W. SMILGA

966

(Einsteinian gravity, which was set up with the goal of

replacing Newtonian gravity, uses the trace of the

energy-momentum tensor instead. Both approaches ar e in

a sense complementary, as far as spherically symmetric

solutions are concerned [7]). According to what has been

said above about conformal scaling, the curvature must

be proportional to the scaling of the momentum. There-

fore, the curvature experienced by the second particle

must be proportional to the traceless part of the energy-

momentum tensor of the first particle, and vice versa.

A curvature tensor that can be set proportional to a

traceless energy-momentum tensor, must itself be trace-

less too. Such a tensor is the Weyl tensor C

R

, which

is the traceless part of the Riemann curvature tensor

W

. From the Weyl tensor, a traceless “gravitation

tensor”

T

can be derived [8]. This tensor can then be

put into relation with the traceless part of the energy-

momentum tensor

.

Examples of traceless energy-momentum tensors,

based on different models of matter, can be found in [8].

Here we simply subtract the trace from the energy-

momentum tensor to make it traceless. This leads to the

field equations of c onformal gravity

1

2

T Tg

G

conf

WG

(18)

with a “gravitational constant” .

conf

Conformal gravity has gained interest in recent years

because it may solve the problems usually associated

with “dark matter” and “dark energy” [7,8] without

additional ad hoc assumptions. Within the scale of our

solar system, conformal gravity is known to deliver the

same results as Einstein’s theory of general relativity,

which is based on the Riemann curvature tensor, rather

than on the Weyl tensor [7]. The problem of “ghosts,”

which has been encountered in “quantized” versions of

conformal gravity [9,10], does not exist for the classical

version.

4. The Gravitational Constant

In [1], the electromagnetic coupling constant

was

calculated from the geometry of the parameter space

associated with an irreducible two-particle state space of

spin-1/2 particles. The same calculation, done for spin-

less particles, results in a coupling constant of 4

.

There is, however, a crucial difference between quan-

tum electrodynamics and gravitation. Whereas in quan-

tum electrodynamics it makes sense to consider an iso-

lated two-particle system, this is an unrealistic configura-

tion in gravitation. There is no way to set up a “neutral”

environment or to “shield” gravitation. Therefore, an ex-

perimental setup for a “scattering experiment” in analogy

to electron-electron scattering must always take into

account the whole environment. This means we have to

take into account at least 1080 heavy particles, which is

the estimated number of protons in the (observable)

universe [11].

A gravitational scattering experiment of an (electri-

cally neutral) particle of, say, the mass of the proton,

includes at first the selection of a second particle from

1080 available particles. This is followed by a transition

from the “incoming” two-particle pure product state to an

irreducible (entangled) two-particle state. Finally, we

have a transition to an “outgoing” two-particle pure pro-

duct state, which is the quantum mechanical description

of measuring the individual momenta of the particles

after the scattering has taken place. Note that there are

two transitions between pure product states and en-

tangled state, but only one selection.

The following is an attempt to quantum mechanically

describe the “selection process.” The selection of a part-

ner particle will be considered as a “transition” from an

“incoming” one-particle state (of the first particle) to a

two-particle state. For the first particle, there are

independent ways to form a two-particle state. Let us

describe the corresponding quantum mechanical transi-

tion amplitude by a state in a 10 -dimensional state

space. Then the states of this state space have to be

normalized by the factor

80

10

80

40

110 .

This normalization ensures that the total transition

probability from a specific incoming (one-particle) state

to an outgoing (one-particle) state, through any interme-

diate two-particle state, equals unity. On the other hand,

the field equations (18) describe the contribution of only

a specific second particle, characterized by its energy-

momentum tensor at a point

, to the curvature of

space-time. Accordingly, the scattering process contains

only the transitions up to the outgoing two-particle pro-

duct state. For this reason, the “selection amplitude”

enters only once. The normalization factor in this ampli-

tude leads to an additional factor to the two-particle

coupling constant 40

0 of 11 .

4

This results in an estimate of the “gravitational cou-

pling constant.” It matches the empirical strength of the

gravitational interaction, which, between two protons, is

37 orders of magnitude weaker than the electromagnetic

interaction (or 43 orders, between two electrons) [12].

This weakness explains why in the experiments of parti-

cle physics the gravitational interaction can be ignored.

5. Quantum Gravity

The field equations (18) describe a classical theory of

gravitation. What, then, is their quantum mechanical

analogue? Since we just have sketched a connection

between quantum theory and classical conformal gravity,

we are able to give an answer to this question: The quan-

tum mechanical basis of conformal gravity is nothing

Copyright © 2013 SciRes. JMP