12 August 2002

Physics Letters A 300 (2002) 603–610

www.elsevier.com/locate/pla

Radiation bursts from particles in the field of compact,impenetrable, astrophysical objects

G. Papinia,b,∗, G. Scarpettab,c,d, V. Bozzac,d, A. Feolid,e, G. Lambiasec,d

a Department of Physics, University of Regina, Regina, SK, S4S 0A2, Canadab International Institute for Advanced Scientific Studies, 84019 Vietri sul Mare (SA), Italy

c Dipartimento di Fisica “E.R. Caianiello”, Universitá di Salerno, 84081 Baronissi (SA), Italyd INFN, Gruppo collegato di Salerno, Italy

e Facoltá d’Ingegneria, Universitá del Sannio, 82100 Benevento, Italy

Received 12 June 2002; received in revised form 12 June 2002; accepted 27 June 2002

Communicated by P.R. Holland

Abstract

The radiation emitted by charged, scalar particles in a Schwarzschild field with maximal acceleration corrections is calculatedclassically and in the tree approximation of quantum field theory. In both instances the particles emit radiation that hascharacteristics similar to those of gamma-ray bursters. 2002 Elsevier Science B.V. All rights reserved.

PACS: 04.62.+v; 04.70.-s; 04.70.Bw; 98.70.Rz

Keywords: Quantum geometry; Maximal acceleration; General relativity; Gamma-ray bursts

Compact, impenetrable, astrophysical objects (CIAOs) [1–4] arise in a model, proposed by Caianiello andcollaborators [5], in which particle accelerations have an upper limitAm = 2mc3/h, referred to as maximalacceleration (MA). The limit can be derived from quantum mechanical considerations [6,7], is a basic physicalproperty of all massive particles and must therefore be included in the physical laws from the outset. This requiresa modification of the metric structure of space–time.

Classical and quantum arguments supporting the existence of a MA have been discussed in the literature [8–10].MA also appears in the context of Weyl space [11], and of a geometrical analogue of Vigier’s stochastic theory [12]and plays a role in several issues. It is invoked as a tool to rid black hole entropy of ultraviolet divergences [13]and of inconsistencies arising from the application of the point-like concept to relativistic particle [14]. MA maybe also regarded as a regularization procedure [15] that avoids the introduction of a fundamental length [16], thuspreserving the continuity of space–time.

* Corresponding author.E-mail addresses: papini@uregina.ca (G. Papini), scarpetta@sa.infn.it (G. Scarpetta), bozza@sa.infn.it (V. Bozza), feoli@unisannio.it

(A. Feoli), lambiase@sa.infn.it (G. Lambiase).

0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0375-9601(02)00886-1

604 G. Papini et al. / Physics Letters A 300 (2002) 603–610

An upper limit on the acceleration also exists in string theory where Jeans-like instabilities occur [17,18] whenthe acceleration induced by the background gravitational field reaches the critical valueac = λ−1 = (mα)−1 whereλ, m andα−1 are string size, mass and tension. At accelerations larger thanac the string extremities becomecasually disconnected. Frolov and Sanchez [19] have also found that a universal critical acceleration must be ageneral property of strings. It is the same cut-off required by Sanchez in order to regularize the entropy and the freeenergy of quantum strings [20].

Applications of Caianiello’s model include cosmology [21], the dynamics of accelerated strings [22], theenergy spectrum of a uniformly accelerated particle [23], neutrino oscillations [24,25], and photons in a cavityresonator [26]. The model also makes the metric observer-dependent,as conjectured by Gibbons and Hawking [27].

The consequences of the model for the classical electrodynamics of a particle [28], the mass of the Higgsboson [29] and the Lamb shift in hydrogenic atoms [30] have been worked out. In the last instance the agreementbetween experimental data and MA corrections is very good for H and D. For He+ the agreement between theoryand experiment is improved by 50% when MA corrections are considered.

MA effects in muonic atoms appear to be measurable [31]. MA also affects the helicity and chirality ofparticles [32].

More recently, the model has been applied to particles falling in the gravitational field of a spherically symmetriccollapsing object [1]. In this problem MA manifests itself through a spherical shell external to the Schwarzschildhorizon and impenetrable to classical and quantum particles [2]. The shell is not a sheer product of the coordinatesystem, but survives, for instance, in isotropic coordinates. It is also present in the Reissner–Nordström [3] andKerr [4] cases. The model therefore seems to preclude the usual process of formation of black holes. These arereplaced by CIAOs. The purpose of this Letter is to study the emission of radiation by a scalar particles in the fieldof a CIAO.

Caianiello’s model is based on an embedding procedure [1] that stipulates that the line element experienced byan accelerating particle is represented by

(1)dτ2 =(

1+ gµνxµxν

A2m

)gαβ dx

α dxβ ≡ σ 2(x)gαβ dxα dxβ ≡ gαβ dxα dxβ,

wheregαβ is a background gravitational field. The effective space–time geometry given by (1) therefore exhibitsmass-dependent corrections that in general induce curvature and violations of the equivalence principle. Thefour-accelerationxµ = d2xµ/ds2 appearing in (1) is a rigorously covariant quantity only for linear coordinatetransformations. Its transformation properties are however known and allow the exchange of information amongobservers. Lack of covariance forxµ in σ 2(x) is not therefore fatal in the model. The justification for this choicelies primarily with the quantum mechanical derivation of MA which applies toxµ, requires the notion of force,is therefore Newtonian in spirit and is fully compatible with special relativity. The choice ofxµ in (1) is, ofcourse, supported by the weak field approximation togµν which is, to first order, Minkowskian. On the other hand,Einstein’s equivalence principle does not carry through to the quantum level readily [33,34], and the same maybe expected of its consequences, like the principle of general covariance [35]. Complete covariance is, of course,restored in the limith→ 0, whereby all quantum corrections, including those due to MA, vanish.

The corrections to the Schwarzschild field experienced by a particle initially at infinity and falling toward theorigin along a geodesic, require thatσ 2(x) be calculated. From (1) andθ = π/2, one finds (in unitsh= c=G= 1)

(2)

σ 2(r)= 1+ 1

A2m

{− 1

1− 2M/r

(−3ML2

r4 + L2

r3 − M

r2

)2

+(

−4L2

r4+ 4E2M2

r4(1− 2M/r)3

)[E2 −

(1− 2M

r

)(1+ L2

r2

)]},

whereM is the mass of the source, andE andL are the total energy and angular momentum per unit of particlemassm [1].

G. Papini et al. / Physics Letters A 300 (2002) 603–610 605

It is now convenient to introduce the adimensional variableρ = r/M and the parametersε = (MAm)−1 andλ= L/M. The analysis of the motion can be carried out in terms of

(3)V 2eff(ρ)= E2

{1+ 1

σ 2(ρ)

[− 1

σ 2(ρ)+ 1

E2

(1− 2

ρ

)(1+ λ2

ρ2σ 2(ρ)

)]}.

Notice thatV 2eff → E2 asρ → 2 andρ → 0, and thatV 2

eff → 1 asρ → ∞. Plots of (3) for different values ofEshow a characteristic step-like behaviour in the neighborhood ofρ = 2 (see Figs. 1 and 2 in Ref. [1]). An expansion

of (3) in the neighborhood ofρ = 2 yields, in fact,V 2eff ∼ E2 + (ρ−2)4

4ε2E4 +O((ρ − 2)5) which has the minimumE2

on the horizonρ = 2. This term vanishes only in the limitE→ ∞ and/or in the limitε→ ∞, for whichAm orMor both vanish and the problem becomes meaningless.

The addition of MA effects does therefore produce a spherical shell of radius 2< ρ < 2 + η, with η� 1. Theshell is classically impenetrable and remains so at higher orders of approximation [4]. The analogous occurrenceof a classically impenetrable shell was first derived by Gasperini as a consequence of the breaking of the localSO(3,1) symmetry [36]. A classically impenetrable shell and shift in horizon also occur in the problem of particlesin hyperbolic motion in a Kruskal plane [24].

The existence of large accelerations in proximity of the shell suggests the possibility of radiation of photons (andgravitons) by bremsstrahlung. It has become recently clear that production of electromagnetic radiation is possibleeven when the acceleration is produced by a gravitational field [37–42]. Two approaches are followed here.

(i) Generalization of Larmor’s formula. A covariant generalization of this formula would replacexµ with itscovariant counterpart which vanishes rigorously along a geodesic in general relativity. It is of course know that evenin Einstein’s theory a charged particle, while trying to adhere to the equivalence principle, does not really followa geodesic in a gravitational field. Deviations necessarily develop an account of damping and of thetail effectdiscussed by De Witt and Brehme [43]. In Caianiello’s model, not only is the equivalence principle violated, butthe MA corrections also introduce deviations from geodetic motion. These are introduced byσ 2(x) which in firstapproximation is given by (2). From the definition (1) ofσ 2(x) one findsgµνxµxν = (σ 2 − 1)A2

m and Larmor’sformula, which applies to particles of arbitrary trajectory, therefore becomes

(4)P −2q2

3A2m

(σ 2 − 1

).

When the MA corrections vanish,σ → 1 andP → 0. Eq. (4) therefore seems suitable to deal with radiation dueto MA. On the basis of (4) and (2) one may expect radiation to be particularly intense whenρ is close to 2, whereσ 2(x) has a singularity. One finds in fact

(5)

P −2q2

3A2m

ε2

ρ7(ρ − 2)3

{λ4(ρ − 2)2

(7− 10ρ + 3ρ2)

+ ρ4[−4+ 4(1+ 2E2)ρ + ρ2(−1− 4E2 + 4E4)]

− 2λ2ρ2(ρ − 2)[10+ (−19+ 10E2)ρ + (

11− 8E2)ρ2 + 2(E2 − 1

)ρ3]}.

However, the particle cannot reach the valueρ = 2 because of its acceleration and ensuing impenetrable shell. Theshell’s radius corresponds to a maximum ofV 2

eff and to the smallest value ofρ the particle can reach. By expandingV 2

eff as a series in the neighborhood ofρ = 2 and keeping terms to sixth order, one finds

(6)

V 2eff ∼

{256E2(ρ − 2)4ε2 + 5

(4+ λ2)2

(ρ − 2)6ε2 + 1024E10ε4

− 16E2(ρ − 2)5[−8

(4+ 2ε2 + ε2λ2) + (

16+ 4ε2 + 3ε2λ2)ρ]} 1

1024E8ε4,

606 G. Papini et al. / Physics Letters A 300 (2002) 603–610

whose maximum is at

ρ1 ={2(768E2 − 240ε2 + 272E2ε2 − 120λ2ε2 + 164E2λ2ε2 − 15λ4ε2)

+ 4E2ε

√6144E2 − 1520ε2 + 1536E2ε2 − 760λ2ε2 + 1152E2λ2ε2 − 95λ4ε2

}(7)× {

3(256E2 − 80ε2 + 64E2ε2 − 40λ2ε2 + 48E2λ2ε2 − 5λ4ε2)}−1

.

The distance of closest approachρ1 is real for

E2 � ε2(1520+ 760λ2 + 95λ4)(6144+ 153ε2 + 1152λ2ε2)−1,

and always positive for small values ofλ (radial infall) andε which are appropriate in many instances. It is usefulto recall at this point the relationship between the adimensional parametersM, E, λ, ε, τ (adimensional time) andthe corresponding (primed) CGS quantities. One has

M = GM ′

c2 , E = E′

m′c2 , λ= L′cGM ′m′ , ε = hc

2GM ′m′ , τ = c3t ′

GM ′ .

If M ′ ∼ 3M� and the falling particle is a proton of energyE′ ∼ 10 MeV, one findsε ∼ 4.7× 10−20 andE ∼ 0.01.For these values of the parameters 2− ρ1 9.7× 10−16.

An expression that givesP as a function ofM, E, λ andε can be obtained by substituting (7) into (5). Itsusefulness is however limited because the parameters involved cannot yet be linked directly to the particle’s

motion in the new fieldgµν . This can be achieved by writingdrds

= drdx0

dx0

ds, with dx0

ds= Eg00

= Eσ2g00

, and using

( drds)2 = E2 − V 2

eff. One gets

(8)

(dr

dx0

)2

= E2 − V 2eff

E2σ 4

(1− 2M

r

)2

.

The desired functionr = r(x0) is obtained by integrating (8) with respect to time. In adimensional variables, onefinds

(9)E

∫ρ dρ

σ 2(ρ)(ρ − 2)√E2 − V 2

eff(ρ)

= −τ + a,

where the negative sign in front ofτ accounts for the fact that the particle approaches the gravitational source asτ

increases. The constanta must be determined by appropriate initial conditions.By assuming that the motion is radial (λ= 0) and for the values ofE andε given above, the l.h.s. of (9) can be

integrated in the neighborhood ofρ1 and yields

(10)2√

2 ln(ρ − ρ1)∼ −τ + a.By requiring thatρ 2.1 at t = 0, one findsa = −1.3272 and (10) becomes

(11)ρ ρ1 + e−(τ+6.5127)/2√

2).

The particle therefore reaches a distance within a 1% ofρ1, in the timet ′ � 4.8 × 10−5 s. However, radiationbecomes particularly intense whent ′ � 1.68× 10−3. If (11) is then substituted into (5), one obtains

(12)|P | ∼ 2q2A2m

3GM ′[2.67× 1042 + 3.85× 1047(t ′ − 1.68× 10−3) + 3.78× 1052(t ′ − 1.68× 10−3)2

].

Radiation becomes appreciable∼1030 erg/s only att ′ � 5×10−4 s and increases very rapidly witht ′. As shown inFig. 1, it already isP ∼ 1051 erg/s att ′ ∼ 10−3 s. Because of the quadratic dependence in both particle charge andmass, small clumps of matter radiate significantly more and at higher frequencies even a certain distance fromρ1.

G. Papini et al. / Physics Letters A 300 (2002) 603–610 607

Fig. 1. Power emitted as a function of time forM ′ = 3M� , E′ = 10 MeV,m′ = 1.7× 10−24 g.

(ii) Tree-level calculations. The second way of estimating the power produced follows the procedure of Crispinoet al. [41]. It treats the electromagnetic field as a scalar field, whose source is the current produced by a particle, alsoa scalar. Quantum field theory at the tree level is then used to describe the electromagnetic field in a Schwarzschildbackground. The method is particularly interesting in the context of Caianiello’s model because it enables acomparison between the massive scalar particle that produces the current and experiences the effective geometryσ 2gµν , and the massless photon that cannot be accelerated and therefore “sees” only a Schwarzschild geometry.The source is the scalar current

(13)j (x)= q√−g u0δ(r − rS)δ(θ − π/2)δ(φ),

normalized by the requirement that∫dσV j (x) = q , wheredσV is the proper 3-volume element orthogonal to

uµ, the four-velocity of the source. The four-velocity componentu0 = Eg00 = Eσ2g00

takes into account the MAcorrections to the motion of the scalar particle, whereas the 3-volume integration is in Schwarzschild space, hencethe

√−g in (13). The scalar electromagnetic field generated by (13) is, at the tree level,

(14)Aωlp = i∫d4x

√−g j (x)u∗ωlp,

where the functions

(15)uωlp(x)=√ω

πRωl(r)Ylp(θ,φ)e

−iωt ,

represent the positive-frequency solutions (ω > 0) in Schwarzschild space–time of the equation∇µ∇µuωlp = 0.The functionsRωl satisfy the equation

(16)e2λR′′ωl +

(2

r+ λ′

)e2λR′

ωl +[ω2 − eλ l(l + 1)

r2

]Rωl = 0,

whereeλ = 1− 2M/r and the primes denote differentiation with respect tor [2]. The emitted power is then [41]

(17)P =∞∫

0

dωω|Aωlp|2T

,

whereT is the time during which the interaction is switched on [44]. Introducing (13) into (14) and using thecoordinateρ − 2 = x, one finds, to leading order,

(18)Aωlp De6.5127/√

2

2M√ω

e(iMω+1/√

2)τ − 1

iMω+ 1√2

,

608 G. Papini et al. / Physics Letters A 300 (2002) 603–610

Fig. 2. Power emitted as a function of time according to Eq. (20). The values ofM ′, E′ ,m′ are as in Fig. 1.

where

D ≡ i 4qE3

A2mM

√πYlp(π/2,0)

(1+ Rωl

).

The contribution of the term−1 on the r.h.s. of (18) is negligible relative to the exponential. The reflectioncoefficientRωl < 1 is also neglected.

Eqs. (17) and (18) give the radiated power as

(19)|P | q2E6e6.512

πA4mM

6

e√

2τ

τ,

where the timeT =Mτ ′ in (17) andτ ′ is the time required for the particle to reach the intense radiation zone ofits trajectory. Though this result vanishes ash→ 0, the correct limit for vanishing MA contributions follows bytaking the limitσ → 1 in (13) and is finite. In CGS units, the power becomes

(20)|P | q2E′6h4e8.21

16πm′G5M ′5e√

2c3t ′/GM ′

t ′ 2.35× 10−110e

1.9×105t ′

t ′.

Despite the enormously small reducing factor in (20), a single proton reaches powers of∼1030 erg/s after onlyt ′ ∼ 2 × 10−3 s, andP ∼ 1051 erg/s is reached fort ′ ∼ 2.3 × 10−3 s, in agreement with the results of (12). Thefrequency distribution ofP is given by

(21)dP

dω |D|2e9.121

4M3

e√

2τ

(M2ω2 + 12)τ,

or, in CGS units, by

(22)dP

dω 104

4π2

h4E′6q2

G4M ′4m′10c8

e√

2c3t ′/GM ′

(G2M ′2ω2

c6+ 1

2)t′ .

It can be seen from Figs. 2 and 3 that the bursts of radiation are sudden and peaked at lower frequencies, thoughthe intensity of radiation also remains high atγ -ray frequencies. Some similarities withγ -ray busters are obvious.The latter have extraordinary large energy outputs∼1051 to ∼1054 erg/s. Their spectra are non-thermal withphoton energies ranging from 50 keV to the GeV region. In CIAO’s case spectra also are non-thermal, in facttypically of the bremsstrahlung type, with intense radiation that extends to the same frequency range of bursters.Significant amounts of low frequency radiation could be absorbed by the progenitor, if this is a black hole, in thecase of busters. Black holes are known to have sizeable absorption for infrared photons [41]. For CIAOs absorptioncould also take place at the shell which, for photons, behaves like a horizon. Nonetheless energy extraction from

G. Papini et al. / Physics Letters A 300 (2002) 603–610 609

Fig. 3. Power spectrum in theγ -ray frequency region and the time interval 2.1 to 2.3 ms. The values ofM ′ , E′ ,m′ are as in Fig. 1.

CIAOs remains extremely efficient. The duration of the intense emission by particles in their proximity is of thesame order of magnitude, 1 ms to 103 s, of the events that are observed. Finally, the lower frequency afterglowsthat are sometimes observed to last several months, may be explained, in CIAO’s scenario, as due to a dip in theeffective potential experienced by scalar particles at higher accelerations [2]. The material trapped in this way maysubsequently leave the relatively shallow attractive area to resume the acceleration and radiation processes at lowerintensities and frequencies.

Acknowledgements

Research supported by MURST PRIN 2001 and by the Natural Sciences and Engineering Research Council ofCanada.

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