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To treat the collision of particles as time evolutional process,
we consider the following slice of spacetime:

region I 


region II 
(9) 

region III 

where and particles collide at . In order to find
apparent horizon on the above slice, we first prepare surfaces with
zero expansion in region I and III, then connect them smoothly by
requiring that the null normal coincides at the junction of region I
and III, . In region I, the surface which have zero expansion
is given by
const. 
(10) 
and its null normal is
In region III, the surface which have zero expansion is given by

(12) 
where is a constant of integration determined by the matching condition
at the junction. For , the function is given by

(13) 
and for ,
where is the Gauss' hypergeometric function. The null
normal of the surface is given by
Matching these surfaces and null normals at the junction , we have
where is the radius of the surface at the junction. From
this, the relation between and can be given parametrically as


(18) 


(19) 
where
.
Figure 1:
The relation between and the horizon radius for
. The intersection of the dotted line and axis is
the time when the apparent horizon appears.
The value of
decreases as increases.

FIG. 1 shows the relation between and for each . We denote the
time when the apparent horizon appears as . The value of
becomes small as increases. For large , we have

(20) 
at .
The intersection of the
const. plane and the surface in region III
is a dimensional sphere, of which expansion is positive and
proportional to .
Thus the surface has negative expansion on
plane and
its curvature on this plane increases with the increase of spacetime dimension .
This leads to the decrease in the distance of two particles at the horizon formation.
The shape of apparent horizons for and are shown in FIG. 2 and FIG. 3.
Figure 2:
The apparent horizon for at
.
The dark line is the horizon at
, and light line is the
horizon at . The unit of the axis is .

Next: Hoop conjecture
Up: Highenergy headon collisions of
Previous: Highenergy particle collisions at
Yasusada Nambu
20020823