Classical Approach to the Theory of Elementary Particles

Transcription

1 Classial Appoah to the Theoy of Elementay Patiles By Yui N. Keilman Abstat: Pesented hee is an attempt to modify /extend lassial eletodynamis (CED) in ode to enable the lassial appoah (the appoah based only on lassial piniples that wee developed befoe the intodution of quantum theoy) to the theoy of elementay patiles. In physis, dualism efes to media with popeties that an be assoiated with the mehanis of two diffeent phenomena. Beause these two phenomena's mehanis ae mutually exlusive, both ae needed in ode to desibe the possible behavios. (Retieved fom " Usually, dualism in physis is explained by the example: wave / patile. This stated with a lassial eletomagneti wave / quantum photon example fo light and poeeded with a lassial patile / quantum wave fo massive patiles. I want to peeive this dualism altogethe as lassial / quantum appoah to the physial objet. This sounds plausible (but we have to emembe that these two appoahes ae "mutually exlusive" and neve intemix them). Taking the idea of lassial appoah to the elementay patiles seiously, we an see that the existing lassial eletodynamis (CED) has to be modified in ways:. Many things wee intodued in CED duing the "quantum ea" in ode to naow the gap between CED and Quantum theoy and also in ode to make CED a sevant of Quantum theoy. Aoding to the lassial piniples that wee established befoe the ise of Quantum theoy, only the eletomagneti field and the field of uent density wee to epesent the physial eality. The potential was not unique and was teated only as a mathematial tool. The Lagangian was also thought of as the unique physial quantity that an not be expessed though potential. Hee I mean the so alled "inteation tems" in the Lagangian that wee intodued to ease the bidge between CED and the Quantum petubation theoy. The CED as we have it now is atually a pat of Quantum theoy. This should be undone (intelligently) to make CED a eal altenative to Quantum theoy (we expet that both theoies will enjoy a patial expeimental onfimation).. The CED (as it was in old times and is now) ontains "singulaities" as models fo the elementay patiles whih deem to be "pointlike". I do not think that infinite quantities exist in physial eality. The tue lassial appoah would be to eplae these singulaities by some extended field stutues. Theefoe CED has to be adially modified. But the way to modify CED will not be the easy one as one may expet. So let us stat with the simplest possible Lagangian:

2 ab d ab g g FaFbd g j a jb () 6 k whee F ik is the eletomagneti field, j k is the field of uent density, and k will be the new onstant of the theoy. We suggest that the onnetion between the fields (also an be alled inteation ) is given by Maxwell's equations with uents (half of Maxwell's system): i ik 4 j F k dive j oth E 4,, j () 4 whih is given as a peliminay ondition befoe any vaiation. The inteation ondition () is an auxiliay ondition with espet to the Lagangian (). The () also povides the onsevation of hage: j a a= (the antisymmety of the eletomagneti field is also a peliminay ondition). The othe half of Maxwell's equations is: * * F, F e F, divh, ote ik ik iklm lm H k (a) This will be the esult of the vaiation poedue late on. Befoe we stat with the vaiation of fields, let us find the enegy-momentum tenso that oesponds to the Lagangian (). The meti tenso in lassial 4-spae is g ik =diag[,-,-,-] (we assume =). Let us onside an abitay vaiation of a meti tenso but on the ondition that this vaiation does not intodue any uvatue in spae. This vaiation is: (3) g ik i k k i whee k is an abitay but small veto. One has to use the mathematial appaatus of Geneal Relativity to hek it with the vaiation (3) the Riemann uvatue tenso emains zeo in the fist ode. Assuming that the ovaiant omponents of the physial fields ae kept onstant (then the ontavaiant omponents will be vaied as a esult of the vaiation of the meti tenso, but we do not use them -- see ()) we an alulate the vaiation of the ation. The vaiation of the squae oot of the deteminant of the meti tenso is: i k g gg ik g (this esult an be found in textbooks on Theoy of Field). The vaiation of the ation beomes: i k i k S g gd T gd ik ik ik g 4 T g F F F F g j j j j g ab ab a ik ia kb ab ik i k a ik 4 6 k k (4) ik The integal in (4) an be tansfomed to the fom T ki gd if we onside that the integal ove some emote losed sufae tuns to zeo due to the smallness of T ik and the integal ove 3-d volume at t and t tuns to zeo due to the assumption: i = at these times. Sine i ae abitay small funtions (between t and t), the equiement S= yields:

3 T (5) ia a We have found the unique definition of the enegy-momentum tenso (4). If we want ation to be minimum with espet to the abitay vaiation of the meti tenso in flat spae then (5) should be satisfied. Let us ewite the enegy-momentum tenso in 3-d fom: T ( E H ) [( j ) ( j) ] 8 k T ( E H E H ) [( j ) ( j) ( j ) ] 8 k 4 T ( E H E H ) j j k 4 T ( E E H H ) j j 4 k Notie, that we did not used Maxwell's o any othe field equations so fa. Substituting (4) in (5) and using the Maxwell's equation () and antisymmety of F ik, we obtain: (4a) a k k k j Fai ja i ji a, j E j & j j H otj (6) A new Dynamis The equation (6) we an all a Dynamis Equation. It is nonlinea equation. It an be shown that any 3-d hypesufae that is eveywhee tangent to the veto J k is a haateisti of the equation (6). The equation (6) is automatially satisfied in vauum (J k =). An anothe possibility (J k ) will be the inside egion of the elementay patile. The bounday between these egions will be a haateisti sufae. Inside the elementay patile the dynamis equation (6) desibes, as we all it, Mateial Continuum. A Mateial Continuum an not be divided into a system of mateial points. The Relativisti (o Newton's) Dynamis Equation of CED, that desibes behavio of the patile as a whole, ompletely disappeas inside the elementay patile. Thee is no foe, no veloity o aeleation inside the patile. A kinemati state of the Mateial Continuum is defined by the field of uent density j k. A wold line of uent j k is not a wold line of a mateial point. That allows us to deny any ausal onnetion between the points on this line. In a onsequene j k an be spae-like as well as time-like. That is in no ontadition with the fat that the bounday of the patile an not exeed the speed of light. If N k is a nomal to the bounday of a patile then N k j k = should be satisfied eveywhee on this bounday. The sala multipliation of N k and (6) gives: 3

4 k F N a j b b a a a ab j j N a b N, j j a a 4 It is lealy seen that the left pat of this equation is definite on the bounday of the patile. So should be and the ight pat. That means that on the bounday with vauum the invaiant density of hage should be zeo. By the equation (6) we have gotten something vey impotant, but we ae just on the beginning of a diffiult and unetain jouney. Let us etae some ou steps.. Ideal Patile (IP) The fist question was: is thee a stati solution to the system of equations (6) plus Maxwell's equations (), (a)? The solution was found: k j R ( z), z n ; j, n z ; j (7) 4 n E kr ( z), z n ; E ( ) kn / z, n z ; H whee z = k o, R o (z)=sin (z)/z, R (z)=sin (z)/z -os (z)/z - the spheial Bessel funtions, is an abitay onstant. The full hage of IP is: n e 4 j d ( ) n (8) k To find the mass we should integate T ove the volume: 8 k m T sin( ) dd d [ E ( j ) ] d k 6 z (9) dz { [ R ( z) R ( z)] z dz ( n) } n k z k z. The seond question was: is thee a stati solution that poses a spin? Afte a long seies of unsuessful attempts it was poven (see Appendix) the theoem that stati spin is impossible with any enegy-momentum tenso if its divegene vanishes eveywhee and it is good on a emote sphee (stati spin is possible in a stati extenal magneti field but the oesponding enegy-momentum tenso is not good at infinity). The seah fo a dependent of time solution with spin gave no esults. Afte these unsuessful attempts it was ealized that the main esult of the equation (6) is the bounday of the patile (haateisti of the oesponding PDE system). We did not expet it duing the deivation of (6). We did not ask fo touble, so to speak, but we got it. The mee existene of the bounday of the elementay patile is a vey impotant hange in ou mathematial appaatus and it equies all possible attention. Now, even befoe we an get the onsevation equation (5) o pefom any othe vaiation, we have to take ae of the bounday of the patile. We have: k i k k i k i k i S T gd T T d T gd T gd i k i i in k i k i k in Sine k is abitay, we ae oming to the onsevation (5) inside and side sepaately, and an additional equiement on the sufae of disuption : ia T Na ontinuous (5a) 4

5 whee N k is a nomal to the sufae. Let us ewite (5a) in 3-d fom. Suppose the bounday of the patile is given in the fom: f(t,y,z)-x =. The ovaiant nomal to this sufae will be (f, t - f, y f, z ). The (5a) podues 4 onditions: 6 E H ( )., ( ) ( ) f& E H n ont j j k 6 ( E H ) f& E H n ( E n) E ( H n) H ont. k (5b) Now we an vay the physial fields J k and F ik keeping in mind that the spae is divided at least in two pats by some disuption sufae. Fo that eason we have to apply a vaiation poedue sepaately fo eah egion. We laim that the peliminay ondition () holds in both these egions, but now we even not sue if it should hold on the bounday itself. In ou system we have unknown independent funtions (4 funtions in J k and 6 funtions in F ik ). These funtions aleady have to satisfy to 8 equations (4 equations in () and 4 equations in (6)). We have only degees of feedom left. We an not vaiate neithe J k no F ik by the staight fowad poedue. Let us employ hee the Lagange method of indefinite fatos. Let us intodue a modified Lagangian: a b ' A ja Fab () 4 whee A k ae 4 indefinite Lagange fatos. Now we have +4=6 degees of feedom and we use them to vay F ik. The vaiation of the field F ik we have to keep zeo on the bounday sine the deivatives of the field ae pesent in (). The Eule's equation gives: Fik Ak i Ai k () inside. The Maxwell's equation (a) follows fom (). We ae not able to vey J k sine thee ae no degees of feedom left, but we aleady have (6) as a dynamis equation. If we do not keep the vaiation of field zeo on the bounday then we will get, in addition to (), the sufae integal: i kl k il A g A g Fikd l 4 Ñ () in whih we want to get id of ( is the bounday sufae of the patile). Let us onside the vauum aound the patile. If we use the same modified Lagangian () in vauum and vay the eletomagneti field in vauum inluding the bounday of the patile, then in addition to the extension of () into the vauum (still, it is not definite whethe () holds on the bounday itself), we will get the sufae integal: i kl k il A g A g Fikd l 4 Ñ (3) Both integals will annihilate if the tangent omponents of a veto potential A k ae ontinuous aoss the patile bounday. The omponent of potential that is pependiula to the bounday of the patile an have a jump. The potential Ak should be hosen so that it satisfies the peliminay ondition (). Using () and (), and not assuming any gauge, we get: 5

6 A A 4 j inside, A A vauum (4) k a a k k k a a k a a a a In addition, as we leaned above, the tangent omponents of potential have to be ontinuous aoss the bounday of the patile. The potential futhe eveals its stange natue. Notie, that the above deivation does not pelude a failue of Maxwell's equations () o (a) on the bounday of the patile. We ame at a vey unusual aangement:. The equations (6), (), and (a) define the solution fo the eleti, magneti, and uent density fields inside the patile.. The same equations define the solution fo the eleti and magneti fields in vauum. 3. Thee ae no diet equiements on the omponents of eletomagneti field and 3 omponents of a uent density on the bounday of the patile (only the omponent of uent nomal to the bounday should be zeo). Instead, we have the 4 nonlinea algebai onditions (5b) on eletomagneti field and uents, and the 3 diet onditions on the tangent omponents of potential (we do not ae ab the omponent of potential that is nomal to the bounday). That means that with the potential we an not solve the physial poblem. Still, the potential is only a mathematial tool (it does not epesent any physial eality in a point of spae (ontay to the eletomagneti field and the field of uent density)). This was the one of the lassial piniples befoe the quantum theoy. Compaison of the two appoahes Above we developed the two possible ways in whih CED an be modified (extended) in ode to appoah the elementay patiles fom the pue lassial point of view:. The "bounday last" appoah. We equested that the "inteation" between the eletomagneti field and the field of uent (expessed by the Maxwell's equation ()) should be the peliminay ondition tue in all the spae. We did not expet any disuption sufae befoe the vaiation. We obtaind the system (), (a), (6) with the onseving enegy-momentum tenso (4), (4a). This system is unlinea due to the equation (6). The haateistis of the system allow a sufae of disuption in a solution. We obtained the stati solution of this system (IP). The whole set of the solutions of the system was investigated puely, but I would like to say that having only Ideal Patile one an not explain muh in the eal wold. The fist eage was to do away with the quantization of the adius of IP. The oesponding patile was alled IP+ and, shue enougth, IP+ violates the enegy-momentum onsevation. I am 9% sue that if we want to ontinue with the "bounday last" appoah, we have to live with this unonsevation. But we ae not questioning the validity of the Maxwell's equations hee. It was mentioned above that IP+ violates the enegy-momentum onsevation. What does it mean? Let us onside two oodinate systems: K the oodinate system in whih the solution IP+ is given, and K' that moves with a speed V in x dietion (we assume =). Using Loentz tansfomation we obtain image of the IP+ solution in K'. If we alulate by integation a global (meaning not onneted to any point of spae) 6

7 onseving linea momentum veto: k k P T d3v t (5) in oodinates K, then we find only time omponent (enegy) is diffeent fom zeo: P. The oesponding integation in K' (ove t'=) will give anothe veto Q k ' with Q ' and Q ' diffeent fom zeo. Now by Loentz tansfomation we an tansfom P k fom K to K' obtaining P k ' (with omponents diffeent fom zeo). If the solution satisfies the onsevation equiements, then we should get: P k '=Q k ' in K'. Let us see what we get fo IP+. m P m z R ( z)os( z), P ; P ' m m V, P ' mv; k m m Q ' ( m mv ) m V, Q ' V ( m m ), m z R ( z ) 3 6k (6) If z=nπ then m=; m stays positive almost at all z; the appoximation is given fo a small V. Fo IP+ it looks like the unonsevation is playing games with the mass.. The "bounday fist" appoah. We state the existene of the bounday fom the vey beginning. We laim that the inteation () is valid inside and side the patile sepaately. We obtained that (), (a), (6) ae valid inside the patile, and (), (a) with a zeo uent ae valid in vauum. We do not question the enegy-momentum onsevation hee. It bings us, in addition, the bounday onditions fo the fields and uents (5a), (5b) whih ae unlinea in natue. In addition, we have the unknown in onventional CED equiement on the tangent omponents of potential on the patile bounday and on infinity. In this appoah the Maxwell's equations an fail on the bounday of the patile (the eletomagneti field and uents an have a jump). IP: Let us obtain a simplest stati solution in the "boundat fist" appoah with eleti hage and eleti field only. We have: k A R ( z) R ( z) bz, z z; j R ( z); 4 z A b, z z ; b R ( z) R ( z); z (7) z E kr ( z), z z; E kb, z z ; z 3 3 m z R ( z) z R ( z) z R ( z) R ( z) ; k Now the position of the bounday z is abitay. In geneal, the eleti field has a jump on the bounday of IP. Aodingly, the total hage has diffeent atual and effetive values (the effetive hage is the one that oesponds to the vauum field of the patile). If z is equal to nπ then the jump disappeas and IP tuns to IP. The IP fully satisfies to the enegy-momentum onsevation law (5), (5a), (5b). The potential A 7

8 is a tangent omponent, so, it should be ontinuous ove the patile's bounday, and should have a easonable value at infinity. If we want the integal () taken ove the infinitely emote sphee to be zeo then the potential has to deease as - o faste. Fo that eason we an always onside two patiles with opposite hages that is vey fa one fom anothe so that the inteation an be negleted, but still, they will make a pope dependene of the potential at infinity. The ontinuity of potential bings up a vey inteesting question: Two patiles does not inteat but the stutue of the one an influene the stutue of the othe though the potential at infinity. Suppose ou patile has a spin dieted along z-axis. Then anothe patile of the simila stutue but with untipaallel spin should exist somewhee in spae (fa, o lose does not matte) in ode to anel the potential at infinity. Magnifient Patile (MP) Hee we dissuss "the bounday fist" appoah. Above we saw the eleti stati solution (IP) that violates Maxwell's equations on the bounday, but (and this is impotant) does not violate the pinipal of minimum ation. We tied to obtain a magneti stati solution (stati spin), but again, it was unsuesslull. Now we ae ompelled that a spin should be looked fo in a steady-state osillating solution. Waning: The MP is not an atual igoous solution (beause we mange to to satisfy the unlinea bounday ondition (5b) only in aveage ove the peiod of osillation) but an inteesting lose shot on it. Sine the potential satisfies the linea equations (4) and the stait fowad bounday onditions (tangent omponents should be ontinuous), it follows that the sum of two solutions of the equations (4) is also a solution (pinipal of supeposition). The touble aises when it omes to the unlinea bounday ondition (5b) fo the eletomagneti field and uents. We will dissuss a solution that ontains 3 pats (we assume that the uents inside the patile ae expessed though the potential k k k j A 4 with exeption of the stati eleti pat; also we assume the Loentz gauge):. A stati eleti pat: e k e A R ( k) R ( k ), j R ( k) in; A k 4 k (8) A is a tangent omponent at the infinitely emote sphee. To anel the potential thee (and make it deease as - ) we have to have anothe patile with α'e'=- αe somewhee befoe the infinity. The oesponding eleti field is:. A stati magneti pat: e E k R ( k ) in; E k (8a) 8

9 A R ( k)sin in; A R ( k ) sin (9) We may do not need to anel this at infinity beause we aleady have a good deease ate. The oesponding magneti field is: H R ( k)os in; H R 3 ( k )os H R ( k) kr ( k) sin in; H R 3 ( k )sin 3. And an osillating pat: (9a) h sin( t), h os( t), z k, k k y ( ) R ( ) Q ( ), y ( ) R ( ) Q ( ) A h R ( z)sin in; A h y ( )sin k A h ( ) R ( z) R ( z) sin in, A h y ( ) y( ) sin k z k k A h ( ) R ( z) R ( z) os in, A h y( ) os k z k k A h ( ) R ( z) R ( z) in, A h y( ) k z () whee Q (x) =-os(x)/x, Q (x)=-os(x)/x^-sin(x)/x ae the seond kind spheial Bessel funtions. The potentials () satisfy (4) and the onsevation ondition: A k k= inside. The (8) and (9) ae witten so that the bounday onditions at aleady satisfied. To satisfy the bounday onditions fo () we have to equest: k k R ( z) y( ); ( ) R R y( ) k z R ; Ri Ri ( z); z k ( zr R ) () The onstant γ is obtained as a esult of the fist two bounday onditions. On this onditions A, A θ, and A φ will be ontinuous (A is not ontinuous but we do not ae ab that). We have to anel the osillating potential at infinity beause the spheial Bessel funtions of both kinds deease as - (not good enough) at infinity. So we have to have anothe patile with δ '=-δ and δ '=-δ (also will be δ'=-δ) with the same oientation of z-axis. Note, that the funtions h(φ-ωt) and h(φ-ωt) should be the same. If we hange the sign of ω then it will be anothe solution. Let us alulate the 9

10 eletomagneti field that oesponds to the osillating potentials (): k E h ( ) R ( z) R ( z) sin in, E k z k E h R ( z) ( ) R ( z) os in, E k z k E h R ( z) ( ) R ( z) in, E k z k H in, H ; H h R ( z) in, H k H h R ( z)os in, H (a) The amazing thing ab this osillating field is: The osillating eletomagneti field inside the patile is pesent, but thee ae no osillating eletomagneti fields in vauum while the osillating potential is pesent. The expession fo γ (see ()) indiates that thee exist the esonane fequenies that oespond to: z= , , ,... (if k and ae given then only fequeny defines z) at whih γ (and the whole solution) beomes infinite. k j A The investigation of the adial uent 4 (see ()) on the bounday shows that it has minimums at z that satisfies the equation: R ( z ) zr ( z) 3 () the solutions of whih ae: z= , , ,.... If we take z satisfying () then fom () we find γ= (the eason why we do not equie j = on the patile bounday will be explained late). We will seek the solution as a supeposition of all the above 3 pats. But ou bounday of the patile is too simplified (it is a stati in time sphee of adius ). We found that we ae not able to satisfy the bounday onditions (5b) on the stati sphee. We think that the eal bounday has to be not a sphee (but lose to it) and has to depend on time. Bound by the spheial oodinate system we wee not able to find the eal bounday (and so the atual solution). Still, we an hope that if we satisfy the bounday onditions (5b) on ou sphee in aveage ove a one peiod of the osillation, then the eal solution should exist somewhee not fa fom ou appoximation. These ae also the easons why we ae seeking the adial uent to be only a minimum on the sphee. Let us take a lose look at the onditions (5b). We have the nomal in spheial oodinates:, t =, n =-, n θ =, n φ =. The time omponent of the ondition (5b) is satisfied beause ( E H n). The phi omponent of (5b) is ontinuous in aveage. The time aveage of the theta omponent of (5b) (the diffeene inside minus vauum) is

11 popotional to: R ( k ) R ( k )sin os Let us take k =π (whih tuns the above expession to zeo) and z= Then k/k.948, ω/k.773. The k still emains indefinite. At these values the adial uent is vey small ompae to its neighboing tems. The time aveage of the adial omponent of the ondition (5b) will be of the fom: A+Bsin θ. Fom A= we will get: α (-e /k / ) δ.688, and fom B= we will get: β δ.954. These onditions will define α and β (we still have the option to hoose the sign of α and β as we like). Afte that we an alulate the effetive hage and magneti moment, and, afte the integation ove 3-spae, the mass and the spin. All these values will be expessed though δ, k, and. These 3 onstants won't allow us to make the esult looking as an eleton o poton (beause it equies 4 onstants). On the top of that the values of the atual solution that satisfies (5b) an shift signifiantly. Still we only hope that this atual solution (MP) does exist, but this peliminay attempt looks pomising. Sine the solution has esonane fequenies we need to intodue a powe dissipation to make a moe ealisti solution. Conlusion: I want to attat attention to the hange of the vaiation poedue: instead of keeping the inteation tem in the Lagangian and vaiating potential (onventional poedue) we define inteation () as an auxiliay ondition befoe the vaiation. Then, vaiating the eletomagneti field itself (not a potential), we use the Lagange method of indefinite fatos with the modified Lagangian (). The impotant esult of this hange of the vaiation poedue is the ontinuity of tangent omponents of the potential (and, onsequently, definite value of them at infinity) is valid in the onventional eletodynamis also. Appendix: The Spin of a Classial Physial System with Continuous (Inluding Fist Deivatives) and "Good" Behaving at Infinity Enegy-Momentum Tenso in Statis is Zeo The flat spae of eal 4-dimensional independent vaiable x k with Loentz meti is assumed. If we have a lassial physial system desibed in that spae and the enegymomentum tenso of that system obeys the onsevation law, and if:. This enegy-momentum tenso is ontinuous in 4-spae and has ontinuous fist deivatives eveywhee in 4-spae (it an have a bake of the seond deivative on some losed 3-d sufae in that spae).. This enegy-momentum tenso is good at teestial infinity (so that the sufae integal ove a 3-sphee of a big adius an be negleted). 3. The enegy-momentum tenso does not depend on time. Then the angula momentum of this system (spin) is zeo.

12 To pove this theoem let us fist onside a onseving veto j k k=. Applying 4-d Gauss theoem sepaately to 4 volume inside and side 3-d losed sufae we an pove that the integal: e j d3v (.a) t does not depend on time. The integation in () goes ove the hypesufae x =. In diffeent oodinates it will be diffeent hypesufae but the integal will have still the same value. It is a good example of a global thing. Let us elaboate on the meaning of a good global. Let us onside two etilinea oodinate systems K and K whih ae onneted by the Loentz tansfomation: We have: e x x' ( x' Vx' ), ( x Vx ), x ( x' Vx' ) x' ( x Vx j' dx' j dx j' ( x', x' ) Vj' ( x', x' ) x' x x ) dx (.a) Hee the integation ove dx and dx 3 supposed to be pefomed but not indiated fo simpliity. In the last integal we just expessed j aoding to the tansfomation of a ontavaiant veto whih tansfoms the same way as oodinates do. Instead of integating ove dx in the last integal we an integate ove dx but we have to fulfill the ondition x =, o x =-Vx and dx = γdx sine we keep x =. We have: e j' (, x' ) dx' j' ( Vx', x' ) Vj' ( Vx', x' ) x' x' Vx' Now suppose that ou distibution of uents does not depend on time x in oodinates K. Sine V is abitay and the Loentz tansfomation (boost) an involve any of the oodinates x,x,x 3 we should onlude that in this ase the integal: j k ' d 3V ' if k,,3 dx' (3.a) This integal diffes fom zeo only if k=. Taking the enegy-momentum tenso we have to be aeful beause the Gauss theoem only applies to a veto. To edue a tenso to a veto we have to intodue a onstant veto e k (fou linea independent onstant vetos an be intodued in a flat spae). Using the onseving veto G k =e i T ik and using the same logi we an pove that if T ik does not depend on time in oodinates K then the integal: T ' ik d3v ' if i k only (4.a) diffes fom zeo only if i=k=. The same way we an pove that in statis the integal: l mn M ' e x' T ' d 3V ' if n only (5.a) ik iklm Now using (4) and (5) we an pove that in statis: i k x T ' d V if i, k, i k ' 3 (6.a) The onsequene of (6) is that M k (spin) equals to zeo in statis. The final onlusion is: An independent of time solution that has a onseving (T ik k=) eveywhee enegy-momentum tenso that is "good" at infinity an not have angula

13 momentum diffeent fom zeo. Refeenes:. Yui Keilman, Phys. Essays,, # p.34, (998). Idem, Phys. Essays, 5, #3, (3) 3