E018024028

RESEARCH INVENTY: International Journal of Engineering and Science ISBN: 2319-6483, ISSN: 2278-4721, Vol. 1, Issue 8 (November 2012), PP 24-28 www.researchinventy.com

Four Force in MOND 1

Manisha R Pund, 2Maya Nahatkar 2

1

Shri Rajendra Junior college, Nagpur, India Priyadarshini College of Engineering, Nagpur, India

Abstract: The transformations of a four force in MOND theory are deduced. They differ to their relativi stic counter parts.

I.

Introduction

The modified Newtonian dynamics (MOND) is a burgeoning theory that proposes a modification of Newton’s second law of dynamics F  m a to exp lain the galaxy rotation problem.When the uniform velocity of rotation of cluster of galaxies was first observed (Oort 1932 and Zwicky 1933), it was not in tune with Newtonian theory of gravity- the galaxies or stars sufficiently away fro m the centre of the cluster move with constant velocities mo re than predicted by the theory. The dark matter was, then postulated to account for this constant large velocity. However, even after seven decades, there is not a convincing evidence of the dark matter. In an attempt to explain the observed uniform velocities of galaxies, Professor Milgro m in 1983 propounded an equation of motion that resulted into a theory which is known as MOND (mod ified Newtonian dynamics). The Newton’s second law of motion now is generalized to

 a F  m    a0

  a , 

(1.1)

which can be considered as modification of Newtonian dynamics, where For

 a  a0

 

 a / a0      1

a  a0 a  a0

.

a  a0 above equation reduces to F  ma which is Newtonian dynamics. The quantity a0 is constant

and has the dimension of acceleration and is evaluated as a0  2  10-8 cm / s 2 (Milgro m 1983a, Bekenstein and Milgrom 1984).Basically the early MOND is non-relativistic and hence will be interesting to probe into the consequences of MOND coupled with Lorentz symmetry. As a first step towards the goal, in this paper, we deduce the transformat ion of four fo rce in the MOND region. It is observed that the transformations to that effect are d ifferent fro m their relativistic counter parts, as is expected.

II.

The Relativistic Analogue Of MOND In Regard To Velocity

The Newton’s second law of motion in the usual notation can be expressed as F

dp d  (mu) , dt dt

In relativity when u is the velocity of a particle with mass m  m(u ) , above gives F  m

(2.1)

du dm u dt dt

Co mparing this equation with its analogue in MOND (1.1), we deduce m

Considering

a

to be a constant

a dm  mu a0 du

k , above linear differential equation reduces to a variable separable form dm du  k  a0   m u   a0

On integration, we obtain

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   

Four Force In MOND m  u  m0  u 0 Noting

m

m0

, above is rewritten as

u2 1 2 c

u  m0   u0

m0 1

u2 c2

Neglecting higher order terms of

  

A

1

 u2  2A u or 1  2    c  u0  

u2 , above yields c2

u2  For real values of

A

  k  a0   , A     a    0 

2 Ac 2 u  2 Ac 2  0 u0

u , we get A

k  a0 u 02 2u 02 i.e.  2 a0 c2 c2

 2u 02  k  a0 1  2  , c  

or

where we assume that

(2.2)

m(u0 )  m0 , u0  0 .

This equation seems to be interesting and deserves physical interpretation. At present we conclude that the conservation of linear mo mentu m is acclaimed for k  0 which corresponds to a rectilinear mot ion of a particle. Also

m  m0 corresponds to k  a0 .

III.

The relativistic analogue of MOND in regard to acceleration

Fro m (1.1) and (2.1),

ma

m u(a  u)  ma, 2c 2

(2.3)

2

where

m

u m0 and   1  2 .  c

Now 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 to

Newtonian dynamics.

Case II: Let a be parallel to u . We consider a  B u , where B is a constant. We investigate the relativistic analogue of MOND in regard to velocity and acceleration as follo ws.

Equation (2.3) imp lies

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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 .  a0  

(2.4)

4. Four force in MOND Considering 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 1     ds dt  c  ds dt c 2 

and

(2.5)

where E represents energy.

The work done dw in mov ing a system through the displacement dr is

dw  F  dr or

dw  (ma)  u dt

If the wo rk done is used in increasing the kinetic energy

T , above gives,

dw dT   Fu dt dt dT dE   Fu dt dt Using above equation, (2.5) becomes

dp4 ma   u .  ds c 2 Thus four force in M OND can be exp ressed by

 ma ma   u   F i   , c 2   c IV.

Fi 

dp i as a four force, ds (2.6)

Transformations of the components of four force F i in MOND

Considering its components under LTS i.e.

F 1 , F 2 , F 3 , F 4 must transform like the components of a four radius vector x i

1 v   F '   F 1  F 4 , F'2  F 2 , F'3  F 3 c  

v  F   F 4  F 1 c  '4

In view of (2.6) in

S'

 v2   1 - 2  ,     c    

frame, we write

26

1 / 2

(2.7)

Four Force In MOND  m' ' a' m' ' a'  u'  ' u' ,    1  F   ,  c2 c 2'  c' 

2

'i

1

2

3

Substituting the values of F , F , F , F

4

fro m (2.6) in (2.7) and by using the result,

2

1

u' 1  v2 / c2 1  u 2 / c2  , uxv c2 1 2 c

we get

m' ' a'x  max 

vm (a y u y  az u z ) (c  u x v ) 2

v2 v2 2 c 1  c2 m  a c 2 m  a , m'' a'  m'' a' y  2 y z z 2 (c  u x v ) (c  u x v ) c2 1 

m' ' a'u' 

c2m  [ax (u x  v)  (a y u y  az u z )] (c 2  u x v )

GMa0 , (see Milgro m (1983), Bekenstein (2004)), we get transformat ions of acceleration r

Using a  | a |  components

m'a'x max mv   (a y u y  a z u z ) , 2 r' r r (c  u x v ) v2 v2 2 c 1 2 c 1 2 m'a' y c ma y m'a' z c ma z  2  2 , , r' r' (c  u x v ) r (c  u x v ) r 2

m'

(a'u' ) mc2  [ax (u x  v)  (a y u y  az u z ) r' r (c 2  u x v )

 ma analogous to Newtonian force mass acceleration relationship, the above transformation Defining F can be rewritten as (N)

(N)

Fx'( N ) Fx  r' r



Fy'( N ) r'

Fz'( N ) r'

v (N) (N) ( Fy u y  Fz u z ) r (c  u x v ) 2

v2 (N ) c 2 Fy  2 r c  uxv

(2.8)

c2 1

v2 (N ) c 2 Fz  2 r c  uxv

(2.9)

c2 1

(2.10)

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Four Force In MOND

and

Fx'( N ) u ' x  Fy'( N ) u ' y  Fz'( N ) u ' z r' 

c2 [ Fx( N ) (u x  v)  Fy( N )u y  Fz( N )u z ] . r (c 2  u x v )

(2.11)

Conclusion When 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 ions in MOND are g iven by the equations (2.6), (2.8), (2.9), (2.10) and (2.11).

Acknowledgement The authors are thankful to Professor T M Karade fo r suggesting the problem.

References [1.]

[2.]

[3.]

[4.]

[5.]

[6.]

Oort J, Bull. Astron. Inst. Neth. 6, 249 1932 Zwicky F, Helv. Phys. Acta 6, 110 1933 Milgro m M, 1983a, Ap J, 270, 365 Milgro m M, 1983b, Ap J, 270, 371.

Milgro m M, 1983c, Ap J, 270, 384.

Bekenstein J, 2004, Phys Rev D, 70, 083509.

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