Is space-time discrete or continuum?

[Boy@n]

Is there a way to know one way or another?

If smallest possible scale is Planck scale, does it mean that space-time is discrete where smallest possible step is Planck's length (PL) and smallest possible time is Planck's time (PT)?

If I move my hand from point A to point B, say 1m exactly in 1s, how many steps have I made? 1m/PL and for each step I spent PT? So, from that viewpoint my hand made those steps with the speed of light?

 

[Chronos]

Tests to date have failed to reveal evidence of quantized space time. One of the early predictions of this model was that high energy photons from distant sources should arrive at Earth at different times than low energy photons. Another was there should be differences in the airy disc between high and low energy photons. Neither effect was detected. The notion is still entertained in some models of loop quantum gravity, but, observational evidence remains lacking.

 

[rbj]

this is musing from an electrical engineer that knows something about discrete modeling of analog or continuous-time systems.

systems that are continuous are often described by continuous differential equations. if the diff eqs. are linear, there is a way (using Laplace Transform) to describe the system exactly and solve for a closed-form solution. but if the system has little non-linear components in them, sometimes the only way to understand the system is to simulate it with a discrete-time approximation. probably the simplest discrete-time approximation is Euler's forward method . now when programming a computer to simulate a physical system will involve turning those differential equations into difference equations, using, say the Euler method. in doing so you will compute dimensionless constants that will serve as coefficients in the discrete-time simulation. the sampling period and other time parameters will play a role in these dimensionless coefficients.

similar to cellular automata, you can take equations of physical interaction, like Maxwell's equations, and Schrodinger, and Einstein's field equation, and continuity equations for conserved quantities of physical stuff, and you can turn them into difference equations that will involve the sampling period (in the time dimension) of the simulation and cell size (in the 3 length dimensions).

now if you were to select a quantization unit in time and length that were essentially the Planck units, but defined so that these constants are removed from the above equations of physical law (which means these constants are set to 1): 4 \pi G = c = \hbar = \epsilon_0 = 1, then when you turn the differential equations into simple difference equations (Eulers method is plenty adequate because the discrete time and cell width are virtually infinitesimally small). but because these constants of nature disappear from expressions of physical law, there are no arbitrary scaling constants in the discrete-time simulator. we don't have Nature taking this quantity of flux density and pulling this number (\frac{1}{\epsilon_0}) out of her butt and scaling it to convert it into field strength. their quantities are equal so then maybe they, the two physical quantities, are the same thing. maybe flux density is the very same thing as field strength in Maxwells equations as well as the discrete-time simulation of it.

but that happens (ditch the arbitrary scaling constants) only if you run your discrete-time, discrete-space simulator with discrete units of the rationalized Planck time and Planck length (where 4 \pi G = c = \hbar = \epsilon_0 = 1 ).

because the Planck scale is soooo... ridiculously tiny, there is no way human beings can ever hope to measure anything around that scale to observe any of this discrete phenomena. way smaller than anything in the atomic or subatomic scale. so i like to pretend it could be true. i don't think there is any way to find out one way or another.

 

[bcrowell]

The notion is still entertained in some models of loop quantum gravity, but, observational evidence remains lacking.

Even if LQG is correct, LQG quantizes area and volume, not length. Length quantization is not possible because a given length can be arbitrarily contracted by Lorentz contractions.

 

[rbj]

Even if LQG is correct, LQG quantizes area and volume, not length. Length quantization is not possible because a given length can be arbitrarily contracted by Lorentz contractions.

don't the Lorentz contractions also affect area and volume? how can it not?

 

[tom.stoer]

Tests to date have failed to reveal evidence of quantized space time. One of the early predictions of this model was that high energy photons from distant sources should arrive at Earth at different times than low energy photons. Another was there should be differences in the airy disc between high and low energy photons. Neither effect was detected. The notion is still entertained in some models of loop quantum gravity, but, observational evidence remains lacking.

It is not even clear whether all models based in discrete spacetime structure do predict deviations from continuum models as tested in these experiments. Afaik LQG as of today does not predict any new disperison relation.

 

[Naty1]

Here are some views:

Space is discrete:

associated with Planck units which clearly implies a discreteness:

...adding one bit of information will increase the horizon of any black hole by one
Planck unit of area, or one square Planck unit. Somehow, hidden in the principles of quantum mechanics and the General Theory of Relativity there is a mysterious connection between individual bits of information and Planck sized bits of area.

so says Leonard Susskind, THE BLACK HOLE WAR, page 154


The following is a paraphrase of an argument for the discreteness of spacetime, made by Smolin in his popular-level book Three Roads to Quantum Gravity.

The Bekenstein bound says there's a limit on how much information can be stored within a given region of space. If spacetime could be described by continuous classical fields with infinitely many degrees of freedom, then there would be no such limit. Therefore spacetime is discrete.


There is no distinction between continuous and discrete:

http://arxiv.org/abs/1010.4354

“The equivalence of continuous and discrete information, which is of key importance in information theory, is established by Shannon sampling theory: of any band limited signal it suffices to record discrete samples to be able to perfectly reconstruct it everywhere, if the samples are taken at a rate of at least twice the band limit. It is known that physical fields on generic curved spaces obey a sampling theorem if they possesses an ultraviolet cutoff.”

http://pirsa.org/09090005/

Spacetime can be simultaneously discrete and continuous, in the same way that information can.

Good discussion in these forums here:

https://www.physicsforums.com/showthread.php?t=391989

https://www.physicsforums.com/showthread.php?p=3558771#post3558771

 

[Naty1]

and one more perspective...
There is a well known contradiction between relativity and Planck length:

[Wikipedia explains the contradiction nicely:

http://en.wikipedia.org/wiki/Double_special_relativity

If Special Relativity is to hold up exactly to this (Planck) scale, different observers would observe Quantum Gravity effects at different scales, due to the Lorentz-FitzGerald contraction, in contradiction to the principle that all inertial observers should be able to describe phenomena by the same physical laws.

Yet the energy density of empty space is believed to have a positive value and apparently this cosmological constant of

"...universal energy density would have the same value for all observers, no matter where or when they made their observations no matter how they moved."

(which is why Einstein called it a cosmological "constant")

 

[martinbn]

Even if LQG is correct, LQG quantizes area and volume, not length. Length quantization is not possible because a given length can be arbitrarily contracted by Lorentz contractions.

Why can't you do the same for a given area by arbitrarily contracting one side?

 

[tom.stoer]

There is a common misconception regarding area-quantization in LQG. The area-operator with discrete spectrum is not a physical Dirac-observable, so it can't be used to classify physical states; or the other way round: its eigenstates carrying discrete area are not physical states. So physical states which are solutions to all three constraints Gauß G, Diffeomorphism D and Hamiltonian H have not yet been constructed (due to H!) and could very well carry continuous area!

Look at a discrete basis like the harmonic oscillator states |n>. They can be used for every problem constructed in an L²[-∞,+∞] Hilbert space. But the conclusion that discrete basis states |n> with discrete energy n+1/2 do exist, does not mean that every physical problem constructed in this Hilbert space must have discrete energy levels (the situation is even more complicated b/c in the case of LQG the operator used to construct the eigenstates is not an observable like energy in the harmonic oscillator case)

So what one has to do is to construct a physical observable "area" and calculate it's spectrum. I do not know whether this has already been achieved, but I guess the Erlangen / Thiemann has done some relevant work, especially for physical constuctions getting rid of diffemorphism invariance using "dust fields" or "obsever fields".

 

[tom.stoer]

and one more perspective...
There is a well known contradiction between relativity and Planck length:

[Wikipedia explains the contradiction nicely:

http://en.wikipedia.org/wiki/Double_special_relativity

Global Lorentz covariance is not a symmetry of general relativity; observers classified according to SR
are not well-defined in GR. Taking diffeomorphism invariance into account everything fits nicely. The strange idea is that 'discrete spacetime' is often treated as equivalent with fixed lattice-like structure. This is wrong. All approaches using discrete models like LQG, CDT do not rely on a fixed structure; of course the is no fixed structure! The structure is a) arbitrary due to diffeomorphism invariance and therefore locally obsever-dependent and b) subject to renormalization and therefore unphysical.

Yet the energy density of empty space is believed to have a positive value and apparently this cosmological constant of

"...universal energy density would have the same value for all observers, no matter where or when they made their observations no matter how they moved."

(which is why Einstein called it a cosmological "constant")

I think that this is not correct. This can be seen quite easily if the cosmological constant becomes part of the energy-momentum tensor T^ab as "dark energy"; of course this term is observer dependent! This is trivial b/c T is subject to local coordinate transformations, and these do affect the metric term cc * g^ab as well. The reason why nobody cares about this is that in cosmology one always uses one fixed reference frame, namely a single solution to the Einstein field equations with highest degree of symmetry. But if you consider a local observer moving with some speed v relative to a fixed background metric (i.e. DeSitter space with cc) this observer would observe a different T', i.e. a different (cc * g^ab)'. Of course the value of cc is not affected, but the local energy density cc * g⁰⁰ is.

 

[phinds]

... and smallest possible time is Planck's time (PT)?

In previous discussions on this forum it has been stated that if time IS quantized, then the quanta is MUCH less than the Plank Time. I do not say that with any authority, I'm just repeating what I've heard here and I do not recall the rationale for that assertion.

 

[julian]

On the issue of quantized geometry and Lorentz contractions..

the basic argument is that one cannot have quantized values as we should be able to reduce them continuously further by performing a Lorentz transformation. The flaw in this argument is that we are not dealing with classical quantities, but rather quantum observables. The resolution then follows from the fact that the length operator, L, in the original frame does not commute with the length operator, L', in the transformed frame: they do not have simultaneous eigenstates - an eigenstate of L is not an eigenstate of L', rather it is a quantum superposition of eigenstates of L'. The eigenvalues of L' will be the same as the (discrete) eigenvalues of L: it is the expectation value of L' that will be Lorentz contracted in a continuous manner.

 

[julian]

There was some controversy about eigenvalues being discrete or continuous:

"Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?" by Dittrich and Thiemann - http://uk.arxiv.org/pdf/0708.1721.

Rebuked by Rovelli in "Comment on "Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?" by B. Dittrich and T. Thiemann" - http://uk.arxiv.org/pdf/0708.2481.

 

[tom.stoer]

Another common misconception is the idea that discrete spacetime or eigenvalues will necessarily violate local Lorentz / Diffeomorphism invariance. Of course this need not be the case b/c
- these symmetries can be recovered in a continuum limit / renormalization flow (like lattice gauge theory)
- violations can be restricted to the unphysical sectors
- discrete eigenvalues for angular momentum in QM do not violate rotational invariance

 

[julian]

On the issue raised above about geometric operators not corresponding to true observables see section III of http://uk.arxiv.org/pdf/gr-qc/9806079 for a counter argument.

 

[julian]

- discrete eigenvalues for angular momentum in QM do not violate rotational invariance

Yep, by exactly the same argument I gave above.

 

[Naty1]

Rovelli, from the above paper, seems to conclude:

... the evidence remains strong towards the conclusion that LQG implies fundamental
discreteness at Planck scale...

 

[akdude1]

well technically there's no definite answer some physicists say that time & motion don't exist that every thing that has ever happened is simply all existing I would suggest Julian Barbour's "The End of Time" if you want to pursue this idea further its a good book

 

[Boy@n]

well technically there's no definite answer some physicists say that time & motion don't exist that every thing that has ever happened is simply all existing I would suggest Julian Barbour's "The End of Time" if you want to pursue this idea further its a good book

I am considering to buy this book (ebook if possible). I've read some reviews, saw his website (with ideas) and description of his paper on Shape Dynamics.

All of it looks very interesting and convincing... Anyone else want to comment on the ideas presented by Barbour, and on his book?

 

[member 11137]

I am considering to buy this book (ebook if possible). I've read some reviews, saw his website (with ideas) and description of his paper on Shape Dynamics.

All of it looks very interesting and convincing... Anyone else want to comment on the ideas presented by Barbour, and on his book?

I didn't read the book but I did read the "Shape Dynamics" document you have linked above.

Your question on discretness is inspiring me some spontaneous thoughts:
A)
- We perceive that our local neighbourhood has what we have called "dimensions" which are in fact independant directions: length, width, hight and time.
- Any concrete progress needs the realization of measurements. This is obliging us to introduce an instrument measuring, e.g. the lengths. But not only that. With that instrument, we must make the choice of a unit, a norm. (My room is x meter long)
- The norm itself introduces a potential discontinuity (because it has a fixed value), except if the mesure x is/can be continouus.

Conclusion: the discretness is mathematical set-dependant!

B)
Mathematics symbolizing what I have tried to described in A) are related to geometry (ds)² = g_ab. dx^a. dx^b). Well, but there are many, many ways to mesure the same length/surface/volume... depending on a method, on the norm we have choosen, on the "form"(topology) of the object we are measuring. The lattice structure that we are in some way projecting into the real world we are living in and which we are trying to measure is something artificial, even if very practical. This is a pragmatic projection of our inner understanding of the world that we are perceiving.

Conclusion: if geometry is the science measuring the geography/topology, then the results (discretness or not: your question) will be highly tool-dependant, even if the topos is itself discontinous (example: if the ladder is longer than the gape, you will never remark that were skying over the gape and risking your life!)

C)
Coming back to the "Shape Dynamics". This is an interesting approach which I relate to the ADM procedure. I am not totally certain to have catched the essence. Does it mean (see § 4.1, § 4.2) that a history (e.g. of a particle) is equivalent to a set of slicing of the 4D spacetime (i.e.: a set of successive hypersurfaces)? If yes, then the equation of motion of the described particle must be itself the constraint imposing the manner how the 4D space must be sliced... (sorry speculation)

 

[akdude1]

I am considering to buy this book (ebook if possible). I've read some reviews, saw his website (with ideas) and description of his paper on Shape Dynamics.

All of it looks very interesting and convincing... Anyone else want to comment on the ideas presented by Barbour, and on his book?

Great book its a little weird starting out but if you keep pushing on the ideas start to sink in I would recommend it. I'll have to look at his other work it looks quite interesting!

 

[Boy@n]

See also this thread:
https://www.physicsforums.com/showthread.php?t=683198
which is about Smolin's new book 'Time Reborn'...

Just got the Kindle version... Looking forward to see how he answers this question.

 

[joris_pixie]

I have always wondered about this myself :)

If spacetime itself would be quantized wouldn't it lead to quantisation of angular momentum and energy ?
Also would it be evidence for the "universe simulation" theory ? ;)

I personally "feel" like it should be quantized ^^

 

[Naty1]

If spacetime itself would be quantized wouldn't it lead to quantisation of angular momentum and energy

those are quantized...h [Planck's constant] is the quantum of action. Discrete energy levels of electron orbitals is an example.

 

[friend]

What would quantized geometry/gravity do to the topology of spacetime? Does QG destroy topology?

 

[Naty1]

What would quantized geometry/gravity do to the topology of spacetime? Does QG destroy topology?

I am sure no expert on topology...but one theory does not necessarily destroy another. If general relativity works pretty well [based on our observations so far] using a continuous pseudo-Riemannian manifold, another theory with discrete spacetime, like quantum gravity, could provide a more wide ranging model...it could extend the range of classical GR to cover black hole and big bang singularities, for example. Or it might just explain those, analogous to how we currrently use quantum mechanics for many small scale phenomena and GR for cosmological scales.

If 'quantum gravity' were to fully describe the quantum behavior of the gravitational field it might or might not be very applicable to large scale gravitational field behavior. The two problems could be connected (as assumed in string theory) or could be separated (as assumed in loop quantum gravity).

 

[friend]

I am sure no expert on topology...but one theory does not necessarily destroy another.

For example, if the metric is quantized, then wouldn't neighborhoods around points be quantized. And if neighborhoods are quantized, then would the Hausdorff property still hold on which manifolds (and therefore GR) rest?

 

[Naty1]

look here, friend, I do not even know what a Hausdorff property is...[really!]
but a quick skim here

http://en.wikipedia.org/wiki/Hausdorff_property

Almost all spaces encountered in analysis are Hausdorff; most importantly, the real numbers (under the standard metric topology on real numbers) are a Hausdorff space. More generally, all metric spaces are Hausdorff.

and here

http://en.wikipedia.org/wiki/Metric_space

In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined.

which suggests to me...No problem! "Don't worry, be happy, mon"

but check those out yourself...

 

[friend]

look here, friend, I do not even know what a Hausdorff property is...[really!]
but a quick skim here

http://en.wikipedia.org/wiki/Hausdorff_property

When you think of Hausdorff, you can think of "house" dorff. This is the property where each point in the topology can be enclosed in a neighborhood that does not include any other point you may choose. Even if you choose points very close together, you can always construct even smaller neighborhoods that exclude the other point you chose. No matter how close you choose the points, they each have their own little house to dwell in.

But if the metric is quantized and with it areas, then you cannot always construct a neighborhood that excludes a close point. The property of Hausdorff could not apply to such a space, and it would not be a manifold. Then since GR is constructed on manifolds, GR would not be applicable, right?

 

[micromass]

What do we mean with "metric" here? A distance function that generates a metric space? Or a metric tensor?

 

[member 11137]

What would quantized geometry/gravity do to the topology of spacetime? Does QG destroy topology?

I have the temptation to ask the question in the converse direction: "Are all topologies" allowed by the GTR of A. Einstein?" If not, what are the acceptable solutions? Does the Cauchy problem (and its solutions) be the key ingredient for answering these questions? Does it not cost (quantized) energy to "go" from one allowed surface to the next one? Does the step between two successive allowed surfaces not be gouverned by the Heisenberg's uncertainty principle (energy x time) versus? Or with other words and, I am afraid at the boarder of science, does not this HUP tells us how the universe is growing and extending, looking its path in an unprobable future which nevertheless occurs?

Let me know if I am wrong; for me topos is related to the forms (thus the surfaces) and the metric is related to the science measuring the distances. We can measure distances, for example, on different surfaces.

 

[tom.stoer]

The question is whether we can answer the question "what it means that geometry is fundamentally discrete".

Suppose CDT is the correct description; it's a rather simple model using (discrete) triangulations of spacetime. Of course we can change the scale and look at spacetime at different resolutions, we can probe the triangulations, gravity and other interactions at different scales; we can calculate physical observables at different scales.

Now the following could happen: when changing scale and zooming to finer triangulations = to higher resolutions the physical answers we get become scale independent. That means that finer and finer triangulations do not have any effect on physical observables (below some "fundamental length").

So we make two observations
1) the theory allows for arbitrary small triangulations, i.e. it has a continuum limit
2) below some length scale physics doesn't change

1) means that the theory is not fundamentally discrete
2) means that it behaves as if it were fundamentally discrete

 

[friend]

The question is whether we can answer the question "what it means that geometry is fundamentally discrete".

Suppose CDT is the correct description; it's a rather simple model using (discrete) triangulations of spacetime. Of course we can change the scale and look at spacetime at different resolutions, we can probe the triangulations, gravity and other interactions at different scales; we can calculate physical observables at different scales.

Now the following could happen: when changing scale and zooming to finer triangulations = to higher resolutions the physical answers we get become scale independent. That means that finer and finer triangulations do not have any effect on physical observables (below some "fundamental length").

So we make two observations
1) the theory allows for arbitrary small triangulations, i.e. it has a continuum limit
2) below some length scale physics doesn't change

1) means that the theory is not fundamentally discrete
2) means that it behaves as if it were fundamentally discrete

I don't see how your description differs from a numerical computer method for solving a differential equation. Do numerical methods have anything to do with the underlying topology on which differential equations are formed?

 

[tom.stoer]

I don't see how your description differs from a numerical computer method for solving a differential equation. Do numerical methods have anything to do with the underlying topology on which differential equations are formed?

My description says that the model uses a discrete spacetime with length scale L, where
1) a continuum limit L→0 is possible, but where
2) a length scale L_min>0 exists below which no physical process can probe any smaller length scale L<L_min
My question to you is whether this means that spacetime is continuous b/c of (1) or whether it is discrete b/c of (2)

 

[friend]

My description says that the model uses a discrete spacetime with length scale L, where
1) a continuum limit L→0 is possible, but where
2) a length scale L_min>0 exists below which no physical process can probe any smaller length scale L<L_min
My question to you is whether this means that spacetime is continuous b/c of (1) or whether it is discrete b/c of (2)

I'd have to know more about what you mean by 2). What is meant by "no physical process can probe"?

 

[tom.stoer]

All I want to say is that it might happen that you have a description which uses continuous variables w/o minimal length, but from which a physical minimal length emerges, which is respected by all processes.

Another example: in LQG the fundamental variables are discrete, but still call for a continuum limit; the spectrum of the area operator is discrete, but this operator is not a physical observable; and as of today there is no proof that all physical observables probing length, area etc. have a discrete spectrum with some minimal length

So for me the relationship between quantization, discrete / continuous variables and discrete / continuous physical entities with / without minimal length is by no means obvious

 

[Natron]

i don't mean to be extremely simple here, but if Planck's constant is considered to discretely divide spacetime, then wouldn't there be a conundrum with gravitational forces over extreme distances? the obscure thing that comes to mind is imagine a complete vacuum of a universe that has 2 grains of sand (classical physical objects) but are placed 15 trillion light years apart, if spacetime is a continuum then these objects will affect each other, if spacetime is discrete, then either one of 2 things would happen, their gravitational forces against each other will be zero, or will be some sort of minimal constant that maintains regardless of the distance they are from each other.

 

[DimReg]

i don't mean to be extremely simple here, but if Planck's constant is considered to discretely divide spacetime, then wouldn't there be a conundrum with gravitational forces over extreme distances? the obscure thing that comes to mind is imagine a complete vacuum of a universe that has 2 grains of sand (classical physical objects) but are placed 15 trillion light years apart, if spacetime is a continuum then these objects will affect each other, if spacetime is discrete, then either one of 2 things would happen, their gravitational forces against each other will be zero, or will be some sort of minimal constant that maintains regardless of the distance they are from each other.

I imagine that since these would be quantum mechanical particles, they would always have some non-zero probability of propagating closer together, and this probability would increase as they get closer together. Further, I doubt that the "forbidden separations" would be positive potential regions, since two particles are falling down a potential gradient when they are gravitating together. That is, I don't think you could even think of the particles as "tunneling" to the next separation.

At this level, it would be difficult to apply your classical intuition.

 

[friend]

When you think of Hausdorff, you can think of "house" dorff. This is the property where each point in the topology can be enclosed in a neighborhood that does not include any other point you may choose. Even if you choose points very close together, you can always construct even smaller neighborhoods that exclude the other point you chose. No matter how close you choose the points, they each have their own little house to dwell in.

But if the metric is quantized and with it areas, then you cannot always construct a neighborhood that excludes a close point. The property of Hausdorff could not apply to such a space, and it would not be a manifold. Then since GR is constructed on manifolds, GR would not be applicable, right?

This all begs the question as to whether neighborhoods in topology are necessarily defined in terms of a metric. I mean the ususal description I've seen in textbooks is that neighborhoods are "balls" of radius r, and r is allowed to be any size. But the radius, r, is a measure of distance. So how do you escape the discrete area nature of a ball if the metric, which measures distance, is quantized? Can you define neighborhoods in topology without reference to a metric?

 

[micromass]

This all begs the question as to whether neighborhoods in topology are necessarily defined in terms of a metric.

They aren't. Please see a basic topology book such as Munkres.

I mean the ususal description I've seen in textbooks is that neighborhoods are "balls" of radius r, and r is allowed to be any size.

This is in a metric space.

Can you define neighborhoods in topology without reference to a metric?

Yes.

 

[Haelfix]

The question is whether we can answer the question "what it means that geometry is fundamentally discrete".

Suppose CDT is the correct description; it's a rather simple model using (discrete) triangulations of spacetime. Of course we can change the scale and look at spacetime at different resolutions, we can probe the triangulations, gravity and other interactions at different scales; we can calculate physical observables at different scales.

Now the following could happen: when changing scale and zooming to finer triangulations = to higher resolutions the physical answers we get become scale independent. That means that finer and finer triangulations do not have any effect on physical observables (below some "fundamental length").

So we make two observations
1) the theory allows for arbitrary small triangulations, i.e. it has a continuum limit
2) below some length scale physics doesn't change

1) means that the theory is not fundamentally discrete
2) means that it behaves as if it were fundamentally discrete

This is an excellent description.

 

[Natron]

my post was mainly based on a discrete minimal scale in which length, time, mass, and energy were based upon. if such a thing were to be considered, such as a minimal energy transfer on which every other transference of energy were a multiple of then we would have to reconsider a classical concept of gravity. let's say space/time was discrete, and that it progressed in small packets. and the advancement transference or adjustments or whatever could not be not on a smaller scale than these packets, then if the classical measurement of gravity were to stand, then at some distance, an object would have to fall below the threshold if minimal discreteness. for example if we determined that the smallest interval of anything were to be 1.0 m/s/kg/liters ^ 2 X 10^(-10000) [hypothetically speaking], then this could be transferred to a force measurement in which was claimed to be minimal. but since the classical definition of gravitational force between massive objects recedes based on distance, then mathematically it should be plausible that two objects can be placed at a distance in which the force enacted upon each other is below that discrete limit. so if spacetime is discrete, then the distance between massive objects should either 1- have a minimal gravitational force that extends throughout the cosmos regardless of distance equal to the minimal force within a discrete system, or 2 - jump to zero at some point and time. simply stating if there is a minimal number, our classical definition of gravity will eventually exceed that minimum regardless of what that number is, so we either have to accept that far flung objects are NOT affected by gravity (between themselves), or explain why they are despite their force being below the minimal barrier of a discrete system.

 

[friend]

They aren't. Please see a basic topology book such as Munkres.

Thank you, micromass. I think I will buy the paper back version of the book. But in the mean time, maybe you could give us a very brief definition of neighborhoods without use of the metric. For me it seems inescapable not to talk about some sort of size associated with neighborhoods, especially when considering concepts of continuity, where the neighborhood is allowed to strink in size to near zero, whatever that means without a metric.

As I recall, and it's been a while, a metric is an added structure to a topology. But once you define a metric on a topological space, it becomes impossible to talk about the size of neighborhoods of points without automatically saying something about their size in terms of the metric. So that if the metric were quantized, then so must be the neighborhoods, and one wonders what happens to the Hausdorff property of any manifold defined on that topology.

Perhaps this is worth a little more time since the thread is concerned with continuity, metrics, and the geometry used in physics.

 

[fanphysics]

The problem of quantisation of space and time is phylosophical problems and it will be solved when the basic principle is found which unifies the concepts of space and time.

 

[Fredrik]

maybe you could give us a very brief definition of neighborhoods without use of the metric.

There's a theorem about open sets in metric spaces that you may be familiar with. It says that if X is a metric space, the following statements are true:

(1) ∅ and X are open sets.
(2) Every union of open subsets of X is open.
(3) Every finite intersection of open subsets of X is open.

This theorem has inspired the following generalization. Let X be any set. A set $\tau$ whose elements are subsets of X is said to be a topology on X if the following statements are true:

(1) $\varnothing,X\in\tau$.
(2) Every union of elements of $\tau$ is an element of $\tau$.
(3) Every finite intersection of elements of $\tau$ is an element of $\tau$.

The pair $(X,\tau)$ is said to be a topological space if $\tau$ is a topology on X.

Suppose that $(X,\tau)$ is a topological space. A subset $E\subseteq X$ is said to be open if $E\in\tau$ and closed if $X-E\in\tau$.

Let $x\in X$ be arbitrary. There are at least two different definitions of "neighborhood of x" in the context of topological spaces:

1. A neighborhood of p is an open set that contains p.
2. A neighborhood of p is a set that contains an an open set that contains p.

 

[WannabeNewton]

As I recall, and it's been a while, a metric is an added structure to a topology. But once you define a metric on a topological space, it becomes impossible to talk about the size of neighborhoods of points without automatically saying something about their size in terms of the metric. So that if the metric were quantized, then so must be the neighborhoods, and one wonders what happens to the Hausdorff property of any manifold defined on that topology.

Topology doesn't care about size e.g. in topology one can show that the unit open ball $B^{n}\subseteq \mathbb{R}^{n}$ is homeomorphic to all of $\mathbb{R}^{n}$. That's the whole point of point-set topology: it removes the structure associated with metric spaces that gives us a notion of distance and size in the primitive geometric sense and instead just deals with neighborhoods in a more abstract sense. Also note that the metrics being spoken of in the context of space-times are pseudo-Riemannian metrics endowed on smooth manifolds, not metrics in the analysis sense. The two are completely different animals.

 

[tom.stoer]

... if the metric were quantized, then so must be the neighborhoods, and one wonders what happens to the Hausdorff property of any manifold defined on that topology.

As I tried to explain quantization (of the metric) does not necessarily lead to discretization (of the space, metric, ...); there are proposals with quantized but continuous gravitational field (in QM both x and p are quantized, i.e. they are operators, but nevertheless x is always continuous and p is only discrete for some eigenvalue problems; nevertheless the Hilberts space is a space of functions u(p) where p is a continuous variable); so again: quantizing the metric does not necessarily imply discretization.

But if there is discretization (either as a result of the quantization procedure or as a starting point put in by hand) then the usual topological properties will not survive. So what? Of course we expect that "quantum geometry" is different from classical one. Where's the problem?

 

[friend]

Topology doesn't care about size e.g. in topology one can show that the unit open ball $B^{n}\subseteq \mathbb{R}^{n}$ is homeomorphic to all of $\mathbb{R}^{n}$. That's the whole point of point-set topology: it removes the structure associated with metric spaces that gives us a notion of distance and size in the primitive geometric sense and instead just deals with neighborhoods in a more abstract sense. Also note that the metrics being spoken of in the context of space-times are pseudo-Riemannian metrics endowed on smooth manifolds, not metrics in the analysis sense. The two are completely different animals.

OK, so now we have two metrics to worry about and whether they are in any way connected to the size of neighborhoods. As I understand it, GR relies on the existence of an underlying manifold, and manifolds seem to rely on a continuous Euclidean metric, per wikipedia.org, which says,

A topological space X is called locally Euclidean if there is a non-negative integer n such that every point in X has a neighborhood which is homeomorphic to the Euclidean space Eⁿ (or, equivalently, to the real n-space Rⁿ, or to some connected open subset of either of two).[1]

A topological manifold is a locally Euclidean Hausdorff space.

A Euclidean space is a space with a Euclidean metric. And this Euclidean metric is continuous as indicated by the word "local". But it is not the pseudo-Riemannian metric of GR, since a pseudo-riemannian metric is not the Euclidean metric. All very confusing. What is the locally Euclidean metric on the manifolds associated with GR if not the pseudo-riemannian metric?

When you write,

in topology one can show that the unit open ball Bⁿ ⊆Rⁿ is homeomorphic to all of R

this only exacerbates the problem I have because it seems every time I read Rⁿ it always seems to be connected to the Euclidean metric. It would go a long way to clear things up for me if that distinction were made obvious with reliable sources.

 

[friend]

As I tried to explain quantization (of the metric) does not necessarily lead to discretization (of the space, metric, ...); there are proposals with quantized but continuous gravitational field...

I'm not aware of any metric quantization procedures that don't assume a result of a discrete spectrum. Maybe you could share some of these efforts with us.

... (in QM both x and p are quantized, i.e. they are operators, but nevertheless x is always continuous and p is only discrete for some eigenvalue problems; nevertheless the Hilberts space is a space of functions u(p) where p is a continuous variable); so again: quantizing the metric does not necessarily imply discretization.

I think we're talking about apples and oranges. There is quantizing fields on a background, and then there is quantizing the background itself. I'm concerned that trying to quantize the background will negate the validity of quantizing fields on the background.

Consider, a typical formulation in quantum mechanics is &lt; x|x&#039; &gt; \,\, = \,\,\,\delta (x - x&#039;). Can the Dirac delta function still be evaluated in a space with a quantized metric? I don't see it.

But if there is discretization (either as a result of the quantization procedure or as a starting point put in by hand) then the usual topological properties will not survive. So what? Of course we expect that "quantum geometry" is different from classical one. Where's the problem?

If you come up with a procedure that ultimately negates the premises, isn't that reducio ad absurdum?