . . . = (S r 2 (r + mG)
Gm 2 G2 m2 ) − = −δij (1 + r r2 = −δij
Gm G2 m2 1+2 + r r2
!
1+ 1− −
Gm r Gm r
!
ri rj + ... r2
ri rj G2 m2 + ... r2 r2
(2)
We will show that using a particular set of Feynman diagrams we reproduce the former with the addition of a long distance quantum correction g00 = 1 − 2
G2 m2 62G2 m¯h Gm +2 2 + + ... r r 15πr 3
1
These corrections have also been considered from the point of view of S-matrix theory in ref. [10]
2
g0i = 0 gij
= −δij
Gm G2 m2 14G2 m¯h + + 1+2 r r2 15πr 3
!
ri rj − 2 r
G2 m2 76G2 m¯h + r2 15πr 3
!
+ (3) ...
G2 m2 76G2 m¯h + r2 15πr 3
!
+ (4) ...
For the Kerr metric, Gm G2 m2 62G2 m¯h +2 2 + + ... 3 r r 15πr ! 2G 2G2 m 36G2 ¯h ~ − + (S × ~r)i + . . . r3 r4 15πr 5
g00 = 1 − 2 g0i = gij
= −δij
Gm G2 m2 14G2 m¯h + + 1+2 r r2 15πr 3
!
ri rj − 2 r
It is of course required that the classical spin-independent terms must be the same for a scalar particle and a fermion. We know of no firm requirement for the spin-independence of the quantum corrections to g00 and gij , but from our calculation they are seen to be identical.
2
Review
The metric is derived from the energy-momentum tensor of a source, using Einstein’s equation as the equation of motion. At lowest order in the fields, the source particle is just a point particle in coordinate space. However, both classical fields and their quantum fluctuations modify the energymomentum of a particle at long distance. These modifications can be found by the consideration of the radiative corrections to the energy-momentum tensor. When these are translated into a metric, they yield the classical nonlinearities and quantum modifications of the metric. Let us review what was found in Ref[9] for the conceptually simpler case of charged particles, as our calculation here will follow the same procedure. In that case the field around the particles was the electromagnetic field and the gravitational interaction was treated purely classically. The masslessness of the photon implies that there are long range fields around a charged particle and these carry energy and momentum. At the same time, in a Feynman diagram calculation of the renormalization of the energy momentum tensor of the charged particle, the masslessness of the photon leads to nonanalytic p 2 terms in the formfactors having the structure ∼ −q , ∼ q 2 ln −q 2 , where q is the momentum transfer, aspwell as analytic terms of order q 2 , q 4 .... It was shown in detail how the ∼ −q 2 terms account for the classical energymomentum of the electromagnetic field and how they exactly reproduce the 3
classical nonlinearities in the metric that are present in the Reissner Nordstrom and Kerr Newman metrics. Nonanalytic terms of the form q 2 ln −q 2 also appear, and when they are included in the equations of motion they produce further corrections in these metrics. Explicit examination shows that these latter are linear in h ¯ - i.e.they are quantum effects. Finally the analytic terms produce only delta functions (or derivatives of delta functions) in the metric, such that they vanish at long distance. Thus we saw that the long distance modifications of the metrics are obtained from the nonanlaytic terms in the formfactors of the energy momentum tensor. The same effects are present in the purely gravitational case. If one expands the energy and momentum of the particle in powers of G, the lowest order result is that of a point particle. However, there is energy and momentum also carried by the gravitational field around the particle and this can be calculated via the one loop Feynman diagrams. Because thepgraviton is massless, there will also be nonanalytic terms of the forms ∼ −q 2 , ∼ q 2 ln −q 2 in the formfactors of the energy momentum tensor. Again these will produce long range modifications of the metric. If we include the relevant dimensionful couplings, this will have the schematic form metric ∼ Gm
Z
∼ Gm
d3 q i~q·~r 1 e 1 − aG~q2 (2π)3 ~q2
s
m2 − bG~q2 log(~q2 ) − cG~q2 + . . . ~q2
1 aGm bG¯h + 2 + 3 + cGδ3 (x) + . . . r r r
(5)
Here a, b, c are dimensionless numbers and further p numerical factors of order unity will be inserted later. We will examine the −q 2 terms in detail and show how they correctly reproduce all the features of the classical metric. For the q 2 ln −q 2 terms we have included the factor of h ¯ that follows from dimensional analysis. The analytic correction to the energy momentum tensor yields the delta function term that is not relevant for the long distance behavior. The nonanalytic terms come from the low energy propagation of gravitons, using the couplings of general relativity. Because these features are independent of the high energy behavior of gravity, they are unambiguous predictions of low energy general relativity. There is also no influence of other possible terms in the gravitational lagrangian, such as R2 or related corrections in the matter lagrangian. These yield only analytic corrections to the formfactor and hence do not provide long distance modifications of the metric. These are behaviors that are well known in the effective field theory of gravity[1]. 4
3
Lowest order
Let us first consider the theory without loop corrections. The metric tensor is expanded as gµν ≡ ηµν + h(1) (6) µν + . . . where ηµν = (1, −1, −1, −1)diag is the usual Minkowski metric and the superscript refers to the number of powers of the gravitational coupling which (1) appear. The dynamical relation which connects hµν and the energy momentum tensor Tµν is the Einstein equation, whose linearized form in harmonic gauge—gµν Γλµν = 0—is 1 2h(1) µν = −16πG(Tµν (x) − ηµν T (x)) 2
(7)
where T = η µν Tµν represents the trace. The metric for a nearly static source is then recovered via the Green function in either coordinate or momentum space 1 d3 yD(x − y)(Tµν (y) − ηµν T (y)) 2 Z d3 q i~q·~r 1 1 = −16πG e (Tµν (q) − ηµν T (q)) (2π)3 ~q2 2
hµν (x) = −16πG
Z
(8)
For a quantum mechanical system, Tµν is given in terms of the transition density < p2 |Tµν (x)|p1 >
and the conservation condition ∂ µ Tµν = 0 together with the requirement that Tµν transform as a second rank Lorentz tensor demands the general (scalar field) form2
i ei(p2 −p1 )·x h 2Pµ Pν F1 (q 2 ) + (qµ qν − ηµν q 2 )F2 (q 2 ) < p2 |Tµν (x)|p1 >= √ 4E2 E1 (10) where we have defined Pµ = 12 (p1 +p2 )µ and qµ = (p1 −p2 )µ . Conservation of energy and momentum requires F1 (q 2 = 0) = 1 but there exists no constraint on F2 (q 2 ). 2
Here we use the conventional normalization for the scalar field < p2 |p1 >= 2E1 (2π)3 δ 3 (~ p2 − p ~1 )
5
(9)
For the case of a point mass m the lowest order form is (cf. Eq. 71) (0) < p2 |Tµν (0)|p1 >= √
1 1 2Pµ Pµ − (qµ qν − ηµν q 2 ) 2 4E2 E1
(11)
while in the case of the spin 1/2 system we have 1 (0) < p2 |Tµν (0)|p1 > = u ¯(p2 ) (γµ Pν + γν Pµ ) u(p1 ) 2 i 1 Pµ Pν − (σµλ q λ Pν + σνλ q λ Pµ ) u(p1 ) = m 4m (12) where we use here the conventions of Bjorken and Drell and have employed the Gordon identity[12]. In either case, for a heavy point mass located at the origin we have the lowest order Breit frame result (0) < p2 |Tµν (0)|p1 >≃ mδµ0 δν0
(13)
The Einstein equation then has the solution h(1) q) = − µν (~ which, using Z
1
8πGm × 0 ~q2 δ ij
1 d3 q i~q·~r 1 , e = 3 2 (2π) q ~ 4πr
Z
µ=ν=0 µ = 0, ν = i + . . . µ = i, ν = j
(14)
d3 q i~q·~r qj irj e = , 3 2 (2π) ~q 4πr 3
(15)
corresponds to the coordinate space result3 h(1) r ) = f (r) × µν (~ with
1
0 δ ij
µ=ν=0 µ = 0, ν = i + . . . µ = i, ν = j
(16)
2Gm r and reproduces the well-known leading order piece of the Schwarzschild solution[13]. In the case of spin 1/2 there is an additional classical component which arises from the spin. Using f (r) = −
(0)
3
< p2 |T0i (0)|p1 >≃ χ†2 ~σ χ1 × ~q
(17)
Here the ellipses represent a very short range component associated with the qdependent piece of Tµν .
6
we find the off-diagonal component of the metric (1)
h0i (~q) = −8πiG
1 ~ (S × ~q)i ~q2
(18)
which corresponds to the coodinate space result (1)
h0i (~r) =
2G ~ (S × ~r)i r3
(19)
and agrees to this order with the Kerr metric[14]. With this basic material in hand we now proceed to the inclusion of loop corrections.
4
Loop Corrections to the Energy Momentum Tensor—-Spin 0
Of course, the lowest order discussion given above is straightforward and familiar, while the purpose of the present paper is determine the nonanap lytic corrections ∼ −q 2 , q 2 log −q 2 to the form factors arising from the higher order gravitational self-interaction. The appearance of such terms was found in I to be associated with the feature that the graviton couples to the (massless) photon, and the same is expected to happen in the case of gravitational self-interaction since the graviton is itself massless. The relevant diagrams are shown in Figure 1 and are similar to their electromagnetic analog considered in I, although the tensor nature of the graviton makes the calculation considerably more tedious. Details of the calculation are given in Appendix A and the results are[1, 5] F1 (q 2 ) = 1 +
−q 2 1 π2 m Gq 2 3 p ) + ... (− log 2 + π 4 m 16 −q 2
1 Gm2 −q 2 7 π 2 m F2 (q 2 ) = − + (−2 log 2 + p 2 ) + . . . 2 π m 8 −q
(20)
As found in the case of the electromagnetically corrected vertex studied in I, we observe that the q 2 = 0 value of the leading form factor F1 (q 2 ) is unchanged from its lowest order value of unity, as required by energymomentum conservation, while the form factor F2 (q 2 ), which is not protected, is modified. Such higher order corrections are to be expected from the feature that gravity is nonlinear and must contain terms to all orders in the gravitational coupling. The momentum space form factors imply a coordinate space structure of the energy momentum tensor which is modified at large distance. Using 7
the integrals listed in Appendix A, we find the correction to the lowest order energy-momentum tensor to be Z
T00 (~r) =
!
d3 q i~q·~r q~2 2 F2 (q 2 ) e mF (q ) + 1 (2π)3 2m
d3 q i~q·~r 7 Gm 2 3 1 e q~ log ~q2 ( − 1) m + πGm2 (− + )|~q| + 3 (2π) 16 16 π 4 2 3Gm 3Gm¯h = mδ3 (r) − − 4 8πr 4π 2 r 5 T0i (~r) = 0 Z d3 q i~q·~r 1 e (qi qj − δij ~q2 )F2 (q 2 ) Tij (~r) = 2m (2π)3 =
Z
=
Z
"
d3 q i~q·~r 7πGm2 Gm e (qi qj − δij ~q2 ) − (qi qj − δij ~q2 ) log ~q2 (2π)3 16|~q| π
7Gm2 = − 4πr 4
#
2Gm¯h ri rj 1 − δij + 2 5 δij 2 r 2 π r
(21)
We have inserted factors of h ¯ where appropriate, although we continue to use c = 1 units. Note that the leading correction to Tµν is classical in nature, since there are no factors of h ¯ . We can show that this effect is generated by the energy and momentum that are carried by the gravitational field—Eq. 16— surrounding the point mass. This field possesses an energy-momentum tensor[11] i h 1 (1) (1) (1) grav 8πGTµν = − h(1)λκ ∂µ ∂ν hλκ + ∂λ ∂κ h(1) µν − ∂κ ∂ν hµλ + ∂µ hνλ 2 1 1 1 (1) (1)σλ λ (1)σ (1) σ (1)λ ∂λ h(1) − µ + ∂λ hσν ∂ h µ − ∂ν hσλ ∂µ h σν ∂ h 2 2 4 3 1 1 σ (1)λχ λ (1)σχ ηµν (∂λ h(1) − ∂λ h(1) ) − h(1) 2h(1) − σχ ∂ h σχ ∂ h 4 2 4 µν 1 (1) + ηµν h(1)αβ 2hαβ (22) 2
in terms of which the classical field correction to the point mass form of the energy-momentum tensor is determined to be grav T00 (r)
=
2 1 3~ ~ (r) − 3f (r)∇ ~ 2 f (r) + . . . = − 3Gm + . . . (r) · ∇f − ∇f 8πG 4 8πr 4
8
Tijgrav (r)
= −
1 3 ~ 1 ~ (r) (r) · ∇f − ∇i f (r)∇j f (r) + δij ∇f 8πG 2 4 1 7Gm2 ri rj 2 ~ − δij + . . . f (r)∇i ∇j f (r) + δij f (r)∇ f (r) + . . . = − 4πr 4 r2 2 (23)
where the ellipses indicate contributions localized about the origin. Obviously Eqs. 21pand 23 are identical, demonstrating the correspondence of the nonanalytic −q 2 terms and the classical field energy, just as found in I for the electromagnetic case. The remaining corrections to Tµν contain an explicit factor of h ¯ and are thus intrinsically quantum mechanical in nature. The ”physics” behind these modifications can be understood in terms of the position uncertainty associated with quantum mechanics, which implies the replacement of the h ¯ . Since for macrodistance r in the classical expression by the value ∼ r + m scopic distances ¯h/m
