**1.4 The Problem of Quantum Gravity**

The quantization of Einstein's gravitational field equations poses one of the still outstanding unsolved problems in theoretical physics. It is nevertheless possible to draw some very general conclusions, obtained from the expression for the gravitational radius /"of a spherical mass *m:*

*r = Gm/c ^{2 }*(1-1)

According to Einstein's gravitational field equations, space begins to curl up if a mass *m *reaches this radius, where the mass becomes a black hole. Applying Heisenberg's uncertainty relations to the mass *m *within the radius *r, *one has:

*mrc ~= h *(1.2)

Because the kinetic energy of a mass gravitationally collapsing to the radius (1.1) would be relativistic, one is permitted to equate the velocity in Heisenberg's relation with the velocity of light. From (1.1) and (1.2), one

obtains *r=r _{p}=*(

*P _{vac} =m_{p}/r_{p}*

large enough to put the mass of the entire universe into a cube with a side length less than one fermi (10~^{13} cm). With the vacuum mass density (1.3), the cosmological constant becomes equal to *r _{p}^{-2}, *which would result in a

radius for the universe equal the Planck length. This impossible result again shows that there must be a mass-compensating effect.

With the space-time at the Planck length made up of densely packed Planck mass black holes, Einstein-Rosen bridges (sometimes called wormholes) would form, connecting regions of space which on a macroscopic scale are separated from each other by arbitrary large distances. But as it was recently shown by Redmount and Suen [6], the conjectured "space-time foam" at the Planck scale might be unstable. If this should turn out to be true, space-time itself would become unstable suggesting that Einstein's gravitational field equations become invalid at the Planck scale.

By adding higher order terms to the Einstein-Hilbert gravitational field Lagrangian, the idea of " Planckions" has come up. Because these higher order terms have to be multiplied by powers of a (length) ^{2}, it has been conjectured that they signify the existence of a new particle with a mass equal the Planck mass. This conjecture and its consequences is treated in substantial detail by Borzeszkowski and Treder [7]. The idea of Planckions as a fundamental particle was independently proposed by Sakharov [8], who called these hypothetical particles "maximons." Sakharov also suggested the existence of hypothetical "ghost particles," to compensate the huge vacuum mass density of a vacuum densely packed with Planck mass maximons.