Is the intermediate point genuine ?

The quadratic form allows one to incorporate some arguments about the boundary conditions both near the stellar surface and infinity, and as a result the special point looks like ``important'' as it yields an extra condition (15). By analogy with nozzle type flows in air dynamics, where critical points exist, we see that a subsonic flow for the inner boundary and a supersonic flow for the outer boundary are satisfied simultaneously simply by having a transonic flow through the critical point. Our question is whether the intermediate point resembles such a point or not ?

As we have already seen in section 2, (5) is a regularised expression for the Lorentz factor , and in contrast, the quadratic form (10) is incomplete in that both and R are coupled with . The quadratic analysis is thus based on having coefficients (of quadratic expression) which are not independent of the variable . To realize this is important because a characteristic point in such a situation is not unique and different characteristic points can be generated.

We may write (10) into a different form
 
where k is an arbitrary constant. Because (20) is identical to (10), so following A95 we still argue that must connect to at a characteristic point. The solutions are
 
To require at the characteristic point, we obtain two conditions,
  
Combining (23) and (24), we obtain
 
By (11), (25) leads to
 
With the use of (4) again, (26) becomes
 
Clearly for a , (27) describes a point other than the intermediate one, and there can be many different points for arbitrary k. Because they are created by having a non-zero k for the same relation (10), they are spurious and therefore have no significance. The easiness to generate all these spurious points roots in the dependence between the coefficients and of the quadratic form, in a similar manner with dependent variables on both sides of a ratio (see LM94). Since the intermediate point is derived on the condition that and R are dependent, we argue that it is created rather than being a genuine one. Once expressing by as an independent variable, i.e., (5), the intermediate point disappears.

© Copyright Astronomical Society of Australia 1997