# Discrete Space, General Relativity, Dark Matter And The Casimir Effect

In an earlier blog, Marcelo discussed Dark Matter and stated that, with the exception of hopes of exotic particles called WIMPs, (Weakly Interacting massive particles) which have not been found, we have no idea what Dark Matter might be.

Fools rush in where physicists, no doubt wisely, do not tread. That said, I wish to propose concepts that may prove helpful with the issue of Dark Matter, relate to the hypothesis of discrete space on the Planck length scale that I discussed two blogs ago, seem to have potential relevance to General Relativity and perhaps quantum gravity, and appear to be testable now using the Casimir effect. I shall predict that the strength of the Casimir effect is weaker in a local gravitational field exactly in the proportion that General Relativity shows that space-time curves locally in a gravitational field.

While the ideas are novel, and of course I am not a physicist, they have the virtue of being simply testable.

What is Dark Matter? If one observes the rotation of galaxies or clusters of galaxies, the outer margins of the galaxies, or clusters of galaxies, are rotating too rapidly to be accounted for by either Newtonian or Einsteinian gravity. There is essentially no doubt about this excess velocity. To explain it, physicists have for years postulated the unknown Dark Matter in the outskirts of a galaxy or cluster of galaxies, in a “squashed sphere” hovering about the galaxy or cluster of galaxies.

My attempt to offer a new line of thought about Dark Matter begins with the postulates of my blog on Discrete Space and Dark Energy, two blogs ago. There, borrowing the general idea from Loop Quantum Gravity, I proposed that on the Planck length scale of 10 to the -34 power centimeters, space is discrete. Again, but not limited by, Loop Quantum Gravity, I supposed that these discrete units of space were tetrahedra that could not interpenetrate, or else space is again continuous. And I supposed that tetrahedra cannot expand. The latter implies that space must “clone” itself, like bacteria, giving rise to an exponential growth of space, hence the start of a theory of Dark Energy. I further proposed that when a new tetrahedron is “born”, new energy comes with the tetrahedron “from nowhere”, thus keeping the energy density of space a constant.

You may cavail at energy entering space units “from nowhere” but we accept energy from nowhere in the Big Bang, with no explanation, so one magic is no worse than the other. In addition, my physicist friends say that with an expanding universe, matter, energy, and gravity, energy is not conserved for the universe as a whole in any case.

Finally, I proposed that space units are quantum degrees of freedom with alternative topologies connecting the nodes of the tetrahedra. Because space has energy, it has mass, thus will decohere at some space scale considering a volume of space a “system” and surrounding space as an “environment. Then quantum space becomes classical because quantum space decoheres.

Now, on to considering a testable hypothesis for Dark Matter.

I begin with General Relativity and the deformation of space in the vicinity of a mass, M. General Relativity speaks in terms of a space “metric”. In the absence of masses, M, space is thought to be flat. One uses two coordinates, dR and dL, where dR is the flat space metric and dL is the curved space metric.

In General Relativity, one can define dL in relation to dR as a massive object, M, is approached from a distance. What occurs is that space is curved into a funnel shape, ever steeper as the center of mass of M is approached.

In terms of dR and dL, as the center of mass M is approached, one can plot dL/dR for each small change in either coordinate. One obtains the funnel described above, where for initial small changes in R, dR, far from M, changes in L, dL are approximately equal to changes in R, dR. That is, space is locally almost flat. As the center of mass, M, is approached, small changes in R, dR, are associated with large changes in L, dL, yielding the funnel described above. In short, as the center of mass is approached, General Relativity states that space is curved in such a way that, to accommodate large changes in the metric, dL, as M is approached, *space is stretched* as the center of mass of M is approached.

How can General Relativity be brought into accord with the hypothesis that space comes in Planck scale discrete units that can neither interpenetrate nor stretch, ie expand?

One possibility seems a bad idea: if space cannot expand, perhaps the stretching of space in the vicinity of a mass, M, requires the generation of more tetrahedra, proportional to the stretching.

This seems a terrible idea, for there is no physical basis then for the “stretching” of space near M.

I turn then to the alternative simple hypothesis. But it has many consequences.

Marcelo pointed out with respect to my blog on Discrete Space, Dark Energy, ...which yielded a Cosmological Constant in the rate at which space clones itself so expands exponentially, that the Cosmological Constant is rich in physical interpretation. Most critically, the Cosmological Constant is identical to the 0 point quantum fluctuations of quantum field theories.

Then I propose a very simple postulate: The local curvature of space is identical with local changes in the 0 point energy of the vacuum, hence with local changes in the Cosmological Constant. In particular, the local stretching of space demanded by General Relativity in the vicinity of a mass, M, is exactly equal to a local proportional *decrease* in the 0 point energy of the vacuum, hence of the Cosmological Constant. This hypothesis proposes a possible underlying quantum “mechanism” for the curvature of classical space in General Relativity.

Why can this hypothesis explain Dark Matter? To my knowledge, vacuum energy is not accounted for in the excess rotational velocity of the outer margins of galaxies or clusters of galaxies.

But if the hypothesis I propose is correct, as the center of mass of a galaxy or cluster of galaxies is approached, the vacuum energy density, ie the 0 point quantum energy, ie the Cosmological Constant, *decreases *in proportion to the “stretching” of space given by the metric of General Relativity. Thus there is less energy per unit of space where it is stretched by the metric near the center of mass of the galaxy than at its outskirts where space is nearly flat. Then if there is less energy near the center of mass and more in the outskirts of the galaxy or cluster of galaxies, by E = MC squared, there is less mass due to vacuum energy near the center of the galaxy, and *more vacuum energy mass* in the outskirts of the galaxy. This is my proposed explanation of the *dark matter* postulated in the peripheries of galaxies or clusters of galaxies.

One can consider a further idea: If the increased mass on the periphery of the galaxy or cluster of galaxies must itself be in an orbit to account for the increased rotational velocity, one can think of wave packets of energy (hence mass) flowing in orbit around the galaxy or cluster of galaxies.

One can consider a second idea: Relax the assumption that the volume of a discrete unit of space cannot expand. Then postulate that the expansion of a unit of space is proportional to its 0 point energy. Then as space curves and 0 point energy per unit space falls, units of space expand proportionally.

**The Casimir Effect**

The hypothesis I put forth can, I believe, be tested by the Casimir Effect which should be weaker in a strong gravitational field that far from such a field. More, the hypothesis of discrete space on the Planck scale may help with an embarrassing infinity in the formulation of the mathematics of the Casimir effect which is “solved” by a questionable “renormalization”.

The Casimir Effect was predicted and confirmed in a setting in which two large conducting metal plates are very close to one another in a vacuum. A close distance is a micrometer or less. Casimir realized that for a quantized field as is common throughout quantum field theory, one could think of the field as a set of quantum oscillators, one at each point in space. Now these oscillators can harbor waves, but the waves must match the boundary conditions of the two metal plates.

The predicted attractive force between the plates, the Casimir Effect, has been measured with two parallel plates and with a plate and a large sphere one of the surfaces of which is near the first plate. It is a well confirmed effect.

But there is a critical problem. If space is continuous, then any possible wavelength meeting the boundary conditions, from a micrometer long to* arbitrarily short *wavelengths, can fit into the vacuum between the plates. The total energy of the Casimir effect should be the sum of the energies of these wavelengths. Each wavelength has an energy which is inversely proportional to its wavelength. The sum is, unfortunately, infinite because wavelengths can become ever shorter and of ever higher energy.

In standard quantum field theory, this is “handled” with a “renormalization” that many physicists are not comfortable with.

Now consider the hypothesis that space is discrete on the Planck length scale. Then there is a natural cutoff of wavelengths at this length scale. No shorter wavelengths can occur and the sum of energies is FINITE. This is, of course, an argument for discrete space, but not a conclusive one. The calculations have been done, reported for a general audience I believe in Leonard Suskind’s “The Cosmic Landscape”. If I remember correctly, with a cut off at the Planck length scale, the energy is still much too high.

The hypothesis for Dark Matter which I suggest above, has an experimental consequence, even without detailed calculations: The Casimir Effect should be less in a strong gravitational field than in a weaker field. This is rather simply testable by measuring the Casimir effect on the surface of the Earth, and in space at different distances from the Earth, including the position of balance between the lunary and the earth’s gravitational fields.

I close with the obvious caveats: I am not a physicist. Physics is a richly interwoven skein of consistent theory, and some crude steps. I am much more likely to be entirely wrong than even partially right. For example, consider the central hypothesis relates local curvature of space to a corresponding decreases in the 0 point energy of a quantum field. This seems radical and may of course be entirely wrong. I do not know the ways the 0 point energy of a quantum field is locally experimentally testable directly. But the Casimir effect would seem to measure the 0 point energy of a local volume of vacuum, so seems a good test of this hypothesis.

The theory I propose in outline seems to contain a conversion of space to matter and energy, for it relates the local curvature of space to a local loss of 0 point energy, hence mass. Thus when space is “stretched” by a nearby mass, M, the energy per volume of space decreases. Then space, matter and energy are all inter-convertable. Since on this proposal, a stretching of space is related to a proportional decrease in energy per unit space, it would be wonderful if “space, matter and energy” are conserved together. I confess I do like this possibility a lot.

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