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Other Discussion Boards => Science & Alternative Science => Topic started by: JRowe on February 25, 2019, 03:38:01 PM

Matter and energy exist, though there are different forms of each.
For matter, this is simple. Obviously you have different states of matter, different elements, but if you break it all down at the end of the day it's just stuff, whether you want to talk about atoms or baryons or quarks. There is of course the interesting quantum question, of whether these end up just being waves if you look at them closely, but given that waves are typically free to dissipate there is plainly something nonwavelike about matter. All of which is a drastic oversimplification of quantum theory, but it merited acknowledgement.
At a basic level we can break all matter down into this stuff. There are interactions with energy that help distinguish it all, but this provides a simple visualisation. On a macro scale it is even easier to see; you can have a solid cube that you drop into water, say, and it cannot occupy the same location as the water. However if the cube is porous, you end up finding that the water and cube essentially may indeed occupy the same location, the water running through the cube, or the cube running through the water, whichever way you want to visualize it.
Energy, now, is where it gets interesting. There are many types of energy. Heat energy, which manifests essentially as the vibration of molecules (also as a form of radiation, that is waves). Sound energy, which really is just an illusion born of human perception, it is again just vibrations that go through matter, but when those vibrations reach our eardrums it is interpreted a specific way by the brain. Again, just vibration. Kinetic energy, obviously, is just the movement of matter. Electrical energy is manifested by the movement of specific types of matter, and this is when we get onto concepts such as chemical energy, potential energy stored that one day could manifest as another type of energy. At a basic level energy and movement are equivalent.
And then we get to light. Though it is modelled as a wave, and it can indeed impart other forms of energy, it does not seem to be as simple as a vibration carried by matter. The trivial resolution to this is the photon, positing that the photon is the particle which carries the wave of light and thus vibrates in such a fashion, but this is unsatisfactory. It does not allow for wavelike behaviour. What does, however, is ironically simple: multiple particles.
Think now of the famous double slit experiment, only instead of photons, we use water. Water waves interfere with each other, create similar patterns to light, but if you then constrain what happens to a single molecule of water going through at a time, you no longer observe the interference pattern. Understanding the double slit experiment, really, is trivially simple. If you limit things so that you can observe a single basic particle, it no longer has anything to interfere with. Only when things are unlimited, when there is no equipment getting in the way to allow you to observe, can the interference occur.
Light, then, is similar. The notion of an 'aether' that carries light must indeed be accurate, it just wasn't as it used to be posited. The light waves themselves are carried by multiple fundamental particles, photons, that only behave the way they do when they are carrying lightwaves. One might say they are synonymous with light, but at the end of the day the two statements are equivalent; energy cannot be created or destroyed, the emission of a photon simply creates the photon by converting it from some other nonphoton potential. Something is done that transforms it into a photon, the addition of 'light' if you will. It then carries this light, and we observe all we do.
The only thing special about light, in this case, is its speed limit. As I hope we all know, anything with mass that approaches the speed of light in vacuum ends up warping spacetime around itself.
So far I'm just expressing what is already known, albeit with a couple of odd bits of focus.
My proposal is thus: there are two building blocks to everything. Movement, and stuff. Movement gives us energy, this 'stuff' gives us matter, though precisely what type of matter depends on energy. There is no more and no less than this. Light is a massless particle with a 'charge' of energy, creating a photon.
This is in line with all we know. The extension is where this gets interesting.
I glossed over photons just now. They are converted from a potential, yes, but what is this? Are lightbulbs haemorrhaging matter without losing mass? We're treading on the toes of thermodynamics with that thought experiment, but it is an illustration. Perhaps it is raw energy converted into matter, we know this is possible, living things do it all the time. The quantum understanding of matter makes this easy; what sets apart a wave of energy and a quantum wave of a fundamental particle of matter is perhaps most simply thought of as some kind of binding that keeps it in place on a macro scale. So long as that can be generated, energy becomes matter.
The third option, though, is the aether response. Instead of a medium which carries a wave, it is a medium composed of stuff already, and the introduction of a lightcharge simply shifts a particle of this into a photon, in the same way a colourcharge defines a quark. If there is such a medium it must be universal. If you want to imagine it as a sea of molecules, a lightcharge essentially transforms one molecule of substance A into substance B with wholly different properties, such as a velocity.
And thus, we reach the conclusion of this. I propose that the medium of light is spacetime itself, that spacetime is composed of another fundamental form of matter. This isn't an alien idea, John Wheeler hypothesised a type of quantum foam, this is simply a generalisation of that.
Location, then, is much like the porous cube mentioned above. You occupy the same space as this primarily noninteracting matter. In a similar fashion, this also means that the speed of light in vacuum is not a property of light, but rather it is the limit to the movement of the fundamental components of spacetime. If a mass is moving at high speeds, it moves faster than the space it occupies can keep up with, of course that is impossible. Relativity begins to make intuitive sense.
I could take this to more speculative ground, analyze why the postulates of relativity are what they are, rather than merely stating it. One could suppose that the particles composing spacetime are locally uniform, thus any force that keeps them together, limiting movement and causing the speed limit, will be constant. In this way, the assumption of the homogeneity of spacetime and the speed of light in vacuum being constant are equivalent. It also tells us the situations in which relativity will break down, and why, taking us a step closed to a unified model.
This is just a lot of theoretical underpinning to the concept of spacetime possessing some form of building block, and by extension the potential for varying concentrations, but I hope it begins to demonstrate the ramifications for even understood science. A lot of topics are treated as big mysteries, or "This is just the way it is," but there are answers to the why as well.

Hi JRowe,
A very nice exposition! I like your idea of a quantum foam. Quantum Field Theory supports this hypothesis, with the idea of quantum virtual pair particles being created and annihilated all the time. This results in an average background energy density which is nonzero, and hence can be interpreted as a medium. Thus, there really is no vacuum on the quantum level. This immediately raises the question in my mind: how does light propagate through this medium? Does it interact with these virtual pairs? QFT says that it should. But I think it would be difficult to try to describe how on a large scale.
If you have a moment, please check out my post on FE Theory. I need your help!

This immediately raises the question in my mind: how does light propagate through this medium? Does it interact with these virtual pairs? QFT says that it should. But I think it would be difficult to try to describe how on a large scale.
Propagation through the medium is the easy part. At a basic level the medium of space time simply acts as a coordinate space, the building blocks being those coordinates. Propagation is just a matter of where a photon exists relative to thsoe coordinates.
The transformation is the interesting topic. Photons would be near unique in this regard, existing and being absorbed instantaneously from its perspective, and given that virtual particles are made/destroyed at the same rate then only something with that property could be made from them.

I must confess I am struggling a bit to fully understand that. If a virtual particle interacts with light then (according to QFT now) then couldn’t the particle reemit it while conserving momentum? If so, then light would not seem to need the same property as the virtual particle. I think maybe that I’m just not getting what you mean though.
Another thought I just had: if a virtual particle interacts in elastic ally with a photon, then it would become onshell, and manifest as real. So we could observe it! I imagine it would look like a cascade that follows the photon’s trajectory.

I must confess I am struggling a bit to fully understand that. If a virtual particle interacts with light then (according to QFT now) then couldn’t the particle reemit it while conserving momentum? If so, then light would not seem to need the same property as the virtual particle. I think maybe that I’m just not getting what you mean though.
Another thought I just had: if a virtual particle interacts in elastic ally with a photon, then it would become onshell, and manifest as real. So we could observe it! I imagine it would look like a cascade that follows the photon’s trajectory.
Light isn't something that the medium interacts with, it is a property that some of it can possess, thus 'creating' a photon.

I must confess I am struggling a bit to fully understand that. If a virtual particle interacts with light then (according to QFT now) then couldn’t the particle reemit it while conserving momentum? If so, then light would not seem to need the same property as the virtual particle. I think maybe that I’m just not getting what you mean though.
Another thought I just had: if a virtual particle interacts in elastic ally with a photon, then it would become onshell, and manifest as real. So we could observe it! I imagine it would look like a cascade that follows the photon’s trajectory.
Light isn't something that the medium interacts with, it is a property that some of it can possess, thus 'creating' a photon.
Hmmm, not sure I’ve ever heard of light being a property of a system before. In QFT, light is represented as a quantum electromagnetic field, which then interacts with the quantum field of a system. The photon that propagates is a local excitation of that field which results from the interaction.
I wonder how one would describe light as a property? Light HAS properties...

Hmmm, not sure I’ve ever heard of light being a property of a system before. In QFT, light is represented as a quantum electromagnetic field, which then interacts with the quantum field of a system. The photon that propagates is a local excitation of that field which results from the interaction.
I wonder how one would describe light as a property? Light HAS properties...
It's a slight abuse of terminology, but it's a useful one to help explain things. Nothing gets created or destroyed, merely converted; before light is emitted, everything that is required for a photon already exists in its vicinity, it just has not been 'converted' to a photon. Thus, light can be viewed as a property, and through possessing that property part of the medium becomes what we call a photon, with all the properties that entails. That's the gist at least.

Hmmm, not sure I’ve ever heard of light being a property of a system before. In QFT, light is represented as a quantum electromagnetic field, which then interacts with the quantum field of a system. The photon that propagates is a local excitation of that field which results from the interaction.
I wonder how one would describe light as a property? Light HAS properties...
It's a slight abuse of terminology, but it's a useful one to help explain things. Nothing gets created or destroyed, merely converted; before light is emitted, everything that is required for a photon already exists in its vicinity, it just has not been 'converted' to a photon. Thus, light can be viewed as a property, and through possessing that property part of the medium becomes what we call a photon, with all the properties that entails. That's the gist at least.
I see, that helps. Since nothing is created/destroyed, but only converted from something else, then in principle isn’t everything is a property? If so, what do we identify that it is a property of?

Hmmm, not sure I’ve ever heard of light being a property of a system before. In QFT, light is represented as a quantum electromagnetic field, which then interacts with the quantum field of a system. The photon that propagates is a local excitation of that field which results from the interaction.
I wonder how one would describe light as a property? Light HAS properties...
It's a slight abuse of terminology, but it's a useful one to help explain things. Nothing gets created or destroyed, merely converted; before light is emitted, everything that is required for a photon already exists in its vicinity, it just has not been 'converted' to a photon. Thus, light can be viewed as a property, and through possessing that property part of the medium becomes what we call a photon, with all the properties that entails. That's the gist at least.
I see, that helps. Since nothing is created/destroyed, but only converted from something else, then in principle isn’t everything is a property? If so, what do we identify that it is a property of?
To an extent, but that aspect was little more than awkward terminology to demonstrate the bigger point; the concept of spacetime as composed of an actual building block.
At a basic level though, everything is just going to be a function of the two things mentioned in the original post. This 'stuff' as the most fundamental type of matter, devoid of charge and mass and anything, and movement which thus gives means to differentiate via the presence of energy. (Though if you want to get even more basic, at a quantum level matter is just energy bound by something so that binding would be the other fundamental in place of matter).

Hmmm, not sure I’ve ever heard of light being a property of a system before. In QFT, light is represented as a quantum electromagnetic field, which then interacts with the quantum field of a system. The photon that propagates is a local excitation of that field which results from the interaction.
I wonder how one would describe light as a property? Light HAS properties...
It's a slight abuse of terminology, but it's a useful one to help explain things. Nothing gets created or destroyed, merely converted; before light is emitted, everything that is required for a photon already exists in its vicinity, it just has not been 'converted' to a photon. Thus, light can be viewed as a property, and through possessing that property part of the medium becomes what we call a photon, with all the properties that entails. That's the gist at least.
I see, that helps. Since nothing is created/destroyed, but only converted from something else, then in principle isn’t everything is a property? If so, what do we identify that it is a property of?
To an extent, but that aspect was little more than awkward terminology to demonstrate the bigger point; the concept of spacetime as composed of an actual building block.
At a basic level though, everything is just going to be a function of the two things mentioned in the original post. This 'stuff' as the most fundamental type of matter, devoid of charge and mass and anything, and movement which thus gives means to differentiate via the presence of energy. (Though if you want to get even more basic, at a quantum level matter is just energy bound by something so that binding would be the other fundamental in place of matter).
So like, are there any descriptions of this “stuff” that is the building blocks of spacetime? I mean is anything known about it, or is it just postulated as an assumption of your model?

So like, are there any descriptions of this “stuff” that is the building blocks of spacetime? I mean is anything known about it, or is it just postulated as an assumption of your model?
Concluded would be a more accurate word than postulated, for the reasons gone into.
All that's used is that it exists, that spacetime is not mere vector. The logical extension from this is to extrapolate a notion of concentration, that coordinate points can be sparser relative to other zones.

So like, are there any descriptions of this “stuff” that is the building blocks of spacetime? I mean is anything known about it, or is it just postulated as an assumption of your model?
Concluded would be a more accurate word than postulated, for the reasons gone into.
All that's used is that it exists, that spacetime is not mere vector. The logical extension from this is to extrapolate a notion of concentration, that coordinate points can be sparser relative to other zones.
Do you have any advice on how I would then formalise spacetime? Usually, when describing properties distinguishable by location, one either uses scalar fields or vector fields. I don’t really know of a third option. Since you’ve explained that space time is not a vector (field?), and I assume it isn’t a scalar field either  since you seemed to imply it was something more involved or fundamental than a vector field, while scalar field is more elementary, how do we model it?
So basically, I’m asking for help pinning down the specifics, so that it can be formalised into FE theory.
Any help you can offer would be greatly appreciated.

Do you have any advice on how I would then formalise spacetime? Usually, when describing properties distinguishable by location, one either uses scalar fields or vector fields. I don’t really know of a third option. Since you’ve explained that space time is not a vector (field?), and I assume it isn’t a scalar field either  since you seemed to imply it was something more involved or fundamental than a vector field, while scalar field is more elementary, how do we model it?
So basically, I’m asking for help pinning down the specifics, so that it can be formalised into FE theory.
Any help you can offer would be greatly appreciated.
it would still be a vector field, especially over uniform stretches. Nonuniform stretches still function basically the same way, but it isn't as though curved spacetime hasn't been modelled, you just need to allow for the possibility for, say, the distance between two parallel straight lines to not always be constant. Just because spacetime has properties beyond just being a direction doesn't prevent it from being modelled, you just have to account for those properties too. That's done with relativity, you just need to use tensors to calculate distance instead so you account for more. This situation's a little more involved, and most of the time you could approximate but at the end of the day all you really need is dt, know the rate of change of the concentration at any location. if it's predictable, and it seems reasonable to me that it would be (the only cause that's worth considering, that is without assuming whole new entities, would be the concentration of nearby locations) then in theory there's no problem.

Oh! This is just general relativity that you are describing. The object that allows one to transform vectors on curved manifolds is called a covariant derivative, and the object used to help transform the tensors is called a Cristoffel symbol. Using these, it is possible to show that parallel lines stay that way on flat geometries, meet on closed geometries, and diverge on open geometries  this is a common problem that we assign our graduate students.
The paths taken by light through these geometries are called geodesics, and require a complicated mix of the dt’s you were talking about. The concentration you reference is the stressenergy tensor in the field equations.
Sorry, I thought your intention was to describe a theory for FE that competes with general relativity. Perhaps this was a clumsy assumption on my part. FE theory cannot survive alongside GR, because GR is consistent with (and predicts) a RE perspective.
Finally, I just want to point out that you described GR in a very intuitive fashion. Have you studied it before? If not, then I’d invite you to take a look. It’s uncommon to see that kind of intuition about such a complicated topic  the mathematics is so involved that extracting intuition is laborious. You may be a natural :)

Sorry, I thought your intention was to describe a theory for FE that competes with general relativity. Perhaps this was a clumsy assumption on my part. FE theory cannot survive alongside GR, because GR is consistent with (and predicts) a RE perspective.
Not just relativity, relativity is not concerned with concentration. This is a generalization without the homogeneity/uniformity of spacetime needing to be assumed. I'm plenty familiar with relativity, that was why it was mentioned, but this goes further than that. As far as relativity is concerned, the concentration referred to above is constant and the distance between parallel straight lines would always be constant, that is the only variation in spacetime would develop from an object's acceleration.
The dt GR uses is dependent on the object, not the local behaviour of spacetime.
GR's greatest flaw is that it isn't concerned with why things happen, only that they do. Assumptions are made because the math requires it, and it's in line with experimental data, but there's no understanding as to where those assumptions originate. That, and it's already accepted that GR is not complete, science never is, and in the same way that Einstein extended Newton, by giving a single formula that, when certain values are small, is identical to Newton, GR can, should and will be extended similarly in such a fashion that, while in certain situations the current math holds, when other variables change so too will the effect. For example, if the concentration of spacetime is included as a factor, one could theoretically encompass QFT as occurring when the sizes of objects nears that of the distance between the 'coordinate points' composing spacetime, for an easy example.
Concentration and curvature are not the same.

Sorry, I thought your intention was to describe a theory for FE that competes with general relativity. Perhaps this was a clumsy assumption on my part. FE theory cannot survive alongside GR, because GR is consistent with (and predicts) a RE perspective.
Not just relativity, relativity is not concerned with concentration. This is a generalization without the homogeneity/uniformity of spacetime needing to be assumed. I'm plenty familiar with relativity, that was why it was mentioned, but this goes further than that. As far as relativity is concerned, the concentration referred to above is constant and the distance between parallel straight lines would always be constant, that is the only variation in spacetime would develop from an object's acceleration.
The dt GR uses is dependent on the object, not the local behaviour of spacetime.
GR's greatest flaw is that it isn't concerned with why things happen, only that they do. Assumptions are made because the math requires it, and it's in line with experimental data, but there's no understanding as to where those assumptions originate. That, and it's already accepted that GR is not complete, science never is, and in the same way that Einstein extended Newton, by giving a single formula that, when certain values are small, is identical to Newton, GR can, should and will be extended similarly in such a fashion that, while in certain situations the current math holds, when other variables change so too will the effect. For example, if the concentration of spacetime is included as a factor, one could theoretically encompass QFT as occurring when the sizes of objects nears that of the distance between the 'coordinate points' composing spacetime, for an easy example.
Concentration and curvature are not the same.
That is not my understanding of relativity, though. Uniformity/homogeneity is not an assumption of GR, and in fact is a consequence of taking the weak field limit. GR is not only geometry, but also provides the explanation for it. This is the right side of the field equations. Solving them explains how energy density  the concentrations  create the geometry. The dt in GR is a description of the changing geometry of space itself  not the object. What one has to do is find the geometry first, and THEN compute the trajectories of the objects through them. The reason why this is so hard is because the process is coupled. Objects (having concentrations) can distort the geometry they are moving through too!
The only assumptions of GR is that 1) the laws do not suddenly change if you shift your coordinate system, and b) there isn’t some second kind of mass that characterizes objects. These assumptions originate from occam’s razor, and form a minimum set one would need. All FE conjectures I have seen have them implicitly.
But you propose a different theory, yes?

But you propose a different theory, yes?
No more different than Einstein was to Newton, it works as an extension, not replacement. The geometry of spacetime as RET posits is exclusively curvature, when it talks about length and time altering it isn't a variation on the concentration of any coordinate points, at least not directly. All that changes is the curve, not the dispersal; this does have a connection, but not in any formalized way, and it is the only time GR comes close. When I talk about uniformity of spacetime, it's things like this; if you're in an inertial reference frame then there is no vehicle by which spacetime might alter. it allows for no alteration beyond acceleration or mass, and as much as there might be issues with the mass side of it, that's more practical than theoretical. Focusing on this, GR is limited in the situations it can apply.
GR posits that the geometry of spacetime can be calculated exclusively by the behavior of the masses within it. This is unjustified and, in my view, simply false. Rather my view is that the behavior of the local region of spacetime must also be taken into account, which has a concentration that might vary independently of the behaviour of matter and energy within it. That is what my post means when it talks about the dt of spacetime.
GR is not wrong, so much as it is incomplete. it cannot tell you why the speed of light is an absolute limit, only that it is. it cannot tell you why it is not yet unified with other theories, only that it doesn't yet work there.

But you propose a different theory, yes?
No more different than Einstein was to Newton, it works as an extension, not replacement. The geometry of spacetime as RET posits is exclusively curvature, when it talks about length and time altering it isn't a variation on the concentration of any coordinate points, at least not directly. All that changes is the curve, not the dispersal; this does have a connection, but not in any formalized way, and it is the only time GR comes close. When I talk about uniformity of spacetime, it's things like this; if you're in an inertial reference frame then there is no vehicle by which spacetime might alter. it allows for no alteration beyond acceleration or mass, and as much as there might be issues with the mass side of it, that's more practical than theoretical. Focusing on this, GR is limited in the situations it can apply.
GR posits that the geometry of spacetime can be calculated exclusively by the behavior of the masses within it. This is unjustified and, in my view, simply false. Rather my view is that the behavior of the local region of spacetime must also be taken into account, which has a concentration that might vary independently of the behaviour of matter and energy within it. That is what my post means when it talks about the dt of spacetime.
GR is not wrong, so much as it is incomplete. it cannot tell you why the speed of light is an absolute limit, only that it is. it cannot tell you why it is not yet unified with other theories, only that it doesn't yet work there.
It is not just mass...it is energy in any form. This is what the stress energy tensor means.
As for inertial frames, GR explains precisely how spacetime would alter  it would be affected by energy densities elsewhere. For example, an inertial frame could encounter a gravitational wave. To say it allows for no alteration beyond acceleration or mass just isn’t true, if I am understanding your statement correctly. There does not need to be mass anywhere, and there will still be a curved geometry.
But I think I am starting to get a feel for your theory. What you want to do is this (please correct me where I’m mistaken):
1: you acknowledge GR as an accurate theory but with limited applicability.
2: you propose to introduce an additional parameterization that measures the “concentration” of spacetime.
3: you propose to demonstrate that this parameter couples with the stress energy tensor to provide a wider predictive landscape for JGR (JRowe’s GR).
So, this is interesting. I have many questions.
a. What is concentrated, exactly. When there is a higher concentration of spacetime, what is there more of? You see, spacetime is a coordinate system. How do you concentrate that? Is there a susbstance that exists? This requires a rigorous definition in order to formally add it to GR.
b. Would the coupling to the stress energy tensor be linear?
c. If yes to b, then it should be easier to add it in. You could even demonstrate that in the weak concentration limit, JGR reduces to GR. That would get scientists’ attention, BTW.
I await with anticipation for your thoughts. It is not easy to find someone here willing to dive in, so I encourage your continued efforts. Kickass so far, man.

As for inertial frames, GR explains precisely how spacetime would alter  it would be affected by energy densities elsewhere. For example, an inertial frame could encounter a gravitational wave. To say it allows for no alteration beyond acceleration or mass just isn’t true, if I am understanding your statement correctly. There does not need to be mass anywhere, and there will still be a curved geometry.
The energy side effect went unmentioned purely because as far as I'm aware, it's purely theoretical, the amount of energy it would take is a huge question mark, and ditto for its properties. Either way though, GR posits that spacetime only changes because of the behavior of things within it.
a. What is concentrated, exactly. When there is a higher concentration of spacetime, what is there more of? You see, spacetime is a coordinate system. How do you concentrate that? Is there a susbstance that exists? This requires a rigorous definition in order to formally add it to GR.
This was the topic of the rest of the thread. The concentration is of, well, those coordinate points; if you want to take Wheeler's hypothesis, more of the building blocks of spacetime. in some places the coordinates are more densely packed, in others more sparsely.
b. Would the coupling to the stress energy tensor be linear?
c. If yes to b, then it should be easier to add it in. You could even demonstrate that in the weak concentration limit, JGR reduces to GR. That would get scientists’ attention, BTW.
It's not weak concentration limit, just if the rate of change of the concentration over the space examined is zero, then you'd get typical GR. That much probably indicates the issues with adding it; you'd have to account for the difference in concentration between the start and end of the motion, and while that would be infinitesimal at any instant in time, it necessitates a separate PDE where the location of whatever's being examined is incorporated, and the relative concentration is calculated via that, and that is probably the tricky part (and it's not like the rest is easy) because while the solution of that PDE could be included directly into GR, to my knowledge it's not even known if there would be a unique solution.
What would need to be added to the formula would basically be in terms of that solution with the partial derivative in terms of time, and the partial derivative in terms of the axis of the direction of motion (so, t and x basically with a suitable coordinate system). Near the Earth's surface I think that'd be mostly negligible, and that certainly matches up with observations, but more accurate analysis of celestial phenomenon is where it becomes important.

As for inertial frames, GR explains precisely how spacetime would alter  it would be affected by energy densities elsewhere. For example, an inertial frame could encounter a gravitational wave. To say it allows for no alteration beyond acceleration or mass just isn’t true, if I am understanding your statement correctly. There does not need to be mass anywhere, and there will still be a curved geometry.
The energy side effect went unmentioned purely because as far as I'm aware, it's purely theoretical, the amount of energy it would take is a huge question mark, and ditto for its properties. Either way though, GR posits that spacetime only changes because of the behavior of things within it.
a. What is concentrated, exactly. When there is a higher concentration of spacetime, what is there more of? You see, spacetime is a coordinate system. How do you concentrate that? Is there a susbstance that exists? This requires a rigorous definition in order to formally add it to GR.
This was the topic of the rest of the thread. The concentration is of, well, those coordinate points; if you want to take Wheeler's hypothesis, more of the building blocks of spacetime. in some places the coordinates are more densely packed, in others more sparsely.
b. Would the coupling to the stress energy tensor be linear?
c. If yes to b, then it should be easier to add it in. You could even demonstrate that in the weak concentration limit, JGR reduces to GR. That would get scientists’ attention, BTW.
It's not weak concentration limit, just if the rate of change of the concentration over the space examined is zero, then you'd get typical GR. That much probably indicates the issues with adding it; you'd have to account for the difference in concentration between the start and end of the motion, and while that would be infinitesimal at any instant in time, it necessitates a separate PDE where the location of whatever's being examined is incorporated, and the relative concentration is calculated via that, and that is probably the tricky part (and it's not like the rest is easy) because while the solution of that PDE could be included directly into GR, to my knowledge it's not even known if there would be a unique solution.
What would need to be added to the formula would basically be in terms of that solution with the partial derivative in terms of time, and the partial derivative in terms of the axis of the direction of motion (so, t and x basically with a suitable coordinate system). Near the Earth's surface I think that'd be mostly negligible, and that certainly matches up with observations, but more accurate analysis of celestial phenomenon is where it becomes important.
I do not think it is a huge question mark, because you can run the equation both ways. You can take a certain manifold and then solve for exactly the energy density needed to produce it. As for physical sources, one might find such energies during supernova events or in the early Universe.
So the coordinates themselves contract. Interesting. What you are describing is similar to Lorentz contraction, except the claim is that you can boost this into a stationary frame.
So there’s is a dynamic application here and a geometric one. Have you begun developing the equations? You seem to have thought through the structures, maybe it’s time to begin writing it down?
I’d love to see what you have so far, and am happy to do some of the mule work of checking limiting cases, locality, conservation properties, etc. These things would need to be checked anyway, and are usually considered by theorists to be a pain. Since I’ve done it many times for my own, I could probably whip through it and send you the calculations for your further development.

I do not think it is a huge question mark, because you can run the equation both ways. You can take a certain manifold and then solve for exactly the energy density needed to produce it. As for physical sources, one might find such energies during supernova events or in the early Universe.
Again, in theory, solving any tensor equation is far from easy and we've not observed any in nearly enough detail to comment, as far as I know.
So the coordinates themselves contract. Interesting. What you are describing is similar to Lorentz contraction, except the claim is that you can boost this into a stationary frame.
The principle is different to curvature, it'd effect space more than it would time. There are no timelike curves, no distortion, it's just a matter of how far it is A to B. No matter how long the distance is, it wouldn't impact the rate at which you progress through time. Thinking about it as any kind of analogue to curvature won't help. Curvature, in this model, is essentially how long it takes you to get from one coordinate point to another; in flat spacetime, there'd be no difference, and the concentration of spacetime purely measures the distance from A to B, and the distance from one coordinate to the next stays constant.
So there’s is a dynamic application here and a geometric one. Have you begun developing the equations? You seem to have thought through the structures, maybe it’s time to begin writing it down?
I’d love to see what you have so far, and am happy to do some of the mule work of checking limiting cases, locality, conservation properties, etc. These things would need to be checked anyway, and are usually considered by theorists to be a pain. Since I’ve done it many times for my own, I could probably whip through it and send you the calculations for your further development.
Did start a while ago, but would need to track it down. Modelling spacetime with a concept akin to diffusion, the logical behaviour of varying concentration, with a lack of viscosity (or at least negligible, though would be capped by the speed of light, but want to get it working in everyday situations before extending it that far) you get a PDE, and observations of the world get us boundary conditions. That was as far as my work got then, and it seemed like there were oversights with the BCs, but discussion ended up impossible with trolls derailing it, and the basic tools like separation of variables were insufficient. Might try tracking it down.
Honest it's never been a priority, as much as it gets brought up, it's never felt like math is going to help make my case, particularly anything advanced. Most people use it more to distract that discuss, if it gets provided they don't talk about it, if it doesn't they crow about it, and either way it tends more to get in the way of talking about principles. My goal's always been more to get interest and consideration, as much as scientists might be interested, the boundary conditions are based on my model of a FE and there's no way to justify any of that in a paper without ensuring no scientist will ever read further. Before it could ever be incorporated into GR I'll lose them.
I don't expect to be the next Einstein or anything, I know when something's beyond my skillset.

If your model’s curvature is how long it would take along a trajectory, then I am confused how it is spacelike and not timelike. It seems to be to be a definition of curvature based on time.
Also, how can it not impact the rate you move through time of the rate is what you’re using to describe the geometry?
Yeah, separation of variables is likely not going to work in this regime. I encourage you to track it down and present it. Forget the trolls, we can work on it. It is critical to establishing the idea and checking to see if it will work.
And to be honest, scientists publish theories all the time that are based on models which they don’t think are correct. It is the exploration and openmindedness that defines science. Heck, I’ve published 1D models for 3D phenomena! You’d be surprised how often it can lead to insights otherwise missed. So I disagree that you’d lose their attention.
Science is free, anything goes. Folks research all sorts of bizarre shit, from teleportation to warp drives to 2D spacetime...
Actual science is not what a lot of folks on this forum say it is.

If your model’s curvature is how long it would take along a trajectory, then I am confused how it is spacelike and not timelike. It seems to be to be a definition of curvature based on time.
Also, how can it not impact the rate you move through time of the rate is what you’re using to describe the geometry?
They're describing different things. There are no curves, timelike or spacelike. Taking graph paper with irregularly spaced squares as an analogue for spacetime, if you curve it into a n shape then the dispersal of those squares doesn't change, but climbing up the curved surface does take a little longer. Ok, that's only because of gravity, but it's the principle.
The situation described in previous posts focuses on the dispersal of coordinate points, assuming that two adjacent points always take the same time to go between. The properties there are exclusively of those coordinate points.
GR doesn't even need the coordinates to exist, it'd lack some explanatory power but it could function with an arbitrary plane described by vector and devoid of building block, and it is focused on the journey from A to B. Assuming fixed distance between coordinate points, it is the journey that becomes harder. The cliche description of relativity is a bowling ball dropped in a blanket, but that kind of curvature only exists by pulling in more of the surroundings, but there's no flow of space caused by acceleration, you don't get drawn towards Concorde or any such thing, when, say, length dilation fundamentally takes the form of there seeming to be more distance to cross, if that was because of coordinates then those coordinate points would need to come from somewhere. That is, those distances can only seem to increase if distances elsewhere decrease, and that doesn't happen. it cannot be the same principle at play. Curvature as a term doesn't really cross over to this extension particularly well, it's more a measure of how difficult it is to move through spacetime. A kind of spatial friction if you will; when you near the speed of light you struggle to progress forward through time so things would seem to take longer, and you wouldn't seem to progress as far as you should be through space (from an external reference point, of course).
Ties back to what the first post mentioned about photons, and the speed of light as a limit being an intrinsic property of spacetime.
Yeah, separation of variables is likely not going to work in this regime. I encourage you to track it down and present it. Forget the trolls, we can work on it. It is critical to establishing the idea and checking to see if it will work.
Will try. Tempted to glance over but go from scratch, the underlying thought process was simple enough, don't want to make the ame mistakes. Should have time in a couple of days.

Oh, I see what your doing now. I didn’t understand that your model keeps the time between two points as invariant. Now it makes sense what you mean. Yes, this is quite different than GR; you are proposing an inherent coordinate system for the Universe as being fundamental, and then working forward.
You lose me with some of the discussion regarding application and GR comparison, but I think that’s just a function of attempting to describe it over a forum, an also me being used to talking about it noncolloquially.
It will be much clearer when we can reference the mathematics as the common language.
I look forward to it :)

It will be much clearer when we can reference the mathematics as the common language.
I look forward to it :)
That can't necessarily be promised, in my experience math helps with prediction more than understanding, and even then not always, particularly on topics like this. Like, my last notes were:
M_{t} + c_{1} M M_{x} + c_{2} M M_{y} + c_{3} M M_{z} = 0
And it's been a while so can't tell you all the details of how that was settled on, know it came from analyzing a number of wave equations used to model similar behaviour and this being the most accurate for inviscid flows, M modelling the concentration at any point, derivatives therefore how that concentration is varying/flowing in one direction, or over time, what's missing is the initial conditions but wanted to go through those again to double check, and that'll give more of a view of my model in general rather than the details of how this works. Like, M_{x}(0,0,0,t) = M_{y}(0,0,0,t)=M_{z}(0,0,0,t)=M_{t}(0,0,0,t)=0, but that's a subset of a separate boundary condition so... (Will properly define origin etc too with the rest, short version under the central pole, z as altitude).
Equally, M_{z}(0,0,z,t)=gz^{2}
First steps at least.

It will be much clearer when we can reference the mathematics as the common language.
I look forward to it :)
That can't necessarily be promised, in my experience math helps with prediction more than understanding, and even then not always, particularly on topics like this. Like, my last notes were:
M_{t} + c_{1} M M_{x} + c_{2} M M_{y} + c_{3} M M_{z} = 0
And it's been a while so can't tell you all the details of how that was settled on, know it came from analyzing a number of wave equations used to model similar behaviour and this being the most accurate for inviscid flows, M modelling the concentration at any point, derivatives therefore how that concentration is varying/flowing in one direction, or over time, what's missing is the initial conditions but wanted to go through those again to double check, and that'll give more of a view of my model in general rather than the details of how this works. Like, M_{x}(0,0,0,t) = M_{y}(0,0,0,t)=M_{z}(0,0,0,t)=M_{t}(0,0,0,t)=0, but that's a subset of a separate boundary condition so... (Will properly define origin etc too with the rest, short version under the central pole, z as altitude).
Equally, M_{z}(0,0,z,t)=gz^{2}
First steps at least.
Don’t worry, I won’t hold you to any promise. Let’s just explore :)
Thanks for the initial equation. What would help the discussion in the future is exposition like the following example:
The horizontal position of an object undergoing projectile motion is modelled as:
x(t)=x0+v0*t
Where x(t) is the x position as a function of time, x0 is the initial x position (the position at time=zero), v0 is the initial x speed, and t is the independent time variable.
In this fashion, every term is detailed for the reader, and it is clear how the equation is used.

M_{t} + c_{1} M M_{x} + c_{2} M M_{y} + c_{3} M M_{z} = 0 (not certain on this without finding my earlier reasoning, will say, meant to essentially function as a diffusion wave equation with Turing patterns and I can't find anything resembling this in a bit of hasty research I've done now)
Where M=M(x,y,z,t).
(x,y,z)=(0,0,0) is essentially the centre of the Earth, a point under the central pole, and each c is a constant to be determined by experimentation.
x,y are between R and R, where R is the radius of the Earth. z between h and h where h is... complicated to explain quickly, but basically the ellipsoid region formed by this (x,y,z) marks the point up to a significant discontinuity (which are after all pretty common when it comes to flows). t>0 signifies time, naturally.
M is essentially the amount of spacetime at a coordinate. The tricky part of this is that we're kind of inventing a coordinate system underlying spacetime, as we can't measure it with itself, but it is strictly mathematical. M_{x} M_{y} M_{z} are the rate of change in that direction, basically the flow. M_{t} is the rate of change in one location over time.
With boundary conditions:
M(x,y,z,t)=M(x,y,z,t)
M_{x}(x,y,z,t)=M_{x}(x,y,z,t)
M_{y}(x,y,z,t)=M_{y}(x,y,z,t)
M_{z}(x,y,z,t)=M_{z}(x,y,z,t)
M_{x}(x,y,0,t)= e^{k1(x+y)}cos(x*n*pi)
M_{y}(x,y,0,t)= e^{k2(x+y)}sin(y*m*pi)
M_{z}(0,0,z,t)=g(hz)^{2}
M(x,y,z,t)=M(x,y,z,t+P)
Each k, m, n are also constants to be found by experimentation.
(Beyond this region, M will be constant. Equally, M integrated over the region (x,y,z) to determine the net 'amount' within will also be constant and not depend on t).
So, basically there's a kind of reflective symmetry over the disk, rotational motion that forms shrinking inwards circular, spiralling motion at z=0, and a straight downwards force following the inverse square law over the centre, with z=9.8 approx. Also M is periodic over time. Though they tie more to my FE model than to GR.
The ideal situation would be some kind of solution, and the c values would determine things like whether M got larger or smaller towards the edge of the boundary, and that they could be approximately determines to give a gauge for the behaviour of spacetime. To be honest I don't expect that to happen, but wanted to give some indication as to the potential long term goals.
M itself admittedly is of less use due to its mathematical construction, but its derivatives give us a more constant rate of change that would have observable results relative to the Earth.
I think that's right at least, some of that's just copying things I noted down years back and I haven't yet tracked down where I got them from so there could be mistakes.

Great! That was fast. My initial thoughts:
1. How can M be a flow rate if the units are not the units of a rate?
2. The boundary conditions have different dimensions in z direction vs x and y.
3. How does one interpret the cosine of a coordinate? What is the cosine of a meter? Is that physical?
4. Your constants c must have units of 1/M.
5. Cosine is an even function, so Mx(x,y*,*)=Mx(x,y,*,*) is possible to satisfy, but sine is an odd function, so I don’t see how one can simultaneously conserve parity while maintaining periodic boundary conditions.
6. By gauge for spacetime, do you mean a measure, or do you mean an extra degree of freedom (e.g., like choosing the Coulomb gauge in electrodynamics)?
Lastly, just want to express how badass it is that you’re bringing these ideas to the forums. It’s not easy putting oneself out there like that. Takes balls and poise. I really appreciate your efforts and am enjoying the discussion.

1. How can M be a flow rate if the units are not the units of a rate?
2. The boundary conditions have different dimensions in z direction vs x and y.
3. How does one interpret the cosine of a coordinate? What is the cosine of a meter? Is that physical?
4. Your constants c must have units of 1/M.
5. Cosine is an even function, so Mx(x,y*,*)=Mx(x,y,*,*) is possible to satisfy, but sine is an odd function, so I don’t see how one can simultaneously conserve parity while maintaining periodic boundary conditions.
6. By gauge for spacetime, do you mean a measure, or do you mean an extra degree of freedom (e.g., like choosing the Coulomb gauge in electrodynamics)?
Lastly, just want to express how badass it is that you’re bringing these ideas to the forums. It’s not easy putting oneself out there like that. Takes balls and poise. I really appreciate your efforts and am enjoying the discussion.
1. M isn't the flow rate, its derivatives are. M is just the raw amount of of concentration.
2. Yep, the region isn't meant to be a sphere.
3. Coordinate times pi, as far as I remember that's the standard way of creating a circle in polar coordinates, using radians.
4. They're dimensionless. M is a function.
5. The presence of z and t would presumably answer that, but looking back I'm not sure those sine/cosine boundary conditions work, they were one of the bits I copied and the x^{2} factor I don't think quite has the effect I wanted. WIll go back and tweak.
6. Gauge as in measure. I try to avoid using technical terms where possible for accessibility's sake.

1. Okay
2. No, I mean the units aren’t the same. Look at Mx and compare to Mz, they have different units.
3. Yes you can do this with radians, but not a length. It’s unphysical to to take the cosine of a length ( or exponentiate it.
4. Look at the equation for a moment. How can you add Mt to c1MMx? The only way is if they have the same units. Just like how you can’t add 5 seconds to 3 meters. So assuming all M’s have the same unit, then c1 must have units of 1/M, otherwise the equation is dimensionally invalid.
5. Got it.
6. Cool.

Coordinates aren't in units. Even if you create a graph where each coordinate is 1m apart, that doesn't change the point (1,1) to (1m,1m). Take the heat equation:
(https://wikimedia.org/api/rest_v1/media/math/render/svg/3edc07e9067b68e6057723653f7c3e7403889598)
u denotes temperature, but there's no attempt to balance the two, particularly given alpha is often set to one. Differentiate it once with respect to time, and you have the same dimension has it being differentiated twice with respect to space; none. Same for taking the cosine, it's not the cosine of 1m*pi, just 1*pi, say.

Coordinates aren't in units. Even if you create a graph where each coordinate is 1m apart, that doesn't change the point (1,1) to (1m,1m). Take the heat equation:
(https://wikimedia.org/api/rest_v1/media/math/render/svg/3edc07e9067b68e6057723653f7c3e7403889598)
u denotes temperature, but there's no attempt to balance the two, particularly given alpha is often set to one. Differentiate it once with respect to time, and you have the same dimension has it being differentiated twice with respect to space; none. Same for taking the cosine, it's not the cosine of 1m*pi, just 1*pi, say.
The heat equation absolutely agrees dimensionally. Alpha is the thermal diffusivity, and has a dimension. It is set to unity when you are solving the problem in units of alpha. But this does not somehow erase the dimensions from the final answer. U has units of kelvin. It’s independent variables are x,y,z. So when you plug in those coordinates  which are physical coordinates of distance  and evaluate the function, you get units of temperature.
Your Mx equation doesn’t give you units of flow when you input the x coordinate. Dimensionally it just doesn’t yield what you want it to.
And there does not exist any equation that evaluates the cosine of a dimensional quantity. A radian is a ratio of lengths, and is hence dimensionless.
Hope this helps!

The heat equation absolutely agrees dimensionally. Alpha is the thermal diffusivity, and has a dimension. It is set to unity when you are solving the problem in units of alpha. But this does not somehow erase the dimensions from the final answer. U has units of kelvin. It’s independent variables are x,y,z. So when you plug in those coordinates  which are physical coordinates of distance  and evaluate the function, you get units of temperature.
Your Mx equation doesn’t give you units of flow when you input the x coordinate. Dimensionally it just doesn’t yield what you want it to.
And there does not exist any equation that evaluates the cosine of a dimensional quantity. A radian is a ratio of lengths, and is hence dimensionless.
Hope this helps!
It's hard to comment on this. Universally everything I saw when studying equations like this made zero mention of units at any stage, for constants and for each term, my conclusion was that they were used simply as coordinates and it was only when all was said and done that it was applied directly to the physical phenomenon as is.
On the units present, M is basically dimensionless in of itself. It's essentially the amount of space in a certain space; that's L/L. Equally, I'll admit when I used the word 'flow' it was inaccurate, that's just how I tend to visualize it; at a basic level a flow is just the description of how something changes, and the derivative is by definition rate of change. Either way you wouldn't get the units for a flow of a fluid.
As far as radians go, can fix that with relative ease without changing the formula.

The heat equation absolutely agrees dimensionally. Alpha is the thermal diffusivity, and has a dimension. It is set to unity when you are solving the problem in units of alpha. But this does not somehow erase the dimensions from the final answer. U has units of kelvin. It’s independent variables are x,y,z. So when you plug in those coordinates  which are physical coordinates of distance  and evaluate the function, you get units of temperature.
Your Mx equation doesn’t give you units of flow when you input the x coordinate. Dimensionally it just doesn’t yield what you want it to.
And there does not exist any equation that evaluates the cosine of a dimensional quantity. A radian is a ratio of lengths, and is hence dimensionless.
Hope this helps!
It's hard to comment on this. Universally everything I saw when studying equations like this made zero mention of units at any stage, for constants and for each term, my conclusion was that they were used simply as coordinates and it was only when all was said and done that it was applied directly to the physical phenomenon as is.
On the units present, M is basically dimensionless in of itself. It's essentially the amount of space in a certain space; that's L/L. Equally, I'll admit when I used the word 'flow' it was inaccurate, that's just how I tend to visualize it; at a basic level a flow is just the description of how something changes, and the derivative is by definition rate of change. Either way you wouldn't get the units for a flow of a fluid.
As far as radians go, can fix that with relative ease without changing the formula.
Hmmm, that is a surprise to me. One of the first things we teach our physics students is something called dimensional analysis, which trains them to always assure that the units balance. Any equation that does not do this is by definition unphysical (and is the worst mistake a student can make on their problem sets). But they are trained for this, of course, and I do not intend criticism of you, only to convey assurance that this is indeed true.
The conceptual difficulty here is that describing the amount of space in a space requires the space to exist in something. It makes no sense to talk about the space volume, i.e., the amount of space in a given volume, or the space density if you prefer, because the space DEFINES the volume. We really need space to exist in something for this to work. In other words, at some point we’ll need to either define some kind of substrate that space exists within, or resort to a geometrical definition of concentration. Either option may be viable, but you should be the one to make the call.

Hmmm, that is a surprise to me. One of the first things we teach our physics students is something called dimensional analysis, which trains them to always assure that the units balance. Any equation that does not do this is by definition unphysical (and is the worst mistake a student can make on their problem sets). But they are trained for this, of course, and I do not intend criticism of you, only to convey assurance that this is indeed true.
The conceptual difficulty here is that describing the amount of space in a space requires the space to exist in something. It makes no sense to talk about the space volume, i.e., the amount of space in a given volume, or the space density if you prefer, because the space DEFINES the volume. We really need space to exist in something for this to work. In other words, at some point we’ll need to either define some kind of substrate that space exists within, or resort to a geometrical definition of concentration. Either option may be viable, but you should be the one to make the call.
I approached on more mathematical grounds, that might be all. I'm familiar with dimensional analysis, just never in applying it to PDEs.
Mathematically it is being modelled with reference to an underlying dimension, just because that's the easiest way to model it. In practise it's essentially geometrical, the same as curvature; it's not like space curves through another dimension, it's just how we refer to it because we're talking in a language invented to point out which tree had the nice fruit up it rather than outline the fundamentals of reality. If we have two identical objects, we can put one in location A, then if the other appears larger at location B and smaller at location C than the original (from the perspective from location A), then B would be defined to have a lower concentration and C to have a higher. Equally, if you were at location B then both would seem smaller, and at C both would seem larger.

Hmmm, that is a surprise to me. One of the first things we teach our physics students is something called dimensional analysis, which trains them to always assure that the units balance. Any equation that does not do this is by definition unphysical (and is the worst mistake a student can make on their problem sets). But they are trained for this, of course, and I do not intend criticism of you, only to convey assurance that this is indeed true.
The conceptual difficulty here is that describing the amount of space in a space requires the space to exist in something. It makes no sense to talk about the space volume, i.e., the amount of space in a given volume, or the space density if you prefer, because the space DEFINES the volume. We really need space to exist in something for this to work. In other words, at some point we’ll need to either define some kind of substrate that space exists within, or resort to a geometrical definition of concentration. Either option may be viable, but you should be the one to make the call.
I approached on more mathematical grounds, that might be all. I'm familiar with dimensional analysis, just never in applying it to PDEs.
Mathematically it is being modelled with reference to an underlying dimension, just because that's the easiest way to model it. In practise it's essentially geometrical, the same as curvature; it's not like space curves through another dimension, it's just how we refer to it because we're talking in a language invented to point out which tree had the nice fruit up it rather than outline the fundamentals of reality. If we have two identical objects, we can put one in location A, then if the other appears larger at location B and smaller at location C than the original (from the perspective from location A), then B would be defined to have a lower concentration and C to have a higher. Equally, if you were at location B then both would seem smaller, and at C both would seem larger.
Okay, so geometrical it is.
Now, the question is how to formalize it in a way that is consistent and testable.
Your idea is quite similar to special relativity, which already accounts for the distortion of objects sizes (Lorentz contraction), and only assumes the speed of light to be constant  a very bare bones set of assumptions.
To adapt this to geometry is the key, and we cannot use the trick of special relativity since you want the time intervals to be concentration independent (the time it takes to travel is adjacent coordinates is invariant). Hence, what then must change are the speeds. This means, for two observers in the same reference frame, they will measure the same object moving at different speeds and having a different size, but will agree on the travel time.
This should be testable. The question is how much concentration do we need to notice a measurable difference in the speed, and can we produce that on Earth?
Any ideas?

To adapt this to geometry is the key, and we cannot use the trick of special relativity since you want the time intervals to be concentration independent (the time it takes to travel is adjacent coordinates is invariant). Hence, what then must change are the speeds. This means, for two observers in the same reference frame, they will measure the same object moving at different speeds and having a different size, but will agree on the travel time.
This should be testable. The question is how much concentration do we need to notice a measurable difference in the speed, and can we produce that on Earth?
Any ideas?
Speed probably would be the best way to directly measure this aspect, yes. Producing's something of a question mark, a way for matter to directly influence is not something I have more than speculation on, but as far as using preexisting concentrations go that might be doable. The question mark would be, like you say, how much of a variation would be needed to observe a difference in speed. That much is what the equation is meant to answer, shoulder it be solved, so precise figures are currently not feasible.
But, so the theory goes, concentrations do increase at higher altitudes. That would be the most practical example of this issue. The biggest problem I see there would be measuring the speed from the more distant observer.

Your altitude boundary condition, Mz=g(hz)^2, has units of s^(2)m^(1). So this means that it is not time invariant. That is, it will not permit the time interval to be the same between adjacent spacetime coordinates.
I recommend using speed in your equations, since that is what must change depending on the observer. Somehow, the resulting speed must change depending on the observer’s position in that Reference frame, and must do so without any time dependence. Something like this:
Mz=R_abcd*g^(ab)*[v^(c)]^2/[(hz)*g]
Where R_abcd is the curvature tensor, g^(ab) is your curvature metric, v^(c) is your fourvelocity vector, and g is local gravity.
The curvature tensor contracts across two spacetime coordinates with the metric to give you the area curvature, and then this contracts with your four velocity to give proper velocity along one axis (z). The proper velocity will be modified by the height (z) above some scale factor (h) and depends on local gravity there.
Try tinkering with this a while. The next stage is to discover transformation equations so that the proper velocity appears different to observers depending on their coordinates in that frame.

Your altitude boundary condition, Mz=g(hz)^2, has units of s^(2)m^(1). So this means that it is not time invariant. That is, it will not permit the time interval to be the same between adjacent spacetime coordinates.
It's g the mathematical constant, not the acceleration.
Try tinkering with this a while. The next stage is to discover transformation equations so that the proper velocity appears different to observers depending on their coordinates in that frame.
Will try to unpack it in a sec.

Your altitude boundary condition, Mz=g(hz)^2, has units of s^(2)m^(1). So this means that it is not time invariant. That is, it will not permit the time interval to be the same between adjacent spacetime coordinates.
It's g the mathematical constant, not the acceleration.
Try tinkering with this a while. The next stage is to discover transformation equations so that the proper velocity appears different to observers depending on their coordinates in that frame.
Will try to unpack it in a sec.
Do you mean Newton’s constant? That’s G=6.7*10^(11) m^3 kg^(1) s^(2).
In this case, Mz has units of m*kg^(1)*s^(2) and is still not time invariant.
Cool, let me know how it strikes you.

Your altitude boundary condition, Mz=g(hz)^2, has units of s^(2)m^(1). So this means that it is not time invariant. That is, it will not permit the time interval to be the same between adjacent spacetime coordinates.
It's g the mathematical constant, not the acceleration.
Try tinkering with this a while. The next stage is to discover transformation equations so that the proper velocity appears different to observers depending on their coordinates in that frame.
Will try to unpack it in a sec.
Do you mean Newton’s constant? That’s G=6.7*10^(11) m^3 kg^(1) s^(2).
In this case, Mz has units of m*kg^(1)*s^(2) and is still not time invariant.
Cool, let me know how it strikes you.
The mathematical constant, no units. It's just there to give the rate of variation, which is connected to the cause of gravity but isn't directly it.

Your altitude boundary condition, Mz=g(hz)^2, has units of s^(2)m^(1). So this means that it is not time invariant. That is, it will not permit the time interval to be the same between adjacent spacetime coordinates.
It's g the mathematical constant, not the acceleration.
Try tinkering with this a while. The next stage is to discover transformation equations so that the proper velocity appears different to observers depending on their coordinates in that frame.
Will try to unpack it in a sec.
Do you mean Newton’s constant? That’s G=6.7*10^(11) m^3 kg^(1) s^(2).
In this case, Mz has units of m*kg^(1)*s^(2) and is still not time invariant.
Cool, let me know how it strikes you.
The mathematical constant, no units. It's just there to give the rate of variation, which is connected to the cause of gravity but isn't directly it.
You mean it is just the number in front? If it gives a rate then it must have units. That’s what a rate means.
The difference between math and physics is that math is pure, it uses no units because it has not yet been applied to something. When you apply it  and do applied mathematics, then all the equations, functions, graphs, numbers, acquire a dimension that represents the application. It is erroneous to develop physical models without units. It is an oxymoron. We do not have the option.
If it is pure mathematics that you wish to do instead, then I can try to help you with that. Just understand that it will not relate to anything physical, and will instead reference previously substantiated mathematics as its assumption set.

You mean it is just the number in front? If it gives a rate then it must have units. That’s what a rate means.
The difference between math and physics is that math is pure, it uses no units because it has not yet been applied to something. When you apply it  and do applied mathematics, then all the equations, functions, graphs, numbers, acquire a dimension that represents the application. It is erroneous to develop physical models without units. It is an oxymoron. We do not have the option.
If it is pure mathematics that you wish to do instead, then I can try to help you with that. Just understand that it will not relate to anything physical, and will instead reference previously substantiated mathematics as its assumption set.
It's always pure until it's applied, getting it working in the abstract is far more important to me because if that can't be done then there's no point making all the tweaks necessary to apply it physically. At the end of the day all that's going to be required to bring units in line are a couple of constants as multiples, and that's as good as trivial. The same goes for Mx and My. Working without them is just a whole lot simpler, and I've seen far too many instances of math being used to bamboozle rather than aid understanding and I'm well aware that making this any more complicated is just going to have people brush it off as pseudoscientific evasion.
M_{z} is the rate of change. g is a constant to give quantity, and in retrospect it probably wouldn't be the 9.8... value anyway as that's be Mt over the direction z, while this is just connected to that. It's not different to any constant, it's just g for gravity because it's defining the value of that quantity, but alone it means nothing.

You mean it is just the number in front? If it gives a rate then it must have units. That’s what a rate means.
The difference between math and physics is that math is pure, it uses no units because it has not yet been applied to something. When you apply it  and do applied mathematics, then all the equations, functions, graphs, numbers, acquire a dimension that represents the application. It is erroneous to develop physical models without units. It is an oxymoron. We do not have the option.
If it is pure mathematics that you wish to do instead, then I can try to help you with that. Just understand that it will not relate to anything physical, and will instead reference previously substantiated mathematics as its assumption set.
It's always pure until it's applied, getting it working in the abstract is far more important to me because if that can't be done then there's no point making all the tweaks necessary to apply it physically. At the end of the day all that's going to be required to bring units in line are a couple of constants as multiples, and that's as good as trivial. The same goes for Mx and My. Working without them is just a whole lot simpler, and I've seen far too many instances of math being used to bamboozle rather than aid understanding and I'm well aware that making this any more complicated is just going to have people brush it off as pseudoscientific evasion.
M_{z} is the rate of change. g is a constant to give quantity, and in retrospect it probably wouldn't be the 9.8... value anyway as that's be Mt over the direction z, while this is just connected to that. It's not different to any constant, it's just g for gravity because it's defining the value of that quantity, but alone it means nothing.
Okay, so this an entirely different ball game then. Developing the pure math is much, much harder.
So the place to begin is by stating your assumptions. What mathematics are you building off of? Are you assuming an arithmetic? If so, is it binary? If you’re assuming the modern fundamental theorem of arithmetic than this has consequences that you need to learn. Otherwise it is very easy to contradict one of them without knowing. Are you using standard algebraic structures? Rings, fields, group theory? How about vector spaces and complex analysis?
If you want to use topology, which I bet you’ll end up needing, then understanding how you develop manifolds and perform algrebra on them is essential.
In summary, I applaud and encourage your pursuit into pure math, but do so with your eyes open  it is a much more abstract and technically demanding task than building a physical model. The development would proceed with a series of lemmas leading to the fundamental proof of the work. Then this would serve as the stepping stone for corollaries, and further lemmas to build a network of consistent proofs that detail your theory.

Did this progress any further?