One can consider the fields inside a spherical electron cavity of radius [R.sub.B] moving in plasma with the

relativistic velocity [v.sub.B] [equivalent] c along x-axis, where [v.sub.B] is the bubble velocity, assuming that the ion dynamics is neglected, i.e., [R.sub.B] [much less than ] c/[w.sub.pi], where Bpi = [(4[pi][e.sup.2][n.sub.i]/[m.sub.i]).sup.1/2] is the ion plasma frequency with ion mass [m.sub.i] and ion density [n.sub.i].

Identifying the

relativistic velocity distribution in equilibrium is essential to understanding the phenomena that involves hyper-energetic particles.

Such a velocity term straddles or couples the two domains, that of the orbiting object and that of the stationary observer and so could sensibly be called the "Coupling Velocity" or possibly the "Relativistic Velocity".

Thus far Relativistic Velocity is only a definition.

There are known many models for Hyperbolic Geometry, such as: Poincare disc model, Poincare half-plane, Klein model, Einstein

relativistic velocity model, etc.

Relativistic velocity is a similar but interestingly different kind of example.

The Einstein

relativistic velocity model is another model of hyperbolic geometry.

[4] Catalin Barbu, Smarandache's Cevian Triangle Theorem in the Einstein Relativistic Velocity Model of Hyperbolic Geometry, http://vixra.org/abs/1003.0254 and http://vixra.org/pdf /1003.0254v1.pdf.

[5] Catalin Barbu, Smarandache's Cevian Triangle Theorem in The Einstein Relativistic Velocity Model of Hyperbolic Geometry, Progress in Physics, University of New Mexico, 3(2010), 69-70.

In this section, we prove Smarandache's minimum theorem in the Einstein relativistic velocity model of hyperbolic geometry.

Van Aubel's Theorem in the Einstein Relativistic Velocity Model of Hyperbolic Geometry.

There are known many main models for hyperbolic geometry, such as: Poincare disc model, Poincare half-plane, Klein model, Einstein

relativistic velocity model, etc.