2. A TEST FOR THE LOCAL INTRINSIC LORENTZ SYMMETRY
Nguyen Vien Tho and Nguyen Quoc Hoan
Hanoi Univerity of Technology

3. Presentation Plan
The parallel between Yang-Mills theory and General Relativity
Interaction of color particle with gauge field
Exact solutions to Y-M’s and Einstein’s field equations
Exact Schwarzschild-like solution for Yang-Mills theory
Effective potential for particles in Schwarzschild-like Lorentz gauge field
Numerical considerations of particle’s orbits
Discussions

4. The parallel between Yang-Mills theory and General Relativity
It is well known that the Yang-Mills theory became a general framework to formulate theories of fundamental interactions.
The idea of Yang & Mills (1954) is to extend the SU(2) internal symmetry to the local SU(2) one: If we want to make a physical theory invariant under local symmetry transformations, it is necessary to introduce compensating fields. The idea was then applied to other unitary internal symmetries (SU(3) and …).

5. The parallel between Yang-Mills theory and General Relativity
Utiyama (1956) observed that the whole idea of principle of equivalence of GR can be reformulated in accordance with the phylosophy of Y-M theory: the gravitational field can be interpreted as a compensating field that cancel all the unwanted effects of local space-time transformations. Utiyama made a direct generalization of the Yang-Mills scheme for the SU(2) group to the Lorentz group.
Local Intrinsic Local Intrinsic
Unitary symmetry Space-Time Symmetry

6. The parallel between Yang-Mills theory and General Relativity
Mathematically, the Yang-Mills fields and the gravitational connections can be represented as connection coefficients on a principal bundle with a structure group (unitary or space-time symmetry groups).
  p = (x, g) = (xm , gi) ; m = 0, 3, i = 1, dim G

7. The parallel between Yang-Mills theory and General Relativity
Therefore the idea of the intrinsic Lorentz symmetry is reasonable, logically and theoretically.
Our purpose: to find an test for this idea ! To verify by considering its consequences
The program:
To find the equations of motion for particles with the Lorentz gauge field
To investigate motions of particles in an Lorentz gauge configuration. It is Schwarzschild-like spherically-symmetric gauge field solution
To make a comparison between Einstein and Yang-Mills approach of the problem

8. Interaction of color particle with gauge field
In accordance to intrinsic unitary symmetries, for example, SU(2), the intrinsic degrees of freedom for particles are described by “color charge” vector. Taking the average of the quantum Dirac equations for a color charge in Y-M field, Wong (1970) obtained the classical limit of equations that describe the motion of a color charge in the SU(2) Y-M field.

9. Interaction of color particle with gauge field
Remark: The Wong’s equations have the form that are a generalization of the equations for electric charges in electromagnetic field to the case of color charges in gauge field.
Color particles interact with gauge field via the vector Ia that describes intrinsic degrees of freedom for particles.
Re-examing Wong ‘s equations with the fibre bundle language of the gauge theories, and with group-theoretical calculations, we have obtained the equations that describe the interaction between the Lorentz intrinsic degrees of freedom with corresponding gauge field:

10. [Nguyen Vien Tho – J. Math. Phys. (2008)]

11. Exact solutions to Y-M’s and Einstein’s field equations
Y-M and Einstein’s field equations are nonlinear differential equations. Exact solutions to them are not easy to find since there exists no general method of solving.
The usual approach is to try some “ansatz” and insert it in the field equations to see if it solves them.
Examples:
For Einstein’s field equations: The Schwarzschild solurtions, the Kerr metrics.
For SU(2) Y-M’s field equations: the Coleman’s plane wave solution, the t’Hooft-Polyakov’s monopole, the Bogomol’nyi-Prasad-Sommerfield (BPS) monopole,…

12. Exact solutions to Y-M’s and Einstein’s field equations
We have paid attention to the following solutions that are obtained by exploiting the parallel between Y-M’s and Einstein’s theory:
Wu & Yang (1976) obtained a solution of gauge field for the Lorentz space-time symmetry group, by extending a solution of the SU(2) Y-M field into the complex region.
Singleton (1995) discovered an exact Schwarzschild-like solution for the SU(2) Y-M field in which the gauge field has also complex components, and singularity at a radius r0 .

13. Exact Schwarzschild-like solution for Yang-Mills theory
With the ansatz:
where
The gauge fields of this form solve the Y-M equations.
[Singleton – Phys. Rev. (1995)]

14. Exact Schwarzschild-like solution for Yang-Mills theory
Applying the method of Wu – Yang, we converted this solution into a static sourceless gauge field for Lorentz group.
Substituting this field configuration into the equations that describe the interaction of particles with Lorentz gauge field, we obtain the following system of equations

16. Effective potential for particles in Schwarzschild-like Lorentz gauge field
It is known that the qualitative informations about the motion of particles in a force field can be extracted from the effective potential.
The effective potential for a particle in the Schwarzschild space-time in Einstein’s theory was investigated. The characteristics of particle’s motions were extracted in comparison with the Newtonian gravity [see, for example, the book of Misner, Thorn, Wheeler: Gravitation ].
We consider what is implications of the Y-M approach of the problem, in parallel along that of Einsteinian and Newtonian theories.

19. Numerical considerations of particle’s orbits
The equations of motion for particles in the Schwarszchild Y-M field allow planar motions that are found in the form:

20. Numerical considerations of particle’s orbits
We rewrite the system in the form of a system of first order differential equations, and apply the (fourth oder) Runge-Kutta method for solving.
The solutions denpend on many parameters, and the initial conditions
Trying with many sets of parameters and different values
of intial conditions, we show that there exit the Kepler-like
orbits with certain set of parameters and initial conditions.

22. Discussions
The main purpose of this work is to try find implications of such fundamental concepts as local symmetry priciple, its extension from unitary to space-time symmetries, to compaire effects from the Yang-Mills approach and the geodesic motion in curve space-time.
It is known that at a radius that larger 10M the motion of particles in Schwarzschild space-time of GR does not differ from the Newtonian gravity. The Yang-Mills approach leads to the same results.
The differences between three approachs (Newtonian, Einstein and Yang-Mills) are at small radii. We recall that in our approach, like for Wong’s equations, the equations of motion is a classical limit of the quantum equations. Therefore, such effects, as tunnelling across potential barrier at Schwarzschild radius, can not described by our approach. This kind of effects that concerns to the “black hole” concept can be described only by Einstein’s GR.