Beginner's Guide to Loll Gravity

[marcus]

quantum gravity is currently the big problem
interest in string approach to quantum gravity has been declining lately (it doesn't have to be permanent, might pick up! but dwindling for the time being)
and interest in Loll gravity approach has been growing (though still only a few papers come out each year. see "Quantum Grafitti" thread for current news)

so why is it a good idea to take a look at Loll gravity? Because quantum gravity is the big problem and interest in Loll approach has been growing.

terminology: it should be called "Ambjorn-Loll" because the first papers were in 1998 by Jan Ambjorn and Renate Loll. Or it should be called "nonperturbative path integral approach to quantum gravity by the method of Causal Dynamical Triangulations", or CDT, but I find it simpler to say Loll gravity. Loll is one syllable.

 

[marcus]

here are the papers if you want to look, these are Loll's:
http://arxiv.org/find/grp_physics/1/au:+loll/0/1/0/all/0/1?skip=25&query_id=291285c13653a251

you can see there are about 50 and up thru 1996, 1997 she was doing Loop Quantum Gravity (notable contrib. abt. spectrum of volume operator) and you can see her gradually getting away from loops and over into triangles, and then having this kind of breakthrough in 1998, making triangle-type gravity ("simplex" gravity) Lorentzian, cutting the simplexes out of specialrelativity 4D space instead of using plain chunks of Euclidean, as people did prior to 1998. well that's the history, you can read the history of the CDT approach by glancing at the arxiv list of Loll papers. part of a long struggle to quantize Gen Rel.

here is a look at present day situation. these are the papers that came out in the past 12 months:
Last 12 months:
http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/past/0/1

the enduring lesson of Gen Rel is that "gravity = geometry"
Gravity is not just a force operating in a fixed geometric framework. It is not just some gravitons running around on a fixed platform space chosen in advance. Gravity is spacetime geometry itself and how that geometry interacts with matter. A fundamental theory of gravity has to explain why spacetime appears 4D at largescale, and say what it's like at smallscale and how it gets hooked up with matter so that matter affects it by putting curvature into it. And quantum gravity means quantum geometry. It means UNCERTAIN geometry where you have quantum operators corresponding to measuring whatever angles and areas and volumes and distances---quantum operators which are the geometric observables measuring all kind of geometric quantities. that's what quantum spacetime geometry (i.e. quantum gravity) has to be if it's going to be the quantum version of Gen Rel.

So we chuck out the notion of a fixed space with forces operating in it, or gravitons running around in it, and we think about all possible 4D geometries as an ensemble, all possible shapes of the universe

to make it manageable we are going to think of bounded-volume, finite life-span universes, so it might be possible to simulate a sample of these various possible spacetimes or "histories" in a finite computer. the idea is to seen how it works with these finite bounded spacetimes and then, if you want, extend it to the infinitely big limit.

also to make it manageable we are going to do it without matter, although putting matter in is in principle straightforward and some preliminary work has been done on that. so for now we are looking at a sort of virtual universe that comes into existence briefly and goes out of existence as a kind blip or burp or "quantum spacetime dynamics fluctuation"

this clearly leaves much to be desired but even with this blip of a universe (without even any matter in it, typically) we can still ask things like what is the largescale dimension? does a familiar macroscopic 4D spacetime emerge? what is spacetime like down at Planck scale? if you take a random walk how quickly do you get lost? how likely are you to wander back? (a way of sensing the dimensionality) if you double the radius of a little region by what factor does the volume increase? might there be microscopic brief wormholes in this spacetime? to the extent one can talk about space-like "slices" what are those slices like, how does their dimensionality behave at various scales?

so even with a simple matterless burp of spacetime there is a lot to explore

 

[marcus]

you can think of a spacetime as a path
it is an evolutionary path taken thru a "space of spaces", or thru the set of all possible spatial configurations, a journey thru the space of 3D shapes.
the idea sounds more complicated than it really is because in mathematics the word "space" is over-used.

anyway a spacetime is a path thru the realm of possible spatial geometries.
and that path could take you from a big bang to a big crunch or it could take you from A to B where you pick the spatial geometry at the beginning and the one at the end, the intial and final shapes of space.

so quantum spacetime dynamics has this aspect of being analogous to a Feynman path integral where you consider all possible paths a particle could take for getting from A to B, and you represent nature's involvement in preferring some over others by WEIGHTING each one with a complex amplitude, and you add them all up. that is a conventional Feynman path integral. And to make it mathematically manageable you might "regularize" things by only considering PIECEWISE LINEAR paths made of line segments because if you let the size go to zero eventually everything is approximately included.

Well Loll does the same thing. that's why it is called a path integral. instead of a piecewise linear path of a particle, what you add up are all possible piecewise linear spacetimes assembled out of these pyramid-like simplex building blocks which have the shape evolving from some initial to some final shape.

I don't know how this is going. this is a first try at a "Beginner's guide to Loll gravity". I may have to edit a lot, or start over. but have to get started sometime so might as well be now.

 

[ohwilleke]

Without matter, it seems like what is really developing so far might better be called "Loll Space" by analogy to Euclidian Space, Minkowski Space, etc. To say that you have "gravity" before throwing matter and its interaction with Space into the mix is a bit premature.

 

[marcus]

the key thing I have to discuss is what the complex amplitude is that is used to weight the various geometries.

in ordinary Feynman path integral it is exp(iS) where S is the classical "action" which tells how busy or wacky the path is.
the number S is something where you look at the path and you calculate S from all the stupid crazy moves the path made in getting from A to B. If it just calmly goes from A to B the way Newton or lagrange thought it would then this number S is minimal or zero.

but Feynman said to think of the particle going along all paths, only out there in the wilderness of wacky paths the "action" number S is increasing rapidly and that makes exp(iS) whirl around and around in a circle and so the different wacky paths CANCEL EACH OTHER OUT.

you have to know that as the number x increases exp(ix) goes around and around the unit circle. and you have to figure out that in a system where the bad paths get multiplied by a quantity that goes around in a circle, the near neighbor bad paths get weights which are opposite on the circle and they cancel each other and IT MAKES THE NICE PATHS WIN

you are thinking of the particle having the freedom to go all different paths from A to B, but Nature expresses her ideas about it by this "action" number S, and she says having S near zero is good and far from zero is silly, so by weighting the sum of histories by exp(iS) we get it to average out.

well you can do Quantum Particle Dynamics that way and also
Quantum Spacetime Dynamics

now an interesting part is that the S NUMBER FOR SPACETIMES COMES FROM EINSTEIN GEN REL EQUATION. the "action" number that you calculate for a possible spacetime history, to tell how kinky weird messed up and un-
Einsteinly it is, so you can put in the weighting exp(iS) and add all the spacetime histories up and have it please Nature. This "action" number S, for spacetimes, is made BY SUBTRACTING THE RIGHTSIDE FROM THE LEFT SIDE OF EINSTEIN GEN REL EQUATION.
so the minimum action case, which nature likes, that will be where the difference between LHS and RHS is zero. Well naturally that would be the right action formula, if we are quantizing Gen Rel, right? It is a nobrainer.

So what we have to look at is how does Loll take the building blocks of some random triangulation and inspect them and count them and do whatever she does, and how does she get the "action" number S for that particular triangulated spacetime.

this is at the core of the computer program that does the simulation. we have to be able to implement the LHS minus the RHS of the main Gen Rel equation, and implement it by a process of counting up various oriented identical triangle-like building blocks which have been stuck together to make a world.

 

[marcus]

Without matter, it seems like what is really developing so far might better be called "Loll Space" by analogy to Euclidian Space, Minkowski Space, etc. To say that you have "gravity" before throwing matter and its interaction with Space into the mix is a bit premature.

hi ohwilleke, as I said they have included matter in some papers but not very extensively or substantially. at the theory level one adds a term to the action. I don't think it is a big deal.

so you can consider it mature or premature or however you want, doesn't make a lot of difference. I would say at the theory level, which is where most theories nowadays are, that essentially MATTER IS ALREADY INCLUDED. one knows how to add a term to the Lagrangian and in some papers, as I remember, something along those lines appears

but with Loll one expects more than just something at theory level, one expects a computer simulation to be run. this is what I am waiting for, as regards matter.

 

[marcus]

Loll's dihedral angles

this "action" S that I was talking about ("Regge form of Einstein action") is something there are a couple of formulas for which are good to understand.

the classical version (before Regge saw how to base it on triangulation and dispense with coordinates) was that you had a metric function g, which gave a volume element and you would integrate this $(R - 2 \Lambda)$ over the whole manifold
where R was the Riemann curvature and Lambda was the cosmological constant.

the idea was that Lambda was a uniform small amount of curvature that nature would put up with and in fact enjoy so that it was only when R got significantly more than this uniform small amount that she took notice.

Remember the idea of minimizing an action integral is that "action" is somehow busy-ness and screwing around and getting off track, minimizing it gets you the classical solution, and the Feynman integral allows these wacky paths or histories to be included but weights them by an amplitude $e^{iS}$ or exp(iS), which still let's the nice cool not overly "active" ones win, by making the extreme goof-offs cancel each other.

So Loll is considering all possible spacetime triangulations T, think of T as a bunch of lists of 4-simplex blocks with directions for gluing them together, T is the necessary lists to implement this particular spacetime geometry by gluing blocks. So in an approximate way she is considering all possible geometries T, and for each T she wants to be able to calculate the action for it: S(T)

this is the formula we have to look at
It is in http://arxiv.org/hep-th/0105267
and it is equation (38) on page 13
and we want to know why is it this.
why does this correspond to the classical integrating the curvature $(R - 2 \Lambda)$ over the whole manifold

and then she immediately translates it to equation (39)
which shows it is simply a matter of COUNTING BLOCKS and it would be good to understand that also.

This is where volumes and dihedral angles come in. Loll blocks are essentially all identical (actually two variants or types of 4-simplex but the spacelike edges are all the same and the timelike edges are all the same).

So the volumes and dihedral angles reduce to a few cases, which are all the same. That means that measuring overall volume reduces to counting, and the same thing happens to CURVATURE. Because curvature is always the DEFECT ANGLE where things meet.

 

[marcus]

in every case the deficit angle at a hinge is going to be $2\pi - number meeting \times dihedralangle$
and when we looked at the 2D case with equilateral triangle blocks, then the dihedral angle just happened to be pi/3
so in that case I wrote the deficit angle
$2\pi - number meeting \times \pi /3$

"hedron" just means "side" so a "dihedral angle" is simply the angle where two sides meet. in a triangle the sides are lines and the dihedral angle is just a vertex angle

Now let's go up to 3D
and there the blocks are TETS
and the sides are triangles
and the HINGES ARE LINES WHERE THE TRIANGLES MEET
and it is at those hinges that we want to be able to add up the
dihedral angles to see if there is the right amount or if there is a deficit
and a deficit will mean that there is positive curvature and that the thing does something analogous to PUCKERING.

So in this case D = 3 and we subract 2 and get D - 2 = 1
and that is the dimension of the hinges, where curvature concentrates in this case. 1-simplexes are lines. so it all checks out.

-----
here in 3D spacetime something new comes in which will carry over to the 4D case so we might as well discuss it and that is that Loll triangulations are "causal" or "Lorentzian" or to put it in more basic language that I like LAYERED.

All the vertices are assigned layers, like integer ticks of a clock, t, t+1, t+2,...

and all the vertices in slice t form a triangulation of that slice by equilateral triangles. A spatial slice like that is 2D, because the spacetime is 3D.

but in this 3D case the BLOCKS are TETS, so we have to look at the tetrahedra and there turn out to be TWO KINDS
there are the normal looking ones with 3 points on the ground and one in the air, or the next story up, and they look somewhat like pyramids but with a triangle base. And also the same thing upside down.
I call this the LEVEL kind and Loll writes them "(3,1)" because 3 points in one layer and 1 point in the other. the distinction between (3,1) and (1,3) is not so important and gets blurred.

The really different kind of tetrahedron is the kind I call TILT and that Loll writes (2,2). It has two points in the t layer and two other points in the t+1 layer, the next one up. It looks like one of those pyramid like LEVEL ones that has been rocked over to one side so it is balancing on an edge.

This business is certainly not real hard, it is pretty simple to keep track of the two kinds. It just adds an extra wrinkle to the counting whether you are doing the 3D spacetime or the 4D spacetime. But I thought I should mention it.

In the 4D case, which we will go to next, you have the same two types of blocks: Level and Tilt.
The blocks are 4-simplexes, so they each have 5 vertexes. and Loll writes the two kinds (4,1) and (3,2)
the (4,1) kind which I call Level, sits on a spatial tetrahedron for a base and has one point up in the air.
the (3,2) kind, which I call Tilt, looks like that except tilted over so that all it rests on is a spatial triangle. There are pictures in the paper we are following.

to be continued
if anyone is reading along, they might like to look at hep-th/0105267
page 8
this has the two pictures of the two types of blocks used to build the 4D spactime

what we need to know is the volumes and dihedral angles of those two blocks. then we can make sense of the action formula (38) and (39) on page 13.

I'll get back to this later

 

[marcus]

now I'll try to say in words what equation (38) on page 13 does.

it is by analogy with 3D spacetime where the blocks are TETS and the hinges are line segments where the sides come together and the angle you want to know is a dihedral angle between two triangular sides of a Tet.

then the action is gotten as the deficit angle when you add up the dihedrals of all the blocks that meet at a given hinge which is a LINE SEGMENT

here is a hard thing to visualize: now we go up to 4D and the SIDES of the block are now TETS and they meet at TRIANGLES
so two tetrahedrons meet at a triangle and the triangle is the hinge and we want to know the dihedral angle these two tets make

And at every triangle in the whole blooming spacetime there is going to be a DEFICIT ANGLE there at that triange which is 2pi minus the sum of all the dihedral angles of all the blocks that meet at that triangle.

IT IS A STRAIGHTFORWARD ANALOG OF 3D SPACETIME and still hard to visualize but you can go back to the 3D case as often as you like and follow the analogy, which is a perfect analogy, and gradually it gets clearer (or it did for me)

So when we calculate the action S, which is the key thing in Loll approach, we go to every triangle and multiply its AREA which Loll calls "volume" (area is just a specialized word for 2D volume) times the deficit angle there. And that is the curvature concentrated at that hinge. And they ADD UP ALL THAT CURVATURE

Now if you look at eqn. (38) on page 13, you see that is exactly what Loll and them are doing. That accounts for the first two terms in (38).
It is two terms because they divide the sum into two parts---one part for the spacelike (SL) triangles and one part for the timelike (TL) triangles.

All spacelike triangles means is all three vertices are in one spatial layer, like all in layer t.
A timelike triangle is one where one of the vertices is in a different layer, like two are in layer t and the third is in layer t+1.

Loll does not precommit to having spacelike and timelike edges the same measured length. So there is this number ALPHA which is usually around 2/3 in the simulations, but which can be adjusted. The squared length of any spacelike edge equals ONE and the squared length of a timelike edge equals minus ALPHA.
This is a diddly detail and at this point it just means that the spacelike and timelike triangles have different areas. So the two parts of the sum in (38) are done separately.

the other two terms, the rest of eqn. (38) is just including this constant curvature LAMBDA the cosmological constant. You just add up the volumes of all the blocks (the 4-simplices that fill the 4D spacetime) and multiply that total volume by Lambda.

So it isn't hard to look at (38) and understand it as the simplicial analog of integrating $R - 2\Lambda$, the classical action.
this is the simplicial version (call it Regge because Tullio Regge discovered how to do Gen Rel using simplices and without coordinates, in 1960) this is the simplicial version of the Einstein action.

You add up all the curvature and subtract off Lambda. or two Lambda: one, two, whatever.

Please have a look at (38). You will see there's a typo that occurs twice. this is a way of checking that you are alert. with preprints, they are free for download but they can have typos. More important have a look to get an idea.

FLIP BACK TO PAGE 4 and look at the classical action (2) and see how analogous!
In (2) we drop the boundary term off, the integral with the K in it. Just take the first integral.

Now we have to see how to get from (38) to (39).
that is a key step because (39) is based entirely on counting blocks.
In 939) there are no summation signs, just these "N" numbers which are the simple block-counts. they are the numbers of various types of simplices comprising the spacetime triangulation T. this is a fast way for the computer to find out the action S(T) for that case of geometry T



So in the simulation the computer is going through geometry after geometry, T after T after T, shuffling the blocks around and generating all different ways spacetime could be. and for each T it can compute the action.

there is another detail, Wick rotation, that comes in. when we get S(T) we might not use the number exp(iS) we might use exp(-S) instead. Another bit of rigamarole. An Italian named Wick. Gian-Carlo Wick born 1909 in Torino. http://books.nap.edu/html/biomems/gwick.html
That is odd, Tullio Regge was born in Torino in 1931. Same home town.

when the computer does a Monte Carlo simulation it is wandering through the realm of different geometries in a kind of random walk and tossing a coin at each junction to see what kind of move to make, shuffling and swapping the blocks around. well when it is doing that kind of random wandering it needs REAL PROBABILITY NUMBERS instead of complex amplitudes. that's one reason why Wick rotating is handy.

but let's forget about Wickery and just focus on getting this action number S which Loll needs, and how this equation (39) comes about.

BTW some authors say hinge and some say "bone" instead. the bones are the simplices of dimension D-2 and they are where the D-1 dimension sides come together and make their dihedral angles and they are where the curvature concentrates in a piecewise flat manifold and they are where around them you measure the deficit angle---and they are the "bones" but I have said "hinges" because it is more intuitive for me.

 

[marcus]

I may have to begin all over again with this "Loll Gravity ABC" to get the organization better. maybe start with more symbols from the beginning.
anyway equation (39) is the central hairy thing, as I see it, so i am going to translate it into words and see if that makes it more understandable.

It is an expression for the action S of a given triangulation.
the triangulation consists of two kind of 4-simplex blocks, level and tipped.
I abbreviate num(LEVEL) and num(TIPPED) for the number of each kind of block.

And there are space-like triangles, SLT, and time-like triangles, TLT. I said earlier what the difference was: a SLT has all its three vertices in one spatial layer, instead of having like 2 downstairs and 1 upstairs.

Loll has calculated and listed all the volumes and dihedral angles for these things. for example on page 6 you see
vol(SLT) = $\frac{\sqrt 3}{4}$

and she puts all the values into equation (39) in one fell swoop, so that you may not immediately recognize where all those $\frac{\sqrt 3}{4}$ and the like come from. So I am going to do it in a slower more plodding fashion and not move so fast. But it will look messier.

what you need to know is that each LEVEL block has 4 dihedral angles where the hinge is a SLT, and it has 6 dihedral angles where the hinge is a TLT. (a 4-simplex has five faces which meet at 10 hinges, analogous to a tetrahedron has four faces which meet at 6 edges or hinges, so the 4-simplex has 10 dihedral angles to account for)

and a TIPPED block has only one SLT hinge and it has 9 TLT hinges (which sort out to 6 odd and 3 even, I have to explain that later). Now when you look at this translated version of equation (39) you will see some coefficients like 4 and 6. And that will be the number of dihedral angles of various kinds.

what I mean by ang(LEVEL, SLT) is the dihedral angle in a level-type block at one of its 4 hinges which is a spacelike triangle SLT.

what I mean by ang(LEVEL, TLT) is the dihedral angle in a level-type block at one of its 6 hinges which is a timelike triangle TLT.

Now we go about adding things up. this is equation (39) with the Lambda terms left out, to put in later. this is a formula for the action S.

(1/i) vol(SLT)[2pi num(SLT) - 4 num(LEVEL) ang(LEVEL, SLT) - num(TIPPED)ang(TIPPED, SLT)]

+ vol(TLT)[2pi num(TLT) - 6 num(LEVEL) ang(LEVEL, TLT) - num(TIPPED){6 ang(TIPPED, TLT, odd) + 3 ang(TIPPED, TLT, even}]

this is what (39) says if you pick it apart and I want to see how this comes from equation (38) immediately before it. For now I am leaving off the Lamba terms at the end, don't worry about the (1/i) term for now.

Equation (38) makes sense to me. Sum over all the SLT: at each SLT take the deficit angle which is ( 2pi - all the dihedral angles of blocks that meet there).

Then sum over all the TLT: and at each take (2pi - all the dihedral angles there).

We can forget for now about the Lambda terms, they are simple. the main idea is AT EACH TRIANGLE ADD IN TWO PI MINUS ALL THE DIHEDRAL ANGLES MEETING THERE.

but now notice that all the dihedral angles of each and every block enter a sum like that SOMEWHERE. so we really don't have to be so finicky how we add things up!

So let's just do the TWO PI part
(1/i) vol(SLT)[2pi num(SLT) -...]
+ vol(TLT)[2pi num(TLT) -...]

that takes care of everything that is not dihedral angles. It is just like in eqn (38): for each SLT put in its volume multiplied by 2pi, so that simply comes to the number of SLTs, times the volume of each, times 2pi. And for each TLT do the same. So again you just get the number of TLTs times the volume of each times 2pi.

Now we put in all the dihedral angles that are at spacelike hinges SLT

(1/i) vol(SLT)[2pi num(SLT) - 4 num(LEVEL) ang(LEVEL, SLT) - num(TIPPED)ang(TIPPED, SLT)]

+ vol(TLT)[2pi num(TLT) - ...]

That is just the NUMBER of leveltype blocks, times 4 (because each block has 4 SLT hinges) times the angle ang(LEVEL, SLT) which a leveltype block always has at a SLT hinge.

and also each tipped type block just has ONE of its ten hinges a SLT kind so we put in the NUMBER of tipped blocks, times one, times the angle you always get at that hinge.

thing is all these blocks are identical, which helps with the "dynamical triangulations" approach. the way Tullio Regge did it at first in his "General Relativity Without Coordinates" the blocks weren't all the same in types like this.

NOW we put in all the dihedral angles that are in LEVEL blocks at timelike hinges TLT. You see we are gradually reconstructing equation (39)


(1/i) vol(SLT)[2pi num(SLT) - 4 num(LEVEL) ang(LEVEL, SLT) - num(TIPPED)ang(TIPPED, SLT)]

+ vol(TLT)[2pi num(TLT) - 6 num(LEVEL) ang(LEVEL, TLT) - ...]

Each leveltype block has 6 of its 10 "edges" or "hinges" being timelike triangles. So that is what the 6 is doing. We are gradually gathering and including all the dihedral angles in sight.

Finally, in tippedtype blocks the 9 TLT hinges are of two sorts which Loll talks about right at the bottom of page 8. Remember that the faces of a block are TETRAHEDRA and the tets can be either (3,1) or (2,2). this is a minor technicality, if they are both (2,2) tets that join at that hinge then I call it "even" hinge
and the angle is ang(TIPPED, TLT, even)
and if it is two different kinds of tet that join at that hinge then I call it "odd" hinge and
the angle is ang(TIPPED, TLT, odd)

Loll lists what all these angles are. I am just using word-like abbreviations for them so I don't have to write a lot of numbers like you see in (39).

So finally putting in those last dihedral angles we have the Loll 4D action that you can see on page 13:

S =
(1/i) vol(SLT)[2pi num(SLT) - 4 num(LEVEL) ang(LEVEL, SLT) - num(TIPPED)ang(TIPPED, SLT)]

+ vol(TLT)[2pi num(TLT) - 6 num(LEVEL) ang(LEVEL, TLT) - num(TIPPED){6 ang(TIPPED, TLT, even) + 3 ang(TIPPED, TLT, odd}]

+ a couple of lambda terms

 

[marcus]

No reason not to tack on the lambda terms, this is the Regge form of the Einstein action so we have to do the equivalent of integrating
$R-2\Lambda$
where R is riemann curvature and Lambda is cosmological constant
and with simplexes that just means adding in the total volume of all the 4-simplex blocks, multiplied by lambda.

S =
(1/i) vol(SLT)[2pi num(SLT) - 4 num(LEVEL) ang(LEVEL, SLT) - num(TIPPED)ang(TIPPED, SLT)]

+ vol(TLT)[2pi num(TLT) - 6 num(LEVEL) ang(LEVEL, TLT) - num(TIPPED){6 ang(TIPPED, TLT, even) + 3 ang(TIPPED, TLT, odd}]

- $\lambda$ [num(LEVEL)vol(LEVEL) + num(TIPPED)vol(TIPPED)]

the leveltype block and the tipped type have different volumes so there are two separate terms.
Now when you compare what I just wrote with equation (39) the way you make the correspondence is with substitutions like this:

vol(LEVEL) =$\frac{\sqrt{8\alpha + 3}}{96}$

vol(TIPPED) =$\frac{\sqrt{12\alpha + 7}}{96}$

What I wrote is exactly the same as what Loll and them wrote except they put in the $\frac{\sqrt{8\alpha + 3}}{96}$ which the volume of the leveltype block actually IS. they had already listed all those volumes and dihedral angles earlier. But I was worried that looking at all the terms with square roots like that the reader might not recognize what it was talking about. So I put in word-like symbols instead.

 

[marcus]

while we are at it, we can list some other volumes and angles, what they actually turn out to be in the 4D spacetime case
these are the volumes of the two types of block that the whole spacetime is made of. these are 4-simplexes with all the spatial links (between two vertices in the same layer) having UNIT length and all the timelike links having SQUARE length MINUS ALPHA.

alpha is a number that is typically around 2/3, but it is allowed to play around with, and when Wickery happens it goes thru the lower halfplane and comes out around - 2/3 so the square length is then plus alpha and everything is euclidean. That is unwanted complication right now. think of alpha as always = 2/3.

$$\text{vol(LEVEL)} =\frac{\sqrt{8\alpha + 3}}{96}$$

$$\text{vol(TIPPED)} =\frac{\sqrt{12\alpha + 7}}{96}$$



Everything is made by sticking together just these two types of block.

The faces of these blocks are TETS and the faces of the tets are
TRIANGLES and these are necessarily of two kinds spacelike SLT and timelike TLT. and each type has its own volume (really an area)


$$\text{vol(SLT)} =\frac{\sqrt{ 3}}{4}$$

$$\text{vol(TLT)} =\frac{\sqrt{4\alpha + 1}}{4}$$




this is described by Loll and them around page 6 thru 8, some calculation is left for the reader to do.

And there are the angles (I have typed them more or less the way they appear in the paper, for easy recognition):

$$\text{ang(LEVEL, TLT)} = \arccos \frac{2\alpha}{2(3\alpha + 1)}$$

$$\text{ang(TIPPED, TLT, even)} =\arccos \frac{ 4\alpha + 3}{4(2\alpha +1)}$$

$$\text{ang(TIPPED, TLT, odd)} = \arccos \frac{-1}{2 \sqrt 2 \sqrt {2\alpha +1} \sqrt {3\alpha + 1}}$$

$$\text{ang(LEVEL, SLT)} = \arccos \frac{-i}{2\sqrt 2 \sqrt {3\alpha +1}}$$

$$\text{ang(TIPPED, SLT)} = \arcsin \frac{-i \sqrt 3 \sqrt{12\alpha + 7}}{2(3\alpha + 1)}$$

Here is the Loll action again:
S = (1/i) vol(SLT){2pi num(SLT) - 4 num(LEVEL) ang(LEVEL, SLT) - num(TIPPED)ang(TIPPED, SLT)}

+ vol(TLT){2pi num(TLT) - 6 num(LEVEL) ang(LEVEL, TLT) - num(TIPPED)[6 ang(TIPPED, TLT, even) + 3 ang(TIPPED, TLT, odd)]}

- $\lambda$ {num(LEVEL)vol(LEVEL) + num(TIPPED)vol(TIPPED)}

 

[Spin_Network]

Marcus, if you could produce an image so that people could evaluate and 'view' attached file? it may be really useful.

There a number of processes Loll can develop for a Fractal transformation from 2-D into 3-D, but I do see a problem for the expanding transformation of 'scale-up'?..but saying that it diverges into an exact Singularity diagram that is constantly used in Blackhole Singularity diagram representations, this is I believe Lolls 'next' task, I have done this from following her early papers, and have found it to be very..very interesting. I have digital images of the process. One problem I encountered is the Rotational Values for the DT process, has a flaw, but then again the way I have resolved this is to asign the Fractal Evolving State into an action/cause, for the Vibrational Modes of ordinary Matter.

The easy explanation without going into detail works like this, on page 26 here:http://arxiv.org/PS_cache/gr-qc/pdf/0506/0506035.pdf
details the templates of 'building-blocks'. If one looks at the complete bits from a 2-D perspective, of 4-D space, one gets a 'Pyramid' view, looking down from a 3-D frame 'us' it looks like the back of an envelope, with no constraints it is just that. Which is ok when there is nothing close by to manipulate the 'framework'. Now the world is 3-D, if one places the single 'template' within the volume of ordinary matter, say a single Proton for instance, then this would not be a 'Straight-line' representation of the Triangle?..the edges would have to have curvature, so the 'close-spherical-packing' of the surrounding Quarks would 'fix' the edges of triangular Space thus, with 'curved' edges, not straight lines?

The Loll dynamics are suitable for non-matter volumes only, so external to matter one gets to active ingredients, an expansive action and a Contracting action, so the process for instance of a 4-D volume collapsing to a 2-D volume, must 'lose' two of the quantities from 4-D to 2-D. Likewise if one chooses to expand a 2-D template, up into a 4-D volume, it has to 'Grow' an extra number of edge's?

Now Fractal Expansion, and Fractal Contraction are not Time-Symetrical, but have corresponding 'identities' that at certain moments can be seen to be..well identical. Think of two fractals that have the same components, if one contracts and one expands, the shape and form will have separate identity correspondance, until for a certain moment you cannot detect which is in expansive mode and which is in contraction mode, even if one fractal is embedded within the other.

Now let's place MATTER inside a SPACE, 3-D matter embedded within a 2-D space. Do the triangles of Space warp the triangles of Matter? or do the Triangles of Matter warp the Triangles of Space? Between 2-D lies 3-D and then 4-D..the edges of 2-D meet 3-D before 3-D meets 4-D, there can be no Geometric Leaps!..for a 'quantum-leap' One has to introduce the only path possible, and you end up with a Penrose Triangle as opposed to a Loll Triangle!

The only path around this is via a Fractal Rotation, the Rotation creates an illusion in surrounding space just long enough re-configure, a Dimensional Transformation, a Quantum Triangulation or Quantum Ranging.

 

[marcus]

Marcus, if you could produce an image so that people could evaluate and 'view' attached file? it may be really useful.
...

The easy explanation without going into detail works like this, on page 26 here:http://arxiv.org/PS_cache/gr-qc/pdf/0506/0506035.pdf
details the templates of 'building-blocks'. If one looks at the complete bits from a 2-D perspective, of 4-D space, one gets a 'Pyramid' view, looking down from a 3-D frame 'us' it looks like the back of an envelope,...

Thanks for the suggestion, Spin Network.
You are probably right that having figures---diagrams, sketches---would improve things a lot. what I wish I knew how to do is copy figures out of Loll papers like the one you mentioned http://arxiv.org/gr-qc/0506035 and paste them in.

Loll and friends have a lot of pictures, and they help. I find myself doodling with a pencil as i read and looking back at some of the diagrams and scribbling on them, adding penciled explanations with lines and arrows. It is nice to have paper print-out that you can mark up.

I went to the article you mentioned (the one about black holes) and looked at page 26, that you were talking about. there is a figure there (Fig. 8) and also on the next page another (Fig. 9). I haven't understood those figures yet. You can almost go thru a Loll paper picture by picture and reading the paper means to understand the pictures. You look at the picture and then you glance down at the text go get help understanding what it is about, and you look back at the picture. gradually it sinks in what it is about. and then you go on to the next picture.

Well I haven't read enough so far in the paper you mentioned. So I don't understand those pictures you may be meaning: Figs. 8 and 9. I just have scarcely looked at those particular ones. Have to get around to it later.

 

[Spin_Network]

Thanks for the suggestion, Spin Network.
You are probably right that having figures---diagrams, sketches---would improve things a lot. what I wish I knew how to do is copy figures out of Loll papers like the one you mentioned http://arxiv.org/gr-qc/0506035 and paste them in.

Loll and friends have a lot of pictures, and they help. I find myself doodling with a pencil as i read and looking back at some of the diagrams and scribbling on them, adding penciled explanations with lines and arrows. It is nice to have paper print-out that you can mark up.

I went to the article you mentioned (the one about black holes) and looked at page 26, that you were talking about. there is a figure there (Fig. 8) and also on the next page another (Fig. 9). I haven't understood those figures yet. You can almost go thru a Loll paper picture by picture and reading the paper means to understand the pictures. You look at the picture and then you glance down at the text go get help understanding what it is about, and you look back at the picture. gradually it sinks in what it is about. and then you go on to the next picture.

Well I haven't read enough so far in the paper you mentioned. So I don't understand those pictures you may be meaning: Figs. 8 and 9. I just have scarcely looked at those particular ones. Have to get around to it later.

I am just about to leave what I am involved in at the moment, and I am going to 'fuse' the Loll principle with Smolin paper of relevence. It will help to give a greater understanding to the overall picture Loll is stating, and I hope it will highlight some Geometric Problems I have come across, I do not understand fully how there can be a Geometric Background in 2-D, and also have a continueous Geometric Path across multiple-dimensions?

So give me a day or two, I am going to use geometric pictorials as the whole paper is based on the principle of a Geometric Base 'template', the triangle.