Self-force and synchrotron radiation in odd space-time dimensions
aa r X i v : . [ h e p - t h ] N ov Self-force and synchrotron radiation in odd space-time dimensions
Edward Shuryak, Ho-Ung Yee, and Ismail Zahed
Department of Physics and Astronomy, State University of New York, Stony Brook, NY 11794 (Dated: November 8, 2018)Classical electrodynamics in flat 3+1 space-time has a very special retarded propagator ∼ δ ( x )localized on the light cone, so that a particle does not interact with its past field. However, this isan exception, and in flat odd-dimensional space-times as well as generic curved spaces this is notso. In this work we show that the so called self-force is not only non-zero, but it matches (in 2+1dimensions) the radiation reaction force derived from the radiation intensity. PACS numbers:
I. INTRODUCTION
This paper deals with a century-old issue, the so called“radiation reaction” force, in classical electrodynamicsand in general relativity. On one hand, it is clear thatenergy and momentum carried away by radiation froman accelerated charge should be compensated by a forcewhich is going to reduce the energy and momentum ofthe particle accordingly. On the other hand, the particlein 3+1 flat space-time does not interact with its own fieldbecause it is fully concentrated on the light cone.The relation to the energy loss in the nonrelativisticdipole radiation and its relativistic extension leads to thewell known Abraham-Lorentz-Dirac force f µ D = 2 e x µ − ¨ x ν ¨ x ν ˙ x µ ) , (1.1)where the dot is a derivative over the proper time d/dτ .Its derivation comes from “the large-distance” discussion,based on the amount of the energy/momentum fluxesthrough some distant surface (large sphere, etc). Al-though in principle such an approach can/was applied forscalar/electromagnetic/gravitational radiations, in somecases it is technically difficult. In particular, the radia-tion and its corresponding ultrarelativistic sources movethrough nearby paths in curved spaces, making the radia-tion calculation highly non-trivial (see e.g. [1]) . It wouldbe more satisfactory logically and much easier practicallyto use some local derivation. In this paper we provide ex-amples in support of a local derivation.We stumbled on this issue while trying to assess thebraking force for a gravitational radiation of an ultra-relativistic particle moving in a particular curved space(the so called thermal or black-hole AdS in 4+1 dimen-sions). This problem is related to the practically impor-tant problem of jet quenching in the quark-gluon plasma.Since the calculations are very different in AdS we rel-egate the analysis of jet quenching by braking radiationto the companion paper [2].The issue we study in this paper is whether one candefine and calculate the self-interaction force , using onlythe particle’s own field related to its past trajectory. Thisquestion was first addressed by Dirac for electrodynamicsin flat 3+1 dimensions [3] and extended to curved 3+1 dimensions by Dewitt [4]. In this work we show that thisis also possible for electrodynamics in odd dimensionalspace-times, namely 2+1 and 4+1 dimensions.The idea that it is possible was inspired by the so calledMSTQW approach in gravitational setting [5, 6]. In itthe remarkable “self-force” expression was suggested ma a = m u b u c Z τ − −∞ dτ ′ u a ′ u b ′ ( 12 ∇ a G bca ′ b ′ (1.2) −∇ b G ac a ′ b ′ − u a u d ∇ d G bca ′ b ′ , )in which the integral is done over proper time and thepast world line of the particle till the regulated presenttime τ − . G is the retarded Green function for the Ein-stein equation with the particle as the source. Note thatthe bracket is just the Chrystoffel force for a gravity per-turbation, induced by the past gravity field of the particleitself.Does this expression (or its simpler analogs) actuallywork? The first obvious try, for electromagnetic radiationin 3+1 flat space-time, produces zero because in flat 4dspace-time the retarded propagator is totally localizedon the light cone ( x α − r α )( x α − r α ) = 0. There aresimply no points on the particle path that can intersectthe past light cone sustained by its present location! (Itproduces the well-known Lienard-Wiechert expression forthe retarded fields we all learned/taught in classical E/Mcourses). This upset can be remedied by realizing thatsuch form of the retarded propagator is not the genericcase, and in fact it is nonzero inside the light cone in otherspace-times. Therefore, we decided to calculate it for thenearby space-time dimensions and see if the results makesense. We also calculate the radiative losses by standardlarge-distance method for comparison.We assume that the motion of the charge is prefixedby some external non-electromagnetic forces and do notdiscuss its origin. We will define and compute the localback-reaction force on a particle originating from its ownelectromagnetic fields. (Of course, this force should be apart of the total force that determines the given chargetrajectory, but we do not specify its effect on the path ).Although the self-force is in general expected to benonlocal in character, and given by the integral over thepast trajectory, the situation is simplified in the highlyrelativistic limit, γ → ∞ . Indeed we will see that in thiscase it is defined by a small range in the proper timeinterval τ ∼ γ , so that the leading γ -contribution de-pends only on the local data of the motion. The resultingleading- γ behavior of the self-force takes a similar form asin (1.1) in terms of local derivatives of the motion, whichwould be the odd-dimensional analogue of the Abraham-Lorentz-Dirac force in 3+1 dimensions. Another featureis that the integrals with the retarded propagator divergeand need to be correctly regulated: we will see this fea-ture explicitly in our analysis shortly. II. THE SELF-FORCE IN 2+1 DIMENSIONS
The case of 2+1 dimensions is the simplest one to con-sider. (Below we will see that it contains all the basicfeatures of the radiation reaction force in any odd space-time dimensions because of some recursive relations be-tween the propagators involved.). As already mentionedin the introduction, the main new feature in odd space-times compared to the conventional flat 3+1 dimensionalspace-time is that the massless retarded propagator froma given source has support inside the light-cone. Thisimplies that at a given moment/position of the charge,the electromagnetic field acting on it obtain some contri-butions from the past trajectory of the charge.A moving charge e with a given trajectory x µ ( τ ) givesa relativistic current j µ ( x ) = e Z dτ δ (3) ( x − x ( τ )) ˙ x µ ( τ ) , (2.1)where τ is the proper time normalized as ˙ x µ ˙ x µ = +1.In the covariant gauge ∂ µ A µ = 0, the Maxwell equationbecomes (cid:3) A µ = j µ , (cid:3) ≡ ∂ µ ∂ µ , (2.2)which has a formal retarded solution as A µ ( x ) = Z d x ′ ∆ R ( x − x ′ ) j µ ( x ′ )= e Z dτ ′ ∆ R ( x − x ( τ ′ )) ˙ x µ ( τ ′ ) , (2.3)using (2.1). ∆ R ( x ) is the massless retarded propagatorin 2+1 dimensions which is given by∆ R ( x ) = θ ( x )2 π θ ( x ) √ x , x ≡ x µ x µ , (2.4)which has support in the entire forward light-cone. Itis clear that only the past trajectory contributes to theelectromagnetic field at a given moment, and we are in-terested in the self-force acting on the moving chargeitself. This amounts to computing the covariant Lorentzforce f µ D = eF µν ( x ( τ )) ˙ x ν ( τ ) , (2.5) where F µν ( x ( τ )) are the field strengths of the retardedelectromagnetic field (2.3) at proper time τ , induced bythe past trajectory of the charge, x µ ( τ ′ ) with τ ′ < τ .As usual, one encounters local divergences in comput-ing (2.5) coming from the region near τ ′ = τ , and onehas to regularize and absorb divergences by renormaliz-ing physical parameters of the moving charge such as itsmass. In this section, we regularize divergences covari-antly by cutting off proper time integral Z τ −∞ dτ ′ → Z τ − ǫ −∞ dτ ′ , (2.6)with a small ǫ >
0, and let ǫ → x ( τ ) of radius r = ǫ and taking ǫ → ∂ µ ∆ R ( x ) = 2 x µ ∆ ′ R ( x ) + δ µ δ ( x )2 π θ ( x ) √ x , (2.7)where ∆ ′ R ( x ) = θ ( x )2 π (cid:18) δ ( x ) √ x − θ ( x ) x √ x (cid:19) , (2.8)the unregularized bare Lorentz force is f µ D = 2 e Z dτ ′ ∆ ′ R ( X ) X [ µ ( τ, τ ′ ) ˙ x ν ] ( τ ′ ) ˙ x ν ( τ )+ e δ [ µ Z dτ ′ δ (cid:0) X (cid:1) π θ (cid:0) X (cid:1) √ X ˙ x ν ] ( τ ′ ) ˙ x ν ( τ ) , (2.9)where X µ ( τ, τ ′ ) ≡ x µ ( τ ) − x µ ( τ ′ ) and [ µ, ν ] = µν − νµ .To identify the local divergences near τ ′ = τ , we expandthe quantities in terms of ( τ − τ ′ ) ≡ ǫ as X µ ( τ, τ ′ ) = ǫ ˙ x µ ( τ ) − ǫ x µ ( τ ) + ǫ x µ ( τ ) + O ( ǫ ) , ˙ x ν ( τ ′ ) = ˙ x ν ( τ ) − ǫ ¨ x ν ( τ ) + ǫ x ν ( τ ) + O ( ǫ ) , X = ǫ − ǫ
12 ¨ x µ ¨ x µ + O ( ǫ ) , (2.10)using the identities such as ˙ x µ ¨ x µ = 0 and ˙ x µ ... x µ = − ¨ x µ ¨ x µ . From these, one obtains after some algebra X [ µ ( τ, τ ′ ) ˙ x ν ] ( τ ′ ) ˙ x ν ( τ ) (2.11)= ǫ x µ ( τ ) − ǫ x µ + ¨ x ν ¨ x ν ˙ x µ ) ( τ ) + O ( ǫ ) , and ∆ ′ R ( X ) = − θ ( ǫ )4 π (cid:18) ǫ − δ ( ǫ ) ǫ (cid:19) + O (cid:18) ǫ (cid:19) , (2.12)so that the proper time integral of the first term in (2.9)near the ǫ = 0 region becomes − e π Z dǫ θ ( ǫ ) (cid:18) ǫ − δ ( ǫ ) (cid:19) ¨ x µ ( τ ) + O ( ǫ ) , (2.13)which is logarithmically divergent. This divergence isproportional to the covariant acceleration ¨ x µ and is read-ily absorbed by renormalizing the mass m ren = m bare − e π log( ǫ ) , (2.14)where ǫ here means a cutoff with a slight abuse of nota-tion. This is equivalent to having a counterterm addedto (2.9) so that the renormalized self-force now takes aform f µ D = − e π lim ǫ → + Z dτ ′ θ ( τ − τ ′ − ǫ ) (2.15) × X [ µ ( τ, τ ′ ) ˙ x ν ] ( τ ′ ) ˙ x ν ( τ )( X ) −
12 ¨ x µ ( τ )( τ − τ ′ ) ! , considering only the first term in (2.9).The finite contribution coming from the piece propor-tional to δ ( ǫ ) in (2.13) is ambiguous up to the value of θ (0). In fact, the second term in (2.9) gives rise to adivergence which is also proportional to θ (0). One canchoose to have θ (0) = 0 removing these ambiguities as achoice of regularization. Alternatively, once we introducea cutoff τ − τ ′ > ǫ they simply don’t appear for any finite ǫ >
0, and hence the limit ǫ → + after taking care of thelogarithmic divergence via (2.14) is insensitive to them.Unlike in 3+1 dimensions where the self-force is thelocal Abraham-Lorentz-Dirac force (1.1), in 2+1 dimen-sions the self-force (2.15) is non-local and receives contri-butions from the entire past tail of the charged particle.This non-locality is due to the peculiar fact that the re-tarded propagator (2.4) has support on the entire forwardpart of the light cone as we noted earlier.However, for ultra-relativistic motion we now showthat the leading γ behavior of (2.15) is completely lo-cal. Let’s denote the n -th derivative of x µ with respectto proper time τ as x µ ( n ) ≡ d n x µ dτ n . In the large γ -limit, x µ ( n ) is of order O ( γ n ). In general, for any combinationsof x µ ( n ) the powers of γ simply add upΠ i x ( n i ) ≤ O (cid:16) γ P i n i (cid:17) , (2.16)where we allow the inequality due to exceptions of somelowest contractions such as ˙ x µ ˙ x µ = 1 and ˙ x µ ¨ x µ = 0.These exceptions will not affect our derivation and con-clusion, because what will be important is that the maxi-mum powers of γ is P i n i . We will prove that the leading γ behavior of (2.15) is coming from the proper time in-terval τ − τ ′ ≡ ǫ ∼ γ : we will first assume this andexpand (2.15) in small ǫ , then show the consistency ofthe assumption in our final results. Using (2.10) and (2.16), one can show the expansion X [ µ ( τ, τ ′ ) ˙ x ν ] ( τ ′ ) ˙ x ν ( τ ) (2.17)= ǫ x µ ( τ ) − ǫ x µ + ¨ x ν ¨ x ν ˙ x µ ) ( τ ) + O ( ǫ γ ) , X = ǫ − ǫ
12 ¨ x ν ¨ x ν + O ( ǫ γ ) , (2.18)Inserting the expansion into (2.15), yields12 ǫ (cid:0) − ǫ ¨ x ν ¨ x ν (cid:1) − ¨ x µ (2.19) − (cid:0) − ǫ ¨ x ν ¨ x ν (cid:1) (... x µ + ¨ x ν ¨ x ν ˙ x µ ) + O (cid:0) γ (cid:1) , In estimating O (cid:0) γ (cid:1) we already used the assumption ǫ ∼ γ . This is justified because of the expression inthe denominator which effectively cutoffs the ǫ -integraland confines it to ǫ ∼ (¨ x ) − ∼ γ . The dominant contri-bution in the large γ limit then comes from the secondline which is O (cid:0) γ (cid:1) , while others are all O (cid:0) γ (cid:1) , so thatthe leading self-force becomes f µ D ≈ e π Z ∞ dǫ (cid:0) − ǫ ¨ x ν ¨ x ν (cid:1) ¨ x ν ¨ x ν ˙ x µ + O ( γ )= − e √ π p − ¨ x ν ¨ x ν ˙ x µ + O ( γ ) , (2.20)which is O (cid:0) γ (cid:1) . This is our 2+1 dimensional version ofthe Abraham-Lorentz-Dirac force.We apply our result to the ultra-relativistic circularmotion x µ ( τ ) = ( γτ, ρ cos( γωτ ) , ρ sin( γωτ )) , (2.21)where γ = (1 − v ) ≪ v = ρ ω ≈
1. Because¨ x ν ¨ x ν = − ρ ω γ ≈ − ω γ , the proper time integral isconfined to | τ ′ − τ | ≈ ωγ ≪ , (2.22)as discussed before, and the leading covariant self-forceis f µ D = − e √ π ωγ ˙ x µ , (2.23)which is longitudinal. The common (non-covariant) lon-gitudinal drag force is ~ f L ≈ − e √ π ωγ ~v . (2.24)Therefore, the external force that is needed to maintainthe circular motion is ~ f ext = m ren d ~xdt − ~ f L , (2.25)and the rate of work done to the particle is given by P = ~ f ext · ~v ≈ e √ π ωγ , (2.26)where we used d ~xdt · ~v = 0 for the circular motion.In section (IV), we will compute the far-field radiationof this synchrotron motion, and find an agreement be-tween the total power radiated at large distance and thework done locally to the charge (2.26). III. LOCAL SELF-FORCE IN 4+1 DIMENSIONS
In this section, we will perform similar computations asin the previous section but in 4+1 dimensional spacetime.We will therefore be brief in explaining the details of thederivation while presenting our results. The Maxwell’sequations sourced by a charge e in the radiative gaugeare solved as before, A µ ( x ) = e Z dτ ′ ∆ R ( x − x ( τ ′ )) ˙ x µ ( τ ′ ) , (3.1)where the retarded propagator in 4+1 dimensions is∆ R ( x ) = − θ ( x )2 π (cid:18) δ ( x ) √ x − θ ( x ) x √ x (cid:19) . (3.2)Note the analogy with (2.4), which in fact stems fromthe generic relationship∆ ( x ) = − π ddx ∆ ( x ) , (3.3)which is readily shown from the momentum space repre-sentation of the retarded propagators and the recursiveproperty of the integer Bessel functions. Modulo normal-izations, (3.3) extends to all odd space-time dimensions.From (3.1) and (3.2), one easily writes down the un-regularized self-force simialr to (2.9) before as f µ D = 2 e Z dτ ′ ∆ ′ R ( X ) X [ µ ( τ, τ ′ ) ˙ x ν ] ( τ ′ ) ˙ x ν ( τ ) , (3.4)where in 4+1 dimensions we have∆ ′ R ( x ) = − θ ( x )2 π (cid:18) δ ′ ( x ) √ x − δ ( x )( x ) + 34 θ ( x )( x ) (cid:19) . (3.5)In (3.4), we have dropped terms that are proportional to δ (cid:0) X (cid:1) due to our regularization scheme that we explainin the following. However, we should caution the readersthat we haven’t checked the regularization scheme inde-pendency of our results, contrary to the previous 2+1dimensional case.We will regularize the divergences appearing in (3.4)by replacing ∆ ′ R ( x ) with∆ ′ ǫR ( x ) = − θ ( x )2 π (cid:18) δ ′ ( x − ǫ ) √ x − δ ( x − ǫ )( x ) + 34 θ ( x − ǫ )( x ) (cid:19) , (3.6)and taking the ǫ → ǫ -expansion reads as X [ µ ( τ, τ ′ ) ˙ x ν ] ( τ ′ ) ˙ x ν ( τ ) (3.7)= ǫ x µ ( τ ) − ǫ x µ + ¨ x ν ¨ x ν ˙ x µ ) ( τ )+ ǫ (cid:18) .... x µ − .... x ν ˙ x ν ˙ x µ + 23 ¨ x ν ¨ x ν ¨ x µ (cid:19) − ǫ (cid:18) x (5) µ − x (5) ν ˙ x ν ˙ x µ −
54 .... x ν ˙ x ν ¨ x µ (cid:19) + O ( ǫ γ ) , where x (5) ≡ d xdτ . Upon inserting (3.7) into (3.4), thedivergences of the self-force are found to be f µ D ∼ e π ǫ ¨ x µ (3.8)+ 3 e π log( ǫ ) (cid:18) .... x µ − .... x ν ˙ x ν ˙ x µ + 23 ¨ x ν ¨ x ν ¨ x µ (cid:19) . The first leading divergence can readily be absorbed intothe renormalized mass m ren = m bare − e π ǫ , (3.9)whereas the nature of the second term of logarithmic di-vergence is unclear to us. We will simply choose ourregularization scheme to remove it minimally. We ex-pect this minimal subtraction to be consistent with thefar-field radiation formulae, although we will only checkthis for the 2+1 dimensional case below.After removing the divergences, the finite contributioncan be computed in the leading γ approximation. Onefinds that the δ ( X ) and δ ′ ( X ) terms in ∆ ′ R ( X ) in (3.4)give us the leading γ contributions to the self-force, f µ D ∼ e π (.... x µ + ¨ x ν ¨ x ν ¨ x µ − .... x ν ˙ x ν ˙ x µ ) + O ( γ ) , (3.10)which is completely local. In deriving the above, one hasto expand X = ǫ − ǫ
12 ¨ x µ ¨ x µ − ǫ
12 (.... x µ ˙ x µ + 2¨ x µ ... x µ ) + O ( ǫ γ ) . However, for circular motion that we are interested in,the longitudinal component of the above force that is re-lated to the rate of work done simply vanishes. Therefore,we are led to seek the next leading term in O ( γ ).The next leading term is quasi-local, that is, it comesfrom the region ǫ ∼ γ as was the case in 2+1 dimensionsbefore. It is given by − e π Z ∞ dǫ (cid:0) − ǫ ¨ x ν ¨ x ν (cid:1) (3.11) × (cid:0) x (5) ν ˙ x ν ˙ x µ + 358 .... x ν ˙ x ν ¨ x µ + 254 ¨ x ν ... x ν ¨ x µ (cid:1) = − e √ π (cid:0) x (5) ν ˙ x ν ˙ x µ + .... x ν ˙ x ν ¨ x µ + ¨ x ν ... x ν ¨ x µ (cid:1) √− ¨ x ν ¨ x ν . Applying the above to the ultra-relativistic syn-chrotron motion, we see that only the first term con-tributes, and we obtain the leading longitudinal force as f µ D ∼ − e √ π γ ω ˙ x µ , (3.12)which leads to the rate of work done to the system, P = e √ π γ ω . (3.13) IV. FAR-FIELD SYNCHROTRON RADIATIONIN 2+1 DIMENSIONS
In this section, we compute the far-field radiation ofthe ultra-relativistic circular motion and check that theresulting total rate of radiation matches the rate of workdone locally to the charge (2.26), which is our first non-trivial consistency check for the “self-force” approach andthe subtractions we have applied.The retarded gauge field from a point-like charge mo-tion is A µ ( t, ~x ) = e π Z t ∗ ( t,~x ) −∞ dt ′ p ( t − t ′ ) − ( ~x − ~x ′ ( t ′ )) dx ′ µ dt ′ , where t ∗ ( t, ~x ) < t, ~x ) and the particle tra-jectory ( t − t ∗ ) − ( ~x − ~x ′ ( t ∗ )) = 0 . (4.1)For the synchrotron motion we are interested in x µ = ( t, ρ cos( ωt ) , ρ sin( ωt ))= ( γτ, ρ cos( γωτ ) , ρ sin( γωτ )) , (4.2)the field strengths can be written explicitly as in the ap-pendix. We will start our discussion with the formulae(6.10), (6.11), and (6.12). We are interested in the far-field asymptotics at r → ∞ , where we introduce a polarcoordinate system ( r, φ ) on the spatial two dimensionalplane. The expression for the total far-field radiation is dEdt = lim r →∞ r Z π dφ T i ˆ n i (4.3)= lim r →∞ r Z π dφ F ( F sin φ − F cos φ ) , where ˆ n = (cos φ, sin φ ) is the unit radial vector.To compute the leading large r asymptotics of the fieldstrengths, it is useful to note t ′∗ = − r + O (1) , ∂t ′∗ ∂t = 11 + v sin( ωt ′∗ − φ ) , (4.4) ∂t ′∗ ∂x = − cos φ v sin( ωt ′∗ − φ ) , ∂t ′∗ ∂x = − sin φ v sin( ωt ′∗ − φ ) , in the r → ∞ limit. These quantities appear in thenumerators of the field strength expressions, and one cansee from the structure of the denominators in the abovethat in the limit v →
1, these quantities are highly peakedaround the angle φ c determined by the condition φ c = ωt ′∗ ( φ c ) + π . (4.5)We note that ωt ′∗ ( φ ) is the azimuthal angle of the particleat the intersection of the past light-cone from ( r, φ ). Thiscondition has a simple geometrical meaning: the lightpulse emitted from one moment of the trajectory is highlycollimated in the direction of the instant velocity, andtravels with the speed of light.Below, we will see that the leading γ contribution is in-deed confined in the small angular range δφ ∼ γ around φ c . For that purpose, it will be useful to have the formula ∂t ′∗ ∂φ = ρ sin( ωt ′∗ − φ )1 + v sin( ωt ′∗ − φ ) , (4.6)so that defining a new convenient angular variable a in-stead of φ a ≡ ωt ′∗ ( φ ) − φ + π , (4.7)we have a relation dadφ = −
11 + v sin( ωt ′∗ − φ ) = − − v cos a (4.8)which can be integrated as a − v sin a = − ( φ − φ c ) , (4.9)using the boundary condition (4.5). As φ runs in therange (0 , π ), a also runs one cycle (0 , π ) monotonicallywith a very steep gradient of order O ( γ ) around a smallregion near a = 0 ( φ = φ c ) that can be seen in (4.8).Therefore, the width δa of the collimated light pulse near a = 0 will translate to the width in φδφ ∼ δaγ , (4.10)near the center φ = φ c . We will see shortly that δa ∼ γ leading to δφ ∼ γ .Another important large r expansion is the one for theproper distance,( t ′ ) − ( ~x − ~x ′ ) = ( t ′ ) − r + 2 rρ cos( ωt ′ − φ ) − ρ , that appears in the denominators in the field strengthexpressions. Since the t ′ integration starts from t ′∗ to −∞ , it is more convenient to shift t ′ integration by t ′ → − t + t ′∗ , (4.11)upon which t ∈ (0 , ∞ ), and the proper distance becomes t − t ′∗ t + 2 rρ sin( a − ωt ) − rρ sin a, (4.12)where we used (4.7) and the definition of t ′∗ (4.1). Notethat t ∗ = − r + O (1) in r → ∞ , and one can considertwo parametric regions of the t -integral: (1) t ≪ | t ′∗ | ≈ r and (2) t ≥ | t ′∗ | ≈ r . For (1), one can clearly neglectthe t term in (4.12) and the proper distance is O ( r ).For (2) the proper distance becomes of order O ( r ), andconsidering that this enters the denominators of the fieldstrength expressions, we see that the region (2) of t in-tegral gives us a sub-dominant large r behavior of fieldstrengths. One concludes that the leading large r value offield strengths arises only from the range (1), and there-fore we can neglect the t term in (4.12) when computingleading large r asymptotics, and the proper distance cansimply be replaced by2 r ( t + ρ sin( a − ωt ) − ρ sin a ) , (4.13)for this purpose.It is straightforward to use (6.10), (6.11), and (6.12)to compute the large r asymptotics of the field strengths.After some algebra, one finds that F = ( F sin φ − F cos φ ) , (4.14)at leading order in r → ∞ , and F → e π vω √ r − v cos a Z ∞ dt ( − cos( a − t ) p t + v sin( a − t ) − v sin a (4.15)+ 12 v sin( a − t ) (cos( a − t ) − cos a )( t + v sin( a − t ) − v sin a ) ) , where we have rescaled t by ωt → t . The above expres-sion is a general result for arbitrary velocity v = ρω andthe angle a (or equivalently φ ).Now, we need to find the leading γ behavior of thetotal radiated power (4.3), dEdt = lim r →∞ r Z π da (cid:12)(cid:12)(cid:12)(cid:12) dadφ (cid:12)(cid:12)(cid:12)(cid:12) − | F | , (4.16)where we changed the angle integration from φ to a forconvenience. By careful inspection of the above integral,it can be shown that the leading γ behavior of O ( γ )arises from the narrow range of ( δt, δa ) ∼ γ around( t, a ) = 0, and one can for example expand the denomi-nator as t + v sin( a − t ) − v sin a ≈ (cid:18) γ + a (cid:19) t − at + 16 t , up to the relevant order. One also expands (cid:12)(cid:12)(cid:12)(cid:12) dadφ (cid:12)(cid:12)(cid:12)(cid:12) = 11 − v cos a ≈ γ + a , (4.17) - - a €€€€€€€€€€ dPda FIG. 1: The angular distribution of radiated power in theleading γ approximation. See (4.9) for the relation between a (more precisely aγ ) and φ . as well as the numerator, and by rescaling the integrationvariables ( t, a ) → γ ( t, a ), the leading γ -piece can easilybe shown to reduce to dEdt = e (2 π ) γ ω Z ∞−∞ da
11 + a | f ( a ) | , (4.18)where f ( a ) = ∂ a Z ∞ dt ( a − t ) q (1 + a ) t − at + t . (4.19)The integral Z ∞−∞ da
11 + a | f ( a ) | (4.20)is (2 π ) √ up to 4 digits numerically. Assuming this nu-merical result to be exact, we obtain for the total powerradiated dEdt = e √ π ωγ , (4.21)which agrees precisely with the rate of work done locallyby the “self-force” (2.26) derived in section II.For completeness, in Fig.1, we plot the angular distri-bution of the radiation power, dPda = a | f ( a ) | , as afunction of a (recall that the true a is given by aγ ). Theplot can be translated to the angular distribution in φ using the relation (4.9). V. SUMMARY
We have shown that, unlike in the 3+1 dimensionalspace-time, the self-force can be defined to be nonzeroin odd-dimensional space-times, as the retarded prop-agators in these cases contain a theta-function part orinside-the-light-cone contributions.Furthermore, in the ultrarelativistic case it can beput in local form, depending only on the instantaneousderivatives of the particle motion. We have explicitly de-rived such expressions for the self-force, in 2+1 and 4+1dimensions. In the former case we have also calculatedthe radiation intensity at large distances and checkedthat it matches the work done by the self-force numeri-cally. We expect the same in 4+1 dimensions with mini-mal subtraction.We think it is perhaps the first instance of an entirelylocal and consistent derivation of the radiation brakingforce. Although the MSTQW approach [5, 6] is 15 yearsold and it had inspired our work, we are not aware of itsexplicit tests in the gravitational setting either. Needlessto say, much more work is needed in order to find the ex-act applicability domain of the “self-force” approach, inflat and curved space-times. In [2] we have suggestedthat the MSTQW equation when adapted to thermalAdS may be of relevance to jet quenching in ultrarela-tivistic collisions such as RHIC and LHC.
Acknowledgments.
This work was supported in parts by the US-DOEgrant DE-FG-88ER40388.
VI. APPENDIX: ALTERNATIVEREGULARIZATION OF THE SELF-FORCE IN2+1 DIMENSIONS
In this appendix, we present another way of regular-izing the self-force in 2+1 dimensions, by averaging theLorentz force f µ D = eF µν ( x ( τ )) ˙ x ν ( τ ) , (6.1)around a small circle of radius r , and taking r → x µ ( τ ) = ( γτ, ρ cos( γωτ ) , ρ sin( γωτ )) , (6.2)where v = ρ ω ≈
1, and will find that the leading γ resultof the finite self-force agrees with the one in section II,which is a confirming check for our results.Starting from the expression of retarded gauge poten-tial (2.3), A µ ( x ) = e Z dτ ′ ∆ R ( x − x ( τ ′ )) ˙ x µ ( τ ′ ) , (6.3)with ∆ R ( x ) = θ ( x )2 π θ ( x ) √ x , x ≡ x µ x µ , (6.4) we need to compute field strengths at radius r from theposition of the charge at τ = 0 in the two-dimensionalspatial plane ~x = ( x , x ). Let us show the computation F = ∂ A − ∂ A in some detail to set-up notationsand procedures, and we will present other componentsof F µν at the end without much details. The expression(6.3) gives us A ( t, ~x ) = e π Z t ′∗ ( t,~x ) − Λ dt ′ p ( t − t ′ ) − ( ~x − ~x ′ ) , (6.5)where ~x ′ ≡ ~x ( t ′ ) is the past trajectory (6.2) parame-terized here in terms of regular time t ′ = γτ ′ , and theintegration starts from t ′∗ ( t, ~x ) < t, ~x ) and the particle trajectory,( t − t ′∗ ) = ( ~x − ~x ′ ( t ′∗ )) . (6.6)The expression itself is infrared divergent and we intro-duced a cutoff Λ, but the field strengths which are phys-ical are completely IR finite as one takes Λ → ∞ at theend, as will be clear in the following. To compute ∂ A ,one needs to evaluate the variation of (6.5) with respectto x → x + δx . The variation will shift both the inte-gration range through t ′∗ and the integrand. The formervariation naively gives us the contribution which is pro-portional to the value of the integrand at t ′ = t ′∗ whichhappens to be divergent due to (6.6). The variation of theintegrand also gives us an integral which is divergent near t ′ = t ′∗ . However, since the original (6.5) is completelyfinite near t ′ = t ′∗ these divergences are mere artifacts ofimproper manipulations, and in fact the two divergencescancel with each other. A better way of handling themis the following. From the expression of A at ~x + δ~x , A ( ~x + δ~x ) = e π Z t ′∗ + ∂t ′∗ ∂~x · δ~x − Λ dt ′ p ( t − t ′ ) − ( ~x + δ~x − ~x ′ ) , (6.7)one shifts the t ′ integral by t ′ → t ′ + ∂t ′∗ ∂~x · δ~x so that thenew t ′ integral starts at the same t ′∗ while the integrandgets additional contributions A ( ~x + δ~x ) = (6.8) e π Z t ′∗ − Λ dt ′ q ( t − t ′ − ∂t ′∗ ∂~x · δ~x ) − ( ~x + δ~x − ~x ′ − ∂~x ′ ∂t ′ ∂t ′∗ ∂~x · δ~x ) , which is free from the divergence near t ′ = t ′∗ . There isalso a shift in the IR cutoffΛ → Λ − ∂t ′∗ ∂~x · δ~x , (6.9)but it can be easily shown that it becomes irrelevant inthe final results, and we omit it in (6.8).Taking difference between (6.8) and (6.5) to first orderin δ~x , one readily computes ~∂A as (after putting t = 0) ~∂A = (6.10) − e π Z t ′∗ −∞ dt ′ ∂t ′∗ ∂~x t ′ − ( ~x − ~x ′ ) + ( ~x − ~x ′ ) · ∂~x ′ ∂t ′ ∂t ′∗ ∂~x (( t ′ ) − ( ~x − ~x ′ ) ) , where we removed Λ to infinity as the final integral isfinite.By similar steps, one obtains ∂ ~A = − e π Z t ′∗ −∞ dt ′ ∂ ~x ′ ∂t ′ ∂t ′∗ ∂t p ( t ′ ) − ( ~x − ~x ′ ) (6.11)+ ∂~x ′ ∂t ′ (cid:16) (1 − ∂t ′∗ ∂t ) t ′ − ( ~x − ~x ′ ) · ∂~x ′ ∂t ′ ∂t ′∗ ∂t (cid:17) (( t ′ ) − ( ~x − ~x ′ ) ) ! . The above (6.10) and (6.11) give us the field strength F i = ∂ A i − ∂ i A , i = 1 ,
2. The expressions for ∂t ′∗ ∂t and ∂t ′∗ ∂~x that appear in the above can be easily obtained fromthe relation (6.6). Finally, F = ∂ A − ∂ A is writtenas F = e π Z t ′∗ −∞ dt ′ ǫ ij ∂ ~x ′ i ∂t ′ ∂t ′∗ ∂x j p ( t ′ ) − ( ~x − ~x ′ ) (6.12) − ǫ ij ∂~x ′ i ∂t ′ (cid:16) ∂t ′∗ ∂x j t ′ − ( x j − x ′ j ) + ( ~x − ~x ′ ) · ∂~x ′ ∂t ′ ∂t ′∗ ∂x j (cid:17) (( t ′ ) − ( ~x − ~x ′ ) ) ! , where ǫ = − ǫ = +1. We will apply the above formu-lae to our case of circular motion (6.2).We then consider a small circle of radius r from theposition of the charge at t = 0, that is, ( ρ, φ , so that a pointon the circle has the coordinate ~x = ( ρ + r cos φ, r sin φ ).We will compute the Lorentz force on the points in thecircle and take an average over φ before taking the limit r →
0. Noting the 3-velocity dx µ dτ ≡ u µ = γ (1 , , v ) at t = 0 (recall v = ρω ), let’s first look at the transversecomponent of the self-force, f = eγ ( F − vF ) . (6.13)Near r →
0, the useful expansions of some quantities are t ′∗ = γ xr + O ( r ) , x ≡ − ( v sin φ + p − v cos φ ) ,∂t ′∗ ∂x = cos φx + v sin φ , ∂t ′∗ ∂x = sin φ − γ vxx + v sin φ ,∂t ′∗ ∂t = γ xx + v sin φ , (6.14)and from these, one can derive that the φ -averaged trans-verse force has a divergence near r → f ∼ e γ vω π Z γ xr dt ′ p ( t ′ − γ rx )( t ′ − γ rx ′ ) ∼ e γ vω π log (cid:18) r (cid:19) = − e π log (cid:18) r (cid:19) ¨ x , (6.15) where x ′ = − ( v sin φ − p − v cos φ ) >
0. In derivingthis, we have used an important expansion( t ′ ) − ( ~x − ~x ′ ) ≈ γ ( t ′ − γ rx )( t ′ − γ rx ′ ) , (6.16)near r → t ′ r ∼ O (1) limit, which gives one factorof γ in (6.15). This divergence is precisely of the samecharacter of the mass renormalization we encountered insection II, and one can absorb it by m ren = m bare + e π log (cid:18) r (cid:19) . (6.17)We are more interested in the leading γ behavior oflongitudinal self-force, f = eγF , (6.18)and using (6.14), averaging over φ and taking r → f = − e γ π Z ∞ dt ′ vω sin( ωt ′ ) q ( t ′ ) − ρ sin (cid:0) ωt ′ (cid:1) + ρ (cid:0) − v cos( ωt ′ ) (cid:1) (sin( ωt ′ ) − ωt ′ ) (cid:0) ( t ′ ) − ρ sin (cid:0) ωt ′ (cid:1)(cid:1) ! , (6.19)where we changed the variable t ′ → − t ′ in the integra-tion. The last step is to find a leading γ behavior of(6.19). Rescaling ωt ′ → t , we have f = − e γ vω π Z ∞ dt sin t q t − v sin (cid:0) t (cid:1) + (cid:0) − v cos t (cid:1) (sin t − t ) (cid:0) t − v sin (cid:0) t (cid:1)(cid:1) ! . (6.20)To study the large γ behavior of the integral in (6.20)replacing v = 1 − γ , one writes the integral after somemanipulations Z ∞ dt ∂ t (1 − cos t ) q t − (cid:0) t (cid:1) (6.21)+ 1 γ Z ∞ dt t sin (cid:0) t (cid:1) + (sin t − t ) cos t (cid:16) t − (cid:0) t (cid:1) + γ sin (cid:0) t (cid:1)(cid:17) , neglecting additional O ( γ − ) contributions. The first in-tegral is completely localized at t = 0 giving a value of −√
3. In the second integral, one can show that a leading O (1) result comes from a range of small t ∼ O (cid:16) γ (cid:17) , andone has an approximation in the denominator, t − (cid:18) t (cid:19) + 4 γ sin (cid:18) t (cid:19) ≈ t γ + 112 t , (6.22)where higher order terms can be shown to be irrelevantin the leading γ contributions. The t term in (6.22)effectively cutoffs the t integral to be t ≤ γ , which wehave seen before in section II (recall the proper time is τ = tγ ). The small t expansion for the numerator up torelevant order is4 sin t sin (cid:18) t (cid:19) + (sin t − t ) cos t ≈ t , (6.23)giving us the second integral1 γ Z ∞ dt t (cid:16) t γ + t (cid:17) = 53 √ . (6.24) In total, the integral in (6.20) gives us ( − ) √ √ + O ( γ − ), so that the leading γ result of f is f ∼ − e ωγ √ π ∼ − e ωγ √ π ˙ x , (6.25)which agrees precisely with (2.23) in section II. [1] I. B. Khriplovich, E. V. Shuryak, Lett. Nuovo Cim. ,911-914 (1973). Zh. Eksp. Teor. Fiz. , 2137-2140 (1973).[2] E. Shuryak, H. -U. Yee, I. Zahed, “Jet Quench-ing via Gravitational Radiation in Thermal AdS,”[arXiv:1110.0825 [hep-th]].[3] P.A.M. Dirac, Proc. R. Soc. A167 , 148 (1938).[4] B.S. Dewitt and R.W. Brehme, Ann. Phys. (N.Y.) , 220 (1960).[5] Y. Mino, M. Sasaki, T. Tanaka, Phys. Rev. D55 , 3457-3476 (1997). [gr-qc/9606018].[6] T. C. Quinn, R. M. Wald, Phys. Rev.