What was the reason spacetime expanded in the first place?

[Aethelwulf]

In an approach recently conjectured by Fotini Markopoulou, geometrogenesis explains that geometry appeared late in the universe when it became sufficiently cool enough and matter dominated atleast the small portion of the universe it does. Her paper as well as others, uncluding the spin network and it's important tie with the triangle inequality is in the references.

Currently it seems we simply believe there was some high potential for a universe to come into existence. Unless we where to think the universe somehow tunneled into existence because of this potential, or there was some mechanism which drove expansion. If we look at the conditions of the universe at big bang, we find that normally we think of regions where spacetime cannot be described by quantum physics because of a singularity. Hawking, in his book a brief history of time, he states that a way to vision this singularity for the big bang is by thinking of matter being infinitely stacked on top of each other. Yet I suppose, one must remember that this is really all done at some ''point''.

If we suppose we try and understand that physical picture of the singularity of the big bang, we can also presume that these particles must be occupying the same space as we presume at the initial conditions there was no dimensions we can really speak about. In the phase space that Markopoulou uses is constructed of what is called a spin network. It has neighbours (N(N-1)=2 for any one particle in a Hilbert Space. They can be thought of forming geometric triangles, and the length of any of the sides must be less than or equal to the other side, obeying the triangle inequality. This is simply just a geometrical form of your usual Uncertainty Principle. If all lengths are reduced to zero, then all edges converge to a single point; it is here I speculate my own extension of the theory saying it is here where particles experience a high instability in space.

So let me explain how this model works. First of all, it seems best to note that in most cases we are dealing with ''three neighbouring points'' on what I call a Fotini graph. Really, the graph has a different name and is usually denoted with something like and is sometimes called the graphical tensor notation. In our phase space, we will be dealing with a finite amount of particles and but asked to keep in mind that the neighbouring particles are usually seen at a minimum three and that each particle should be seen as a configuration of spins - this configuration space is called the spin network. I should perhaps say, that to any point, there are two neighbours.

Of course, as I said, we have two particles in this model , probably defined by a set of interactions (an approach Fotini has made in the form of on-off nodes). In my approach we simply define it with an interaction term:



I have found it customary to place a coupling constant here for any constant forces which may be experienced between the two distances made in a semi-metric which mathematicians often denote as .

If are adjacent vertices and is the set of edges in our phase space, (to get some idea of this space, look up casual triangulation and how particles would be laid out in such a configuration space), then



It so happens, that Fotini's approach will in fact treat as assigning energy to a graph



which most will recognize as an expection value. The Fotini total state spin space is



Going back to my interaction term, the potential energy between particles or all -particles due to pairwise interctions involves a minimum of contributions and you will see this term in Fotini's previous yet remarkably simple equation.

is the complete graph on the - vertices in a Fotini Graph i.e. the graph in which there is one edge connecting every pair of vertices so there is a total of edges and each vertex has a degree of freedom corresponding to .

Thus we will see that to each vertex there is always an associated Hilbert space and I construct that understanding as



From here I construct a way to measure these spin states in the spin network such that we are still speaking about two particles and by measuring the force of interaction between these two states as



where the is the unit length. The angle between two spins in physics can be calculated as



Thus my force equation can take into respect a single spin state, but denoted for two particles as we have been doing, it can describe a small spin network



with a magnetic coefficient on the spin structure of the equation and is the unit matrix.

I now therefore a new form of the force equation I created with an interaction term, as I came to the realization that squaring everything would yield (with our spin states)





Sometimes it is customary to represent the matrix in this form:



As we have in our equation above. The entries here are just short hand notation for some mathematical tricks. Notice that there is a magnetic moment coupling on each state entry. We will soon see how you can derive the Larmor Energy from the previous equation.

Sometimes you will find spin matrices not with the magnetic moment description but with a gyromagnetic ratio, so we might have



The compact form of the Larmor energy is and the negative term will cancel due to the negative term in my equation





The part of the Larmor energy is in fact more or less equivalent with the spin notation expression I have been using , except when we transpose this over to our own modified approach, we will be accounting for two spins.

We can swap our magnetic moment part for and what we end up with is a slightly modified Larmor Energy



This is madness I can hear people shout? In the Larmor energy equation, we don't have we usually have ?

Well yes, this is true, but we are noticing something special. You see, is really



This is the angle between two vectors. What is again? We know, this, is calculates the angle between two spin vectors again as



So by my reckoning, this seems perfectly a consistent approach.

Now that we have derived this relationship, it adds some texture to the original equations. If we return to the force equation, one might want to plug in some position operators in there - so we may describe how far particles are from each other by calculating the force of interaction - but as we shall see soon, if the lengths of the triangulation between particles are all zero, then this must imply the same space state, or position state for all your -particle system. We will use a special type of uncertainty principle to denote this, called the triangle inequality which speaks about the space between particles.

As distances reduce between particles, our interaction term becomes stronger as well, the force between particles is at cost of extra energy being required. Indeed, for two particles to experience the same position requires a massive amount of energy, perhaps something on the scale of the Planck Energy, but I have not calculated this.

In general, most fundamental interactions do not come from great distance and focus to the same point, or along the same trajectories. This actually has a special name, called Liouville's Theorem. Of course, particles can be created from a point, this is a different scenario. Indeed, in this work I am attempting to built a picture which requires just that, the gradual seperation of particles from a single point by a vacua appearing between them, forced by a general instability caused by the uncertainty principle in our phase space.

As I have mentioned before, we may measure the gradual seperation of particles using the Lyapunov Exponential which is given as



and for previously attached systems eminating from the same system, we may even speculate importance for the correlation function



where calculates the distance. Indeed, you may even see the graphical energy in terms maybe of the Ising model which measures the background energy to the spin state - actually said more correctly, the background energy



acts as coefficient of sigma zero. Thus the energy is represented by a Hamiltonian of spin states



Now, moving onto the implications of the uncertainty principle in our triple intersected phase space (with adjacent edges sometimes given as , there is a restriction that is even and none is larger than the sum of the other two. A simpler way of trying to explain this inequality is by stating: must be less than or equal to , less than or equal to , and less than or equal to .

It actually turns out that this is really a basic tensor algebra relationship of the irreducible representions of according to Smolin. If each length of each point is necesserily zero, then we must admit some uncertainty (an infinite degree of uncertainty) unless some spacetime appeared appeared between each point. Indeed, because each particle at the very first instant of creation was occupied in the same space, we may presume the initial conditions of BB were highly unstable. This is true within the high temperature range and can be justified by applying a strong force of interactions in my force equation. The triangle inequality is at the heart of spin networks and current quantum gravity theory.

For spins that do not commute ie, they display antisymmetric properties, there could be a number of ways of describing this with some traditional mathematics. One way will be shown soon.

Spin has close relationships with antisymmetric mathematical properties. An interesting way to describe the antisymmetric properties between two spins in the form of pauli matrices attached to particles and we can describe it as an action on a pair of vectors, taking into assumption the vectors in question are spin vectors.

This is actually a map, taking the form of



This is amap of an action on a pair of vectors. In our case, we will arbitrarily chose these two to be Eigenvectors, derived from studying spin along a certain axis. In this case, our eigenvectors will be along the and axes which will always yield the corresponding spin operator.



with an abuse of notation in my eigenvectors.

It is a 2-form (or bivector) which results in



This is a result where and do not commute.




[1] - http://arxiv.org/abs/0801.0861
[2] - http://arxiv.org/abs/0911.5075
[3] - http://fqxi.org/community/forum/topic/376
[4] - http://arxiv.org/abs/gr-qc/9505006

[Aethelwulf]

No one has answered my little theory so I thought I'd add some new thoughts I have had when thinking about the converging of two particles to their neighbouring points in this model. We should remind ourselves, that there are three neighbours which form a triangle in our phase space. Our original phase space constructed of Fotini's approach for a pairwise interaction which had the value . It is still quite convienient not to involve any other particle yet, just our simple two-particle system; more specifically, two quantum harmonic oscillators. It seems like a normal approach according to Fotini to assume the energy of the system as a pair of interactions given as where where is the set of interactions. Using this approach, I construct a Hamiltonian for myself which has the physics of describing the convergence of two oscillations into a single seperation neighbouring point/position. First I begin with the simple form of the Hamiltonian



Where is the Hermitian Operator. This equation describes the Hamiltonian of our pairwise interactive system which can be exchanged for particle satisfying, say for example, position and particle in position . These two particles form a side of the triangle three particles gives you your triangulation, so if we invoke the idea of two particles converging to a single point, space position then it will follow this tranformation . Before I do this, since I am working in a phase space with potentially the model known as the spin network, it might concern me then to change the energy term in the Hamiltonian for which is just the Ising Energy. So our Hamiltonian would really look like:



Now, for a Hamiltonian describing two particles converging to the adjecent edge we should have



As one of a few possibilities. There are six possible solutions in all for different coordinates. The spins in our space is assigning energy to our particles , in fact perhaps a very important observation of the model we are using, is that energy is assigned to points in this space we are dealing with. In fact, as has been mentioned before, if are adjecent vertices and are the neighbouring edges, then on each edge there is some energy assigned in our Hilbert Space. It seems then, you can really only deal with energy if there are really adjecent vertices and neighbour edges to think about. Remember, I am saying that it might be possible to state that the uncertainty principle could have tempted spacetime to expand, but this was because there was really no spacetime, no degree's of freedom for energy to move in -- which seems to be the way nature intended. So if there are no degrees of freedom, we cannot really think about energy normally in our model, since we define energy assigned to points in a Hilbert Space, which deals with a great deal more particles/points. But for this thought experiment, we have chosen two particles, and another possible position for convergence, so the equation



Actually looks very innocent. But it cannot happen in nature, not normally. Nature strictly refuses two objects to converge to a single point like . One way to understand why, is the force required to make two objects with angular momentum to occupy the same region in space. I won't recite it again right below my OP, but my force along a spin axis could determine such a force, or atleast, the force required to do so - which would in hindsight even seem impractical thinking about it... But it does give us some insight into what kind of conditions we might think about mathematically if somehow the singularity of the big bang can be overcome with some solution. In my force equation with the spin between two vectors, would state that as the angle between the vector closed in to complete convergence, the force should increase exponentially. I haven't came to an equation which describes this exponential increase, however, I do know that this is what experimentation would agree on.

The same is happening in our Hamiltonian. The force equation, with it's rapid increase of energy is proportional to the Hamiltonian experiencing an increase of energy from the spin terms through it's crazy transformation . In field theory, this would be the same as saying that the distortions of spacetime of some quantum field(s) are converging to a single point in spacetime.

[Aethelwulf]

Sorry, the last one should have been



But I am sure you get the drift. Anyway, going one way, the convergence of two particles into a single point, we can surely think about it the opposite way: particles emerging from a single point.



Which basically says that two pairwise interactions have diverged from a single point, occupying the -coordinates.

[Bogie]

I hate it to see that you are not yet being engaged by those who might grasp what you are saying so in the interim let me ask a question that might bring in others if only to show they have a better grasp than I do, lol.

Correct me if I'm wrong but are you using a beginning point in space and time where all dimensions of your spin triangles are zero resulting in all particles being stacked in the same space, a point space occupied by multiple spin networks at the creation point?

Coincident with that initial condition then are you saying that there is the uncertainty principle in play corresponding to inherent instability of extreme temperatures in the phase space (the initial point)?

Do I understand you to say that this uncertainty then can provide the force for separation and therefore expansion by causing vacua to appear between particles giving volume to the spin networks? Can you elaborate on this?

Can you correct any misunderstanding as I have stated it and explain the progression of the expansion, i.e. is the vacua equivalent to dark energy?

Are you considering preconditions, i.e. do you have a scenario of how the stacking of the spin networks into a point-space came about, or is it a singularity, or is it simply 'something from nothing"? *

[Aethelwulf]

I hate it to see that you are not yet being engaged by those who might grasp what you are saying so in the interim let me ask a question that might bring in others if only to show they have a better grasp than I do, lol.

Correct me if I'm wrong but are you using a beginning point in space and time where all dimensions of your spin triangles are zero resulting in all particles being stacked in the same space, a point space occupied by multiple spin networks at the creation point?

Exactly. These multiple spin networks are highly unstable - they are literally occupying the same point with angular momentum which is equally forbidden by the Pauli Exclusion if there was any fermions in the mix.

Coincident with that initial condition then are you saying that there is the uncertainty principle in play corresponding to inherent instability of extreme temperatures in the phase space (the initial point)?

I haven't quite made that connection, but I couldn't say it was wrong. In the high energy epoch, we are dealing with very volatile conditions. More so maybe because these high temperatures are linked to no geometry yet it is concerned with jam-packing particles into the smallest point possible, giving rise to a new instability, the uncertainty principle.

Do I understand you to say that this uncertainty then can provide the force for separation and therefore expansion by causing vacua to appear between particles giving volume to the spin networks? Can you elaborate on this?

You understand right. Spacetime did not really exist at big bang. If everything arose from a point, then imagine your phase space as being made of a single point. As temperatures cooled, then geometry appeared - then so must have our spin network - which is the ability to organize particle spins in phase space. You wind back the hands of time to the very first point, the very first instant, is like reducing the lengths of your triangle to zero to let all neighbouring particles occupy the same point and this is why it shoud indicate a violation of the uncertainty principle.

[QUOTE=Bogie;2010963]Can you correct any misunderstanding as I have stated it and explain the progression of the expansion, i.e. is the vacua equivalent to dark energy?

Are you considering preconditions, i.e. do you have a scenario of how the stacking of the spin networks into a point-space came about, or is it a singularity, or is it simply 'something from nothing"?

Yes, I have considered preconditions. The beginning where these particles where stacked infinitely high, could have only existed for a very short period of time, maybe the kind of time we associated to a small virtual quantum fluctuation. Whatever the singularity was, by this theory, it did not exist for very long. Whatever existed before the singularity (if anything did) may have existed a lot longer. Maybe eons.

[Aethelwulf]

About the dark energy part, sure it is allowed... see my theory does not explain continued expansion, it only explains why it expanded in the first place.

[Aethelwulf]

In fact, you might ask, what kind of preconditions give rise to singularities? This question may give clues to what happened before big bang. As I said, the initial singularity could not exist in its current state long without any degrees of freedom... but what happened before this may have existed for eons.

[Aethelwulf]

Damn... I said pauli uncertainty... there is no such thing hahaha

Pauli exclusion principle, stating that no two fermions can exist in the same energy state.

[Aethelwulf]

Hey, why is no one responding to my OP? lots of Op's have been anwered with less...



... *shakes shoulders*

[Aethelwulf]

This person, whoever wrote this was on the right track. They assume that the Heisenberg Uncertainty principle dominated the early universe. I bet even my idea above using causal triangulation to explain an instability of space would have impressed perhaps?

http://assa.saao.ac.za/features/cosm...lyUniverse.pdf

As they explain, general relativity breaks down, things take infinite values and a lot of this has to do with the fact we don't know how to apply our theory. If my theory is correct, it will be the first of a kind at applying quantum mechanics where physics usually struggles with interpretations.

[Aethelwulf]

Let's study this equation a bit more:



What we have in our physical set-up above, is some particle oscillations which presumably, under a great deal of force, being measured to converge to position . In our phase space, we are using the triangulation method of dealing with the organization of particles. At we may assume the presence of a third spin state, let's denote it as which seems to be a favourable way to mathematically represent the spin state of a system, meaning quite literally, ''the spin state at vertex z''. [1] Let us just quickly imagine that at any positions, to make any particle move to another position where a particle is already habiting it requires a force along a spin axis. (I can't stress enough this is not what happens in nature), this is only a demonstration to explain things better later. Sometimes working backwards, from maybe illogical presumptions can lead to a better arguement. The calculation to measure the angle between two spin states is



Thus my force equation can take into respect a single spin state, but denoted for two particles so you may deal with either spin respectively.



But perhaps, more importantly, you may decompose the equation for both particles. Let us say, particle is in position/vertex and particle is in position , meaning our final spin state is . In the force equation, making all lengths of your phase space go to zero, means that your are merging your spin state's together. Hopefully this can be intuitively imagined, but here is a good diagram: http://en.wikipedia.org/wiki/File:Spin_network.svg provided by wiki. If we stood in the z-vertex, and made the xy-vertices merge to the zth then obviously the lengths of each side would tend to zero. This means, whilst the force between particles may increase by large amounts, the angle between the vectors also goes to zero. The unit length, or unit vector which seperates particles from an origin on an axis will also tend to zero. Indeed, if you draw a graph, and make the -axis the two lengths of both particles and , where the origin is vertex spin state then by making the lengths go to zero would be like watching the axes shrink and fall into the origin. So when complete convergence has been met, the force equation has been mangled completely of it's former glory. We no longer have an angle seperating spin states, nor can we speak about unit vectors, because they have shrank as well. Using a bit of calculus, we may see that



Then naturally it follows that the force once describing the seperation of particles no longer exists, because anything multiplied by zero is of course zero. Here we have violated some major principles in quantum mechanics. Namely the uncertainty principle and for the fact that particles do not converge like this. By making more than one particle occupy the same space is like saying that either particle will have a definate position and this of course from the quantum mechanical cornerstone, the uncertainty principle is forbidden. May we then speculate that the universe was born of uncertainty? Uncertainty has massive implications for statistical physics. In the beginning of the universe, most physicists would agree that statistical mechanics will dominate the quantum mechanical side... quantum mechanics is afterall a statistical theory at best. Perhaps then, no better way to imagine the beginning of the universe other than through the eye's of Heisenberg?

[1] - http://www.math.bme.hu/~gabor/classwork/ising.pdf

[Aethelwulf]

Below Planck Lengths, the Uncertainty Principle dominates. When the universe was a point at our universes finite past, time as we know it did not exist. But because we are inferring on things below the Planck Time, there is a very large uncertainty in energy - could my own analysis of the spin networks and the triangulation of spacetime be hinting at an uncertainty which is also related?


At below planck lengths, geometry as it is understood by General Relativity breaks down. The planck lengths are derived using dimensional analysis. Another way to state this, is that the Schwartzchild radius of a black hole is equal to the Compton wavelength at the planck scale thus a photon trying to probe this would gain no information at all.

For a quick comparisson, the Classical Electron Radius is in fact times larger than the Compton Wavelength. The Compton Wavelength is where h is Plancks Constant and it has a value of . The Compton Wavelength itself has a value for the electron as value varies with different particles) and is a measure itself of the wavelength of a particle being equal to a photon (a particle of light energy) whose energy is the same as the rest-mass energy of the particle.

Basically, all particles have a wavelength. Photon's can never be at rest but the energy of a photon can be low enough to have it's wavelength match any particle who is at rest. It's often seen in the eye's of many scientists as the ''size'' of a particle. Actually, a more accurate representation of the size of an object would be the Reduced Compton Wavelength. This is just when you divide the Compton Wavelength by and it gives a smaller representation for the mass of a system.

Furthermore, if a photon could measure a planckian object, it could actually create a new class of particle called a Planck Particle - it would distort that space so badly that the photon would be gobbled up and no measurement could be performed. This is due to the Uncertainty Principle if my memory serves

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

Interestingly, Brian Greene has speculated on sub-Planckian existences. Whilst the Planck lengths could be fundamental, we don't know this for fact. He said:

"the familiar notion of space and time do not extend into the sub-Planckian realm, which suggests that space and time as we currently understand them may be mere approximations to more fundamental concepts that still await our discovery.”

Which is interesting, because if anyone actually follows my own speculations and contentions, I have been wheeling the idea that space and time could certainly not be fundamental since in the very beginning, there was no geometry (space-time) - not in the sense that GR deals with it.

So, as we have seen, the geometry imposes a very strong connection between time and energy at below Planck Lengths, pretty much approaching the singular state of existence. If the spin networks is the correct approach, then maybe the uncertainty relationship I deduced are both actually hinting at the same phenomenon. This might be the way in how to treat quantum mechanics very early on in the universes history.

[Aethelwulf]

... I was asking for this to be closed now; but I realized I actually do have something more to say.

[Aethelwulf]

There is one major inconsistency with my idea - no one has brought it up which I find quite odd. I have been attempting to explain the expansion of space via the conditions set for a singularity - a confined region, a point in which energy is stacked up to infinity - and thereby using quantum mechanics, namely the uncertainty principle to state that there needed to be extra degree's of freedom .

Well, there is one big problem with this: quantum mechanics does not allow for singularities, and the main reason why is because of the uncertainty principle! That's like trying to fight a fire with fire in my approach?

I must admit, I am surprised no one even mentioned this, all this time. A lot of what I have said is true however - in the beginning, there was indeed a lot of uncertainty. If we bring quantum mechanics into the picture there simply cannot be any singularities because of the connetcion of momentum and location that must exist for every object - if you try and squeeze an object into a space too small, it will resist that squeeze. This is why physicists say a singularity cannot exist...

But wait a minute!!!

Isn't that what my theory has been saying all along?

Yes it is. I think we can still deal with a singularity, but I don't think the singularity was there for

1) There forever, and
2) Was even there that long

We must infer that the beginning of time was a very special case of violating the uncertainty principle. It almost hardly never happens in nature, but the beginning of existence itself might require a little violation. One which was quick, one which keeps to the laws of quantum mechanics but tries to accommodate for a singularity. What fascinates me, is, that is the big bang is the true representation of the universe, then scientists have thrown out the suggestion that quantum mechanics cannot explain singularities because you can't squeeze matter into regions of spacetime without expecting a violation to occur. But perhaps we have overlooked the question that maybe the violation needed to occur in the first. The singularity afterall, makes no physical sense from GR, therefore why should it make any lawful sense in QM?

The uncertainty principle surely did dominate the early framework - and I think we should prepare our minds to begin thinking that maybe violations like this had to occur - afterall, it was only for a short time - maybe a time equivalent to the lifespan of a virtual particle. But what of the infinite singularity? Well it may not exist in that form today - the singularity that once existed has been shifted, not of an infinitely dense point, but of an infinite expansion of space with an infinite amount of energy and matter contained therein.

[Swift]

... I was asking for this to be closed now; but I realized I actually do have something more to say.

We'll go with closed for now. Once you finish with the Big Bang theory, you can come back to this (maybe)