To investigate the CM energy of the collision on the horizon of the BTZ black hole, we have to derive the 2 + 1 dimensional "4-velocity" component of the colliding particle in the background of the 2 + 1 dimensional BTZ black hole.
For a time-like geodesic, the affine parameter can be identified with the proper time, and thus, from (7), we can solve the 2 + 1 dimensional "4-velocity" components [??] and [??] (where the dot denotes a derivative with respect to the proper time now) as
Thus we get the square of the 4-velocity radial component:
Here we have obtained all nonzero 2 + 1 dimensional "4-velocity" components for the geodesic equation.
where [u.sup.[mu].sub.1] and [u.sup.v.sub.2] are the "4-velocity" vectors of the two particles (u = ([??], [??], [??])).
Then we find the Hamiltonian in its two forms, with the help of 4-velocity and the generalized momentum, and substitute the Hamiltonian into Hamilton equations to verify the motion equations.
In the coordinates [x.sup.[mu]] = (ct, x, y, z) the Lagrangian Depends on the coordinates [x.sup.[mu]] , on the 4-velocity of substance motion [u.sup.[mu]] = [cdx.sup.[mu]]/ds (where c-the speed of light, ds indicates the interval for the moving substance unit), on 4-potential [D.sub.[mu]] of gravitational field and 4-potential [A.sub.[mu]] of electromagnetic field and on metric tensor [g.sub.[mu]v] of the reference frame.
However, the specified quantities in the first approximation are independent from the 4-velocity of the substance unit.
We shall note that from the definition of 4-velocity [u.sup.[mu]] = [cdx.sup.[mu]]/ds and of the interval ds = [square root of ([g.sub.[mu]v] [dx.sup.[mu]] [dx.sup.v])] follows the standard relation [u.sub.[mu]] [u.sup.[mu]] = [c.sup.2].
where [u.sup.[mu]] d[x.sup.[mu]]/d[tau] is a typical component of the test-mass 4-velocity
In the case at hand, the relevant field is proportional to the earth's 4-velocity in the vicinity of the earth.
(23) The relevant component of the stress-energy tensor is [T.sup.00] = [rho][U.sup.0][U.sup.0] with [[summation].sub.[mu]] [[summation].sub.v] [U.sup.[mu]] [g.sub.[mu]v] [g.sub.[mu]v] [U.sup.v] = -1 for the 4-velocity [U.sup.[mu]] due to the -+++ signature of the metric tensor.