# Conjecture on the non-homogeneity of all matter

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After more thought on black hole singularities being coordinate singularities (or if I'm using the term wrong, rather: singularities that disappear depending on where you view them from), I figure that the solution that makes the most sense is that, uh...

Say you're outside a black hole and that most of its mass is in the singularity, but not all of it is. As you pass the event horizon and approach the singularity, suppose that rather than the singularity disappearing, that more and more of its mass appears as "normal matter" outside the singularity, which itself becomes less massive. You could approach it "forever" as it expands spatially the closer you are to it, and more of its mass would expand out of it until you realize that you're surrounded by a universe that came from the "shrinking" singularity that you're still chasing.

In order for that to be possible, the mass distribution of a black hole cannot be uniform or homogeneous or whatever. There would not be a hard boundary between outside and inside it (other than the event horizon, which is a precise boundary but there is no physical wall of matter or energy there). It would be distributed along something that looks like f( r ) = 1/r or 1/r2, with the density at 0 undefined (representing the singularity), and the density approaching infinity as r approaches 0.

Extrapolating this idea from black holes to all matter, we get the following conjecture:

- All mass is non-homogeneous in terms of energy or mass distribution.

- All mass has a singularity at its center.

Basically this would mean that the concentration of any distinct quantity of mass is greatest at its center, and tapers off to blend seamlessly into the surrounding nothingness, rather than there being a distinct boundary between mass and surrounding space. Depending on how you look at the mass, it could be that it has no size and 100% of its mass is contained in a singularity, or half of its mass is, or just a tiny fraction of its mass is contained in the singularity, yet that still represents infinite density for that small mass.

We can extrapolate further and imagine that any mass can be described as a distinct unit in the same way. On the smallest scale, all particles could be viewed as individual masses with individual singularities. On a larger scale: If you were far enough away or warped space in the right way, all of Earth could be viewed as a combined mass with most of its matter contained in one singularity at its center. If you were outside the universe, most of it would be in one singularity, with some of its mass outside the singularity (and each particle of that outside mass containing its own singularity).

Then since we're speculating without restraint anyway, why not conjecture that all fundamental forces are due to non-homogeneity of geometry, IE. curvature of spacetime. Just as large-scale curvature effects gravity, small-scale curvature may effect electromagnetism and/or nuclear force.

Thrown in there is the idea that any mass might be described as a particle, depending on how and from where you viewed it. Thus, particles might be defined as an observer-defined quantization of matter into individual indivisible components. Then, just as a universe might be fully contained in a singularity, or might "spill out" into something with size (eg. a black hole) and divisible mass, so too might an elementary particle be a singularity or a divisible mass, depending on how it is viewed.

A simplification of this idea might be:

- All mass results in space-time curvature (already accepted with general relativity?)

- The point of maximum curvature of any curve in spacetime is always a singularity. (There are no "gentle bumps" in spacetime.)

Any related or contradictory ideas or evidence? Thanks.

Edited by md65536
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Then since we're speculating without restraint anyway, why not conjecture that all fundamental forces are due to non-homogeneity of geometry, IE. curvature of spacetime. Just as large-scale curvature effects gravity, small-scale curvature may effect electromagnetism and/or nuclear force.

You can write the classical action for the electromagnetic force, the weak and the strong (and more general Yang-Mills type theories) in terms of curvature tensors related to connections in principle bundles or on associated vector bundles.

In short, yes all known forces have something to do with curvature. Gravity is to do with the curvature associated with the frame bundle of a pseudo-Riemannian manifold. Alternatively, you can think about the tangent bundle as the associated vector bundle. Either way, as these bundles come for free once you have the structure of a (smooth) manifold it is customary to talk about the curvature of the manifold and not these bundles.

When dealing with Yang-Mills like theories extra structure is needed and you don't talk about the underlying space-time as being curved. Instead you have to think in terms of principle bundles over the space-time manifold.

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In short, yes all known forces have something to do with curvature. Gravity is to do with the curvature associated with the frame bundle of a pseudo-Riemannian manifold. Alternatively, you can think about the tangent bundle as the associated vector bundle. Either way, as these bundles come for free once you have the structure of a (smooth) manifold it is customary to talk about the curvature of the manifold and not these bundles.

Way over my head but I'll have to research those topics if I ever try to develop the theory, thanks.

A smooth manifold would have no singularities? So it all works without them, and may not (or may) work with them?

Basically this would mean that the concentration of any distinct quantity of mass is greatest at its center, and tapers off to blend seamlessly into the surrounding nothingness, rather than there being a distinct boundary between mass and surrounding space.

This might mean that there are no abrupt edges to matter. The matter at a table's edge doesn't end there but carries on to exist (in a superficial form) through all of space. The hard edge that we experience might be similar in some way to an event horizon, dividing the matter into a volume where light and other matter interact with it, and a volume where they don't.

The physical presence of matter would coincide with its effect on spacetime curvature. There wouldn't need to be a distinction between things like "The matter is over here but it curves spacetime way over there." Any matter would "fill" the curvature that it causes to spacetime. Just as a single molecule has a tiny but calculable gravitational effect in a location a light year away, that molecule's tiny speculative energy density should be calculable at the same location.

This might relate to the aspect of the holographic principle that all matter in a volume maps to all points on a holographic surface. Basically: Any matter would exist everywhere at once, but it is only fully "experienced" at a small location where its concentration becomes infinite. All matter existing everywhere at once is also compatible with the idea that the universe can be fundamentally described as a singularity, with time and space being emergent observational effects.

Edited by md65536
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This might mean that there are no abrupt edges to matter. The matter at a table's edge doesn't end there but carries on to exist (in a superficial form) through all of space. The hard edge that we experience might be similar in some way to an event horizon, dividing the matter into a volume where light and other matter interact with it, and a volume where they don't.

This might relate to the aspect of the holographic principle that all matter in a volume maps to all points on a holographic surface. Basically: Any matter would exist everywhere at once, but it is only "experienced" at a small location where its concentration becomes infinite. All matter existing everywhere at once is also compatible with the idea that the universe can be fundamentally described as a singularity, with time and space being emergent observational effects.

I am really interested in these kinds of ideas. I find that people have a hard time with them, though, because it comes more naturally to them to think of objects as external to each other and themselves as external to situations as they perceive them. I'm not sure what it would take, cognitively speaking, to enable people to be able to think in terms of internal configurations of intersecting entities, but if anything I think it would render mathematics impotent (saying this without sufficient breadth of knowledge about mathematics, btw). As far as I know, all mathematical logic relies on a boolean-type notion that a set of elements must be mutually exclusive unless they themselves are sets with their own mutually exclusive elements/contents. But otherwise, why couldn't the table be inside the chair and the chair inside the table at the same time?

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I am really interested in these kinds of ideas. I find that people have a hard time with them, though, because it comes more naturally to them to think of objects as external to each other and themselves as external to situations as they perceive them. I'm not sure what it would take, cognitively speaking, to enable people to be able to think in terms of internal configurations of intersecting entities, but if anything I think it would render mathematics impotent (saying this without sufficient breadth of knowledge about mathematics, btw). As far as I know, all mathematical logic relies on a boolean-type notion that a set of elements must be mutually exclusive unless they themselves are sets with their own mutually exclusive elements/contents. But otherwise, why couldn't the table be inside the chair and the chair inside the table at the same time?

Well, there's all sorts of abstract mathematics that can be used to describe real and unreal things. Singularities are simple things in math but they don't make sense given an assumption of a fundamental continuous geometry of the universe. However, if you explore some ideas that hint that geometry is not fundamental, then you can ignore geometry for awhile and imagine other things. For example, topology can be used without requiring geometry. It is still math.

If we assume that geometry is fundamental and any mathematical description of the universe must fit within that geometry, then singularities are a problem.

If we assume that geometry emerges from some other fundamental description of the universe, then whatever is consistent might be possible. Singularities might be a convenient means for simple consistency, and thus might be common.

I think math is necessary because without it, we must use other conceptual tools such as language and spatial reasoning, and I think we would then limit our understanding to what we've seen or experienced. Is it possible to figure out a universe that may exist fundamentally "underneath" 3- or 4-D geometry, using spatial reasoning that's based on that geometry?

So yes, I suppose if you remove the mathematical/geometrical restrictions that says a table can't intersect a chair etc, you can work with more abstract ideas... but we have to be careful about what we claim because words like "inside" may only have a precise definition with respect to a given geometric representation.

I would think that studying topology would be the best way to precisely contemplate junk abstractly without being confined to our habit of thinking geometrically???

Edited by md65536
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So yes, I suppose if you remove the mathematical/geometrical restrictions that says a table can't intersect a chair etc, you can work with more abstract ideas... but we have to be careful about what we claim because words like "inside" may only have a precise definition with respect to a given geometric representation.

In a sense, I can say I'm "inside a building" because I define the inside as being the area that starts when I walk through the door and can no longer see the outside walls. However, in another sense, "inside the building" could be taken to refer to insulation, wiring, plumbing, or other things that are not visible without penetrating some surface of the building. I suppose that's a bad example because it is contextual, but take the example of a cloud instead. From a far distance, a cloud can appear to have a very well-defined perimeter that is opaque to light-penetration. However, as you approach the cloud in an aircraft, it is never really clear exactly when you penetrate the surface to enter the "interior" of the cloud. In fact, when it's foggy I usually think about being inside a cloud from some other perspective.

With physical force-fields, I think the issue becomes more fundamental. If you are standing on the surface of the Earth, you are clearly in its gravity-well (field). But the moon is also in the Earth's gravitational field, so when you are standing on the moon are you in both fields at the same time? Likewise, are we on Earth in both the sun's gravity field AND the Earth's? What about the moon's? The tides suggest that we are. But still we can say that we're "on Earth" but not "in the Sun" or "in the moon" right?

But why? What is it about crossing into the atmosphere from outer space that justifies saying that one is now "on Earth" instead of "in outer space?" Since the moon has no atmosphere, can we then say that we are not yet "on the moon" if we are 10m above the surface even though we can fly in a plane 20,000ft above sea level on Earth and say we are still "on Earth?" Likewise, if the moon is just as much part of the Earth's gravity-well as is Mt. Everest, why shouldn't we say we are "on Earth" when on the moon? Doesn't the intersection of the gravity-wells make the Earth-moon system akin to what a molecule is vis-a-vis its constituent atoms?

If anything, it seems to me that the boundaries of massive bodies are assumed on the basis of the intersection between electromagnetic and electrostatic forces and gravitational forces. If you were to go by gravitation alone, would there be any basis for delimiting the gravitational field according to how the protons and electrons within it are arranged? Couldn't these other forces just be seen as "internal topology" within a gravity field? Likewise, couldn't the sun and planets be viewed as "internal topology" of a single solar gravity field? This is all admittedly counter-intuitive, but I wonder how accurate intuition is when dealing with objective physical relations. Certainly intuition always seems to be misleading at the sub-atomic level.

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A smooth manifold would have no singularities? So it all works without them, and may not (or may) work with them?

Doing general relativity very carefully in the presence of singularities usually means you have to cut them out, so that you remain working with smooth manifolds. But I would not yet let that over worry you.

If we assume that geometry is fundamental and any mathematical description of the universe must fit within that geometry...

It seems that way. Physics is geometry.

If we assume that geometry emerges from some other fundamental description of the universe, then whatever is consistent might be possible. Singularities might be a convenient means for simple consistency, and thus might be common.

Classical electromagnetism has singularities. The famous one is due to the electron self energy. This is all cured when passing to a quantum description. It is expected that the singularities in general relativity can be cured in a similar way. So, singularities probably represent our misunderstanding of the physics and our holding on to classical ideas where we should not.

I would think that studying topology would be the best way to precisely contemplate junk abstractly without being confined to our habit of thinking geometrically???

Topology is very important in understanding physics and geometry. Classical gravity is very geometric, passing to a quantum description would need many ideas from topology.

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With physical force-fields, I think the issue becomes more fundamental. If you are standing on the surface of the Earth, you are clearly in its gravity-well (field). But the moon is also in the Earth's gravitational field, so when you are standing on the moon are you in both fields at the same time? Likewise, are we on Earth in both the sun's gravity field AND the Earth's? What about the moon's? The tides suggest that we are. But still we can say that we're "on Earth" but not "in the Sun" or "in the moon" right?

Tides are caused by gravitational gradient, not magnitude (the sun's gravitational pull is stronger but the moon's gradient is steeper due to it being closer and thus the moon's effect on tides is greater).

Gravitational fields intersect or overlap. The gravitational pull of the sun and moon and Earth summed together would be the same as summing the gravitational pull of all of the particles that make up the 3 bodies. This is similar to the above idea that any mass can be considered a particle, from the right viewpoint. From some locations, Earth's gravitational pull is identical to treating Earth as a point mass (right?). I think that from the right locations, all physical aspects of Earth can be made indistinguishable from a point particle with the right properties.

The meaning of "on earth" or "on the moon" is semantics. I don't think it needs to be defined to precisely describe gravity. However, it does bring up a good point: We know and can measure that gravitational fields intersect, but there is no evidence that mass or energy overlaps the same way. In fact it most certainly doesn't. The mass-energy distribution of a mass probably could not be 1/r or 1/r^2 because the integral of those over all of space is infinite. Unless we attribute vacuum energy or dark matter to this theory (which wouldn't work anyway cuz neither of those are infinite), we would probably require that the total mass is finite and in fact it should be equal to existing mass measurements of whatever mass we're considering. So the speculative "influence over area" of mass energy does not have the same drop-off function as the space-time curvature drop-off function of a mass. (Does that make sense?)

So this idea certainly doesn't magically unify the forces. Mass energy and the spacetime curvature due to said mass would need different distributions. Why would that be? The distribution of the mass of a particle would have to drop off at a high enough rate that we could consistently observe that all the mass appears to be within a fixed boundary, in all observations in all of our history (whether it be planets or tables or particle experiments).

UNLESS... I'm confusing a gravitational force function (proportional to 1/r^2) with a spacetime curvature function (which I have no understanding of). Googling it, I see that "space is nearly flat for weak gravitational fields". My conjecture is that this is not true, for a very small volume within any mass: At this point, spacetime is infinitely curved.

Is it possible to "pinch" spacetime in a very small volume, such that:

1. It is infinitely curved at many points (wherever there is a mass particle), but

2. On average it is "nearly flat". On average, the curvature is exactly as GR predicts.

???

It seems that way. Physics is geometry.

I think that eventually, physics will explain all of geometry, and will also transcend it and explain things that are "more fundamental" than geometry.

Isn't this already partly true? Can (some) thermodynamic systems be described without geometry? Or the holographic principle?

Classical electromagnetism has singularities. The famous one is due to the electron self energy. This is all cured when passing to a quantum description. It is expected that the singularities in general relativity can be cured in a similar way. So, singularities probably represent our misunderstanding of the physics and our holding on to classical ideas where we should not.

Topology is very important in understanding physics and geometry. Classical gravity is very geometric, passing to a quantum description would need many ideas from topology.

I think it would make sense if singularities exist in the geometry of space (just like they do in math), but only in the geometry and are "cured" in a topological description of the universe.

Is the quantum description of electrons a geometrical one? I suppose it would also make sense if singularities disappear from the geometry with a new or more sophisticated description (this is really all over my head; I don't know what I'm saying).

But... I think that singularities in the geometry are too convenient a thing to assume they're not really there.

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Tides are caused by gravitational gradient, not magnitude (the sun's gravitational pull is stronger but the moon's gradient is steeper due to it being closer and thus the moon's effect on tides is greater).

I've never thought about that, but it makes sense.

Gravitational fields intersect or overlap. The gravitational pull of the sun and moon and Earth summed together would be the same as summing the gravitational pull of all of the particles that make up the 3 bodies. This is similar to the above idea that any mass can be considered a particle, from the right viewpoint. From some locations, Earth's gravitational pull is identical to treating Earth as a point mass (right?). I think that from the right locations, all physical aspects of Earth can be made indistinguishable from a point particle with the right properties.

How can you be sure that gravitation doesn't cancel itself out to some extent, even relative to a satellite far away? E.g. supposedly the center of the planet is weightless due to gravity canceling itself out in all directions, but then are solar and lunar gravity still present as the dominant gravitational forces there? Actually, that's a bad example. The issue would be whether the cancellation of gravity in the center of the planet gradually decreases as you move away from the center, and then does it also gradually decrease from the lagrangian points between the Earth, sun, and moon. In other words, when is gravity "pure" and not "net gravitation" that results from all interacting fields? I suppose this is what "spacetime curvature" refers to.

The meaning of "on earth" or "on the moon" is semantics. I don't think it needs to be defined to precisely describe gravity. However, it does bring up a good point: We know and can measure that gravitational fields intersect, but there is no evidence that mass or energy overlaps the same way. In fact it most certainly doesn't. The mass-energy distribution of a mass probably could not be 1/r or 1/r^2 because the integral of those over all of space is infinite. Unless we attribute vacuum energy or dark matter to this theory (which wouldn't work anyway cuz neither of those are infinite), we would probably require that the total mass is finite and in fact it should be equal to existing mass measurements of whatever mass we're considering. So the speculative "influence over area" of mass energy does not have the same drop-off function as the space-time curvature drop-off function of a mass. (Does that make sense?)

I'm trying but I forgot what it means to find an integral. Is that related to taking the derivative of something to find the amount of change over time by knowing the rate?

So this idea certainly doesn't magically unify the forces. Mass energy and the spacetime curvature due to said mass would need different distributions. Why would that be? The distribution of the mass of a particle would have to drop off at a high enough rate that we could consistently observe that all the mass appears to be within a fixed boundary, in all observations in all of our history (whether it be planets or tables or particle experiments).

How does the rate of drop off determine the ability to observe anything? Doesn't that mix objective with subjective?

Is it possible to "pinch" spacetime in a very small volume, such that:

1. It is infinitely curved at many points (wherever there is a mass particle), but

2. On average it is "nearly flat". On average, the curvature is exactly as GR predicts.

???

Usually people say that gravity is negligible at the (sub)atomic level, because they're comparing it with the other forces that are relatively strong despite the minute masses involved. However, I don't know of any claims about the minimum volume of an electron and/or why it couldn't be so small that its gravity would be strong if anything could get miniscule-y close to it. I think there are supposedly other sub-electron particles that constitute it, though, and these are probably theorized as being bonded by stronger forces that gravity. Since electrons don't seem to stay in one place for any continuous amount of time, though, I don't know how they could ever maintain close enough proximity to anything to have their gravity play a determinant role.

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How can you be sure that gravitation doesn't cancel itself out to some extent, even relative to a satellite far away? E.g. supposedly the center of the planet is weightless due to gravity canceling itself out in all directions, but then are solar and lunar gravity still present as the dominant gravitational forces there? Actually, that's a bad example. The issue would be whether the cancellation of gravity in the center of the planet gradually decreases as you move away from the center, and then does it also gradually decrease from the lagrangian points between the Earth, sun, and moon. In other words, when is gravity "pure" and not "net gravitation" that results from all interacting fields? I suppose this is what "spacetime curvature" refers to.

I'm sure that gravity does cancel itself out as you suggest.

At a Lagrange point between the the Earth and moon, the overlapping spacetime curvature due to the Earth (curving in one way) and the moon (curving in the other way) combine to result in flat spacetime.

I don't think there's any distinction between "pure" and "net" gravitation: It's all net, whether you're describing a system with one atom, or a trillion stars.

I'm trying but I forgot what it means to find an integral. Is that related to taking the derivative of something to find the amount of change over time by knowing the rate?

Yes, it's the opposite of a derivative.

In this case, if you take the total mass energy of say a particle, then the derivative of that mass would represent the density of mass across space -- a mass distribution function. The integral of the mass distribution function would be the total sum of the mass.

I may not be describing that with the right terminology.

How does the rate of drop off determine the ability to observe anything? Doesn't that mix objective with subjective?

I'm not sure. However: We can make observations of mass and determine where that mass is. If I'm saying "that mass is actually spread out across all of space", that claim has to somehow match observed reality.

One way to do that is the conjecture that any such "spread out" mass somehow appears as a particle: The spread out mass appears to be only in one not-spread-out place.

Another way is if the mass density falls off at such an extreme rate that any mass energy outside of some distinct boundary is undetectable and unobservable and negligible, and thus all the mass appears to be within that hard boundary.

Edit: I think the first option is true. I think that the mass distribution of a mass depends on how it is observed. For example, from far enough a way, a table should appear to have a uniform mass density, no matter how it is measured. From closer up (or on a smaller scale), that uniform mass will separate into particles, each of which appear to have uniform mass density. If you get smaller/closer then the particles can be broken up into smaller particles. I suppose at some point there is a quantum limit to this. However, I suspect that any apparent uniform mass density that appears from any of these points of reference is not fundamental; it is only an observational side-effect.

Usually people say that gravity is negligible at the (sub)atomic level, because they're comparing it with the other forces that are relatively strong despite the minute masses involved. However, I don't know of any claims about the minimum volume of an electron and/or why it couldn't be so small that its gravity would be strong if anything could get miniscule-y close to it. I think there are supposedly other sub-electron particles that constitute it, though, and these are probably theorized as being bonded by stronger forces that gravity. Since electrons don't seem to stay in one place for any continuous amount of time, though, I don't know how they could ever maintain close enough proximity to anything to have their gravity play a determinant role.

Exactly... If we allow tiny singularities of infinite spacetime curvature inside any particle, then we may be able to model reality such that the singularities "average out" to form fairly flat spacetime on a large scale (effecting gravity), but have much stronger effects on very small scale (effecting EM and/or nuclear forces).

If we assume that spacetime curvature is actually smooth, then there's no way that I know of to make the same curvature "weak" enough to explain gravity, and "strong" enough to explain the other forces, and still maintain a gentle curvature for small masses.

Edited by md65536
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I'm sure that gravity does cancel itself out as you suggest.

At a Lagrange point between the the Earth and moon, the overlapping spacetime curvature due to the Earth (curving in one way) and the moon (curving in the other way) combine to result in flat spacetime.

I don't think there's any distinction between "pure" and "net" gravitation: It's all net, whether you're describing a system with one atom, or a trillion stars.

Right, but as you said about the moon's effect on the tides being due to the gradation of field-strength instead of absolute strength, "flat" spacetime could never really be flat insofar as there must always be some degree of "net force" acting on any object or particle above absolute zero, right? I suppose what I'm saying now is that an object's inertia is like it moving under its own gravity in the form of momentum. That sounds wrong because you think of gravity as an inherent force that emerges from mass itself whereas momentum is due to an energetic push, but I think you could view spacetime curvature (i.e. net gravitation) as including momentum, which could explain why geodesics for photons are different than those for satellites, no?

Yes, it's the opposite of a derivative.

In this case, if you take the total mass energy of say a particle, then the derivative of that mass would represent the density of mass across space -- a mass distribution function. The integral of the mass distribution function would be the total sum of the mass.

I'm still struggling to understand this. What kind of mass are you saying is distributed and how?

I'm not sure. However: We can make observations of mass and determine where that mass is. If I'm saying "that mass is actually spread out across all of space", that claim has to somehow match observed reality.

I thought you were just saying that the force-fields of constituent particles had no defined boundary. You can't observe a magnetic field using photons, presumably, because photons don't bounce of them but you can observe it using another magnetic field. But how do you define where the magnetic fields begin to interact and where not? If two bar magnets were suspended in a perfect vacuum devoid of any other forces or energies, would their magnetism eventually draw them together? If all energy were taken out of any system with any amount of attractive force, the force would have to collapse the system into a singularity, wouldn't it, unless there was some counteracting force to prevent it?

One way to do that is the conjecture that any such "spread out" mass somehow appears as a particle: The spread out mass appears to be only in one not-spread-out place.

But like a cloud or the sky itself, the effect is due to compound light-interference through a collection of particles that would be transparent in smaller amounts or concentrations. You can't technically define the edge of a cloud without setting an arbitrary level of relative humidity to denote the boundary, right? Otherwise a cloud is just a lump in humidity that fills the entire sky, no?

Another way is if the mass density falls off at such an extreme rate that any mass energy outside of some distinct boundary is undetectable and unobservable and negligible, and thus all the mass appears to be within that hard boundary.

In practice, I agree with you. But being "undetectable and unobservable and negligible" are relative and subjective, right? I supposed "detectable by any means" would be objective, but how can "negligible" be more than subjective?

Exactly... If we allow tiny singularities of infinite spacetime curvature inside any particle, then we may be able to model reality such that the singularities "average out" to form fairly flat spacetime on a large scale (effecting gravity), but have much stronger effects on very small scale (effecting EM and/or nuclear forces).

That is interesting. Maybe that could be how photons get absorbed by electrons, i.e. due to gravitation of the electron at a very minuscule level.

If we assume that spacetime curvature is actually smooth, then there's no way that I know of to make the same curvature "weak" enough to explain gravity, and "strong" enough to explain the other forces, and still maintain a gentle curvature for small masses.

I think the key to dealing with this issue would lie in how atoms and other particles maintain volume. The empty space of an atom could be viewed as making it "buoyant" and thus resistant to condensation to the point of be susceptible to collapse under the gravity of its constituent particles. Maybe the nuclear forces are not so much holding the nucleons together as they are preventing them from collapsing under their own microgravity. This is getting very speculative, though, so I don't know how unfavorable the referees will get when they read this kind of thinking.

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Right, but as you said about the moon's effect on the tides being due to the gradation of field-strength instead of absolute strength, "flat" spacetime could never really be flat insofar as there must always be some degree of "net force" acting on any object or particle above absolute zero, right? I suppose what I'm saying now is that an object's inertia is like it moving under its own gravity in the form of momentum. That sounds wrong because you think of gravity as an inherent force that emerges from mass itself whereas momentum is due to an energetic push, but I think you could view spacetime curvature (i.e. net gravitation) as including momentum, which could explain why geodesics for photons are different than those for satellites, no?

I think spacetime can be flat at a point (or over a volume of points), iff at that point the gravitational pull from all mass balances out to a net force of 0.

If the magnitude of gravitational force is directly related to curvature, then a gravitational gradient would involve the difference in curvature at different points, not the "steepness" of the curvature at those points. The main thing that affects tides is that the force of gravity from the moon at the point on Earth that is closest to the moon, is stronger from the force at the point on Earth farthest from the moon. This means a difference (gradient) in the curvature of space across the volume of the Earth.

You could have uniformly curved spacetime (ie. be in a uniform gravitational field) and tides shouldn't happen, but that doesn't mean that spacetime there is flat.

I don't think of gravity as an inherent force, but rather a kind of inertial motion in curved space.

Geodesics are "paths of zero acceleration", so satellites do not follow geodesics as they're constantly being accelerated due to gravity (inward gravitation balances outward inertia in a perfect orbit).

I'm still struggling to understand this. What kind of mass are you saying is distributed and how?

I'm speaking of any mass, specifically when viewed as a single indivisible unit (a particle, basically).

I'm saying that its distribution isn't uniform, but that it is infinitely dense at a point in its center. This describes a singularity.

Consider for example a spherical drop of water of radius R. Its mass distribution might be something like

f( r ) = { 0, r > R

1, r < R

This is a uniform distribution within its radius.

A distribution like f( r ) = 1/r^2 has a singularity at its center, but unfortunately it describes infinite mass.

So I'm not sure what type of distribution would fit what I'm proposing.

f( r ) = sin^2( r ) / r^2 would not be infinite (I think)... and it may capture a "wave nature" of matter... but unfortunately it has areas of negative mass-energy density which doesn't make sense to me, and it is not strictly decreasing which I assume matches reality.

f( r ) = 1 / C^r for some constant C might work, but I don't see how this has any connection with observations and reality).

I thought you were just saying that the force-fields of constituent particles had no defined boundary.

No... I think that's already accepted by all(?) gravitational theories. I'm talking about the uniformity of the mass that cause these gravitational fields, not the field itself.

You can't observe a magnetic field using photons, presumably, because photons don't bounce of them but you can observe it using another magnetic field. But how do you define where the magnetic fields begin to interact and where not? If two bar magnets were suspended in a perfect vacuum devoid of any other forces or energies, would their magnetism eventually draw them together? If all energy were taken out of any system with any amount of attractive force, the force would have to collapse the system into a singularity, wouldn't it, unless there was some counteracting force to prevent it?

Yes. 1/r (magnetism) and 1/r^2 (gravity) approach 0 as r approaches infinity, but they are non-zero for any finite r. They have no finite boundary, though far enough out they become negligible relative to nearer interstellar matter... or possibly even vacuum energy (dark energy?))

Yes, I think that as long as the objects didn't have inertia exceeding escape velocity, they should collapse.

But like a cloud or the sky itself, the effect is due to compound light-interference through a collection of particles that would be transparent in smaller amounts or concentrations. You can't technically define the edge of a cloud without setting an arbitrary level of relative humidity to denote the boundary, right? Otherwise a cloud is just a lump in humidity that fills the entire sky, no?

Yes, I suppose that's a good analogy. You could say that the boundary of the cloud is the volume in which all its water molecules are contained, but since the air around the cloud contains water molecules too, there's not a hard boundary. It's like a smooth transition between higher and lower humidity. You could define a precise boundary using some humidity limit.

But like you said earlier, from far away a cloud may look like a solid thing (perhaps with uniform density), but as you move into it (as with fog), the boundary of the cloud appears to change with your location. This is what I think happens with masses or particles.

In practice, I agree with you. But being "undetectable and unobservable and negligible" are relative and subjective, right? I supposed "detectable by any means" would be objective, but how can "negligible" be more than subjective?

I think the important things would be

1. It has to be consistent with all existing observations (subjective negligibility I suppose)

2. It has to be consistent with a sensible theoretical description (which might predict objective detectability)

That is interesting. Maybe that could be how photons get absorbed by electrons, i.e. due to gravitation of the electron at a very minuscule level.

I think the key to dealing with this issue would lie in how atoms and other particles maintain volume. The empty space of an atom could be viewed as making it "buoyant" and thus resistant to condensation to the point of be susceptible to collapse under the gravity of its constituent particles. Maybe the nuclear forces are not so much holding the nucleons together as they are preventing them from collapsing under their own microgravity. This is getting very speculative, though, so I don't know how unfavorable the referees will get when they read this kind of thinking.

I think we're safe in the Speculations forum as long as we're speculating and not asserting claims!

This conjecture wasn't meant as an explanation of gravity... to me it would only need to be consistent with existing gravitational theory.

The effect of infinite spacetime curvature within a particle might be related to gravity but I personally wouldn't call it gravity because it might confuse things (nuclear force is similar in ways to gravity but different enough to avoid calling it gravity).

Yeah... the issue of explaining volume of particles is key. I think the answer is that what we observe (of the subatomic world or the macroscopic world or anything), as far as volumes and empty space etc go, is not a fundamental aspect of the universe. Reality depends on how it is observed. That might open up some theoretical possibilities, but doesn't really help us figure out which are right :/

CRAZY IDEA:

- The mass distribution of a point mass is 1/r^2 but it is scaled by some infinitesimal factor, such that the mass is only apparent where the density is infinite (at the location of the singularity, though I don't know how it would then appear to have volume).

- This distribution represents "uniform mass density" throughout the entirety of a 2-dimensional universe...

1/r^2 means that a spherical shell of any radius would contain the same amount of mass as a shell of any other radius.

If we remove a dimension or 2 (time and distance, specifically), we might be able to describe a 2-D geometry in which all 3-D spheres of different radii become identical structures in the 2-D world. In this case, the mass then is uniformly distributed across the entire 2-D universe.

This is "nice" because it fits (vaguely) with the holographic principle, which among other things suggests that any point in a 3D volume maps to all points on a 2D holographic surface. This would allow a particle to be uniformly distributed across this surface.

Question: Is it possible to modify f( r ) = 1/r^2 such that an infinite 3D volume integral becomes finite? Would the "infinitesimal factor" have to be non-constant? I will try to ask in a math forum...

Edited by md65536
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I think spacetime can be flat at a point (or over a volume of points), iff at that point the gravitational pull from all mass balances out to a net force of 0.

If the magnitude of gravitational force is directly related to curvature, then a gravitational gradient would involve the difference in curvature at different points, not the "steepness" of the curvature at those points. The main thing that affects tides is that the force of gravity from the moon at the point on Earth that is closest to the moon, is stronger from the force at the point on Earth farthest from the moon. This means a difference (gradient) in the curvature of space across the volume of the Earth.

I got that about the tides, but this is a clearer explanation. As for the different sources of gravity balancing to net zero, this is what I was thinking about when I started saying all that stuff about inertia/momentum and gravity being the same. Just think about a point of net zero gravity? If you have an object at that point, that object is not being pulled in any direction by gravity, correct? So is it completely still? Probably not. If not, how is its motion/momentum different from gravitational force? I.e. think of Einstein's example of the acceleration of an elevator contributing to the net gravity inside the elevator. That elevator's momentum is thus, in effect, a contributor to net gravity no different than the planetary gravity it intersects with. Still, there is an intuitive difference to me between momentum/acceleration force and gravitation generated by mass.

I don't think of gravity as an inherent force, but rather a kind of inertial motion in curved space.

But if the curvature is the "net product" of all intersecting forces, gravitational as well as motion-acceleration, can you not "dissect" the situation for constituent gravities derived from multiple sources? This seems a bit like using a prism to separate out different colors of light from sunlight.

Geodesics are "paths of zero acceleration", so satellites do not follow geodesics as they're constantly being accelerated due to gravity (inward gravitation balances outward inertia in a perfect orbit).

I thought "geodesic" just referred to any path taken by an object or particle purely due to its own inertia. Here's a quote from wikipedia:

geodesics are defined to be (locally) the shortest path between points in the space. In the presence of an affine connection, geodesics are defined to be curves whose tangent vectors remain parallel if they are transported along it.The term "geodesic" comes from geodesy, the science of measuring the size and shape of Earth; in the original sense, a geodesic was the shortest route between two points on the Earth's surface, namely, a segment of a great circle.

I'm speaking of any mass, specifically when viewed as a single indivisible unit (a particle, basically).

I'm saying that its distribution isn't uniform, but that it is infinitely dense at a point in its center. This describes a singularity.

Consider for example a spherical drop of water of radius R. Its mass distribution might be something like

f( r ) = { 0, r > R

1, r < R

This is a uniform distribution within its radius.

A distribution like f( r ) = 1/r^2 has a singularity at its center, but unfortunately it describes infinite mass.

Ok, I get it. You're basically describing the shape of a spacetime 'dent' in geometric terms I can't completely interpret because of my limited math skills. 1/r^2 is basically the weakening of a gravitational field or photon emission as you move away from the source, right? 1/r^3 would be the nuclear force, right? (at first I thought it was a magnet but you said that is 1/r). But you're just talking about a singularity where force can approach infinity as the radius becomes infinitely small.

f( r ) = sin^2( r ) / r^2 would not be infinite (I think)... and it may capture a "wave nature" of matter... but unfortunately it has areas of negative mass-energy density which doesn't make sense to me, and it is not strictly decreasing which I assume matches reality.

f( r ) = 1 / C^r for some constant C might work, but I don't see how this has any connection with observations and reality).

Could the negative mass-energy density refer to repulsion instead of attraction? Could that be responsible for the erratic tunneling behaviors, etc. of electrons? Sorry if I'm making no sense. I'm barely able to make sense of what you're saying and why.

No... I think that's already accepted by all(?) gravitational theories. I'm talking about the uniformity of the mass that cause these gravitational fields, not the field itself.

Right, but what is "mass" as a cause for field-force? I mean, the atom contains the protons, which supposedly have mass but do they have volume? Is it possible to measure the volume/size of the nucleus of an atom? What's more, since the protons can basically not exist as mass without the electrons, do you call the volume of the atom as a whole a "uniform mass" that causes gravity? If so, why do you define the gravitational field in relation to the electrostatic field? Aren't they both just fields of force that emerge from the constituent particles? The negative charge of the electrons of a stable atom/molecule extends slightly beyond the positive charge of the nucleus, right? Why would that electrostatic field define its boundary as a particle more so than its gravitational field, however far or close that extends in isolation from other atoms/molecules?

Yes, I suppose that's a good analogy. You could say that the boundary of the cloud is the volume in which all its water molecules are contained, but since the air around the cloud contains water molecules too, there's not a hard boundary. It's like a smooth transition between higher and lower humidity. You could define a precise boundary using some humidity limit.

Right, but it's an arbitrary boundary chosen for subjective reasons, like focussing a camera lens.

Yeah... the issue of explaining volume of particles is key. I think the answer is that what we observe (of the subatomic world or the macroscopic world or anything), as far as volumes and empty space etc go, is not a fundamental aspect of the universe. Reality depends on how it is observed. That might open up some theoretical possibilities, but doesn't really help us figure out which are right :/

Reality doesn't depend on how it is observed. Our knowledge/perception of reality depends on how we perceive and know it. This discussion, to me, is about the ontology of boundaries. What constitutes a boundary and why? Is it ultimately adequate, however pragmatically useful, to define the boundaries of objects in terms of apparent surfaces visible due to light reaching a certain level of reflection or refraction/diffraction? When an object is immersed in another in a different phase, it is easy to claim that the surface of the object is its boundary, even though that may not be the case at the atomic level, right? At that level, it's the energetic motion of the particles that define them in terms of how close they can get to each other before being deflected, right? Two repellant magnets can get closer to each other before deflecting if they are moving more forcefully toward each other than if they have very little force. But yet it's the distance of deflection that defines the boundary of the repellant fields, no?

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Wait a sec... I've been on crack this whole time.

The infinite integral of 1/x is divergent.

The infinite integral of 1/x^2 is convergent, EXCEPT for the singularity at x=0.

So the "fall-off" of function f( r ) = 1/r^2 is fine... it doesn't imply an infinite mass,

unless we allow r=0. Suddenly the singularity is not so convenient :/

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Just think about a point of net zero gravity? If you have an object at that point, that object is not being pulled in any direction by gravity, correct? So is it completely still? Probably not. If not, how is its motion/momentum different from gravitational force? I.e. think of Einstein's example of the acceleration of an elevator contributing to the net gravity inside the elevator. That elevator's momentum is thus, in effect, a contributor to net gravity no different than the planetary gravity it intersects with. Still, there is an intuitive difference to me between momentum/acceleration force and gravitation generated by mass.

Force (such as gravity) on a mass results in acceleration. Net 0 gravity means no acceleration, but it still has its velocity independent of this (v can be 0 relative to other locations).

Yes, a frame's acceleration provides a force equivalent to gravity, but not its momentum.

I thought "geodesic" just referred to any path taken by an object or particle purely due to its own inertia. Here's a quote from wikipedia:

Sure, but satellites are accelerated by gravity. If they weren't, they would fly off on their own inertia, and follow the path that light would: a geodesic that curves along spacetime.

Ok, I get it. You're basically describing the shape of a spacetime 'dent' in geometric terms I can't completely interpret because of my limited math skills. 1/r^2 is basically the weakening of a gravitational field or photon emission as you move away from the source, right? 1/r^3 would be the nuclear force, right? (at first I thought it was a magnet but you said that is 1/r). But you're just talking about a singularity where force can approach infinity as the radius becomes infinitely small.

Yes, 1/r^2 describes the magnitude of gravity and other things as well, such as the density of a fixed amount of energy stretched across a spherical surface.

The initial conjecture is that there's a mass density singularity in any mass.

This suggests that the density would taper off to fill all of space, instead of having an abrupt edge between zero mass density and some finite mass density.

I think a mass density singularity would imply a spacetime curvature singularity.

It *might* be that the mass density and the spacetime curvature could be described using the same functions. Otherwise they'd have to be discussed separately.

Yes, I think that infinite spacetime curvature could imply infinite force.

That's 3 separate but highly related things (mass, curvature, force).

Could the negative mass-energy density refer to repulsion instead of attraction? Could that be responsible for the erratic tunneling behaviors, etc. of electrons? Sorry if I'm making no sense. I'm barely able to make sense of what you're saying and why.

Negative mass makes no sense to me. All I can say is that it probably has no relation to reality, but I don't know for sure. I'd rather avoid it, but if I couldn't avoid it, it might not destroy the theory (it probably would tho).

I'm not actually making sense! I proposed a function with sin(x) in it because the integral of that is finite, but that's because sin(x) has periodically negative values. Sin(x)/x is a bad guess anyway, because the limit as x approaches 0 is not infinite, which is what I'd want for a singularity of infinite mass density.

Reality doesn't depend on how it is observed.

Sure does! Lengths, and time, depend on an observer's relative velocity and gravitational field. Quantum mechanics deals with observers differently depending on interpretation, with variations from "observing reality affects it" to "differently observable realities exist in superposition."

Invariable aspects of reality don't depend on how it is observed. I equate that with "fundamental" aspects of the universe, with observer-dependent things being "emergent". Some aspects of geometry (eg. the curvature of space) are observer-dependent. I believe that all geometry is ultimately emergent.

Our knowledge/perception of reality depends on how we perceive and know it. This discussion, to me, is about the ontology of boundaries. What constitutes a boundary and why? Is it ultimately adequate, however pragmatically useful, to define the boundaries of objects in terms of apparent surfaces visible due to light reaching a certain level of reflection or refraction/diffraction? When an object is immersed in another in a different phase, it is easy to claim that the surface of the object is its boundary, even though that may not be the case at the atomic level, right? At that level, it's the energetic motion of the particles that define them in terms of how close they can get to each other before being deflected, right? Two repellant magnets can get closer to each other before deflecting if they are moving more forcefully toward each other than if they have very little force. But yet it's the distance of deflection that defines the boundary of the repellant fields, no?

There are and may need to be different boundaries or fields for the different forces. Those would determine how it interacts with other matter or energy. I think that the precise location of mass-energy of a quantum of matter describes a boundary of mass (which may be different from interactions).

You're going into topics that are beyond my knowledge or reasoning, though.

Edited by md65536
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Yes, a frame's acceleration provides a force equivalent to gravity, but not its momentum.

But momentum through spacetime curvature produces acceleration, no?

Sure, but satellites are accelerated by gravity. If they weren't, they would fly off on their own inertia, and follow the path that light would: a geodesic that curves along spacetime.

they don't follow the same path as light because they curve more than light does. Light only orbits where escape velocity is C.

Yes, I think that infinite spacetime curvature could imply infinite force.

That seems as impossible to me as the possibility that light contains infinite energy and infinite speed (instantaneity).

Negative mass makes no sense to me.

I agree except in terms of negative gravitation. What's so implausible about repellant gravity, especially considering how electrons behave around the nucleus?

There are and may need to be different boundaries or fields for the different forces. Those would determine how it interacts with other matter or energy. I think that the precise location of mass-energy of a quantum of matter describes a boundary of mass (which may be different from interactions).

And how exactly is that "precise location" determined then?

You're going into topics that are beyond my knowledge or reasoning, though.

I had that with you since a few posts back:) We respond where we deem our response relevant enough to share, no?

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But momentum through spacetime curvature produces acceleration, no?

No, I don't think so.

Do you mean that something moving along a geodesic changes direction? I don't think that's true either. The object might appear to curve from other viewpoints, because space itself is curved, but the geodesic is essentially a straight line in curved space. From the viewpoint of the object, the geodesic IS straight... it APPEARS always straight all along the geodesic. The object would not experience any acceleration.

> they don't follow the same path as light because they curve more than light does. Light only orbits where escape velocity is C.

I don't think this is a correct or meaningful way to describe it. "Orbiting" requires moving off of a geodesic (continuously, I suppose).

> That seems as impossible to me as the possibility that light contains infinite energy and infinite speed (instantaneity).

I don't see how that is the case.

My theory would have to have that the predicted behavior of light is no different than our accepted predicted behavior of light.

This is possible, because the singularities are inside particles, and "out of reach" of light. Light either misses the particle or is absorbed or redirected by the particle, but the light never reaches the singularity. This may in turn explain things like refraction, where to light, a "bumpy surface consisting of particles" can behave as a smooth flat surface to light which acts like a wave.

Basically, the infinite spatial curvature would have an "infinite" effect only for a radius of 0. It would quickly fall off, on the scale of subatomic particles or something like that. It would not be allowed to cause any sort of inconsistency -- if it could then the theory is wrong.

> I agree except in terms of negative gravitation. What's so implausible about repellant gravity, especially considering how electrons behave around the nucleus?

Well, for negative gravitation, you'd have to have spacetime that curves in the opposite way from "normal", so that this "repellent source" causes length expansion and time speeding up. I don't know of anything that would be explained by this, or any reason to suggest it's possible, or any aspect of reality that seems related.

It might also involve a paradox, such as "Spacetime that is flatter than perfectly flat."

> And how exactly is that "precise location" determined then?

I don't know, but experiments and observations and derivations and theories have calculated the size of various particles.

Observations of mass and light and various interactions and measurements can determine where particles are.

It's subject to the uncertainty principle, but I don't know if that relates to all this.

> I had that with you since a few posts back:) We respond where we deem our response relevant enough to share, no?

Yes, but I feel compelled to try to respond to all questions, even when my answers are just a guess! Much of what I'm saying is nonsense, I'm sure! lol

The problem with integrating 1/r^2 at r=0 makes me feel like I've hit a wall with being able to reason about this topic.

Sorry for the inline quotes... I got lazy.

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I think that eventually, physics will explain all of geometry, and will also transcend it and explain things that are "more fundamental" than geometry.

Isn't this already partly true? Can (some) thermodynamic systems be described without geometry? Or the holographic principle?

It is true that many ideas from physics have been put to good use in geometry and low dimensional topology. Now, the question of if physical ideas are enough to describe all of geometry is very much open. The trouble as I see it is that there has been so much cross-fertilisation between geometry and physics it is now hard to separate the two clearly.

Is the quantum description of electrons a geometrical one?

Ideas from geometry do appear in both the semi-classical and quantum theories.

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It is true that many ideas from physics have been put to good use in geometry and low dimensional topology. Now, the question of if physical ideas are enough to describe all of geometry is very much open. The trouble as I see it is that there has been so much cross-fertilisation between geometry and physics it is now hard to separate the two clearly.

I suppose for me the biggest clue would be an answer to: Is "our" geometry the only one that works?

If I am one unit of distance away from two other locations that are also one unit away from each other, that forms an equilateral triangle, with an angle of 60 degrees in "flat" geometry. Exactly 6 of these can fit around me in a circle. Why exactly 6? In curved space it could be 4, or some other number... could that curved space be transformed into a flat geometry and have everything work, only with a different value for pi, etc?

Then if you could say derive pi from c without using any of Euclid's axioms or postulates, or express geometry in terms of entropy or something, then I think you could show that geometry is the way we perceive it because of some fundamental universal constants, and not due to the fundamental nature of our particular geometry.

In other words if geometry is shown to be subjectively determined by universal constants, that would answer the question, no?

Another way might be to find a way to deduce Euclid's postulates from physics.

I need to go back to school!!!

Edit: I suppose you would also need to show that the other direction (deducing fundamental universal constants from geometry) is not valid or something.

Edited by md65536

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