Re: Relativistic Paradox

A rigorous, quantitative answer to this question would require a full
time-dependent solution of the non-linear Einstein eqns for matter and
fields, and does not appear very useful to the questioner, besides being
beyond my personal competence in GR.  Note that since the question is
really about gravitational scattering, considerable work (especially in
the post-Newtonian approximation, a step beyond the Newtonian) has been
done and is in the literature.  I am far from an expert in this subject,
but I believe the following to be substantially correct: 


1.  As the question is stated, neither object is compact enough to be a
BH, considered in its own rest frame. 

2.  The formation of an event-horizon (1-way surface, the characteristic
feature of a BH) is a Lorentz-invariant property of a region of
spacetime.  This follows from the fact that a light signal cannot
propagate outward across it, and no finite Lorentz transformation can
alter that property. 

3.  The standard Schwarzschild solution (assuming the masses are non-
rotating) for a BH is described in the rest frame of the mass, and is
static. 

4.  Considering items (1)-(3), we cannot expect the formation of an
event-horizon in the spacetime around a moving mass, even if the
"relativistic mass"  due to its uniform motion is very large, because in
its rest frame there will be no event-horizon, hence none in any Lorentz
frame. 


This is the essential answer to the question:  a Lorentz transformation
cannot turn a non-BH into a BH.  An L.T., after all, just shows us what a
mass looks like if we view it from a uniformly moving co-ordinate system. 


5.  The notion of "relativistic mass" has gone out of fashion in recent
years, because it does not seem to lead to simplicity and clarity.  While
a moving object can have a very large kinetic energy, and thus apparently
be a strong source of gravitation, the standard static solution certainly
does not apply, and so the relationship that it gives between
gravitational radius and mass are not valid.  A good exercise for a
student would be to apply a Lorentz transformation (a pure velocity trf,
say along the z-axis) to the particle 4-momentum and to the curvature and
stress-energy tensors, and see how they behave. Nothing like a black hole
will result! 

6.  In the limit of a small ultra-relativistic object passing an object
which is not a black hole, the dynamics are essentially like those of the
gravitational deflection of a light ray, and the same answer should be
obtained. 


From an intuitive, geometric point of view, an object slightly off to the
side does not feel the G field of the approaching mass until it has
essentially reached its closest point of approach, because the G field is
retarded and propagates at the speed of light--only slightly faster that
the object, if the latter is ultrarelativistic.  On the other hand, if
the more massive object were actually a BH in its rest frame, then a
particle passing relativistically would be captured if it passed through
the event-horizon, even though in the non-rotating, spherically symmetric
(in the BH rest frame) case the event-horizon would be Lorentz-contracted
into a pancake as seen in the rest frame of the passing particle.  From
this it seems that the transverse acceleration does actually rise in the
limit as the relative velocity approaches c and the "pancake" gets
arbitrarily thin.  The duration of the transverse force seen by the
passing particle must decrease in such a way that the deflection, as
measured in the rest frame of the more massive body, approaches a
constant value, that of a light ray, as v approaches c, at least in the
situation where one body is much more massive and energy losses due to
gravitational radiation are negligible. 


7.  Regarding the question about two colliding black holes, the total 
energy E (in units where the speed of light c=1), seen in the center-of-
mass system, would be

                            E = M1 + M2 + T,

where M1 and M2 are the rest mass-energies of the two components, and T
is the kinetic energy (KE) in the CM.  The KE of the two would go into the
total mass-energy of the surviving object, except for that lost due to
gravitational radiation. (In extreme cases the gravitational energy lost
can be large, even larger than T.  A limit on the maximum possible loss
due to radiation can be obtained from the area theorem ["Second Law of
Black Hole Dynamics", see Misner, Thorne & Wheeler, below], which says
that the total area of the event horizons in any BH interaction cannot
decrease.)  In any case, although the linear momentum of the final state
might be small or even zero, its rest energy (ie, mass) would still be 
large. 

Much more information about the above subjects can be found in standard
works like Misner, Thorne, & Wheeler, "Gravitation", or S. Weinberg's
text on general relativity.  Some other references which may be useful (I
have only looked up the abstracts, and not actually read any of these to
be sure how relevant they are): 


Braginsky, Caves, and Thorne, Phys.Rev. D 15, 2047-2068, 1977. 

"Analogy Between General Relativity and Electromagnetism for Slowly Moving
Particles in Weak Gravitational Fields",
Edward G. Harris, Am.J.Phys. 59: 421-425, 1991.

"Approximately Relativistic Interactions",
Kennedy, F.J. Am.J.Phys 40:63-74 1972.

"Gravitational Deflection of Fast Particles and of Light",
Koltun, D.S. Am.J.Phys. 50:527-532, 1982.

"Gravitational Deflection of Relativistic Massive Particles",
Silverman, M.P. Am.J.Phys. 48:72-78, 1980.