The Simple Solutions of Four Actual Problems. of General Theory of Relativity.

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1 The Simple Soltions of For Atal Problems of General Theory of Relativity. H Changwei Room 81, No.17,Lane 1769, Pdong Wlian Road, 19 Shanghai China,1-8818, hhangwei5@yahoo.om.n Abstrat: It is qite ompliated and diffilt generally to solve the problem of the general theory of relativity. We apply the eqations of qantitative effet of general theory of relativity and kinemati priniple of doble-mass whih is new-proposed to solve simply for atal problems: gravitational red shift of spetral line, delay of radar eho, perihelion preession of planet, defletion of light in gravitational field, and the reslts are idential with the formlae that is derived by general method. Key words: General theory of relativity, Eqation of qantitative effet, Kinemati priniple of doble-mass, For atal problems We know that the spae-time of general relativity is twisted into Riemann geometri ontinm by the matter with mass. So the formlae of problems of general relativity are qite ompliated generally and are diffilt to solve. Then whether these problems an be simplified by new thinking and way, below we will make a try for it. 1, The eqations of qantitative effet of relativity theory and kinemati priniple of doble-mass The general relativity onsidered that the nit length dr or nit time or mass m an vary with speed : 1 (1) 1 dr 1 dr 1 dr () m m (1 ) m () 1, dr and m are the proper nit time length and mass. The (1), () and () are the eqations of qantitative effet in the speial theory of relativity. The general theory of relativity onsiders that the nit time and length an vary with gravitational potential, whih may be allate throgh the priniple of eqivalene and onservation of energy: in isolated gravitational field of a heavenly body with mass M, let a objet with mass m falls free towards the heavenly body from where infinite far, the beginning veloity is zero, and its veloity is when the objet is r away from the heavenly body and where the gravitational potential is (it is the zero where infinite far), then 1

2 1 m m,namely 1 (4) Sbstitte (4) into (1), () and (), we obtain (G is the gravitational onstant): 1 1 GM 1 r (5) dr GM (6) r 1 dr 1 dr 1 dr m m 1 (1 ) m GM 1 m r (7) The (5), (6) and (7) are the eqations of qantitative effet in the general theory of relativity. The (7) shows that the mass of objet an be inreased by the addition of gravitational potential. So an say there are two masses: the proper mass m and the effet mass m. The qantitative relation of two masses is deided by (7), they an not interhange as between kineti energy and potential energy. Now we deompose a motion into proper motion of proper mass and effet motion of effet mass, then what have relations between that two? First the system of proper motion does not be hanged by effet motion, while is hanged the diretion. For example, the system of proper motion of planet is the ellipse, the existene of planet effet mass does not hange the form of ellipse, while makes whole ellipse rotated slowly, namely the preession; otherwise the mass is diretly proportional to the energy, so the ratio of energy is eqal to the ratio of mass for two objet. Generally, the ratio of energy is eqal to the ratio of speed sqare; while the movement of heavenly body, whih do not inlde the movement that is ased by the exploder of elestial body, is the movement in the gravitational interation, the proper and effet motion of a objet are the motion in same gravitational interation, their energy are the ability of making work at this ondition, so the ratio of speed or rote of two mass motion is eqal the ratio of two mass or energy. It is alled kinemati priniple of doble-masses. Kinemati priniple of doble-masses: The objet motion in gravitational field an be deomposed into proper motion of proper mass and effet motion of effet mass; the system of proper motion does not be hanged by effet motion, whih hanges only that diretion; the ratio of the veloity (or rote) of effet motion to the veloity (or rote) of proper motion is eqal to the ratio of two masses (or energies), and is approximately. Whether the kinemati priniple of doble-masses an be established, whih shold grond on whether it an be idential with fats. The following, it is sed to allate the perihelion preession of planet and defletion of light in gravitational field, the reslts are idential with the approximate formlae that is derived by the general theory of relativity, whih shows that it is reasonable.

3 , The derivation of the formla of perihelion preession of planet Abot the perihelion preession of planet, above had pointed ot the effet motion has only relevane to preession. Here the effet energy is as extra kineti energy of anglar diretion, whih makes the angle that the vetor radis rotates is not, and is when the planet aomplishes a period s ellipti motion, the is jst the angle of preession. Both of two ation are same diretion and step between kineti energy of anglar diretion of the extra and proper motion, then the preession angle an be derived simple: to derive the ratio of anglar diretion kineti energy between preession and proper motion, then the preession angle an be derived in proportion as the ratio when the planet aomplishes a period s ellipti motion. When the effet motion does not be onsidered, the anglar diretion motion is made by anglar diretion kineti energy of proper motion. First we derive the proportion of the anglar diretion kineti energy to the sm of energy. For irlar orbit, all of kineti energy are the anglar diretion kineti energy, whih is half of potential energy, or the anglar diretion kineti energy is 1 the sm of energy. For ellipti orbit, part of kineti energy beome the radial diretion kineti energy, whih has not relevane to anglar diretion motion. when the planet lies to the aphelion, its kineti energy is the least, and is GMm ( a ) ( G is gravitational onstant; M is the solar mass; m is the planet mass; a is the half length of long axis; is the half of foal length); when the planet lies to the perihelion, its kineti energy is the largest, and is GMm, so the average kineti energy of ellipti motion is: ( a ) GMm GMm 4 a a a 1 e, ( e is the eentriity); while the kineti energy of GMm irlar motion with radis a is, whih is 1 e time as mh as the average kineti a energy of ellipti motion. Therefore the anglar diretion kineti energy of proper motion is approximately 1 e E ( E is the sm of energy of proper motion). Aording to the kinemati priniple of doble-masses, the anglar diretion kineti energy of planet preession is 1 e E E, while the anglar diretion kineti energy of proper motion is, the ratio of them is 6 a / T 1 e 4 a 1 e 1 e T 1 e, so the angle of preession is (T is the time that the planet goes one period), whih is idential with the formlae that is derived by the general theory of relativity.

4 , The derivation of defletion of light in gravitational field Look on the sketh map, if it were withot the gravitational field, the photon wold move along level line MAN; and atally the photon moves along rve ABC, whih is an approximate line and the angle between ABC and MAN is very small atally; the O is the mass entre of heavenly body; R=AO is radis of the heavenly body; MN EF DC OG. When the photon reah an arbitrary point B on the ABC, FBC=αis the total angle of light defletion. Bease the ABC is an approximate line and the light defletion is orred hiefly nearby the point A, therefore the line AB is in line with the rve BC and its tangent line on point B approximately, when the point B is qite far away the point A. Then ABE= FBC=, tg AE BE R r R sin, r sin When r os 1 tgtg, bease the angle of defletion of light is very small, the is eqal arately enogh the total angle of light defletion that the photon passes throgh the whole gravitational field from the point A, at this time, R r sin (8) The (7) shows that althogh the photon has not stati mass and has only dynami mass, yet it an prode the effet of general relativity, and may apply the kinemati priniple of doble-mass. Then the photon s motion an be deomposed into proper motion and effet motion, its proper motion is linear motion with nvaried veloity; its effet motion does not hange the speed of proper motion and hanges only the diretion of that, so it is always perpendilar to the diretion GM of proper motion, the energy ratio between two motions is ( M is the mass of the r heavenly body, r is the length of OB), whih is a vale in a moment. Now we onsider another ondition: let the mass entre of a heavenly body with same mass and radis BG is in the point G, 4

5 and the photon passes levelly the point B, in this time, the BF indiates the light veloity; the BD indiates the speed of effet motion, aording to the kinemati priniple of doble-masses, obtain: BD GM, the defletion angle is jst the. That is to say, the kinemati priniple of BF BG doble-masses an be apply at point B: sbstittes level diretion for the diretion of light; sbstittes total defletion angle for the defletion angle in a moment; sbstittes r sin for r. Then sing the (8), the total defletion angle that the photon passes trogh the whole gravitational field from point A is: BD GM GM tg BF r sin R. The moved loi of light before and 4GM after point A are symmetri, so the total defletion angle is:, whih is idential with the R approximate formlae that is derived by the general theory of relativity. 4, Gravitational red shift of spetral-line and gravitational delay of light The (5) shows that the time is inflened by gravitational potential. Therefore, the gravitational red shift of spetral-line an be nderstood: the proper freqeny v is invariable, only bease the lok in gravitational field goes slower than the lok where is withot gravitational field, therefore sing lok measres same photon, the freqeny of photon in gravitational field is higher than that in withot gravitational field. Einstein onsiders that the freqeny of photon is as the nmber of tiks of the lok per nit time [1], so the freqeny of v photon v 1 v. Then sing the spae-time standard where the photon 1 is loated measres the photon whih moves towards the diretion that the gravitational potential is less, the spetral-line of photon is going red shift. Abot the gravitational delay of light, whih is analyzed and solved omprehensively in the book Gravitation and Spaetime [], whih points ot that the ase of gravitational delay of light is: defletion of light in gravitational field and the light s veloity is lowered by gravitational field. The rote s addition that is ased by the former is very little, is a two-level amendment, and an be omitted, the major is the latter. In the general theory of relativity, the light veloity is also invariable, bt the time-spae standard is not nited in the gravitational field, if we measre the light veloity of some point in the gravitational field sing respetive time-spae standard, the light veloity will be onstant, while if we measred the light veloity of gravitational field sing the spae-time standard that is far away the gravitational field, the light veloity wold be lowered. In this book, the atal light s veloity is allated ot by the field eqation of general GM theory of relativity throgh many steps, the reslt is: 1, (here =1), then the time r of gravitational delay an be derived by the methods of alls. In fat, applying the (5) and (6) an derive the light s veloity in the gravitational field, whih is to transfer qantitative veloity nit dr / into proper veloity nit dr / : 5

6 1 dr dr / / 1 (1 ) dr GM Then proper light s veloity is: ( 1 ) (1 ). From this, the time of r gravitational delay of light an be derived by the methods of alls. The eqations of qantitative effet in the general theory of relativity are sed to solve simply for atal problems of the general theory of relativity on the basis of flat spae, whih don t mean that the spae-time is flat, bease when sing the general theory of relativity solves these problems, the eqation is linearized, whih an be onsidered as that the rved spae is flatted. Bt a probability an not be removed: the spae-time shold be flat, and the Riemann spae is only a mathematial model; the proper physial qantity in relativity theory, whih have not relevane to speed and gravitational potential, are Newtonian physial qantity in absolte spae-time theory; only the relativity theory has amended qantitatively Newtonian physial qantity, whih is refleted by eqations of qantitative effet. / Referenes [1], A. Einstein, Relativity The Speial and The General Theory,London, Methen & CO. LTD, 1955,P19 [], H.C.Ohanian and R. Rffini,Gravitation and Spaetime,Beijing, Siene Press,6, P (in Chinese) 6