4 Dimensional Mass

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

4 Dimensional Mass

Postby ICN5D » Thu Jun 12, 2014 7:17 pm

How does this work? If a 3D object with a certain density has a certain mass, it seems like a 4D object would be heavier. But, if building a 4D object with the same density, how does that even apply to 3D weight as we know it? It's an infinite stack of 3D masses! Surely it can't be infinitely heavy. Does 4D mass take on a whole new characteristic not seen or felt in 3D ? Even trying to think about how a 2D sheet would have mass is just a strange, for me.

We could assume elementary particles are higher than 3D, but mostly 3D. They're only stacked up in a 3D lattice, in the way we see it. But, because they actually extend into these higher dimensions, they can also be stacked this way. One only needs to develop the technology required to stack subatomic particles in a higher dimension, without having them topple over ( for all we know ! ). We could then build a real life duocylinder, or perhaps build a magnetic coil in the shape of one, utilizing some effect of physics yet to be discovered. Using higher dimensional space that's there by default sounds like a next step in a technological era. This is where, according to some law, technology sufficiently advanced enough only seems magical. A machine that extends into higher dimensions of space would no doubt seem like beyond-comprehensible ultra advanced super technology. Maybe we have yet to discover how a magnetic coil in the shape of a (((II)I)((II)I)) could bend space and let us travel to the stars.
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Re: 4 Dimensional Mass

Postby quickfur » Fri Jun 13, 2014 9:32 pm

Mass is proportional to the number of unit particles in a given object, whereas density is mass per unit bulk (n-dimensional volume). But bulk is a dimension-dependent quantity: it has different units in every dimension! So one should not confuse 2D density with 3D density (or density in any other dimension): they have different units! In 2D, bulk is measured in terms of unit distance squared -- if we standardize on meters, say, then 2D bulk has units m2. But in 3D, bulk is measured in terms of m3. Thus, in 2D, density is measured in terms of units per m2, that is, m-2, whereas in 3D, density is measured in terms of units per m-3, that is, m-3. Since these two quantities have different units, they are incomparable. Similarly, with 4D density we're dealing with m-4, which is again incomparable with the other two densities.

Calculating mass from density involves multiplying with the bulk of the object, which, again, is a dimension-dependent quantity. A 2D object has bulk measured in terms of m2, whereas a 3D object has bulk measured in terms of m3. So it's invalid to compare 2D bulk with 3D bulk: the units are incompatible! Similarly, you can't meaningfully compare 3D bulk with 4D bulk, because their units are different.

Note, however, that mass itself is a dimensionless quantity, so it's something that could be applied across dimensions. But you can't meaningfully derive mass from density without getting into trouble across dimensions, since then the dimensionality of space enters into the equation and you end up with incomparable units.

As to what mass is, that's a pretty deep philosophical question. :P Technically, we can't measure mass directly -- we can infer its value based on other measurements, such as density (which unfortunately is dimension-dependent), weight (which is force-dependent -- so when in orbit you're weightless, but that by no means implies that you're massless!), acceleration (deducing mass by measuring trajectories), inertia, etc.. In fact, one of the mysteries of the universe is why inertial mass -- that is, an object's resistance to change in velocity -- is identical to gravitational mass -- the object's response to the force of gravity.

Consider this: mass is just one of the scalar properties of an object, but (electrical) charge is another scalar property. For example, you can in theory have an object that consists of protons without any electrons, and have a positively-charged thing. Now consider how a generic object X responds to some generic force F. Presumably, any given object X "feels" the force F to a certain degree, depending on some inherent characteristics it has. A positively-charged particle responds to the electrical force from a negatively-charged particle differently from, say, a negatively-charged particle. And the amount of reaction you get depends on how positively/negatively-charged it is. A highly-positive object will respond very strongly to an electromagnetic field F, whereas a neutral (zero charged) object doesn't respond at all. So then, it would seem that an object X has a number of "parameters", that characterize how much it responds to certain forces. If X has electrical charge q, for example, then it would "feel" an external electrical force in a way proportional to q. So one would expect that gravity, being a distinct force from inertia (via, say, a rocket engine), should be associated with a distinct parameter on X than inertia. The object X should have an inertial mass M_i, which tells us how strongly it resists changes in velocity, and a distinct gravitational mass M_g, which tells us how strongly it reacts to the force of gravity, just as it has a distinct parameter q (i.e., electrical charge), that tells us how strongly it reacts to an electromagnetic field.

However, what we find is that M_i = M_g always. This is a very strange observation, since on the surface, inertia and gravity are two unrelated things, so why should all objects respond the same way to gravity as they do to inertia, whereas they don't respond to the electromagnetic force the same way (you almost never find q=M_i or q=M_g)? The fact that inertial mass is always equal to gravitational mass is a strange coincidence that suggests that perhaps inertia and gravity aren't different things after all. Perhaps they are just two aspects of the same thing! This is what led Einstein to discover general relativity, where this coincidence is explained by postulating that gravity is simply curvature in space itself, and thus objects under the influence of gravity are actually travelling in a straight line just as before, except that due to the curvature of space, this "straight line" is actually crooked. In other words, gravity changes the trajectory of objects not by exerting some external force on them, but by changing the local meaning of "straight line" in the object's frame of reference! So, from the object's point of view, its path is still dictated by the law of inertia just like before, and so it will react according to its inertial mass M_i. That's why M_i = M_g: gravity isn't a different force that operates on the object, but just a bending of space that changes the inertial behaviour of the object, that's why the object reacts according to its inertial mass, instead of some other quantity!

But then if we're talking about the curvature of space, then we're back to dependence on the dimensionality of space -- because after all, the kinds of ways that space can curve depends on how many dimensions it has. So what is mass??! Intuitively speaking, it's the amount of "stuff" in an object -- but what does that mean across different dimensions of space? It seems intuitively obvious, but if you look at it carefully, it's actually quite mysterious! Elementary particles like electrons have mass -- but, as far as we know, they have no discernible shape or structure, so they seem to be point-masses. Which is another mystery, because it would appear that they are 0D dimensionless points, yet they have non-zero mass! Which then begs the question, is an electron in 3D different from an electron in 4D? Is mass a quantity that can be meaningfully compared across dimensions? For example, if 4D electrons are different from 3D electrons, then wouldn't it be meaningless to compare their masses, since a 4D electron can't exist in 3D space and therefore it's impossible to weigh them both to see which one is more massive? But if they are comparable -- that is, if electrons exist in both 3D and 4D, then they would have to be the same object, just confined to a different dimension of space; in which case, one may ask, can an electron exist in spaces of arbitrary dimensions? Can it exist outside of space itself as an independent entity? No matter what the answers are, it seems far from clear what "mass" means across different dimensions, and it's not obvious at all whether they are even comparable!
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Re: 4 Dimensional Mass

Postby ICN5D » Sat Jun 14, 2014 2:36 am

quickfur wrote:In fact, one of the mysteries of the universe is why inertial mass -- that is, an object's resistance to change in velocity -- is identical to gravitational mass -- the object's response to the force of gravity.



Yeah, I imagine our worldline through time closely resembling a top-down view of driving along the highway. Both accelerating in a straight line and taking a corner at constant speed have identical curves, and produce the same result, acceleration. So, in that case, it would seem as if space really is bent into a curving path, which produces the omnipresent acceleration field we feel on earth. Technically speaking, we're accelerating straight up skywards at all times, which is a very strange notion. We only cancel gravity by falling, sliding back down the curving sheet like some colossal hyper-waterslide.

Which can only mean one thing, in my mind. I firmly believe, according to this idea, that we are in motion at all times ( no pun intended ). We are travelling at a fixed velocity through the dimension(s) of time, the speed of light. When we begin to move in space, our worldline bends away from our direction along time. By moving in space, we simply rotate our 4D vector away from full time-ward, removing our displacement a little. Going around this time-bend is identical to taking a sharp corner on the highway. We were in motion to begin with, and simply rotated our 2D vector more eastward, and less north.

I feel that our 3D slice now-moment is a speed of light reaction of energy fields, travelling as a thin sheet along a higher dimensional universe. This thin sheet cuts along the Grand Unified Toratope, whose surface describes the intercept points and evolution sequence, of all 3D intercepting particles in the universe. At least that's how I see it!




But, lately, I've been nurturing some new ideas on particles and how they appear to us. It stems from my understanding of rings, duorings, triorings, and such. Ring is simple to grasp, it's a 1D edge of a circle. Represented in 1D, we only see two points along a line. It looks this way in all 1D slices, no matter the angle.

Now, take a duoring: here is a 2D closed surface that curves into 3 and 4D. But, it only intercepts a 2D plane as four locations of 0D points, in a square. So, here is an object of decent size that bends in a such special way, it's hardly in the dimensions it has.

The trioring is the edge of a triocylinder, product of three solid disks. It's edge is a 3D closed sheet that curves into 4,5,and 6D. It intercepts a 3D plane in only 8 locations of a 0D point, in the corners of a cube! So, here is another high dimension shape, that has a decent size, that intercepts a 3D hyperplane in the most minimal way possible. Inflating this structure with a 3D sphere makes the tritiger, ((II)(II)(II)), with a sphere in the XYZ hyperplane, stuck to a trioring in the other, orthogonal WVU hyperplane.

And then there's a di-trioring, a trioring inflating a trioring! Each point in the 2x2x2 cube array gets replaced with another, smaller 2x2x2 cube array of 8 points, making a 4x4x4 cube of 64 points of 0D. This is a 3D closed surface that curves into 4,5,6,7,8,and 9 dimensions. It's the edge of a (torus^3)-prism, and I shall render it one day in it's full 45x45x45x45x45x45 sextuple-oblique midsection!


Maybe subatomic particles are in some way a clifford torus-type energy ripple on a closed surface? Which sounds like what string theory is describing to some degree: convergent wave patterns of vibration across an N-dimensional clifford torus-type shape. A high-D complex clifford torus could the compactified dimensions impregnated into a 4D space time manifold.

Furthermore, I feel that each force of nature has its own hyperplane manifold. There's a medium-sized electroweak universe stuck to ours, and is how electrons and weak nuclear bosons communicate. We have a 4D gravity universe, which conducts the gravitational force ( and is the weakest and largest). And, then there's the nuclear strong universe, which is the smallest extent, but highest yield. Where the end result of each multiplied together is all that we see around us, everyday. Hmmm, pure speculation .....
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