Updates from 2018 - “Spectral Relativity”
PolymathicPerfraction: Spectral Relativity and the #ManyBodyProblem
Simulated particles systems have three challenges:
1) Continuity problems: requires the Hamiltonian & Conservation of Energy
- high magnitude force calculations (force splatting)
- infinitessimal-to-zero values resulting in discontinuities in the gradient calculation
- non-smooth values used for force calculation
- performance trade-offs
- infinitessimal-to-zero values resulting in discontinuities in the gradient calculation
2) Boundary Problems:
- the specific solution you choose impacts the conservation of energy for hamiltonian
- “fixing” this violation of the hamiltonian is the primary challenge with boundary problems
3) Distribution of mass/charge
- particle systems aren’t point-like
- addressing electrical configurations of ionic/covalent bonding is difficult
- this results in local relativistic effects for some elements (mercury)
4) Special Relativity
- Brute force approaches to relativity
- challenges with discrete computations of a continuous system,
- where relativistic effects occur over a large scale of distance
- challenges with discrete computations of a continuous system,
- Interesting to explore: do the relativistic effects found in mercury electrons impact interactions b/w some electrons in large, highly regular crystals?
general means of disaggregating/deconstructing behavior of physical systems:
- particle position/momentum
- energy equilibrium
2) Subgroupings of components (of the hamiltonian)
3) Application of Spectral analysis to hamiltonian components
- FFT(d(.)) == d(FFT(.))
4) inference of the structure of interdepenency of variance from expected outcome
- hamiltonian components as distributions (with max/min, moments, variance)
- more useful with momentum
- spatial distribution of entropy and kinematics of entropy changes over time
5) Stochastic Calculus
- analyzing possible short-term trajectories of particles (and phase space
- short-term trajectories a la “instantaneous impact” from Lie Algebra
7) Special-relativistic inferences
- examine impact of relativistic parameters on deformation of phase space
- again, “instantaneous impact” from Lie Algebra
- when crossing relativistic inferences with the spectral decomposition of hamiltonian, more information should be produced
8) mechanics of continuity
- all of the above must be connected, yet all preserve energy equilibrium
then the system must have ideal paths in phase space (at least in classical mechanics)
- measures of phase space path deviation (in aggregate) must be connected
- for analyzed observations or simulated systems
- this has to be reconciled with the classical Hamiltonian…
- measures of phase space path deviation (in aggregate) must be connected
9) Concept of Linear Hull (& Hamel Basis?)
- (this is incorrect, i’m talking about something else)
- as applied to the Solution Space of Phase Space
- examine the phase space curves and connections between curves
for all configurations of the system
- how and why do these phase space curves follow along common paths,
while being “chaotic”?
- i.e. any point along a curve represents a potential starting configuration of the system for which the phase space curve will be a subset
- then, for all points in phase space, which points are connected as curves?
- are all these phase space curves disjoint?
- if they are disjoint, how do rational & algebraic numbers transform into values along the curves?
- the systems are differential & thus the rationality/algebraicity of values along paths in phase space feed in and out of each other (this definitely connects to Lie Algebra and E8)
- how and why do these phase space curves follow along common paths, while being “chaotic”?
- and other number theory concepts as applied
- differential operator and linear hull…
- quotient spaces & number theory (does this lead to E8)
- examine the phase space curves and connections between curves for all configurations of the system
In which, I convolute ideas with those I’ve recently acquired from #ExteriorAlgebra with:
Hamiltonian, conservation of energy and phase space
spectral components of position/momentum in #DoublePendulums
- conserve/reconstruct continuity
- gradient descent applied towards observed #Divergence from expected paths through phase space
Why spatial invariance of physical laws and composition of aggregate quantities like #Entropy necessitates interesting extrapolations
Detection of local/global #Convexity & #Concavity on combinatorial multivariate surfaces in high-dimensional cost functions and how this applies to efficient feature prioritization in #MachineLearning algorithms
DifferentialForms, as applied to the anticipation & detection of expected/actual dissonance
from true conservation of energy.
- The #FreeAlgebra, as applied to the differential operator, to create a direct summation of tensor products, where #Algebraicity, connected quotient spaces, and #Continuity are preserved. This constructs something similar to a #Manifold, mapping low dimensional spaces to higher dimensional surfaces of the differential operator or 1-forms, 2-forms, etc.
Other new ideas.
Content from 2016
Ars Technica recently posted an article on an exciting new development in dark matter and this piqued my interest on the subject. Another article I just saw on Gizmodo teases a Nobel-worthy revelation on gravity waves. I’ve had a lot of questions about dark matter and gravity that my current understanding of physics can’t answer. Again, yet another subject where my understanding of math limits me.
Nobel Worthy Gravity Waves
The Gizmodo article mentions a possible Nobel-worthy discovery of gravity waves. Apparently a binary blackhole merger has been discovered, which is one of the few events that would produce gravity waves detectable by current means.
TL;DR; Gravity is Wierd
As far as I can see, understanding gravity necessitates a shift in how we model space. I’ve used a few thought experiments to help simplify the important pieces. Here’s a few of the more profound questions escaping me at the moment. I wish I knew more of the math and physics to understand this more completely.
Update: I started looking further into some of the questions that I asked and found that these are indeed very relevant issues to be discussed. Frustratingly, I have again found that my intuition is particularly accurate here - frighteningly so, considering I dropped out of university physics. Yet, at the moment, I lack the mathematics to pursue further inquiry, even though I suck it up like a sponge - it’s easy because every new term maps to a familiar, but unnamed concept I’ve explored on my own. For most of the questions that I’ve asked, the answers have not yet been conclusively determined.
I may update some of the questions in the future, to map my exploration to existing concepts. I used the word ‘frighteningly’ above for a reason. In particular, de Sitter Invariant Special Relativity represents an approach to answering many issues that confounded me in this article. Doubly Special Relativity was another interesting topic because some of the approaches used model spacetime with ‘momentum space’ instead of position space. I didn’t mention this in the article, but I got the feeling certain issues with relativity can only conclusively be resolved using momentum, yet I couldn’t really elaborate on why that was. Like I said, it was a feeling. Quantum Geometry is being developed to resolve how interactions are processed at lengths relevant to the Planck Length, though it’s implemented differently for each theory of Quantum Gravity.
My point is, someone with my education shouldn’t be able to understand much of this, beyond the typical summarization of high level information from A Brief History of Time. Yet, dipping into the math and specific theories, I’m able to pick some of this up without anyone guiding me along.
And I now understand why Quantum Gravity is so difficult. The approaches are so diverse, varied and intricate. The experimental evidence for Quantum Gravity isn’t there, nor will it be anytime soon, so all we have to go on is theory, hoping that we’ll get indirect clues from further experiments. The biggest missing element is the quantum structure of space. If we understand that, we’ll understand the mechanism of gravity. The biggest mathematical challenge is that, for analogy, whereas General Relativity is an Initial Value Problem (IVP), formulating Quantum Field Theory in curved spacetime is like a meta-IVP. It’s like an IVP wrapping another IVP, where the mathematic structures to use to resolve the problem are undeterminable. That’s my intuition speaking. Resolving the math is so challenging because the same spacetime curvature can ‘look’ different from each point.
Is there a distinction between gravity and the space it curves?
Or is gravity literally the ripple in space? That would imply space is simpler than it seems. That’s a bummer.
Does gravity cause the path of gravitation itself to curve?
This makes sense. If gravitons exist, they should follow similar rules as photons, right? It’d be weird if most bosons, like photons, followed the curvature of spacetime, but others didn’t at all. It’d make sense if each boson interacted in it’s own way with the curved nature of spacetime, yet still adhered to the same features of local space.
And it’d just be weird to have a set of bosons, that interacts via a completely different kind of space. Your model would have to account for multiple models of space: one relativistic and another, who knows. It’s a lot simpler if all the bosons are in the same space.
In other words, what I’m trying to ask is –
If gravitons follow curved spacetime, how does this affect other bosons?
And this is the question I got stuck on tonight: If a gravity wave is moving in the exact same direction as several parallel beams of photons, how are they affected? The photons that are moving just outside of parallel to the gravity wave would certainly be affected and the more their trajectory is altered, I can say confidently that the influence of gravity on them would only increase.
However, this would seem to break some kind of directional or spatial invariance of gravity, relativity or spacetime. Why is it that the photon whose trajectory exactly matches a gravity wave not be affected, while all adjacent photons with the same origin would be greatly affected? They would all assume and adhere to a change in distribution, causing a doppler effect. Except for this one beam of photons that wouldn’t be affected at all. This would be a violation of continuity in this function.
Here’s Some Gravitational Lensing
Notice how the objects seem to move away from the black hole as it first approaches. That is happening because rays that would otherwise not be directed towards you are following the curved spacetime. Here’s an image describing how light bends around massive objects.
Gravitational Lens Geometry
In the presence of one mass, light will follow a path that only curves in one direction. After passing the object, the light will gradually return to traveling in a straight line, but it only curves towards the gravitational source. It doesn’t curve back the other way as it returns to a straight path.
Thought experiment: traveling with a gravity wave
If you imagine a uniformly distributed narrow beam of protons, all traveling perfectly parallel to each other and traveling in sync with a gravity wave, you’ll get an idea of how the resulting distorted distribution of photons would look a bit odd. Imagine the photons beginning from various adjacent sources and they are all parallel, but that the gravity wave was emitted from a single source, so there are slightly varied angles of incidence to each photon. Some photons lie in the crest of the gravity wave and some in the trough.
For the most part, the photons that lie in the crest of the wave get their trajectory bent slightly inward. And those that lie on the trough of the trajectory get bent slightly outward (i think). So you can imagine that the resulting distrbution of photons, in this idealized scenario, would look parabolized. That is, except for the beam of photons that is exactly in line with the gravity wave, which may be technically impossible under real conditions. This beam, informed by my current understanding of general relativity and curved spacetime, could not be slowed down because it’s traveling at the speed of light and that is constant. It is taking a straight path through spacetime that has been curved, but such that the curved spacetime does not affect it’s apparent trajectory. And thus, this beam of light would not be distorted in any way by the gravity, but would stick out like a sore thumb in the parabolized distribution of photons.
It’s a violation of continuity because of a problem with invariance. Unless my understanding of this is incorrect, which is likely, then this center beam of photons must also be slowed down to some degree, but I can’t resolve how. And thus, the photons traveling in crest of the wave would slow down a bit, apparently. The photons traveling in the trough of the wave would speed up a bit, so that all the photons would be uniformly distorted.
However, this thought experiment has been set up with idealized conditions, similar to those for general relativity. In this problem, the originating mass for the gravity wave is point-like and can apparently alter its mass with no change in position, producing a perfectly circular gravity wave, which is impossible in nature. Also, in nature, it would be impossible to produce a stream of photons that are exactly in line with the origin of the gravity wave, unless you could ensure that it was lined up down to the plank length, but I’m not sure I really understand that.
If gravitons alter the spacetime that other bosons operate on …
Then how do you even resolve that dependency graph? One would need to take precedence over another … or something. In other words, if none of the bosons alter spacetime, then one can assume that spacetime is fairly consistent, except for the exception of matter that moves, but that’s really not that hard to resolve.
All the particles operate on the same space. If the space changes, all particles, for the most part, react accordingly and at the same time. There are some exceptions maybe with the propagation of gravity. Here you can’t really have quantum gravity.
However, if gravitons alter the spacetime that these other bosons are using, how does one deal with that dependency graph that arises? If you view bosons as units in streams of information that modify states of other particles, then what stream of information do gravitons make up? Do they comprise a stream to the surrounding space? How so? Would that mean space itself is composed of particles too? Would it stream information all those spatial particles simultaneously? That gets complicated. How do you deal with deformed space if the plank length is constant? Does that mean that space is voxelized?
How do scalar bosons, like the Higgs, differ from Gauge bosons?
I don’t really know enough to elaborate on this, but there should be some fundamental differences in how scalar bosons work.
If energy is mass, does gravitational energy interact with space or mass in transit?
If gravity travels at the speed of light and force is mass, and if all matter exhibits gravitational force on all other matter, then what happens to the gravitational energy between two massive objects moving away from each other, whose gravitational energy takes significant time to reach the other object?
Would space curve for the objects in between, because of the gravitational energy in transit? If this gravitation energy does cause space to curve while it’s in transit, does the source of the gravity affect how space curves? Or is the force expressed on space simply an aggregate of all the forces being transferred through that space?
Another Thought Experiment
Even with the limits of relativity, one can imagine situations where the gravitational force expressed on two objects could never reach the other side, AFAIK. E.G. if large two objects, both traveling toward each other at half the speed of light, relative to a nearby observer, then miss each other and continue to travel at nearly half the speed of light in opposite directions, it could take nearly an eternity for the force of gravity to exert influence on the other object.
How do we preserve conservation of energy with quantum gravity?
So, it’s clear that quantum gravity has some complex issues to resolve. Otherwise, there are loopholes under which the conservation of energy would be violated. This may be an indication that gravity operates via a non-local mechanism. Or, in other words, we need more dimensions to explain it mathematically. We can be fairly sure that the force that gravity exhibits travels at the speed of light over space, but does that mean all of the mechanisms of gravity operate over our 3-space?
What is the nature of these other dimensions?
So if gravity operates via mechanisms in other dimensions, how does the speed of light apply there? Is there some similar limit? What is the structure and nature of these dimensions and why do they not seem to affect existing observations? Have we not noticed them before because the other 3 forces do not interact with these dimensions? Is this a kind of compartmentalization that’s required to prevent quantum gravity from creating dependency issues? By this, I mean it may be mathematically simpler if gravity operates on a different space than the space it manipulates.
The universe is composed of streams of information
I found it interesting to think about photons, bosons, etc as streams of information. That is, we’re not looking at a star, but we’re looking at a stream of photons and imagining the star that’s producing them. And when we look at the night sky, were receiving long, unbroken streams of photons for each star in the sky. These streams aren’t just directly to us, but are more like a radial stream of photons from the star.
It’s useful to think of other quantum phenomena in the same way i think. Yet it’s really hard to imagine gravity as we know it in this way. If objects in motion require spacetime to determine their trajectory, what stream of information could provide the spacetime with the information needed to distort? We know that gravity is not the result of the interaction of two masses, but instead the interaction of two masses through curved spacetime.
When the observer is directly in line with a black hole and a large galaxy further behind it, you can see Einstein Rings. In this generated image, as it zooms further in, you can even see that light from the observer’s region of space is being redirected back towards the viewer in alternating rings. For more information, check out the wikipedia entry.
Is the speed of light truly constant for all conditions?
The main issue I have with all this is that gravity, which theoritically also travels at c, the speed of light, also effects the medium through which it flows. And as I’ve shown above, light is always slowed down by gravity. This is because the only value for which light will not curve is a value that is exactly in line with gravity’s influence and this is impossible. Therefore, light will always be affected in some way and slowed down by gravity. Technically, the light doesn’t slow down, but it’s flowing through more “space,” as it’s affected by gravity. And if light is “slowed down” by a gravity wave, then it would always be slowed down in some way by the net influence of gravity on space.
Is gravity really the sum of effects on curved spacetime?
Which begs the question: are two areas of space functionally equivalent if they have the same net distortion of space from any number of gravitational sources? That is, if you can ensure that two areas of space have the exact same net component for gravity, even though that gravity is caused by different sources, is the effect on masses in that area of space exactly the same.
I’m fairly confident that it’s not. If there is some structure to spacetime, than you could not produce two regions with identically composed spacetime without providing the exact input from masses. That is, just because you have created a net equivalent of gravitational forces doesn’t mean that your spacetime is functionally equivalent in all respects.
Example: you’ve got two planets. One is huge and the other is small. You’re in a spaceship in the middle. Measure the net force of gravity on your spaceship. Now, increase the mass of the planets, but move them further away, so that the net force of gravity is the same. Is the state of spacetime that your ship occupies identical to it’s previous state? No, for a dozen reasons. And I think getting this to work to control the state of spacetime would be pretty difficult.
Does gravity lose energy to dispersion?
If gravity is also similarly “slowed down” in the same way that light would be, does that mean that a gravity wave loses energy as it propagates through space?
If the apparent speed of light varies from place to place…
– and I’m pretty sure this section is shit, but I left it in anyways –
How does this affect how we make observations. Then the speed of light itself becomes a kind of initial value problem, where you can’t pin it down because it’s impossible to measure the net forces distorting local spacetime.
But then, what is c, really? If the measurement of this constant can’t be understood as accurate because it would require measuring the net distortion of local spacetime, then how does this affect the other laws of physics?
Like I said, I almost threw this out, but it’s interesting to think about the dependencies between various laws of physics. Which are dependent on c? Would the be dependent on local or universal c? And how does this affect the conclusions we’ve reached using statistic modeling of the cosmos? And it’s an interesting thought experiment to examine the dependencies of theories in physics.
Does spacetime have memory?
If spacetime is distorted by mass, and particularly, if spacetime is composed of some quantum structure, is this structure permanently changed by the presence of mass? Is local space directly connected to the mass that occupies it?
How to extend the notion of singularity to two or three points?
So if a singularity for a black hole is a point so massive it creates an event horizon from which light cannot escape, are there extensions for systems with two or three such points? If you have a binary black hole system, what happens as the event horizons merge? Would there be a singularity that “streched” across space, forming a line? This seems impossible, but what would the space between these singularities be like?
And if there was a trinary black hole system that was collapsing, it seems like an extremely interesting situation could emerge, where a “triangular” region of space is formed, surrounded by event horizons linking each singularity. Learning more about what happens here might tell us more about the nature of space when a singularity forms, since prior to that the massive object has volume and will be decomposed into a singularity in a chain reaction.
Dark Matter’s Incoherent Doppler Shift
The Ars Technica article discusses unexpected low-energy gamma rays, whose sources can’t be determined. It is suspected that extremely rare interactions with dark matter are triggering these gamma ray bursts.
The article mentions an interesting inverse correlation in the doppler shift in dark matter suspected gamma ray bursts (GRB) in relation to the observed doppler shift from other sources of matter. Perhaps not an inverse correlation per se, but a lack of coherence with the doppler shift normally exhibited by normal matter we observe. Most matter in our galaxy is spiraling inwards towards the center and moving, on average, at specific speeds in relation to us. Because of the difference in speed, we observe the doppler shift and because the motion of all matter in our galaxy is more or less synchronously spiraling inwards, we observe a predictable, smooth doppler pattern for normal matter.
Yet, dark matter doesn’t interact with matter, except through gravity. The only reason we suspect it exists is because our observations don’t match our cosmological models. They mostly match; that is, on a micro scale, our models for gravity seem to work. But on the cosmic scale, there seems to be an enormous amount of missing mass out there than we can’t account for. Personally, I’m not so convinced this exists at all, instead feeling that there’s something subtly wrong with our models.
This research looked into the doppler shift of the GRB’s, which is around 3.5 KeV, whereas normal light signal is between 1.5eV and 3.5eV. Because of the speed difference between the sun and dark matter, measuring the doppler shift requires sensing very low energy differences. But new satellites will be launched with gear that should be capable of doing this.
This research is significant because we may have observed direct evidence of dark matter that lines up with our expectations. Hopefully, the frequency of the GRB’s we see are spatially coherent with the distortion that our models predict for dark matter’s effect on mass in each area of our galaxy.
Basically, dark matter’s average motion is distinct from the average motion of the galaxy and this is what allows us to distinguish the doppler shift data. GRB’s shift towards blue when we would expect normal matter to shift red, which means it’s moving away from the majority of matter (that we’re moving.
My Thoughts on Dark Matter
My own uneducated thoughts: I get the feeling that there’s something subtly off with our model of the universe. We see that there’s some general variance that’s spatially distributed with the observations that accompany our data. That is, our observations work in smaller systems, but on a larger scale, there’s this huge disparity in the motion we see in galaxies and clusters. We suspect that dark matter causes galaxies to cluster and even to form in the first place, but simply because we can’t explain it otherwise.
But the stars in nearby galaxies? They seem to cohere to our model’s predictions, at their interactions with local stars do. IMO, there’s something missing from our model of gravity or our understanding of the structure of space.