Fully Classical Quantum Gravity
Thomas C Andersen*
NSCIR - Ontario, Canada
(Dated: May 16, 2017)
It’s an experimental fact that quantum objects in the ground state do not radiate
electromagnetic energy, but what are the limits on our knowledge of the gravitational
equivalent of this? In semiclassical gravity it is the expectation values of quantum
particle positions that form the source for the Einstein equations, thus a particle or
atom in a ground state emits no gravitational radiation. Here we instead assume
a fully classical quantum gravity - the internal components of objects in a pure
quantum state are assumed to classically radiate gravitational waves. The eﬀects of
this theory of microscopic gravity on the measured properties of the hydrogen atom,
along with possibilities to experimentally measure the eﬀects of atomic or nuclear
scale gravitational radiation are explored.
Essay written for the Gravity Research Foundation 2017 Awards for Essays on Gravitation.
NSCIR 8 Bruce St. Bx 841 Thornbury Ontario, N0H 2P0, Canada
The quantum gravity problem remains unsolved in physics today. There are many pos-
sible solutions proposed, but almost all of them suppose the existence of the graviton. The
graviton should have the same energy relation as the photon:
There not only exists no experimental conﬁrmation of this relationship for gravity, it
is also widely known that an experiment to detect a single graviton is well beyond the
capabilities of any present or future realizable experiment. Gravity may simply be a non
quantum eﬀect. Rosenfeld in 1963 is still very much relevant
There is no denying that, considering the universality of the quantum of
action, it is very tempting to regard any classical theory as a limiting case to some
quantal theory. In the absence of empirical evidence, however, this temptation
should be resisted. The case for quantizing gravitation, in particular, far from
being straightforward, appears very dubious on closer examination.
OTHER CLASSICAL GRAVITY THEORIES
Semiclassical gravity can be summarized as a classical gravitational ﬁeld coupled to quan-
tum matter ﬁelds. While semiclassical gravity is widely thought of as a workable limiting
approximation until a quantum theory of gravity is discovered, there are researchers who
treat semiclassical gravity as a real possibility and hence in need of experimental tests.
The semiclassical equations for quantum gravity are as from Møller and Rosenfeld:
gµν Rµν −1
2gµν gµν R=8πG
c4gµν hΨ|Tµν |Ψi(2)
While seemingly straightforward, semiclassical gravity has subtleties, especially in deter-
mining the quantum expectation value (see Appendix A of Bahrami).
Another classical treatment of quantum gravity comes from Roger Penrose with the
Gravitization of Quantum Mechanics where he posits that gravity connects not to the
expectation value, but rather directly to each superposed quantum state. Gravitation causes
collapse as the gravitational ﬁeld of multiple superposed states becomes too energetic.
FULLY CLASSICAL QUANTUM GRAVITY
Fully classical quantum gravity (FCQG) uses Einstein’s equations as given,
gµν Rµν −1
2gµν gµν R=8πG
c4gµν Tµν (3)
with the coupling to microscopic matter being on some assumed sub-quantum level,
where particle positions always have a deﬁnite value, as in for instance de Broglie - Bohm
mechanics. Of course if one uses Bohmian mechanics in its entirety, then gravitation is
also quantized, and particles will not radiate from their ground states. We thus assume here
that quantization does not apply to gravity at all, that particle trajectories are real and that
they interact directly and classically using the laws of Einstein’s general relativity. In many
ways it is similar to the program of stochastic electrodynamics(SED), in that classical
ﬁelds couple directly to sub - quantum particle motions. Indeed if one is to assume a SED
like explanation of quantum behavoir, then gravity should also be treated classically.
GRAVITATIONAL RADIATION FROM ATOMS AND NUCLEONS
Ashtekar for example elucidates the need for a quantum theory of gravity by citing
Einstein in 1916:
...Nevertheless, due to the inner-atomic movement of electrons, atoms would
have to radiate not only electro-magnetic but also gravitational energy, if only in
tiny amounts. As this is hardly true in Nature, it appears that quantum theory
would have to modify not only Maxwellian electrodynamics, but also the new
theory of gravitation.
Using instead Rosenfeld’s position that we must rely on experiment to show the need for
quantum gravity, consider the energy loss rate of a circa 1916 style Bohr planetary hydrogen
atom in the ground state, using Eddington’s formula for the gravitational energy radiated
by a two body system (in the approximation that one mass is much heavier):
dE/dt(atom) = −32Gm2
5c5=−10−43eV /s (4)
Which even over the age of the universe amounts to an energy loss due to gravitational
waves for a hydrogen atom in the ground state of only 10−25eV . Why was Einstein worried
about such a small rate of gravitational energy loss for a hydrogen atom? In contrast the
electromagnetic lifetime of the classical hydrogen atom is about 10−11swhich of course
helped lead to the discovery of quantum mechanics.
As a comparison to the above estimate, a quantum mechanical prediction of the lifetime
of the 3p−1sstate for emitting a graviton is about 1.9×1039s , which is within a
few orders of magnitude of the fully classical estimate above.
This energy loss is of no experimental signiﬁcance. So we can conclude that the stability
of atomic orbitals is not an experimental indication of a need for quantum gravity. In other
words we cannot experimentally determine if atoms radiate gravitational waves continuously
Gravitational Radiation from Within Nuclei
The Sivram - Arun paper Thermal Gravitational Waves is an expansion of Wein-
berg’s results in his 1972 book. Both calculate the gravitational wave (GW) emission
from nuclei passing each other thermally in an astrophysical hot plasma (stars). In fully
classical quantum gravity we make the additional assumption that gravitational waves are
also produced by nucleon motion inside each individual nucleus, even in the ground state,
greatly increasing GW emission and making it happen at any temperature, since it arises
from internal nucleon movements within each nucleus. Calculating an estimate for the GW
emission would depend on the model one uses for the nucleus. The Fermi gas model of the
nucleus assumes that the nucleons are free to move inside the potential well of the nucleus.
Since we are assuming that gravity is fully classical, we can use the same calculations as
that of Weinberg and Sivram to arrive at an estimate of gravitational wave emission from
nucleons inside nuclei.
A GW Nuclear Emission/Absorption Model
Taking the calculation of Weinberg to nuclear material, Sivaram ﬁnds a rate of 10−16eV /s
per neutron (using their neutron star calculation). Fully classical quantum gravity would
then suggest that the Sun emits about 1022 watts of 1022 Hz gravitational wave energy, as
opposed to the 109watts at a lower atomic frequency that Weinberg calculates from plasma
Another way to arrive an estimate for GW emission in nuclei is to treat a nucleus as having
several nucleons moving in it at some typical internal velocity. The speed of nucleons is given
by their kinetic energy in the Fermi gas model with a peak momentum of about 250M eV/c.
Using only one pair of these peak energy nucleons and setting r= 1fm, Eddington’s formula
for a bar of mass 2 nucleons, spinning at a nuclear 1023hz, predicts an emission rate of about
While these two approaches to calculate the GW emission of a nucleus in the fully classical
model diﬀer by several orders of magnitude, GW emission rates near these levels hint that
such eﬀects (or perhaps more likely a lack of eﬀect) might be measurable in the lab.
Experiments might need to use diﬀerential absorption eﬀects to arrive at results. Ab-
sorption models are harder to quantify, as the cross section estimate is quite uncertain due
to unknown detailed information on particle substructure.
Within this fully classical quantum model each nucleon will have its own characteristic
spectrum of nucleon - frequency gravitational waves, depending on the structure and size
of the atomic nucleus. Experiments similar to those done to look for ’big G’ could use
dissimilar materials for the masses whose force of attraction is to be measured. It’s notable
that experiments to determine Newton’s constant G have had great diﬃculty obtaining
consistent results. Most measurements of G do not agree with each other to within the
errors carefully determined by the experimenters.
Another experimental avenue would be to search for GW interaction eﬀects between the
bulk of the earth and masses in a lab of dissimilar materials.
EMISSION/ABSORPTION PARAMETER SPACE
FIG. 1 is a sketch of allowed emission and absorption parameters. Some - but not all -
combinations of emission and absorption parameters are ruled out by experiment. Towards
the upper left of the image limited absorption combined with higher emission would mean
that the stochastic background of gravitational waves would be too energetic, having for
example energy greater than the baryonic mass in the universe. The phrase ’stability of
nuclei’ refers to the experimental fact that nuclei live for billions of years. On the right a
ruled out region exists where absorption cross sections are not physically likely. The top
FIG. 1. Nuclear frequency gravitational wave emission and absorption. The elusive nature of grav-
itational wave detection means that even fully classical quantum gravity cannot be experimentally
ruled out. The frequency of the gravitational waves is that of nucleons (ω≈1022 Hz ).
line shows a calculation for the gravitational wave emission rate of a proton due to parton
(quark) motion. ’Nuclear emission (high)’ refers to the Eddington emission rate for a heavy
nucleus, while the lower nucleus emission rate is calculated assuming thermal Coulomb GW
emission inside each nucleus.
Due to the weak nature of gravitational eﬀects on subatomic particles, even fully classical
gravity cannot be experimentally ruled out at this time. Quantum gravity experiments that
are possible with today’s technology are very rare, this proposal represents an opportunity
to test one of the tenants of quantum gravity.
Null results from experiments as described here will be able to constrain the allowed
parameter space of a fully classical theory of microscopic gravity, thus suggesting that gravity
needs to be quantized.
These tests are also a test of the ubiquity of quantum mechanics. With a non null result
the conceptual foundations of quantum mechanics would be in question, as gravity might
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