Saturday, September 06, 2008

Mass-Parity-Distance Invariance

During July and August, I took a break from China and Tesol and Groovy, visiting my home country, New Zealand. I hoed into a copy of Roger Penrose's The Road to Reality, and came up with an idea to explain Dark Energy...

Negative Mass
Negative mass is usually defined in such a way that Einstein's equivalence principle still holds, where gravitational mass is proportional to inertial mass. This results in some bizarre effects. But while reading Penrose's book, I got an idea on how to define negative mass so that all the positive matter and all the negative fly off in two opposite directions at the Big Bang, with the equivalence principle still holding.

The key is how we calculate the (scalar) distance with respect to some mass. For positive matter, we would continue to use the positive solution to the formula where we square root the sum of the squares of the three spatial coordinates. But we'd introduce an invariance, known as the Mass-Distance Invariance, where we'd use the negative solution to the square root for scalar distances measured with respect to negative masses.

Some consequences of this invariance are:
  • The same vector values for velocity and acceleration would be used for negative mass as for positive mass, but their scalar values would depend on whether positive matter was referenced, or negative matter. Negative matter would use negative speeds and, to indicate increasing speeds, negative acceleration values.

  • A positive-valued g-force (created by positive matter) would still mean attraction for positive matter, but repulsion for negative matter. However, a negative g-force (created by negative matter) would mean attraction for negative matter, but repulsion for positive.

  • When calculating the (scalar) gravitational force between two objects, the square of the distance between them would always be positive, but a positive force is attraction, and a negative force is repulsion. This means two negative masses attract, as do two positive masses, but positive and negative masses repel each other.

  • Such scalar values for force involving negative matter would use negative distance again when calculating energies, resulting in negative energies. Penrose mentions negative energies mess with quantum mechanical calculations, but in the real Universe, this would be OK because positive and negative energies would be partitioned off due to the gravitational effects of the Big Bang.

Therefore, when calculating scalar values in the negatively-massed side of the Universe, we'd use (1) negative distances, (2) multiplied by positive time to give negative-valued speed, (3) multiplied by positive time to give negative acceleration values to indicate increasing speeds, (4) multiplied by negative mass to give positive-valued scalar forces to indicate attraction, (5) multiplied by negative distances to give negative values for energy.


Picturing All This
When picturing such a scenario using the common "matter bends space which moves matter" 2D curved-space picture to model the 3+1D reality in general relativity, the positive matter would be on top of the sheet sinking downwards as before, but the negative matter would be under the sheet, to indicate negative distances, floating upwards, to indicate the negative mass. We can then visualize positive and negative matter each self-gravitating, but repelling each other.


Gravitons
The positive matter would act via left-handed gravitons as before, but the negative matter would act via right-handed gravitions. Penrose, in his description of Twistor Theory, says that there's a problem in the calculations getting left-handed and right-handed gravitons to interact with each other to enable graviton plane polarization, similar to what's possible with electromagnetism. But in my theory, it would be a requirement that left-handed and right-handed gravitons don't interact in any way. This enables both attractive gravity and repulsive gravity to operate at different scales in the same spacetime.

This graviton-handedness has a counterpart in neutrinos, reponsible for the vast excess of matter over antimatter in the observable Universe. So we need to follow the lead of Charge-Parity-Time (CPT) Invariance, and likewise introduce parity invariance, resulting in what I'm now calling Mass-Parity-Distance Invariance, or MPD-invariance.


Dark Energy
Observational evidence of such MPD-invariant negative matter would be an expected after-effect of the inflation of the very early Universe. The modified version of the Big Bang is that the Universe's overall zero energy fractures into equal Planck-distance-separated positive and negative amounts in the first quantum instant of the Universe, then their respective gravitational fields repelled the positive and negative away from each other, resulting in a Big Bang in two different directions along one spatial axis. The actual reason for the Big Bang can therefore be explained by quantum effects.

After the faster-than-light inflation stopped, the right-handed gravitons from the negative matter would be travelling towards the positive matter at the speed of light only, resulting in a time lag between inflation ending and the gravitational repulsion of the negative mass beginning to affect the positive mass with a renewed expansion. This is exactly what happened after about 10 billion years, what's called Dark Energy.


Negative-Frequency Electromagnetism
The photon would behave differently to the graviton. Planck's famous equation states photon energy equals Planck's constant multiplied by the frequency. Negative-energy photons would then have negative frequency, but for a photon this is not the same as changing the handedness (helicity), because photons have both electric and magnetic vectors. Both left-handed and right-handed photons have positive energy, and can polarize. Photons of negative energy/frequency, whether left-handed or right-handed, would have their electric and magnetic vectors swapped around.


Antimatter
Negative matter and antimatter are two separate concepts. Matter and antimatter created from positive energy in normal particle interactions would both have positive mass, similarly negative mass for negative energy. But a virtual particle-antiparticle pair in a vacuum would not only have an overall charge of zero, but also an overall energy of zero, one of the pair having positive energy, the other, negative. Perhaps the particle has negative mass, or perhaps the antiparticle does. This fact could provide a solution to the "hierarchy problem", there no longer being any need for supersymmetric particles to adjust quantum energy values.

The first quantum event of the Big Bang would determine how much energy, positive or negative, is in each side of the Universe. The left-handed gravitons and left-handed neutrinos go one way, their right-handed counterparts, the other. So one half of the Universe is matter with positive mass, the other half, antimatter with negative mass. One spatial dimension of the Universe is thus different to the other two, with homogeneity and isotropy being more local effects.

An alternative shape of the Universe is a four-partitioned one, where positive matter, positive antimatter, negative matter, and negative antimatter fly off in 4 different directions on a plane. This can be visualized with the 2-D saddle-shape for a hyperbolic Universe, with positive matter on top of the sheet, its matter going one way and its antimatter the other, both down the saddle on each side, and negative matter underneath the sheet, its matter and antimatter each flying off up the saddle, at ninety-degree angles to the positive matter and antimatter.


It's been two decades since I finished my undergrad maths degree, and I haven't used it since, so I'm rusty. And although I basically followed the maths in Penrose's book, I didn't get all the intricacies of manifold calculus and bundles and Langrangians. If anyone out there fills my wordy explanation of MPD-Invariance with numbers, let me know if it works or if it's rubbish. But there's more follow-on ideas I've had...

The Universe as Two Complex Planes
There's an eiry similarity between the well-known Charge-Parity-Time (CPT) Invariance and my proposed Mass-Parity-Distance (MPD) Invariance. I think it suggests a certain structure to the Universe alluded to by Penrose in his Twistor Theory. He suggests the Universe can be modelled as three complex planes (i.e. 6 real dimensions), the "imaginary" dimension being as physically real as a "real" one. But elsewhere Penrose says if there are only 4 observational dimensions of spacetime, we shouldn't try to model them with 11 or 26 dimensions. I'd suggest the Universe can be modelled as only 2 complex planes to match the 4 observational dimensions of spacetime. The extra 2 dimensions required by Penrose's model could come from the fractal dimensions created by those 2 complex planes.

A curve on a complex plane usually has a (Hausdorff) dimension of 1, but fractal curves have a dimension higher than 1, but less than or equal to 2. Only very special fractals, such as the Mandelbrot set and Julia sets, have Hausdorff dimension of 2. If there exist on any complex plane an (aleph-zero-)infinite number of concentric Hausdorff-dimension-2 sets, then I suspect the plane itself would have Hausdorff dimension 3. The union into a manifold of two such complex planes would have Hausdorff dimension 6, while only having topological dimension 4, thus satisfying Penrose's minimal number of dimensions to model our Universe.

We can create such an arrangement on both our complex planes by relating them together using an uncertainty relation. Because the Mandelbrot and Julia sets are the only sets I know of with Hausdorff dimension high enough to be valid in this model, I'll use the Mandelbrot set as an example. The basic set is only one connected curve on the complex plane, but when a computer calculates it, many circles of various colors are usually displayed to reflect different accuracies of calculation. These circles are concentric. Although only the infinitely accurate Mandelbrot set normally has any mathematical significance, when relating two complex planes together in an uncertainty relationship, the curve generated from each accuracy level takes on significance.


Relationship Between CPT and MPD Invariances
Mass would be modelled as one of the fractal dimensions, while charge modelled as the other. The two invariances, CPT and MPD, both of them having parity (i.e. space reflection) included, bear a vague resemblance to the requirements for 2n-D real manifolds to be treated as n-D complex manifolds under the Newlander-Niremberg theorem, in this case 4 real dimensions as 2 complex planes. One plane, required to be CPT-invariant, would have time as one dimension, say, the real. The imaginary dimension would be a dimension of space, and the fractal dimension, charge. The other complex plane, required to be MPD-invariant, would have the other two dimensions of space for its real and imaginary dimensions, and mass for its fractal dimension.

Planck's constant defines the uncertainty relationship between time (i.e. the real dimension of one complex plane) and energy (i.e. a proxy for mass, the fractal dimension of the other complex plane). This would be the uncertainty relationship that makes the complex planes have (Hausdorff-)dimension-3.

The other dimensionless constants of nature could be interpreted as observational coordinate mappings between dimensions on these two complex planes. The speed of light is a mapping between the time and space dimensions on the same plane. Newton's gravitational constant is a mapping between the (fractal) mass dimension and a space dimension. Coulomb's constant is a mapping between the (fractal) charge dimension and space dimension. The three space dimensions wouldn't need mapping between one another, as their differences from one another are only apparent in the helicity of the graviton and neutrino. So the four dimensionless constants would be sufficient mappings for the two planes.


Everyday Observation
I've ignored the forces without an infinite range (the strong and weak forces) in this model. The basic difference between MPD-invariant gravity and CPT-invariant electromagnetism is that in gravity, like masses attract while unlike ones repel, whereas in electromagnetism, like charges repel while unlike ones attract. The logical effect of this (ignoring finite-range forces) is that gravity's masses are real numbers, while charges are polar.

So we have two complex planes, each with three dimensions, i.e. real-imaginary-fractal. The first has Time-Distance-Charge, the second, Distance-Distance-Mass. Perhaps, in our own everyday observation of these planes, the charge, having polar (i.e. 0 or +1 or -1) values only, doesn't require its total dimensional freedom to operate, and only needs a Planck-distance portion of the (fractal) charge and (imaginary) distance dimensions. So the second complex plane "takes" the excess distance dimension from the first plane to create 3D-space, and the mass, having aggregative values, also "takes" the excess fractal freedom of the charge. So we end up with 1D-time, 3D-space, polar-charge, and aggregative-mass.


If I had the time, I'd be looking at the maths for relating two complex planes together, each with (Hausdorff-)2D fractal curves, using an uncertainty relationship, trying to derive relativity axioms and asymmetric time and such stuff. But I've now got other demands on my time, hence this blog entry. If my description rings a bell with anyone, let me know how it goes.

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