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Fully Classical Quantum Gravity

Thomas C Andersen*

NSCIR - Ontario, Canada

(Dated: May 16, 2017)

ABSTRACT

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

*tandersen@nscir.ca

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INTRODUCTION

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:

Egraviton =~ν(1)

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[10]

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[7].

The semiclassical equations for quantum gravity are as from Møller[8] and Rosenfeld[10]:

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[2]).

Another classical treatment of quantum gravity comes from Roger Penrose with the

Gravitization of Quantum Mechanics[9] 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.

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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[3]. 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)[5], 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[1] 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[6] formula for the gravitational energy radiated

by a two body system (in the approximation that one mass is much heavier):

dE/dt(atom) = −32Gm2

er4

hω6

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

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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[13] [4], 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

or not.

Gravitational Radiation from Within Nuclei

The Sivram - Arun paper Thermal Gravitational Waves[12] is an expansion of Wein-

berg’s results in his 1972 book[13]. 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[12] (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

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conditions only.

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

10−9eV/s.

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[11].

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

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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.

DISCUSSION

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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then be determined to be outside of the realm of quantum mechanics.

[1] Abhay Ashtekar, Martin Reuter, and Carlo Rovelli. From General Relativity to Quantum

Gravity. arXiv:1408.4336 [gr-qc], 2014.

[2] Mohammad Bahrami, André Großardt, Sandro Donadi, and Angelo Bassi. The

Schrödinger–Newton equation and its foundations. arXiv:1407.4370 [quant-ph], 2014.

[3] David Bohm. A suggested interpretation of the quantum theory in terms of "hidden" variables.

I. Physical Review, 85(2):166–179, 1952.

[4] Stephen Boughn and Tony Rothman. Aspects of graviton detection: graviton emission and

absorption by atomic hydrogen. Classical and Quantum Gravity, 23(20):5839–5852, 2006.

[5] Luis de la Peña, Ana María Cetto, and Andrea Valdés Hernández. The Emerging Quantum.

2015.

[6] Arthur S. Eddington. The Propagation of Gravitational Waves. Proceedings of the Royal

Society of London A: Mathematical, Physical and Engineering Sciences, 102(716):268–282,

1922.

[7] André Großardt. Newtonian self-gravity in trapped quantum systems and experimental tests.

arXiv:1702.04309 [quant-ph].

[8] C. Møller. Les theories relativistes de la gravitation. Colloques Internationaux CNRS, Paris,

91, 1962.

[9] R Penrose. On the Gravitization of Quantum Mechanics 1: Quantum State Reduction. Found

Phys, 44:557–575, 2014.

[10] L ROSENFELD. ON QUANTIZATION OF FIELDS. Physics, Nuclear Co, North-holland

Publishing, (February):353–356, 1963.

[11] G Rosi, F Sorrentino, L Cacciapuoti, M Prevedelli, and G M Tino. Precision measurement of

the Newtonian gravitational constant using cold atoms. Nature, 510(7506):518–21, jul 2014.

[12] C. Sivaram. Thermal Gravitational Waves. arXiv:0708.3343, 2007.

[13] Steven Weinberg. Gravitation and cosmology : principles and applications of the general theory

of relativity. Wiley, 1972.