Gravimagnetic shock waves and gravitational-wave experiments
aa r X i v : . [ g r- q c ] J a n Gravimagnetic shock waves and gravitational-waveexperiments
Yu.G. Ignat’evKazan State Pedagogical University1 Mezhlauk Str., Kazan 420021, Russia
Abstract
Causes of the unsatisfactory condition of the gravitational-wave experiments are discussedand a new outlook at the detection of gravitational waves of astrophysical origin is proposed.It is shown that there are strong grounds for identifying the so-called giant pulses in the pul-sar NP 0532 radiation with gravimagnetic shock waves (GMSW) excited in the neutron starmagnetosphere by sporadic gravitational radiation of this pulsar.
In the history of physics of the 20th century, I suppose, there is no such a grave experimental problem(except the controlled thermonuclear fusion problem) that, being solved for over thirty years bydifferent research groups as the gravitational wave (GW) detection problem. Although much meansare used to solve it, no sufficiently convincing possitive results have been obtained. What are thereasons for this situation? An error in the gravitational theory? The experimentalists’ incapability?Are there realizable opportunities to detect gravitational radiation in the visible future? Is the GWdetection problem worth studing? We will try to answer these questions in the present paper. Asany other radiation detection problem, this one also splits into two independent problems: (1) “GWsources” and (2) “GW detectors”. It is important not to forget to join both branches by solving aconcrete experimental problem.
The average power of gravitational radiation from a source is calculated by the formula [9] L GW = G c < ... t ik... t ik >, (1)where G is the gravitational constant, c is the velocity of light, t ik = Z ρ ( x i x k − δ ik r ) dV (2)is the reduced quadrupole moment of the source; dots mean time derivatives. There are two param-eters of interest in the GW detection problems: the GW magnitude (i.e. the deviation from the flatmetric h ik = g ik − η ik ) and the GW frequency, ω . The GW energy flow is expressed in terms of theseparameters by the formula [9]: ct = P = c πG (cid:20) ˙ h + 14 ( ˙ h − ˙ h ) (cid:21) . (3)1Throughout the paper we use the metric signature ( − , − , − , +1)). From (1) - (3) follow estimat-ing formulae for the gravitational radiation power and magnitude: L GW = GE q ω c ; (4) L GW = c ω R G h , (5)where E q is the energy of quadrupole ole oscillations of the source of the characteristic frequency ω ( L = ωE q is the quadrupole oscillation power), R is the distance from the source to an observer. In particular, a useful formula follows from (4) – (5): hh = E q N c , (6)where h is the gravitational potential of the source of the total mass M : h = GMc R . (7)According to (6) and (7)), the ratio ofGW magnitude to the Newtonian gravitational potential of thesource is of the order of the ratio of the quadrupole energy of the source oscillations to its completeenergy at rest, E = M c . It is obvious that always E q < E , and E q /E ≪ M ⊙ = 2 · g) and adistance of 1 pc (3.26 light years = 3 · cm)1, from Eq. (7) we obtain : h · (pc /M ⊙ ) = 4 , · − . For a mass of 1 kg at a distance of 1 m: h · (m kg − ) = 7 , · − . Therefore for an A-bomb explosion with (∆
M/M ∼ − ), under the condition that the wholeexplosion energy turns into the quadrupole oscillations energy, at a distance of 1 m (!) from theepicentre, we get the GW magnitude h ∼ − . If one makes all atoms of a compact graser oscillatein the optic range (the radiation energy is about ( ~ ν ∼ l ∼ h ∼ − .GW sources can be divided into two classes: (1) stable (quasistable) sources, which cannot bedestroyed during the GW radiation process; (2) catastrophic sources, being destroyed in the GWradiation process. A graser represents a source of the first type, an A-bomb a source of the secondtype. For first type sources the quadrupole oscillation energy cannot exceed the binding energy ofthe source as a whole, unlike second-type ones, which are sources for one occasion. For example,close binaries and quadrupole oscillations of neutron stars are first-type astrophysical sources, andSupernovae are second-type ones. The nearest stars’ distance is about 1,3 pc.
2s mentioned above, stable radiation sources are subject to the condition E kin < E b , (8)where E kin is the inner kinetic energy of separate parts of the source, E b is their binding energy.Since it is always E q E kin, the condition (8 takes the form E q E b . (9)Hence the upper limit of GW magnitude from such sources can be obtained from the formula h < h E b M c . (10)For astrophysical sources the binding energy is essentially that of gravitational attraction. Let ∆ M bethe part of the mass of an astrophysical object performing quadrupole oscillations. Its gravitationalbinding energy is E b < G ∆ M · Ml , (11)where l is the chracteristic size of the system. Thus for the upper limit of a GW magnitude fromsuch a source Eqs. (10) and (11) give: h h r g l ∆ MM , (12)where r g = 2 GM/c is the gravitational radius of the radiating system. Consider first a source of total mass M , which consists of two parts, so that the second part ∆ M performs a free motion in the gravitational field of the system (rotation or free fall). Let ω be acharacteristic frequency of this process . Equating the centrifugal and free fall accelerations, weobtain the well-known relation GM = ω l , (13)which connects the characteristic size of the system with its characteristic frequency.Now let the gravitational attraction in the system be held by the forces of pressure (for stellarquadrupole oscillations). Equating these forces, we get the hydrostatistic balance condition | ~ ∇ P | = ρ GMl , (14)where P is the pressure and ρ is the density. Using the known relation dP = v f dρ , where v f is thevelocity of sound, from Eq. (14) we obtain: lv f ≈ M G. (15) Evidently the order of magnitude of this quantity coincides with the frequency of gravitational radiation from thesystem. v f /l ≈ ω is the system proper oscillation frequency. Therefore for systems supported by theforces of pressure we return to the estimate of Eq. (13). Thus for stable astrophysical GW sources Eq. (13) has a universal nature if omega is understoodas a characteristic frequency of the system oscillations .From the law (13) we can estimate the radiation characteristics of the collapsing objects, collidingstars and the like. It follows from this law that a maximum radiation frequency can be achieved forobjects close to the gravitational collapse condition. In this case, by (12), the maximum magnitudeof radiated GW is achieved (see [1]). For objects of masses of the order of the Solar mass ( r g = 2 , ω max ∼ c/r g ≈ sec − . For intense astrophysical sources of gravitational radiation, this radiation is the basic mechanismof quadrupole oscillation energy loss. Therefore, more rigorously, such sources should be calledquasistable. The gravitational radiation power of a system of two orbiting gravitating masses m1and m2 is calculated from the known formula [10] L g = − dEdt = 32 G m m ( m + m )5 c r , (16)where r is the separation of the centres of mass. The energy balance leads to the mass approachinglaw [10]˙ r = 64 G m m ( m + m )5 c r , (17)Its integration yields a formula for the time t needed for the mass centres to approach to a distanceof r from r : t = 5 c G m m ( m + m ) ( r − r ) . (18)Further for simplicity we will study a pair of equal stars, setting m = m = M , r = 2 R where R is the stellar radius, i.e. we will calculate the time until the catastrophic stellar collision, τ (thelifetime). Then from (18) – (18) we get: τ = 5 c ( l − R )384 G M and for l ≫ R τ ≈ (cid:18) lr g (cid:19) lc . (19)The gravitational radiation frequency of a binary system increases with time; the ratio of the fre-quency shift per period ∆ ω to the radiation frequency ω is, by order of magnitude,∆ ωω ∼ (cid:16) r g l (cid:17) . (20)Figs. 1 and 2 show the dependence of the gravitational radiation power of a binary system andthe radiated GW magnitude on the distance between the stars. Fig. 3 shows the binary lifetimeversus their separation for stars with the masses m = m = M ⊙ .4igure: 1. Gravitational radiation power of a binary L g (erg/s)vs. distance between the stars l (km). Figure: 2.
GW magnitude h from a binary vs. its size l (km), ata distance of 1 kpc Let us estimate the probability of
GW detection from a close binary system in the Galaxy atgiven rotation period, assuming that the Galaxy age is of the order of 1 · years. Further, weassume that the average stellar number density in the Galaxy is of the order of 0 ,
120 stars/pc [15],the Galactic volume is 300 kpc [16], then the number of stars in the Galaxy is about 0 , · .Besides, we take into account that approximately half of the stars are in binary systems [17]. Thenthe possibility of existence of a binary with a prescribed lifetime τ is proportional to the ratio τ /t ,where t is the age of the Galaxy. In the columns of Table 1 corresponding to system lifetimessmaller than 1 year, the detection probability of such systems in experiments lasting 1 year is shown.Evidently for such systems the probability of detection in a year-lasting experiment coincides withthat for a binary having a lifetime of 1 year. The number of such systems in the Galaxy is estimatedto be of the order of one. 5igure: 3. Lifetime of a binary τ (seconds) vs. its size l (km) The presented data show that at the instant preceding the catastrophic collision, the gravitationalradiation power from the binary is of the order of Supernova luminosity. Thus, a stellar collision ina close binary is an event whose scale is of the order of a Supernova explosion. As mentioned above,in the Galaxy the probability of GW detection from a binary with a lifetime of the order of1 year isclose to one. This means that catastrophic phenomena with energy release of the order of 1 · erg/s should happen once a year. However, in reality such phenomena happen once in 40 to 80 yearsin the Galaxy [19]. A reason for such a discrepancy is that in a close binary with a lifetime of theorder of1 year the stars’ separation is about 12000 km.Therefore for such a system to exist it is necessary that both stars be at least white dwarfs. Butin this case even in a stellar collision the energy released is 4 orders of magnituded smaller thanthat of a Supernova explosion. Thus for a catastrophic collision of this scale it is at least necessarythat one of the components be a neutron star, while the second one is a white dwarf. The existenceprobability of such systems in the Galaxy is much smaller. Note that it is difficult to understandthe experimental programmes intended for registration of GW from the binaries with periods of theorder of a few seconds. According to Table 1, their lifetime does not exceed 5 years, and in this caseit would be more reasonalble to wait these 5 years and to detect the gravitational radiation from acatastrophic collision: its power is higher by at least 13 orders and the GW magnitude is greaterby 3 orders (!), as follows from Table 1. However, at least in the last 10 years nobody detectedcatastrophic events on such a scale at distances smaller than 15 kpc.Since the pulsars are identified with Supernovae remnants, the average frequency of Supernovabursts may be estimated from the data on pulsars spreading in the Solar neighbourhood. Thus,at distances within about 1 kpc, on the whole, about 20 pulsars are observed. Table 2 shows thedata on the pulsars nearest to the Solar system3. As follows from this Table, almost all the pulsarsare younger than 108 years. Therefore, it may be stated that the observed pulsars are remnants ofSupernovae which exploded in the last hundred million years. It gives 1 flash per 50 years, coincidingwith the estimate of Ref . [9]. It seems likely to be also close to the average frequency of catastrophiccollisions in close binaries. 6able 1: Characteristics of gravitational radiation from a close binary ∗ l τ ω T
136 34,3 12,1 7,10 3,07 0,38 0,068 0,012 0,003 0,001 L g h P N h ⊕ P ⊕ ∗ ) Here and henceforth, the figures in parantheses indicate the order of magnitude (a(b) = a · b ); thequantities h and P (the gravitational radiation flow density) are calculated at a distance of 1 kpc from thebinary; T = 2 π/ω is the GW period; N is the expected number of binaries in the Galaxy; R = < R > is theexpected distance to the binary in kpc; h ⊕ is the expected GW magnitude on the Earth; P. is the expectedgravitational radiation flow density on the Earth. The quantities T and τ are given in seconds, ω in sec − , l in km, L g in erg/sec, P and P ⊕ in W/cm . Table 2:
Data on pulsars located at distances smaller or of the order of 1 kpc from the Sun
No Pulsar(name) Distance(kpc) P(sec) ˙
P /P (years1 MP 0031 0,21 0,94 7,1(7)2 MP 0450 0,33 0,553 NP 0532 2,0 0,033 2,5(3)4 MP 0628-28 0,170 1,24 1,6(7)5 CP 0809 0,19 1,29 2,5(8)6 AP 0823+26 0,38 0,53 1,0(7)7 PSR 0833-45 0,5 0,089 2,3(4)8 CP 0834 0,43 1,27 5,9(6)9 PP 0943 0,30 1,09810 CP 0950 0,10 0,253 3,5(7)11 CP 1133 0,16 1,188 1,0(7)12 AP 1237+25 0,20 1,382 4,6(7)13 PSR 1451-68 0,40 0,263 ¿2,8(6)14 HP 1508 0,26 0,740 4,7(6)15 CP 1919 0,42 1,337 3,2(7)16 PSR 1929+10 0,27 0,227 6,2(6)17 JP 1933+16 3 0,359 1,9(6)18 AP 2016+28 0,47 0,558 1,2(8)19 PSR 2045 0,38 1,962 5,7(6) ÷ T ∼ ÷ h ⊕ ∼ (1 ÷ · − , P ⊕ ∼ − ÷ · − W/cm . GW with such parameters canhardly be detected in the coming decades. In this situation it only remains to hope for a case, rareand simultineously dangerous for the Earth, of a Supernova burst or a catastrophic end of a closebinary. There is, however, one more class of stable astrophysical GW sources — quadrupole oscillations ofneutron stars. Table 3 shows the calculated parameters of gravitational radiation from neutron stars[9] (columns 1 ÷ ∼ (0 , ÷ · s − ; h ⊕ ∼ − , P ⊕ ∼ · − W/cm by the energyofstellar quadrupole oscillations of the order of 10 ÷ erg and the whole gravitational luminosityof L g ∼ · The cause of the unsatisfactory condition in the GW detection problem is, in the author’s opinion,the originally chosen erroneous way ofits solution — the programme ofcreating GW detectors. DirectGW detection can be realized either due to their tidal effect on a nonrelativistic (solid-state) detector,or due to their relativistic effect on a detector having a relativistic component (a laser ray). In bothcases the GW effect on a detector (test-body displacement or laser ray deviation) is propotional tothe GW magnitude. And the expected GW magnitudes from astrophysical sources are extremelysmall (see, for example, [9]).The existing GW detection programmes are generally meant for astrophysics sources of two types:1 . Supernovae; 2 close binaries. In the first case one may expect GW magnitudes about 10 − ÷ − with the radiation in a wide frequency range with the characteristic frequency of the order of 10 sec − , in the second case magnitudes about 10 − ÷ − with a fixed frequency in the range of0 , ÷
10 sec − . Due to the very small expected GW magnitudes on the Earth, the experimentalprogrammes intended for direct GW detection inevitably come across the problem of noise of externalthermal and quantum nature. This struggle is already in its third decade and requires the creationof high-precision deeply cooled detectors.On the other hand, it is well-known that even such weak-magnitude GW carry rather a highenergy: in the above examples, this energy is of the order of1 W/cm2 in the first case and about10 − ÷ − W/cm in the second case. Electromagnetic signal detection on such a power level hasno problems. Therefore, the GW detection problem should be solved in a different way: by lookingfor specific electromagnetic signals from GW effect upon matter in those regions of the Galaxy wherethe gravitational radiation intensity is high. Setting the problem in such a way, we should above allstudy the GW effect on plasma-like media. The corresponding studies, carried in the eighties mainlyin the Kazan school of gravitation, revealed a number of specific electromagnetic reactions of plasmato GW. In Refs. [2]— [5] the effect of plane GW (PGW) on plasma-like media was investigated by8he methods of relativistic kinetic theory in the approximation of negligible back reaction of matteron the PGW:(8 πG/c ) ε ≪ ω . (21)where ω is the GW characteristic frequency, ε is the matter energy density. These papers haverevealed a number of phenomena of interest, consisting in induction of longitudinal electric oscilla-tions in the plasma by PGW. In spite of the strictness of the results obtained in [2]–[5], the effectsdiscovered have very little to do with the real problem of GW detection. Moreover, the above cal-culations show a lack of any prospects for GW detectors based on dynamical excitation of electricoscillations by gravitational radiation. There are two reasons for that: the smallness of the ratio m G/e = 10 − and the small relativistic factor h v i /c of standard plasmalike systems. The GWenergy conversion coefficient to plasma oscillations is directly proportional to a product of thesefactors.However, the situation may change radically if strong electric or magnetic fields are present inthe plasma. In Ref. [6], where the induction of surface currents on a metal-vacuum interface by aPGW was studied, it was shown that the values of currents thus induced can be of experimentalinterest. In [7], on the basis of relativistic kinetic equations, a set of magneto-hydrodynamics (MHD)equations was obtained, which described the motion of collisionless magnetoactive plasma against thebackground of a PGW of arbitrary magnitude in drift approximation and it was shown that, providedthe propagation of the PGW is transversal, there arises a plasma drift in the PGW propagationdirection.In Ref. [1] an exact solution of the relativistic MHD equations in the PGW background ofarbitrary magnitude was obtained and, on its basis, a fundamentally new class of sufficiently nonlinearthreshold effects was discovered, named GMSW (“jimmysway”) - gravimagnetic shock waves. The PGW metric of the polarisation e+ is described by the expression [9]: ds = 2 dudv − L [ e β ( dx ) + e − β ( dx ) ] , (22)where β ( u ) is an arbitrary function (the PGW magnitude), while L ( u ) (the PGW background factor)obeys the ordinary second-order differential equation L ′′ + L ˙ β = 0; (23) u = 1 / √ t − x ) is the retarted time and v = 1 / √ t + x ) is the advanced time. Let there be noPGW at ( u β ( u ) | u = 0; L ( u ) | u = 1 , (24)while the plasma be homogeneous and at rest: v v ( u ) | u = v v ( u ) | u = 1 / √ v (cid:12)(cid:12) u = v (cid:12)(cid:12) u = 0; ε ( u ) | u ; p ( u ) | u = p (25)9 p = p ( ε ) is the plasma pressure, v k is its dynamic velocity vector) and a homogeneous magneticfield is directed in the ( x , x ) plane: H ( u ) | u = H cos Ω; H ( u ) | u = H sin Ω; H ( u ) | u = 0; H α ( u ) | u = 0 , (26)where Ω is the angle between the axis Ox (the PGW propagation direction) and the magnetic field H direction. The conditions (26) correspond to the vector potential : A v = A u = A = 0; A = H ( x sin Ω − x cos Ω); ( u . (27)The exact solution of the relativistic MHD equations against the metrics background (22). Theexact solution of the relativistic MHD equations against the metrics background (22) obtained in [1]satisfies the initial conditions (24) - (26) and is determined by the governing function :) obtained in[1] satisfies the initial conditions (25) - (27) and is determined by the governing function :∆( u ) Df = 1 − α ( e β − , (28)where α is a dimensionless parameter: α = H sin Ω4 π ( ε + p ) . (29)This solution contains a physical singularity on the hypersurface Σ : u = u ∗ :∆( u ∗ ) = 1 − α ( u ∗ )( e β ( u ∗ ) −
1) = 0 , (30)where the plasma and magnetic field energy densities tend to infinity and the dynamic velocity of theplasma as a whole tends to the velocity of light in the PGW propagation direction. In this case theratio of the magnetic field energy density and the plasma energy tends to infinity. This singularity is agravimagnetic shock wave (GMSW, [1]), spreading in the PGW propagation direction at a subluminalvelocity. According to Eq. (30), necessary conditions for the occurrence of the singularity are β ( u ) >
0; (31) α > . (32)An extrimely important fact is that a singular state is even possible in a weak PGW ( | β | ≪ α ≫ u = u ∗ : β ( u ∗ ) = 1 / (2 α ) . (33)In particular, for a barotropic equation of state ( p = kε , 0 k < ε = ε Λ − ν ; (34) v v = 1 √ L ν ∆ ν ; (35)10 u v v = ∆ − (cid:2) Λ − ν + (∆ − L − e − β cot Ω (cid:3) ; (36) H = H Λ = (cid:0) cos Ω + L Λ − ν e β sin Ω (cid:1) , (37)whereΛ = L ( u )∆( u ) , ν = 2 k − k > , and H = 12 F ik F ik is the electromagnetic field invariant, (squared magnetic field strength in the frame of referencecomoving with the plasma).It follows from (34) - (37) that if β >
0, the plasma moves in the GW propagation direction( v = 1 / √ v u − v v ) >
0) and if β <
0, in the opposite direction. The effect is maximum in thePGW propagation direction, that is, perpendicular to that of the original magnetic field strength,and vanishes in the direction parallel to the magnetic field strength.In the case of strictly transversal PGW propagation (Ω = π/ Ox the plasmadrift vanishes, and the component of the plasma physical 3-velocity in the Ox direction v is v = c v u − v v v u + v v = c − ∆ Λ ν Λ ν . (38)The components of the total (including the magnetic field) EMT of the magnetoactive plasma, T αk ( α = 1 ,
4) have a hydromagnetic structure: T αk = ( E + P ) v α v k − P δ αk , (39)where ε H = P H = H π ; E = ε + ε H ; P = p + P H , (40)where P and E are the total pressure and energy density of the magnetoactive plasma. There arisesan energy flow in the plasma in the direction Ox : T = ε + p L (∆ − Λ ν − ν + α e β ) . (41)The parameter ν takes in these formulae the following values in the two extreme cases: ν = (cid:26) k = 0;1; k = 1 / . (42)For a weak GW | β ( u ) | ≪ L ( u ) = 1 + O ( β ) ≈ , (43)the expressions (37), (41) and (38) take the form v c = 1 − ∆ m m ; 11 = 14 ( ε + p )(1 + α )(∆ − n − H H = 1∆ m ( u ) ; (45)where the coefficients m and n take integer values for nonrelativistic ( k = 0) and ultrarelativistic( k = 1 /
3) plasma: k = 0; m = 2; n = 4; k = 1 / m = 3; n = 2 . (46) Since on the singular hypersurface (30) ∆( u ) = 0, the energy densities of the plasma and the magneticfield tend to infinity, and the velocity of the plasma as a whole tends to the speed of light, the totalenergy of the magnetohydrodynamic shock wave and its flow in the GW propagation direction tendto infinity. The singular state emerging in the plasma due to the PGW violates the basic assumption(21) of the weaknesss of GW interaction with the plasma. In a more complete self-consistent problemincluding gravitation, the back reaction of the shock wave upon the PGW should lead to PGW energyloss and its magnitude damping up to the valuesm ax | β | < / α . (47)Thus a GMSW is an effective mechanism of a gravitational wave energy pumping over into plasma[1]. A rigorous solution of the problem of PGW energy transformation into the shock wave energy isonly possible by studying the self-consistent set of the Einstein equations and the MHD equations.Ref. [1] suggested a semiquantitative solution of this problem on the basis ofa simple modelof energo–ballance. Due to its extreme importance, we do not restrict ourselves to [1] and returnto a more complete study of the problem of energy transmission from a GW to magnetoactiveplasma. However, instead of solving the Einstein equations, we make use of their consequence, theconservation law of the total momentum of the system “plasma + gravitational waves”. Clearly,this model is only approximate and cannot replace a rigorous solution to the Einstein equations.According to [10], an arbitrary gravitational field provides the conservation of the system’s totalmomentum P i = 1 c Z ( − g )( T i + t i ) dV (48)where t ik is the energy-momentum pseudotensor of the gravitational field and the integration coversthe whole 3-dimensional space. Let us take into account that the above solution is plane-symmetricand only depends on the retarded “time” u . Consequently the integration over the “plane” ( x , x π/ P isnonzero, we obtain the conservation law of the surface density of the momentum P : P = 1 c + ∞ Z −∞ ( − g )( T i + t i ) x = Const . (49)Let the right semispace x > x < u ∈ [0 , u f ], where t f = √ u f is the gravitational pulse duration. Sincethe integral in Eq. (49) is conserved all the time, let us consider it at t <
0, when the GW hasnot yet reached the magnetoactive plasma, and t f > t >
0, when the GW has reached the plasma.Taking into account that the vacuum solution depends only on the retarded time, we get for theintegral in Eq. (49): u f Z t = t/ √ Z ( T + t ) du + u f Z t/ √ t , (50)where t = t ( β ( u )); t = t ( β ( u )), β ( u ) is the vacuum magnitude of the PGW, β ( u ) is thePGW magnitude with allowance for interaction with the plasma. Transferring one of the integralsto the left-hand side of Eq. (50), we arrive at the relation u Z t = u Z ( T + t ) du, (51)where the variable u = t/ √ > any positive values.A similar law may be written for the plasma total energy; in this case instead of Eq. (51) weobtain: u Z t = u Z ( T − E + t ) du, where E is the total energy density of the unperturbed plasma. Since the relation (51) must be valid at any values of the variable u , the corresponding local relationshould hold: T ( β ) + t ( β ) = t ( β ) , (52)i.e. a local conservation law of the energy flow density should hold, as was assumed in Ref. [1]. Itshould be pointed out that the local conservation law (52) is a direct consequence of the solutionstationarity, i.e. the solution dependence on the retarted time u = ( ct − x ) / √
2. There are two factorspreventing the solution in a rigorous model from being stationary: (1) PGW interaction with the13lasma; (2) the boundary conditions on the surface x = 0. In accordance with the approximation(21), we introduce a small dimensionless parameter χ [1]: χ = πG ( ε + p )(1 + α ) c ω ∼ ω g ω , (53)where ω is the characteristic GW frequency, ω g = 8 πG E c . The approximation (21) is equivalent to the condition χ ≪ . (54)Under the condition (54) the GW velocity tends to that of light, thus providing the required solutionstationarity even in inhomogeneous plasma [24]. Let β = Const > β ∗ . Let us introduce one more dimensionless parameter, the first GMSWparameter ξ : ξ = χ /β ∼ E / E GW (55)where E GW = β ω c / (4 πG ) . Thus, the parameter ξ is of the order oft he ratio of the total magnetoactive plasma energy to thevacuum GW energy.Making use of the solution of the MHD equations in the case of strict transversal PGW propa-gation (Ω = pi/
2) as well as the expression of the total plasma EMT (44) and that for the energyflow of a weak PGW (3), we reduce Eq. (52) to the form˙ β + V ( β ) = β ∗ , (56)where V ( β ) = χ [∆ − n ( β ) −
1] ( ν = 1) (57)is a function of β ; ˙ β is now a derivative in the dimensionless “time” s = √ ωu/c .Introduce the relative PGW magnitude: q = β/β ; q ∗ = β ∗ /β . (58)Then Eq. (56) may be rewritten in the form˙ q + V ( q ) = ˙ q ∗ , (59)where V ( q ) = ξ [(1 − Υ q ) − n −
1] (60)and a new dimensionless parameter has been introduced:Υ = 2 α β (61)14the second GMSW parameter). The total energy conservation law leads roughly to the same result.Eq. (59) may be treated as an equation with respect to the variable q . On the other hand, (59)completely coincides in its form with the energy conservation law of a 1-dimensional mechanicalsystem described by the canonical variables { q ( s ) , ˙ q ( s ) } [23], where V ( q ) is the potential, ˙ q is itskinetic energy and ˙ q ∗ = E is its total energy. Fig. 4. shows the qualitative form of the potential V ( q ). Figure: 4. Potential V ( q ) of Eq.(59) Two points on the potential curve (A and B) correspond to any positive value of E . These pointsare the system trajectory turning points. No real system states exist under the potential curve V ( q ).At the point q = q c = Υ − ; ( β = 1 / α ) (62) V ( q c ) → ∞ . (63)To analyze the system behaviour, let us suppose that the moment s = 0 corresponds to the frontedge of the GW, while β ∗ ≈ β sin s ⇒ q ∗ ≈ sin( s ) . (64)Thus, provided the initial conditions (24) are satisfied, the system always starts from the point S along the line ( AB ) towards A (for β > q in the system is q ( A ). This is the smallest root q − = q ( χ, Υ , E ) of thealgebraic equation V ( q ) − E = 0 . (65)The maximum attainable PGW magnitude in the system, β max , is β max = q − β . (66)Thus q − coincides in its sense with the “PGW magnitude damping factor” γ introduced Ref. [1].Solving Eq. (65), we obtain the required root q − : q − = 1Υ " − (cid:18) q ∗ ξ (cid:19) − /n . (67)15rom (65) it follows that always q − Υ − , (68)and also, as E → q − ≈ ˙ q ∗ ξ → . (69)With increasing E this magnitude grows and for E → ∞ it reaches the value ( q = q c ): β max → β Υ . (70)After the turning point the GW magnitude diminishies, reaching negative values. For s → + ∞ s → + ∞ β ′ → β ′∞ = Const < β ∼ β ′∞ u → −∞ ;the metric (22) degenerates ( g → , g → −∞ ); the only nonzero components of the curvaturetensor take the following form due to the Einstein equations (see (23)): R u u = ( L ) ′ β ′∞ exp(2 β ′∞ u ) → − R u u = ( L ) ′ β ′∞ exp(2 β ′∞ u ) → + ∞ . Thus, as s → + ∞ , a true singularity is formed in the system. It is easily verified that in this case H → ε → V → − c . The plasma in the final state moves to meet the original GW direction.this reverse of the plasma needs a more detailed self-consistent analysis. Let us pass to a more detailed study on the selfconsistent motion of the system. From Eq. (59) weobtain the differential equation dq/ds = ± p q ∗ − V ( q ) (71)where the plus sign is chosen before and the minus sign after the turning point q − . It is helpful tosolve and analyze Eq. (71) using the new dimensionless variables: ∆( β ) and S = Υ s ≡ √ ωu (72)Substituting into (71), for example, q ∗ ( s ) = sin s , we reduce it to the form d ∆ /dS = ∓ p cos S/ Υ − V (∆) , (73)moreover, the initial condition is to be fulfilled:∆(0) = 1 . (74)Figs. 5–11 show some results of numerical integration of Eq. (73) with the initial condition(74). An analysis of the formulae describing GMSW and numerical calculations make it possible todiscover a number of general laws of the GMSW excitation process in homogeneous and isotropicplasma under the condition that the PGW propagation is strictly transversal:16. A GMSW is completely described by three nonnegative dimensionless parameters: the param-eter k in the plasma equation of state, the first ( ξ ) and the second (Υ) GMSW parameters.2. Necessary conditions for GMSW excitation are (31) and (32):Υ > . (75)3. The only criterion of strong GW absorption is, according to (69), a large value of the secondGMSW parameter (Υ):Υ ≫ . (76)4. Under these conditions a maximum response of the plasma to GW is achieved when the valuesof the first GMSW parameter are small: ξ ≪ . (77)5. The plasma response to GW is a single pulse, and the shock wave stage is always replaced by areverse stage, when the plasma turns back. Simultaneously its density, pressure and magneticfield strength fall off.6. The ultrarelativistic ( k = 1 /
3) plasma response is much greater (by approximately 2 orders)than that of a plasma with the nonrelativistic equation of state ( k = 0), and in ultrarelativisticplasma the pulse duration is also slightly greater.7. The profiles of the plasma response at sufficiently large values of the second GMSW parameter(Υ >
5) actually coincide on the S scale. This means that in the conventional time scale t thepulse duration is inversly proportional to the second parameter, or more precisely∆ τ ≈ π ω Υ = T π Υ , (Υ > , (78)where T is the GW period.8. With Υ < ∼ τ T / π/ (2 ω ) . (79)9. A decrease in the first GMSW parameter causes a rapid increases in the response (roughlypropotional to 1 /ξ ); simultaneously increases the pulse duration approximately by a factor of2.10. A maximum response is achieved at the instant S ≈ Here and further on, speaking of a maximum response of the plasma, we mean its energy characteristics: theplasma energy flow density and the magnetic field energy density. E Σ , is f or ultrarelativistic plasma of the order of E Σ ∼ cH / ( ωξ Υ) . (80)12. The shock wave energy is taken from the GW energy, so under the conditions (76) and (77)the GMSW is an effective mechanism of gravitational waves energy transformation into otherforms of energy.Figure: 5. Relative GW magnitude in ultrarelativistic plasma, q ( S ) ; everywhere Υ = 10 . 1 —- ξ = 1 ; 2 — ξ = 0 , ; 3 —- ξ = 0 , . Figure: 6.
Physical velocity of plasma in GW field, v ( S ) /c : 1 —-nonrelativistic, 2 —- ultrarelativistic equation of state; Υ = 10; ξ = 0 , . Physical velocity of plasma in GW field, v ( S ) /c : 1 – Υ = 10; 2 – Υ = 100 . Everywhere ξ = 0 , . The lines practicallycoincide. Figure: 8.
Dimensionless density of ultrarelativistic plasma energyflow, T ( S ) = 4 πGT /β ω c : 1 —- Υ = 10 ;2 —- Υ = 100.Everywhere ξ = 0 , . The lines practically coincide. Magnetic field strength lg H ( S ) /H in GW field, 1—- k = 0 ,2 —- k = 1 / ; Υ = 10 ; ξ = 0 , . . Figure: 10.
Magnetic field strength in GW field for ultrarelativis-tic plasma, H ( S ) /H : 1 —- Υ = 10 ; 2 —-
Υ = 100 . Everywhere ξ = 0 , . The lines practically coincide. Magnetic field strength in GW field for ultrarelativis-tic plasma, lg H ( S ) /H : 1 —- ξ = 1 ; 2 —- ξ = 0 , ; 3 —- ξ = 0 , . Everywhere Υ = 10 . Figure: 12.
Magnetic field strength in GW field for ultrarelativis-tic plasma, H ( S ) /H : 1 —- Υ = 1 ; 2 ——-
Υ = 10 ; 3 ——-
Υ = 100 . Everywhere ξ = 0 , . The lines 2 and 3 practicallycoincide. In [1] it was shown that in the magnetospheres ofneutron stars performing quadrupole oscillations,large values of the second GMSW parameter are realized. That is, the necessary condition for GMSWexcitation (76) is fulfilled. Let us study this problem in more detail. The electron number density ina pulsar magnetosphere, n e ( r ), which is necessary for calculating the parameter Υ, may be obtainedby dimensional estimation from the Maxwell equations [12]: n e ( r ) ∼ H ( r ) / (4 πer ) . (81)21urther, as is known from (see [15]), the pulsar period slowing-down rate t is connected with thepulsar parameters as follows: t ≈ c M P π H R , (82)where R is the neutron star radius, M is its mass, H is the magnetic field strength at the stellarsurface, P is the rotation period. This formula gives for the pulsar NP 0532 N P H ≈ · G (see Table 3) according to the known slowing-down rate of this pulsar. Actually, as pointed out inRef. [11], the magnetic field strength at the surface of NP 0532 is somewhat smaller than the valueobtained on the basis of (82), and is of the order of 10 G. Further on we use this value. In thefigures shown below (unless specially indicated) the following values of the parameters are adopted: R = 1 , · cm, β ( R ) = 10 − and the magnetic field in the magnetosphere is assumed to be dipole: H ( r ) ∼ ( R/r ) .Table 3. GMSW in a neutron star magnetosphere r g /R δM T n τ n Em/ ∆ R L g / ∆ R R H ( R ) √ ∆ R α Υ E m ∗ Comments to Table 3. The data placed in columns 1 - 7, 9 and 12 are taken from the book [9]. δM = M/M ⊙ is the neutron star mass related to the Solar mass; R is the star radius in km; T n isthe neutron star eigen-oscillation period in the basic quadrupole mode (in milliseconds); τ n is theoscillation damping time (in seconds); ∆ R = h ( δR/R ) i is the root-mean-square relative magnitudeof the neutron star oscillations; E m is the oscillations kinetic energy in erg; L g is the the star’sgravitational luminosity in erg/sec. The GW magnitude value at the neutron star surface, β ( R ), isassumed to be equal to 10 − . H ( R ) is the magnetic field strength in gauss (G). The data placed incolumns 8 ÷
11 are calculated using Eqs. (3), (82) and (81) for the observed Crab pulsar (NP 0532)parameters; P = 0,033 s, t = 2500 years. The data placed in the last line of the table (columns 2 ÷ α and Υ are given for the magnetosphere near the stellar surface.If the magnetic field of a neutron star is described as that of a dipole, then the geographic angleΘ (counted from the magnetic equator) will be connected with the above angle Ω by the relationΩ = π/ − Θ. Therefore the GMSW excitation condition depends on the angle Θ:sin Θ < − α | β | ∼ − Υ − . Thus, in the magnetosphere of a neutron star (or a Supernova) a GMSW can be excited in thevicinity of the magnetic equator, similarly to pulsars, with a knife radiation pattern. In this region,as was demonstrated by the above examples, the gravitational radiation can be absorbed almostcompletely by shock wave excitation. Fig. 13 shows the radial dependence of the GMSW parameters22agnetic equator plane (Θ = 0) of a neutron star magnetosphere with the above parameters R , H ( R )and β ( R ) . According to Table 3, in the case of NP 0532 such values of the parameters β ( R ) and R correspond to the gravitational radiation power L g ≈ , · erg/sec.Figure: 13. GMSW parameters in a neutron star magnetospherefor a dipole radial dependence of the magnetic field strength. Theline corresponds to lg ξ ; the points correspond to lg Υ . As is seen from Fig. 13, the region favourable for the GMSW formation lies in the range 6 R ÷ R ,i.e., where the local magnetic field strength is 3∆10 ÷ G. When the GMSW pulse passes,these local values increase by a factor of 10 to 30. Thus a neutron star in whose magnetospherea GMSW zone is formed, is able to radiate GW only from its magnetic poles, like pulsars with apencil radiation pattern. In this case the probability of direct GW detection from such sources isdrastically decreased. However, GMSW open another way for GW observation. A formed GMSWcarries above all strong magnetic fields. They move from the neutron star in the magnetic equatorplane, and therefore should lead to an increased pulsar magnetic bremsstrahlung intensity at themoment when the GMSW front passes. Thus anomalous electromagnetic radiation flashes in thepulsar radiation should be observed at moments when quadrupole oscillations are excited.The total magnetic bremsstrahlung intensity of a relativistic electron is proportional to squaredmagnetic field strength [10]: I = 2 e H ~p / (3 m c ) , (83)where ~p = m~v p − ~v /c is the electron momentum.Therefore the curves H ( S ) shown in Figs. 9–12, actually describe the time dependence of themagnetosphere magnetic bremsstrahlung intensity, i.e. the local electromagnetic response to thegravitational radiation of the neutron star. Such a response might be detected by an observer at restplaced in the magnetosphere and screened from the electromagnetic radiation coming from otherregions. The situation is more difficult with a total response of the magnetosphere to the GW,detected by a distant observer. We will later return to this problem. In all further figures the magnetosphere is considered in the magnetic equator plane.
23s was mentioned above, the response of a homogeneous magnetoactive plasma even to strictlyperiodic gravitational radiation has the form of a single pulse. But even if it were not the case, theresponse of a neutron star magnetosphere to a GW would still have the same form. Indeed, a shockwave (GMSW), emerging after the excitation of quadrupole oscillations of a neutron star, shouldthrow the equatorial sector of the magnetosphere away into the interstellar space. For the next pulseto be formed, the magnetosphere should restore. The necessary time for its restoration is of theorder of ∆ t ∼ l/v s where l is the characteristic size of the magnetosphere and v s is the velocity ofsound. For a typical neutron star magnetosphere ∆ t ∼ According to Table 3, neutron star eigen-oscillation periods vary depending on the stellar mass inthe range of 0,3 to 1,2 ms. Therefore the local duration of GMSW pulses should, by (79), satisfy thecondition∆ τ < · − ÷ · − s, (84) (84)i.e., be shorter than 70 to 300 microseconds. Fig. 14 shows the dependence of the GMSW pulse localduration on the radial coordinate r, ∆ τ ( r ), calculated according to Eq. (78).Figure: 14. The dependence of local pulse duration ∆ τ (in mi-croseconds) on the distance form the star centre r/c (in microsec-onds). The duration was calculated by Eq. (78). The blackcircles mark the boundaries of a region where the GMSW effectis sufficiently well-developed. Outside this region the result is ofa formal nature. On the left boundary of the region the pulseduration should quickly grow up to the values (84). As is seen from Fig. 14, the actual local pulse duration in the region where the GMSW mechanismis fairly effective, ranges from 1 to 10 µ s. Thus in this region ∆ τ ∼ − r the following condition isfulfilled with a large spare:∆ τ c ≪ r, (85)24ustifying the use of the GMSW formulae for describing an inhomogeneous magnetosphere.An observer out of the neutron star magnetosphere would detect the magnetic bremsstrahlungfrom the magnetospheric electrons during the whole time while the local pulse passes through themagnetospheric region r − < r < r + where favourable conditions for GMSW development are realized,namely, (76), (77). With a certain caution these conditions may be specified: ξ < , r − and Υ > r + ). So the size of the GMSW zone is∆ r = r + − r − . If ∆ r <
0, a GMSW zone does not appear in the neutron star magnetosphere at all. Since, as wehave seen, the GMSW pulse spreading velocity is very close to c , the whole magnetic bremsstrahlungdetected by remote observer will be concentrated in the time “window” of duration ∆ T ∆ T = ∆ r/c = t + − t − . (86)where t ± = r ± /c are the instants when the GW leading front reaches the upper and lower boundariesof the GMSW zone. Near its boundaries r − and r + the GMSW is poorly developed (in the firstcase the first GMSW parameter is too large, in the second case the second parameter is too small).Therefore the intensity of the electromagnetic signal is small near the boundaries of the window,while in its medium domain a radiation maximum (large Υ and small ξ ) is achieved. The form ofthe signal itself is yet to be calculated. Fig. 15 shows the dependence of the window width on themagnetic field strength and the GW magnitude.Figure: 15. Dependence of the GMSW existence range ∆ T (in µs ) in the magnetosphere of a neutron star of radius R = 1 , · cm on the magnetic field strength H ( R ) , (related to G) andthe GW magnitude β ( R ) . The thin line corresponds to β ( R ) =5 · − , the points to β ( R ) = 7 , · − , the thick line to β ( R ) = 10 − . Note that the local density of the bremsstrahlung intensity W ( t, r ) is determined by the local val-ues of the squared magnetic field strength, H ( t, r ), and the local electron number density in the25agnetosphere, n e ( t, r ). For ultrarelativistic electrons by [10] it is W = 2 e H m c (cid:18) Emc (cid:19) n e , where E is the electron energy. Further we will assume that the size of a local pulse is much smallerthan both the characteristic scale of the magnetospheric inhomogeneity r and the window width∆ r . Thus the retarded time u is a quick variable and the radial coordinate r is a slow variable.Then a GMSW may be described by the formulae for homogeneous plasma, where it is necessary touse the local values of the GMSW parameters, ξ ( r ) and Υ( r ). Mean while in the exact stationarysolutions there arises a weak dependence on the radial coordinate r , i.e. the solution will be weaklynonstationary and the nonstationarity will show it selfin the form of a functional dependence of theGMSW solutions on the local values of the parameters, e.g.,∆( u ; r ) = 1 − Υ( r ) q ( u ; r ) , etc. Thus, from the particle number conservation law and the solution stationarity it follows: L n e ( r, t ) v v ( u ) = Const ≈ √ n e ( r ) , (87)where n oe ( r ) is the unperturbed electron number density in the magnetosphere. Taking into accountthat En e = ε/ n e ( r, t ) = n e ( r )∆ − / ( u ); (88) E ( r, t ) = E ( r )∆ − / ( u ); (89)where E ( r ) is the unperturbed energy of magnetospheric electrons and H ( u ) = H ( r )∆ − ( u ) . (90)Thus we get the following relation for the magnetic bremsstrahlung of the ultrarelativistic magneto-sphere: W ( r, t ) = W ( r )∆ − / ( u ) , (91)where W ( r ) is the magnetic bremsstrahlung intensity density for an nonperturbed magnetosphere.Eq. (91) needs a relativistic correction taking into account the plasma motion: the radiation densityshould be multiplied by the relativistic factor (1 − v /c ) − / . The net result is W ( r, t ) = W ( r )∆ − /
12 (∆ / + ∆ − / ) . (92)Thus W ( r, t ) ∼ W ( r )∆ − . Note a large value of the exponent of the governing function ∆( u ),whichleads to a large steepness of the local magnetic bremsstrahlung pulse. Integrating Eq. (92) overthe GMSW zone near the magnetic equator, we obtain a formula for the variation of the completemagnetospheric magnetic bremsstrahlung resulting from a GW pass:∆ J ( t ) = 2 π Θ r + Z r − Φ(∆) W ( r ) r dr, (93)26hereΦ(∆) = 12 ∆ − / (∆ / + ∆ − / ) −
1; (94)Θ is the angle of the knife radiation pattern. This formula completely describes the shape of thesignal to be detected by a remote observer. The expression in the square brackets in (93) is notablynonzero only in the domain of the local GMSW pulse, i.e. in the domain0 < S π ⇔ ct − π/ ( ω Υ( r )) r < ct. (95)Therefore the integral (93) tends to zero for t < r − /c and t > r + /c + π/ ( ω Υ( r )). The observedpulse duration is formally determined by these limits. However, since, as noted above, near the zoneboundaries the GMSW is poorly developed, the actually observed pulse duration (more precisely, itshalf-width) can turn out to be smaller than this value. Without solving the problem of the observedpulse shape, let us estimate its magnitude in its medium domain r − /c < t < r + /c, (96)when the pulse local duration is much smaller than the window width:∆ τ ≪ ∆ T. (97)Under these conditions the integrand in Eq. (93) is δ -like, therefore with a good precision thefollowing estimate is valid:∆ J ( t ) ≈ π < W ( r ) r > c Θ ω Υ S − Z S + Φ(∆( S )) dS, (98)where S ± = ω Υ( t − r ± /c ) , < W ( r ) r > and the value of r ∗ is determined from the equation r ∗ = ct − / (( ω Υ( r ∗ )) . (99)Thus ∆ J ( t ) ∼ π Θ ω Υ( r ∗ ) W ( r ∗ ) r ∗ c ∆ T h ∆ − i , (100)where h ∆ − i = 1∆ S S − Z S + ∆ − ( S ) dS ; (101)∆ S = ∆ T ω
Υ. Thus, by order of magnitude, the intensity change of the magnetic bremsstrahlung(in its maximum) as a result of the GMSW excitation is equal to a product of the unperturbedmagnetospheric radiation intensity in the GMSW zone by the dimensionless factor h ∆ − i (which canreach 10 − ). We will return to a calculation of the observed pulse shape in our next paper. Here werestrict ourselves to estimates of the complete energy of a GMSW pulse (Fig. 16).27igure: 16. The radial coordinate ( r ) dependence of the pulseenergy E pulse = 4 πr E Σ (in erg) by Eq. (80). The GMSW effect issufficiently well-developed only in the range , < lg ( r/R ) < , ( , · cm < r < , · cm). Outside this region the resultsare of clearly formal nature. The pulse energy should rapidly fallnear the boundaries of this range. There naturally arises the question of a source ofpulsars’ quadrupole oscillations. A possible energysource for such oscillations might be represented by explosive nuclear reactions with heavy hyperonslike n + n ←→ p + Σ + taking place in neutron stars cores at densities over 10 g/cm [13]. The pres-ence of strong magnetic fields should lead to an asymmetry of the explosions, i.e. to the quadrupolemoment excitation. For such processes to occur in a neutron star, it must be sufficiently young.Numerical simulations of the process of a neutron star cooling shows [14] that after a Supernovaexplosion the neutron star temperature falls approximately by an order of magnitude in 10 years.Consequently, GMSW should be sought in radiation from sufficiently young pulsars formed no earlierthan 10000 years ago. A pulsar with the required parameters does exist: it is the famous pulsar in the Crab nebula, NP0532, life-time is less than 1000 years (the 1054 Supernova). This pulsar is the youngest of allknown ones (and consequently the hottest), it has the shortest period (at least among the closestpulsars, enumerated in Table 2): T = 0 ,
033 sec. What is suprising is that the radio emission of thispulsar contains anomalies which can be with a large degree of confidence identified with the GMSW.Namely: there are single irregular, the socalled giant pulses (on the average a pulse in every 5 to 10minutes) [11]. The radiation intensity in the giant pulses is a few tens of times higher (by roughly afactor of 60) than in common pulses. But the most interesting is that the duration of the giant pulsesis no more than 9 · − s, i.e., almost by 2(!) orders shorter than that of the common pulses fromNP 0532 ( τ ∼ · − s). The common pulse duration, as is easily seen, is about 1/7 of the NP 0532rotation period, so that the common pulses are clearly explained geometrically by pulsar rotation.28he giant pulse duration is 300 times shorter than the pulsar rotation period, and consequently theexistence of the giant pulses has not yet found any satisfactory theoretical model.However, the giant pulses are easily explained by the GMSW, and their duration is not relatedto the pulsar rotation period, or an angle of the knife radiation pattern, but to its eigen-oscillationperiod, T . A comparison of the NP 0532 giant pulse duration with that of a GMSW pulse (84)shows a striking coincidence. Indeed, for the NP 0532 pulsar, as known from the annihilation lineshift in the γ radiation spectrum (400 keV instead of 511 keV), the gravitational redshift is known[21, 22]:∆ E/E = M G/Rc = 0 , . Then from Table 3 we find the pulsar mass: M = 1 , M ⊕ and the corresponding neutron starradius: R = 12 km. According to Eq. (79) and Table 3, the GMSW pulse duration for the NP0532 pulsar should be about 87 microseconds, while the observed NP 0532 giant pulse duration isapproximately 90 microseconds ?? (!).For the Crab pulsar t = 2 , · years ?? ; then, setting δM = 1 , R = 12 km, we find from(82): H ∼ , · G. The angle Θ of the knife radiation pattern is connected with the observedpulse duration τ by the relation Θ = 2 πτ /P . For the pulsar NP 0532 this angle is 0 , ≈ o .Assuming the complete pulsar luminosity in the continuous spectrum to be about 5 · erg/s, wefind the giant pulse intensity recalculated for the whole neutron star surface: L giant ≈ · erg/s . As has been pointed out above, the real magnetic field strength on the pulsar NP 0532 surface is 10 G[11]. To explain the observed giant pulse emission power, one needs GW magnitude values on thestellar surface of the order of 10 − . Note that, according to Fig. 15, the window width ∆ T ≈ µ scorresponds observed GMSW pulse should be about 90 µ s, which again precisely coincides with theobserved giant pulse duration!The indicated magnitude β ( R ) corresponds to the gravitational radiation power of the order of4 · erg/s and the neutron star oscillations energy about E m ≈ · erg. In this case the neutronstar surface oscillation magnitude is about 1 cm. Taking into account that in the whole lifetime ofNP 0532 (1000 years) approximately 7 · giant pulses have been emitted, we get an estimate of theenergy carried away from the neutron star by GW for the whole time of its existence: E = 2 , · erg. It is 10 − of the rest energy of this neutron star, which completely agrees with the assumptionof a permanent rebuilding of its core. Thus, to a high degree of confidence we can state that thegiant pulses observed in the pulsar NP 0532 radiation are optical manifestations of gravimagneticshock waves (GMSW) excited by the gravitational radiation of the neutron star corresponding tothe pulsar NP 0532 [20]. Besides NP 0532, among all known pulsars only PSR 0833 seems to be able to emit (but more seldom)giant pulses. Other pulsars are too old for it. Therefore it is necessary to concentrate the main efforton observations of these two pulsars. It should be stressed that there is no other mechanism ableto accelerate a shock wave to subluminal velocities. Therefore an investigation of the giant pulsespectrum in the X-ray range, aimed at discovering a violet shift in the radiation spectrum, is ofutmost importance. A comprehensive study of the giant pulses (their shapes and instantaneous29pectrum) will allow one not only to verify the existence of gravitational radiation, but also to getadditional information on the neutron stars structure and the processes in their interior. In turn itis necessary to study the GMSW pulse formation in detail theoretically.
Acknowledgement
The author is thankful to S.V. Sushkov for help in carring out the calculations and to N.A. Zverevafor translating the article into English.