Astroparticle Physics
Volume 23, Issue 1, February 2005, Pages 117-129

Tachyonic synchrotron radiation from γ-ray pulsars

Roman TomaschitzCorresponding Author Contact Information, E-mail The Corresponding Author

Department of Physics, Hiroshima University, 1-3-1 Kagami-yama, Higashi-Hiroshima 739-8526, Japan

Received 28 July 2004; 
accepted 25 November 2004. 
Available online 9 December 2004.


Superluminal radiation emitted by electrons orbiting in strong magnetic fields is investigated. We show that electrons gyrating in the surface fields of rotation-powered neutron stars can radiate superluminal quanta (tachyons) via the synchrotron mechanism. The tachyonic luminosity of γ-ray pulsars is inferred from COMPTEL and EGRET observations, and so is the magnetospheric electron population generating this radiation. In the surface fields, electromagnetic synchrotron radiation in the γ-ray band is suppressed by a quantum cutoff, but not so tachyonic γ-radiation. This provides an exceptional opportunity to search for tachyon radiation, unspoiled by electromagnetic emission. Estimates of the superluminal power radiated and the tachyonic count rates are obtained for each of the seven established γ-ray pulsars, the Crab and Vela pulsars, as well as PSR B1706−44, Geminga, PSR B1055−52, B1951+32, and B1509−58. Detection mechanisms such as tachyonic ionization and Compton scattering are analyzed with regard to superluminal γ-rays.

Keywords: Superluminal curvature radiation; Tachyonic γ-rays; Longitudinal γ-radiation; Pulsar magnetospheres

PACS classification codes: 95.30.Gv; 41.60.Ap; 97.60.Gb

Article Outline

1. Introduction
2. Tachyonic synchrotron radiation in strong magnetic fields
3. Tachyonic γ-rays from pulsar magnetospheres
4. Conclusion

1. Introduction

When considering superluminal quanta, we may try a wave theory or a particle picture as starting point. The latter has been studied for quite some time, but did not result in viable interactions with matter [1], [2], [3], [4], [5] and [6]. Here, tachyons will be modeled as wave fields with negative mass-square, coupled by minimal substitution to subluminal particles. Interaction with matter is indeed the crucial point, we will maintain the best established interaction mechanism, minimal substitution, by treating tachyons like photons with negative mass square, a real Proca field minimally coupled to subluminal matter [7] and [8].

We will work out a specific example, tachyonic synchrotron emission from electrons gyrating in the surface fields of γ-ray pulsars. The detectors used to collect the γ-ray fluxes operated in adjacent or partly overlapping energy bands with very different mechanisms, ionization [9], Compton scattering [10], and electron-positron pair creation [11], and we will discuss the respective cross-sections with regard to tachyonic γ-rays. We will calculate the energy stored in the electron population gyrating in the surface magnetic fields, and relate the slope of the electronic power-law distribution to the frequency scaling of the tachyon flux. We will conclude from the break frequencies of the observed fluxes that the gyration energies can reach the low TeV range, and comment on cosmic ray acceleration in the magnetospheres.

In Section 2, we study superluminal synchrotron radiation from electrons subjected to high magnetic field strengths. We will introduce the tachyonic counterparts of certain quantities familiar from electromagnetic synchrotron radiation, such as critical frequencies, break frequencies and critical field strengths, and explain how they relate to the spectral densities, the tachyon mass, and the electronic Lorentz factor. In Section 3, we study the synchrotron power radiated in tachyonic γ-rays by the magnetospheric electron populations of γ-ray pulsars. We calculate the synchrotron power in the high-magnetic-field limit as well as the tachyonic count rates and average the tachyonic spectral densities with electronic power-law distributions. The tachyonic γ-ray luminosity is inferred from the measured fluxes and all that goes with it, such as the electronic power-law indices and the energy range of the gyrating electrons. In Section 4, we present our conclusions.

2. Tachyonic synchrotron radiation in strong magnetic fields

The superluminal spectral densities can be readily assembled from the general formalism of tachyonic radiation theory [12] and [13], which will not be repeated here, but we restate the densities in Figs. (2.1), (2.2), (2.3), (2.4), (2.5), (2.6), (2.7) and (2.8). There is a low-and a high-frequency regime, separated by the break frequency ωb = γmtc, where γ much greater-than 1 is the Lorentz factor of the radiating ultra-relativistic source, typically a circularly orbiting electron, and mt is the tachyon mass. The latter has the dimension of an inverse length, being a shortcut for mtc/planck constant over two pi. The subsequent spectral densities only apply in the range ω less-than-or-equals, slant ωb. There is also high-frequency radiation beyond this break frequency, but this needs second quantization, as the high magnetic field strengths considered here lead to an exponential cutoff of the classical radiation. In the low-frequency regime, however, quantum effects are perturbative, cf. after (2.10).

The transversal radiation can be split into two linear polarization components,

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The lower plus-sign in (2.2) refers to the perpendicular-polarization, and q is the tachyonic charge of the electron. The tachyon mass, mt ≈ 2.15 keV/c2, and the ratio q2/e2 ≈ 1.4 × 10−11 of tachyonic and electric fine structure constants can be inferred from Lamb shifts in hydrogenic systems [7]. The spectral functions can be traced back to Airy functions,

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where the argument ξ is a shortcut for

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and ωb = eB/(γmc) is the gyrofrequency, cf. after (2.9). As mentioned, we restrict to frequencies ω less-than-or-equals, slant ωb. The parameter κ will turn out to be the expansion parameter for the power radiated, cf. Section 3. We will consider the κ → 0 asymptotics, attained in high magnetic fields. The spectral density of the longitudinal radiation reads,

View the MathML source
with the spectral function L0(ξ) defined in Figs. (2.3) and (2.4). The ascending series and asymptotic limits of F0, G0 and L0 can be found in Ref. [13].

We split the densities Figs. (2.2) and (2.5) into a linear and a curvature component,

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where the densities pT,lin and pL,lin stand for the tachyon radiation generated by a charge in linear uniform motion (in the ultra-relativistic limit),

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The curvature radiation subtracted in (2.6) can be read off from Figs. (2.2) and (2.5),

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where the upper minus-sign refers to View the MathML source. The Lorentz factor of the radiating charge enters the transversal linear density pT,lin via ωb. It also enters in pL,lin, again by ωb, which is the cutoff frequency for uniform motion; a uniformly moving ultra-relativistic charge can only radiate frequencies ω less-than-or-equals, slant ωb, cf. Ref. [12]. The three densities View the MathML source in (2.6) are safely positive definite, but not so the terms View the MathML source and pL,curv in (2.8), which are generated by the orbital curvature and oscillate for large ξ.

We consider κ much less-than 1, cf. (2.4), but still large enough that κγ2 much greater-than 1. Since γ much greater-than 1 in the ultra-relativistic limit, we can assume View the MathML source, so that the mass term in the denominator of the spectral densities can be dropped for frequencies View the MathML source. Moreover, 0 less-than-or-equals, slant ξ(ωless-than-or-equals, slant 1 in the interval View the MathML source, cf. (2.4), so that View the MathML source and ξ(ωb) = 0. Within this interval, the spectral functions ξ2/3F0, ξ2/3G0 and L0 are of O(1) . Outside this range, we find ξ much greater-than 1 for View the MathML source, so that the spectral functions can be replaced by their asymptotic limits. There are three frequencies helpful to understand the qualitative behavior of the spectral densities,

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where the three proportionality constants pertinent to the indicated choices of ω are of the same order. The densities decay ∝ ω for ω much less-than mtc. The longitudinal density attains its maximum at ω ≈ mtc. The transversal density also has its maximum at ω ≈ mtc, followed by a minimum at about View the MathML source. The spectral functions (2.3) are composed of (anti-) derivatives of Airy functions, and admit straightforward analytic continuation [13]. The argument ξ(ω) in the spectral functions can still be used in the high-frequency regime, ω greater-or-equal, slanted ωb, if we replace the parentheses in (2.4) by an absolute value. We then find ξ(ωb/κ) = 1, where the ratio ωb/κ is independent of the tachyon mass and coincides with the critical photon frequency, ωc = (3/2)ωBγ3. At about this frequency, the analytically continued transversal spectral density attains a second maximum, like the electromagnetic counterpart, before the exponential decay sets in. In actual fact, the classical densities are exponentially cut off before this high-frequency maximum is reached. This is a quantum effect arising in the κ much less-than 1 limit, due to the high magnetic field strength.

We shortly summarize the notation and the constants. Gyroradius and gyrofrequency relate as R ≈ c/ωB, where ωB = eB/(γmc), and 1G · e ≈ 2.998 × 10−7 GeV cm−1. We write γ = E/mc2, so that eB ≈ E/R, where E and m denote energy and mass of the gyrating electron or positron, other possible source particles like protons and heavier nuclei will not be considered. We use the Heaviside–Lorentz system, so that e2/(4πplanck constant over two pic) =: αe ≈ 1/137 and q2/(4πplanck constant over two pic) =: αq ≈ 1.0 × 10−13 are the electric and tachyonic fine structure constants. We restore the mass unit in the above densities, mt → mtc/planck constant over two pi ≈ 1.09 × 108 cm−1, so that the break frequency reads ωb = γmtc2/planck constant over two pi. Finally, αq/αe ≈ 1.4 × 10−11 and mt ≈ m/238 ≈ 2.15 keV/c2.

When calculating the power radiated, cf. Section 3, we will use κ in (2.4) as expansion parameter,

View the MathML source
In the surface magnetic fields of γ-ray pulsars, κ much less-than 1 as well as κγ2 much greater-than 1 usually apply, cf. Table 2. The opposite limit, κ much greater-than 1, is realized in planetary magnetospheres and supernova remnants [13]. The surface fields are quite comparable to the critical field, Bc = m2c3/(eplanck constant over two pi) ≈ 4.414 × 1013 G, cf. Table 1. The gyrofrequency and the ratio of critical and break energy relate to the magnetic field ratio as

View the MathML source
This notation is kept close to electromagnetic synchrotron radiation [14], [15], [16] and [17], as is the whole formalism, of course. However, the spectral densities (2.2) applicable below the break frequency do not have an electromagnetic counterpart, only the upper frequency range has, but the high-frequency radiation requires second quantization in strong fields. Quantum corrections [18] are negligible for frequencies much smaller than the electron energy, which is the case below the break frequency. In the opposite limit, planck constant over two piω/E much greater-than 1, the classical spectral peak near the critical frequency is wiped out by an exponential cutoff, as is the electromagnetic synchrotron radiation [19]. Quantum effects also lead to an attenuation of the classical radiation in the cyclotron limit, for slowly gyrating source particles. The quantization of tachyonic cyclotron and synchrotron radiation will be discussed elsewhere.

Table 2.

Setting the scales: break energy, critical energy and electronic Lorentz factor determining the shape of the tachyonic spectral densities, cf. (2.9)

κ κγ κγ2 εc/E κlc D (kpc)
Crab 1.67 × 10−5−k 0.0326 6.37 × 101+k 2.5 × 102+k 6.7 × 101−k 2.0
Vela 1.86 × 10−5−k 0.0365 7.12 × 101+k 2.3 × 102+k 1.4 × 103−k 0.3
B1706−44 2.04 × 10−5−k 0.0400 7.81 × 101+k 2.1 × 102+k 2.3 × 103−k 1.82
Geminga 3.96 × 10−5−k 0.0775 1.51 × 102+k 1.1 × 102+k 5.75 × 104−k 0.16
B1055−52 5.75 × 10−5−k 0.113 2.20 × 102+k 7.3 × 101+k 4.9 × 104−k 1.53
B1951+32 1.29 × 10−4−k 0.253 4.94 × 102+k 3.3 × 101+k 8.7 × 102−k 2.50
B1509−58 4.08 × 10−6−k 8.00 × 10−3 1.56 × 101+k 1.0 × 103+k 1.5 × 103−k 4.3

Input parameters as in Table 1. The expansion parameter for the tachyonic powers and count rates is the ratio εb/εc of break and critical energy, denoted by κ, cf. Figs. (2.4) and (2.10). The κ-expansions in Section 3 apply for κ much less-than 1 and κγ2 much greater-than 1, which defines constraints on the electronic Lorentz factor, γ ≈ 1.96 × 103+k. The product κγ occurring in the averaged tachyonic spectral densities (3.12) is independent of the electron energy. A large ratio εc/E of critical and electron energy indicates that electromagnetic synchrotron radiation and the resulting radiation damping is suppressed by a quantum cutoff. κlc is the expansion parameter for gyration in the light cylinder fields, cf. Table 1. For κlc to be small, very high gyration and TeV γ-ray energies are required, which are not observed in the pulsed emission. These light cylinder fields and the surface fields of millisecond pulsars exemplify the crossover between the asymptotic regimes studied here and in Ref. [13]. The pulsar distance D(kpc) is taken from Ref. [26].

Table 1.

Entries as defined in Section 2: electron energy E(GeV) = 10k (input), surface field B (input), gyroradius R, tachyonic break energy εb(keV) ≈ 4.2 × 103+k, critical energy εc

B (1012G) R (cm) εc (GeV) εmin (keV) P (ms) Blc (G)
Crab 3.8 8.8 × 10−7+k 2.5 × 102k+2 1.7 × 10k/2+1 33.40 9.4 × 105
Vela 3.4 9.8 × 10−7+k 2.3 × 102k+2 1.8 × 10k/2+1 89.29 4.4 × 104
B1706−44 3.1 1.1 × 10−6+k 2.1 × 102k+2 1.9 × 10k/2+1 102.45 2.7 × 104
Geminga 1.6 2.1 × 10−6+k 1.1 × 102k+2 2.65 × 10k/2+1 237.09 1.1 × 103
B1055−52 1.1 3.0 × 10−6+k 7.3 × 102k+1 3.2 × 10k/2+1 197.10 1.3 × 103
B1951+32 0.49 6.8 × 10−6+k 3.3 × 102k+1 4.8 × 10k/2+1 39.53 7.3 × 104
B1509−58 15.5 2.15 × 10−7+k 1.0 × 102k+3 8.5 × 10k/2 150.66 4.2 × 104

One may envisage −2 < k < 4 as typical range for the continuous parameter k labeling the electron energy. The transversal and longitudinal spectral peaks are located at the tachyon mass (2.15 keV), the minimum of the transversal tachyonic energy density is at εmin. The surface fields B and periods P are taken from Ref. [26], and Blc is the field strength at the light cylinder. With one exception, the listed pulsars are detected in hard γ-rays, with power-law spectra extending beyond 1 GeV: Crab (PSR B0531+21), Vela (PSR B0833−45), PSR B1706−44, Geminga (PSR B0633+17), PSR B1055−52, and PSR B1951+32. The γ-ray spectrum of the high-magnetic-field pulsar PSR B1509−58 is soft, cut off at about 10 MeV. The γ-ray fluxes of these pulsars are quoted in Section 3.

To get an overview regarding orders of magnitude, we define the shortcuts E0 = E[GeV] and B0 = B[G], and write the preceding formulas as scaling relations in these dimensionless numbers. For instance, View the MathML source and ωB[GHz] ≈ 29.98/R[cm]. Lorentz factor and expansion parameter scale as γ ≈ 1957E0 and View the MathML source, respectively. We also note View the MathML source and κγ2 ≈ 2.42 × 1014E0/B0. The energies attached to break and critical frequency and to the transversal spectral minimum as defined after (2.9), εb,c,min = planck constant over two piωb,c,min, scale as

View the MathML source
Some other numerical relations are needed to connect the foregoing to synchrotron fluxes from pulsars, cf. Section 3. The surface magnetic fields are inferred from the period and period derivative via View the MathML source. The light cylinder field is obtained by a rescaling of the surface field, Blc = (Rns/Rlc)3B. Here, Rns ≈ 106 cm and Rlc = cP/(2π) are the neutron star and light cylinder radii, respectively, so that Rlc/Rns ≈ 4.77P [ms]. Some of these quantities are listed in Table 1 and Table 2. There is a voltage gap, ΔU = BlcRlc, or ΔU[V] ≈ 1.43 × 109Blc[G]P[ms], between surface and light cylinder. This potential drop is an order of magnitude estimate, essentially on dimensional grounds (1G ≈ 299.8 V/cm), which may even be substantially reduced by pair production. ΔU is of the order of 1014–1016 V for the known γ-ray pulsars [20], [21] and [22]. This gap is not related to synchrotron radiation, but there may be an important implication for cosmic ray acceleration [23]. Electrons cycling in the surface fields can acquire energies in the low TeV range, cf. Section 3. When subjected to a voltage of the indicated magnitude or even a few orders less, they are accelerated into the 1021 eV region, unless slowed down by electromagnetic radiation loss. In the surface field, electromagnetic synchrotron radiation is suppressed by a quantum cutoff, which annihilates the classical radiation peak [19]. However, tachyonic synchrotron radiation below the break frequency (vanishing in the zero-mass limit) is not affected by this cutoff. When the synchrotron electrons or any other charged particles spiral toward the light cylinder, driven by the electric voltage, the synchrotron radii increase, but the electromagnetic radiation loss may still be much smaller than the classical estimates on curvature radiation along the magnetospheric electric field lines would suggest. There is no radiation damping by tachyon radiation, as the Green function outside the light cone is time symmetric and the radiated energy is drained from the absorber field [8] and [12].

3. Tachyonic γ-rays from pulsar magnetospheres

We will study the tachyonic synchrotron power radiated by electrons gyrating in the surface magnetic fields of γ-ray pulsars, based on the spectral densities Figs. (2.1) and (2.5). The parameters of the individual pulsars are listed in Table 1 and Table 2. The radiated power and the tachyonic count rates calculated in Figs. (3.1), (3.2), (3.3), (3.4), (3.5), (3.6), (3.7), (3.8), (3.9) and (3.10) are stated in Table 3 for each of the known γ-ray pulsars, parametrized by the gyration energy. In Figs. (3.11), (3.12) and (3.13), we will average the tachyonic spectral densities with an electronic power-law distribution, and then proceed with a phenomenological discussion of the observed γ-ray fluxes, including the detection mechanisms of the γ-ray counters.

Table 3.

Superluminal power and tachyonic count rates

P(GeV s−1) = PT + PL N(105 s−1) = NT + NL
Crab PT ≈ 3.2 × 102k/3+2 + 1.1k + 2.3 NT ≈ 5.1 + 3.05 × 10k/3 − 8.5 × 10k/2−1
PL ≈ 3.1 + 1.1k − 4.8 × 10−2k/3−4 NL ≈ 5.1 − 1.2 × 10k/2−1 + 1.1 × 10k−3

Vela PT ≈ 3.0 × 102k/3+2 + 1.1k + 2.3 NT ≈ 5.1 + 2.8 × 10k/3 − 8.1 × 10k/2−1
PL ≈ 3.15 + 1.1k − 5.1 × 10−2k/3−4 NL ≈ 5.1 − 1.15 × 10k/2−1 + 1.1 × 10k−3

B1706−44 PT ≈ 2.8 × 102k/3+2 + 1.1k + 2.4 NT ≈ 5.1 + 2.7 × 10k/3 − 7.7 × 10k/2−1
PL ≈ 3.2 + 1.1k − 5.45 × 10−2k/3−4 NL ≈ 5.1 − 1.1 × 10k/2−1 + 1.1 × 10k−3

Geminga PT ≈ 1.8 × 102k/3+2 + 1.1k + 2.5 NT ≈ 5.1 + 1.7 × 10k/3−5.5 × 10k/2−1
PL ≈ 3.3 + 1.1k − 8.5 × 10−2k/3−4 NL ≈ 5.1 − 7.9 × 10k/2−2 + 1.1 × 10k−3

B1055−52 PT ≈ 1.4 × 102k/3+2 + 1.1k + 2.6 NT ≈ 5.1 + 1.3 × 10k/3 − 4.6 × 10k/2−1
PL ≈ 3.4 + 1.1k − 1.1 × 10−2k/3−3 NL ≈ 5.1 − 6.5 × 10k/2−2 + 1.1 × 10k−3

B1951+32 PT ≈ 8.2 × 102k/3+1 + 1.1k + 2.8 NT ≈ 5.1 + 7.8 × 10k/3−1 − 3.1 × 10k/2−1
PL ≈ 3.6 + 1.1k − 1.9 × 10−2k/3−3 NL ≈ 5.1 − 4.4 × 10k/2−2 + 1.1 × 10k−3

B1509−58 PT ≈ 8.2 × 102k/3+2 + 1.1k + 2.0 NT ≈ 5.1 + 7.8 × 10k/3 − 1.7 × 10k/2
PL ≈ 2.8 + 1.1k − 1.9 × 10−2k/3−4 NL ≈ 5.1 − 2.45 × 10k/2−1 + 1.1 × 10k−3

Input parameters as in Table 1 and Table 2, the parameter k refers to the electron energy, E = 10k GeV. The transversal (unpolarized) and longitudinal powers PT,L(GeV s−1) include linear and curvature radiation, cf. Section 3. The corresponding tachyonic count rates are denoted by NT,L(105 s−1). For comparison, the superluminal power transversally and longitudinally radiated by an electron in straight uniform motion with the same Lorentz factor reads PT,lin(GeV s−1) ≈ 1.6k + 4.95 and PL,lin(GeV s−1) ≈ 1.6k + 5.3, respectively [13]. The curvature radiation is obtained by subtracting these linear powers from PT,L. The linear count rates NT,L,lin approach a constant value of 5.1 × 105 s−1 in the ultra-relativistic limit.

We start with the integration of the transversal spectral density (2.1). The power radiated by a single gyrating electron can be split into linear polarization components,

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Alternatively, we may decompose PT into a linear and a curvature component according to Figs. (2.6), (2.7) and (2.8),

View the MathML source

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so that View the MathML source. The power stemming from the linear transversal density (2.7) is readily calculated,

View the MathML source
which is the leading order in the ultra-relativistic 1/γ-expansion of the first integral in (3.2), cf. Ref. [12]. We have restored the natural units, mt → mtc/planck constant over two pi; the tachyonic fine structure constant αq is defined before (2.10). The second integral in (3.2) gives the transversal curvature radiation,

View the MathML source
The lower plus-sign in (3.4) refers to the perpendicular-polarization, and PT,curv is the total transversally polarized curvature radiation, cf. (3.2). The expansion parameter κ, the ratio of break and critical frequency, is defined in (2.10), and γE ≈ 0.5772.

The longitudinal radiant power can likewise be decomposed into a linear and a curvature term, cf. (2.6),

View the MathML source

View the MathML source
An elementary integration gives the linear power,

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up to O(γ−1) like in (3.3). The κ-expansion of the longitudinal curvature radiation reads,

View the MathML source
The total power radiated is P = PT+PL, with the polarized powers PT,L, cf. Figs. (3.2) and (3.5).

The count rates are composed like in Figs. (3.2) and (3.5), with the spectral densities divided by planck constant over two piω; we write the transversal count as NT = NT,lin − NT,curv, where View the MathML source. The linear count is constant in leading order, in the ultra-relativistic regime that is,

View the MathML source
The contribution of the orbital curvature to the transversal count is

View the MathML source

View the MathML source
where we used αqmtc2/planck constant over two pi ≈ 3.27 × 105 s−1, cf. before (2.10). The longitudinal count rate, NL = NL,lin − NL,curv, is composed of the linear count, NL,lin not, vert, similar NT,lin, and the curvature term

View the MathML source

View the MathML source
In Table 2, the expansion parameter κ is listed for the individual pulsars, parametrized by the electronic gyration energy. In Table 3, we list the above expansions parametrized accordingly, which gives an overview as to how the magnetic field strengths and electron energies impact the weight of the linear and curvature terms. The range of electron energies (k-interval) for these expansions to apply can be read off from Table 2 for each pulsar.

To connect to γ-ray fluxes from magnetospheric electron populations, we need to know the tachyonic spectral densities generated by a non-singular electron distribution. To this end, we average the densities Figs. (2.2) and (2.5) with an electronic power-law, dn(γ) = s dγ, of index s > 1. The ultra-relativistic electronic Lorentz factors range in a finite interval, γ1 less-than-or-equals, slant γ less-than-or-equals, slant γ2, with γ1 much greater-than 1; the normalization constant A relates to the electron count in this interval as stated in (3.12). The transversal density is averaged as

View the MathML source
and the longitudinal average left angle bracketpL(ω)right-pointing angle brackets is defined in the same way. We will focus on the number densities View the MathML source and left angle bracketnL (ω)right-pointing angle brackets = left angle bracketpL(ω)right-pointing angle brackets/(planck constant over two piω), since scaling exponents for γ-rays are usually defined with regard to differential number counts. We restore the natural units, mt → mtc/planck constant over two pi, and use the fine structure constant αq = q2/(4πplanck constant over two pic) and the scaling variable View the MathML source, to find

View the MathML source

View the MathML source
This frequency scaling, the leading order asymptotics of the integrals (3.11), is valid in the range View the MathML source and for electron indices s > 1. The amplitude A of the electronic power-law depends on the electronic source number ne. The product κγ of expansion parameter and Lorentz factor is independent of γ, cf. Figs. (2.4) and (2.10); thus we may take both factors at γ1 and conclude that the transversal number density overpowers the longitudinal one, since View the MathML source. In fact, the ratio of the longitudinal and transversal densities (3.12) reads,

View the MathML source
with s > 1. The constant κγ is listed in Table 2 for the respective pulsars, and the tachyon energy, Et = planck constant over two piω, is meant in MeV units. In the frequency bands studied in the subsequent examples of γ-ray pulsars, this ratio is well below 10−2, so that we will focus on the transversal radiation only. The transversal densities View the MathML source in (3.12) also apply to electron indices 1/3 < s < 1, as one may expect, but this requires a further restriction on the scaling range, a limit on the upper cutoff of the electron density, γ2 much less-than γ1/κ(γ1). For these indices, the radiation is likewise overwhelmingly transversal.

We denote the energy of the tachyonic γ-rays by Et as in (3.13). The tachyon flux, F = dN/dEt, is derived from the normalized differential number count dN = (4πD2)−1left angle bracketnT(ω)right-pointing angle brackets dω, where View the MathML source stands for the total (unpolarized) transversal radiation, cf. (3.12), and D is the distance to the pulsar, cf. Table 2. Concerning units, we choose View the MathML source, where Et is in MeV, and the measured proportionality factor and scaling exponent will be specified below for the enumerated pulsars. These units will be used throughout, mostly without mentioning. The scaling exponent relates to the electron index as α = 1/3 + s, cf. (3.12), and the dimensionless number density can be retrieved from the flux and the distance, left angle bracketnT(ω)right-pointing angle brackets = 4πD2planck constant over two piF; we will infer the electronic source count ne by equating View the MathML source in (3.12) with

View the MathML source

In the following case study of the seven established γ-ray pulsars, the observational input will be the amplitude and the scaling exponent of the measured flux, View the MathML source, the energy band for a scaling exponent to apply, as well as the spectral breaks where different power-laws join. We will determine the electron distributions generating the radiation in the respective frequency bands, in particular the electronic source count, that is, the number of gyrating electrons generating the tachyon flux or a power-law component of it. When inferring the electronic source number from the measured fluxes, we have to take into account that only a fraction αq/αe (ratio of tachyonic and photonic fine structure constants, cf. before (2.10)) of the tachyon flux passing through the detectors is absorbed. This requires a rescaling of the electron count ne by αe/αq ≈ 7.3 × 1010, where ne is based on the tachyon count in the detectors. That is, the electronic source count ne is calculated, via Figs. (3.14) and (3.12), from the observed flux F, but the actual tachyon flux is by a factor of αe/αq larger than the observed one, so that the actual number of synchrotron electrons in the surface magnetic field is Ne ≈ neαe/αq.

The three telescopes employed in measuring the fluxes in the various energy bands cited below operated with very different detection mechanisms, but the rescaling of the observed fluxes withαe/αq applies to all of them. In the OSSE detector [9], the basic interaction was ionization generating scintillations in NaI(Tl) crystals. COMPTEL [10] was a counter based on Compton scattering, and the EGRET telescope [11] counted γ-rays by conversion into electron-positron pairs. The crucial point here is, that the cross-sections for ultra-relativistic photoionization, photonic Compton scattering and photonic pair production also apply to transversal tachyons, with the mentioned rescaling by αq/αe. The tachyon mass drops out in the dispersion relation, k2 = ω2/c2 + (mtc/planck constant over two pi)2, at γ-ray energies, which only makes an overall rescaling of the respective photonic cross-sections [24] necessary. This rescaling also shows in the non-relativistic limit, derived for tachyonic ionization in Ref. [25], where we also studied the effect of the tachyon mass in the low-energy (soft X-ray) regime. The same rescaling of the Compton cross-section (Klein–Nishina formula) can be traced back to classical Thomson scattering, that is, to the acceleration of an electron by the incoming tachyonic wave field, which triggers electromagnetic radiation [8]. Despite the fact that the photonic cross-sections of the three mentioned processes scale with higher powers of the electric fine structure constant, the tachyonic counterpart is obtained in all three cases by an αq/αe-rescaling, for transversal γ-rays at least; this is further discussed in the Conclusion.

Apart from the electronic source count and the total gyration energy, View the MathML source, stored in these electrons, we will determine their tachyonic γ-ray luminosity, the power transversally radiated in a given frequency band,

View the MathML source
where the interval boundaries Et,1,2 are in MeV units. The conversion factors used in the sequel are 1 TeV ≈ 1.602 erg and 1 kpc ≈ 3.086 × 1021 cm. The tachyon energy Et is given in MeV and the electron energy E in GeV, parametrized by 10k like in the tables, and the flux F is in units of cm−2 s−1 MeV−1, cf. (3.14).

We start with the Crab pulsar. The flux in the 30–4000 MeV interval scales as View the MathML source, cf. Refs. [27] and [28], suggesting an electron index s ≈ 1.72. As pointed out after (3.14), the observed flux has still to be rescaled by a factor of αe/αq ≈ 7.3 × 1010, as most tachyons pass unnoticed through the detectors. The cited interval boundaries correspond to electron energies of 100.85 and 103.0 GeV, respectively. There is a spectral break at 30 MeV, followed by a second power-law, View the MathML source, in the 0.12–30 MeV band, cf. Refs. [29] and [30]. This flux component is generated by electrons gyrating with energies from 10−1.55 to 100.85 GeV, with power-law index s ≈ 1.92. The tachyonic spectral densities above and below the spectral break relate to the amplitudes of the respective electronic power-laws as View the MathML source and View the MathML source, cf. (3.12). The normalization constants of the electron distributions in the adjacent intervals above and below the break energy (of 100.85 GeV) read Aα=2.05 ≈ 690ne and Aα=2.25 ≈ 36.9ne, respectively. We find ne ≈ 1.4 × 1035 in the upper interval (where View the MathML source) and ne ≈ 2.0 × 1037 below the break energy (where View the MathML source). These electron counts are calculated by equating flux and number density, cf. Figs. (3.14) and (3.12), and they still have to be renormalized; the actual count in the 100.85−103.0 GeV range reads View the MathML source, and View the MathML source is the count below the break energy, in the 10−1.55−100.85 GeV interval. The total electronic energy in the surface magnetic field is calculated as indicated before (3.15); we find View the MathML source above the break energy, and View the MathML source below. The respective electron populations produce the tachyonic luminosity View the MathML source in the 30–4000 MeV band, and View the MathML source in the 0.12–30 MeV interval, cf. (3.15). These gyration energies and luminosities are based on the observed flux F; the actual tachyon flux is by a factor of αe/αq higher, so that we have to replace the View the MathML source by the renormalized counts View the MathML source.

Remarks. There is no superluminal radiation damping that could slow down the gyration, as the tachyonic Green function is time symmetric. The energy radiated stems from the oscillators of the cosmic absorber [25] and [31]. The absorber field breaks the time symmetry and renders the interaction causal and non-local [8]. The latter has implications on energy conservation, as the radiated energy is drained from the absorber. For this reason, the search for missing negative mass-squares suggested in Ref. [4] fails, where bubble-chamber events were reanalyzed, assuming a local but otherwise unspecified interaction of tachyons with matter, discounting the time symmetry of the Green function outside the light cone. Had there been detection in these experiments, in the relativistic framework in which they were interpreted, this would have been tantamount to causality violation. The radiation mechanism scrutinized here, based on a Proca field with negative mass-square minimally coupled to subluminal matter, is non-relativistic. It implies the absolute spacetime defined by the absorber [5] and [6], even though the Lagrangians are still covariant. Finally, electromagnetic synchrotron radiation would lead to radiation damping, but is suppressed by a quantum cutoff, cf. after (2.11).

There is a further spectral break at 0.12 MeV, where the spectrum flattens to View the MathML source, cf. Ref. [29], and then continues as an unbroken power-law even into the soft X-ray band [32] and [33]. The radiation below 0.12 MeV is not accessible in the ultra-relativistic limit because of the moderate electronic Lorentz factors; the spectral densities would have to be recalculated, with the Nicholson asymptotics replaced by Debye’s approximation of large-order Bessel functions [34]. At the opposite end, above 4000 MeV, there is some indication that the radiation terminates in a steep spectral slope, and no pulsed emission is detected in the TeV region [28]. The fluxes used here and in the subsequent examples refer to the pulsed emission, and they are phase averages. Phase-resolved spectra exist for the Crab and Vela pulsars and Geminga. As for the Crab pulsar, the phase-dependent exponents α range from 1.42 to 2.65 in the 30–4000 MeV band [27], and from 2.16 to 2.41 in the 0.12–30 MeV interval, cf. Ref. [29]. The respective phase-resolved electron indices, s = α − 1/3, are usually larger than 1, and they never fall below 1/3, cf. after (3.13).

The Vela pulsar radiates a flux of View the MathML source in the 30–2000 MeV interval, cf. Refs. [27] and [35], generated by electrons in the 100.85–102.7 GeV range with power-law index s ≈ 1.29. The normalization of the electron distribution in this range is A ≈ 6.32ne, and the electronic source count reads ne ≈ 5.9 × 1033 or Ne ≈ 4.3 × 1044 if renormalized. These electrons store an energy of Ee[erg] ≈ 0.125Ne and generate the number density View the MathML source, which results in a tachyonic γ-ray luminosity of Lt[erg/s] ≈ 10.4Ne in the 30–2000 MeV band. The phase-resolved exponents of this pulsar range from 1.38 to 2.21, cf. Ref. [27]. Pulsed TeV radiation is not detectable [36].

γ-Radiation from PSR B1706−44 has been detected from 50 MeV up to 20 GeV, cf. Ref. [37]. A flux View the MathML source is observed in the 50–1000 MeV interval, corresponding to electron energies from 101.1 to 102.4 GeV. This is the only example where the (phase-averaged) electron index drops below 1, s ≈ 0.94. The electronic power-law normalization in the indicated GeV range is Aα=1.27 ≈ 0.160ne, and the tachyonic number density scales as View the MathML source in the 50–1000 MeV interval. There is a spectral break at 1000 MeV, followed by a second flux component, View the MathML source, cf. Ref. [37], which applies up to 20 GeV, suggesting electron energies up to 103.7 GeV in the surface field. This flux is generated by an electronic power-law of index s ≈ 1.92 and amplitude Aα=2.25 ≈ 1.54 × 105ne in the 102.4–103.7 GeV range. The tachyonic number density in the 1–20 GeV interval reads View the MathML source. By the way, the frequency scaling is obtained by substituting planck constant over two piω/1 MeV for Et. Both electronic power-laws admit the same source number, ne ≈ 9.2 × 1033, valid for s ≈ 0.94 as well as s ≈ 1.92; the renormalized source count for each of these slopes is accordingly Ne ≈ 6.7 × 1044. These electron populations below and above the break energy of 102.4 GeV contain a gyration energy of View the MathML source and View the MathML source, respectively; they produce a tachyonic luminosity of View the MathML source in the 50–1000 MeV band, and of View the MathML source above the spectral break.

Geminga generates the flux View the MathML source in the 30–2000 MeV band, corresponding to electron energies between 100.85 and 102.7 GeV, cf. Refs. [27], [38] and [39]. This flux allows us to infer the electron index, s ≈ 1.09, and the tachyonic number density, View the MathML source. The electronic power-law normalization in the mentioned GeV interval is A ≈ 0.649ne, with source count ne ≈ 7.9 × 1032. We find the renormalized electron count, Ne ≈ 5.8 × 1043, the total energy of these electrons, Ee[erg] ≈ 0.16Ne, as well as their tachyonic γ-ray luminosity in the 30–2000 MeV interval, Lt[erg/s] ≈ 9.6Ne. The power-law indices of the phase-resolved flux range from 1.27 to 1.89, cf. Ref. [27].

PSR B1055−52 radiates the flux View the MathML source in the 70–1000 MeV band [40], generated by electrons cycling with energies from 101.2 to 102.4 GeV. In this range, we infer an electronic power-law distribution of index s ≈ 1.25, normalization Aα=1.58 ≈ 6.65ne, and source number ne ≈ 4.4 × 1033, which produces the tachyonic number density View the MathML source in the 70–1000 MeV interval. Above the spectral break at 1000 MeV, the flux scales as View the MathML source up to at least 4000 MeV, cf. Ref. [40]. This means electron energies up to 103.0 GeV in the surface field, an electronic power-law index s ≈ 1.71 above the electronic break energy of 102.4 GeV, and a tachyonic number density of View the MathML source in the 1000–4000 MeV interval. The electronic power-law normalization in the 102.4−103.0 GeV range reads Aα=2.04 ≈ 1.15 × 104ne, with ne ≈ 1.5 × 1033. Below the electronic break energy, we count View the MathML source electrons, which store a total energy of View the MathML source and radiate a power of View the MathML source in tachyonic γ-rays in the 70–1000 MeV band. Above the break energy, there are View the MathML source electrons gyrating in the 102.4–103.0 GeV range, which add up to an electronic energy of View the MathML source and produce a tachyonic luminosity of View the MathML source in the 1000–4000 MeV interval.

γ-radiation from PSR B1951+32 has been detected in a wide band, from 10 MeV to 20 GeV, cf. Refs. [41] and [42], generated by electrons gyrating in the 100.38–103.7 GeV range. The flux scales as View the MathML source, cf. Ref. [41], seemingly without a spectral break in this band. We find the electron index s ≈ 1.41, the electronic power-law normalization A ≈ 13.2ne, and the tachyonic number density View the MathML source, from which we infer the electronic source count ne ≈ 6.2 × 1034 or Ne ≈ 4.5 × 1045 when renormalized. The gyration energy of these electrons is Ee[erg] ≈ 0.25Ne, and their tachyonic radiation power in the 10 MeV–20 GeV interval amounts to Lt[erg/s] ≈ 2.9Ne. A spectral break must occur below 10 MeV, according to the upper flux limits derived in Refs. [43] and [44]; no pulsed TeV radiation has been found [45].

PSR B1509−58 is not detected in hard γ-rays, possibly due to a quantum cutoff inflicted by its high magnetic field, but this needs further investigation in second quantization, cf. after (2.11). The observed flux, View the MathML source, extends from 0.2 to 10 MeV, cf. Refs. [46] and [47], and is produced by electrons with energies in the 10−1.3–100.38 GeV range. We infer an electronic power-law of index s ≈ 1.35 and normalization A ≈ 2.25ne, where ne ≈ 1.75 × 1036, as well as a tachyonic number density of View the MathML source in the mentioned MeV band. We note the electron count in the surface field, Ne ≈ 1.3 × 1047, the energy stored in these electrons, Ee[erg] ≈ 6.5 × 10−4Ne, and their tachyonic luminosity, Lt[erg/s] ≈ 0.82Ne, in the 0.2–10 MeV interval. Above 10 MeV, there is a steep spectral break [47], and no pulsed TeV radiation is observed [48]. At the lower end, at 0.2 MeV, there is likewise a spectral break, where the spectrum hardens, admitting a spectral index of about α ≈ 1.3 in the hard X-ray band [49] and [50], but this radiation is not accessible in the ultra-relativistic synchrotron limit [34].

4. Conclusion

We have given a quantitative and in part phenomenological discussion of superluminal synchrotron radiation in strong magnetic fields. We will further comment on the interaction of tachyons with matter, on the basis of the three detectors used to infer the magnetospheric electron populations and their tachyonic γ-ray luminosity. The ionization in the scintillation crystals of the OSSE counter [9] happens in the relativistic regime. OSSE operated above a 50 keV threshold, so that the tachyon mass of 2 keV can be neglected in the dispersion relation, given that only the squared energies enter, cf. before (3.15). Therefore, the relativistic photonic ionization cross-section [24] can be used for transversal tachyons, if properly rescaled. The only change necessary for tachyonic X-rays or γ-rays above 50 keV is a rescaling with the ratio of tachyonic and electric fine structure constant. This rescaling can be traced back to the non-relativistic limit, which depends on the tachyon mass and applies in the soft and hard X-ray bands [25]. The longitudinal relativistic cross-section has to be calculated from scratch without reference to a zero-mass limit [51]. The tachyon flux generated by synchrotron radiation in the surface fields of γ-ray pulsars is overwhelmingly transversally polarized, cf. (3.13), in contrast to the low-magnetic-field limit [13], and therefore we have not attempted a quantitative study of the longitudinal fraction beyond some basic estimates.

Compton scattering and pair creation were employed in the COMPTEL and EGRET detectors [10] and [11]. The tachyonic Thomson cross-section, the non-relativistic classical limit of Compton scattering, was derived in Ref. [8], but a quantum mechanical version is still lacking, especially if the incident tachyonic X-rays have energies close to the tachyon mass. In the relativistic regime, for γ-rays above 0.75 MeV (the threshold of the COMPTEL counter), the transversal cross-section is obtained by a rescaling of the Klein–Nishina formula with αq/αe, the same rescaling that applies to the Thomson and ionization cross-sections. The EGRET detector was a spark chamber recording electron-positron pairs produced by γ-rays scattered in tantalum foils. The conversion of tachyons into electron-positron pairs has not been studied in any limit and context as yet. However, one can reckon from the foregoing that the cross-section for the conversion of transversal γ-rays is just the Bethe–Heitler formula rescaled with αq/αe; the fluxes collected by EGRET were in the 20 MeV–30 GeV range, so that the tachyon mass is negligible in the dispersion relation. This rescaling of the EGRET fluxes is also suggested by cross-calibration with the COMPTEL detector, to avoid discontinuities in the band overlap.

Tachyonic synchrotron radiation above the break frequency has not been discussed here, cf. after (2.9) and Ref. [13]. This requires second quantization in strong magnetic fields, as quantum corrections in the high-frequency regime cannot be treated perturbatively. Above the break frequency, the classical radiation theory is severely modified by a quantum cutoff, which results in exponential decay of the spectral densities at the critical energy, where the classical spectral functions are peaked. This quantum cutoff may well be the reason that no pulsed TeV radiation is detected from the known γ-ray pulsars.


The author acknowledges the support of the Japan Society for the Promotion of Science. The hospitality and stimulating atmosphere of the Tata Institute of Fundamental Research, Mumbai, and the Institute of Mathematical Sciences, Chennai, are likewise gratefully acknowledged.


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