I’m getting ready for the FFP10 meeting later this month. In reading the abstracts of those who will be giving talks or posters, I came upon “Analyses of the 2dF deep field” by Chris Fulton, Halton Arp and John G. Hartnett. The abstract is about the relationship between low redshift and high redshift astronomical objects. The claim is that some quasars have redshifts that do not give their true distance; instead, they are much closer. Looking on arXiv finds: The 2dF Redshift Survey II: UGC 8584 – Redshift Periodicity and Rings by Arp and Fulton.

If these high and low redshift objects actually are related, this places doubt on the Hubble relation. In addition, when low and high redshift objects appear to be related, their redshifts are related by quantum values . From observations, Arp has proposed that quasars evolve from high to low redshift, and finally become regular galaxies.

Now for quasars to have redshifts that differ from their true distances implies that their redshifts are determined gravitationally; that is, what we are seeing is partly the redshift of light climbing out of a gravitational potential. And if these redshifts are quantized, this gives a clue that the structure inside the event horizon of a dark hole is not a simple central singularity but instead there must be repetitive structure.

In a classical dark hole, the region inside the event horizon can only be temporarily visited by regular matter. Even light cannot be directed so as to increase its radius in this region. Let’s refer to this region as the “forbidden region” of the dark hole as it is near the central singularity. For the classical dark hole, this includes everything inside the event horizon. We will be considering the possibility that the forbidden regions of a dark hole occur as infinitesimally thin shells, and that between these shells, light can still propagate outwards:

Forbidden regions shown in red.

Event Horizons as Quantum amplitudes

If we were looking for a quantum mechanical definition of the inside of a dark hole, we could define it as the region where particles have a zero probability of moving outwards. We could say that the transition probability for the particle moving outwards is zero. However, in quantum mechanics probabilities are defined as the squared magnitudes of complex amplitudes. The way we compute transition probabilities is from complex transition amplitudes. If the transition amplitude between two states is zero, we say that they are “orthogonal”. Zero transition amplitudes correspond to points where a sine wave mechanics is zero; at these points, deviations to either side give nonzero transition amplitudes:

How to get zero probabilities from nonzero in QM.

In quantum mechanics, it’s never the case that making steady changes to a state or pair of states causes their transition amplitude to become zero and then remain zero. Instead, upon obtaining zero transition amplitudes by arranging for a parameter to continuously change, we find that continuing to change the parameter causes the transition amplitude to become nonzero. For this reason, from basic quantum mechanical principles, it would be surprising that the forbidden regions of a dark hole should have more than an infinitesimal thickness.

For the parameter that continuously changes, so as to create a forbidden region, we can propose the flux density of gravitons. This will be maximal at the central mass concentration, and then decrease, approximately following a 1/r^2 law, as one moves away from the center. Along this line, my paper that won an honorable mention at the annual gravity essay contest, provides calculations showing that this flux has to interact with itself in order to match tests of general relativity, see The force of gravity in Schwarzschild and Gullstrand-Painlevé coordinates, 0907.0660.

My paper on Spin Path Integrals

Now the paper I’m discussing at FFP10, Spin Path Integrals and Generations, proposes that the spin-1/2 elementary fermions have internal structure. The idea is that the left (or right) handed electron oriented with spin in the (1,1,1) direction, has three components, R, G, and B, which we will assume are oriented in the +x, +y, and +z directions. This behavior appears only at extremely short times; we don’t have sufficient energies in our experiments to observe it. However, the three degrees of freedom create three orthogonal particles which we do observe as the three generations. The paper discusses various examples of this. If you do all this, it turns out that spin and position end up with similar Feynman path integral behavior. This is a way of unifying the internal and external degrees of freedom of the elementary fermions.

I chose the +x, +y, and +z directions because they are taken from the three “mutually unbiased bases” of the spin-1/2 algebra, that is, the algebra of the Pauli spin matrices, the Hilbert space of two dimensions. Mutually unbiased bases are the finite dimensional version of complementary variables such as position and momentum. This is why they are used to make spin path integrals and position path integrals have similar behavior; we are taking the ideas behind position and momentum and applying it to spin.

Mutually unbiased bases are defined by requiring that transition probabilities be equal. Thus, if we are to include gravitation in this sort of theory we must assume that the effect of gravitons is to change the transition probabilities.

The paper is about the transition amplitudes and long term propagators for spin moving between orientation in the +x, +y, and +z direction (or any other three perpendicular directions). This is a model of the left or right handed electron; to get a complete model of the electron we have to allow for transitions between them. How can we modify these transition probabilities in order to model the effect of gravity?

From my paper, it’s clear that the (gravity modified) transition probabilities between +x, +y, and +z will have to stay the same. Thus we will assume that gravity modifies the transition probabilities between the states contributing to the left handed electron, say +x, +y, and +z, and the states contributing to the right handed electron, -x, -y, and -z.

If these transition probabilities are modified (but the ones among the + or – states remain unchanged), the requirement that we use a complete set of mutually unbiased bases amounts to our modifying the six states so that, for example, the +x, +y, and +z are moved closer together, and the -x, -y, and -z are moved farther apart.

Effect of gravity is to warp the MUBs.

The paper doesn’t discuss this, but the natural interpretation of the +x, +y, and +z states is that the particle is moving at some speed in these various directions. Since the maximum particle speed is c, that speed must be c sqrt(3). This result is similar to that of Feynman’s checkerboard model of the Dirac equation in 3+1 dimensions. See around equation (42) of Peter Plavchan’s informal paper Feynman’s Checkerboard, the Dirac equation, and spin.

With this interpretation, the effect of warping the mutually unbiased bases is to effect the maximum speeds of the particle in the +(1,1,1) and -(1,1,1) directions. For the illustration above, the particle speed in the -(1,1,1) direction has been slightly increased while the speed in the +(1,1,1) direction has slightly decreased. For a dark hole, this would correspond to different speeds for the radial inward -(1,1,1) direction as opposed to the radial outward +(1,1,1) direction.

These calculations are easy to do with the particle’s spin oriented parallel or antiparallel to the gravitational force. To obtain more arbitrary directions, one uses linear superposition in a manner similar to how spin-1/2 can be written in terms of spin-up and spin-down.

Gravity as changes to velocity
These sorts of ideas about gravity, that it should be interpreted as a modification of the natural velocities of particles with differences between inbound and outbound particles, is used in the important paper by Andrew J. S. Hamilton and Jason P. Lisle, The River Model of dark Holes, Am.J.Phys.76:519-532,2008. This paper models rotating and non rotating dark holes as a river of “space” that is sucked into the dark hole. The river defines a velocity at each point in space and from this one can derive the various properties of dark holes.

The non rotating coordinates used in Hamilton and Lisle’s papers are Gullstrand-Painleve, the same used in my paper on flux gravity. These coordinates, along with their rotating (and charged) generalizations, are unique in that they allow general relativity to be rewritten in terms of David Hestenes’ geometric algebra. This amounts to getting rid of the tensors general relativity is usually defined with, and replacing them with functions of Dirac’s gamma matrices. This is a particularly useful version of general relativity because Dirac’s gamma matrices are used to model the elementary fermions. The Cambridge geometry (geometric algebra) group has many papers giving calculations of electrons in dark hole coordinates using these methods.

Redshifts are sort of multiplicative; to compare them we look at ratios of (1+z). For example, if an object has an intrinsic redshift (due to gravitation) of (1+y), and it is at a distance or moving with a velocity that gives a redshift of (1+x), then its total redshift is (1+x)(1+y).

The claim of the redshift quantization folks is that redshifts are quantized according to a factor of (1+z) = (1 + 0.23) . The factor 0.23 is suspiciously close to 2/9, the factor (prominent in my spin path integral paper above) which Marni Sheppeard and I call “that damned number”, which I’ve assumed comes from a sum over infrared divergences. These arise when considering particles with very small energies or very long distances.

Quasar redshift quantization

Quasar redshifts are observed by emission and absorption lines. In order for these to be gravitational rather than due to distance, the light we see, and the atoms responsible for the emission and absorption, have to originate deep inside event horizons of dark holes. As far as arranging for the light to escape, this is not too difficult: Since the forbidden regions are thin, light need only tunnel through it in order to carry the imprint of the inner layer.

Inside the most central forbidden region, matter is not constrained to fall to the singularity so this region will consist of normal matter. Provided temperatures are sufficiently low, normal matter can exist in this region. This matter will emit light, providing emission lines, and imprint light with absorption lines. The light then works it way out of the quasar.

For all this to happen, the temperature of the dark hole has to become sufficiently low that normal matter can exist deep with the forbidden regions. This can only happen if the dark hole is sufficiently cool. Arp’s observation is that new quasars have high redshift so the implication is that they are cool as they are ejected, and then heat up as they age.

A dark hole is heated by matter falling into it, so a cold dark hole would be a good candidate for eventually heating up and running through the quasar sequence of quantized redshifts. When the dark hole attracts sufficient matter to form a new galaxy, it has been sufficiently heated that its quantum structure becomes hidden.

As far as evidence, it might be possible to find a quasar with two sets of emission / absorption lines. This would happen when a dark hole has cooled just sufficiently to expose two different forbidden regions. This would be observed as a single quasar with two different redshifts.

Cosmology

The cosmology implied by this model is one where the effects of the big bang and dark energy are due to changes in the background level of graviton flux. The effect of gravitons on matter is to change the probability amplitudes in such a way that, at low levels, the probability of a transition is increased. The transitions influenced (raised) are those that cause the particle to move in the direction from which the graviton flux arrived.

So over cosmological time, the internal clocks on particles is sped up. In a flat universe, the frequency of light is not changed with time; there is no stretching of space to redden it. However, the clocks of the observer do change and an observer will see ancient light reddened according to how much time has gone by.

It should be noted that when using Schwarzschild coordinates, the clocks of particles are slowed by the presence of a gravitating body. But in the velocity model, this is due to the velocity of those particles (near the speed of light) rather than the gravitons per se. To see particle clocks slowed by gravity, one can instead use Gullstrand-Painleve (GP) coordinates.

In GP coordinates, the particle speeds depend on the direction. For particles on a radial axis, the modified particle speeds are modified in such a way that the difference in velocity between an outgoing particle and an incoming one is still 2c (as in free space). In such a case, a particle moving back and forth between two points will be slowed down, relative to free space. The analogous example given in freshman physics theory classes is that when you travel one direction at 60+v mph, and travel back at 60-v, you will arrive later than someone who travelled at speed 60 both directions.

At the foundation of quantum mechanics particles with mass m have a de Broglie frequency proportional to its mass. If we are to increase the frequency over cosmological time, this is the same effect as increasing the mass. So a cosmological theory compatible with the evidence given in this post would be one where the masses of the elementary particles increase over time.

As it turns out, just such a cosmology theory is propounded by none other than Halton Arp. It’s called the “variable mass theory” and is mentioned in the paper I referenced above, Evolution of Quasars into Galaxies and its Implications for the Birth and Evolution of Matter.

The variable mass theory was originated by Jayant Narlikar. With Hoyle, he developed what is also called “Hoyle-Narlikar” theory. This was sufficiently long ago that it is difficult to find review articles on it. A more recent version is the quasi-steady state cosmological model (QSSC) Cosmology and Cosmogony in a Cyclic Universe, J.Astrophys.Astron.28:67-99,2007 (0801.2965), by Narlikar, Geoffrey Burbidge and R.G. Vishwakarma.

In theories where the internal clocks of particles changes with the background graviton flux, it is also the case that the speed of light (as measured by the particles for propagation between two fixed points) will decrease with time. This is the subject of Louise Riofrio’s cosmology.

Weird Gravitational effects

Finally, I would be remiss without mentioning that theories of gravity that involve graviton flux imply that when planets are aligned, gravity should be a little stronger. This effect, however, can be canceled by absorption of gravitons. It’s not clear what the net effect should be, however, strange behavior of pendulums and gravity measuring devices during total eclipses of the sun have been seen, and not seen. See my blog post, The Moon’s Subtle Influence.