To Find Different Gravity Laws

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Elfreda Garrison
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1 T Fin Diffn ai Laws Pal A Ma д Mninsa Appli Psis LLC inna A 8 Absa T a iffn ainals f plin ais ai ls T ipan is lk in spifi pis in siabl alia s ls T q is w is in iss spp l aliain In sal f s ls i is n b in s laws an is infs a a inapabl f nain ai was assin a is M s bina plsas ipl a nsal bai a pi in f np aiainal analis an insis Ein a is wi pinal aa pi b pial spaaf ajis a Libain pins nsiin sabili an n if is a pa f p analial pfan Wn ainin Libain pins n ll is nsi as a nsan wil Nwnian ai S f s laws ipl a is is n ln ali b a n ll a in nas pains inas a as a fnin f i If is nn wi a iffn l w a a as awa s f s ls f pl i alii F is a pi insis inas nw ls f ain ai siabl f lpin aan spa ppl is a i allw wap i nps I Inin T a iffn ai laws a is f iffn appliains an ins T q is fin if s ls a n an if s wa wl b a asnabl s nsa s ls as wll as pssibl lp f ls psss anlai an i aa Obisl Nwnian aiain is s ppla an i as wk inl wll wi siain plans ns asis an sallis Is flaws w is a if Einsin is an a ai was is Nwnian aiain is an llipial paial iffnial qain an s n ai s f ai was M Einsin wl qi i a saisfis a wa paial iffnial qain f isn f ai was T bis slin is a Nwnian is pssibl wn is i i wiin ain nsains a a b in? Cn inkin in s spaa ls is a sl aspiall saisf fa-fil bna niins bas pn Nwnian aiain Ta is ai sl anis a infini Tis iial sss a law sl a s f f ins fnin wi isan saisf is bna niin T isn f ai was a n b as paln as n wl p f ai was a a ans f allwin si ns wi s i la as wll as pn insanans spnss wn a spna pls Einsin i is bas fl a sl b a i la in s ns H ap na a ai law [] a wl a ai was In is iniial l nl was a ai islf was fin fnin f ai In ws ai wl b slf-ssain an slf-fin Hn ai pns pn islf Einsin fl is was n apabl an an is iw in a i psnain f spa-i nin Psiiss a lk a is as s a a ai ns iss an i saisfis a wa qain a inls all f f al ins T ns is nall siplifi sl in a ain ianal an all s ff f ns ianal a in Ts s wl ipl a ai in n iin f aain bwn w bis l als a an ipa in an iin in n f i lina anla n Hw if nl ianal lns a s analiall is pssibili will n appa Tis is sa w fink [ ] s f H n nl fins a ai law a s aspiall saisf Nwnian ai b als inls ffs f Einsin s T f Rlaii Tis is a bi nibin if is aiainal law al T aj ipliain is a an nw ai law s saisf sa niins as д CEOp@ninsaap Cpi b as Pblis wi pis

2 Nwnian ai in a is n ai ff a infini isans B as an iss a inls anla n H ai is n nl a lina aain bwn w bis b a bis will in anla n bwn a basiall a nsqn f fa ain a swa inls ial lkin Wa is als ial is a is a in fa bas pn sp f b f ins as a fnin f sp f li Wa is f ins is a Nwnian aiain is bas pn isan bwn w aai bis fink iplis a w bis a aai b a ai is als a fnin f li a b an aai f as wll as anla n T s qins fink nas an fil Tis is ai fil a pnsas wi in a na an sppsl ab sp f li T pbl is a w a nfnal nw nin a ans al w w lak aninfl aa in i alia n an f s is an ls If a wa qains is inl in l ai sl als b a fnin f i Tis qis s pss ali a ai is la in s ann wi i M fninal psnain will inl spaial inas as wll as a fnin f i Hw wl is infln anis in ainin ajis lain n a wl nall ap as a nsan b nw wl b la a fnin f i? In ws an n lain wl bas f isn f ai wl als b a fnin f i Tis sss n a i inas as wi i wi an fas a ipa nais Tis will qi s p nsiain all nsan is pin bas i wl appa ila nsain f n as wll as n In s f an aa s [-6] pf an pin wi a sns n a lsp H was abl ain s qani f aiai n bas pn lain a a in sa T ails a n la ab is insnain T pin is a lks a sa s lain an sal inass s aiain n ll s s a is is n a is i f sa a s a sp f li H n sas a is is n aal sa lain H s lsp an i s a w sa s al psiin iss T sns piks p s infain an s s a al s a Ea als insanansl H n s sin in w plas sns a s f lain f sa H sns as a lw al b sill a ain s is pl as wll as n ls wi n la apabl planain a sa is sw piin infain f a f lain Tn as la Lani pfs sa pin an lw an bl ais sa ni pinal sls Aain f als a pplin wi n la planain In b ass b insias pf ilin wi i s an ps T iss ab f ns is iffil sl Pkln pf sal in pins On f s inl nain a aiainal pls a s a spn sn b a Tsla fil an a sal illin ls U sal lass Pkln [7] ipli a ai pls li as a a sp na 6 is sp f li Tis is a nsal an ilas nsains f nninal wis T pin is a Einsin sss a naal liis f sp wl b sp f li w isn f s apabiliis is w plin Ma [8] f apl fin T f Rlaii wi pails a fas an sp f li an in s siains n iaina als wiin isan inains All f s pins a f ins isian b q ains n w fin an alia a ai law Tis will a b pf in a ann a allws s s npa nl Tis is f ins ain spifi niins a wl s fink s ai law an w i ipas nninal wis Obisl w n a a sns ain sp f li ab s niins Hw s siains a b ain II Diss T pi inin is lp a asnabl s l ain s iffn aiainal laws Obisl is wl b ib in finin a iffn slin f plainin ak a S ns ss a a iffn law wl pnsa f all f iffns a s as anla in f ain as wiin spial alais Aiinall iw is if ak a iss i as n lial ani a If in a fas an sp f li an llis wi ak a wl a a sal nsqn wi isass sls pn a spa il In ws if ak a was al w wl n b abl la bl abl a w affinal all Ea

3 A aiainal Mls L s isss s ls T Nwnian ai law asss a is an aain a as bwn w spaa bis I is bas pn asss f w bis an spaain isan bwn s bis T aiainal fil ain n n f bis is la ass f bis as wll Of s is isan as a bwn b bis an nal is in aial ni iin is a nsan al an is ass In is l ai s a l la isans isappas a infini T a nsiabl nbs f ls f aiain Effs b is a [9] an s a inl in fllwin abl Tabl I Diffn aiainal Laws a a sp f niins f ins ai Law Asspins aiainal Rl Nwnian aiain an w an Winb s Rl w w an fink s ai an -ai an Ma s ifiain f fink an k k k k Ma s ai law w an i i T fink aiainal l inls an aai f bwn w bis as wll as ain an anla n Al is a in f sp f li niins aiainal law is n nl a fnin f isan bwn w bis b als a lai li fnint fnk l inls / B Ts Mls A asnabl s wl inl spifi ins a pssss nsal aj pfan n sl in laiis [ ] F apl Libain pins [ ] a i as a lai in in pas spa

4 psnain f si -b pbl si in sns a aiainal fil als wi w la bis an a i b i a sall salli spa af s n a appiabl ai is infln In ali is is n a lai b a a nll pin in aiainal fil Ts pins inia a ai aains bwn w ain bis pnsa f a as wll as in f i b If i was a lai i wl b infini s as iiin b a isan w isan s Ts Libain Laanian pins wl b in f iffn ai ls On as ais q a a fi Libain pins Nwnian aiain Of s llina pins a nsi as nsabl wil w a nsi as sabl Cnlss ls a lk a s pins an s a assss analial ans na n sabili al wi s nsabl pins T aj iss w is a fw sallis bis a all a s lains s i a b f al s if a aiains a s lains if s pins a is bas pn ls b in O bial analis i is a infini b if w lii asspin a a l s aspiall appa Nwnian aiain s isss a ala sl In s analsis in nsiin aiinal kinai s f ain laii slins sis sl in llipial ajis b ajis a n s H analial ajis a a spial-lik ff w a i fqn iss F iws f is insia i is iffil a s ajis an is bas i ns nai siain wn is s p wn n i is n la ab w aais an b sib alisiall Iniil sls sl b s s ajis s s l als b pa f in f pins II Analsis Tis sin will nninal wis as wll as aiains a ipa ajis la-lik bai Aain bji is fin s paila ins b n nssail n spifi sls A Pliina Rsls f Obis an Tajis Cnsi a w-b pbl Nwnian ai W will ass a a w spaa bis n a is sall la wil sn n is lail sall T la n is ass b saina an ffs f sn b a nsi lail sall wi sp la b Wi s asspins aial an anla n qain is fin as F a F a ˆ ˆ ˆ T sbsips f fs in LHS a n iais b aall aial an aial f iins spil Diais a fnins f i T aial f is bas pn aiainal aain bwn w bis M sn qain asss aial f aniss f a f s bis In is pbl aial in is an as iffn bwn isans w bjs T pbl an b n in wi s finiins w µ is al ass f b bis + Tis is nsi as aiainal aain f is pbl as fllws Clal aial aiain isappas wi a nsan a is anla n p ni ass s saisf is qain f aial alain Ts sn qain aniss A is pin a aiabl is

5 sl bas pn an ins fnin f ais siplif pbl an in i iais wi sbsiins f pbl Tis sls in Wn s a sbsi in ab qain f aial n sls a O wi s siplifiains 5 T slin f is ina iffnial qain nsiin a i ln l an nii is s s l C 6 T ipan f is qain is a nii plas a ifian l Basiall sall b as ab la b wi a ila bi if is if nii is psii an lss an bi is an llipial bi wi aj b la a n f fal pins in llipial bi If nii is a an bi is pbli an i las saps aiainal pll awa f la b Obisl is sl pns pn iniial li niins an kini n bf inain B fink s Tw-B Pbl B applin aiainal law aial an anla n bs In is pbl l li f spaaf b ass al a ba asspin as a nsan siplif s qains Tis is n as sipl as in nninal wis w ia is pi a lainsip f i iai f anla ans Tis wl b s i a lainsip f aial iais an a pssibl slin Tis is finin a slin b fis finin anla li p ln n F Tis sss a anla n is an ins fnin f i an a ais is sqa F anla n an b al as a fnin f i wi wl infln ajis Unfnal is n al wa n i in an anla fnin T is an wa w lk a is

6 w L U ins fnin f isan f s qains b Siplifin is ini bs Tis is a nnlina qain pa nninal wis bas f nina On pssibl slin is s n s fnin wi iniial al pbl [ 5] Wi is slin bs In is pbl inal als a bas pn iniial anl final anl Aiinal s sl inl i iniial als bna al niins T is a pbl wi a n s fnin Tis is ass a Appni A C aiainal Hpbli Paial Diffnial Eqain Ml Einsin s f laii iplis a aiainal l s saisf wa qains s as Tis is in s-sipls f f s an qain If w lk a is nl as a ns qain n f a ai l will inl s fs a inl ansin s T w-b pbl wl lk siila s T pli is a nninal wis will s ais as a fnin f anl inpnn f i b nw wi is i wi l als inl s as as wll ann b in s a f in a ai as s an wl n s likl b pnn pn a spifi anla pfn Tis ff f s s is inas wi i an pssibl appa an aspi al U s s ils s s Tis pis an in pin F apl an plans an bina plsas s as 9+6 [] a nsal ain ab pia bis Tis l b an inas a as a fnin f i H la apl inias a nsal aj in l b as b lss f n in nn sa b nain aiainal was T iss is w la i spaial inas an spiall anla ans Tis is sin a will n appn a n i pi T nl pin is a n a b s w sai a sn is n a nla f i bas i sl a a i sfa pa Wn ask wa is a sa? His spns was a a sa is a ain a ns spa-i nin in n! T pin is ab ass nin in

7 n b w an w la i i anla n n f a a ass? Tis is bn al f nl b i ss pin a las ais iss D A Rlaiisi Pbl Tis pbl [6] ains ans in bi laiisi ans T nin qain is Tis qain is nnlina T slins i f insias ss aiinal s wi fnin an slin bi is a insa f nain a s aj a sial kinks aln aj Tis s n s asnabl f a nais pspi an slin sl iniil b a s In ws n wl p a nii i an U Appni A f an iniial al slin sl wl lk lik s Tis sws ipa f ais n nina an a is sl pi a sin sl N a as isan inass ff ns lw is al in s f a in fa In ws f la isans sl n p s an ipa n an bi an s ffs a nl paln a lw bis A in f isan bas pn bi wi an wi laiisi ffs is s s Tis sl alas iffns f laiisi bis wi bnin bis In b f s qains n a nina is qi la wi als is s qains lw als Hw i sill a b wwil ain aiainal ffs L s ass a is a ila bi a a nsan isan f a l f bi T isan an bas pn alisi ffs an b in Hw sl a b aninlss p f an llipial bi T sls a Clal inas in isan iiniss laiisi ffs f is p f bi an iniil n wl p sa sls f an llipial bi U Appni B a ps-slin f w-pin bna al pbl wl lk lik s s s

8 T pbl wi w-pin bna-al pbl is a n s fnin ann b sablis bas pn npin Tiall si al f θ = π f ls wi a ls llipial ila bi sls in a lai f n s fnin Ts w an nl lk a a fiiis siain w θ is f f π/ π/ saisf qin a n s fnin s a n pins Wi is sii asspin w an ak a siila spns as in as f iniial al pbl T iffn f s ajis is s s s s s s An iffn f a l f aj is sa If is was n li sls wl b iffn f a paabli sai lin pbli aj bas f n pin in aial iin E Pliina Rsls f Libain Pins Tis pbl is all si -b pbl L s lk a psnain f Libain pins Nwnian aiain F -inal Rsi T-b pbl kinais a F an F F W ifis i iais an a lns f a Casian ina ss an aiainal law qis an w T al is isan bwn aj bis bas pn sbsip an i b µ is nali wi f la bis L s als ass a F als a psn fs a as wll T n ps is fn b liplin fis qain n wi an ina in i Tis is als ai wi lipli b an b als ina as a fnin f i T slin n qain is E E On is las qain is is pin a i s n appa pliil Hw i a an la aiains wi an f s F apl if ai ans wi i n n wl p a n wl als b a fnin f i Hw q is w n s f as a fnin f i nsiin nsain

9 U Calls f aiains Libain pins psn fi pins a a lal ini an ai als bas pn ai law spifiain T llina pins a nsi as nsabl wi ans a psn lal ais wil w ianla pins a sabl a an b nsi as lal ini als f nain sabili Wa s is an? If nsabl a salli pla a llina pins wl qi s ppl ainain is lain was sabl pins wl qi lw ppl ns Wa is f nn is a nsiabl analsis as bn pf alia s pins w a fw sallis psiall a s lains pinall alia s isss T pbl is if s fink s law qains an as fllws an Tis appas as a in a is a fnin f / Tis is a plia siain wi aiinal ss-plin s Finin n is als iffil as wll as finin aal Libain pins If s pis ps in n f is nai l i is l pl M s s a pi insis n w ain lains a sabl ss nsabl lal bis If aiainal law inls a wa qain a siila in is s as a fnin f i an T aiainal ain lks lik s s an w Al lss pl inif n bs s s E E In b f s ass n is n ln a nsan b inls a in fa f sp f li in fis siain an is a fnin f i in sn siain Tis la siain l plain w llina pins a nsabl qiin s f sain kpin Likwis sp f li fa sss a as spaaf inass infln f pia bis ai iiniss an as lss f a naiain pbl f inflnin spaaf Clal w a w iffn pssibl s wi na iws bas pn sp f spaaf Tis p f siain is ial b fnins alia s aiainal laws F apl knwl ns b fn ain nais a s Libain pins s if s aiinal s a al n If n n is aiainal law sl b nsi as inali F sns f n ain as a fnin f i bas pn is infln ns b nfi If n n i jp nl a a qains a n ali an a ai l f T f Rlaii a waan s f insiains

10 I Cnls Tis insiain sss a if iffn aiainal ls is sl b s in a l b s s s spaa ls In s f s siains ls qi ina n as a fnin f i F Libain pins isn f i a Rlaiisi l a plain nsabl llina pins S slins f sipl pbls a aaial isss a n a sipl n iial slins If anin s f Nwnian aiain las wa iabl an sipl aaial psnains w if w a a wi ak a an nknwns in fa-aba is nsal pnnn will w b apabl f insiain pssibl ls a a b qi? Rfns [] H A Ln H Wl an H Minkwski Einsin T Pinipl f Rlaii D Pbliains In 95 [] fink O D aiain an Caiain El Sinifi Cpan Sa Ci ISBN X 6 [] fink O D Elani Raain an T f Rlaii El Sinifi Cpan Sa Ci Ws iinia 997 [] a SM 99 Fal finiin f asali an s ais alilan Elnais [5] a SM T f f i alilan Elnais 9- [6] a S M Sk O an Skin MO Epinal ifiain f s Inain f Naal Psss alilan Elnais - 8 [7] Pkln E an Mans S f Li Inain wi ai Iplss an Masns f Sp f ai Iplss - ai-spn Inains T an Epin Eis - ianni Mans an ln A Rbsn Bna Bks ISBN [8] Ma P A Fas Tan Li Tal ss Einsin l N alilan Elnais Ma/n [9] Ma P A aiainal Sks Sk Was An Ei Spa Ppl SAP nal an STAIF II Albqq Nw Mi [] Ma P A Plsa Bai a a ipa a F Spa Ppls STAIF II Albqq Nw Mi [] Ma P A A Tial Sl F Tw-B Bina Plsa Clsial Manis Pbl STAIF II Albqq Nw Mi [] Bain R H An Inin Maais an Ms f Asnais ISBN AIAA Eain Sis 987 [] Bak R M L j Asnais- Appliains an Aan Tpis Aai Pss Nw Yk an Lnn 967 [] Fian B Pinipls An Tniqs f Appli Maais Appli Maais Sis n Wil & Sns In 956 [5] Winb H F Paial Diffnial Eqains wi Cpl aiabls an Tansf Ms Blaisll Pbli Cpan 965 [6] Nlsn R A Hanbk n Rlaiisi Ti Tansf Eiin O Appni A Slin f Iniial al Bna-al Pbls n s Fnins Tis pbl lks a niins w iniial als a fin a iniial i Tis s n a nsains an bi an b an llipial ila pbli paabli bi If s n s fnin f pbl f slin f iniial al pbl lks lik s In all f s pbls θ is nsi as f iniial niins b swn f plnss T las w s nsi iniial al an is iai is wi sp anla an in Tis bs s

11 O s N a is is iffn f qain f sibin ajis a inl llipial ila an paabli s bas f bas pn ffiin f s θ Tis wl infln nii finiin nsanin iniial niins L s lk a s wi s ails T aj will b fin if anl a iniial psiin is sl an infln iniial aial li Tpiall f llipial ila bis wl sl iniial aial li as a pin f bi s ai ini lain w ffiin f wl isappa Tis sls s s O ps f nii bs Tis snl pns pn iniial niins f iniial ais On is basis lainsip saisfis nninal wis Appni B Slin f Tw-Pin Bna-al Pbls n s Fnins Tis aj is li in a wan lk a bis a pa sls S bis wl inl ila an llipial bis F is pbl sls will b bas pn an iniial al an al af sa π wi sa al If s n s fnin f pbl f f T slin f iniial al pbl lks lik s s s f f T pbl is a slin qis a sa al s wn θ θ is as wll as wn i is a Unfnal al f = π ann b s fin n s fnin

From thriving city cycle paths to calm and tranquil car-free routes, the UK offers some of the most stunning and varied bike rides around.

From thriving city cycle paths to calm and tranquil car-free routes, the UK offers some of the most stunning and varied bike rides around. Vii i ii 195 0 i Wi T f l W Wi l W ilw li l p l i i v i, F f i qil f. UK ff i bik k i l p v p f li lp j i S, p Bii x l w w lv, w k ff i i v f f i w i l v l pbli l p. l k fi b il v l l w i f i f T f f i.