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The polarisation of gravitational synchrotron radiation
Volume 42A, number 1
PHYSICS LETTERS
THE POLARISATION OF GRAVITATIONAL
6 November 1972
SYNCHROTRON RADIATION
V.R. KHALILOV, Ju.M. LOSKUTOV, A.A. SOKOLOV and I.M. TERNOV Department
of Physics, Moscow
State University,
USSR
Received 19 September 1972 In the linear approximation we show that the gravitational radiation induced by ultrarelativistic charges, moving in a magnetic field or in the field of a circularly polarised electromagnetic wave does not have the primary linear polarisation contrary to the case ofthectromagnetic synchroton radiation.
Recently, in connection with Weber’s experiments [I] the question of the gravitational radiation of different cosmic objects have been widely discussed. A large place, in this discussion, is given to the question of the polarisation features of the gravitational radiation [2]. But, theoretically, the polarisation properties of the gravitational radiation were not studied in detail. We shall consider these properties of gravitational radiation by the example of ultrarelativistic charges *, moving in a homogeneous and constant magnetic field or in the field of an electromagnetic wave, which is circularly polarised. In both cases, in a special coordinate system, the clasical trajectories of the charges are circles with the radii R, = /3E/%H in the magnetic field and R2 = (e0e/m002) (moc2/E) in the field of an electromagnetic wave. Here E is the energy of the particle, with rest mass m. and with charge q, H is the strength of the magnetic field, fl= u/c, IJ is the velocity of the particie, e is the amplitude of the electric vector of the wave, o is the wave frequency. Using the methods of quantum theory, given, for example in [4-71 and omiting detailed calculations, performed by standard methods [8,9], we shall write the final expression for the intensity of the gravitational radiation by the ultrarelativistic particles with ex* The gravitational radiation by a relativistic charge, moving in a magnetic field was studied in the classical theory in ref. [ 31. But the question of the polarisation properties of the gravitational radiation was not considered in ref. [ 31. ** From the analysis of the angular distribution it follows, that the field of the radiation is concentrated near the plane of the classical trajectory of the charge in a narrow cone with the axis, directed along the motion of the charge with angular dimensions A0 - moc2/E.
pansion in two linear polarisations (on the analogy of “0” and “rr” components of the synchrotron radiation**) _
where y = f n (1 --/$) %, n is the harmonic number, R is the instantaneous radius of curvature, K is the gravitational constant and the functionsfi and f2 are
fi
(2)
= Y{-K,(Y)+~Ky(Y)tS~*,~(x)dr} Y
(K,,3 (y) and K, (y) are Macdonald functions). These function characterise the spectral distribution of the intensity of the radiation. From the formulas (1) and (2) it is seen, that in the ultrarelativistic case (/3 - l), the emission of gravitational radiation, as in the case of electromagnetic radiation, takes place mainly at the high harmonics with nextr % (Elmuc2)3. After integrating the expressions (1) and (2) over whole spectrum, we obtain easily: w, = WI t w1 =?3 (Kmic/R2) (E/moc2)4 Attention should be paid to the circumstance,
(3)
W = W,
that, 43
Volume 42A, number 1
PHYSICS LETTERS
6 November 1972
References contrary to the synchrotron electromagnetic radiation, where W2/Wl = 7, the gravitational radiation, described [II J. Weber, Phys. Rev. Lett. 22 (1969) 1320; 24 (1970) 276. J.A. Tyson and D.H. Douglass, Phys. Rev. Lett. 28 (1972) [21 by a tensor interaction does not have the primary 991. linear polarisation. Thus, as it is likely, that the main Zh. Eksp. i Teor. [31 V.I. Pustovoit and M.N. Gertzenshtein, role in gravitational radiation of charges is played by kineFiz. 42 (1962) 163. matic effects, we may assume the obtained results to 141 D.D. Ivanenko and A.A. Sokolov, The quantum theory of fields (in Russian, Moscow, 1952). have a wider applicability. It may well be that Weber’s Ju. S. Vladimirov, Zh. Eksp. i Teor. Fiz. 45 (1963) 251. [51 experiment corroborates this circumstance, though at 161 D.D. Ivanenko and A.A. Sokolov. Vestn. Mosk. University, present the mechanism of gravitational radiation is III, No. 8 (1947) 103. being discussed. (71 A.A. Sokolov, Vestn. Mosk. Univ. III, No. 9 (1952) 9. radiation (in Russian, Moscow 1966). ISI The synchrotron [91 I.M. Ternow, W.G. Bagrow and A.M. Khapaew, Ann. Physik
44
22 (1968)
25.
PHYSICS LETTERS
THE POLARISATION OF GRAVITATIONAL
6 November 1972
SYNCHROTRON RADIATION
V.R. KHALILOV, Ju.M. LOSKUTOV, A.A. SOKOLOV and I.M. TERNOV Department
of Physics, Moscow
State University,
USSR
Received 19 September 1972 In the linear approximation we show that the gravitational radiation induced by ultrarelativistic charges, moving in a magnetic field or in the field of a circularly polarised electromagnetic wave does not have the primary linear polarisation contrary to the case ofthectromagnetic synchroton radiation.
Recently, in connection with Weber’s experiments [I] the question of the gravitational radiation of different cosmic objects have been widely discussed. A large place, in this discussion, is given to the question of the polarisation features of the gravitational radiation [2]. But, theoretically, the polarisation properties of the gravitational radiation were not studied in detail. We shall consider these properties of gravitational radiation by the example of ultrarelativistic charges *, moving in a homogeneous and constant magnetic field or in the field of an electromagnetic wave, which is circularly polarised. In both cases, in a special coordinate system, the clasical trajectories of the charges are circles with the radii R, = /3E/%H in the magnetic field and R2 = (e0e/m002) (moc2/E) in the field of an electromagnetic wave. Here E is the energy of the particle, with rest mass m. and with charge q, H is the strength of the magnetic field, fl= u/c, IJ is the velocity of the particie, e is the amplitude of the electric vector of the wave, o is the wave frequency. Using the methods of quantum theory, given, for example in [4-71 and omiting detailed calculations, performed by standard methods [8,9], we shall write the final expression for the intensity of the gravitational radiation by the ultrarelativistic particles with ex* The gravitational radiation by a relativistic charge, moving in a magnetic field was studied in the classical theory in ref. [ 31. But the question of the polarisation properties of the gravitational radiation was not considered in ref. [ 31. ** From the analysis of the angular distribution it follows, that the field of the radiation is concentrated near the plane of the classical trajectory of the charge in a narrow cone with the axis, directed along the motion of the charge with angular dimensions A0 - moc2/E.
pansion in two linear polarisations (on the analogy of “0” and “rr” components of the synchrotron radiation**) _
where y = f n (1 --/$) %, n is the harmonic number, R is the instantaneous radius of curvature, K is the gravitational constant and the functionsfi and f2 are
fi
(2)
= Y{-K,(Y)+~Ky(Y)tS~*,~(x)dr} Y
(K,,3 (y) and K, (y) are Macdonald functions). These function characterise the spectral distribution of the intensity of the radiation. From the formulas (1) and (2) it is seen, that in the ultrarelativistic case (/3 - l), the emission of gravitational radiation, as in the case of electromagnetic radiation, takes place mainly at the high harmonics with nextr % (Elmuc2)3. After integrating the expressions (1) and (2) over whole spectrum, we obtain easily: w, = WI t w1 =?3 (Kmic/R2) (E/moc2)4 Attention should be paid to the circumstance,
(3)
W = W,
that, 43
Volume 42A, number 1
PHYSICS LETTERS
6 November 1972
References contrary to the synchrotron electromagnetic radiation, where W2/Wl = 7, the gravitational radiation, described [II J. Weber, Phys. Rev. Lett. 22 (1969) 1320; 24 (1970) 276. J.A. Tyson and D.H. Douglass, Phys. Rev. Lett. 28 (1972) [21 by a tensor interaction does not have the primary 991. linear polarisation. Thus, as it is likely, that the main Zh. Eksp. i Teor. [31 V.I. Pustovoit and M.N. Gertzenshtein, role in gravitational radiation of charges is played by kineFiz. 42 (1962) 163. matic effects, we may assume the obtained results to 141 D.D. Ivanenko and A.A. Sokolov, The quantum theory of fields (in Russian, Moscow, 1952). have a wider applicability. It may well be that Weber’s Ju. S. Vladimirov, Zh. Eksp. i Teor. Fiz. 45 (1963) 251. [51 experiment corroborates this circumstance, though at 161 D.D. Ivanenko and A.A. Sokolov. Vestn. Mosk. University, present the mechanism of gravitational radiation is III, No. 8 (1947) 103. being discussed. (71 A.A. Sokolov, Vestn. Mosk. Univ. III, No. 9 (1952) 9. radiation (in Russian, Moscow 1966). ISI The synchrotron [91 I.M. Ternow, W.G. Bagrow and A.M. Khapaew, Ann. Physik
44
22 (1968)
25.