GRAVITATION EXPLAINED BY THE THEORY OF INFORMATONS

Transcription

1 GRAVITATION XLAIND BY TH THORY OF INFORMATONS Antoine Ake * ABSTRACT The theoy of infomatons explains the avitational inteations by the hypothesis that infomation is the substane of avitational fields The onstituent element of that substane is alled an infomaton The theoy stats fom the idea that any mateial objet manifests itself in spae by the emission of infomatons: anula mass and eney less entities ushin away with the speed of liht and ayin infomation about the position ( -infomation ) and about the veloity ( -infomation ) of the emitte In this atile the avitational field is haateised; the laws of avito-eletomanetism ae mathematially dedued fom the dynamis of the infomatons; the avitational inteations ae explained as the effet of the tend of a mateial objet to beome blind fo flows of infomation eneated by othe masses; and avitons ae identified as infomatons ayin a quantum of eney INTRODUCTION Daily ontat with the thins on hand onfonts us with thei substantiality An objet is not just fom, it is also matte It takes spae, it eliminates emptiness The amount of matte within the ontous of a physial body is alled its mass The mass of an objet manifests itself when it inteats with othe objets A fundamental fom of inteation is avitation Mateial objets (masses) ation at a distane on eah othe: they attat eah othe and if they ae fee, they move to eah othe alon the staiht line that onnets them Aodin to the lassial theoy of fields, the avitational inteations an be desibed by intoduin the avitational field : eah mateial objet manifests its substantiality in spae by eatin and maintainin a veto field and eah objet in that field expeienes a tendeny to hane its state of motion The field theoy onsides the avitational field as the mathematial entity that mediates in the avitational inteation This is futhe developed by Olive Heaviside () and Ole Jefimenko () In the theoy of avito-eletomanetism (GM) they desibe the avitational field statin fom the idea that it must be isomophi with the eletomaneti field This implies that it should be haateized by two vetoial quantities that ae analoue to espetively the eleti field and the maneti indution B, and that the elations ovenin these quantities should be analoue to Maxwell s laws * antake@skynetbe - wwwantoineakenet Gavitation explained by the theoy of infomatons Antoine Ake

2 Within the famewok of eneal elativity, GM has been disussed by a numbe of authos () It is shown that the avitational analoues to Maxwell s equations (the GM equations) an be deived fom the instein field equation Althouh GM desibes the avitational phenomena in a oet and oheent manne, it doesn t eate laity about the tue natue of the ation at a distane In the ontext of GM, the avitational field is a puely mathematial onstution that doesn t povide insiht in the mehanisms that ae at the base of the physial laws In this pape we develop the idea that, if masses an influene eah othe at a distane, they must in one way o anothe exhane data We assume that eah mass emits infomation elative to its manitude and its position, and is able to intepet the infomation emitted by its neihbous In this way we popose a physial foundation of GM by intoduin infomation as the substane of the avitational field xpliitly, we stat fom the idea that the avitational field of a mateial objet an be explained as the maosopi manifestation of the emission by that objet of mass-, eneyand anula entities ushin away with the speed of liht and ayin infomation about the position ( -infomation ) and the veloity ( -infomation ) of the emitte Beause they tanspot nothin else than infomation, we all these entities infomatons In the postulate of the emission of infomatons, we define an infomaton by its attibutes and detemine the ules that oven the emission of infomatons by a point mass that is anhoed in an inetial efeene fame The fist onsequene of that postulate is that a point mass at est - and by extension any mateial objet at est - an be onsideed as the soue of an expandin spheial loud of infomatons, that - in an abitay point - is haateised by the vetoial quantity is the density of the flow of -infomation in that point That loud of infomatons an be identified with the avitational field and the quantity with the avitational field stenth in A seond onsequene is that the infomatons emitted by a movin point mass, onstitute a avitational field that is haateised by two vetoial quantities:, the density of the -infomation flow, and B, the density of the -infomation loud We show that the elations - aisin fom the dynamis of the infomatons - between these two quantities (the laws of GM) ae the avitational analoues of the laws of Maxwell- Heaviside Next we explain the avitational inteation between masses as the eation of a point mass on the distubane of the symmety of its own avitational field by the field that, in its diet viinity, is eated and maintained by othe masses And finally we examine the emission of eney by an aeleatin mass Gavitation explained by the theoy of infomatons Antoine Ake

3 I The ostulate of the mission of Infomatons With the aim to undestand and to desibe the mehanism of the avitational inteation, we intodue a new quantity in the asenal of physial onepts: infomation We suppose that infomation is tanspoted by mass and eney less anula entities that ush thouh spae with the speed of liht () We all these infomation aies infomatons ah mateial objet ontinuously emits infomatons An infomaton always aies - infomation, that is at the oot of avitation The emission of infomatons by a point mass (m) anhoed in an inetial efeene fame O, is ovened by the postulate of the emission of infomatons: A The emission is ovened by the followin ules: The emission is unifom in all dietions of spae, and the infomatons divee with the speed of liht ( 8 m/s) alon adial tajetoies elative to the loation of the emitte dn N &, the ate at whih a point-mass emits infomatons, is time independent and dt popotional to its mass m So, thee is a onstant K so that: N & K m The onstant K is equal to the atio of the squae of the speed of liht () to the lank onstant (h): 5 K,6 k s h B We all the essential attibute of an infomaton its -spin The -spin of an infomaton efes to infomation about the position of its emitte and equals the elementay quantity of -infomation It is epesented by a vetoial quantity s, the -spin veto : s is points to the position of the emitte All -spin vetos have the same manitude, namely: s 6 6,8 m s K η 9 ( η,9 k s m with G the avitational onstant) 4 π G s, the manitude of the -spin-veto, is the elementay -infomation quantity We nelet the possible stohasti natue of the emission, that is esponsible fo noise on the quantities that haateize the avitational field So, N is the aveae emission ate Gavitation explained by the theoy of infomatons Antoine Ake

4 4 II The avitational Field of Masses at Rest The avitational field of a point mass at est In fi we onside a point mass that is anhoed in the oiin of an inetial efeene fame O It ontinuously emits infomatons in all dietions of spae The infomatons that pass nea a fixed point - defined by the position veto - have two attibutes: thei veloity and thei -spin veto s : e and s e K η K η Z s O m Y X Fi The ate at whih the point mass emits -infomation is the podut of the ate at whih it emits infomatons with the elementay -infomation quantity: N& s m η Of ouse, this is also the ate at whih it sends -infomation thouh any losed sufae that spans m The emission of infomatons fills the spae aound m with an expandin loud of - infomation This loud has the shape of a sphee whose sufae oes away - with the speed of liht - fom the ente O, the position of the point mass - Within the loud is a stationay state: beause the inflow equals the outflow, eah spatial eion ontains an unhanin numbe of infomatons and thus a onstant quantity of - infomation Moeove, the oientation of the -spin vetos of the infomatons passin thouh a fixed point is always the same - The loud an be identified with a ontinuum: eah spatial eion ontains a vey lae numbe of infomatons: the -infomation is like ontinuously spead ove the volume of Gavitation explained by the theoy of infomatons Antoine Ake

5 5 the eion That loud of -infomation suoundin O onstitutes the avitational field * o the -field of the point mass m Without inteuption ountless infomatons ae ushin thouh any - even vey small - sufae in the avitational field: we an desibe the motion of -infomation thouh a sufae as a ontinuous flow of -infomation We know aleady that the intensity of the flow of -infomation thouh a losed sufae that spans O is expessed as: m N& s η If the losed sufae is a sphee with adius, the intensity of the flow pe unit aea is iven by: m 4 π η This is the density of the flow of -infomation in eah point at a distane fom m (fi ) This quantity is, toethe with the oientation of the -spin vetos of the infomatons that ae passin nea, haateisti fo het avitational field in that point Thus, in a point, the avitational field of the point mass m is defined by the vetoial quantity : N& m m s e 4 π 4 π η 4 π η This quantity is the avitational field stenth o the -field stenth o the -field In any point of the avitational field of the point mass m, the oientation of oesponds to the oientation of the -spin-vetos of the infomatons who ae passin nea that point And the manitude of is the density of the -infomation flow in that point Let us note that is opposite to the sense of movement of the infomatons * The time T elapsed sine the emeene of a point-mass (this is the time elapsed sine the emeene of the univese) and the adius R of its field of avitation ae linked by the elation R T Assumin that the univese - sine its beinnin (,8 yeas ao) - unifomly expands, a point at a distane fom m uns away with speed v: v H R T H is the Hubble onstant: H 4,7 T m / s millionliht yeas Gavitation explained by the theoy of infomatons Antoine Ake

6 6 Let us onside a sufae-element ds in (fi,a) Its oientation and manitude ae ompletely detemined by the sufae-veto ds (fi,b) ds ds e n ds e n ds α Fi,a Fi,b By dφ, we epesent the ate at whih -infomation flows thouh ds in the sense of the positive nomal and we all this sala quantity the elementay -flux thouh ds: dφ ds dsosα Fo an abitay losed sufae S that spans m, the outwad flux (whih we obtain by inteatin the elementay ontibutions dφ ove S) must be equal to the ate at whih the mass emits -infomation Thus: Φ ds This elation expesses the onsevation of -infomation in the ase of a point mass at est The avitational field of a set of point-masses at est We onside a set of point-masses m,,m i, m n that ae anhoed in an inetial fame O In an abitay point, the flows of -infomation who ae emitted by the distint masses ae defined by the avitational fields,,,, m η i n dφ, the ate at whih -infomation flows thouh a sufae-element ds in in the sense of the positive nomal, is the sum of the ontibutions of the distint masses: n n dφ ( i ds) ( i ) ds ds i So, the effetive density of the flow of -infomation in (the effetive -field ) is ompletely defined by: n i i i Gavitation explained by the theoy of infomatons Antoine Ake

7 7 We onlude: The -field of a set of point masses at est is in any point of spae ompletely defined by the vetoial sum of the -fields aused by the distint masses Let us note that the oientation of the effetive -field has no lone a elation with the dietion in whih the passin infomatons ae movin One shows easily that the outwad -flux thouh a losed sufae in the -field of a set of anhoed point masses only depends on the spanned masses m in : Φ min ds η This elation expesses the onsevation of -infomation in the ase of a set of point masses at est The avitational field of a mass ontinuum at est We all an objet in whih the matte in a time independent manne is spead ove the oupied volume, a mass ontinuum In eah point Q of suh a ontinuum, the aumulation of mass is defined by the (mass) density ρ G To define this sala quantity one onsides a volume element dv that ontains Q, and one detemines the enlosed mass dm The aumulation of mass in the viinity of Q is defined by: dm ρ G dv A mass ontinuum - anhoed in an inetial fame - is equivalent to a set of infinitely many infinitesimal mass elements dm The ontibution of eah of them to the field stenth in an abitay point is d, the effetive field stenth in, is the esult of the inteation ove the volume of the ontinuum of all these ontibutions It is evident that the outwad -flux thouh a losed sufae S only depends on the mass enlosed by the sufae (the enlosed volume is V) ds S ρg dv η That is equivalent with (theoem of Ostoadsky) (4) : V div ρ G η Gavitation explained by the theoy of infomatons Antoine Ake

8 8 This elation expesses the onsevation of -infomation in the ase of a mass ontiuum at est Futhemoe, one an show that: ot, what implies the existene of a avitational potential funtion V fo whih: adv Rest mass and elativisti mass III The avitational Field of movin Masses ZZ v m O Y X O Y X Fi In fi, we onside a point mass that moves with onstant veloity v v ezalon the Z-axis of an inetial efeene fame O At the moment t, it passes thouh the oiin O and at the moment t t thouh the point We posit that N & - the ate at whih a point mass emits infomatons in the spae onneted to O - is detemined by its est mass m and is independent of its motion: dn N & Km dt That implies that, if the time is ead on a standad lok anhoed in O, dn - the numbe of infomatons that duin the inteval dt by a - whethe o not movin - point mass is emitted in the spae onneted to O, is: dn K m dt We an the spae-time also onnet to an inetial efeene fame O (fi ) whose oiin is anhoed to the point mass and that is unnin away elative to O with the veloity v v ez We assume that t t when the mass passes thouh O (t is the time ead on a standad lok in O and t the time ead on a standad lok in O ) We detemine the time that expies while the movin point mass emits dn infomatons Gavitation explained by the theoy of infomatons Antoine Ake

9 9 An obseve in O uses theefoe a standad lok that is linked to that efeene fame The emission of dn infomatons takes dt seonds The elationship between dn and dt is: dn K m dt To detemine the duation of the same phenomenon, the obseve in O an also ead the time on the movin lok, that is the standad lok linked to the inetial efeene fame O Aodin to that lok, the emission of dn infomatons takes dt seonds (x, y, z; t) - the oodinates of an event onneted to O - and (x, y, z ; t ) - the oodinates of the same event onneted to O - ae elated by the Loentz-tansfomation (5) : x ' x x x' y ' y y y' z ' z vt z z' + vt' t ' v v t z t' + z' t The elationship between dt and dt is: dt dt' with v So: and: dn dt' m N& K m dt K m K dt' dt' dn dt N& K m ' K m with m m, the elativisti mass Conlusion: The ate at whih a point mass, movin with onstant veloity elative to an inetial efeene fame O, emits infomatons in the spae linked to O, is detemined by its elativisti mass if the time is ead on a standad lok that is anhoed to that mass The field aused by a unifom etilinea movin point mass In fi 4,a, we onside aain a point mass with est mass m that, with onstant veloity v v e z, moves alon the Z-axis of an inetial efeene fame O At the moment t, it passes thouh the oiin O and at the moment t t thouh the point It is evident that: Gavitation explained by the theoy of infomatons Antoine Ake

10 O z v t m ontinuously emits infomatons that, with the speed of liht, ush away with espet to the point whee the mass is at the moment of emission We wish to detemine the density of the flow of -infomation - this is the -field - in a fixed point The position of elative to the efeene fame O is detemined by the time independent Catesian oodinates (x, y, z), o by the time dependent position veto axis θ is the anle between and the Z- ZZ Z v θ ' O Y θ ' X O Y O Y X X (a) Fi 4 (b) Relative to the inetial efeene fame O, that is anhoed to the movin mass and that at the moment t t, oinides with O (fi 4,b), the instantaneous value of the density of the flow of -infomation in is detemined by: ' m 4 πη ' ' Indeed, elative to O the point mass is at est and he position of is detemined by the time dependant position veto ' o by the Catesian oodinates (x, y, z ) So, the -field eneated by the mass is detemined by The omponents of ' in O X Y Z, namely: ' x ' m ' m ' m x' ' y ' y 4 πη z ' z' ' 4 πη ' 4 πη ' detemine in the densities of the flows of -infomation espetively thouh a sufae element dy dz pependiula to the X -axis, thouh a sufae element dz dx pependiula to the Y -axis and thouh a sufae element dx dy pependiula to the Z -axis Gavitation explained by the theoy of infomatons Antoine Ake

11 The -fluxes thouh these diffeent sufae elements in, o the ates at whih - infomation flows thouh it ae: ' m x' x' dy' dz' dy' dz' 4πη ' ' y' ' z' m y' dz' dx' 4πη ' m z' dx' dy' 4πη ' dz' dx' dx' dy' The Catesian oodinates of in the fames O and O ae onneted by (5) : x x y y z ' z v t z z And the line elements by: dx dx dy dy dz' dz Futhe : : ' sin θ So elative to O, the -infomation fluxes that the movin mass sends - in the positive dietion - thouh the sufae elements dydz, dzdx and dxdy in ae: m 4πη m 4πη m 4πη ( sin θ ) ( sin θ ) ( sin θ ) x dy dz y dz dx ( z z ) dx dy In O: And in O : x + y + ( z ), ' + z x' + y' z' and x + y sinθ and x ' + y' sinθ ' ' os z z θ We expess in funtion of x, y and z: ' ( z z ) ( z z ) x + y + sin θ + ( ) sin θ( ) + os θ sin θ Gavitation explained by the theoy of infomatons Antoine Ake

12 Sine the densities in of the flows of -infomation in the dietion of the X-, the Y- and the Z-axis ae the omponents of the -field aused by the movin point mass m in, we find: x y z m 4πη m 4πη m 4πη ( sin θ ) ( sin θ ) ( sin θ ) x y ( z z ) So, the -field aused by the movin point mass in the fixed point is: w m e 4πη 4πη ( sin θ ) m ( sin θ ) We onlude: A point mass desibin - elative to an inetial efeene fame O - a unifom etilinea movement eates in the spae linked to that fame a time dependent avitational field, the -field in an abitay point, points at any time to the position of the mass at that moment and its manitude is: m 4πη ( sin θ ) If the speed of the mass is muh smalle than the speed of liht, this expession edues itself to that valid in the ase of a mass at est This non-elativisti esult ould also been obtained if one assumes that the displaement of the point mass duin the time inteval that the infomatons need to move fom the emitte to an be neleted ompaed to the distane they tavel duin that peiod The oientation of the field stenth implies that the spin vetos of the infomatons that at a etain moment pass thouh, point to the position of the emittin mass at that moment Fom this onlusion on the dietion of the -field, one an dedue that the movement of an objet in a avitational field is detemined by the pesent position of the soue of the field and not by its liht-speed delayed position Gavitation explained by the theoy of infomatons Antoine Ake

13 The emission of infomatons by a point mass desibin a unifom etilinea motion In fi 5 we onside a point mass m that moves with a onstant veloity v alon the Z-axis of an inetial efeene fame Its instantaneous position (at the abitay moment t) is Z v θ s m θ Fi 5 The position of, an abitay fixed point in spae, is defined by the veto The position veto - just like the distane and the anle θ - is time dependent beause the position of is onstantly hanin The infomatons that - with the speed of liht - at the moment t ae passin nea, ae emitted when m was at Bidin the distane took the time inteval Δt: t Duin thei ush fom to, the mass moved fom to : v t -, the veloity of the infomatons, points in the dietion of thei movement, thus alon the adius - s, thei -spin veto, points to, the position of m at the moment t This is an impliation of ule B of the postulate of the emission of infomatons and onfimed by the onlusion of The lines ayin s and fom an anle θ We all this anle, that is haateisti fo the speed of the point mass, the haateisti anle o the haateisti deviation Gavitation explained by the theoy of infomatons Antoine Ake

14 4 The quantity s s sin( ) - efein to the speed of its emitte - is alled the haateisti -infomation o the -infomation of an infomaton We note that an infomaton emitted by a movin point mass, tanspots infomation efein to the veloity of that mass This infomation is epesented by its avitational haateisti veto o -index s that is defined by: s s - The -index is pependiula to the plane fomed by the path of the infomaton and the staiht line that aies the -spin veto, thus pependiula to the plane fomed by the point and the path of the emitte - Its oientation elative to that plane is defined by the ule of the oksew : in the ase of fi 5, the -indies have the oientation of the positive X-axis - Its manitude is: s s sin( ), the -infomation of the infomaton We apply the sine ule to the tianle : sin( ) v t sinθ t It follows: v s s sinθ s sinθ s v is the omponent of the dimensionless veloity pependiula to s Takin into aount the oientation of the diffeent vetos, the -index of an infomaton emitted by a point mass movin with onstant veloity, an also be expessed as: v s s 4 The avitational indution of a point mass desibin a unifom etilinea motion We onside aain the situation of fi 5 All infomatons in dv - the volume element in - ay both -infomation and -infomation The -infomation efes to the veloity of the emittin mass and is epesented by the -indies s : s v s s Gavitation explained by the theoy of infomatons Antoine Ake

15 5 If n is the density in of the loud of infomatons (numbe of infomatons pe unit volume) at the moment t, the amount of -infomation in dv is detemined by the manitude of the veto: s v s n s dv n dv n dv And the density of the the -infomation (haateisti infomation pe unit volume) in is detemined by: s v s n s n n We all this (time dependent) vetoial quantity - that will be epesented by B - the avitational indution o the -indution in : - Its manitude B detemines the density of the -infomation in - Its oientation detemines the oientation of the -vetos s of the infomatons passin nea that point So, the -indution aused in by the movin mass m (fi 5) is: B v s v n ( n s ) N - the density of the flow of infomatons in (the ate pe unit aea at whih the infomatons oss an elementay sufae pependiula to the dietion of movement) - and n - the density of the loud of infomatons in (numbe of infomatons pe unit volume) - ae onneted by the elation: N n With N s, we an expess the avitational indution in as: v v B ( N s ) Takin into aount (): We find: w m 4πη ( sin θ ) This quantity is also alled the oavitational field, epesented as K o the yotation, epesented as Ω Gavitation explained by the theoy of infomatons Antoine Ake

16 6 B m 4πη ( sin θ ) ( v ) We define the onstant ν 9,4-7 mk - as: And finally, we obtain: B ν m 4π ν ( sin θ ) η ( v) B in is pependiula to the plane fomed by and the path of the point mass; its oientation is defined by the ule of the oksew; and its manitude is: B ν m 4π ( sin θ ) v sinθ If the speed of the mass is muh smalle than the speed of liht, this expession edues itself to: ν m B ( v) 4 π This non-elativisti esult ould also be obtained if one assumes that the displaement of the point mass duin the time inteval that the infomatons need to move fom the emitte to an be neleted ompaed to the distane they tavel duin that peiod 5 The avitational field of a point mass desibin a unifom etilinea motion w A point mass m, movin with onstant veloity v v e zalon the Z-axis of an inetial fame, eates and maintains a loud of infomatons that ae ayin both - and -infomation That loud an be identified with a time dependent ontinuum That ontinuum is alled the avitational field of the point mass It is haateized by two time dependent vetoial quantities: the avitational field (shot: -field) and the avitational indution (shot: - indution) B - With N the density of the flow of infomatons in (the ate pe unit aea at whih the infomatons oss an elementay sufae pependiula to the dietion of movement), the -field in that point is: Also alled: avito-eletomaneti (GM field) o avito-maneti field (GM field) Gavitation explained by the theoy of infomatons Antoine Ake

17 7 w N s m 4πη ( sin θ ) The oientation of leans that the dietion of the flow of -infomation in is not the same as the dietion of the flow of infomatons - With n, the density of the loud of infomatons in (numbe of infomatons pe unit volume), the -indution in that point is: B n s ν m ( v) 4π ( sin θ ) One an veify that: div ot div 4 otb B B These elations ae the laws of GM in the ase of the avitational field of a point mass desibin a unifom etilinea motion If v <<, the expessions fo the -field and the -indution edue to: m ν m and B ( v) 4πη 4 π 6 The avitational field of a set of point masses desibin unifom etilinea motions We onside a set of point masses m,,m i, m n that move with onstant veloities v,, v i,, v n in an inetial efeene fame O This set eates and maintains a avitational field that in eah point of the spae linked to O, is haateised by the veto pai (, B ) - ah mass m i ontinuously emits -infomation and ontibutes with an amount i to the -field at an abitay point As in we onlude that the effetive -field in is defined as: i - If it is movin, eah mass m i emits also -infomation, ontibutin to the -indution in with an amount B i It is evident that the -infomation in the volume Gavitation explained by the theoy of infomatons Antoine Ake

18 8 element dv in at eah moment t is expessed by: ( B dv ) ( B dv i i ) Thus, the effetive -indution B in is: B B i The laws of GM mentioned in the pevious setion emain valid fo the effetive -field and -indution in the ase of the avitational field of a set op point masses desibin a unifom etilinea motion 7 The avitational field of a stationay mass flow The tem stationay mass flow indiates the movement of an homoeneous and inompessible fluid that, in an invaiable way, flows elative to an inetial efeene fame The intensity of the flow in an abitay point is haateised by the flow density J G The manitude of this vetoial quantity equals the ate pe unit aea at whih the mass flows thouh a sufae element that is pependiula to the flow in The oientation of J G oesponds to the dietion of that flow If v is the veloity of the mass element ρ G dv that at the moment t flows thouh, then: J ρ v So, the ate at whih the flow tanspots - in the positive sense (defined by the oientation of the sufae vetos ds ) - mass thouh an abitay sufae ΔS, is: i J ds We all i G the intensity of the mass flow thouh ΔS Sine a stationay mass flow is the maosopi manifestation of movin mass elements ρ dv, it eates and maintains a avitational field And sine the veloity v G of the mass element in eah point is time independent, the avitational field of a stationay mass flow will be time independent It is evident that the ules of also apply fo this time independent -field: G G G S G div ρ G η and ot what implies: adv One an pove (6) that the ules fo the time independent -indution ae: divb what implies B ota and otb ν J G Gavitation explained by the theoy of infomatons Antoine Ake

19 9 This ae the laws of GM in the ase of the avitational field of a stationay mass flow 8 The avitational field of an aeleated point mass 8 The -spinveto of an infomaton emitted by an aeleated point mass In fi 6 we onside a point mass m that, duin a finite time inteval, moves with onstant aeleation a a ez elative to the inetial efeene fame OXYZ At the moment t, m stats - fom est - in the oiin O, and at the moment t t it passes in the point Its veloity is thee defined by v v ez a t ez, and its position by z a t v t We v suppose that the speed v emains muh smalle than the speed of liht: << Z e v a s θ m e ϕ θ e O θ Y X Fi 6 The infomatons that duin the infinitesimal time inteval (t, t+dt) pass nea the fixed point (whose position elative to the movin mass m is defined by the time dependant position veto ) have been emitted at the moment t t t, when m - with veloity v v ez v( t t) e z - passed thouh (whose position is defined by the time dependant position veto ( t )) Δt, the time inteval duin whih m moves fom t to is the time that the infomatons need to move - with the speed of liht ()- fom to We an onlude that t, and that v v( t t) v( t ) v a Between the moments t t and t t + Δt, m moves fom to That movement an be onsideed as the esultant of a unifom movement with onstant speed v v( t ) and t a unifomly aeleated movement with onstant aeleation a Gavitation explained by the theoy of infomatons Antoine Ake

20 In fi 6,a, we onside the ase of the point mass m movin with onstant speed v alon the Z-axis At the moment t t t m passes in and at the moment t in ' : ' v t Z θ ' v ' θ ' s ' m θ Fi 6,a The infomatons that, duin the infinitesimal time inteval (t, t + dt), pass nea the point - whose position elative to the unifomly movin mass m at the moment t is ' defined by the position veto - have been emitted at the moment t when m passed in Thei veloity veto is on the line, thei spin veto s points to ' : ' v t v Z " a θ " " s θ " " m θ Fi 6,b In fi 6,b we onside the ase of the point mass m statin at est in and movin with onstant aeleation a alon the Z-axis At the moment t t t it is in " " and at the moment t in : a( t) The infomatons that duin the infinitesimal time inteval (t, t + dt) pass nea the point (whose position elative to the unifomly aeleated mass m is - at the Gavitation explained by the theoy of infomatons Antoine Ake

21 moment t - defined by the position veto ") have been emitted at the moment t when m was in Thei veloity veto points to, thei spinveto s to " " To detemine the position of, we onside the tajetoies of the infomatons that at the moment t ae emitted in the dietion of, elative to the aeleated π efeene fame OX Y Z that is anhoed to m (fi 6,; α θ) Z s " α m Y Fi 6, Relative to OX Y Z these infomatons ae aeleated with an amount a : they follow a paaboli tajetoy defined by the equation: a z' tα y' y' os α At the moment t t + Δt, when they pass in, the tanent line to that tajetoy uts the Z -axis in the point, that is defined by: " z ' " a( t) a That means that the spinvetos of the infomatons that at the moment t pass in, point to a point on the Z-axis that has a lead of on ", the atual position of the mass m " And sine " " a( t) a " " " " " +, we onlude that: a In the inetial efeene fame OXYZ (fi 6), s points to the point on the Z-axis detemined by the supeposition of the effet of the veloity () and the effet of the aeleation (): ' " v a + + Gavitation explained by the theoy of infomatons Antoine Ake

22 The aie line of the spinveto s of an infomaton that - elative to het inetial fame OXYZ - at the moment t passes nea foms a haateisti anle with the aie line of its veloity veto, that an be dedued by appliation of the sine-ule in tianle (fi 6): sin( ) sin( θ + ) v a We onlude: sin( ) sin( θ + ) + sin( θ θ + ) Fom the fat that - the distane tavelled by m duin the time inteval Δt - an be neleted elative to - the distane tavelled by liht in the same inteval - it follows that θ θ + θ and that So: v a sin( ) sinθ + sinθ We an onlude that the spinveto s of an infomaton that at the moment t passes nea, has a omponent in the dietion of - its veloity veto - and a omponent pependiula to that dietion It is evident that: v a s s os( ) e ssin( ) e s e s( sinθ + sinθ ) e 8 The avitational field of an aeleated point mass The infomatons that, at the moment t, ae ushin nea the fixed point - defined by the time dependent position veto - ae emitted when m was in (fi 6) Thei veloity is on the same aie line as Thei -spin veto is on the aie line Aodin to 8, the haateisti anle - this is the anle between the ayin lines of s and - has two omponents: - a omponent ' elated to the veloity of m at the moment ( t ) when the onsideed infomatons wee emitted In the famewok of ou assumptions, this omponent is: v( t ) sin( ') sinθ - a omponent θ" elated to the aeleation of m at the moment when they wee emitted This omponent is, in the famewok of ou assumptions: a( t ) sin( ") sinθ Gavitation explained by the theoy of infomatons Antoine Ake

23 The maosopi effet of the emission of -infomation by the aeleated mass m is a avitational field (, B ) We intodue the efeene system ( e, e, eϕ ) (fi 6), the field in, is defined as the density of the flow of -infomation in that point That density is the ate at whih -infomation pe unit aea osses in the elementay sufae pependiula to the dietion of movement of the infomatons So is the podut of N, the density of the flow of infomatons in, with s, thei spinveto: N s Aodin to the postulate of the emission of infomatons, the manitude of s the elementay -infomation quantity: 6 s 6,8 m s K η and the density of the flow of infomatons in is: N& N& K m N 4 π 4 π 4 π Takin this into aount and knowin that ν, we obtain: η is m 4 π η e m { 4 π η ν m v( t )sinθ + a( t )sinθ} e 4 π B, the avitational indution in, is defined as the density of the loud of - infomation in that point That density is the podut of n, the density of the loud of infomations in (numbe pe unit volume) with s, thei -index: n s B The -index of an infomaton haateizes the infomation it aies about the state of motion of its emitte; it is defined as: s s And the density of the loud of infomatons in is elated to N, the density of the N flow of infomatons in that point by: n N s ( N s ) So: B n s And with: m 4 π η e m { 4 π η ν m v( t )sinθ + a( t )sinθ} e 4 π, Gavitation explained by the theoy of infomatons Antoine Ake

24 4 we obtain: B ν m 4 π ν m v( t )sinθ + a( t 4 π { )sinθ} e ϕ IV The Laws of the avitational Field - The Laws of GM In the spae linked to an inetial efeene fame O, the avitational field is haateised by two time dependent vetos: the (effetive) -field and the (effetive) -indution B In an abitay point, these vetos ae the esults of the supeposition of the ontibutions of the vaious soues of infomatons (the masses) to espetively the density of the flow of -infomation and to the loud of -infomation in : N s and B n s The infomatons that - at the moment t - pass nea with veloity ontibute with an amount ( N s ) to the instantaneous value of the -field and with an amount ( n s ) to the instantaneous value of the -indution in that point - s and s espetively ae thei -spin and thei -index They ae linked by the elationship: s s - N is the instantaneous value of the density of the flow of infomatons with veloity at and n is the instantaneous value of the density of the loud of those infomatons in that point N and n ae linked by the elationship: N n 4 Relations between and B in a matte fee point of a avitational Field In eah point whee no matte is loated - whee ρ G (x, y,z;t) ( x, y, z; t) - the followin statements ae valid J G In a matte fee point of a avitational field, the spatial vaiation of obeys the law: div Gavitation explained by the theoy of infomatons Antoine Ake

25 5 This statement is the expession of the law of onsevation of -infomation The fat that the ate at whih -infomation flows inside a losed empty spae must be equal to the ate at whih it flows out, an be expessed as: So (theoem of Ostoadsky) (4) : ds S div In a matte fee point of a avitational field, the spatial vaiation of B obeys the law: divb This statement is the expession of the fat that the -index of an infomaton is always pependiula to its -spin veto s and to its veloity Y θ s X Q s Z Fi 7 In fi 7, we onside the flow of infomatons that - at the moment t - pass with veloity w in the viinity of the point An infomaton that at the moment t passes in is at the moment (t + dt) in Q: Q dt In, the instantaneous value of the density of the onsideed flow of infomatons is epesented by N, the instantaneous value of the density of the loud that they onstitute by n, and the instantaneous value of thei haateisti anle by θ We intodue the oodinate system XYZ: s s s ex and s s sin( ) ez The ontibution of the onsideed infomatons to the -indution in is: n s Fom mathematis (4) we know: B Gavitation explained by the theoy of infomatons Antoine Ake

26 6 divb v div( n s ) ad ( n) s + n div( s ) ad ( n) s beause ad(n) is pependiula to s Indeed n hanes only in the dietion of the flow of infomatons, so ad(n) has the same oientation as : nq n ad ( n) Q s ds n div( s ) Aodin to the definition: div( s ) We alulate the dv double inteal ove the losed sufae S fomed by the infinitesimal sufaes ds dzdy whih ae in and in Q pependiula to the X-axis and by the tube whih onnets the edes of these sufaes dv is the infinitesimal volume enlosed by S It is obvious that: s ds div( s ) dv Both tems of the expession of div ae zeo, so div, what implies (theoem of Ostoadsky) that fo evey losed sufae S in a avitational field: B B ds S B In a matte fee point of a avitational field, the spatial vaiation of and the ate at whih B is hanin ae onneted by the elation: B ot This statement is the expession of the fat that any hane of the podut n s in a point of a avitational field is elated to a vaiation of the podut N s in the viinity of that point We onside aain and B, the ontibutions to the -field and to the -indution in the point of the infomatons whih - at the moment t - pass with veloity in the viinity of that point (fi 7) w s N s N s e x and B n s n n s sin( ) ez We investiate the elationship between ot ad ( N ) s } + N ot( s ) and { B n w s s + n Gavitation explained by the theoy of infomatons Antoine Ake

27 7 The tem { ad ( N) s} desibes the omponent of ot aused by the spatial vaiation of N in the viinity of when emains onstant N has the same value in all points of the infinitesimal sufae that, in, is pependiula to the flow of infomatons So ad(n) is paallel to and its manitude is the inease of the manitude of N pe unit lenth With N N, N Q N + dn and Q dt, ad(n) is detemined by: It follows: N N Q dn ad ( N ) Q dt dn dn ad ( N ) s s s dt dt Fom the fat that the density of the flow of infomatons in Q at the moment t is equal to the density of that flow in at the moment (t - dt), it follows: If N (t) N, then N (t - dt) N Q (t) N + dn The ate at whih N hanes at the moment t is: N N ( t) N ( t dt dn ) dt dt And sine: N n dn N n : dt t We onlude (I): ad ( N ) s n s Y q p s Q Q s Z s Fi 8 θ X Gavitation explained by the theoy of infomatons Antoine Ake

28 8 The tem { N ot( s )} desibes the omponent of ot aused by the spatial vaiation of θ - the oientation of the -spinveto in the viinity of - when N emains onstant At the moment t, ( θ ) - the haateisti anle of the infomatons that pass in - diffes fom ( θ ) Q - the haateisti anle of the infomatons that pass in Q If ( θ ) θ, than ( θ ) Q θ + d( θ ) (fi 8) Fo the alulation of ot( s ), we alulate s dl alon the losed path Qqp that eniles the aea ds Qp dtp (Q and qp ae paallel to the flow of the infomatons, Qq and p ae pependiula to it s dl N ot( s ) N ez ds s N sin{ + d( )} Qq ssin( dt p ) p e z Fom the fat that the haateisti anle of the infomatons in Q at the moment t is equal to the haateisti anle of the infomatons in at the moment (t - dt), it follows: If ( θ ) (t) θ, then ( θ ) (t- dt) ( θ ) Q (t) θ + d( θ ) The ate at whih sin( θ ) in hanes at the moment t, is: {sin( )} sin( ) sin{ + d ( )} d{sin( )} dt dt And sine N n, we obtain (II): sin{ + d( )} sin( ) s N ot ( s ) N s { n s sin( ) ez} n dt Combinin the esults (I) and (II), we obtain: ot ad ( N ) s + N ot ( s n ) ( s s B + n ) B The elation ot implies (theoem of Stokes (4) ): In a avitational field, the ate at whih the sufae inteal of B ove a sufae S hanes is equal and opposite to the line inteal of ove its boundey L: B Φ dl ds B ds S S b Gavitation explained by the theoy of infomatons Antoine Ake

29 9 The oientation of the sufae veto ds is linked to the oientation of the path on L by the ule of the oksew Φ B ds is alled the b-flux thouh S b S 4 In a matte fee point of a avitational field, the spatial vaiation of B and the ate at whih is hanin ae onneted by the elation: otb This statement is the expession of the fat that any hane of the podut N s in a point of a avitational field is elated to a vaiation of the podut n s in the viinity of that point We onside aain and B, the ontibutions to the -field and to the -indution in a point, of the infomatons that - at the moment t - pass nea with veloity (fi 8) And we note fist that w s N s N s e x and B n s n n s sin( ) ez s s ( ) w s ex and that s e y We investiate the elationship between otb { ad ( n) s } + n ot( s ) and N s s + N Fist we alulate otb : otb { ad ( n) s } + n ot( s ) The tem { ad ( n) s } desibes the omponent of otb aused by the spatial vaiation of n in the viinity of when θ emains onstant n has the same value in all points of the infinitesimal sufae that, in, is pependiula to the flow of infomatons So ad(n) is paallel to and its manitude is the inease of the manitude of n pe unit lenth With n n, n Q n + dn and Q dt, ad(n) is detemined by: nq n dn ad ( n) Q dt Gavitation explained by the theoy of infomatons Antoine Ake

30 The veto { ad ( n) s } is pependiula to het plane detemined by and s So, it lies in the XY-plane and is thee pependiula to Takin into aount the definition of vetoial podut, we obtain (fi 8): dn dn ad ( n) s s e ssin( ) e dt dt Fom the fat that the density of the loud of infomatons in Q at the moment t is equal to the density of that loud in at the moment (t - dt), it follows: If n (t) n, then n (t - dt) n Q (t) n + dn The ate at whih n hanes at the moment t is: And, takin into aount that n N n t n t dt n t n t ( ) ( ) ( ) Q ( dn ) dt dt dt N n, we obtain (I) n N ad ( n) s ssin( ) e s sin( ) e The tem { n ot ( s )} is the omponent of otb aused by the spatial vaiation of s in the viinity of when n emains onstant The fat that s Q s at the moment t, follows fom the fat that, at that moment, ( θ ) - the haateisti anle of the infomatons that pass in - diffes fom ( θ ) Q - the haateisti anle of the infomatons that pass in Q If ( θ ) θ, than ( θ ) Q θ + d( θ ) Fom the definition of otf (4), it follows (fi 9): ot( s ) s dl s sin( ) p s sin{( ) + d( )} qq d sin( ) e e s e ds dt p dt Y e s Q Q s θ q X s p Z Fi 9 Gavitation explained by the theoy of infomatons Antoine Ake

31 Fom the fat that the haateisti anle of the infomatons in Q at the moment t is equal to the haateisti anle of the infomatons in at the moment (t- dt), it follows that if ( θ ) (t) θ, then ( θ ) (t- dt) ( θ ) Q (t) θ + d( θ ) The ate at whih sin( θ ) in hanes at the moment t, is: {sin( )} sin( ) sin{ + d ( )} d{sin( )} dt dt Futhe : ( {sin( )} os( ) ) and n N Finally, we obtain (II): n ot( s ( ) v ) N sos( ) e Combinin the esults (I) and (II), we obtain: otb N { s sin( ) + N ( ) s os( ) } e Next we alulate : t N s s + N N s ex ( ) + N s ey Takin into aount: e x os( ) e sin( ) e and e y sin( ) e + os( ) e we obtain: N ( ) N ( ) s os( ) + N s sin( ) e + s N s e t t sin( ) + os( ) Fom the fist law of the avitational field, it follows that the omponent in the dietion of e of is zeo Indeed N We know (4): ad ( N ), so: N ad ( N ) s s os( ) (III) Gavitation explained by the theoy of infomatons Antoine Ake

32 We detemine div( s ) s ds (IV) Fo that pupose, we alulate the double dv inteal ove the losed sufae S fomed by the infinitesimal sufaes ds whih ae in and Q pependiula to the flow of infomatons (pependiula to ) and by the tube whih onnets the edes of these sufaes (and that is paallel to ) dv dtds is the infinitesimal volume enlosed by S: s ds s dsos( ) s dsos{ + d ( )} d{os( )} ( ) s s sin( ) θ dv ds dt dt So (IV): ( ) N div( s ) N s sin( ) Aodin to the fist law of the avitational field (V): div Substitution of (III) and (IV) in (V): div( N s ) ad ( N) s N div( s ) div N s os( ) + N s ( ) sin( ) So, the omponent of In the dietion of e is zeo, and: N ( ) { s sin( ) + N s os( )} e Conlusion: Fom en follows: otb This elation implies (theoem of Stokes): In a avitational field, the ate at whih the sufae inteal of ove a sufae S hanes is popotional to the line inteal of B ove its boundey L: Φ B dl ds ds S S e Gavitation explained by the theoy of infomatons Antoine Ake

33 The oientation of the sufae veto ds is linked to the oientation of the path on L by the ule of the oksew Φ ds is alled the e-flux thouh S e S 4 Relations between and B in a point of a avitational field The volume-element in a point inside a mass ontinuum is in any ase an emitte of - infomation and, if the mass is in motion, also a soue of -infomation Aodin to, the instantenuous value of ρ G - the mass density in - ontibutes to the instantaneous value of div ρ in that point with an amount G ; and aodin to 7 the instantaneous η value of J G - the mass flow density - ontibutes to the instantaneous value of otb in with an amount ν J (7) G Geneally, in a point of a avitational field - linked to an inetial efeene fame O - one must take into aount the ontibutions of the loal values of ρ G ( x, y, z; t) and of ( x, y, z; t) This esults in the enealization and expansion of the laws in a mass fee J G point By supeposition we obtain: In a point of a avitational field, the spatial vaiation of obeys the law: ρ div G η In inteal fom: Φ ds ρg dv η S G In a point of a avitational field, the spatial vaiation of B obeys the law: divb In inteal fom: Φ b B ds S In a point of a avitational field, the spatial vaiation of and the ate at whih B is hanin ae onneted by the elation: ot In inteal fom: B B Φ dl ds B ds S S b Gavitation explained by the theoy of infomatons Antoine Ake

34 4 4 In a point of a avitational field, the spatial vaiation of B and the ate at whih is hanin ae onneted by the elation: otb In inteal fom: B dl ν ds ν J ds ds ν J G ds ν t t S S S S J G Φ i These ae the laws of Heaviside-Maxwell o the laws of avitoeletomanetism V The inteation between masses 5 The inteation between masses at est We onside a set of point masses anhoed in an inetial efeene fame O They eate and maintain a avitational field that is ompletely detemined by the veto in eah point of the spae linked to O ah mass is immesed in a loud of -infomation In evey point, exept its own anhoae, eah mass ontibutes to the onstution of that loud Let us onside the mass m anhoed in If the othe masses wee not thee, then m would be at the ente of a pefetly spheial loud of -infomation In eality this is not the ase: the emission of -infomation by the othe masses is esponsible fo the distubane of that haateisti symmety Beause in epesents the intensity of the flow of - infomation send to by the othe masses, the extent of distubane of that haateisti symmety in the diet viinity of m is detemined by in If it was fee to move, the point mass m ould estoe the haateisti symmety of the - infomation loud in his diet viinity: it would suffie to aeleate with an amount a Aeleatin in this way has the effet that the exten field disappeas in the oiin of the efeene fame anhoed to m If it aeleates that way, the mass beomes blind fo the -infomation send to by the othe masses, it sees only its own spheial - infomation loud These insihts ae expessed in the followin postulate 5 The postulate of the avitational ation A fee point mass m in a point of a avitational field aquies an aeleation a so that the haateisti symmety of the -infomation loud in its diet viinity is onseved A Gavitation explained by the theoy of infomatons Antoine Ake

35 5 point mass who is anhoed in a avitational field annot aeleate In that ase it tends to move We an onlude that: A point mass anhoed in a point of a avitational field is subjeted to a tendeny to move in the dietion defined by, the -field in that point One the anhoae is boken, the mass aquies a vetoial aeleation a that equals 5 The onept foe - the avitational foe Any distubane of the haateisti symmety of the loud of -infomation aound a point mass ives ise to an ation aimed at the destution of that distubane A point mass m, anhoed in a point of a avitational field, expeienes an ation beause of that field, an ation that is ompensated by the anhoae - That ation is popotional to the extent to whih the haateisti symmety of the own avitational field of m in the viinity of is distubed by the exten -field, thus to the value of in - It depends also on the manitude of m Indeed, the -infomation loud eated and maintained by m is moe ompat if m is eate That implies that the distubin effet on the spheial symmety aound m by the exten -field is smalle when m is eate Thus, to impose the aeleation a, the ation of the avitational field on m must be eate when m is eate We onlude: The ation that tend to aeleate a point mass m in a avitational field must be popotional to, the -field to whih the mass is exposed; and to m, the manitude of the mass We epesent that ation by F and we all this vetoial quantity the foe developed by the -field on the mass o the avitational foe on m We define it by the elation: F m A mass anhoed in a point annot aeleate, what implies that the effet of the anhoae must ompensate the avitational foe This means that the distubane of the haateisti symmety aound by must be anelled by the -infomation flow eated and maintained by the anhoae The density of that flow in must be equal and opposite to It annot be othewise than that the anhoae exets an ation on m that is exatly equal and opposite to the avitational foe That ation is alled a eation foe This disussion leads to the followin insiht: ah phenomenon that distubs the haateisti symmety of the loud of -infomation aound a point mass, exets a foe on that mass Gavitation explained by the theoy of infomatons Antoine Ake

36 6 Between the avitational foe on a mass m and the loal field stenth exists the followin elationship: F m F So, the aeleation imposed to the mass by the avitational foe is: a m Considein that the effet of the avitational foe is atually the same as that of eah othe foe we an onlude that the elation between a foe F and the aeleation a that it imposes to a fee mass m is: F m a 5 Newtons univesal law of avitation In fi we onside two point masses m and m anhoed in the points and of an inetial fame m eates and maintains a avitational field that in is defined by the -field: This field exets a avitational foe on m : m 4 π η e F m m e m 4 π η R e F m F m e Fi In a simila manne we find F : F m m e 4 π η F This is the mathematial fomulation of Newtons univesal law of avitation Gavitation explained by the theoy of infomatons Antoine Ake