Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year.
RESEARCH INVENTY: International Journal of Engineering and ScienceISBN: 2319-6483, ISSN: 2278-4721, Vol. 1, Issue 8 (November 2012), PP 24-28www.researchinventy.com Four Force in MOND 1 Manisha R Pund, 2Maya Nahatkar 2 Shri Rajendra Junior college, Nagpur, India 1 Priyadarshini College of Engineering, Nagpur, IndiaAbstract: The transformations of a four force in MOND theory are deduced. They differ to their relativi sticcounter parts. I. Introduction The modified Newtonian dynamics (MOND) is a burgeoning theory that proposes a modification ofNewton’s second law of dynamics F m a to exp lain the galaxy rotation problem.When the uniform velocityof rotation of cluster of galaxies was first observed (Oort 1932 and Zwicky 1933), it was not in tune withNewtonian theory of gravity- the galaxies or stars sufficiently away fro m the centre of the cluster move withconstant velocities mo re than predicted by the theory. The dark matter was, then postulated to account for thisconstant large velocity. However, even after seven decades, there is not a convincing evidence of the darkmatter. In an attempt to explain the observed uniform velocities of galaxies, Professor Milgro m in 1983propounded an equation of motion that resulted into a theory which is known as MOND (mod ified Newtoniandynamics). The Newton’s second law of motion now is generalized to a F m a a, (1.1) 0 which can be considered as modification of Newtonian dynamics, a a / a0 a a0where . a0 1 a a0For a a0 above equation reduces to F ma which is Newtonian dynamics. The quantity a0 is constantand has the dimension of acceleration and is evaluated as a0 2 10-8 cm / s 2 (Milgro m 1983a, Bekensteinand Milgrom 1984).Basically the early MOND is non-relativistic and hence will be interesting to probe into theconsequences of MOND coupled with Lorentz symmetry. As a first step towards the goal, in this paper, wededuce the transformat ion of four fo rce in the MOND region. It is observed that the transformations to thateffect are d ifferent fro m their relativistic counter parts, as is expected. II. The Relativistic Analogue Of MOND In Regard To VelocityThe Newton’s second law of motion in the usual notation can be expressed as dp d F (mu) , dt dtIn relativity when u is the velocity of a particle with mass m m(u ) , above gives F m du u dm (2.1) dt dtCo mparing this equation with its analogue in MOND (1.1), we deduce a dm m mu a0 duConsidering a to be a constant k , above linear differential equation reduces to a variable separable form dm du k a0 m u a0 On integration, we obtain 24
Four Force In MOND A m u k a0 , A m0 u 0 a 0 m0Noting m , above is rewritten as u2 1 2 c A 1 m0 u u2 2A u m0 u or 1 2 c u2 0 u0 1 c2 u2Neglecting higher order terms of , above yields c2 2 Ac 2 u2 u 2 Ac 2 0 u0For real values of u , we get 2 2u 0 k a0 u2 A i.e. 2 0 c2 a0 c2 2u 0 2 1 2 , k a0 c or (2.2) where we assume that m(u0 ) m0 , u0 0 .This equation seems to be interesting and deserves physical interpretation. At present we conclude that theconservation of linear mo mentu m is acclaimed for k 0 which corresponds to a rectilinear mot ion of aparticle. Also m m0 corresponds to k a0 . III. The relativistic analogue of MOND in regard to accelerationFro m (1.1) and (2.1), m u(a u) ma ma, (2.3) 2c 2 2 m0 uwhere m and 1 2 . cNow we consider fo llo wing two cases,Case I: Let a be perpendicular to u. Then a u 0 and a a0 , follo ws fro m (2.3). A lso (1.1) goes over toNewtonian dynamics.Case II: Let a be parallel to u . We consider a B u , where B is a constant. We investigate the relativisticanalogue of MOND in regard to velocity and acceleration as follo ws.Equation (2.3) imp lies 25
Four Force In MOND m0 (m / c 2 )( Bu u) m Bu 0 u 0 B u. 3 This simp lifies to a 1 2. a (2.4) 04. Four force in MONDConsidering four mo mentu m p i (p, p 4 ) , we write d p m ( a / a0 ) a dp mμa or . ds c ds c dp 4 d E dt dE 1and (2.5) ds dt c ds dt c 2 where E represents energy.The work done dw in mov ing a system through the displacement dr is dw dw F dr or (ma) u dtIf the wo rk done is used in increasing the kinetic energy T , above gives, dw dT Fu dt dt dT dE Fu dt dtUsing above equation, (2.5) becomes dp4 ma u . ds c 2 dp i Thus four force in M OND can be exp ressed by Fi as a four force, ds ma ma u Fi c , c 2 (2.6) IV. Transformations of the components of four force F i in MONDConsidering its components F 1 , F 2 , F 3 , F 4 must transform like the components of a four radius vector x iunder LTS i.e. 1 v F F 1 F 4 , F2 F 2 , F3 F 3 c 1 / 2 (2.7) 4 v v2 F F 4 F 1 , 1 - 2 c c In view of (2.6) in S frame, we write 26
Four Force In MOND m a m a u 2 i u F c , , 1 2 c 2 c 1 2 3 4Substituting the values of F , F , F , F fro m (2.6) in (2.7) and by using the result, 2 u 1 v2 / c2 1 u 2 / c2 1 , c2 1 2 uxv cwe get vm m ax max (a y u y az u z ) (c u x v ) 2 v2 v2 c2 1 c2 1 2 m a y 2 c 2 m a , m a c m a (c u x v ) (c u x v ) y z 2 z c2m m au [ax (u x v) (a y u y az u z )] (c 2 u x v ) GMa0Using a | a | , (see Milgro m (1983), Bekenstein (2004)), we get transformat ions of acceleration rcomponents max max mv (a y u y a z u z ) , r r r (c u x v ) 2 v2 v2 c 1 2 2 c 1 2 2 ma y c ma y ma z c ma z 2 , 2 , r (c u x v ) r r (c u x v ) r (au ) mc2 m [ax (u x v) (a y u y az u z ) r r (c 2 u x v )Defining F (N) ma analogous to Newtonian force mass acceleration relationship, the above transformationcan be rewritten as (N) Fx( N ) Fx v (N) (N) ( Fy u y Fz u z ) (2.8) r r r (c u x v ) 2 v2 c2 1 Fy( N ) (N ) c 2 Fy 2 (2.9) r c uxv r v2 c2 1 Fz( N ) c 2 Fz (N ) 2 (2.10) r c uxv r 27
Four Force In MOND Fx( N ) u x Fy( N ) u y Fz( N ) u zand r c2 [ Fx( N ) (u x v) Fy( N )u y Fz( N )u z ] . (2.11) r (c 2 u x v ) ConclusionWhen a is perpendicular to u , MOND boils down to Newtonian mechanics and for a parallel to u yields to(2.4), the relativistic analogue of MOND in regard to velocity and acceleration. The four fo rce transformat ionsin MOND are g iven by the equations (2.6), (2.8), (2.9), (2.10) and (2.11). AcknowledgementThe authors are thankful to Professor T M Karade fo r suggesting the problem. References[1.] Oort J, Bull. Astron. Inst. Neth. 6, 249 1932[2.] Zwicky F, Helv. Phys. Acta 6, 110 1933[3.] Milgro m M, 1983a, Ap J, 270, 365[4.] Milgro m M, 1983b, Ap J, 270, 371.[5.] Milgro m M, 1983c, Ap J, 270, 384.[6.] Bekenstein J, 2004, Phys Rev D, 70, 083509. 28