Next: Summary and discussion
Up: Highenergy headon collisions of
Previous: Time slicing and apparent
Now we examine the difference of the horizon formation for various
spacetime dimension using the hoop conjecture. The hoop conjecture
gives the criterion of black hole formation in the 4dimensional general
relativity [5]. It states that an apparent horizon forms when and only when
the mass of the system gets compacted into a region of which circumference satisfies

(21) 
As
is the circumference of the 4dimensional Schwarzschild horizon,
we can expect that the criterion of black hole formation in the
dimensional Einstein gravity is given by

(22) 
where
is the Schwarzschild radius of dimensional
spacetime. This criterion was implicitly used to
estimate the total cross section for black hole production
via nonheadon collisions [2].
To calculate the ratio and , we must specify the definition of
the mass of the system. In this paper, we use total energy as
the mass of the system. The circumference is defined as minimum
length which encloses two particles. We take the loop as shown in
FIG. 4 and calculate by taking the limit
.
reduces to which is the twice the distance of two particles.
The value of at is shown in TABLE I. As increases,
the value of decreases and the mass must be compacted into
the region with smaller circumference than
to produce a black hole.
This reflects the decrease in
with increase in .
Figure 3:
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 .

Figure 4:
The closed loop to calculate the circumference.
We calculate by taking
.

The value of at is also shown in TABLE I.
This result can be written as

(23) 
where
. The dimensional gravitational constant
is related to the Planck energy as

(24) 
Using this formula, Eq.(23) becomes


(25) 
If the Planck energy is TeV scale, is
and
becomes
. Thus the mass does not need to be
compacted into a small region of which circumference is
to produce a black hole.
Table 1:
The value of and at for
.



























Next: Summary and discussion
Up: Highenergy headon collisions of
Previous: Time slicing and apparent
Yasusada Nambu
20020823