Two World-theories (neither one especially stringy)

[marcus]

there is a common element in the last 3 CDT papers. since they carefully repeat it each time maybe we should listen extra carefully:

---from hep-th/0404---

Causal dynamical triangulations are a framework for defining quantum gravity nonperturbatively as the continuum limit of a well-defined regularized sum over geometries. Interestingly, and in complete agreement with current observational data is the fact that the physical cosmological constant \Lambda in dynamical triangulations is necessarily positive...

---from hep-th/0411---

Causal Dynamical Triangulations constitute a framework for defining quantum gravity nonperturbatively as the continuum limit of a well-defined regularized sum over geometries. We reported recently on the outcome of the first Monte Car..."

---from hep-th/0505---

"In the CDT approach, quantum gravity is defined as the continuum limit of a regularized version of the nonperturbative gravitational path integral . The set of spacetime geometries to be summed over is represented by a class of causal four-dimensional piecewise flat manifolds (“triangulations”). Every member T of the ensemble of simplicial spacetimes can be wick-rotated to a unique Euclidean piecewise flat geometry, whereupon the path integral assumes the form of a partition function ...

All geometries share a global, discrete version of proper time. In the continuum limit, the CDT time \tau becomes proportional to the cosmological proper time of a conventional minisuperspace model..."

 

[marcus]

... compared to networks and spin foams...

you prompted me to compare CDT with spin foams and it occurred to me that they look rather alike, there is a (superficial?) visual resemblance

but CDT is less dependent, I believe, on coordinates
I think of Georg Riemann around 1850 offering a way to do geometry without prior commitment to a particular metric---but only with some coordinate patches: a shapeless "manifold"---Riemann had to have coordinates.

And then around 1950 came Tulio Regge offering a way to do "general relativity without coordinates" (I think that was his paper's title or something like it).

so then you did not even need to make a prior commitment to coordinate patches (and the DIMENSION always sneaks in as the number of coordinate functions, the "n" in the local resemblance to Rⁿ.)

So CDT as a Regge offspring makes less prior commitments. It doesn't have an essential need for coordinates because the "geometry is all in the gluing". the geometry is all in how the many identical cells are stuck together.

and although one chooses at the outset cells of a particular dimension (for instance simplices which are pieces of Minkowski space) when they are glued together it turns out that the dimension can be determined in some empirical ways (e.g. by running diffusion) and it does not need to come out exactly 4, or even to be an integer. it is, after all, not a DIFFERENTIABLE manifold that one makes by gluing the cells together. this seems very strange to me but it does look like one is going further in the direction of not having prior commitments to anything.

and yet they are able to define a transfer matrix and a hamiltonian and to calculate and to do computer simulations, which a lot of people working with stuff on differentiable manifolds do NOT seem able to do. so it is paradoxical----they throw out---they make less prior commitment to structure---and yet they seem able to calculate MORE rather than less----like they even get statistics about their spacetimes----having made so many of them that they must record statistics.

utrecht interests me, now 't Hooft and Robbert Dijkgraaf, and Renate Loll, and Jan Ambjorn are there and the last 3 of them are invited speakers at the October "Loops 05" conference. I am thinking that Utrecht work will be fairly noticeable at this year's conference

 

[selfAdjoint]

It's an interesting and I believe deep toplogical question how much you actually predetermine the coordinate stuff with this "gluing"? There was always this uneasy feeling that AJL had presetermined pseudo-Riemannian with their "causality" - see also causal sets which seems to do that without the gluing.

Back in the grad school day, I was always impressed by the fact that the left side of the Riemann-Roch index theorem was pure toplogy and the right side was analytic. Of course since then we have the same thing much bigger with Atiyah-Singer. And look at the run BRST has had! Save your Jordan curves boys, topology will rise again!

 

[marcus]

It's an interesting and I believe deep toplogical question how much you actually predetermine the coordinate stuff with this "gluing"? There was always this uneasy feeling that AJL had presetermined pseudo-Riemannian with their "causality" - see also causal sets which seems to do that without the gluing.

Back in the grad school day, I was always impressed by the fact that the left side of the Riemann-Roch index theorem was pure toplogy and the right side was analytic. Of course since then we have the same thing much bigger with Atiyah-Singer. And look at the run BRST has had! Save your Jordan curves boys, topology will rise again!

I'll talk around the issues you raise here, which are interesting ones, though unable to address them head-on. Another way to think of the gluing of 4-simplices together is as carving up the S³ x R carcass.

AJL start by putting a topological S³ x R into the computer.

for technical reasons they specify that there are going to be, say, 80 time steps and some total number of simplexes, like 200,000.
this S³ x R gets repeatedly divided up into around 200,000 4simplexes, and they takes their statistical measures of what results.
and then they start again, using a different total number, like 300,000

their idea is that the CDT theory is the continuum limit

they say spacetime is not discrete (or that they have seen no indication of a fundamental discreteness, in that sense they are at odds with LQG which has seen a discrete spectrum of area and volume operators)

they say they have seen no indication of a minimum length

their spacetime, their universe, is a continuum limit of quantum geometries that are established in a way that does not use coordinates or metrics or connections, but only uses simplexes (instead of coordinates)

but they do not think the world is made of simplexes, that is just what they use to describe a sequence of finer and finer geometries.

now I have to think how to relate what I've just said to the issues you raised

 

[marcus]

I'll talk around the issues you raise here, which are interesting ones, though unable to address them head-on. Another way to think of the gluing of 4-simplices together is as carving up the S³ x R carcass.

AJL start by putting a topological S³ x R into the computer.
...

they put S³ x R into the computer and carve it up, in the case where they are using 4-simplexes as the basic relational cell (I think of the 4-simplex, which is a tiny piece of minkowski space, as an ATOM OF RELATIONSHIP, or as an atom of spacetime relation and causality)

in the cases where they have used 2-simplexes or 3-simplexes as the basic relational buildingblock, then they used a different topological carcass to carve up. they used S¹ x R, or else S² x R.

now the interesting thing, or one very interesting, is that if you don't use the right carving rules (dont make it causal or Lorentzian but only euclidean) then you can get any dimension. Like with S² x R
dividing up into 3-simplexes it does not have to come out 3D. You can get a beast that is 2D or that is INFINITE dimensional. unbounded (infinite in the limit) anyway.
the infinite dimensionality comes from too much connectivity

like think of the Earth surface with the N hemisphere carved into 360 skinny triangles each one with its vertex at the N pole. this is too connected because every spot is in the same triangle as the N pole. One has to have a carving rule that makes this kind of thing unlikely.

the point I am making is that DIMENSIONALITY MUST EMERGE and it is not predetermined by the fact that you started with a topological space that is
S³ x R
(because a mere topological space does not have preordained dimension, and when it gets some structure that permits dimension to be defined then that dimension does not have to be the same at every point)
and the dimensionality that emerges is also not predetermined by the fact that we happen to be using 4-simplexes to carve the S³ x R up in.

Because the topological space can be partitioned into 4-simplexes in such a way as to give very high dimension or very low dimension to various locations in it (the S³ x R carcass doesn't have any geometry until the 4-simplexes come and partition it)

 

[marcus]

I had better quote AJL about this, because it is a key point, and I will repost the basic CDT reading list in case anyone wants to refer to the original papers

This is from the introductory paragraph of "Emergence of a 4D World", reference 2 below:

"... a particular case of the more general truth, not always appreciated, that in any nonperturbative theory of quantum gravity dimension will become a dynamical quantity, along with other aspects of geometry. (By dimension we mean an effective dimension observed at macroscopic scales.)..."

Dimension is just another aspect of geometry. AJL are working with a topological continuum which has no prior geometry and they put the geometry on in a quantum way, and dimensionality emerges, and it is different at different scales, and down near Planck scale it tends to be around D = 2, or even less. But happily enough at big scale, where we like to build our houses and go snow-boarding etc, it is right around D = 4.

Here is the short list of CDT references

1.
http://arxiv.org/hep-th/0105267
Dynamically Triangulating Lorentzian Quantum Gravity
J. Ambjorn (NBI, Copenhagen), J. Jurkiewicz (U. Krakow), R. Loll (AEI, Golm)
41 pages, 14 figures
Nucl.Phys. B610 (2001) 347-382
"Fruitful ideas on how to quantize gravity are few and far between. In this paper, we give a complete description of a recently introduced non-perturbative gravitational path integral whose continuum limit has already been investigated extensively in d less than 4, with promising results. It is based on a simplicial regularization of Lorentzian space-times and, most importantly, possesses a well-defined, non-perturbative Wick rotation. We present a detailed analysis of the geometric and mathematical properties of the discretized model in d=3,4..."

2.
http://arxiv.org/abs/hep-th/0404156
Emergence of a 4D World from Causal Quantum Gravity
J. Ambjorn (1 and 3), J. Jurkiewicz (2), R. Loll (3) ((1) Niels Bohr Institute, Copenhagen, (2) Jagellonian University, Krakow, (3) Spinoza Institute, Utrecht)
11 pages, 3 figures; final version to appear in Phys. Rev. Lett
Phys.Rev.Lett. 93 (2004) 131301
"Causal Dynamical Triangulations in four dimensions provide a background-independent definition of the sum over geometries in nonperturbative quantum gravity, with a positive cosmological constant. We present evidence that a macroscopic four-dimensional world emerges from this theory dynamically."

3.
http://arxiv.org/abs/hep-th/0411152
Semiclassical Universe from First Principles
J. Ambjorn, J. Jurkiewicz, R. Loll
15 pages, 4 figures
Phys.Lett. B607 (2005) 205-213
"Causal Dynamical Triangulations in four dimensions provide a background-independent definition of the sum over space-time geometries in nonperturbative quantum gravity. We show that the macroscopic four-dimensional world which emerges in the Euclidean sector of this theory is a bounce which satisfies a semiclassical equation. After integrating out all degrees of freedom except for a global scale factor, we obtain the ground state wave function of the universe as a function of this scale factor."

4.
http://arxiv.org/abs/hep-th/0505113
Spectral Dimension of the Universe
J. Ambjorn (NBI Copenhagen and U. Utrecht), J. Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)
10 pages, 1 figure
SPIN-05/05, ITP-UU-05/07

"We measure the spectral dimension of universes emerging from nonperturbative quantum gravity, defined through state sums of causal triangulated geometries. While four-dimensional on large scales, the quantum universe appears two-dimensional at short distances. We conclude that quantum gravity may be "self-renormalizing" at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction."

 

[marcus]

...This is from the introductory paragraph of "Emergence of a 4D World", reference 2 below:

"... a particular case of the more general truth, not always appreciated, that in any nonperturbative theory of quantum gravity dimension will become a dynamical quantity, along with other aspects of geometry. (By dimension we mean an effective dimension observed at macroscopic scales.)..."

Dimension is just another aspect of geometry. AJL are working with a topological continuum which has no prior geometry and they put the geometry on in a quantum way, and dimensionality emerges, and it is different at different scales, and down near Planck scale it tends to be around D = 2, or even less. But happily enough at big scale, where we like to build our houses and go snow-boarding etc, it is right around D = 4.

Here is the short list of CDT references...

Now #5 on the short list has appeared

5.
http://arxiv.org/hep-th/0505154
Reconstructing the Universe

and we have another new title in the "in preparation" or "to appear" category, which is the first CDT paper AFAIK about BLACK HOLES.

This is to be a collaboration of Renate Loll with BIANCA DITTRICH.

The two of them co-authored a CDT paper back in 2002 but since then Dittrich has been doing stuff with Thiemann "Master Constraint" program and also with partial-observable, relational-time. In effect she has been helping Thiemann rake his chestnuts out of the fire.

Dittrich work on Master Constraint has been major. She is an interesting case because she is both active currently in LQG/LQC and also has a significant interest in CDT.

Also it seems like a high priority job to see how CDT deals with black holes
(we already know how CDT does the big bang non-singularity, or seem to know, all the AJL simulations portray that, but what AJL show is a vacuum state universe so far with no black holes)

the Dittrich-Loll paper to appear is called
Counting a Black Hole in Lorentzian Product Triangulations

It would not surprise me if Dittrich moved from AEI-Potsdam to Utrecht. The Utrecht group seems to me to be growing.

[EDIT in case anyone is curious the 2002 paper of Dittrich-Loll was
http://arxiv.org/abs/hep-th/0204210 ]

 

[marcus]

Basic incompatibility between CDT and LQG (removable?)

the basic incompatibility is that LQG is based on a smooth continuum and CDT on an unsmooth. CDT is the limit of piecewise flat manifolds, like a Feynman path is a superposition of piecewise linear paths and the result is not differentiable

neither CDT nor LQG is DISCRETE. but in LQG the underlying space is a differentiable manifold (smooth although without prior metric)
and in CDT the underlying space is not smoothly coordinatized, it is NOT a differentiable manifold. it is all jig-jaggy piecewise flat. and it does not have any coordinates.

in 1850 Riemann invented how to do a continuum without prior metric (but with smooth functions as coordinates, the point is the patches)

in 1950 Tulio Regge invented how to do a continuum and write Einstein Gen Rel on it WITHOUT COORDINATES, no differentiable functions.

So this is a major disconnect---it looks as if Renate Loll is A HUNDRED YEARS MORE ADVANCED, in the history of mathematics, than, say Lee Smolin and Carlo Rovelli. this could be serious.

But maybe it can be fixed. Maybe one can do LQG on a series of finer and finer piecewise flat manifolds. simplicial manifolds. whatever.
(here "flat" means 4D minkowski, the usual special relativity 4D flat)

the thing that Renate Loll can say (and the Utrecht gang have their knives out and mean business these days) is what I quoted a post back:
in any nonperturbative theory of quantum gravity dimension will become a dynamical quantity, along with other aspects of geometry.

Because LQG is built on a smooth manifold of some chosen dimension, with exactly 3 coordinate functions, or exactly 4, it is so-to-say stuck with that choice of dimension. But a piecewise flat manifold can be so wrinkled and kinky and crumpled and frazzled that it loses track of what its dimension is.
at least at small scale and maybe the real world is like that

this gives the Utrecht people a strong card to play

 

[marcus]

Now #5 on the short list has appeared

5.
http://arxiv.org/hep-th/0505154
Reconstructing the Universe

...

Here is the first paragraph of Reconstructing the Universe:

"Nonperturbative quantum gravity can be defined as the quest for uncovering the true dynamical degrees of freedom of spacetime geometry at the very shortest scales. Because of the enormous quantum fluctuations predicted by the uncertainty relations, geometry near the Planck scale will be extremely rugged and nonclassical. Although different approaches to quantizing gravity do not agree on the precise nature of these fundamental excitations, or on how they can be determined, most of the popular formulations agree that they are neither the smooth metrics g_mu,nu (or equivalent classical field variables) of general relativity nor straightforward quantum analogues thereof. In such scenarios, one expects the metric to re-emerge as an appropriate description of spacetime geometry only at larger scales."

Renate Loll can write English better than most of us nativespeakers, or else it is an AJL team phenomenon. Anyway the style of the AJL writings is usually strong and clear. This helps (them at least).

Now that String seems less promising, seems off track in fact, the other approaches to Quantum Gravity which have so far been mostly at peace under the big Loop tent, are becoming more disposed to controversy among themselves.

Here right away at the beginning, the AJL writer is playing the strong card against LQG. The suggestion is that LQG cannot be right because it is too smooth at small scale. How can LQG be right? since it is built on a smooth differentiable manifold of fixed dimension, and uses "classical field variables" equivalent to the metric, namely the connections. Actually LQG has made a lot of progress and will have accomplished a lot even if it eventually turns out to have been a PILOT STUDY for something else. It has tackled the big bang and inflation and black holes and produced a lot of new ideas like Freidel/Starodubtsev and for instance lately Perez/Rovelli about the Immirzi parameter. Probably there is some stuff it DOES get right. Thiemann has started confirming details of Bojowald's LQC big bang using the full LQG theory etc. Plenty is happening in LQG. But the AJL writer hints how can LQG be fundamentally correct? since "because of the enormous quantum fluctuations predicted by the uncertainty relations, geometry near the Planck scale will be extremely rugged and nonclassical."

Well maybe that is not as aggressive as it sounded to me. Maybe everybody inside the big Loop tent feels that he or she is enough rugged and nonclassical. Maybe spin networks are that. So one can hear this opening paragraph different ways.

There may be some fire at the Potsdam conference in October.
I think String has dwindled so that there is less of a common rival and less of a collective spirit. So people and ideas can wrangle with each other more openly, which I guess is traditional in science so must be OK

 

[selfAdjoint]

I'll talk around the issues you raise here, which are interesting ones, though unable to address them head-on. Another way to think of the gluing of 4-simplices together is as carving up the S³ x R carcass.

AJL start by putting a topological S³ x R into the computer.

for technical reasons they specify that there are going to be, say, 80 time steps and some total number of simplexes, like 200,000.
this S³ x R gets repeatedly divided up into around 200,000 4simplexes, and they takes their statistical measures of what results.
and then they start again, using a different total number, like 300,000

their idea is that the CDT theory is the continuum limit

they say spacetime is not discrete (or that they have seen no indication of a fundamental discreteness, in that sense they are at odds with LQG which has seen a discrete spectrum of area and volume operators)

they say they have seen no indication of a minimum length

their spacetime, their universe, is a continuum limit of quantum geometries that are established in a way that does not use coordinates or metrics or connections, but only uses simplexes (instead of coordinates)

but they do not think the world is made of simplexes, that is just what they use to describe a sequence of finer and finer geometries.

now I have to think how to relate what I've just said to the issues you raised

Two responses to this:

I. Maybe CDT will wind up doing with LQG what the lattice does with the standard model: achieve geniune if spotty non-perturbative results while the lion's share of the physics is being done n-loop perturbative with the continuum theory. (n being a small integer).

II. Topology has lots of more general things than polyhedra (which is what triangulated manifolds are). CW-complexes, ANRs, lots of things in graduated sequences of generality. And there are all those lovely categories.

 

[marcus]

Hi selfAdjoint,
before I forget I must say I am looking out for a paper by Bianca Dittrich and Renate Loll that applies CDT to Schwarzschild black holes.

It has not been posted yet. but curiously I just now encountered a citation to it in an article by Arundhati Dasgupta
http://arxiv.org/find/gr-qc/1/au:+Dasgupta_A/0/1/0/all/0/1

Right at the end of his recent article about black holes (mostly from LQG and related perspective), actually it is the very last sentence of his conclusions, he says:

" There is a discretisation of the Schwarzschild space-time using dynamical triangulation techniques in [20], it shall be interesting to obtain the entropy in that formalism. "

and his [20] is
"[20] B. Dittrich, R. Loll, Dynamical Triangulations of Black Hole Geometries, in preparation. "

However in the AJL article they give a different title, by coincidence it is also their reference [20]
"[20] B. Dittrich and R. Loll: Counting a black hole in Lorentzian product triangulations, preprint Utrecht, to appear. "

So far we know Bianca Dittrich mainly from her work with Thiemann, in particular on the Master Constraint programme.

 

[marcus]

Two responses to this:

I. Maybe CDT will wind up doing with LQG what the lattice does with the standard model: achieve geniune if spotty non-perturbative results while the lion's share of the physics is being done n-loop perturbative with the continuum theory. (n being a small integer).

II. Topology has lots of more general things than polyhedra (which is what triangulated manifolds are). CW-complexes, ANRs, lots of things in graduated sequences of generality. And there are all those lovely categories.

I am not so optimistic about LQG. I value its contribution enormously it has beat a track into the bush. unearthing interesting stuff like the Immirzi parameter and focusing attention on form theories of gravity as in the case of Freidel/Starodubtsev (who almost seem to have a background indep perturbative approach, resolving singularities like BH and BB, finding an automatic generic mechanism for inflation. many of its discoveries will probably persist and mutate in other contexts. BUT. my intuitive feeling is that CDT is now entering an exponential growth phase and remember

CDT HAS a hamiltonian, and looks like it HAS a reasonable chance of correct classical and semiclassical limiting behavior. and it also has the
DYNAMIC DIMENSION card.

and it looks more like a Feynman path integral to me. So I am apprehensive about the longrange prospects of Loop. I see that it can be a very valuable set of PILOT STUDIES, but I see a possibility that it could be eclipsed by CDT.

of course there is always the possibility of convergent evolution and the eventual proof of an equivalence theorem mapping one model onto the other.

(sorry, I have just been idly speculating, usually a waste of time----what we need to do, I suspect, is not speculate but quickly understand everything we can about CDT as it is right now)

 

[Mike2]

I am not so optimistic about LQG. I value its contribution enormously it has beat a track into the bush. unearthing interesting stuff like the Immirzi parameter and focusing attention on form theories of gravity as in the case of Freidel/Starodubtsev (who almost seem to have a background indep perturbative approach, resolving singularities like BH and BB, finding an automatic generic mechanism for inflation. many of its discoveries will probably persist and mutate in other contexts. BUT. my intuitive feeling is that CDT is now entering an exponential growth phase and remember

CDT HAS a hamiltonian, and looks like it HAS a reasonable chance of correct classical and semiclassical limiting behavior. and it also has the
DYNAMIC DIMENSION card.

If all there is so far is space without particles, how do they avoid scale invariance since there seems to be nothing with respect to measure distance with? What is small distance compared to large distance in a universe without particles to measure with respect to?

 

[selfAdjoint]

Here's a new paper http://www.arxiv.org/abs/hep-th/0505165

by Mohammed Ansari & Fotini Markopoulo:

We rewrite the 1+1 Causal Dynamical Triangulations model as a spin system and thus provide a new method of solution of the model

with lots of pictures.

 

[marcus]

Here's a new paper http://www.arxiv.org/abs/hep-th/0505165
by Mohammed Ansari & Fotini Markopoulo:
with lots of pictures.

selfAdjoint, thanks for catching this. I will add it to the surrogate sticky links thread. Good for Fotini for venturing into a new field herself and for getting her gradstudent Ansari into a promising research line at the ground floor

If you like pizza, look at footnote #3 on page 8.

 

[marcus]

I checked the list of grad students at Perimeter. It includes two that have recently co-authored CDT papers:

Tomasz Konopka
Mohammad Ansari

and one we know of from some posts here by John Baez and Kea, namely
Artem Starodubtsev

If I were a grad student, i would want to be in Utrecht (with Ambjorn and Loll), but if not there then maybe Perimeter would not be too bad because they seem to stay engaged. I wish some Utrecht code could be transplanted to a Canadian computer

 

[nightcleaner]

Hi

This is great stuff. I wish I had more time here.

Marcus, do you think the mathematics of fractals as demonstrated by the Mandlebrot and Julia sets will apply to these CDT fractal dimensions? I am still looking for my books on fractals.

Thanks,

Richard

 

[marcus]

... do you think the mathematics of fractals as demonstrated by the Mandlebrot and Julia sets will apply to these CDT fractal dimensions? ...

I can only guess that yes, some of what has been learned about fractal sets WILL be applicable to spacetime at very small scales, but since i am not very knowledgeable about those things, I would not know what to expect.

It seems to me that some things about the familiar beautiful fractals would NOT apply. They are self-similar (proportions the same at all scales) and so their dimensionality is the same all the way down. But the CDT people seem to be saying that spacetime is almost just ordinary cliche 4D spacetime at large scale but gets more frizzy as you go down scale
so that at small sizes it gets quite cheesy and flakey.
So this is not self-similar or scale-invariant behavior at all.

It's early days for understanding these things (at least for me)
and also CDT could be wrong.
I like it that AJL have not been afraid to go ahead with something that is radically innovative and doesn't even have an underlying smooth continuum (such as strings live in, or such as the paraphernalia of LQG is built on)

instead of a smooth differentiable continuum, they have an extremely kinky continuum, without even coordinate patches. this is a moral satisfaction to me and resonates with my deep inner perversity, thus having a calming effect. it almost makes me happy. I hope you too

 

[nightcleaner]

Yes, quite so.

I suspect the universal set is not discrete in itself, but any observer is a limited system and therefore the interaction of the observer with the universe is limited...hence the fact that, to any observer, the universe appears to have limits. These limits necessarily receed, as the observer develops.

R

 

[marcus]

Nightcleaner, I am coming to recognize the paper I call "Dynamically..."
as a tutorial.

(short for hep-th/0105267 "Dynamically Triangulating Lorentzian Quantum Gravity)

Lorentzian Q. G. is one of their old names for CDT. they finally settled on CDT permanently in 2004. But when you take that change in terminology into account, then the title tells you what it is. This is a HOW TO do it paper. they take you thru the 3D case partly because it is easier and once you have been thru the 3D case the 4D case feels better.

A tutorial type paper is one that it is worthwhile working thru at least part of it, equation by equation.

For someone who likes triangles and tetrahedra, you may at first be confounded by the fact that there are two kinds of triangles spacelike and timelike and they have different areas!

For example look on page 6, equation (4) near the bottom.

area of spacelike triangle = \frac {\sqrt{3}}{4}

I suspect it was Rafael Sorkin who made up the word "bones" for the D-2 simplexes. So if you are in 3D case, the "bones" are just line segments, the edges of the tets. the reason I think this is because he grew up in Chicago.

anyway a spacelike triangle is just an equilateral with sides equal ONE.
so naturally the area = \frac {\sqrt{3}}{4} we did this in middleschool or 9th grade

BUT THE SQUARE LENGTH OF A TIMELIKE EDGE IS MINUS ALPHA.

In CDT you allow the timelike length to be an imaginary number oi veh oi veh, and you give it some freedom so that its square length does not have to be exactly minus one, but can be - \alpha

So imagine a timelike triangle, in the sandwich, with its base in one spacelike layer. So its base has length one! OK but the other two sides are timelike. so the square of one of those sides is - \alpha.

It is an ISOSCELES triangle with the two equal sides IMAGINARY (whoops ) so what is the HEIGHT? well you just do pythagoras and square the hypoteneuse (minus alpha) and subtract 1/4 (the square of the side)

the square of the height = - \alpha - 1/4 = - \frac{4\alpha + 1}{4}.

the height = i \frac {\sqrt{4\alpha + 1}}{2}.

now the base = 1, remember it is spacelike and all in one layer,
so one half base times height = \frac {\sqrt{4\alpha + 1}}{4}.

where I dropped a factor of i because I thought you might not be watching.
and hey, an area or volume should be a real number.

 

[marcus]

what I was calling "height" in the previous post was just the height of a timelike triangle (timelike means it spans two layers, we worked with layers before, the CDT universe is foliated in spacelike layers)

but suppose now we have a TETRAHEDRON of the (3,1) sort, that has a spacelike base that it is sitting on. an equilateral triangle with side one.
And it has three timelike isosceles triangles as sides.

now what is the (timelike) height of that tetrahedron?

well you draw a picture and see that it is the vertical leg of a right triangle whose squarehypoteneuse is minus alpha and whose other square leg is 1/3
so (ignoring a factor of i) the height of the tetrahedron is
\frac {\sqrt{3\alpha + 1}}{\sqrt{3}}.

and the volume of a cone or pyramid is always 1/3 the base area times the height, and the base of this tet is just \frac {\sqrt{3}}{4}

cause its an equilateral triangle! so you multiply the base area times the height and the sqrt{3} cancels and you get
\frac {\sqrt{3\alpha + 1}}{4}
and one third of that (because it is a pyramid type thing) is

\frac {\sqrt{3\alpha + 1}}{12}

now let's see if that is what AJL say. YES! just look at their
equation (5)
so these are the stretches they have you do before anything aerobic happens. it is a tutorial.

 

[selfAdjoint]

I suspect it was Rafael Sorkin who made up the word "bones" for the D-2 simplexes. So if you are in 3D case, the "bones" are just line segments, the edges of the tets. the reason I think this is because he grew up in Chicago.

Is there some sly angeleno dig at Chicago here? What is the connection of Chitown with 'bones"? And what about LA then? I saw Chinatown, and it was (loosely) based on real events.

 

[marcus]

americans like short visual-type words
(this is one reason much of our language is still strong even if the populace gets fat and fatuous)

even when they are doing misguided empty physics, american physicists will think up good words to call things

please remember I am not an angeleno, I do not do angeleno "digs"

I am proud that Rafael Sorkin grew up in Chicago.

Look at the list of organizers of the Loops 05 conference!
Almost no one there was born in the States!
Rafael Sorkin is one of the very few.

Theoretical physics in the US has lost ground and gotten off track probably. When we hear about really new developments they seem to be coming from Utrecht! It didnt used to be that way, IIRC.

I think "bones" is a good strong-visual monosyllable coinage, for something that needed a name. (the D-2 simplices are important in Regge calculus)

i don't know who named them bones. But I would bet an american
and Sorkin in 1975 wrote
"Time-evolution in Regge calculus" which I have not seen.
So I don't know who coined it, but he is as likely as anyone i can think of.

there is also an Englishwoman, ruth williams. she might have.

It was not a German, who would have said
\mathfrak {Knochen}
instead

Sometimes I think you are sensitive about the Midwest. We coasters like the Midwest. We listen to Garrison Kiellor Prairie Home Companion.
and sometimes i can't tell if you are joking, like when you say angeleno (which to N calif ears sounds derogatory)

 

[selfAdjoint]

Marcus I think the "bones" came from the fact that what you have been calling a tet (for tetrad) is also known as a vierbein (= "four bones" or perhaps "quadruple bone"). In string theory where they do 1+1 GR on the string worldsheet, sometimes you see the term "zweibein" for the corresponding thing, and sometimes by a stated abus de langage they still say vierbein.

 

[marcus]

Marcus I think the "bones" came from the fact that what you have been calling a tet (for tetrad) is also known as a vierbein (= "four bones" or perhaps "quadruple bone"). In string theory where they do 1+1 GR on the string worldsheet, sometimes you see the term "zweibein" for the corresponding thing, and sometimes by a stated abus de langage they still say vierbein.

words are fascinating, arent they? and they are soaked in history which is fascinating too

In German Bein is "leg"
and a "Dreibein" is the same as a Tripod (word constructed the same way, as "three-leg" or "three-footed thing")

there is no connotation of "bone" as far as I know

and yet Bein does certainly sound like bone!

I suppose that Dreibein is an ancient German word, like the Greeks had Tripods for burning incense, and in Homer you have catalogs of X number of goblets and Y number of gold plates and Z number of Tripods, catalogs of things that one gave to the priest of apollo or in recompense to appease wrath etc.

I could be wrong. I will try to look it up

 

[marcus]

It is important to notice that in simplicial manifolds Regge calculus a "bone" has nothing obvious to do with a Bein or a Vierbein or Dreibein.

where the basic building block is a 4-simplex, the bones are the TRIANGLES that form the sides of the Tetrahedra that form the faces of the 4-simplex.

where the basic building block is a 3-simplex, or tetrahedron, then the bones are the EDGES of the triangles that form the sides of the 3-simplex.

The bones are, by definition, the (D-2)-simplexes.


By contrast, a "Dreibein" at some point in a manifold can be pictured as a little X,Y,Z reference frame made of 3 tangent vectors. You can stick out your thumb and 2 fingers to make XYZ axes and give the idea.

In a 4-manifold, a "Vierbein" at some point is the analogous thing made of 4 tangent vectors.

The reason that the concept bone is a natural is that in 3D Regge calculus, the edges play the same role and the triangles do in 4D.

IN 3D, To make the Regge version of the Einstein action one is measuring the DEFICIT ANGLE of the tetrahedra around some EDGE

IN 4D one is measuring the deficit angle of the 4-simplexes which are joined around some TRIANGLE.

THE BONE IS WHERE THE THINGS COME TOGETHER THAT YOU ADD UP THEIR DIHEDRAL ANGLES TO FIND OUT THE DEFICIT ANGLE

the reason you need the "bone" concept is so that you can speak in general about 3D and 4D and 5D gravity all at once, and so you can treat them as analogs of each other.

the deficit angle of the simplexes joined around a bone is, of course, how Regge discovered he could measure curvature and in this way, using simplexes, he could implement General Relativity.

it is only gradually dawning on me how beautiful this approach is

 

[nightcleaner]

Hi Marcus

I am way behind on my reading and there are strawberries to put in and trees to plant, as well as dogs, rabbits, and a horse to care for. The good news is the sun has made an appearance and I have managed to get my laptop setup on my friend Peg's table, a rather crowded place, but with a nice view of the bird feeder, apple trees, sauna, and horse eating hay in the pasture. I have coffee, and a quiet idyllic setting for contemplation of the mysteries of alpha. Oh, lucky man.

I am reading the triangulation paper, Dynamically Triangulating Lorentzian Quantum Gravity, arXiv:hep-th/0105267v1, and I even think I can follow some of it. The idea that there are time-like and space-like edges, faces, volumes and so on, seems contrary to spacetime equivalence, but if it produces results I am willing to play along.

I am currently trying to understand Figure 1. Part (a) and part (b) seem to me to be identical tetrahedrons except for a small rotation. Part (a) is said to be a (3,1) tetrahedron, and part (b) is said to be a (2,2) tetrahedron. I would have thought (3,1) and (2,2) are (space,time) notations, but I can't seem to make sense of the figures from that perspective. Perhaps you or selfAdjoint or someone will help me find what it is I am missing here.

I have to use daylight to do chores, but hope to return to this tonight.

Thanks,

Richard



(later...)

Hi Marcus and all

I am working offline while the rice cooks.

Returning to the triangulation paper, I see that a point is defined as having volume one. This seems to me a generalization of the idea of volume, but the authors say that it is conventional and so I accept it, even though my sense of geometry tells me that a point has no volume at all. Then in eq. (3) Vol(space-like link)=1 and Vol(time-like link) = sqrt(alpha). Here a link is also given the attribute of volume. Further in equation (4) a volume of a space-like and a time-like triangle is defined. So the authors have generalized the use of the term volume to apply to point-like objects, line-like objects, and surface-like objects. I am not sure what the purpose of this generalization is. Then the authors go on to introduce the idea of space-like and time-like bones. Evidently they need a new term other than line, edge, and link to describe spacetime connectivity. The distinctions between these terms seem vague to me, but perhaps all will be made clear.

Richard

 

[marcus]

...

I am way behind on my reading and there are strawberries to put in and trees to plant, as well as dogs, rabbits, and a horse to care for. The good news is the sun has made an appearance and I have managed to get my laptop setup on my friend Peg's table, a rather crowded place, but with a nice view of the bird feeder, apple trees, sauna, and horse eating hay in the pasture. I have coffee, and a quiet idyllic setting for contemplation of the mysteries of alpha. Oh, lucky man.

...

The picture of happiness. the laptop and the coffee make it complete. I was curious about what kind of tree you'd be planting, whether fruit or a windbreak line of trees, or for stove-wood. People who plant trees think ahead. My wife and I planted a dozen redwoods in a creekbed near here that is owned by various bureaucratic agencies who seem to have forgotten it exists.

I am reading the triangulation paper, Dynamically Triangulating Lorentzian Quantum Gravity, arXiv:hep-th/0105267v1, and I even think I can follow some of it. The idea that there are time-like and space-like edges, faces, volumes and so on, seems contrary to spacetime equivalence, but if it produces results I am willing to play along.

I am currently trying to understand Figure 1. Part (a) and part (b) seem to me to be identical tetrahedrons except for a small rotation. Part (a) is said to be a (3,1) tetrahedron, and part (b) is said to be a (2,2) tetrahedron. I would have thought (3,1) and (2,2) are (space,time) notations, but I can't seem to make sense of the figures from that perspective. Perhaps you or selfAdjoint or someone will help me find what it is I am missing here.

Yes! I too am willing to play along. BTW that "Dynamically..." paper has the hard nuts and bolts. I have only calculated a very few of the volumes and checked a few of the sines and cosines. there is a limit to how much of that I want to do---the humble nittygrit as opposed to understanding the ideas. maybe it is a swinging pendulum: gnaw on the hard details a little while and then visualize and think about what it might mean.

the (3,1) and (2,2) refer to the two ways a tetrahedron can sit in the foliation. Foliation means "leaf-ing" or layering. their spacetime is made of layer upon layer of spacelike leaves or sheets connected by layers of tetrahedra.

there is also the (1,3) tetrahedron which is just the (3,1) upsidedown standing on its point with its base in the next layer up

figure 1 is talking about the D = 3 = 2+1 case. So the spatial sheets are 2D and they are divided up into actual triangles (this is the one case where a "triangulation" is actually what it sounds like)

time is just the discrete numbering of spacelike leaves---an integer layer-count

in between any two 2D spacelike leaves is filled in by a layer of tetrahedrons. The triangles that triangulate a layer of space are the BASES of (3,1) and (1,3) tetrahedrons. In that way the tetrahedrons actually CONNECT THE TWO TRIANGULATIONS of two adjacent spacelike sheets. A mosaic of tetrahedra in the 3D bulk sandwich, joins the mosaic of triangles covering each 2D leaf. I don't know whether to call these 2D things leaves (which "foliation" as in leafy foliage suggests) or sheets, so I may alternate.

The (3,1) and (2,2) notation refers to the number of vertices in each of two adjacent spatial sheets. A (2,2) tetrahedron has one top ridge (with two endpoints) and one bottom ridge (with two endpoints). With two spacelike sheets each covered by triangles, and a the tetrahedrons fitted in between to make a perfect connection between the two triangulations, then every side of every triangle must be the top ridge or bottom ridge of some (2,2) tetrahedron

Now you can erase the paper-thin spatial leaves, if you want, and just imagine a 3D spacetime that is PACKED SOLID WITH TETRAHEDRA, that is all the approximating 3D triangulation is. It is a 3D spacetime packed solid with tets but don't forget that the packing was arrived at in a special way that respects causality. If you like dectective work you could look at the tets and find the spacelike sheets and DEDUCE the plan of each one.

Each space-sheet consists of events which could have CAUSED events in the next layer up, and which could have BEEN CAUSED by events in the next layer down.

So this particular way of packing 3D spacetime solid with tets EMBODIES in it a primitive idea of causality. that things cause other things and not the other way around. that is the reason for the layers

 

[nightcleaner]

Hi Marcus I'm online for an hour or so if you want to try real-time on this BB.

I'll work in edit and refresh the screen every few.

Causality seems to be an important item in this landscape of idylls. I read the image of foliation like a book. Write what you want on the pages. If you are clever, you can make a flip-vid (ok now I KNOW that one is a first use!).

Causality is placed in the idyll by two hands, one provides the sequence of the pages, the other provides something clever written upon them. Now we are considering fractal patterns in the idyll. A flip-book of fractals, chosen to foliate into a semblance of motion. The pages do not move and the patterns do not move, but the observer flips through the sequence...



Richard

 

[marcus]

Something to keep in mind. the CDT spacetime is not made of simplexes but is the CONTINUUM LIMIT of approximating mosaic spacetimes with smaller and smaller building blocks.

the quantum mechanics goes along with this process of finer and finer approximation. so at each stage in the process of going for the limit you have an ensemble of many many mosaic geometries

so there is not just one continuum which is the limit of one sequence of mosaics (mosaic = "piecewise flat", quite kinky manifold, packed solid with the appropriate dimension simplex)
there is a quantum jillion of continuums each being the limit of a quantum jillion of sequences of mosaics.

or there is a blur of spacetime continuums with a blur of different geometries and that blur is approximated finer and finer by a sequence of simplex manifold blurs

BUT DAMMIT THAT IS TOO INVOLVED TO SAY. So let us just focus on one of the approximating mosaics. Actually that is how they do it with their computer model. they generate a mosaic and study it and measure things, and then they randomly evolve it into another and study that, one at a time, and in this way they get statistics about the set of possible spacetime geometries. One at a time. One definite concrete thing at a time. Forget about the blur.

I am working offline while the rice cooks.

Returning to the triangulation paper, I see that a point is defined as having volume one. This seems to me a generalization of the idea of volume, but the authors say that it is conventional and so I accept it, even though my sense of geometry tells me that a point has no volume at all. Then in eq. (3) Vol(space-like link)=1 and Vol(time-like link) = sqrt(alpha). Here a link is also given the attribute of volume. Further in equation (4) a volume of a space-like and a time-like triangle is defined. So the authors have generalized the use of the term volume to apply to point-like objects, line-like objects, and surface-like objects. I am not sure what the purpose of this generalization is. Then the authors go on to introduce the idea of space-like and time-like bones. Evidently they need a new term other than line, edge, and link to describe spacetime connectivity. The distinctions between these terms seem vague to me, but perhaps all will be made clear.
...

yes, in mathtalk there is always this tension between wanting to use general terms so that you can JUST SAY IT ONCE and have that apply to all cases and all dimensions, and on the other hand wanting to use very concrete words that mean something to the reader.
So they have decided to use the word VOLUME to mean length of 1D things, and area of 2D things, and ordinary volume of 3D things, and hyperspace volume of 4D things.


the word BONE deserves a whole essay by itself. As you know the world evolved by itself without the help of any divine creator. However God did create mathematics (he just didnt bother to create the world, that happened spontaneously by some curious accident for which no one is responsible).
so by his infinite grace and glory and kindness, God arranged that in a 3D spacetime, packed solid with tetrahedra, the CURVATURE of the frigging thing can actually be measured by counting the tetrahedra around each LINE SEGEMENT. this sounds like a damnable lie but Regge discovered it, essentially. this is a CDT version of a truth Regge found in 1950.

And again by the infinite grace and mercy of the Creator (of mathematics) in a 4D spacetime the curvature of the frigging thing can be measured by counting the 4-simplexes around each TRIANGLE.

So to GENERALIZE the terminology, as mathematicians believe the Lord wishes them to do, so that they don't have to say edge in one case and triangle in another case, they invented the word BONE. a bone is the
D-2 thing.

so in the 3D case, the bone is the 1D thing
and in the 4D case, the bone is the 2D thing (the triangle)

I have a hard time imagining in 4D how a bunch of simplexes can surround a triangle.

but in 2D I can picture how 5 or 6 or 7 equilateral triangles can surround a POINT (which is the bone in 2D)
and obviously if the count is exactly 6 then the manifold is flat at that point, and if the count is 5 then the manifold has positive curvature at that point etc etc etc.

and in 3D I can picture how various numbers of tets can surround an edge (which is the 3D bone), and how that relates to curvature

but like I say it is hard for me to imagine in 4D how various numbers of 4simplexes can surround a triangle, which is the bone in 4D

my chorus is doing a concert in a few hours, so I must practice some