# Introduction To Loop Quantum Gravity

1. Nov 25, 2004

### marlon

INTRODUCTION TO LOOP QUANTUM GRAVITY, everything you ever wanted to know...

In Loop Quantum Gravity, also referred to as LQG, the attempt is made to introduce the concept of quantum gravity. This is the unification of the General Theory of Relativity and the Quantummechanics. It is a very well established fact that gravitation and quantummechanics both have totally different foundations, which makes it very difficult to unify them at “first sight”. On the one hand position is uncertain in QM due to the Heisenberg-principle, while this is never the case in GTR. On the other hand, there is no metrical connection between space and time in QM, similar to the space-time-continuum of GTR. This leads to the fact that there is no curvature of space nor time in QM

In order to quantize the GTR we need gauge-fields, curved on a manifold just like in GTR. These gauge fields then need to be quantized just like other fermionic fields are quantized in QFT. When following this procedure, one needs to obey the following two laws at any point and time :

1) diffeomorfism invariance (this is the general covariance of GTR)
2) gauge invariance (like in QFT, invariance of gauge fields under local symmetries)

Basically these two laws ensure us that we have background independence so that we can choose any metric we want in order to describe the manifold. The different possible frames on that manifold must yield the same physical equations at any point on the manifold, that is the covariance (just like in GTR). These diffeomorfisms from one possible metric to another make sure that the physical laws remain the same when metrics are interchanged.

More specifically one needs to describe a manifold. Great mathematicians like Gauss and Riemann have taught us that this is done by the socalled connections. A metric describing a manifold is the most familiar example of a connection, i.e. the socalled metrical connection. There are other options though, like in GTR the connections are not metric-functions but they are gauge fields.

Next question is, how do we study some manifold ? What system can be followed in order to describe how objects behave on some chosen manifold ?
Well, we want background independence, so we must be able to chose any metric or connection we want in order to describe our manifold we are working on. In the early stages of LQG all possible metrics were used in order to implement this concept of back-ground-independence. A certain physical state was then represented as a probability-density containing all these metrics. This way of working was not very practical and in the mid-eighties it was even replaced by a description based upon the set of connections instead of all possible metrics.

Now, how does a connection work ? Well suppose you are on the manifold at a certain point A. Then you want to move in some direction on the manifold along a loop to that starts in A and comes back to A, like a circle. In order to describe this transition in mathematics, one uses the concept of parallel transport of tangent vectors. In order to be able to talk about such things as vectors, we need a reference frame that we can choose as we please because of the two laws mentioned above. Take a frame in A then make a very little step along the loop and look how this chosen frame has changed its position during the movement. Then complete the same procedure until you get back in A after completing the loop. Ofcourse it is not useful to look at the movement of the frame at every intermediate step along the loop. Actually one can integrate out the evolution of the frame over the entire trajectory that is followed from A to A.

When we start in A we actually take a tangent vector. This is an element of the tangent space of the manifold at point A. The transformation that is used to go from a point A on the manifold to the tangential space is called a projection. This tangent space can be turned into a socalled the Lie-Algebra, containing vectors written in terms of differentials, and provides the description for the movement from A to A along the loop. Now the operations that can be executed on the elements of a Lie-Algebra, like the identity or rotations, can be found in the Lie-Group.

As stated in the above paragraph it will be the intention to map elements from the Lie-Algebra to the Lie-Group. To be more specific : suppose we look at some vectors from the Lie-Algebra at A and we parallel transport them along the loop back to A. Now, we see for example that these vectors have rotated 45 degrees during their transport. This 45-degree-rotation is an element from the Lie-Group and the map between these two concepts gives us some idea on how vectors behave when replaced along some chosen loop on the manifold. Thus, yielding in a system to describe the manifold itself. It is proven that if you exponentiate Lie-Algebra-elements, you get the Lie-Group-elements.

More specifically, we take the frame around some loop and integrate all the differential motions of this frame during it’s transport. It is this integral that is exponentiated in order to get the corresponding group-element. In the Lie-Algebra, the group-element has a certain representation like a matrix. It is the trace if this matrix that is considered because the trace is a scalar and it will be the same for all reference frames. The map between the Lie-Algebra element and the Lie-Group element is called a Wilson Loop. Basically it “tries to feel” the metric by parallel transporting a Lie-Algebra element along a loop and “measuring” how this element changes it’s position with respect to the original position, after the loop is completed. Thus yielding a Lie-Group element.

The reason why we can ultimately speak about integrations and so on, is because initially everything is considered to be very very small. We work in terms of differential motions, which add up into the total motion between A and A. We use the Algebra’s in order to talk in terms of differentials d. As we move the frame along some "d(loop)" it experiences some "d(rotation)."

Now, once we have established such a relation, we can calculate the total movement by exponentiating the two differentials of the Lie-Algebra. The d(loop) ofcourse yields a transformation that describes the trajectory of the loop, while the d(rotation) yields the total rotation that has been undergone by the transported vector.

The main consequence of Loop Quantum Gravity is the fact that our space-time-continuum is no longer infinitely divisible. In LQG space has a “granular” structure that represents the fact that space is divided into elementary space-quanta of which the dimensions can be measured in LQG. The main problem of QFT is the fact that it relies on the existence of some physical background. As stated one of the main postulates of LQG is the fact that we need background independence. The diffeomorfisms give us the possibility to go from one metric to another and the physical laws must remain the same. Basically some physical state in LQG is a superposition over all possible backgrounds or in other words a physical state is a wavefunction over all geometries.

In String Theory, the main “competitor” when it comes to quantumgravity starts from the fact that there must be some kind of fictitious background space, thus actually undoing the aspects of general relativity. All calculations can then be made with respect to this background field and in the end the background independence must “somehow” be recovered. LQG starts from a totally different approach, though. We start from the knowledge we have from General Relativity, thus no background field, and we then try to rewrite the entire Quantum Field Theory in a certain way that no background-field is needed.

How to implement this nice background-independence in QFT has already shortly been introduced, i.e. The Wilson Loop and more generally the spin networks :

The map between the Lie-Algebra element and the Lie-Group element is called a Wilson Loop. Basically it “tries to feel” the metric by parallel transporting a Lie-Algebra element along a loop and “measuring” how this element changes it’s position with respect to the original position, after the loop is completed. Thus yielding a Lie-Group element.

The strategy is as follows : in stead of working with one specific metric like in “ordinary” QFT, just sum up over all possible metrics. So QFT should be redefined into somekind of pathintegral over all possible geometries. A wavefunction is then expressed in terms of all these geometries and one can calculate the probability of one specific metric over another. This special LQG-adapted wavefunction must obey the Wheeler-DeWitt equation, which can be viewed at as some kind of Schrödinger-equation for the gravitational field. So just like the dynamics of the EM-field is described by the Maxwell-equations, they dynamics of the gravitational-field are dedeterminedy the above mentioned equation. Now how can we describe the motion of some object or particle in this gravitational field. Or in other words, knowing the Maxwell equations, what will be the variant of the Lorentz-force ???

This is where the loops come in. First questions one must ask is :

Why exactly them loops ?

Well, let’s steal some ideas from particle physics... In QFT we have fermionic matter-fields and bosonic force-fields. The quanta of these force-fields or the socalled force-carrier-particles that mediate forces between matter-particles. Sometimes force-carriers can also interact with eachother, like strong-force-mediating gluons for example. These force carriers also have wavelike properties and in this view they are looked as excitations of the bosonic-forcefields. For example some line in a field can start to vibrate (think of a guitar-string) and in QFT one then says that this vibration is a particle. This may sound strange but what is really meant is that the vibration has the properties of some particle with energy, speed, and so on, corresponding to that of the vibration. These lines are also known as Faraday’s lines of force. Photons are "generated" this way in QFT, where they are excitations of the EM-field. Normally these lines go from one matter-particle to another and in the absence of particles or charges they form closed lines, aka loops. Loop Quantum Gravity is the mathematical description of quantum gravity in terms of loops on a manifold. We have already shown how we can work with loops on a manifold and still be assured of background-independence and gauge-invariance for QFT. So we want to quantize the gravitational field by expressing it in terms of loops. These loops are quantum excitations of the Faraday-lines that live in the field and who represent the gravitational force. Gravitons or closed loops that arise as low-energy-excitations of the gravitational field and these particles mediate the gravitational force between objects.

It is important to realize that these loops do not live on some space-time-continuum, they are space-time !!! The loops arise as excitations of the gravitational field, which on itself constitutes “space”. Now the problem is how to incorporate the concept of space or to put it more accurately : “how do we define all these different geometries in order to be able to work with a wave function ?”

The Wheeler-DeWitt equation has solutions describing excitations of the gravitational field in terms of loops. A great step was taken when Abhay Ashtekar rewrote the General Theory of Relativity in a similar form as the Yang-Mills-Theory of QFT. The main gauge-field was not the gravitational field. No, the gravitational field was replaced by the socalled connection-field that will then be used to work with different metrics. In this model space must be regarded as some kind of fabric weaved together by loops. This fabric contains finite small space-parts that reflect the quantization of space. It is easy to see that there are no infinite small space regions, thus no space-continuum. Quantummechanics teaches us that in order to look at very small distance-scales, an very big amount of energy is needed. But since we also work in General Relativity we must take into account the fact that great amounts of energy concentrated at a very small scale gives rise to black holes that make the space-region disappear. By making the Schwardzschildradius equal to the Comptonradius we can get a number expressing the minimum size of such a space-region. The result is a number that is in the order of the Planck Length.

Now how is space constructed in LQG ? Well, the above mentioned minimal space-regions are denoted by spheres called the nodes. Nodes are connected to eachother by lines called the links.

By quantizing a physical theory, operators that calculate physical quantities will acquire a certain set of possible outcomes or values. It can be proven that in our case the area of the surface between two nodes is quantized and the corresponding quantumnumbers can be denoted along a link. These surfaces I am referring are drawn as purple triangles. In this way a three-dimensional space can be constructed.

One can also assign a quantumnumber which each node, that corresponds to the volume of the grain. Finally, a physical state is now represented as a superposition of such spin-networks.

regards
marlon, thanks to marcus for the necessary information and corrections of this text

REFERENCE : maestro Carlo Rovelli “Loop Quantum Gravity”
Physics World, November 2003

Last edited by a moderator: Jul 26, 2007
2. Nov 27, 2004

### marcus

marlon,
congratulations on a great sticky thread!
though you are the primary author, I'm assuming it is OK
for others to contribute. So I would like to bring in some
bibliography----some more links to online reading, besides
the Rovelli article that you already have

I believe you plan on continuing your essay, when studies permit,
and hope that others' contributions don't interfere with
your writing future chapters.

best regards,

====
hi, I just saw your next message #3 that you posted. I will reply here to save space. That is a good point about keeping the level Introductory.
I will keep that in mind and concentrate on adding just a small amount of bibliography (unless you get around to it before I do) which is the
more accessible sort. Actually that makes sense for several reasons including the fact that more technical articles can have a shorter shelf life!
the technical methods can get old and be replaced while the basic intuitions
stay useful longer. hope your mainstream QFT studies are going well.
BTW this sticky is really nice to have. thanks again!

Last edited: Nov 27, 2004
3. Nov 27, 2004

### marlon

Marcus, thanks for the reply...

It can only be a good thing that others contribute but i am convinced that we need to keep the level basic enough in this sense that i wanna move up the "difficulty-scale" gradually. It would be a bad thing if we were to discuss high-level papers because i think most of us (including myself) will not be able to follow this up and we would get discouraged and drop the subject. I will continue this matter and i would suggest that we follow the content of Rovelli's book which is online at his website.You have given the reference to it...

regards
marlon

4. Nov 28, 2004

### marcus

I am very much looking forward to your continuing the essay, marlon!
I will restrain my tendency to talk too much, so as not to crowd.
BTW just yesterday in the mail was delivered the copy of Rovelli "Quantum Gravity" which I ordered from Amazon. I am very happy with the book and
have been reading it instead of being at computer.

I am only sad that it is so expensive----70 dollars. You have to be rich, or be willing to splurge. Or you have to be in graduate school and need it for a class, as textbook. In US the textbooks are all very expensive, so 70 dollars is fairly normal.

Anyway Rovelli is a good writer and Cambridge Press did a good job, with the editing and just the physical production----nice paper, nice binding, nice feel, and printing. So it is a pleasure to own: at least for me.

But to save money it certainly makes sense to print off the free draft copy at Rovelli site. Even just the first 3 or 4 chapters and some appendices---or whatever you find the most accessible parts and most relevant for you.

Marlon, why not give some online bibliography yourself? It would be a refreshing change (I am always doing the librarian work) and I would enjoy seeing your picks and how you organize it. (If you do not want to, I will not shirk the job, but maybe you would like to list intro-level links?)

5. Nov 28, 2004

### jeff

I sincerely hope you enjoy your new book, which I know you will. I was leafing through it at the U of T bookstore. I want to point out two things carlo says in the introductory bit.

1) That any correct quantum gravity theory must be able to calculate amplitudes for graviton-graviton scattering, and that he hopes that lqg will one day lead to a theory that can.

2) That he knows that GR must almost certainly be an effective field theory that is modified at higher energies so that lqg can't be correct. Thus he says he views lqg basically as a laboratory for investigating certain fundamental issues in quantum gravity.

As far as your sticky goes, would you be bothered if I corrected it?

6. Nov 28, 2004

### marcus

I believe you are mistaken, jeff. Carlo does not say these things in the introductory bit.
At least I looked in the first part of the book, and used the index to search the rest, and could not find any statements of the kind.

It would be nice to have some page references, if you have any more would-be paraphrases from Rovelli----even sweller of you to provide actual quotes. Since a paraphrase can often mislead as to what was said in the original.

Thanks for your kind wish as to the book! Indeed it is surprising me. I was not expecting this much, since I had read much of the last year's draft version.

BTW if you pick up a copy either at library or store and can give me some actual page reference (whether or not in the first 50-or-so pages, anywhere in the book will do) where he says these things 1. and 2. that you state, that would be most helpful of you and I will be very interested to read the actual passages and think about it. If he does say something like that my eye somehow missed it.

7. Nov 28, 2004

### jeff

We'll, I don't have the book on hand, but...

In rovelli's dec 30 2003 draft, he says on page ix entitled "PREFACE"

"What we need is not just a technique for computing, say, graviton-graviton scattering amplitudes (although we certainly want to be able to do so, eventually)"

On page 5 of the same draft,

"The einstein-hilbert action might very well be a low energy approximation of something else. But the modification of the notions of space and time has to do with the diffeomorphism invariance and the background independence of the action, not with it's specific form."

Be this as it may, jim bjorken in the forward of carlo's book states quite plainly that effective field theory has taught us that GR must be viewed as just an effective field theory, and it's difficult to believe that carlo would've allowed such a statement if it fundamentally contradicted his position.

Btw, did you notice that carlo writes (probably in the preface) that thiemann is publishing a book on the more mathematical aspects of lqg?

8. Nov 29, 2004

### marcus

the Mexican Loop and String Show (21-27 November)

Oh I see.
I thought you were talking about the actual book. that you said you were browsing in the bookstore.

but you apparently meant the draft, from 2003, which is available online.

there's been considerable up-dating and revision. so one should be specific which
=============

Meanwhile, maybe readers of this thread would be interested in the Loop and String lineup of talks at the conference that just finished in Mexico (at the Quintana Roo beach resort in sight of the island of Cozumel)

A lot of the lectures were by top people both string and loop, and they were rather much introductory. The conference aimed at being a "school" to bring more people in. And to introduce stringies to loop research and viceversa.

I thought the lineup of who the organizers wanted to talk about the various hot topics was enlightening. So since it could be instructive, I will copy it here:

http://www.nuclecu.unam.mx/~gravit/EscuelaVI/courses.html [Broken]

--quote--
COURSES AND INVITED TALKS

Courses:

A. Ashtekar (PSU, USA): Quantum Geometry

A. P. Balachandran (Syracuse, USA): Quantum Physics with Time-Space Noncommutativity

P. T. Chrusciel (Tours, France): Selected Problems in Classical Gravity

R. Kallosh (Stanford, USA): De Sitter Vacua in String Theory and the String Landscape

A. Peet (Toronto, Canada): Black Holes in String Theory

C. Rovelli (Marseille, France): Loop Quantum Gravity and Spinfoams

Plenary talks:

J. D. Barrow (Cambridge, UK): Cosmological Constants and Variations

M. Bojowald (AEI, Germany): Loop Quantum Cosmology

A. Corichi (ICN-UNAM, Mexico): Black Holes and Quantum Gravity

A. Linde (Stanford, USA): Inflation and String Theory

O. Obregon (U. Guanajuato, Mexico): Noncommutativity in Gravity, Topological Gravity and Cosmology

A. Perez (PSU, USA): Selected Topics on Spin Foams

L. Smolin (PITP, Canada): Loops and Strings

R. Wald (U. Chicago, USA): Topics on Quantum Field Theory

Short talks:

E. Caceres (CINVESTAV, Mexico): Wrapped D-branes and confining gauge theories

A. Guijosa (ICN-UNAM, Mexico): Far-from-Extremal Black Holes from Branes and Antibranes

H. Morales (UAM, Mexico): Semiclassical Aspects and Phenomenology of Loop Quantum Gravity

D. Sudarsky (ICN-UNAM, Mexico): Spacetime Granularity and Lorentz Invariance

L. Urrutia (ICN-UNAM, Mexico): Synchrotron Radiation in Lorentz-Violating Effective Electrodynamics

---endquote---

Last edited by a moderator: May 1, 2017
9. Nov 29, 2004

### matthias

marlon...got any extra info on LQG???

great introduction btw...

lola

10. Nov 29, 2004

### jeff

You want to play games? Fine with me.

11. Nov 29, 2004

### nightcleaner

This is a project I've been working on, and I'd very much like to know what the participants on this thread think. Thanks, nc

Abstract and prospectus, Spacetime at the Planck Scale
This is an abstract and prospectus for additional research. The proposal would use computational techniques such as those described in Stephen Wolfram's New Kind of Science as an exploratory probe of events at the Planck scale. Authors are currently recruiting mathematicians and physicists to mentor and contribute to the work. We still need someone who can design the NKS experiments.

In this work in progress, we describe a mechanism by which four space-time dimensions are reduced to the classical view of three space-like dimensions arrayed in the customary orthagonal basis with one time-like dimension which can be thought of as permeating the space-like dimensions. The time-like dimension is shown to appear to be unique to a moving observer, and preserves the appearance of freedom of choice as one perspective in a structure which can also be viewed from other perspectives as competely deterministic.

The Einstein-Minkowski principle of space time equivalence taken in the strongest sense creates a powerful model for investigation of the relationship between general relitivity and quantum mechanics. We begin by defining the Planck Sphere (here named to be consistant with the Planck length and Planck time) as a three dimensional volume filled by a radient event at the speed of light in one Planck time. Thus the radius of the Planck Sphere is equal to one Planck length and is equal to one Planck time, making a three dimensional model which can be used in a perspective sense to portray events which occur at the Planck scale in four dimensions.

After describing the features of the model, we go on to propose that computational graphing techniques similar to those used by Stephan Wolfram in his book A New Kind Of Science be developed to explore the evolution of the Planck Sphere in Kepler dense packed space up to the scale of the fine structure constant, thereby showing the geometric origins of mass and charge. The first step in this process is to define a viable space-time lattice structure, which we believe we have done by defining the Planck Sphere as an element in a Kepler stack. The next step in this process is to develop a rational algorithem to simulate events on the Planck scale. This may be accomplished by applying what we know of cosmogeny and of physics near singularities. As a first approximation we advance the conjecture that expansion from the Planck scale will recapitulate cosmogeny. We carry through the first steps in this approximation to demonstrate a mechanism for early inflation in the burgeoning universe.

References:
[PDF] On quantum nature of black hole space-time: A Possible new source of intense radiation DV Ahluwalia - View as HTML - Cited by 11 ... spheres of fluctua- tions. The one that may be called a Schwarzschild sphere, and the other a Planck sphere. The sizes of these ... International Journal of Modern Physics D, 1999 - arxiv.org - ejournals.wspc.com.sg - arxiv.org - adsabs.harvard.edu

[PDF] The Quantum structure of space-time at the Planck scale and quantum fields S Doplicher, K Fredenhagen, JE Roberts, CM Phys - View as HTML - Cited by 242... In the classical limit where the Planck length goes to zero, our Quantum spacetime ...components are homeomorphic to the tangent bundle TS 2 of the 2–sphere. ... Communications in Mathematical Physics, 1995 - arxiv.org - arxiv.org - adsabs.harvard.edu

[PDF] Inflationary theory and alternative cosmology L Kofman, A Linde, V Mukhanov - Cited by 9 ... the large scale structure observed today were generated at an epoch when the energy
density of the hot universe was 10 95 times greater than the Planck density ... The Journal of High Energy Physics (JHEP) - iop.org - arxiv.org - physik.tu-muenchen.de - adsabs.harvard.edu - all 7 versions »

[PDF] Physics, Cosmology and the New Creationism VJ Stenger - View as HTML ... 10. -43 second time interval around t. = 0, if it was confined within a Planck sphere as big bang cosmology implies. The. universe ... colorado.edu

200411290100GTC
Richard T. Harbaugh
Program Director
Society for the Investigation of Prescience

12. Dec 1, 2004

### luco_jash

Hello Marlon

i will thank you for the nice clear introduction on loop quantum gravity. I am planning to do my thesis on this subject and i would like to keep in touch with all the specialists here in order to get more info. I am just starting to know this field...

bye...Luco

13. Dec 8, 2004

### Rothiemurchus

The challenge for string theorists and LQG theorists is to explain why the vacuum energy exists at 10^120 J/m^3 ( there is no reason to think there is anything wrong with the QM calculation) but does not curve space-time.How can
quantum gravity be proved if gravity is not understood on its own yet?

14. Dec 8, 2004

### marcus

!

gotta be something wrong with it

15. Dec 8, 2004

### marcus

beg your pardon Rothie but that is a crazy amount of energy
maybe QFT can come up with a mechanism that cancels all or most of it out, or find some reason to say that it doesnt really exist----maybe QFT already has.

but that density of energy, not canceled out and real enough to cause gravity, is simply incredible (at least to me). commonsense persuades me that there must be something wrong with any theory that predicts it

And there is some reason to be hopeful, because QFT is still formulated in an unrealistic way: using a fixed spacetime framework. Reformulating it in a background independent version might possibly get rid of that huge vacuum energy.

BTW just to have a basis for comparision, the astronomers' dark energy estimate is currently around 0.6 joule per cubic km. In joules per cubic meter (the units you were using) that comes to:

0.6 x 10-9 joule per cubic meter.

16. Dec 8, 2004

### Rothiemurchus

I am aware of the cosmological evidence.But the problem is this:
the energy that can be experimentally associated with the Casimir force
is greater than the cosmological observation (10^-6 Newtons/m^2 net force
at 10^-7 m plate separation - i think but i'm not sure,that this is at
least 10-7 J/m^3).So, the plates involved in
measurements of the Casimir force must somehow, switch on vacuum energy,locally.And what sort of effect would a galaxy have on the vacuum energy?

Last edited: Dec 8, 2004
17. Dec 9, 2004

### marcus

Rothie, I will try to respond---tell me if I am making a mistake. I do not believe that the experimental existence of the Cas. force proves that the
QFT calculation of a huge vacuum energy is correct.
what I think is true is that there is some normal vacuum energy density and that between two conducting plates it is LESS namely

$$\text{energy density betw. plates = usual vacuum energy density} -\frac{\hbar c \pi^2}{720 d^4}$$

the QFT calculation of the usual vacuum energy density is bad or dubious, but the Casimir effect does not depend on this, it depends on the fact that the energy density between plates is LESS by the amount shown, which QFT does calculate successfully!, and which depends on the inverse fourth power of the separation distance.

So I say that I believe the QFT calculation of the Casimir effect and I like the Casimir effect, and this is consistent with not believing the huge vacuum energy which QFT calculates, which is roughly 120 OOM wrong---or actually different people try to fix it different ways and say different things, but anyway wrong.

Last edited: Dec 9, 2004
18. Dec 9, 2004

### marcus

If I calculate right, this is what the energy density has to be in order that the force turn out
what one usually sees for the Casimir effect

$$\text{force divided by area} = -\frac{\hbar c \pi^2}{240 d^4}$$

Last edited: Dec 10, 2004
19. Dec 28, 2004

### marlon

https://www.physicsforums.com/journal.php?s=&journalid=13790&action=view#DUALITY%20:%20STRING%20THEORY%20PART%203 [Broken]

check out my journal if you are interested in an introductory text on string theory and dualities

regards

Last edited by a moderator: May 1, 2017
20. Jan 22, 2005

### marlon

https://www.physicsforums.com/journal.php?s=&journalid=13790&action=view [Broken]

Check out my journal. I posted a link to the paper that John Baez will be using for his speech on monday on LQG...very nice introduction...

regards
marlon

Last edited by a moderator: May 1, 2017