Comptive Studies of Lw of Gvity nd Genel Reltivity No. of Comptive hysics Seies pes Fu Yuhu (CNOOC Resech Institute, E-mil:fuyh945@sin.com) Abstct: As No. of comptive physics seies ppes, this ppe discusses the sme points of lw of gvity (including impoved lw of gvity) nd genel eltivity: they e ll the effective theoies of gvittionl intection, nd cn be used to solve the poblems of dvnce of plnety peihelion nd deflection of photon ound the Sun; nd the diffeent points of them: genel eltivity cn confom to genel eltivity pinciple while lw of gvity cnnot confom to this pinciple, nd lw of gvity cn solve the constint poblems in gvittionl field (such s smll bll olls long the inclined plne) while genel eltivity cnnot solve these poblems. These two theoies cn lso ovecome thei own wek points by lening fom ech othe's stong points, nd jointly pesent the new explntion tht the dvnce of plnety peihelion is the combined esult of two motions: the fist ellipticl motion decided by lw of gvity cetes the peihelion, nd the second votex motion cetes the dvnce of peihelion in which the dvnce pmetes cn be decided by genel eltivity. The common defect of these two theoies is tht they cnnot tke into ccount the pinciple of consevtion of enegy, comping with the esult tht pplying the pinciple of consevtion of enegy to deive Newton's second lw nd lw of gvity, the futhe topic is pplying the pinciple of consevtion of enegy to deive the elted fomule nd equtions of genel eltivity. Key wods: Comptive physics, lw of gvity, impoved lw of gvity, genel eltivity, sme point, diffeent point, pinciple of consevtion of enegy Intoduction In efeence [, the concept of comptive physics is poposed. As one of the seies ppes of comptive physics, this ppe discusses the comptive studies of lw of gvity nd genel eltivity. Minly include: two theoies' sme points nd diffeent points; two theoies' joint ppliction; nd two theoies' common defect nd the wy of impovement. Sme points of lw of gvity (including impoved lw of gvity) nd genel eltivity The fist sme point: they e ll the effective theoies of gvittionl intection. Lw of gvity is the fist effective theoy of gvittionl intection expessed by mthemticl fomul. It evels the ules of the motions of celestil bodies, nd it is widely used in the fields of stonomy nd stodynmics. The discoveies of Comet Hlley, Neptune nd luto in the histoy of science e ll the exmples of the successful pplictions of lw of gvity. Newton lso explins tht the tide phenomenon is cused by gvittionl pull of the Moon nd the Sun. Fo most of the phenomen of gvittion, lw of gvity is sufficiently ccute. Mny schols believe tht genel eltivity is the moe ccute effective theoy of gvittionl intection thn lw of gvity; howeve fte seies of impovements to lw
of gvity, this clim hs been questioned moe nd moe; but in ny cse, genel eltivity hs hd substntil impct in histoy. The second sme point: ll of them cn be used to solve the poblems of dvnce of plnety peihelion nd deflection of photon ound the Sun. The initil two expeimentl tests nd veifies of genel eltivity e the poblems of dvnce of plnety peihelion nd deflection of photon ound the Sun. Especilly duing the totl sol eclipse of My 9, 99, the mesued deflection of light is vey good geed with the theoeticl pediction of genel eltivity, fo this eson, thee hd been the senstionl effect in the wold. Mny schols lso believe tht these two poblems cnnot be solved with lw of gvity, howeve, the el fct is tht both poblems cn lso be solved with lw of gvity. As well-known, by mens of lw of gvity, the field equtions of genel eltivity cn be deived. Similly, by mens of genel eltivity, the impoved lw of gvity cn lso be deived. In efeences [-4, jointly pplying lw of gvity nd genel eltivity, the impoved fomul of univesl gvittion cn be deived, which my be ewitten s follows. As discussing the poblem of plnet s movement ound the sun ccoding to the genel eltivity, the following eqution cn be given 3GMu u" u () p c whee, u ;G gvittionl constnt;m mss of the Sun;c velocity of light; p - hlf noml focl chod. ue to the centl foce, the obit diffeentil eqution (Binet s fomul) eds F h u ( u" u) () m whee, h is constnt. Substituting Eq.() into Eq.(), we hve F 3GMu mh u ( ) (3) p c The oiginl lw of gvity eds F u (4) Fo Eq.(3) nd Eq.(4), comping the tems including u, we hve h GMp Substituting h into Eq.(3), it gives 4 3G M mpu F u (5) c
Substituting u into Eq.(5), the impoved lw of gvity eds F 3G M mp (6) 4 c In efeence [5, by using this impoved fomul, the clssicl mechnics cn be used to solve the poblem of dvnce of plnety peihelion nd the poblem of gvittionl deflection of photon ound the Sun, while these solutions e the sme s given by genel eltivity. iffeent points of lw of gvity (including impoved lw of gvity) nd genel eltivity The fist diffeent point: genel eltivity cn confom to genel eltivity pinciple while lw of gvity cnnot confom to this pinciple. In fct, fo lw of gvity, mthemtics is only tool, the thn the stting point nd bsic pinciple; while fo genel eltivity, mthemtics is not only tool, but lso the stting point nd bsic pinciple. The second diffeent point: lw of gvity cn solve the constint poblems in gvittionl field (such s smll bll olls long the inclined plne) while genel eltivity cnnot solve these poblems. The odiny schols gee tht ll the poblems tht cn be solved by lw of gvity, cn lso be solved by genel eltivity; while, some poblems tht cn be solved by genel eltivity, my not be solved by lw of gvity. Howeve, this is not the el cse, some poblems tht cn be solved by lw of gvity, my not be solved by genel eltivity. The eson fo this is tht genel eltivity cn only solve the fee-pticle motion in gvittionl field, so genel eltivity cnnot solve the constint poblems in gvittionl field, nd t pesent these poblems cn only be solved by using lw of gvity o impoved lw of gvity. In efeence [4, 5, fo the exmple of smll bll olls long the inclined plne, the impoved lw of gvity nd impoved Newton s second lw e deived with pinciple of consevtion of enegy. follows. follows. The esults suitble fo this exmple with the constnt dimension fctl fom is s The impoved lw of gvity eds F.99989 The impoved Newton s second lw eds.0458 F m The esults suitble fo this exmple with the vible dimension fctl fom e s Supposing tht the impoved Newton s second lw nd the impoved lw of gvity with the fom of vible dimension fctl cn be witten s follows: k ; F m, u
F /, u ( u x H ). k ; whee: u is the hoizon distnce tht the smll bll olls Afte the vlues of k, k e detemined, the esults e s follows 8 3 8.850 u,.70 u The esults of vible dimension fctl e much bette thn tht of constnt dimension fctl. 3 Jointly pesent the new explntion of the dvnce of plnety peihelion with these two theoies Afte comping we cn find tht these two theoies cn lso ovecome thei own wek points by lening fom ech othe's stong points, nd jointly pesent the new explntion tht the dvnce of plnety peihelion is the combined esult of two motions. This poblem hs been bsiclly solved in efeence [6, nd it cn be ewitten s follows. Mny schols believe tht genel eltivity does not end the studying fo poblem of dvnce of plnety peihelion, becuse thee e mny fctos ffecting the dvnce of plnety peihelion, theefoe it still needs to continue to study this issue. Although the explntion of genel eltivity fo the dvnce of plnety peihelion is esonbly consistent with the obseved dt, becuse its obit is not closed, whethe o not it is consistent with the lw of consevtion of enegy hs not been veified. Fo this eson, jointly pplying lw of gvity nd genel eltivity, new explntion is pesented: The dvnce of plnety peihelion is the combined esult of two motions. The fist ellipticl motion cetes the peihelion, nd the second votex motion cetes the dvnce of peihelion. In the motion of plnet-sun system, unde the ction of gvity, the plnety obit is closed ellipse, nd consistent with the lw of consevtion of enegy. Menwhile, the plnet lso pticiptes in the votex motion of sol system tking the Sun s cente; the long-tem tend of the votex is the futhe topic, but in the shot-tem my be consideed tht due to the ineti the plnety peihelion will un cicul motion in votex nd led to the dvnce of peihelion, thus lso without cting ginst the lw of consevtion of enegy. Bsed on the esult of genel eltivity, the ppoximte ngul velocity of dvnce of peihelion is given. Accoding to genel eltivity, the vlue of dvnce of plnety peihelion eds 3 4 (7) T c ( e ) whee: c is the speed of light; T,, nd e e obitl peiod, semi-mjo xis nd eccenticity espectively. Accoding to Eq.(7), tking the Sun s cente, the ngul velocity of dvnce of plnety peihelion is s follows 3 4 (8) 3 T T c ( e )
Accoding to Keple's thid lw, it gives T 3 4 GM whee: G is the gvittionl constnt, nd M is the sol mss. Then Eq. (8) cn be ewitten s 3/ 3/ 3G M (9) 5/ c ( e ) Accoding to this expession we cn see tht, the ngul velocity of dvnce of plnety peihelion is invesely popotionl to 5/, nd the velocity of dvnce of plnety peihelion is invesely popotionl to 3/. Fo the esults of Eq.(7), thee e smll diffeences comped with ccute stonomicl obsevtions, so we sy tht esults of Eq.(8) nd Eq.(9) e the ppoximte ngul velocities of dvnce of peihelion bsed on the elted esults of genel eltivity. Now the otte tnsfomtion in Ctesin coodinte system is pplied to deive the plnety obit eqution including the dvnce of peihelion. In the plnet-sun system, tking the sol cente s the oigin of coodinte, the plnety obit eqution eds ( x k) y b whee: k is the semi-focl length of ellipse. Accoding to the otte tnsfomtion in Ctesin coodinte system, it gives x x' cos y' sin y x' sin y' cos whee: is the ngle of ottion (nmely the ngle of dvnce), t. Thus, fte consideing the votex motion, the plnety ottion obit eqution is s follows ( x'cos y'sin k) ( x'sin y'cos ) b 4 The common defect of these two theoies inciple (lw) of consevtion of enegy is the most impotnt pinciple (lw) in physics. Afte comping we cn find tht the common defect of these two theoies is tht they cnnot tke into ccount the pinciple of consevtion of enegy, while mny schols hve not pid ttention to this defect.
The wy to eliminte this common defect is to deive lw of gvity nd elted fomuls nd equtions of the genel eltivity with the pinciple of consevtion of enegy. The comping with the esult tht pplying the pinciple of consevtion of enegy to deive Newton's second lw nd lw of gvity, the futhe topic is pplying the pinciple of consevtion of enegy to deive the elted fomule nd equtions of genel eltivity. Now we deive the oiginl Newton's second lw nd lw of gvity. Above ll, in this section only Newton's second lw cn be deived, but we hve to pply lw of gvity t the sme time, so we pesent the genel foms of Newton's second lw nd lw of gvity contined undetemined constnts fistly. Assuming tht fo lw of gvity, the elted exponent is unknown, nd we only know the fom of this fomul is s follows F whee: is n undetemined constnt, in the next section we will deive tht its vlue is equl to. Similly, ssuming tht fo Newton's second lw, the elted exponent is lso unknown, nd we only know the fom of this fomul is s follows ' F m whee: is n undetemined constnt, in this section we will deive tht its vlue is equl to. As shown in Figue, supposing tht cicle O denotes the Eth, M denotes its mss; m denotes the mss of the smll bll (teted s mss point ), A O is plumb line, nd coodinte y is pllel to AO. The length of AC is equl to H, nd O C equls the dius R of the Eth. We lso ssume tht it does not tke into ccount the motion of the Eth nd only consideing the fee flling of the smll bll in the gvittionl field of the Eth (fom point A to point C). Figue A smll bll fee flls in the gvittionl field of the Eth
Fo this exmple, the vlue of v which is the sque of the velocity fo the smll bll locted t point will be investigted. To distinguish the quntities clculted by diffeent methods, we denote the vlue given by lw of gvity nd Newton s second lw s v,while v denotes the vlue given by lw of consevtion of enegy. ' Now we clculte the elted quntities ccoding to lw of consevtion of enegy. Fom lw of gvity contined undetemined constnt, the potentil enegy of the smll bll locted t point is s follows V ( ) O ' Accoding to lw of consevtion of enegy, we cn get ( ) And theefoe O' A mv' ( ) O ' v GM [ ( R H ) ' O' Now we clculte the elted quntities ccoding to lw of gvity nd Newton s second lw. Fo the smll bll locted t ny point, we hve dv / dt We lso hve dy dt v Theefoe vdv dy Accoding to lw of gvity contined undetemined constnt, long the plumb diection, the foce cted on the smll bll is s follows gives F O' Fom Newton's second lw contined undetemined constnt, it gives F ( ) m / ' Then we hve GM ( ) O' GM vdv { ( R H y) / ' } / ' dy Fo the two sides of this expession, we un the integl opetion fom A to, it
v y p / ' ( GM) 0 ( R H y) / ' dy v ( GM) / ' { [( R H y) / ' / ' y p 0 } v / ' ( GM) [ ( / ' ) ( / ') O' ( R H) ( / ') Let v, then we should hve: / ', nd ( / ' ) ; these two ' v equtions ll give: ', this mens tht fo fee flling poblem, by using lw of consevtion of enegy, we stictly deive the oiginl Newton's second lw F m. Hee, lthough the oiginl lw of gvity cnnot be deived (the vlue of my be ny constnt, cetinly including the cse tht =), we ledy pove tht the oiginl lw of gvity is not contdicted to the lw of consevtion of enegy. Now, we deive the oiginl lw of gvity by using lw of consevtion of enegy. In ode to elly deive the oiginl lw of gvity fo the exmple of fee flling poblem, we should conside the cse tht smll bll fee flls fom point A to point (point is lso shown in Figue) though vey shot distnce Z (the two endpoints of the intevl Z e point A nd point ). As deiving the oiginl Newton's second lw, we ledy ech v GM [ ( RHZ) ( RH) ' ' whee: R H Z O' ' Fo the eson tht the distnce of Z is vey shot, nd in this intevl the gvity cn be consideed s line function, theefoe the wok W of gvity in this intevl cn be witten s follows W F v Z R H Z) ( Z whee, F v is the vege vlue of gvity in this intevl Z, nmely the vlue of gvity fo the midpoint of intevl Z. Omitting the second ode tem of W ( R H Z ( Z RH RZ HZ ) 4 ( Z) / ), it gives As the smll bll fee flls fom point A to point, its kinetic enegy is s follows mv ' ' ( R H) ( R H Z) [ ( R H RH RZ HZ )
Accoding to lw of consevtion of enegy, we hve W mv' ' Substituting the elted quntities into the bove expession, it gives ( R H) ( R H Z) [ ( R H RH RZ HZ ) ( R H Z RH RZ HZ ) To compe the elted tems, we cn ech the following thee equtions / Z ( RH) ( RHZ) / All of these thee equtions will give the following esult Thus, we ledy deive the oiginl lw of gvity by using pinciple of consevtion of enegy. Comping with the esult tht pplying the pinciple of consevtion of enegy to deive Newton's second lw nd lw of gvity, the futhe topic is pplying the pinciple of consevtion of enegy to deive the elted fomule nd equtions of genel eltivity. 5 Conclusions In comptive physics, bsed on the comptive method, we cn discuss the sme points nd diffeent points of diffeent physicl lws; nd then discuss how diffeent physics lws cn len fom ech othe; nd the common defet of some physics lws cn be eliminted with the effective wy of comptive method. Thus, it seems tht the comptive physics will hve good development pospects. Refeences Fu Yuhu. Expnding Comptive Litetue into Comptive Sciences Clustes with Neutosophy nd Qud-stge Method, Neutosophic Sets nd Systems, Vol., 06, 8~ Fu Yuhu, Impoved Newton s fomul of univesl gvittion, Zinzzhi (Ntue Jounl), 00(), 58-59 (in Chinese) 3 Fu Yuhu, New Newton Mechnics Tking Lw of Consevtion of Enegy s Unique Souce Lw, Science Jounl of hysics, Volume 05, Aticle I sjp-30, ges, 05, doi: 0.737/sjp/30 4 Fu Yuhu. New Newton Mechnics nd Relted oblems, LA LAMBERT Acdemic ublishing, 06 5 Fu Yuhu. Solving oblems of Advnce of Mecuy s eihelion nd eflection of hoton Aound the Sun with New Newton s Fomul of Gvity, vix:507.064 submitted on 05-07-
6 Fu Yuhu. New Explntion of Advnce of lnety eihelion nd Sol System s Votex Motion. See: Unsolved oblems in Specil nd Genel Reltivity. Edited by: Floentin Smndche, Fu Yuhu nd Zho Fengjun. Eduction ublishing, 03. 49-5