Is Space-Time a Continuum or a Physical Object?

[marcus]

TD, given that "discrete" seems semantically problematic what to do? You've suggested one way to turn. Another might be to say "the way to handle semantic problems is to have fun with them : ^)
google translator says the Greek word
χασμα
means "gap"

The LQG researchers refer to LQG having a smallest nonzero area eigenvalue as "THE AREA GAP".

So how about calling QG theories that have a smallest geometric eigenvalue like that by the adjective "METROCHASMIC"?

Or would "GEOCHASMIC" be better?

It just means "has gappy geometric observables" and of course in particular smallest nonzero eigenvalues for some of them.

 

[martinbn]

I think it's worth noting that the converse is not necessarily true: that is, points of spacetime being a countable set does not necessarily mean spacetime must be discrete. The rational numbers are a countable set, but they are also continuous (at least with the standard ordering on them).

That's not how I understand continuous. The rational numbers are not continuous because they are not complete, Cauchy sequences of rational numbers need not be convergent. The real numbers are defined exactly because the rationals are not continuous.

 

[rrogers]

Thank you for introducing me to this wonderful paper. I said space is continuous by definition because I don't know what the stuff in between would be called if space is discrete.

Just say that when one interprets the GR equation symbols as calculus limits on the smaller scales; the equations are stochastic. In other words the classical GR equations are summaries/extreme cases of some stochastic equations where the independent variables happen to be random at a small scale.
Having said that I haven't given up on the geometry at larger scales. My reasoning is that if the QM equations twist and curve like they are on a surface/substrate then a geometric approach or interpretation is entirely reasonable. My case in point is the causal cone of light bending when passing by a massive object. The QM equations have to have "geometric" modifications to explain that.
As far as continuity/discreteness goes: I haven't given up hope that someone will think of a way to experimentally establish whether the "real" universe is Aleph 0,1, or even 1/2 or 2 (which is my favorite). The fact that on the smallest scale the equations turn stochastic doesn't determine whether the the large scale effects aren't Aleph 1 (or higher) and isomorphic to the equations describing curved surfaces.

 

[PeterDonis]

The rational numbers are not continuous because they are not complete

There are different definitions of "continuous". The one I was using is that "continuity" is the property that, between any two rational numbers, there is always another rational number; there is no such thing as a "minimum" interval between two rational numbers. So if possible measurement values were represented by rational numbers, there would be no minimum possible measurement value.

You're right that this definition of "continuous" does not guarantee other properties that might be desirable, such as convergence of all Cauchy sequences.

 

[Torbjorn_L]

Wow, so to you GR is quantizable and discrete.

Yes and no respectively.

• For the first, obviously to any physicist GR can be expressed as an effective [quantum] field theory at low energies and large scales:

"It’s often said that it is difficult to reconcile quantum mechanics (quantum field theory) and general relativity. That is wrong. We have what is, for many purposes, a perfectly good effective field theory description of quantum gravity. It is governed by a Lagrangian

(1) S = ∫d4x−g−−−√⎛⎝M2plR+c1R2+c2R2μν+c3M2plR3+…+L_matter⎞⎠

This is a theory with an infinite number of coupling constants (the ci and, all-importantly, the couplings in L_matter). Nonetheless, at low energies, i.e., for ε ≡ E²M²_pl≪1, we have a controllable expansion in powers of ε. To any finite order in that expansion, only a finite number of couplings contribute to the amplitude for some physical process. We have a finite number of experiments to do, to measure the values of those couplings. After that, everything else is a prediction.

In other words, as an effective field theory, gravity is no worse, nor better, than any other of the effective field theories we know and love.

The trouble is that all hell breaks loose for ε∼1. Then all of these infinite number of coupling become equally important, and we lose control, both computationally and conceptually."

[ https://golem.ph.utexas.edu/~distler/blog/archives/000639.html ]

That gravitons are discrete doesn't mean that the field (or spacetime) is.

• As for the latter, I was careful to point out that GR describes spacetime as smooth!

[And after listening to Nima Arkani-Hamid's latest lecture, on today's physics and how constrained it is, over the weekend I have come to appreciate this even more. Spacetime is likely smooth, because the universe is large.

It goes like this:

Nima described how we can extract 3D and 1/r^2 forces from knowing about relativity and QM, as well as that only spin 0, 1/2, 1, 3/2 and 2 particles will be seen. The 3/2 slot is the only remaining unrealized degree of freedom in the semi-classical physics, and it is covered by supersummetry.

_If_ it is taken, according to Nima this correspond to curled up "quantum" dimensions of length 1. Since they share (entangle and decohere, I guess) quantum fluctuations, they regulate the vacuum energy to open up for a large, non-planck scale, universe. (See Chronos for some arguable observations consistent with this.)]

 

[torsten]

For me, it is a dogma that the spacetime in quantum gravity has to be discrete. As far as I know, there is no experiment showing the discreteness. It is interesting that a smooth manifold has much to do with discrete structures.
I will only mention a few one: (for more details, see my essay at FQXi http://fqxi.org/data/essay-contest-files/AsselmeyerMalu_FQXIessay201_1.pdf and http://fqxi.org/data/essay-contest-files/AsselmeyerMalu_AsselmeyerMa.pdf)
1. A smooth manifold is determined by a discrete set of information (the handlebody structure)
2. Areas and volumes can be discrete in hyperbolic 3-manifolds (smooth) by using Mostow rigidity.
3.The smooth structure of a 4-manifold can be very wild and is determined by an infinite discrete structure, a tree.

These are only a few points. Ok, currently our theory does not include gravity but we are working on this and it looks promising.

 

[marcus]

... It is interesting that a smooth manifold has much to do with discrete structures.
...

That's a good observation! And you mention some interesting examples of discrete structure. We should emphasize the distinction between structure (primarily geometry and topology in this case) and any underlying point-set which the theory might have to represent "spacetime".

I suppose that there can be a theory of geometry in which there are physical objects: tables, chairs, cell-phones, exploding fire-crackers, fried chicken, clocks etc.
and in which there are angles, distances, areas, volumes describing and relating these physical objects. And there might be no underlying point-set in the theory. The only things like coordinates in the theory would then be measurements of relative positions and angles etc as they correlate with the clocks.

A point-set representing "spacetime" might, in other words, be excess baggage.
I don't know of any experiment showing the physical existence of points of space. All I know about are measurements of relative position etc...that is, of geometric relations between things, events IOW.

If the geometric structure is, as you suggest, discrete, then I would expect the results of those measurements (the quantum observables) to have discrete spectra. Just as so many other quantum observables have discrete spectra.

That might or might not be the case. However it could be the case that the observables have discrete outcomes without our being required to imagine any underlying spacetime point-set (discrete or smooth or what-you-will). IOW the point-set is excess baggage. One may then only introduce it into the mathematical formalism as a convenience if that turns out to be useful.

 

[julcab12]

.. There has been a lot of formulation going on the structure of spacetime -- gravity as thermodynamics, LQG, Casual sets, Casual dynamical sets and holography. Each has it's own special way of representing and expressing reality. We do not know what kind of reality might be happening beyond. Or such ultimate fundamental 'thing' exist. Hence, Discrete or continuous. But we can offer clues or at least a sense of directional path that nature is trying to tell us(hypothetically). So, we construct a model that agrees with reality in a way that is effective or serves as a reflection. We introduced a vision that became the backbone on how to create a model of reality. Newton has that first picture of spacetime (space, time, particle moving in empty space/stationary/rigid Newton space) then Faraday and Maxwell adds fields/electromagnetism(that vibrate and expressed dynamics of deformity) to the picture. Later on. Einstein added another field (gravitational field--well a dynamic type of Newtonian space) but he didn't stop there! he look at the picture and integrate a concept of relativity of spacetime as a single block as part of the whole picture of the field. Then quantum mechanics--Dirac/Heisenberg etc kicks in gave a picture of particle as quantum objects/probabilistic that also resembles a field which is discrete, Despite the fact that the equations has a naturally occurring continuous dynamic as part of the formulation but not the structure itself.

Ok. We know that the electromagnetic field has discreteness-- flickering particles and all it's weirdiness. It's is also natural that the gravitational field/spacetime 'might' have an minimal structure - granular -- LQG guys! I don't know if the direction is true but i know this guy Rovelli follows that same effective path or picture.

http://www.nature.com/news/theoretical-physics-the-origins-of-space-and-time-1.13613#reality

 

[PeroK]

There are different definitions of "continuous". The one I was using is that "continuity" is the property that, between any two rational numbers, there is always another rational number; there is no such thing as a "minimum" interval between two rational numbers. So if possible measurement values were represented by rational numbers, there would be no minimum possible measurement value.

You're right that this definition of "continuous" does not guarantee other properties that might be desirable, such as convergence of all Cauchy sequences.

Re the question of measurements: of length, say. It might be that there is a minimum measurable length and, perhaps, all measurements would be an integer multiple of that.

Or, perhaps any rational number is a possible measurement.

But, it's not clear to me how you could get an irrational number as a direct measurement. You might infer it (circumference of a circle from a rational radius). How could an irrational number be the direct result of a measurement? Is it possible?

Or, alternatively, could you define the length of something as the limit of a sequence of hypothetically more and more accurate measurements? Then lengths would typically be irrational.

 

[marcus]

.. There has been a lot of formulation going on the structure of spacetime -- gravity as thermodynamics, LQG, Casual sets, Casual dynamical sets and holography. ..

http://www.nature.com/news/theoretical-physics-the-origins-of-space-and-time-1.13613#reality

Julcab, thanks for the NATURE NEWS article! It covers a lot of different approaches to the quantum theory of space-time geometry currently being worked on, and it's written in popular wide-audience style. Zeeya Merali is a talented journalist, I've seen other articles by her and they're not bad.

But we should try to do is be clear about the terminology. discrete and continuous are basic terms out of point-set topology. A discrete (topological) space means something definite. It means something definite for a function f: X --> Y to be continuous. One normally does not apply the term "continuous" to sets or topological spaces. The expression "continuous space" is not used because it does not mean anything.

So if you look at Torsten's post, just now, you see he does not use the word "continuous" applied to spaces. He talks about "smooth" manifolds, a well-defined concept out of differential geometry. It's a condition on the coordinate mappings, smooth means that certain *functions* are infinitely differentiable---their derivatives are defined to all orders.

So discrete and continuous are not opposites and they apply to different classes of objects. There is the term "connected" in point-set topology. Sometimes when people say "continuous" they really mean connected. (can't be broken into two disjoint open sets)
The set of real numbers < 0 or >1 is not connected. It consists of the real line with usual topology, with the closed interval [0, 1] removed.
R \ [0,1] is not connected, however notice that if you pick any two points in it you can always find a point between them. Infinitely many points, in fact.
And you can make it even more disconnected and remove another closed interval.
If you pick any two points in R \ [0,1] \ [2,3] you can always find infinitely many points between them.

The rational numbers with the usual topology (which they get as a subspace of the reals) are not connected (easy to divide them into two disjoint open sets) but they are also not discrete. (any point is an accumulation point)
It does not make sense to call the rationals "continuous" because a "continuous space" or "continuous set" is not meaningful. It doesn't have a commonly recognized meaning in mathematics.

 

[marcus]

...
Or, alternatively, could you define the length of something as the limit of a sequence of hypothetically more and more accurate measurements? Then lengths would typically be irrational.

Maybe not. For example if Loop QG or something like it is right, the area observable has discrete spectrum. There's a practical problem of how you prepare many copies of a system that instantiates the same physical area. But what they are saying is if you could measure very small physical areas you would always be getting answers from the same discrete set. There is a minimum positive area measurement but the larger areas you can get, though they are discrete, are not simply multiples of it.

 

[torsten]

That's a good observation! And you mention some interesting examples of discrete structure. We should emphasize the distinction between structure (primarily geometry and topology in this case) and any underlying point-set which the theory might have to represent "spacetime".

I suppose that there can be a theory of geometry in which there are physical objects: tables, chairs, cell-phones, exploding fire-crackers, fried chicken, clocks etc.
and in which there are angles, distances, areas, volumes describing and relating these physical objects. And there might be no underlying point-set in the theory. The only things like coordinates in the theory would then be measurements of relative positions and angles etc as they correlate with the clocks.

A point-set representing "spacetime" might, in other words, be excess baggage.
I don't know of any experiment showing the physical existence of points of space. All I know about are measurements of relative position etc...that is, of geometric relations between things, events IOW.

If the geometric structure is, as you suggest, discrete, then I would expect the results of those measurements (the quantum observables) to have discrete spectra. Just as so many other quantum observables have discrete spectra.

That might or might not be the case. However it could be the case that the observables have discrete outcomes without our being required to imagine any underlying spacetime point-set (discrete or smooth or what-you-will). IOW the point-set is excess baggage. One may then only introduce it into the mathematical formalism as a convenience if that turns out to be useful.

In principle I agree with: the discrete spektra is the most important point, see the smooth Schrödinger equation.
I also agree that the point is not fundamental in GR. If I remember correctly, it is the famous hole argument of Einstein
(see http://plato.stanford.edu/entries/spacetime-holearg/). It was resolved by assuming that matter fully (and only) generates gravity. But one can go a step further: what if matter is itself a part of spacetime (as a special 3-manifold)? Then there is only spacetime and spacetime is not only excess baggage.

 

[PeterDonis]

It might be that there is a minimum measurable length and, perhaps, all measurements would be an integer multiple of that.

I agree this is possible; but if it is true, then the measurement results do not form a continuous set, so if we have two length measurements that differ by the smallest measurable amount, then it makes no sense to ask what is "between" the measured points. That was the point of my original post along these lines.

 

[member 11137]

Continuous or not, it seems to be a question of scala and of motion. Just think about the propeller of a plane. If it doesn't move, you see for example three or four wings. When it turns, you just see a kind of circle. All this is inspiring a link between a stroboscopic principle and the fact that an object seems to be continuous or not...

 

[jambaugh]

I believe it is a mistake to even ask the question, or posit an answer to questions about the nature of space-time as if it were a real object. This mistake is natural given the beauty and elegance of the geometric representation of Einstein's GTR. But remember it is only a description. The equivalence principle led Einstein to equate a dynamical force of gravity to a curved space-time geometry in which there is no dynamical force, only geodesic motion. Note that this "equating" is a two way street. We cannot observe geometry. We can only observe a relative relationship between geometry and a dynamical gravitational force (in which inertial and gravitational mass are equivalent). We do not observe space-time geometry, or space-time anything. The fact that we can excise the dynamical force of gravity by suitable choice of geometry makes it easy to say "There is no force of gravity, only geometry" but that would not be scientifically correct. We should rather say "The force of gravity is only defined relative to a choice of geometry both of which are relative." One might as easily say there is no geometry but that makes formulating theories harder.

We use a concept of space-time geometry as a scaffolding to describe the behavior of objects and events. Those are the "real things". Space-time itself is but a mental construct (a useful one and perhaps a uniquely necessary one but none-the-less...) and as such should not be argued about as if it were a physical object. Is it continuous? I find using a continuous one more useful for physics. You are free to choose a discrete one if you like.

(Side note: I think this "error" is also leading many down a dead end channel to attempt to "quantize space-time" in various quantum gravitation approaches. I'm just not smart enough to back this opinion up with a good journal article. ;))

 

[julcab12]

We use a concept of space-time geometry as a scaffolding to describe the behavior of objects and events. Those are the "real things". Space-time itself is but a mental construct

...I'm just curious thought. Even visual information -- mental construct. Almost/all of our experiences are interactions fundamentally. Objects are also interactions and events. Spacetime is not just description. Whatever it is. It should have some structure.

http://iai.tv/video/spacetime-and-the-structure-of-reality

 

[PeterDonis]

We should rather say "The force of gravity is only defined relative to a choice of geometry both of which are relative."

I'm not sure this is quite right. First, GR does not define a "force of gravity relative to a choice of geometry". It says "gravity" is not a force at all. In other words, GR adopts a definition of "force" that is physically different (and, IMO, more reasonable) than the definition in Newtonian theory: a force in GR is something that causes proper acceleration, i.e., something that is felt as weight. An object moving solely under gravity is weightless, feeling no force; so GR says gravity is not a force. None of this says anything about "geometry"; it's all about what, physically, we should denote by the term "force".

Also, there is only one choice of geometry that eliminates gravity as a force, i.e., that respects the physical definition of "force" that GR uses. However, there is nothing requiring us to call the thing we are talking about "geometry" or "spacetime geometry". That name is chosen because the math we use is the math of Riemannian geometry (more precisely, pseudo-Riemannian geometry), but that doesn't mean the physical interpretation has to be the same as it is in ordinary geometry.

We use a concept of space-time geometry as a scaffolding to describe the behavior of objects and events. Those are the "real things". Space-time itself is but a mental construct (a useful one and perhaps a uniquely necessary one but none-the-less...) and as such should not be argued about as if it were a physical object.

Even in the context of classical GR, I don't think this is quite right either. The whole point of GR is that the geometry of spacetime (or whatever word you want to use to denote that thing) is dynamical; it interacts with matter and energy. IMO that makes it a physical object, not just a mental construct.

In quantum gravity, spacetime has to be a physical object, because it is built out of simpler physical objects, just like atoms are built out of simpler objects. But that's going beyond GR.

 

[jambaugh]

...The whole point of GR is that the geometry of spacetime (or whatever word you want to use to denote that thing) is dynamical; it interacts with matter and energy. IMO that makes it a physical object, not just a mental construct.

That is the *model* we use in describing GR but it is not quite what the theory states. The theory states that the dynamical behavior of physical object will be such that... (fill in here with what you said). " But you cannot observe space-time itself in order to say if it does or does not dynamically interact with anything. It is just like the aether of the aetheric forms of SR where clocks are "really slowed by their motion through the aether" and measuring rods "are really shortened by their motion through the aether" and so the aether "must be physical".

Recall that in non-relativistic physics x is an observable but t is not... it is a parameter. Whether you ask if t is "really continuous" is meaningless in that theory. When we relativize we don't make t more "space-like" we make x more "time-like" in that it too ceases to be meaningful as an observable, instead becoming a parametric quantity.

Space-time is parametric. The geometry we overlay let's us distinguish between "free geodesic motion" and "motion under the influence of a dynamical force" but that dividing line is arbitrary and relative. Choosing the geometry is like selecting the gauge condition. The constraint surface of the gauge condition isn't physically real but it is physically relevant to the predictions of how a physical object behaves.