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/c² (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=(Gh/c³)^0.5 = 10^-33 cm and m=m_p=(hclG)^0.5~10^-5 g, which are just the Planck length r_p and Planck mass m_p. With space curling up at the gravitational radius and with Heisenberg's uncertainty relation a measure of the zero-point energy, the zero-point energy would have to be cut-off at the Planck length. This means that the vacuum should be densely occupied with Planck mass black holes. This is essentially the result obtained by Hawking [4] in applying Feynman's path integral quantization to Einstein's gravitational field equations. The idea that the Planck length provides a natural cut-off for the zero-point energy of the vacuum has been also advocated by Wheeler [5]. If the zero point energy is cut-off at the Planck length, it would result in a mass density of the vacuum equal to:

P_vac =m_p/r_p³ =c⁵/G^zh ~= 10⁹⁵g/cm³ (1.3)

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.