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View Full Version : If space were discrete, what shape are the units?

WaxRubiks
2018-Mar-06, 09:46 PM
If space were discrete, what shape are the units?

Shaula
2018-Mar-06, 10:01 PM
What shape is a joule?

WaxRubiks
2018-Mar-06, 10:14 PM
What shape is a joule?

by 'unit' I mean compartments/segments etc.

PetersCreek
2018-Mar-06, 10:19 PM
Why would it have an intrinsic unit shape?

WaxRubiks
2018-Mar-06, 10:29 PM
Why would it have an intrinsic unit shape?

well, then if it were a zero dimension location, I suppose another question would be what is the arrangement of these points through the system.

Strange
2018-Mar-06, 10:42 PM
There are multiple approaches to quantising space-time. Here is one that (kinda) answers the question: https://en.wikipedia.org/wiki/Causal_dynamical_triangulation (https://en.m.wikipedia.org/wiki/Causal_dynamical_triangulation)

grant hutchison
2018-Mar-06, 11:43 PM
Why would it have an intrinsic unit shape?Actually, how could it have an intrinsic shape of any kind, if it were the smallest possible unit of space? To describe any sort of shape we'd need finer coordinates than the smallest unit.

Grant Hutchison

ShinAce
2018-Mar-07, 04:55 AM
If time was absolute, we might be able to answer the question. However, once you realize that you need to quantize space and time along with matter and energy, you realize that the structure doesn't have a well define shape such as 'cubic' or 'spherical'. It becomes a mathematical structure where time is part of the shape itself, and thus it becomes impossible to describe it as a shape anyone would be familiar with.

This is the domain of quantum gravity.

Copernicus
2018-Mar-07, 05:53 AM
It is actually very simple, but it has not been figured out.

dgh64
2018-Mar-16, 06:03 AM
Actually, how could it have an intrinsic shape of any kind, if it were the smallest possible unit of space? To describe any sort of shape we'd need finer coordinates than the smallest unit.

Grant Hutchison

Just because it's the smallest unit of physical space that can exist, doesn't mean we can't come up with numbers (which are abstract, after all) that are smaller.

I just did some quick googling and a thought experiment.

Here's an interesting page on the space-filling polyhedra (http://mathworld.wolfram.com/Space-FillingPolyhedron.html). My first thought is that it should be one (and only one) of these shapes. If the smallest unit of space were some other shape (a sphere, for example) that would mean that in between all the "space" are voids full of "not space".

My second thought is that the units of space should have rotational symmetry. If space were made from triangular prisms, for example, then its structure along one axis would be different than its structure along another axis, like the grain in a piece of wood. If that were the case, then there should be some experiment we could do that would show which direction (perpendicular or parallel) a particle is moving relative to the grain. This is also why I discount all the ways in which 2 or more polyhedra can be combined to fill space -- you always end up with one direction behaving differently than another. The laws of physics all work the same way regardless of the orientation of the experiment, so clearly the cube is the best choice.

That's not quite good enough, however -- if space were made of cubes like Minecraft, there should STILL be some effect from whether a particle was moving diagonally vs. orthogonally. The only shape that has rotational symmetry for all angles in all directions is a sphere, but that brings us back to the first point: what are all the voids between spheres full of, if it isn't "space"?

WaxRubiks
2018-Mar-16, 06:10 AM
What occurred to me years ago was that if space was discrete, and if the compartments had shape, then light might appear to travel at greater speed in one direction than another. eg, if space were made of cubes then light might appear to cover more distance going diagonally rather than perpendicular to the cube surfaces...as a simple view of space-time discretion.

gzhpcu
2018-Mar-16, 05:37 PM
I thought space is quantized to the Planck length.

grant hutchison
2018-Mar-16, 05:49 PM
Just because it's the smallest unit of physical space that can exist, doesn't mean we can't come up with numbers (which are abstract, after all) that are smaller.It's not the number I was talking about, but the physical edge length of these hypothetical space "voxels", which are always shorter than the longest dimension of the relevant solid.
If the diameter of the solid denotes the smallest thing we can measure, then how can it have edges that are shorter than that measure? If the edges are the shortest thing we can measure, then the solid they define is not actually the smallest unit of space. (We can, of course, bring this together if we take the edge of a cube as the smallest measure, and the volume of the cube as the smallest volume - but that's just putting in what we want out.)

So I think that (unless space turns out to be made of neatly aligned cubes) while we might potentially be able to talk about how these small units pack, we couldn't reasonably talk about a precise shape.

Grant Hutchison

Strange
2018-Mar-16, 10:29 PM
I thought space is quantized to the Planck length.

That seems to be a common misconception. There is no evidence space is quantised. From what I have seen, in most theories based on space being quantised it is at a scale much smaller than the Planck length.

cjameshuff
2018-Mar-16, 11:57 PM
That seems to be a common misconception. There is no evidence space is quantised. From what I have seen, in most theories based on space being quantised it is at a scale much smaller than the Planck length.

A related misconception is that the Planck units are the smallest possible quantities. The Planck energy is equivalent to the total energy produced by a 1 GW power plant operating for just under 2 seconds, or the energy released by about 460 kg of TNT.

Copernicus
2018-Mar-19, 10:35 PM
What occurred to me years ago was that if space was discrete, and if the compartments had shape, then light might appear to travel at greater speed in one direction than another. eg, if space were made of cubes then light might appear to cover more distance going diagonally rather than perpendicular to the cube surfaces...as a simple view of space-time discretion.

I'm thinking that if we measure distance by the time and speed of light, it might be difficult to distinguish this.

Copernicus
2018-Mar-19, 10:38 PM
Just because it's the smallest unit of physical space that can exist, doesn't mean we can't come up with numbers (which are abstract, after all) that are smaller.

I just did some quick googling and a thought experiment.

Here's an interesting page on the space-filling polyhedra (http://mathworld.wolfram.com/Space-FillingPolyhedron.html). My first thought is that it should be one (and only one) of these shapes. If the smallest unit of space were some other shape (a sphere, for example) that would mean that in between all the "space" are voids full of "not space".

My second thought is that the units of space should have rotational symmetry. If space were made from triangular prisms, for example, then its structure along one axis would be different than its structure along another axis, like the grain in a piece of wood. If that were the case, then there should be some experiment we could do that would show which direction (perpendicular or parallel) a particle is moving relative to the grain. This is also why I discount all the ways in which 2 or more polyhedra can be combined to fill space -- you always end up with one direction behaving differently than another. The laws of physics all work the same way regardless of the orientation of the experiment, so clearly the cube is the best choice.

That's not quite good enough, however -- if space were made of cubes like Minecraft, there should STILL be some effect from whether a particle was moving diagonally vs. orthogonally. The only shape that has rotational symmetry for all angles in all directions is a sphere, but that brings us back to the first point: what are all the voids between spheres full of, if it isn't "space"?

Why is something better than nothing?

Shaula
2018-Mar-20, 01:10 AM
A more common way to try to describe spacetime structure in quantum gravity theories is in terms of a spin network which describes a spatial geometry. CDT and the LQG use exactly this method. The closest you see to a unit cell of spacetime is when you evolve these spin networks that encode spatial geometries in time. Then you end up with a spin foam which is a series of planar surfaces in n+1 dimensional space where the edges trace out permissible evolutions of spin networks akin to the history or potential histories of spacetime structure.

gzhpcu
2018-Mar-20, 03:52 PM
That seems to be a common misconception. There is no evidence space is quantised. From what I have seen, in most theories based on space being quantised it is at a scale much smaller than the Planck length.How about Quantum Loop Gravity?

Space's structure prefers an extremely fine fabric or network woven of finite loops. These networks of loops are called spin networks (https://en.wikipedia.org/wiki/Spin_network). The evolution of a spin network, or spin foam (https://en.wikipedia.org/wiki/Spin_foam), has Planck length (https://en.wikipedia.org/wiki/Planck_length), that is approximately 10−35 metres. Therefore, scales smaller than the Planck scale do not exist. from Wikipedia

Strange
2018-Mar-20, 06:28 PM

I did say "most" :) (And I may well be wrong ...)

Copernicus
2018-Mar-20, 07:02 PM
from Wikipedia

It may be that we may be able to get some information of the granular universe from the Planck length, but there may be other scales much smaller and much larger that are also necessary for getting the full information about the structure of space and space-time.

mkline55
2018-Mar-20, 07:26 PM
Shape is a characteristic of the model. Choose a model, and maybe someone can define the shape. The question as stated is unclear. It's equivalent to asking if a line is made of segments, what is the length of each segment? In general you might say it's the minimum length of interest to the model. If you were measuring galactic distances, then meters are useless. The minimum useful distance is perhaps a light year. If you are measuring streets, then micrometers are useless, as well as light years, but maybe meters or centimeters are the least useful measurement.

Copernicus
2018-Mar-20, 07:38 PM
Shape is a characteristic of the model. Choose a model, and maybe someone can define the shape. The question as stated is unclear. It's equivalent to asking if a line is made of segments, what is the length of each segment? In general you might say it's the minimum length of interest to the model. If you were measuring galactic distances, then meters are useless. The minimum useful distance is perhaps a light year. If you are measuring streets, then micrometers are useless, as well as light years, but maybe meters or centimeters are the least useful measurement.

I was just saying, if space is granular, that the granularity may not be independent of the very largest and very smallest components. I obviously don't know much about string theory, so I may be way off base here, but it always seems the strings are on the scale of Planck length. Maybe some of the variables are Hubble sphere length. I assume that String theory has some very compelling elements, but there also seem to be some very compelling shortfalls. It seems to be if we really understood the universe, the mass of the proton and electron, and the value of Planck's constant and other elementary aspects should just fall right on a platter.

Shaula
2018-Mar-21, 12:39 AM
I obviously don't know much about string theory, so I may be way off base here, but it always seems the strings are on the scale of Planck length. Maybe some of the variables are Hubble sphere length.
Worth noting that while strings are small there is a duality that allows you to swap r and 1/r provided you also interchange the winding and momentum quantum numbers. So, in a sense, they can also be understood by thinking of them as large!

Copernicus
2018-Mar-21, 01:22 AM
Worth noting that while strings are small there is a duality that allows you to swap r and 1/r provided you also interchange the winding and momentum quantum numbers. So, in a sense, they can also be understood by thinking of them as large!

That always mystifies me. Why swap r and 1/r I am sure it is because I don't know enough. They wouldn't even have the same units.

Shaula
2018-Mar-21, 01:30 AM
That always mystifies me. Why swap r and 1/r I am sure it is because I don't know enough. They wouldn't even have the same units.
Perhaps sloppy wording on my part - strictly you replace the distance R with 1/R, but keeping the units. So a circle of radius 2m would be replaced by a circle of radius 0.5m.

See https://en.m.wikipedia.org/wiki/T-duality

Copernicus
2018-Mar-21, 03:55 AM
Perhaps sloppy wording on my part - strictly you replace the distance R with 1/R, but keeping the units. So a circle of radius 2m would be replaced by a circle of radius 0.5m.

See https://en.m.wikipedia.org/wiki/T-duality

Does this have anything to do with parallel vs perpendicular acceleration on certain geometries?

Shaula
2018-Mar-21, 04:05 AM
Does this have anything to do with parallel vs perpendicular acceleration on certain geometries?
Not as far as I know. It is just a mathematical duality, albeit an important one as it is linked to M-theory. T dualities linked up type IIa and IIb versions of string theory, plus the two heterotic versions. S dualities made the other links.

gzhpcu
2018-Mar-21, 07:13 PM
It may be that we may be able to get some information of the granular universe from the Planck length, but there may be other scales much smaller and much larger that are also necessary for getting the full information about the structure of space and space-time.Doesnt that depend on the scientific model? In GR, there is no quantification. QLP has a spin network. Similar to elementary particle shape: point, string, or what?
We only work with models and have no idea of what is really out there.

Copernicus
2018-Mar-22, 01:08 AM
I don't know if this is what happening, but some of the large number theories may work that way. For example one of the large numbers is 10^40. If one takes one part in 10^40 of the mass of a neutron, and calculate a Compton wavelength of this particle it is of the size of the Hubble sphere. Kind of like the smaller the R the bigger the 1/R

Copernicus
2018-Mar-22, 01:11 AM
Doesn’t that depend on the scientific model? In GR, there is no quantification. QLP has a spin network. Similar to elementary particle shape: point, string, or what?
We only work with models and have no idea of what is really out there. Right! When I looked at shape of granular space-time. Depending on how I looked at it, it could be absolutely nothing, a ring, a hollow sphere, a solid sphere, a torus, or parts of a cuboctahedron.
As far as I am concerned, it is like a hall of mirrors in many different levels of dimensions.

loglo
2018-Mar-23, 01:19 AM
All quantum gravity theories I know of only quantize space and not spacetime. Their staring point is ADM 3+1 GR or they use the more modern Ashtekar type variables (frame fields and connection 1-forms). Time is usually left to emerge from the dynamics of the Hamiltonian somehow.

In LQG and similar theories the spin networks are equivalence classes (of embeddings of graphs) so are in effect superpositions of shapes. These are used to build the quantum area and volume operators which would eventually produce space. So its quantum all the way up and looking for a specific unit shape seems to miss the point of quantising space in the first place.

gzhpcu
2018-Apr-03, 08:40 PM
String theory claims that every point in spacetime is actually a tiny 6D world with the structure of a Calabi-Yau manifold.