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# Hoop conjecture

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 4-dimensional 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 4-dimensional 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 non-head-on 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 .

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.

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