General Relativity
The mathematics of curved spacetime.
We have now defined all possible 2D rotations within 4D \HH flat spacetime, standard model, quantum mechanics, matter and antimatter, we must now extend it to 8D \OO curved spacetime to further define gravity, dark matter and dark energy.
Matter and antimatter being shown by the alternating blue squares (each square representing 4D \HH standard model). Dark matter being represented by the alternating red squares and dark energy everything beyond the light-like and vacuum-like \HH, so stretching the universe in all directions. Note that the 4D \HH standard model particles of matter, have no connection to dark matter except via 4D \pm \infin light-like and vacuum-like rotations, dark energy via 4D \pm \infin light-like and vacuum-like linear stretching.
Thus dark matter must be fermionic like matter, and dark energy bosonic similar to light. Similar to what we had previously, there can only be vacuum-like connectivity between matter and dark matter, and light-like connectivity between matter and dark antimatter.
Here we see the universe can be far more than 4D, however it must follow both wave-like behaviour and grouped in rings of matter, antimatter, dark matter and dark antimatter. Here the inner box is the \OO ring defining general relativity for all the four \HH sub universes. The outer box is the \SS ring that is now an ecletic mix of the original four forms of matter. Dark energy being the 4D eqivalent of bosons like the 2D electromagnetism, strong and weak forces. All of these 4D forms of matter and energy having real and virtual forms exactly the same as the 2D versions.
Spacetime: Ring Groups
Dark energy is not limited to 4D stretching, as is dark matter to 4D rotations. Which is where the spin of the particles comes into play, as 4D transforms would be Spin 2, 8D as Spin 3, etc. With the dimensional size of translation related to D_{n} = 2^{s}.
So we now define special relativistic flat spacetime as the D_{sr} = 2^{s} that defines the matter we are using as reference (aka the observer), and inner looking quantum mechanics as D_{qm} \le 2^{s-1}, and outer looking general relativity as D_{gr} \ge 2^{s+1} as the observed. This will become important as we will show later, to how many dimensions we set as relating to special relativistic flat spacetime as that will determine the way we interpet the particles within it.
The reason I have been using the \HH, \OO and \SS terminology, is that we already have a clear mathematical process to go from one to the other. That is known as the Cayley-Dickson construction. The progression of causality can now be described with a phase velocity of the overall wave, and it is that which defines the speed of light.
General Relativity: Wave-Like
We now have defined what matter, antimatter, dark matter, dark antimatter and dark energy are, as well as how energy can transfer between each other via rotations, 2D \CC rotations withing \HH spacetime for special relativity and standard model, extended to \HH rotations within \OO spacetime for general relativity. Simultaniously this explains why dark matter and dark energy are unable to directly interact with either matter or antimatter. Causality is now the net direction of all of the various spacetime elements combined.
- Spin 0 (Position): Ring of \RR rotations within \CC spacetime, separating space and time, thus also creating mass and momentum, light and vacuum energy.
- Spin 1 (Velocity): Ring of \CC rotations within \HH spacetime, creating the differences between electrons and quarks, electromagnetism, strong and weak forces.
- Spin 2 (Acceleration): Ring of \HH rotations within \OO spacetime, creating the differences betweek matter and dark matter fermions, as well as gravity and other forms of dark energy bosons.
- Spin 3 (Jerk): Ring of \OO rotations within \SS spacetime.
Each red peak defining + \infin and each blue peak defining - \infin, finally the green area as the connecting 0 points. (\hat{x}, \hat{y}) axis representing the complex nature of either light and vacuum energy, or matter and dark matter, the only difference being the dimensional scale. The height or \hat{z} axis is what we describe as higgs field that defines scale or total energy within the universe.
Because of this repeated structure, there must always be the same number of real and virtual bosons defining the rotations. Each of these rotations can now be clearly mathematically defined by the same rules of hypercomplex numbers and connectivity between them from the Cayley-Dickson construction graphs. This also will including the ramifications of loss of various operations such as commutativity, associativity as the size of the hypercompex number grows. This may well be the source of causality, as the larger the dimension of operation, the more unidirection the operation becomes.
If we now look at the above wave-like view of general relagtivity, there is no such thing as a singularity, as the wave moves forward it continuously chages matter to dark matter, then to antimatter and dark antimatter before returning again back to matter describing a full rotation around the ring. The green portions being the point where no matter or spacetime exists, just a scalar field. So as causality progresses, shown by the phase velocity of the overall wave, matter must be be created in more and more complex forms in an expanding universe where the amplitude is increasaing, and matter must be destroyed into simpler forms in a decaying universe where the amplitude is decreasing.
Spacetime is now flat (special relativity) when looking at like for like matter, aka matter within the same spacetime. The various levels of curvature (general relativty) are now what define how we look at higher dimensional objects. Quantum mechanics and Hilbert spaces are how we inward to the lower dimensional components. One major point is this is not just related to Acceleration (spin 2), but now can include Jerk (spin 3), Snap (spin 4), Crackle (spin 5), Pop (spin 6) and beyond. With Velocity (spin 1), being the boson that acts along the flat plane, spin 0 Higgs as inward curl and scale, and spin 2 and beyond the outward curl and scale.
Just like \HH matter is a mix of electrons and three quarks, or antimatter as a mix of positrons and three anti-quarks. \OO matter when seen as like for like flat universe would consist of electron and seven quarks, if seen from \HH would appear to be the standard electron and three quarks, with four quarks invisible as dark matter to us. However inside that \HH dark matter universe we would see it with one electron and three colour quarks as normal. The one dark matter electron being converted to quark during the conversion from \HH to \OO perspective of matter in the same way the Cayley-Dickson construction converts one real number part to imaginary number. This way, when viewing the universe as flat, there is always one time dimension, along with singular electron and neutrinos, with varying numbers of quarks and strong force colours. For example a \OO universe would have 7 quarks and strong force that follows SU(7) grouping.
The effect of growing complexity of matter is to convert the original energy that makes up electrons to that of quarks, so while the nucleus grows in mass, the number of electrons availible to create atoms and molecular bonds must decrease. During high energy explosions such as supernova, some matter is rotated to become dark matter possibly through jerk acceleration converting the negative energy portion of the cubic curve. This dark matter is not free of the originating matter that created it, aka \HH, but instead must be \OO matter. This has the good side effect of nullifying the information paradox as no information held by any matter that either originally made up or later falls into a black hole is lost, just held in the rotations of higher and higher dimentional objects. Any radiation that escapes the black hole mearly being the equivalent of unwinding of one of those rotations, keeping all forms of energy and momentum conservation laws intact.
This could explain the differences between stellar and supermassive black holes, as they may relate to different number of dimensions, stellar black holes as they are created from a 4D source are most likely \OO events, and supermassive black holes being \SS or larger events that can only be created by multiple black holes combining. To understand the supermassive black holes, they have an inside and outside, separated by a light-like event horizon (shown as \pm \infin lines on the first graph) the outside must be \OO so have matter and dark matter components, the inside will be something similar just hodge dual of the outside. So the \OO equivalent difference between matter and dark matter in \HH universes. Something akin to dark matter that acts with SO(3) rotational grouping of dark matter component universes, to singular matter universe in the same way we have three quarks to one electron that makes up atoms in our \HH universe.