On the Relativistic Forms of Newton's Second Law and Gravitation


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1 On the Relativistic Foms of Newton's Second Law and avitation Mohammad Bahami,*, Mehdi Zaeie 3 and Davood Hashemian Depatment of physics, College of Science, Univesity of Tehan,Tehan, Islamic Republic of Ian, PO Box Faculty of Science, Isfahan Univesity, Isfahan, Islamic Republic of Ian, PO Box Faculty of Science, Univesity of Sistan and Baluchestan, Zahedan, Islamic Republic of Ian, PO Box Coesponding autho Tel: Fax: Abstact: We found out that foce in natue cannot be infinite and by deiving Loentz facto of foce, we wee able to obtain the elativistic fom of the gavitation law and the second law of Newton. These elativistic equations ae compatible with special elativity. We also popose a pocedue fo solving the dak matte poblem in which the validity of these equations can be tested. Keywods: avitation law, Second law of Newton, Relativistic mass. PACS: Kd; Sf.
2 Intoduction Fundamental constants have an impotant ole in the physics. Thei simplest and most impotant ole is tansfoming popotional elationships into equations. The significance of fundamental constants is not limited to this, but also these constants have physical concepts, and by combining them, significant physical quantities could be obtained. Fo example, the space (length)time elation is as l t = ε 0 μ 0 o l t = c The special theoy of elativity says that thee is in natue an ultimate speed c, and no entity that caies enegy o infomation can exceed this limit. Within this theoy, the constant c is not exclusively about light; instead it is the highest possible speed fo any physical inteaction in natue. Fomally, c is a convesion facto fo changing the unit of time to the unit of space []. In fact, above equation epesents a univesal and fundamental elation between space and time in which popotionality constant of this elation is a combination of fundamental constants of physics. We can extend the postulate pointed out by above equation to othe base quantities and povide simila elations among the thee base quantities including time, space (length) and mass. Theefoe, we establish ou theoy on the postulate that states: popotionality constant of the elationships between the base quantities including space, time and mass is a combination of the fundamental constants of physics. The thee fundamental equations, which can be deived fom these thee base quantities, ae l = c (Eq. ) t Whee c = 3 08 m/s m l = c (Eq. ) c Whee =.3 07 kg/m m t = c3 (Eq. 3) c3 Whee = kg/s We call above equations the "elativity equations". Accoding to special elativity, we can claim that Eq. states that the ultimate limit of the space to time atio o velocity is c in natue and that can neve be eached, and also coesponding phenomenon to this equation in natue is light. Thee ae easons that make us intepet Eq. and Eq.3 like Eq.. The fist eason is the symmety, which is consideed as a poweful tool in physics. The second eason is that the values of popotionality constant in these two equations like Eq. ae a vey lage numbe. Theefoe, we can claim that Eq. and Eq.3 state that the ultimate limit of the mass to space and mass to time atios ae c / and c 3 /, espectively, in natue and that can neve be eached. We know that when the velocity (i.e. l/t) of a body appoximates the velocity of light, the time dilation and length contaction take place (Eq.4). Based on symmety and since Eq. and Eq.3 show the spacetime elation with mass, we can wite simila equations as follows: t = γ v t 0 (Eq. 4) ; whee γ v = ( v c ) t = γ m l t 0 (Eq. 5) ; whee γ m l = t = γ m t t 0 (Eq. 6) ; whee γ m t = m l ( c ) m t ( c 3 ) In Eq.4, we inseted that vey familia quantity of the velocity (v) instead of the atio of spacetime (l/t). But, in the Eqs.56, thee is no coesponding quantity in physics fo the atio of masstime (m/t) and atio of massspace (m/l). Fo this eason, we diectly inset them in the equations. Eq.5 and Eq.6 indicate
3 spacetime coodinates of the obseves in pesence of the mass. Fo example, Eq.5 shows that the shote the distance of the obseve o the clock fom the cente of the mass (the lage m/l), the moe time gets slowe. Since the value of the ultimate limit of the mass to space atio is lage, slowing down of the time caused by the pesence of the mass fo massive bodies such as planets and stas is of impotance. Fo eath planet, the neaest distance fom the cente of the mass is its suface, and in this condition space (l) becomes equal with the adius of the eath. If thee is a mass coodinate which applies to Eq., time will stop in the suface of such a mass accoding to Eq.5. On the othe hand, since time stops in the suface of black holes [], Eq. should be its coodinates. As a esult, the coesponding phenomenon to Eq. in natue is black hole. If Eq. is the base of special elativity, then on the basis of symmety, Eq. should also be the base of a new elativity (as you see late). If we also inset Eq. instead of t in Eq.3, Eq. will be obtained. So, t in Eq.3 is the distance fom the cente of the mass (l) which is stated accoding to the time in which light tavels this distance. As we explain late, this can means that the speed of gavity must be exactly equal to that of light [3]. The obvious question now is: Can Eq.3 be the base of a new elativity like Eq. and? The answe is negative. The fact is that Eq.3 can be obtained fom the othe two equations, and theefoe we cannot elate any special phenomenon o a new elativity to this equation (In fact γ m/l and γ m/t ae equivalent). We will elaboate a little moe on the quantity of m/t and its impotance late. Note that the elativistic equations of mass cannot be obtained fom Eq. and Eq.3 because these two equations show the elation between spacetime and mass. In fact, when mass is influenced by the foce, it has meanings and can be measued. So, to obtain the elativistic equation of mass, we should fist obtain the elativistic equation of foce. Relativistic foms of gavitation and newton's second law Speed and foce ae two vey impotant quantities in physics one of which is the basis of the special elativity and the othe is the basis of the geneal elativity. Symmetically, if speed has the ultimate limit of c, foce also should have an ultimate limit. If we wite the foce based on the base quantities, we will get m. l [F] = (Eq. 7) t. t If we inset Eq. and Eq.3 in Eq.7, the ultimate limit of the foce is obtained as F = m t. l t = c4 c3. (c) F = c4 (Whee =. 044 N) In fact, if the extemity of l/t and m/t is placed in Eq.7, the value of the foce obtained will be the extemity of the foce. Foce cannot be infinite, because volume o mass of a body cannot be zeo o infinite, accoding to Eq.. Thee is no expeimental sample in natue that can match an infinite mass. The coesponding phenomenon with ultimate limit of the speed is light. In contast, the coesponding phenomenon with ultimate limit of the foce should be in natue (suface of) black hole because this limiting value is a diect consequence of Eq.. In ode to obtain elativistic equation of foce, we should fist obtain the Loentz facto of the foce (γ F ). The stating point in ou discussion is the geneal fom of foce equation as F = m l t t To find a elativistic expession fo foce, we stat with the new definition F = m l t 0 t 0 3
4 Hee, t 0 is the time measued by an obseve at est with espect to body; thus that measued time is a pope time. Using Eq.4 and Eq.6 we can then wite F = m l = m t l t m l = γ t 0 t 0 t t 0 t t m t γ v 0 t t Howeve, since m l/ t. t is just the foce F acts on the body, we have m l F = γ F (Eq. 8) ; whee γ t t F = γ m t γ v If we multiply the Loentz facto of masstime atio (γ m/t ) and speed (γ v ), we will get this equation γ F = (Eq. 9) ml t + ( ) ( v c ) m t ( c 3 ) Eq.9 shows the geneal fom of Loentz facto of foce fo thee types of foce in Newtonian mechanics i.e. gavitation law, Newton s second law and centipetal foce, and depending on the type of foces acting on a system, it can be simplified. Note that ml/t in Eq.9 shows the expession of foce, such as ma fo Newton s second law. Now, we will see how beautifully Eq.9 shows the elativistic fom of the thee foces and thei deviation fom classical equations..the gavitational foce (F g ): fo a stationay body (v=0) with mass m that is being influenced by the nonvaiable gavitational foce (m/t=0) esulting fom a massive body with mass M at constant distance, γ F will be γ F = + ( mm ) So, the elativistic fom of the Newton s law of gavitation will be accodingly F g = mm + ( mm ) (Eq. 0) Figue shows plots of the gavitational foce (F g ) as calculated with the coect definition (Eq.0) and the classical appoximation, both as functions of mm/. Note that on the left side of the gaph the two plots coincide at lowe foces; but, on the ight side of the gaph, at foces nea c 4 /, the two plots diffe significantly. As Fig. attests, the expession mm/ in Eq.0 can have any quantity (moe o less than c 4 /); but foce (F g ) can neve become equal to c 4 /. This equation shows that only when the value of foce is equal to its own extemity i.e. c 4 / that mm/ is infinite, namely; If mm F g c4 On the othe hand, the value of mm/ can be infinite only if M= o =0. As peviously mentioned, Eq. does not allow the mass o volume of a body is equal to infinite o zeo, espectively; this means that we cannot each the ultimate limit of foce. 4
5 Figue. The elativistic (solid cuve) and classical (dashed line) equations fo the gavitational foce (F g ), plotted as a function of mm/. Note that the two cuves blend togethe at low values of mm/ and divege widely at high mm/. This figue also indicates that foce neve eaches its ultimate limit, i.e. c 4 /, unless mass (M) is infinite (c 4 /= N). When a velocity appoximates the velocity of light, the time and length quantities ae affected (time dilation and length contaction). Now, the question is what quantity is affected when foce appoximates its own ultimate limit? Response: quantity is mass. In classical physics, the gavitational est mass of a body detemined by its esponse to the foce of gavity via F g =m 0 M/. If we solve Eq.0 fo m and inset F g =m 0 M/ in it, the tue elation of elativistic gavitational mass will be obtained as m = m 0 m0m (Eq. ) (Whee m m 0 ) The above equation states that if the gavitational foce acting on a body be equal to c 4 /, the mass of the body becomes infinite. A seemingly contadictoy point that can esult fom the compaison of Eq.0 and Eq. is how mm/ can assume any value in Eq.0, but m 0 M/ in Eq. should be less than c 4 /. The answe to this question is hidden in this point that m is the elativistic mass and m 0 is the est mass. When the value of m 0 M/ appoximates the ultimate limit of foce, m and as a esult mm/ can assume any value accoding to Eq.; yet, the elativistic fom of the gavitational law (Eq.0) assues us that at any state, the value of foce does not exceed c 4 /. Relativistic fom of gavitation law is only deviated fom the classical fom when a body is exposed to intense gavity.. The centipetal foce (F ): fo a body with mass m which moves in a cicle (o a cicula ac) at constant speed v (m/t=0), γ F will be γ F = + ( mv ) ( v c ) In this situation the elativistic fom of centipetal foce is Whee F = mv + ( mv ) ( v c ) (Eq. ) c 4 If F = c4 v = c 5
6 The pevailing view in physics is that to incease an object s speed to c would equie an infinite amount of enegy; thus, doing so is impossible. But, accoding to the above equation, the velocity of bodies cannot each that of light because we cannot have access to the ultimate value of foce. Conside a body with mass m otating aound the moe massive cental body with mass M in a cicula obit with constant velocity. In this situation, the magnitude of the centipetal foce (F ) is equal to the magnitude of the gavitational foce (F g ). As above equations tells us, the geate the gavitational foce between two bodies, the moe apidly the body otates aound its obit; and when the magnitude of this foce eaches its ultimate value, i.e. c 4 / the velocity of the otating body equals that of light. On the othe hand, we showed that if the value of the mass of the cental body (M) is equal to infinity, the magnitude of the gavitational foce eaches its ultimate point. Since such a thing is impossible, i.e. the value of foce can neve each its ultimate point; Eq. expesses that the velocity of a otating body neve eaches that of light. The elativistic fom of Newton s second law shows a simila situation fo motion with speed c. Also, it is clea that the equation indicating the elativistic mass fo the otating body is the vey Eq.. 3. Newton s second law (F N ): If we define that the acceleation of a paticle at any instant is the ate at which its velocity is changing at that instant, i.e. a=v/t, the second law of Newton (F N =ma) will be F N =mv/t. As the appeaance of this equation shows, the value of the two quantities of v and m/t is not equal to zeo fo acceleated motion, and as a esult, γ F will be γ F = mv t + ( c 4 ) ( v c ) ( c 3 ) v=at γ F = + ( ma c 4 ) ( at c ) ( c 3 ) So, the elativistic fom of the Newton s second law will be accodingly o F N = mv t + ( c 4 mv t ) ( v c ) ( ma F N = + ( ma c 4 ) ( at c ) ( c 3 ) (Eq. 3) ) (Eq. 4) c 3 As a test, if the magnitude of the foce is equal to c 4 / in Eq.3 and Eq.4, the velocity and acceleation of the body at the infinite time (whee the value of m/t appoaches zeo) will be equal to c and zeo, espectively.; namely, Eq. 3 Eq. 4 F= c4 v c F= c4 at c + + m t c 3 m t c 3 = t v = c = t a = 0 In ode fo a stationay body to each the speed of light, it should be acceleated using ultimate foce. Based on the elativistic equations of foce (Eq.0), we can neve each the ultimate foce, meaning that nobody can each the speed of light. Now, the question is: what type(s) of deviation fom the second law of Newton do Eq.3 and Eq.4 show? To answe this question, let us solve these two equations fo v and a sequentially 6
7 m t c v = 3 m Ft + c (Eq. 5) m t c a = 3 m F + t c m v (Eq. 6) (whee a = t ) and compae them with thei two coesponding classical equations in Newtonian mechanics. v = Ft m (Eq. 7) and a = F m (Eq. 8) (whee a = v t ) The velocity and acceleation cuve pedicted fom the elativistic and classical equations of Newton s second law, as a function of time fo a body of mass kg and with a foce of N on it, is plotted in Figue. Accoding to this figue, the elativistic equations show two geneal deviations fom classical equations: one elates to the vey shot time in which the value of m/t appoximates the ultimate value of c 3 /, and the othe elates to long time in which the value of velocity appoximates the ultimate value of c. The fact is that Eq.5 and Eq.6 eject some of the hypotheses on which Newton s mechanics is based. Newtonian mechanics states that foce acts on a body instantaneously and the velocity of the acceleating body inceases linealy with time so that its value can be infinite in infinite time (Eq.7). Also, accoding to Newtonian mechanics, due to the constant foce F, the body's acceleation is constant and does not change with time (Eq.8). Fo example, if a foce of N acts on a kg body, accoding to Eq.7 and Eq.8, the body's velocity at 036 s, s, and 7y will be equal to 036, and m/s (which is moe than speed of light), espectively; and duing all these time peiods, the acceleation of the body is constant and equal to m/s. In contast, elativistic Eq.5 and Eq.6 pedict that duing these times, the body's velocity should be equal to 0,, and m/s (which is less than speed of light) and the value of acceleation should be equal to 0,, and 0.49m/s, espectively. In ode to undestand why the elativistic equations of Newton s second law show such a deviation fom classical equations at small time intevals, we assume that we want to act on a black hole of mass m=kg with a foce of N. Eq. shows that the adius of this black hole should be equal to m. On the othe hand, we know that foce is tansfeed at the speed of light. Theefoe, fo the foce to be tansfeed fom the suface of this black hole to its cente (i.e. the cente of its mass), a time equal to s is equied. As a esult, fo the time intevals less than this value, the black hole has not felt any foce (no foce has acted on it) and its velocity and acceleation is zeo. Since nobody can be changed into a black hole, this time inteval cannot be less than o equal to s fo a body of mass kg. As a geneal conclusion, the value of m/t cannot each the ultimate value of c 3 / fo anybody; and theefoe, fo a body of mass m, the minimum time inteval equied fo the foce to act on it, should always be lage than m/c 3 (t>m/c 3 ) (this means that the speed of gavity is exactly equal to that of light [3]). As the solid cuve in Fig.a and c attests, fo times less than s, velocity and acceleation of the body with a mass of kg must equal zeo. Note that we did not use the quantity of m/t in the elativistic equations of gavitation law and centipetal foce because these two foces ae employed fo situations when foce is fully tansfeed to bodies. 7
8 Figue. (a,b) The v(t) cuves and (c,d) the a(t) cuves foa body of mass kg and with a foce of N on it. The cuves in Fig.a and c disappea in the cuves of Fig.b and d because the scale is lage. Note that the solid and the dashed lines show the cuves pedicted by the elativistic (Eqs.56) and classic equations (Eqs.78) of the second law of Newton, espectively. The second deviation is impotant when we want to study the velocity and acceleation of an acceleating body ove long time. When we say that a body is moving at an acceleation of m/s, it means that a value equal to m/s is added to the body's velocity in each second; and accoding to the Newtonian mechanics, the velocity of a body must be moe than that of light ove long time (the dashed line in Fig.b). In contast, the elativistic equations of foce do not give such pemission to the body (the solid cuve in Fig.b). When the velocity of the acceleating body appoximates that of light, the inceasing ate of velocity is educed instant by instant in ode to pevent the velocity fom eaching the speed of light. Theefoe, by the passage of time, the acceleation of the body stats to educe (indicated by the solid cuve in Fig.d) and in this state, its value deviates fom that pedicted by Eq.8 (the dashed line in Fig.d). The question in Newtonian mechanics is: what will a body's acceleation be when a foce F acts on it? But, accoding to elativistic mechanics, the coect question is: how much will the velocity of a body with mass m be at the time t when foce F acts on it, because acceleation changes with time? It can be shown at what values of t; the classical equation of the second law of Newton is tue. If m t c3, m Ft c In this case, the elativistic Eq.5 and Eq.6 ae changed to classical Eq.7 and Eq.8. If the above unequals ae combined, we attain the following unequal showing at what time inteval the classical fom of the second law of Newton is tue: m mc c 3 t (If F c4 t 0) F The moe the value of foce F, the geate is the deviation fom classical equation; and we will witness full deviation fom the second law of Newton when foce F=c 4 / act on a body. This misconception may 8
9 aise hee that if Eq.5 and Eq.6 ae solved fo m, and equation F=m 0 a is inseted in it, the equation showing the inetia elativistic mass (like the condition in which we obtained the gavitational elativistic mass) can be obtained as follows: v c m = m 0 + m 0 t m o m = m 0 0v t c 3 c 4 + m 0 t at c m 0a c 3 Whee a=v/t. The above equation shows that when the velocity of the acceleating body equals that of light, the body's mass becomes zeo. The fact is that this eduction in mass is not eal due to the change of acceleation with time, and esults fom measuement. Conside a peson, holding a balance with a ball on it, is locked up in a small box. Also, assume that this box moves at an acceleation ate of 9.8m/s, and the value of the mass is shown as kg by the balance. Ove long time, when the velocity of the box appoximates that of light, the acceleation of the box is educed ove the passage of time, and as a esult, the balance shows a smalle numbe fo the peson. While, we know that the eal mass of the body is not changed (as the peson in the box sees it) because the numbe shown as the mass of the body on the balance depends on the acceleation of the box. The above equation points to this fact, too. In fact, velocity (special elativity) only affects space and the time of the obseve, while it is only the gavity that affects the mass. As a esult, only if the gavitational foce affects the acceleating body, does its mass eally incease accoding to Eq. (it becomes elativistic). 4. avitational acceleation (a g ) and otational speed (v ): In classical mechanics, the equations of gavitational acceleation and otational speed is given by Mm Mm = ma g solving fo a g a g = M = mv solving fo v v = M In contast, in ou theoy the elativistic foms of two above classical equations is obtained fom equating Eqs.0, and 4 as mm + mm ma g = + ma g a gt c c 3 solving fo a g m t c a g = 3 (Eq. 9) M + tc mm + mm = mv + mv v c solving fo v v = M γ v (Eq. 0) As it was aleady explained, Eqs.9 and 0 show two geneal deviations fom thei classical foms; one in a vey shot times, whee the speed of gavity is impotant, and the othe in a vey long times, whee the instantaneous speed of body is compaable with light speed (c). The afoesaid equations do not show fesh and new deviations fom themselves, because they can be easily foeseen fom the special elativity. Howeve ou theoy foecasting anothe deviation as well, whee not only can popose a new explanation fo the dak matte, but also it can be taken into account as a test to study ou theoy coectness and validity as well. This deviation whose esouce is the elativistic mass, is explained with 9
10 an example. Take a alaxy into consideation whee a supemassive black hole (with mass M bh ) in its cente and otating n stas aound the alaxy cente (Fig.3). In Newtonian mechanics, the obital speed of the stas is given by Fo m 0, : v, = M bh Fo m 0, : v, = M ; M = M bh + m 0, Fo m 0,3 : v,3 = M 3 ; M = M bh + m 0, + m 0, And so on up to the nsta Fo m 0,n : v,n = M n n ; M n = M bh + m 0, + m 0, + + m 0,n Note that since the value of v is much less than c fo the stas otating aound the cente of galaxies, theefoe, elativistic effects esulting fom special elativity, ae ignoed. Eq. showing the fact that a sta s mass is incease in the gavitational field, and this incease should be taken into account in calculations. In fact, Eq. causes the eal mass of stas to be moe than ou estimations. In this situation, ou theoy pedicts the following equations fo the obital speed v of the stas: Fo m 0, : v, = M bh Fo m 0, : v, = M ; M = M bh + m = M bh + m 0, m0,m bh Fo m 0,3 : v,3 = M 3 ; M = M bh + m + m = M bh + And so on up to the nsta m 0, m0,m bh + m 0, m0,m Fo m 0,n : v,n = M n n ; M n = M bh + m + + m n = M bh + m 0, m0,m bh + + m 0,n m0,n Mn n Whee m 0 and m efe to the inetia and gavitational masses, espectively. Inceasing the mass of a sta aising fom Eq. otating nea the alaxy cente is not so consideable. Fo example, the value of m 0,S M bh / fo the sta S with mass m 0,S =3 0 3 kg otating in m distance in the cente of milky way galaxy aound the supe massive black hole,i.e. Sagittaius A * with mass M bh = kg, is equal to N [5], which is much smalle than c 4 / (= ). If we move out fom the cente of galaxy, both the numbe of stas and the value of M x ae inceased, and when the above equations fo a lage numbe of stas in a typical galaxy is implemented (fo example, in the milky way galaxy n>0 ), 0
11 then the elativistic mass eveals its effect on the incease in M total value, and subsequently, an incease in the obital speed (v ) of the stas. In othe wods, applying Eq. to all stas in a galaxy causes the mass of the whole galaxy to be geate than that pedicted by the classical fom of gavitation law. Figue 3. A typical galaxy with supemassive black hole (M bh ) at its coe and a lage numbe of otating stas. Obital motion of the planets of the sola system obeys the Newton s laws so that with an incease in the distance fom the cente of it, otational speed of the planets aound the sun deceases. But in galaxies, fo clustes of stas which ae scatteed in the fa distances fom the galaxy cente, the obseved otational speed emains unchanged with an incease in the fa spaces [4]. The only explanation of this issue is that the galaxy should have much moe mass than what we see. Since this matte cannot be seen, it is called dak matte. It has been yeas that physicists have been uselessly seaching fo explanation of dak matte, and no satisfying esults have been obtained so fa. The obvious question now is  is it possible that the elativistic gavitational mass is dak matte? Thee ae some confiming and pomising evidences in this egad:  No expeimental evidence has been indicated so fa that the "missing mass" is fomed fom an odinay matte (including the fundamental paticles as well as the lage objects). We only know that the dak matte inteacts only gavitationally with the common paticles, namely, it should only be mass. These findings compaing with ou theoy stating that A dak matte is not anything except the mass incease of a sta when setting in the gavitational field is quite compomising.  eneally, empiical evidences suggest that the existence of dak matte is impotant in galaxies and thei clustes [4]; exactly whee the value of m 0 M/ to c 4 / atio becomes significant, because of the existence of a lage quantity of mass including supe massive black holes and a lage numbe of massive stas. In the sola system, the value of gavitational elativistic mass and theefoe the dak matte is not consideable. 3 The galaxy otation cuves indicating that: only the stas obital speed otating in the most exteio obits out of the galaxy cente ae incompatible with Newtonian mechanics [4]. As it was shown by stating an example, the mass inceasing amount aising fom Eq. is not consideable fo the stas otating nea the galaxy cente, consequently foecasting an accuate Newtonian mechanics obital speed fo these stas. But, when moving towads the galaxy exteio pat, the impotance of elativistic mass is
12 inceased, causing the exteio stas feel a lage mass out of the galaxy inteio pat. This means that the stas obiting the galaxy cente in thei outemost obits ae held with a geate gavitational foce. Consequently, these stas can otate at a highe speed compaed to ou pedictions based on Newtonian mechanics (In othe wods, the stas should move at a geate speed to emain in its own obit). Wheeas the numbe of stas ae so numeous in the galaxies, theefoe applying the afoementioned equations, equiing a lage volume of datum about the stas mass and thei distance up to the galaxy mass cente and doing so numeous numbe of calculation by supe compute. Although, ou theoy has some similaities with geneal elativity, such as pediction of gavitational time dilation and making distinction between the gavitationl active mass (M), gavitationl passive mass (m), and inetia mass (m 0 ), but thee ae two impotant diffeences between them, the fist one is singulaity which in contay to the geneal elativity, thee is not in ou theoy because mass to adius atio of a body, and thus its density accoding to Eq. can neve be infinite. Second, ou theoy does not conside the cuvatue of spacetime caused by the gavitational field, unlike geneal elativity. In contast, geneal elativity does not take into account the incease of the mass of an object aising fom Eq.. If it is poved that Eq. is coect, in this case, geneal elativity must incopoate the spacetime cuvatue caused by the elativistic mass into thei equations. Howeve, it cannot be moe discussed with cetainty, until ou theoy test is passed. Conclusions Based on ou postulate, we found out that the massspace (m/l) and masstime (m/t) atios cannot be infinite in natue; and by deiving Loentz facto of foce, we wee also able to obtain the elativistic fom of the gavitation law, the second law of Newton and centipetal foce (table ). We also poposed that these equations may be able to explain dak matte poblem. In fact, the dak matte is a eal test of the validity of ou theoy. Table shows elativistic foms of thee type of foce and its elativistic mass equations Equation Classical fom Relativistic fom avitation law F = m M F = mm + ( mm ) Centipetal foce Newton s Second law F = m v F = ma F = mv + ( mv ) ( v c ) ma F = + ( ma ) ( at c ) ( c 3 ) Acknowledgments We thank to D. M. Hajian, J. Khanjani,. Hosseinzadeh and E. Faamazi fo thei help and thei useful guidance. Refeences [] Wheele, J. A. & Taylo, E. F. Spacetime physics. (WH Feeman, 966).
13 [] Rindle, W. Relativity: special, geneal, and cosmological. Vol. (00). [3] Tang, K. et al. Obsevational evidences fo the speed of the gavity based on the Eath tide. Chinese Science Bulletin 58, (03). [4] Sofue, Y. & Rubin, V. Rotation cuves of spial galaxies. axiv pepint astoph/ (000). [5] Matins, F. et al. On the natue of the fastmoving sta S in the galactic cente. The Astophysical Jounal Lettes 67, L9 (008). 3
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