Keywords: variational methods, Lagrangian, lightlike particle, Fermat principle, geodesics
International Journal of Physics, 2013 1 (1),
pp 14.
DOI: 10.12691/ijp111
Received December 20, 2012; Revised January 24, 2013; Accepted March 04, 2013
Copyright © 2014 Science and Education Publishing. All Rights Reserved.
1. Introduction
In general relativity light is assumed to propagate in the vacuum along null geodesics in a pseudoRiemannian manifold. Geodesic equations can be obtained using a variational method for the integral of motion. In a general case the geodesic is extremal in a set that includes isotropic and nonisotropic curves with close to zero length of spacetime path ^{[1]}. Besides the geodesics principle there exists Fermat principle for stationary gravity fields. In ^{[1]} a variational method without the violation of isotropy of the path of the lightlike particle is proposed and it is shown that it gives equations identical to those which follow from the Fermat principle. For the Gödel space these methods give solutions being different from the null geodesic.
In the present paper the class of metrics extended by metrics with zero coefficient corresponding to the time coordinate is considered. We also find a solution by the proposed method of energy variation for a generalized Gödel space and compare it with that obtained in ^{[2]}.
Our results here lead us to a search to find additional criteria for the choice between the null geodesic principle and the variational method without violation of isotropy of the path.
2. Fermat and Geodesics Principles
For a stationary spacetime with appropriate metric tensor the Fermat principle is formulated ^{[3]} as follows
 (1) 
where is the first variation and is element of spatial distance along the ray:
 (2) 
Here is the time coordinate and indices are the space coordinates.
The null geodesics equations with affine parameter and the fourvelocity vector are written as
 (3) 
where are Christoffel symbols.
For a static spacetime the Fermat principle and the geodesics equations coupled with equation of null geodesic condition
 (4) 
give the same solutions ^{[4]}.
3. Lagrangian for a LightLike Particle
In the principle of energy variation for a lightlike particle’s motion, the pseudoRiemannian metric with coefficients defined by a transformation
 (5) 
is written in form
 (6) 
where is some quantity, which is assumed to be equal 1 and regarded as the energy of the lightlike particle with . Solving this equation for under condition gives two solutions
 (7) 
Even if one solution in accordance with making definitions will be .
With and even if for one the energy takes form
 (8) 
In both cases for the free moving particle the lagrangian is
 (9) 
Its derivatives give the canonical momenta
 (10) 
and the forces
 (11) 
4. Equations of Motion Produced by Energy Variation and Fermat Principles
In energy variation principle equations of motion are found from the principle of action
 (12) 
where , are values of the affine parameter of the points, which are linked by trajectory of particle. Taking into account (9), variation of leaves interval to be lightlike, and an application of standard variational procedure yields EulerLagrange equations
 (13) 
With we obtain
 (14) 
For the EulerLagrange equations give
 (15) 
Let’s show that for the stationary spacetime the above equations are identical to those which follow from the Fermat principle. Denoting
 (16) 
and comparing this expression with (7) we write
 (17) 
Therefore, variation (1) is equivalent to variation of integral
 (18) 
The EulerLagrange equation, corresponded to the time coordinate, will be
 (19) 
Since the energy is assumed to be constant along critical curve, its differential is zero. The metrical coefficients in case of the stationary field doesn't depend on time, therefore we have. Thus, expression (19) is idential to equation (14). For the space coordinates, the equations are follows:
 (20) 
We condition by appropriate choice of the affine parameter the constant value of . Then, the second term in the left part of equations will be vanishing and they shall be identical to (13).
5. Flat Expanding Space
Let's find by the proposed method the extremal isotropic curves in the flat expanding space, which corresponds to the cosmological model. With rectangular coordinates it is described by metric
 (21) 
where is the length scale factor.
Equation (14) gives
 (22) 
where overdot denotes derivative with respect to time. The EulerLagrange equations for the cyclic coordinates yield constants of motion
 (23) 
Extracting the derivatives with respect to the spacelike coordinates from this equation and substituting them in (22) we obtain
 (24) 
This equation has a solution, which with is written in form
 (25) 
where is constant. Substitution in equation (23) gives
 (26) 
The condition (4) will be
 (27) 
It yields and components of the fourvelocity vector turn out to
 (28) 
 (29) 
They conform to solution of equations of the null geodesics in FRLW spacetime ^{[5]}.
6. Generalized Gödel SpaceTime
The stationary solution of the Einstein field equation with cosmological constant found by Gödel describes gravity field of the rotating uniform dust matter. With coordinates the line element for generalized Gödel metric is written in form
 (30) 
where are constants.
The canonical momenta (10) for cyclic coordinates are the constants of motion. They are written in form
 (31) 
 (32) 
 (33) 
These equations with condition of null geodesic
 (34) 
yield components of the fourvelocity vector:
 (35) 
 (36) 
 (37) 
 (38) 
where function is
 (39) 
The canonical momenta must satisfy the condition
 (40) 
since only in this case the radial velocity exists. The singularity takes place with
 (41) 
Let's compare fourvelocity obtained above with solution of the null geodesics equations ^{[2]}. For the affine parameter, being proportional to the coordinate time with corresponding constants , the null geodesics equations yield
 (42) 
 (43) 
 (44) 
 (45) 
where function is
 (46) 
Though with
 (47) 
both methods give identical space curve, the coordinate velocity along it will be different.
7. Conclusions
For the static spacetime all three variational methods give the same solutions, also Fermat principle for stationary gravity fields is identical to the principle of variation of energy of lightlike particles. Though expression for energy is different in cases of zero and nonzero metrical coefficients that does not influence the form of the canonical momenta and forces as well as the expressions for equations of motion. For the flat expanding space, a curve corresponding to the critical value of integral of the photon energy coincides with geodesic, while for the generalized Gödel spacetime these solutions are different.
Only experiment can establish which of the principles, geodesics principle or energy variation, conforms better to physical reality. If it will be demonstrated experimentally that with some physical conditions a motion of material test particle corresponds to the spacetime, for which these methods give different solutions for the light, then experimental determination of the light trajectory in the same physical conditions allows to make choice between these principles.
Acknowledgement
I am grateful to Vish Viswanathan for help in preparing of this manuscript.
References
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