David J Reynolds and Bryan W Doughty

david @ isoprano.net

Experiments using the inverse beta process to detect solar neutrinos have discovered that there is a significant shortfall in the number of neutrinos detected compared to the amount expected from the standard solar model. It would appear, therefore, that there is either a mistake in our understanding of the method of energy generation in stars in general and our sun in particular (the standard solar model) or an error in the underlying physics of solar neutrino emission. In this paper the basic physics underlying the propogation of neutrinos through a gravitational field is examined and an explanation proposed.

The energy source of the sun has long been assumed to be nuclear fusion. The basic reaction is the creation of an alpha particle, two positrons (e+) and two neutrinos (v) from the fusion of four protons:

4p -> a + 2e+ + 2v (1)

This reaction takes place in a number of steps known as the proton-proton chain:

Number Reaction Solar Terminations / %

1 p + p -> 2H + e+ + v 99.75

2 p + e- + p -> 2H + v 0.25

3 2H + p -> 3He + g

4 3He + 3He -> 4He + 2p 86

or

5 3He + 4He -> 7Be + g

6 7Be + e- -> 7Li + v

7 7Li + p -> 2 4He 14

or

8 7Be + p -> 8B + g

9 8B -> 8Be* + e+ + v

10 8Be* ->2 4He 0.02

However, it is difficult to test our understanding of solar processes since the region in which thermonuclear fusion is assumed to take place, that is the suns core, is hidden by thick layers of cooler material. Of the particles released by these reactions only one, the neutrino, can escape into space. The study of solar neutrinos is therefore the only practicable way we have, at present, to study the processes by which the sun (and other stars) shine.

Neutrinos may be captured by a process known as the inverse beta process, an example of which is the inverse of the electron capture decay of 37Ar:

v + 37Cl -> 37Ar + e- (2)

This process was used in the Brookhaven solar neutrino detector the results from which (and subsequent experiments) have given rise to an anomaly1. The results show an apparent solar neutrino production rate of only about a fifth of that expected from the standard solar model. There are thus two possibilities, on one hand there could be an error in our understanding of solar processes whilst on the other there could be an error in the underlying physics. If our understanding of solar energy production is correct, and I believe it is , then it follows that there must be an error in the accepted physics of solar neutrino emission.

Let us remind ourselves of the conventional view of the effect of gravity on light. Consider a photon emitted by an atom in a uniform gravitional field described by the acceleration, g. This photon possesses a gravitational mass E0/C2, falling through a distance, d, the photon gains energy:

(3)

Where E is the actual energy, E0 is the emission energy and C is the speed of light. We can also describe the energy in terms of Plancks constant, h:

(4)

where f is the frequency of the light. We can now compute the frequency shift:

(5)

Which has apparently been verified by experiment.

It is implicit in this calculation that the frame of reference defined by the composite gravitational field provided by all relevant matter (that is matter which has gravitational influence over the coordinate in question) in the universe, that is the inertial frame, is the same as the frame of reference defined by the space through which the photon is travelling. It is here postulated that this is not so and that the inertial frame is distinct from the frame of reference defined by the space through which the photon is travelling. In this case, the acceleration of space itself, what we call gravity, is defined relative to the inertial frame. If this second postulate is applied then the behaviour of light in a graviational field is subtly different and it is this difference that could provide an explanation for the apparent shortfall in detected solar neutrinos.

Going back to our photon let us change views and think about a wave of wavelength l and frequency f. The speed of light can be expressed:

(6)

but we are now in a gravitional field so how will this affect our lightwave ? Well, gravity is a spatial distortion which attracts all matter with an acceleration, g. It also attracts light but, in the conventional view, shifts its frequency and wavelength, but leaves its speed unchanged.

Applying the new postulate, it can be shown that light accelerates under the influence of gravity. This can be viewed as the space in which the light wave is travelling, accelerating at a rate of g and the propogation of the light wave operating with respect to the frame of reference defined by the composite gravitational field rather than the space through which it is travelling. If this is true then the space through which the photon is travelling will move distance, dl, during the period of a single wave. This allows us to compute an increase in the wavelength:

(7)

which is a Doppler shift, just like the conventional view predicts. However, now the increase in energy is manifested as an increase in the unitary energy per wave (in other words Plancks constant) rather than an increase in the frequency of the radiation which now remains constant. The inference of this is that the speed of light has now increased, accelerating under the influence of gravity in the same way as matter. We can express this increase as

(8)

If we now subsitute equation (7) into equation (8) we obtain a new expression for the speed of light:

(9)

which rearranges to give:

(10)

If we now replace the lf term, which refers to the free space (g=0) value of the speed of light, with C0 :

(11)

which we can extend to the general case:

(12)

where is the angle between the gravitational acceleration vector and the light velocity vector.

We now have derived a second term in the expression for the speed of light which depends both on the graviational acceleration and on the frequency of the radiation. But is this view consistent with the observed evidence, after all the Doppler shift predicted by the conventional view has been confirmed by experiment.

Well, the expression for frequency in equation (5) is an inference from wavelength based on the implicit assumption that the speed of light is unaffected by gravity. The theory advanced here predicts the same shift in energy manifested as an increase in wavlength rather than frequency. Therefore the Doppler shift evidence will support both points of view since if Plancks constant is assumed fixed and the speed of light were assumed constant any increase in the wavelength would be viewed as an apparent increase in frequency.

For example, in reference 2, Pound and Rebka confirmed the Doppler shift for gamma ray photons, of frequency 3.5 x 1018 Hz, over a distance of 22.5 m, to be the predicted Df/f=2.45 x 10-15 to within 10 per cent. The shift predicted by this theory gives a value of Dl/l=2.45 x 10-15 which, given the assumption of fixed speed of light and fixed Plancks constant, would appear to be an increase in frequency of Df/f=2.45 x 10-15.

Note that for the purposes of the previous discussion the definition of the speed of light in a vacuum as a constant has been ignored.

How would this affect a neutrino ? Wave particle duality dictates that a neutrino will , in its own frame of reference, have a frequency related to its rest mass via Plancks constant. It is known that the rest mass of a neutrino is very small and has been determined to be less than 60eV although how much less is not known.

Equation (12) shows that the suns gravitational field will cause a situation in which the speed of light becomes negative for outward bound particles below a critical rest mass according to:

(13)

where E is the rest mass and h refers to the free space value of Plancks constant. Note that since this is a per wave cycle effect it depends on the rest mass rather than the total energy. Physically this would be a situation where the space through which the particle was travelling moving towards the centre of the sun faster than its wave aspect can form, effectively turning the wave back to front and reversing the motion of the particle.

The result of this would be that the ability of the neutrino to escape the sun and be detected here on Earth may be directly related to where the originating thermonuclear reaction takes place. Neutrinos failing to escape will oscillate inside the sun according to condition (13) until liberated by solar extinction. It can also be determined that the rest mass of the neutrino would have to be very low for there to be a significant effect from the suns gravity.

It has been proposed that the anomlay in solar neutrino production is due to an error in the basic physics of neutrino emission rather than any significant error in our understanding of solar energy production. The conventional view of the behaviour of light and matter in a gravitational field has been modified such that the speed of light is dependent on the prevailing gravitional field and on the frequency of the associated wave. In this case it has been shown that particles below a critical rest mass will be unable to escape from the core of the sun and will remain oscillating inside the sun according to condition (13).

Note that this also provides an elegant solution to the EPR Paradox (since the net frequency of a created particle pair will be zero) and also explains the, conventionally unexplained, extra gravitational deceleration experienced by interplanetary space probes.