Trinormal Vector Of The Worldline Of A Particle

DOI : 10.17577/IJERTV1IS6395

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Trinormal Vector Of The Worldline Of A Particle


I J E R T ,


We would like to publish a paper in your Esteemed Journal. We

are sending herewith manuscript of the paper in duplicate. Also the same paper we e-mailed in your email address. Kindly consider the paper for publication.

Thanking you,

Prof. Prakash B Lugade

Professor in Mathematics,

P V P I T ,Budhghoan College of Engineering, Tal:- Miraj,Dist:- Sangli

Res.: Harihar Ganesh nagar Budhghoan Tal:-Miraj,Dis:- Sangli (MS)-416304


1P.B. and 2D.B. Unde Department of Mathematics,

1Dr. J.J. Magdum College of Engineering, Jaysingpur.Dist.Kolhapur Pin-416 101

2P.V.P.I.T., Budhgaon Sangli, Pin 416 304 Maharashtra (INDIA)


In this paper the trinormal vector field Ra and bitorsion scalar B of the worldline of a particle are expressed in terms of the third and lower order intrinsic derivatives of the flow vector ua. The applications of Ra and B are seen for relativistic continuum mechanics.

Introduction :

The expression for Trinormal Ra :

We adopt the signature of the metric tensor as ( +, -, -, – ). The timelike tangent field ua to the worldline is chosen to satisfy

uaua = 1

The vector field Pa is a spacelike vector orthogonal to ua. It is called as principal normal. The spacelike unit vector fields Qa and Ra are called respectively the binormal and the trinormal fields which are orthogonal to velocity as well as acceleration. For the tetrad denoted by (ua,Pa,Qa,Ra) the rheotetrad formulae (Relativistic Serret Frenet formulae) are (Synge 1960, Davis 1970)

u a = KPa (A)

P a = Kua+TQa (B)

Q a = -TPa+BRa (C)

R a = – BQa (D)

The tetrad is invented in Relativistic continuum mechanics by Pirani (1956). The tetrad is related to velocity vector ua, principal normal Pa, binormal Qa and trinormal Ra. Thus tetrad is denoted by (ua, Pa, Qa, Ra). For the tetrad Synge (1960) has introduced first, second and third curvture K,T,B, respectively. Synge(1960) also given the relativistic serret Frenet formulae, Gursey (1957) given the explicit expression for binormal Qa interms of derivatives of velocity vector ua. In this paper the expression for trinormal Ra and bitorsion B in terms of derivatives of ua are given.

The orthonormal conditions are

uaua= -PaPa=-QaQa=-RaRa=1

and uaPa = uaQa=uaRa=PaQa=PaRa=QaRa=0

An overhead dot denotes the covariant differentiation along the flow vector ua, thus P a = Pa;bub

Following Gursey (1957), the explicit expression for bionormal Qa is

Qa =

1 ( – K' u a – K2ua) (E)

KT a K

On covariantly differentiating the expression for trinormal Qa along the flow vector ua, we obtain.

Q a=

1 ( a – La + Mua – NK2ua) (F)



L = T'


2K' ,



M = K2

T'K' K' K2


N = K' T'


Substituting Equations (A), (F) in Eq (C) yields on rearrangement,


1 [a – La + (M+T2) u a – NK2ua] (G)


This is the expression for Ra interms of the third and lower order intrinsic derivatives of the flow vector ua,

The expression for Bitorsion B :

K2 = -u au a

T2= 1

K 2

(K4 K2 – aa)

To evaluate B, we use (Eq. F) in RaRa=-1 and after simplification we obtain


1 [9K2K 2 – (K +K3-KT2)2 – (2K T+KT )2 a ] (H)

K2T2 a

on exploiting the identities

uaua = 0

uaa = K2

u au a = – K2

aua =

aa = K2(K2-T2)-k 2

au a

= 1 (u au


a) = -KK

au a = K2(T2-K2) – KK

aa = 3K3K – K3T T -K K

Thus in equation (G) we have accomplished the expression for bitorsion scatar field in terms of product of ua, úa, a, a and K,T. The quite completed nature of Ra and B is obvious from the equations predicted by Vishweshwara (1988).

Theorem :

Remark : a is a linear combination of ua,ua, a iff B=0

Soln : Necessary Part

a = KTBRa + La – (M+T2)u ' a – NK2ua (by Eq. G)

For linear combination of ua,u a, a ,

KTB = 0

As K 0, T 0, B = 0

Sufficient part : if B = 0 then

a = La – (M+T2)u ' a NK2ua.

It is linear combination of a, ua, ua, of vanishing Bitorsion

Physical Significance : The path of classical gravitationally self interacting spin partical with Frenkel-weissenhoff constraints (Geonner 1967) has vanishing bitorsion. The Equation of path is linear combination of a, ua, ua.

m2 b

8Gm m2

'b i a a i a

a =


u' u'b

u a +

15 s2

u u'b

(u ' u ' iu + ) – 3u ' iu

Where m is the mass of the particle, G is the gravitational constant and s is spin of the particle of mass m.

Conclusion :

In this paper trinormal vector field Ra and bitorsion B are expressed interms of linear combination of derivatives of ua. The expression of bitorsion can be studied for non- geodesic flows in Relativistic continuum mechanics. According to the exhaustive and magnificent survey of the exact solutions of Einsteins field equations by Kramer, Hertz, Maccallum and Stephani (1980), right from the inception of general relativity in 1916, the properties of the models are confined to the study of geodesic paths. Only few models with non-geodesic paths are reported in their book. This provides the motivation for this paper which explores non-geodesic flow with torsion as well as bitorsion.


  1. Synge, J.L. (1960) Relativity; The General Theory North-Holand Publishing company, Amsterdam PP 10-15

  2. Davis, W.R. (1970) Classical Fields, Particles, and the theory of Relativity.

    Gordon and Breach Science Publishers New York P.20

  3. Gursey, F. (1957) Relativistic Kinematics of a classical Point Particle in Spinor Form. IL NUOVO CIMENTO 5(4) PP, 784-809

  4. Vishveshwara, C.V. (1988) Black Holes; A slanted overview in Highlights in Gravitaton and Cosmology ed. IYER, B.R. etal. Cambridge University, Press PP 312-326

  5. Goenner, H., Gralewski, U. and Westpfahl, K. 1967. Gravitational self forces nd Radiation Losses of classical spin particles (first approximation) zeitschrift fur physik 207 PP 186-208

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