#### Abstract

Is there experimental evidence that Einstein’s relationship $E = \hbar\nu$ holds for gravity? This paper explores the consequences of the non existence of the graviton – that classical gravitational radiation is emitted by all objects in quadrupole motion. The effects of this on the measured properties of the hydrogen atom, along with possibilities to experimentally measure the effects of atomic or nuclear scale gravitational radiation is explored. Experiments similar to those that are measuring ‘big G’ might be able to detect the presence of such stochastic compton – like frequency background gravitational waves. Experiments in this class look feasible with today’s technology, and are thus important tests of quantum gravity.

#### Introduction

Planck’s constant is involved in all quantum mechanical interactions. Much of quantum mechanics can be derived from the relation of the energy of an electromagnetic interaction to its wavelength:

(1) $E = \hbar\nu$ .

This quantum of action, first derived from electromagnetic phenomena, is assumed to be ubiquitous in physics.  Thus the strong and weak forces are also supposed to be governed by this quantum of action using force carriers that are not photons. Actual quantum experiments are harder to do on the weak and strong force, but things like the range of these forces and their overall behaovoir are modelled well using quantum physics. All is good so far.

One of the biggest problems in physics is the quantum gravity problem. There are many possible solutions proposed to this problem, but almost all of them suppose the existence of the graviton. The graviton should have the same energy relation as the photon:

(2) $E_{graviton} = \hbar\nu$

This is an assumption. There not only exists no experimental confirmation of this relationship for gravity, its also widely known that an experiment to detect a single graviton is well beyond the capabilities of any present or future experimentalist.

Of course if $E_{graviton} != \hbar\nu$ then quantum mechanics is incomplete, as the quantization of a field requires that the quantum of action is described by energy relations similar to those above. Given the success of quantum mechanics, $E_{graviton} = \hbar\nu$ is assumed to hold.

#### Gravitational radiation from atoms and nucleons

Einstein in 1916 wrote:

“...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.”

Why did Einstein worry about something (gravitational waves from atoms) that would effect the lifetime of an atom on very long time scales (of order billions of times the age of the universe) vs the tiny amount of time that a classical hydrogen atom would radiate away its EM energy? Furthermore, many orbitals of the hydrogen atom have no quadrupole moment whatsoever. (The shape of the orbitals of the hydrogen atom were not known in 1916 though).

If we look at the energy loss rate of a 1916 style Bohr ‘planetary’ hydrogen atom in the ground state, using Eddington’s [ref 1] formula for the gravitational energy radiated by a two body system (in the approximation that one mass is much heavier):

(3) $dE/dt(classical\:gravity\:bohr\:atom) = -\frac{32Gm_e^2r_h^4\omega^6}{5c^5} = (10^{-43} eV/s)$

Which even over the age of the universe amounts to an energy loss due to gravitational waves for a hydrogen atom of only $1e^{-20} eV$ . This calculation was available to Einstein – whether he performed it or not. Why was he worried about such as small rate of energy loss for a hydrogen atom? In contrast the classical electromagnetic lifetime of the classical hydrogen atom is about $10^{-11}s$ which of course helped lead to the discovery of quantum mechanics.

As a comparison to the above extremely simple estimate, a more formal measurement of the lifetime of the $3_p - 1_s$ state lifetime for emitting a graviton is $1.9e^{38} s$, which compares to within a few orders of magnitude with my estimate above converted to a lifetime of about $10^{42} s$. See the Problem Book in Relativity and Gravitation, problem 18.18. (Solutions given)

This is nevertheless some energy loss, and further gravitational radiation would be expected from the quarks confined to the proton in the hydrogen atom, where a similar calculation using proton dimension, mass and frequencies results in a energy rate of

(4) $dE/dt(classical\:gravity\:proton) = -\frac{32Gm_e^2r_h^4\omega^6}{5c^5} = 1e^{-6}eV/s$

Which is much higher, about  an eV per week per proton. Furthermore this naïve calculation is only likely accurate to within a few orders of magnitude. The proton would not simply radiate energy however as protons tend to exist in the neighbourhood of other protons.  Indeed since all protons are the same, the spectrums of these gravitational emissions would line up and protons would lie in a bath of stochastic gravitational radiation, picking up and losing energy via gravitational radiation, in much same manner as an atom in a gas neither gains nor loses energy on average.

#### Experimental Proposal

If one supposes that protons have their own bath of stochastic gravitational waves to survive in, then experiments to detect the effects of this small amount of radiation might be difficult. If one looks instead at the gravitational radiation of high Z nuclei, then we can get different effects – each element would have its own characteristic spectrum of gravitational waves. Thus experiments similar to those done to look for ‘big G’ might be able to obtain different results by using dissimilar materials for the masses whose force of attraction is to be measured. It is notable that experiments to determine Newton’s constant G have had great difficulty of obtaining consistent results. Most measurements of G do not agree with each other to within the errors determined very carefully by the experimenters. (references on big G searches).

There might be shielding effects that could be measured.

Measuring the torque. The iron (lead would work better) plate is made to be the same density as aluminum by drilling holes or using meshes, etc. Standard physics implies torque should be zero in such a symmetric setup.

The aluminum and lead ‘shielding panels’ would both be the same volume and mass (lots of holes drilled in the lead one). If the lead panel shields the lead more than the aluminum one, then a net torque would be seen. Similar to a fifth force experiment (Ref – in mendely – Search for an intermediate-range composition-dependent force coupling to N-Z).
The aluminum and lead plates in the experiment below might interact differently with the 4 lead test masses. The idea is that the stochastic gravitational radiation from the lead test masses will interact weakly with the plate, creating a small repulsive force that results in less attraction than in the case with the aluminum plate in experimental sketch below.
What is the expected value of the torque assuming that there exists a stochastic gravitational wave background idealized as being in two frequency bands? The stochastic gravitational frequencies of the aluminium nuclei are assumed to not match those of the lead. What is the power involved? Treat an lead nucleus as having several nuclei orbiting it at the radius of the nucleus at some internal velocity of an lead nucleus.

Lead nucleus – speed of nucleons is (20 MeV kinetic energy) and say one pair is radiating Gravitational waves: r = 1 fm, so

I get about $10^{-25} watts$ or so. (using this) . The rough model is that nucleons are moving about in the nucleus, and at times have a quadrupole motion, which is on the order of a bar of mass 2 nucleons, spinning about a fm apart at the $10^{23} hz$ of the nucleon rotational period in a fermi gas model nucleus. (Note that the Sivaram and Arun paper about thermal gravitational radiation from neutron stars shows about a billion times less than this, since it deals with nuclei passing each other at relatively slow speeds and large distances).

Using Pressure = E/c , where E is in Watts/metres^2 and 1e-25 watts per nucleon emitted, with complete absorption. (the cross section is assumed about the physical size of the nucleon, which is also the gravitational wavelength). This comes out to 10^-10 newtons.

Taking 1e-25 watts – which is 10e-7 eV/second, calculate the pressure between two 10kg masses 0.1 metres apart – about  10^-10 newtons. So for the experiment outlined above, we get forces in the range of 10^-10 Newtons, and for the Aluminum side, a much smaller force – since there is no interaction with the plate. The difference is about equal to the last digit in ‘big G’ experiments. Thus current big G experiments may be effected by this effect, as they don’t setup the experiment in a neutral manner, but seek to measure the force. In other words the various vacuum chambers (some constructed of Aluminium, some from stainless steel) and other experimental details like the composition of the test masses may be affecting the outcome of these big G experiments.

In order to get an adequate cross section for these short wavelength gravity waves we need a thickness of material such that a nucleus is in any line of sight through the iron. Assuming that the cross section is about the physical size of the nucleus that works out to about 10cm requirement for the mass sizes. i.e. the effect will maximize when the masses are 10 cm in size and the plate is also about 10cm in size. (http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/crosec.html#c1)

The normal attractive Gravitational force between 2 10kg masses at 0.1m apart is 6.7e-7 Newtons. So the size of the effect proposed here comes in at something like the uncertainty in Big G measurements.

This force created by simply assuming that the nucleus can be treated classically for gravitational waves. The nucleons generate GWs which are can be absorbed by another nucleon of the same kind.

Almost all quantum physics experiments done to date have used electromagnetic interactions.

http://physics.stackexchange.com/questions/10582/the-energy-of-a-graviton

#### What about the ultraviolet catastrophe ?

Does that not prove the existence of gravitons?

On the electromagnetic side, classical physics predicts that hot objects will instantly radiate all their heat as electromagnetic waves. This is one way to appreciate the Ultraviolet Catastrophe. Yet even the accepted 1905 electromagnetic collapse of the hydrogen atom is in question as the field of Stochastic Electrodynamics has shown. (Timothy H. Boyer 2015 arXiv)

Since gravitational radiation is ‘the same as electromagnetic radiation’, a similar effect should happen for gravitational waves. But it does not in the model presented here. Look at the classical radiation of gravitational waves by a hydrogen atom – $(10^{-43} eV/s)$ . Hardly a catastrophe. Even in the subatomic realm other physics gets in the way of having an ultraviolet catastrophe via gravitational wave radiation. One way to look at this avoidance is that the laws of quantum mechanics in the electromagnetic (and strong) structure of the micro-world physically save the ultraviolet catastrophe from happening in the GR world. So GR does not need quantization to avoid run away ultraviolet effects. See also a 1984 paper by Lee Smolin:

Because a state of thermal equilibrium involving gravitational
radiation cannot be reached in a finite time from a generic
initial configuration there is no ultraviolet catastrophe
for gravitational radiation. That is to say, we cannot
argue, as did Planck and Einstein for the case of the
electromagnetic field, from the existence of an equilibrium
state involving radiation and matter to the necessity that
the energy of the field is carried and transferred to matter
in quanta satisfying $E = \hbar\nu$ .

Another more speculative way of looking at this is that electromagnetism arose from GR as an emergent phenomena as run away ultraviolet effects tried to happen in the early universe. Its a case of electromagnetism arising as a defence mechanism against a ultraviolet catastrophe. Order from chaos and all that.

Classical physics predicted that hot objects would instantly radiate away all their heat into electromagnetic waves. The calculation, which was based on Maxwell’s equations and Statistical Mechanics, showed that the radiation rate went to infinity as the EM wavelength went to zero, “The Ultraviolet Catastrophe”. Plank solved the problem by postulating that EM energy was emitted in quanta

#### Emergent Quantum Mechanics

In many theories of emergent quantum mechanics, quantum behaviour emerges from an underlying media. {references}. In other words the underlying media for quantum mechanics is not governed by quantum mechanics. Since this author is proposing that quantum effects emerge from classical general relativity,  gravity should not be quantized, $E_{graviton} != \hbar\nu$, and any quadrupole motion should produce gravitational waves.

1. A.S. Eddington, Proceedings of the Royal Society (London) A 102, 268 (1923).
4.