Physics Subtly Implies no Geometry for Space

[Reiku]

Is thinking...

That the origin of matter and energy is that of the curvature of spacetime itself. Have a sufficiently large enough curvature and you can create matter and energy, this is what Einstein's equations predict..

But equally, matter especially is a late low energy phenomena; the appearance of matter is therefore described by the low energy epoch called Geometrogenesis. Maybe matter did not appear from the big bang, but energy did. Einstein once taught us that there is really no such thing as matter - it was all just various forms of energy.

If this is the case, maybe matter is not fundamental. Maybe energy is. Equally, maybe geometry is not fundamental so a final theory will not be concerned with matter or geometry, this would mean it is not concerned with space as we usually recognize it every day in our lives.

The latter has already been questioned by Fotini Markoupoulou.

So if matter comes from geometry and eminated from the curvature of a reasonably early universe, then were did space itself come from?

I think Fotini is correct. She believes that the final theory of physics will be concerned with no geometry. That space essentially does not exist.


So maybe space, matter and geometry are all concerned with the same topic, in that we will find that none of them will describe our final theory. We shall tackle some of these idea's later - such as whether a no-space no-time model can fit into this idea. The idea that timelessness exists would imply in relativity that no space really exists either, since both are unified relativitistically. This is in the same kind of light I treated my no-time no-energy Cosmological Problem. The very fact that we seem to have no Global description of time given in the Wheeler de Witt equation must imply an undefined non-conserved energy for a universe. Thus the absence of geometrical space would imply there is no geometrical time, that the vanishing time derivative in the WDW-equation gives us light to assume other idea's, such as the no-energy postulation as well. It is a cosmological assumption that all the negative particles equally cancel out the positive charges in the vacuum.

So, just like feet on a centipede, they are all conjoined by one body.

The absence of space, must imply the absence of matter and geometry, would include the absence of time which would naturally assume the absence of energy. So maybe energy is not as fundamental as we think?

Maybe the four ''fundamental'' ingredients for a vacuum like ours, those including space, time, matter and energy are all but fascets of one theory hinting at an illusion.

I will get Fotini's paper later today and I will talk about this further.

 

[Reiku]

First of all, we should tackle what kind of theory geometrogenesis implies: It means the materialistic organization space; the configuration of matter giving rise to geometrical bodies.


If Markopoulou is right, then it seems she predicts something quite odd usually - the notion that space, is not a fundamental object, that in fact, physics seems to be hinting at a space with no geometry.

What we are challenged with is rooted from a relaltivistic discipline. It teaches our mind to think of time as being a dimension of space, the fourth dimension of space to be exact. It is an imaginary leg which is orientated 90 degrees from the real leg on the spacetime triangle. Time is as invariant as space was, then that neither of them could be seperated. It also implied time was primal, that it had some kind of unique meaning in the beginning of everything.

However, even with these predictions that Einstein's theory made on these things, there was an unusual glitch in his equations. Some of them, those concerned with global measurements have no time description. Called Timelessness in general relativity, it is considered a ''problem'' for physicists. If the timeless solutions to Einstein's equations is implying that the universe is timeless, then there are some interesting symmetries which are obtained from other sources of work.



If all the negative matter cancels out every positive peice of matter, such as the zero-energy universe. In particular interest, it is in fact the Minkowski metric (the four dimenensional manifold) which implies zero energy. The reason why the Minkowski metric implies the null energy condition [1] will require only a few steps. We begin with an equation which has been selected because it is a result of the Schwarschild's metric

$$E = Mc^2 - \frac{Gm^2}{2R}$$

According to this equation, when $$M=0$$ the energy is also zero, but you obtain the metric itself. So this is classed as the null energy condition.

Interestingly though, I have already elaborated on a time-energy problem concerning how if there is a vanishing time then there is no way to translate a universal symmetry with any energy, the equivalent to energy being conserved.

Alone, we have seen that time and energy have some problems if they are to be understood fully.


But as I mentioned earlier, Fotini has described a universe without space. Albeit, she does this to achieve merit in her opinion that time can be saved, but she still raises an interesting argument in favor of Geometrodynamical models. [2],[3]

The first paper linked to, involves the spaceless theory, but makes mention of her quantum graphity model. The second paper is on the paper which first speculated quantum graphity.

It is really from this point I have been wondering whether the fact that physics seems to be implying a no-space, no-time and no-energy universe, (where energy and matter in this sense can be freely exchanged) there is still the evidence that Geometrogenesis predicts matter as a low energy phenomenon, one which appeared when there was some geometry involved.


Julian Barbour seems to be important to mention here.

He constructed a theory which would describe motion from a timeless theory [4]. He tackles his questions with a series of simple equations which seem to contain the relevant data without the description of time. He assumes that we should concentrate on the right hand side of the equation which does not contain the time variable and deal with observable quantities, which he says, is the way physics should work.

One of his final equations are described as:

$$v_i=\frac{\delta d_i}{\delta t} = \sqrt{\frac{2(E-V)}{\sum_i M_i(\delta d_i)^2}} \delta d_i$$

He rids the equation of the time description by using the fact that the speed of a body is not the ratio of it's displaceent to an abstract time increment but to which involves displacements of all the bodies in the system. By doing this, he rids his use of time describing motion. Most interesting of all, is that his theory predicts that time is no longer measured by particular individual motions, but by a sum of all the motions.


It might seem like a stretch in believing that all the factors which make our theory, those being of space, matter, energy and time as being illusions, yet I say that in hindsight that the fact all these things are called into question more of as a clue. It might be one which could hold some merit if one wanted to construct a theory of everything, but if it states that time, energy, space and matter are all absent, how do you approach such a theory?

It seems that we require new parameters that do not vanish. Here I am taking use of Julian Barbour's approach in creating an equation which ridded itself of a time variable, but also contained dynamics which would still successfully decribe the system.

Perhaps our final approach will be one which is similar.

We might find that all those ingredients are insufficient to describe everything, so we must consider alternatives.





[1] http://arxiv.org/pdf/gr-qc/0605063v3.pdf

[2] http://fqxi.org/data/essay-contest-files/Markopoulou_SpaceDNE.pdf

[3] http://arxiv.org/pdf/0801.0861v2.pdf

[4] http://www.platonia.com/nature_of_time_essay.pdf


ps. I have more to state, more thoughts on these illusions of physics. I will share them as the week passes us by.

 

[Reiku]

Maybe just as a side note, a few weeks back I created a mass-flow equation. The design was to basically measure a mass-flow rate, which necesserally includes the passage of a time over some part of a geometry.

The equation which was of interest to me, was a mass flow equation I frivoulously derived with no merit behind why it was derived, only to achieve the dimensions required. I arrived at this equation:

$$\dot{\chi} = (\frac{\partial \mathcal{L}}{\partial \dot{q_i}} \cdot d_) \nabla^2$$

The matter field $$\chi$$ can be thought of any matter field. You may freely chose to replace $$\chi$$ with any or all the matter fields, each piece of matter with their own individual clocks.

$$\sum^{k}_{k=1} \dot{M}_k \hat{S}_{k} = \nu_i$$

where $$\nu$$ is the net rate of flow of entropy. A classical field version again of a mass flux is given by

$$\dot{\chi} = (\frac{\partial \mathcal{L}}{\partial \dot{q_i}} \cdot d_) \nabla^2$$

where the dot represents a multiplication, not a dot product, which has been mistaken before.

Plugging in $$\nu_i$$ gives

$$\sum^{k}_{k=1} \dot{\chi}_k \hat{S}_{k} = \sum^{k}_{k=1} (\frac{\partial \mathcal{L}}{\partial \dot{q_i}} \cdot d_)_k \nabla^2 \cdot \hat{S}_k = \dot{M}_{k}(\hat{-K_B \sum_i P_i In P_i})_k$$

where $$P_i$$ is the probability.

Assuming that information can actually leave a universe, this equation works for it describes the mass flux in and out of the system.

Using Julian Barbours approach, where he considers each particle with a mass $$M$$ as having recording the ticking of time. There maybe atleast one matter field we may use in physics to describe a mass flow/flux.

If so, we have a mass flux with individual constituents with their own dynamical clocks. For particles like an electron, this is not a bad idea. To me, it seems clear that one can derive his timeless equations and be plugged in to the existing values.

 

[khan]

Light following a geodesic as predicted by general relativity best explains and predicts what is observed. Non-geometric Newtonian gravity and refractive gravity both do not accurately predict the bending of light as it passes a massive object like the sun but space-time curvature does.

A unified field theory based on geometry is then required.

Albert Einstein wrote what appears to be an interesting paper on the unification of gravity and electromagnetism:

http://www.alexander-unzicker.de/rep0.pdf

Unified Field Theory of Gravitation and Electricity
Albert Einstein
translation by A. Unzicker and T. Case

[...]

The applied method can be characterized as follows. First, I looked for the formally most simple expression for the law of gravitation in the absence of an electromagnetic field, and then the most natural generalization of this law. This theory appeared to contain Maxwell’s theory in first approximation.
In the following I shall outline the scheme of the general theory (§ 1) and then show in which sense this contains the law of the pure gravitational field (§ 2) and Maxwell’s theory (§ 3).

If what we call gravity is actually the curvature of space-time, and it appears to be so, then it seems likely that the path to unifying gravity with the other fundamental forces of nature would entail a geometric approach.

:shrug::shrug::shrug:

 

[khan]

Heisenberg uncertainty is a form of the geometric Cauchy Schwarz inequality law - AKA the triangle inequality. There is also a time-energy uncertainty derived from the Heisenberg uncertainty principle.

http://en.wikipedia.org/wiki/Cauchy–Schwarz_inequality#Physics

Such hopeful clues exist and possibly they will help us to discover the geometric laws of the unified field theory

 

[AlphaNumeric]

The equation which was of interest to me, was a mass flow equation I frivoulously derived with no merit behind why it was derived, only to achieve the dimensions required. I arrived at this equation:

$$\dot{\chi} = (\frac{\partial \mathcal{L}}{\partial \dot{q_i}} \cdot d_) \nabla^2$$

The matter field $$\chi$$ can be thought of any matter field. You may freely chose to replace $$\chi$$ with any or all the matter fields, each piece of matter with their own individual clocks.

$$\sum^{k}_{k=1} \dot{M}_k \hat{S}_{k} = \nu_i$$

where $$\nu$$ is the net rate of flow of entropy. A classical field version again of a mass flux is given by

$$\dot{\chi} = (\frac{\partial \mathcal{L}}{\partial \dot{q_i}} \cdot d_) \nabla^2$$

where the dot represents a multiplication, not a dot product, which has been mistaken before.

Plugging in $$\nu_i$$ gives

$$\sum^{k}_{k=1} \dot{\chi}_k \hat{S}_{k} = \sum^{k}_{k=1} (\frac{\partial \mathcal{L}}{\partial \dot{q_i}} \cdot d_)_k \nabla^2 \cdot \hat{S}_k = \dot{M}_{k}(\hat{-K_B \sum_i P_i In P_i})_k$$

where $$P_i$$ is the probability.

Assuming that information can actually leave a universe, this equation works for it describes the mass flux in and out of the system.

The equation has been thoroughly shredded, so much so you even admit it was just something you randomly made up one drunken evening, yet here you are inserting it into ever more elaborate claims.

Why are you piling on more and more claims on an equation which is nonsense, both physically and mathematically? Just because you've heeded one of several comments about it, ie you now say its not a dot product but a scalar product, doesn't make it mathematically meaningful, it's still nonsense.

On January 19 you make a load of excuses why no one should pay attention to it, yet on February 3rd you're still talking about it?

This is why people think you're actively dishonest. You complain how James and I shouldn't be giving your 'equation' too much attention, it was just the result of a drunken evening, yet you then go and post it again and again.

And your desperate desire to throw in more and more expressions for things gives your ignorance away, namely in your expression for entropy. Ignoring the fact you suddenly produce probabilities from nowhere, without justification, definition or derivation, you write In, ie that is a capital i and then a lower case n. It's a mistake someone utterly unfamiliar with entropy or even basic calculus might make if they just looked at the equations because, as with many fonts, there's little distinction between a capital i and a lower case L, ie I vs l.

It isn't $$In$$, it is $$\ln$$. It's the standard expression for the natural logarithm, which is lower case l followed by lower case n. There's even a LaTeX symbol for it, \ln . The fact you don't even know notation so basic it's used in A Level classes shows, yet again, the laughable nature of your attempts to look like you're doing something meaningful.

If you insist on posting BS at least be clear about it. Don't try to dress it up as if you're working on developing published material or formal mathematics. How many years of this are you going to do?

 

[Reiku]

The equation has been thoroughly shredded, so much so you even admit it was just something you randomly made up one drunken evening, yet here you are inserting it into ever more elaborate claims.

Why are you piling on more and more claims on an equation which is nonsense, both physically and mathematically? Just because you've heeded one of several comments about it, ie you now say its not a dot product but a scalar product, doesn't make it mathematically meaningful, it's still nonsense.

On January 19 you make a load of excuses why no one should pay attention to it, yet on February 3rd you're still talking about it?

This is why people think you're actively dishonest. You complain how James and I shouldn't be giving your 'equation' too much attention, it was just the result of a drunken evening, yet you then go and post it again and again.

And your desperate desire to throw in more and more expressions for things gives your ignorance away, namely in your expression for entropy. Ignoring the fact you suddenly produce probabilities from nowhere, without justification, definition or derivation, you write In, ie that is a capital i and then a lower case n. It's a mistake someone utterly unfamiliar with entropy or even basic calculus might make if they just looked at the equations because, as with many fonts, there's little distinction between a capital i and a lower case L, ie I vs l.

It isn't $$In$$, it is $$\ln$$. It's the standard expression for the natural logarithm, which is lower case l followed by lower case n. There's even a LaTeX symbol for it, \ln . The fact you don't even know notation so basic it's used in A Level classes shows, yet again, the laughable nature of your attempts to look like you're doing something meaningful.

If you insist on posting BS at least be clear about it. Don't try to dress it up as if you're working on developing published material or formal mathematics. How many years of this are you going to do?

Well, I posted that specific post just to see if you would bite.

Didn't take much now did it.

 

[Reiku]

As for the ''In'' part, you might as well call it basic, which it is. It's obviously my latex.

 

[Reiku]

Light following a geodesic as predicted by general relativity best explains and predicts what is observed. Non-geometric Newtonian gravity and refractive gravity both do not accurately predict the bending of light as it passes a massive object like the sun but space-time curvature does.

A unified field theory based on geometry is then required.

Albert Einstein wrote what appears to be an interesting paper on the unification of gravity and electromagnetism:

http://www.alexander-unzicker.de/rep0.pdf



If what we call gravity is actually the curvature of space-time, and it appears to be so, then it seems likely that the path to unifying gravity with the other fundamental forces of nature would entail a geometric approach.

:shrug::shrug::shrug:

We also require that time takes to move from one place to another, but if timelessness exists, then the same dichotemy presents itself.

Space is not fundamental. Geometrodynamics proves this well.

 

[James R]

Well, I posted that specific post just to see if you would bite.

Didn't take much now did it.

Are you saying you knowingly posted something you knew to be false, in order to throw out some troll-bait?

Please explain your comment in detail, because you may be banned for this.

 

[AlphaNumeric]

Well, I posted that specific post just to see if you would bite.

Didn't take much now did it.

Firstly I didn't even reply for several days, so it's not like I'm chomping at the bit to point out your ignorance. Secondly I don't believe you, given you have repeatedly posted nonsense and only admitted it when your back is against the wall because so many problems have been highlighted.

It's always some excuse with you. You can't say "I was wrong, I don't understand this stuff", instead you come out with "It was the result of a drunken evening" or "I typed up something I'd printed out ages ago and forgot it wasn't me who wrote it" or "I was troll baiting".

As James has said, 'troll baiting' in that form is itself trolling. Whatever your motivation, be it to try to con others into thinking you're not ignorant or be it to elicit a response from me, you post knowingly false material which you try to present as viable discussion. You're spreading misinformation. Regardless of whether or not you do it deliberately it's extremely dishonest and shows you're willing to mislead others just to entertain yourself.

Such actions make your complains about other people being mean or unkind hypocritical. You're willing to mislead people to give yourself a chuckle. Mind you, if you find humour in your actions I think you should take a long look at yourself. Actually I think you should do that anyway.

As for the ''In'' part, you might as well call it basic, which it is. It's obviously my latex.

Excuses excuses. It's a mistake common to people who haven't really worked with logs before. If this were a once off I'd give you the benefit of the doubt but your 'latex errors' are so frequent and basic they cannot all be typos, they are further evidence of how little you actually do.

 

[Reiku]

Are you saying you knowingly posted something you knew to be false, in order to throw out some troll-bait?

Please explain your comment in detail, because you may be banned for this.

No, I wasn't troll baiting at all.

I am still under the impression the equation could have merit without the limit I previously had on it.

Now, it has been with difficult task for me to actually get alphanumeric to engage in a topical debate. Because of this, I wondered what would do it.

It has been a previous thought of mine that the equation could hold merit and as you will see, the equation has been applied to something in that post. I wondered how long it would be before alphanumeric actually took a bite of the subject.

I still want him to try and participate in the OP, not wholey with the topics which he thinks he is impervious to mistakes, which is demonstratably false.

 

[Reiku]

Plus, how can I derail my own thread... or more to the point, why?

 

[Reiku]

We also require that time takes to move from one place to another, but if timelessness exists, then the same dichotemy presents itself.

Space is not fundamental. Geometrodynamics proves this well.

Let me expand on this, the other day my taxi arrived and I quickly wrote the above.

A particle takes time to move from one point to another, but timelessness persists in GR, and we are told to take it seriously by some scientists. One however is applied to a local time whereas the other is more of an application to a global time, described by a Global Wave Function.

Global Time has problems. It simply does not exist. However papers I have read suggest that perhaps the absence of a Global Time is often taken for granted and that perhaps the absence of a Global Time must indicate some kind of singularity. Actually, there are existing proofs of this (if memory serves, by George Ellis).

Now geometry, is all fine and well in General Relativity but we have failed to find an appropriate quantum theory of General Relativity, which must imply that GR breaks down as you get to the understanding of High Energy Physics.

In light of this, Geometrodynamics states a very obvious truth then. That Geometry is a concern of a low energy epoch where matter dominated the universe (the place where fixed clocks can tick off time according to Julian Barbour). The high energy epoch is were geometry no longer exists and the geometrical vision of Einstein is under threat.

 

[Reiku]

Heisenberg uncertainty is a form of the geometric Cauchy Schwarz inequality law - AKA the triangle inequality. There is also a time-energy uncertainty derived from the Heisenberg uncertainty principle.

http://en.wikipedia.org/wiki/Cauchy–Schwarz_inequality#Physics

Such hopeful clues exist and possibly they will help us to discover the geometric laws of the unified field theory

That is actually a very interesting approach! Some one who uses their imagination I see

The reason why that approach could be important, is because the uncertainty principle dominates the condition known as the Big Bang, the applications are huge. I even speculated that maybe space (required) to expand initially not because of any negative energy density, but maybe because particles where so ''confined'' that to remain as compact at the big bang would almost certainly violate the Uncertainty Principle.

Mind you, how do you speak of geometry still, at the big bang?

Dare I say, maybe big bang is what is crippling the unified approach rather than GR necesserily meaning to break down. GR may not need to break down, if our approach did not involve a big bang. There are already scores of problems concerning the big bang.

 

[AlphaNumeric]

No, I wasn't troll baiting at all.

Now you're contradicting yourself.

I am still under the impression the equation could have merit without the limit I previously had on it.

Except it had been explained to you why it was flawed in several ways and you yourself told us not to pay any attention to it. So why are you posting it again? Are you now claiming people should pay attention to it? You can't have it both ways.

Now, it has been with difficult task for me to actually get alphanumeric to engage in a topical debate. Because of this, I wondered what would do it.

There can't be any 'topical debate' if you're dishonest and posting knowingly false things.

I still want him to try and participate in the OP, not wholey with the topics which he thinks he is impervious to mistakes, which is demonstratably false.

Where did I claim to be impervious to mistakes? Stop being so dishonest and lying so blatantly. Doing so only demonstrates you aren't really looking for a discussion, you just want an excuse to spout nonsense. Sorry, I'm not going to feed your desire to delude yourself.

 

[Reiku]

Now you're contradicting yourself.

Except it had been explained to you why it was flawed in several ways and you yourself told us not to pay any attention to it. So why are you posting it again? Are you now claiming people should pay attention to it? You can't have it both ways.

There can't be any 'topical debate' if you're dishonest and posting knowingly false things.

Where did I claim to be impervious to mistakes? Stop being so dishonest and lying so blatantly. Doing so only demonstrates you aren't really looking for a discussion, you just want an excuse to spout nonsense. Sorry, I'm not going to feed your desire to delude yourself.

I took the flaw that the limit kind of redefines the world line in such a way that it cannot be good form. I've accepted that. But the rest of your quibbles lie marginely on the fact I had no basis for its derivation, I only sought to balance dimensions.

Anyway, since you won't participate I have no choice now but to block you. I've tried and failed.

 

[AlphaNumeric]

I took the flaw that the limit kind of redefines the world line in such a way that it cannot be good form. I've accepted that. But the rest of your quibbles lie marginely on the fact I had no basis for its derivation, I only sought to balance dimensions.

I gave multiple reasons why it was mathematically nonsense. That alone should have been enough for you not to try to build on it. Instead you ignored them all and ignored the fact you have absolutely no physical justification for such an expression to mean anything by your own admission. You repeatedly complained that James and I shouldn't be slating you for the equation because you'd just made it up one drunken evening. Now you're building on it, trying to pass off your work as something related to actual physics. That's hypocritical, dishonest and deceitful.

Anyway, since you won't participate I have no choice now but to block you. I've tried and failed.

What is there to participate in? When you have fundamental flaws in your claims pointed out you make excuses and tell people not to take any heed of the claims but then you turn right around and start piling more and more things on top of those claims.

How can anyone participate in an honest discussion if you're not only unable to be honest but are actively deceitful? You can't complain I'm not willing to play ball when I'm the one using actual maths/physics pointing out the fundamental problems in your claims, claims you admit are the product of a drunken evening.

You don't want people to talk about your 'work' when they are showing it to be flawed but then you want to talk about your work. You want to be immune from criticism or correction, which simply isn't going to happen.

 

[Reiku]

I gave multiple reasons why it was mathematically nonsense. That alone should have been enough for you not to try to build on it. Instead you ignored them all and ignored the fact you have absolutely no physical justification for such an expression to mean anything by your own admission. You repeatedly complained that James and I shouldn't be slating you for the equation because you'd just made it up one drunken evening. Now you're building on it, trying to pass off your work as something related to actual physics. That's hypocritical, dishonest and deceitful.

What is there to participate in? When you have fundamental flaws in your claims pointed out you make excuses and tell people not to take any heed of the claims but then you turn right around and start piling more and more things on top of those claims.

How can anyone participate in an honest discussion if you're not only unable to be honest but are actively deceitful? You can't complain I'm not willing to play ball when I'm the one using actual maths/physics pointing out the fundamental problems in your claims, claims you admit are the product of a drunken evening.

You don't want people to talk about your 'work' when they are showing it to be flawed but then you want to talk about your work. You want to be immune from criticism or correction, which simply isn't going to happen.

How about participating in a discussion of the OP, and divert your attention from the equation. It is not the only thing that has been discussed.You act like it is.

 

[Reiku]

In an approach according to a paper I linked previously in the past, it was customary as a reparamaterization of the Wheeler de Witt equation to reconfigure time as $$\chi = \tau$$, so the matter field was now acting as a clock.

The Total Hamiltonian for a universe is given by the WDW-Hamiltonian

$$H_{WDW}=H_T=H_{\phi, h_{\mu \nu}} - H_0$$ [1]

where $$h_{\mu \nu}$$ is a metric perturbation. The equation of interest is

$$|\psi> = |\phi_a> |\chi_{\phi, h_{\mu \nu}}>$$ [2]

Using a proceedure of seperation of variables, we obtain

$$(H+V(a)|\psi_a> = E|\psi>$$ [3]

and

$$(H + V(\chi)_{\phi h_{\mu \nu}})|\chi_{\phi, h_{\mu \nu}}> = E|\chi_{\phi h_{\mu \nu}}>$$ [4]

The potential terms in these equations are equal to

$$V(a) = a^2-g^2a^4$$ [5]

and

$$V(\chi) = \chi^2 - g^2\chi^4$$ [6]

Going back to eq. [4] we have

$$(H+(\chi^2-g^2\chi^4))|\chi_{\phi, h_{\mu \nu}}> = E|\chi_{\phi h_{\mu \nu}}>$$ [7]

Because of continuity of eigenvalues, one should state that as $$(\chi_{\phi, h_{\mu \nu}}) \rightarrow \phi$$ then the interaction term $$(\chi^2-g^2c^4) \rightarrow 0$$ goes to zero.

A potential solution to this equation is

$$|\psi> = \phi + \frac{1}{E - H_0} \cdot (\chi^2- g^2\chi^4)|\psi>$$ [8]

where $$\cdot$$ is multiplication. Whilst this is a potential solution, a problem exists. A serious problem in the form of a singularity exists in $$(E-H_0)$$ since $$E$$ is an eigenvalue of $$H_0$$.

There is one distinct way of making this singularity to normally disappear. One way is to make the denominator slightly complex $$i\epsilon$$. Instead, I am going to use a different approach.

To avoid the singular existance of this mathematical object, we are going to substitute $$E$$ for a result of the Schwartzchild Metric

$$Mc^2 - \frac{Gm^2}{2R}$$ [9]

This changes our equation in such a drastic way, it rids us of the singularity in our equation. No longer does $$(E-H_0)$$ hold, because the null energy condition states that as $$M=0$$ all we are left with is the metric.

So we have a correspondance between $$(Mc^2 - \frac{Gm^2}{2R})$$ and $$H_0$$ in which if you imply a nullified energy condition you obtain a solution to avoid the singularity problem.

Interestingly, it does not only imply that our matter field $$\chi$$ no longer has any mass, but it also means we have no radiation fields either.

Let's look at the original Hamiltonian in a different way, now accounting for systems of many particles if one desired but for simplicity we will assume particles $$i$$ and $$j$$.

For $$N$$-particles (2 in our case), we can have

$$\{ \sum_{i} H(i) + \sum \frac{1}{r_{ij}} \} = E \psi$$

where $$r_{ij}$$ calculates the distance between $$i$$ and $$j$$. If we represent this in a Hilbert Space, the configuration space looks like

$$H= \sum_i H_i + \sum_{k \in I} h_k$$

where $$h_k$$ is a hermitian operator. $$I$$ is the set of interactions $$\{i,j\} \equiv k$$ and naturally can form a direct tensor Hilbert space $$\mathcal{H}_3 = \mathcal{H}_1 \otimes \mathcal{H}_2$$.

This is already beginning to be like a theory which deals with no geometry. Indeed, a Hilbert Space are often called ''points'' which describe abstractly the configuration space. This is the similar approach Markoupoulou attempts to make in her Graphity model. She takes Hilbert Spaces and describes their interactions in such a model without resorting to geometry.