Gravity. David Barwacz 7778 Thornapple Bayou SE, Grand Rapids, MI David Barwacz 12/03/2003

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1 avity David Bawacz 7778 Thonapple Bayou, and Rapid, MI 495 David Bawacz /3/3 Uing the concept dicued in the peceding pape ( ), I will now deive a fomula fo gavity. The fomula will educe to Newton fo object in cloe poximity and to the MOND fomula fo object elative to the cente of the galaxy. If the eade i unfamilia with MOND, a quick intenet each would be helpful. MOND theoy baically alte the gavitational field fom one ove quaed to one ove when the field tength i vey mall. It ha been enomouly ucceful in pedicting the motion of ta elative to the cente of galaxie without the aumption of dak matte. It i an empiical fomula with no bai in phyic. Aume we have two lage object and that thei time both ead at the ame time, which i the tat of the obevation, in an obeve efeence pace. We will aume that ou cyclic ytem i vey lage, (the Milky Way galaxy fo example) and ou object ae the eath and moon. Unde thee containt we can conclude that the pace time adiue K and K both cale, when caled to a efeence pace, to eentially the ame value which we hall call K. We wite the following equation fo both. Object Object Pc + mc eq Pc + mc eq c T K eq c T K eq. 4.3 Whee we ue the Univeal Invaiant dicued peviouly ( ). Now we take the quae oot of each equation uing the Diac matice.

2 α Pc+ βm c eq. 4.4 α Pc+ βm c eq 4.5 α ct K + β eq 4.6 α β + ct K eq. 4.7 P and ae vecto in geneal, but we can implify them to one dimenion. ince we only have two object we make one axi lie on the line connecting the object. We aume that thee two object ae at et elative to ou obeve at time, o we need only concen ouelve with one component of P which alo will lie on the ame axi. i imply the ditance between the object in thei epected pace. Now we multiply equation 4.4 by equation 4.7. If we aume that the total enegy fo both object emain contant ove the length of any meauement, then thi i no diffeent then aying x 4. The eult of the multiplication i: P c+ T mc K Thee object ae maive object and T emain mall ove the time of ou obevation. Fom ou peviou dicuion, thi could liteally be million of yea. K i eentially We will eliminate the tem Tma it i vey mall compaed to K. Thi leave P c K. We can evaluate the change in P to get: K P c Now fom equation 4.7 α β T We can now vapoize the matice by quaing and then taking the quae oot to get: ± T but T T

3 Thi leave u with: K T P whee we ae uing T c Now we convet eveything to ou common efeence fame (ee peviou pape) uing the following elation. Remembe that K imply become K by the etiction we et. P m V m m m V ef ef ef T Tef ef t ef t and time ae ditinguihed ince they may be diffeent. ef. The efeence enegy fo pace ef t Tef T and ef ef ubtituting yield V K m T ef ef t ef ef ef Fma dividing by 4 c and ubtituting Newton definition of Foce, mm We get F whee k K and ef K i a contant. K i the pace time ditance fom the oigin of the otating ytem. In the above dicuion I abitaily ued T but I could have ued + T and got the oppoite foce. Uing the oppoite ign would yield anti-gavity. Quantum mechanic i jut a eult of the ame cyclic natue of pace/time and a deciption of a quantum object mut include thee anti-gavity tate. Fo all object in cloe poximity ( a ditance that could be atonomical elative to the adiu of a galaxy), K i eentially contant and we ee that the gavity equation educe to Newton fomula. When decibing the motion of a ta elative to the cente of ma of a galaxy, i the patial adiu, and K i the pace/time ditance.

4 Aume that K i equal to. It cetainly won t be le than. Then the gavitational fomula become: mm F k which i functionally equivalent to the MOND fomula. We can get a value fo k, by conideing ou own ola ytem. k K. i jut ou local gavitational contant 8 3 cm g in cg unit ec If we aume Then k.55 K i the ditance to the cente of the galaxy (85 pc) o 3 The MOND gavitational field equation can be witten:.6 cm, am a whee a i the field tength, a i the MOND contant (detemined empiically) i the local gavitational contant, M i the ma of the galaxy and i the ditance to the cente. Conveting my gavitational fomula to a field equation yield: km a quating to MOND: am k M implifying yield: a k M o a km We know, we etimated k and g M (the viible ma of the galaxy) i about We calculate that:

5 a a i an eluive value becaue of the numbe of unknown, howeve all the liteatue 8 etimate it to be on the ode of, jut a we calculated. ummay Although thee may vey well be a fom o fom of dak matte, the exitence of dak matte in t neceay to explain the motion of ta in a galaxy. Thee definately in t a need fo 9 % of the matte in a galaxy to be dak. The value of the local gavitational contant deceae with deceaing ditance to the cente of the galaxy, appoaching zeo at the cente. One would expect then a egion of emptine at the cente, thee i nothing to hold matte thee. The nuclea funace inide ta i diven by the foce of gavity. If gavity i le nea the cente of the galaxy, it would be eaonable to expect ta to bun coole (given the ame ma) nea the cente. Mot ta do in fact bun cool nea the cente. The peent explanation i that they ae old ta having eentially buned out. With thi theoy they could vey well be young ta buning lowe.