During July and August 2008, 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 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 might 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.
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.
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 8 billion years, what's called Dark Energy. Though I suspect for it to have its observed strength and timing, the observable Universe would be a very tiny proportion of the actual Universe. Just as our sun is one of about 100 billion in the Milky Way, and our galaxy one of about 100 billion in the observable Universe, so our observable Universe could also be a 100-billionth of the actual Universe.
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.
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. 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.
MPD/CPT symmetry duality
There's an eiry similarity between the well-known Charge-Parity-Time (CPT) Invariance and my proposed Mass-Parity-Distance (MPD) Invariance. 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.
Under presently known laws of physics, the Universe isn't a self-contained system because it looks different at its largest scale than it does at its smallest. The entire Universe might simply be contained in a speck of dust in another one with different laws. Perhaps the CPT/MPD symmetry duality suggests the Universe looks exactly the same at its largest scale as at its smallest.
So perhaps gravity and electromagnetism started off as exactly the same force. Gravity is simply what we see when we're looking outwards to the edge of the Universe, and electromagnetism is what we see when we're looking inwards to the smallest scale of the Universe. On the inside looking outwards, there's only one instance to look at, but on the outside looking in, we see many instances. From the inside looking out, it looks like MPD-symmetric positive and negative Mass obeying the laws of gravity, but from the outside looking in, it looks like CPT-symmetric positive and negative Charge obeying electromagnetic laws. The other forces, those with finite ranges, would fork from the electromagnetic under low energies, as explained by current theories.
How real is the negatively massed side of the Universe? If the only way we detect its presence is via Dark Energy at the expected strength and distribution, then surely it's no more real than quantum-based possibilities that never actualized in our positively-massed side of the Universe. Quantum uncertainty would therefore exist at the largest scale of the Universe, as well as the smallest. The main actualized effect is that such Dark Energy causes one of the spatial dimensions in our positively-massed Universe to have a downwards direction, perhaps a complement to one of the dimensions in our spacetime having a timelike arrow. It might be difficult to comprehend this, but 500 years ago people knew one dimension of space had a downwards direction, and Copernicus got into a lot of trouble for suggesting it didn't.