Mordred

I also recaall during that thread I had also mentioned the unlikelyhood that the universe exists I a BH or WH with much the same reasons I posted here. I gave you the info on Poplowskii so you can study the problem and make an informed opinion for yourself. Poplowskii ran into several of the problems I mentioned above. He took a considerable effort to try and address these problems. However he also couldn't cover early large scale structure formation and as more and more data came in from Planck he has slowed down much of his work.

I would like to touch upon a particular section of the above. In mathematics we are confronted by sets that are infinite in quantity etc. A good example would be all the possible paths a lightbeam can follow (geodesics.) So there is a technique that is handy called compactification. Any infinite mathematical object/set etc will contain both infinite and finite parts. So in the example above we can compactify the possible paths to a finite probability weighted likelihood . This is via the possible paths of least resistance, it is a weighted probability based on kinetic vs potential energy relations. In QFT there is two boundaries Dirichlet and Neumann boundaries these correspond to many of the PDE and ODE (partial differential) and (Ordinary differentials) the Direchlet being the PDE'S. So lets provide a QFT application to all the above. Now particles can pop in and out of existence all the time but lets stick to the Feyman path integrals. The external lines on a Feyman diagram are observable, this is an amount of "action" of 1 or more quanta. This involves coordinate change, the path of that external line will be a probability weighted likelihood. Now what about the internal lines these are not observable Operators but the unobservable propagators. Where the VP term is often applied, but its better to simply consider it as part of the field. (particles being field excitations). So in essence there can be an infinite possible number of particles however there is a likelihood of the number depending upon the available energy potential of the fields involved. Under QFT this is the purpose of the creation and annihilation operators. Unlike QM the field is treated as an operator. There is a few examples of how we can apply mathematics to deal with functions we use to model our universe with infinite possible values but restrict those functions to the finite portions. Those two boundaries are involved in gauge group such as SU(2) and U(1) Now another compactification technique is wicks rotation. If you take say a waveform and rotate a mirror image of that waveform there will be a point where the two will meet. This provides a definitive finite coordinate or graph point. As you expressed an interest in the mathematics I felt this will be something you would find of interest As a visual aid here is an example of a Feymann diagram the external lines on the diagonals are the observable (on shell particles) the internal wavy line is the propagators. [math]\array{e^+ \searrow &&\nearrow P^-\\&\leadsto &\\ e^-\nearrow &&\searrow P^+}[/math]

With what little data we have if the mass distribution is much the same as ours then so too will be the curvature term. No the observable portion we have now is the furthest we have ever seen. In the past you would see less and less the further back in time the observer is placed. The observable portion is our region of past and present causality. The cosmocalc is far more flexible than this but I focussed the calcs on the particle horizon which is the cosmological event horizon. Here is the range of the observer horizon back in time. I went from from today bottom row to the CMB last surface at redshift 1090. The calc can go earlier but it does reach a limit ie doesn't handle inflation. [latex]{\small\begin{array}{|c|c|c|c|c|c|}\hline T_{Ho} (Gy) & T_{H\infty} (Gy) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}[/latex] [latex]{\small\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline S&T (Gy)&D_{par}(Gly) \\ \hline 1090.000&0.000373&0.000856\\ \hline 768.343&0.000668&0.001561\\ \hline 541.606&0.001183&0.002822\\ \hline 381.779&0.002074&0.005052\\ \hline 269.117&0.003606&0.008973\\ \hline 189.701&0.006228&0.015819\\ \hline 133.721&0.010698&0.027712\\ \hline 94.260&0.018302&0.048276\\ \hline 66.444&0.031211&0.083704\\ \hline 46.837&0.053094&0.144544\\ \hline 33.015&0.090158&0.248752\\ \hline 23.272&0.152887&0.426844\\ \hline 16.405&0.258995&0.730635\\ \hline 11.564&0.438378&1.247998\\ \hline 8.151&0.741396&2.127725\\ \hline 5.746&1.252327&3.620922\\ \hline 4.050&2.109877&6.148142\\ \hline 2.855&3.531250&10.399216\\ \hline 2.013&5.813076&17.448904\\ \hline 1.419&9.228712&28.800505\\ \hline 1.000&13.787206&46.278944\\ \hline \end{array}}[/latex] Stretch is thee inverse of the scale factor, which will also correspond to the CMB blackbody temperature which is also the inverse of the scale factor.

Yes we should see variations as one approaches the cosmological event horizon. Let toy model this, by descriptive as most likely few will understand the math but if requested I will happily provide. Lets assume for a moment the a universe portion just on the outside our observable. Now if this portion has a different mass density this will cause a response by the nearby regions within our observable universe. A higher or lower mass density will result in a net flow of mass, this in turn affects the temperature as well as redshift. The direction of flow will be that which supports an equalization of mass density between the two regions. We don't see this as far as we can measure. The temperature distribution of the CMB for example is incredibly uniform. This uniformity also limits a rotating universe. The upper bounds on the rotational speed is such that it needs to be low enough to explain why we haven't bee able to detect a rotating universe. However that's just a side note. We can safely assume the regions of shared causality just outside our observable portion but within another observable portion say just at the edge of our observable portion is of the same uniform distribution. As we can see regions that can be affected by regions we cannot see via its shared causality that within this limit we can infer that it is much the same as our observable. Beyond any possible shared causality with our observable portion we have absolutely no clue.

Gravity always attracts hence a one directional force. A BH will always be attracted to nearby celestial objects. Here is the thing, whenever you have a continuous attractive force you get mass density distributions of a gradient. Hence the Poisson equations which describe the Newton limit to gravity is a gradient potential (a type of vector field). A field with no inherent directional components on average is a scalar field. Now how can this be detectable, the answer is via temperature. Under the gas laws, when you have a gradient field where every particle species is involved in particular. Which spacetime does, (lets use spacetime instead of gravity). Then you have a non uniform mass distribution, which has several effects. Higher mass density means higher temperature, it also means that light paths are also affects which we see as redshift in particular gravitational redshift in this instance. This also causes distortions and is described via a curvature term under the FRW metric. This is literally what is being described by Universe curvature. It is the mean free path null geodesic of light that is being described as curved. Spacetime crvature caused by higher mass density is also curved and thus also causes distortions such as gravitational lensing. Now we have this handy object in space, the CMB. This object had a particular significance, it provided us via WMAP to seek those distortions to decide on a curvature k for the universe in general. This turned out to be significantly flat, so flat that two or more lightbeams in parallel will remain in parallel as they approach us. Positive curvature those beams will converge, negative curvature they will diverge. This is the supportive evidents that we are not under some gradient gravitational potential as well as these curvature terms also apply under GR (though care must be taken on the choice of metric signature lol -+++ or +++-. Evidents for temperature gradients is easy enough, as you approach a star for example the temperature rises, well this also occurs with an accretion disk hence the jets. This shows there is also a mass density gradient. Recall the above mention of WMAP and you can add Planck to that list. Much of the BH universe models literally lost a huge popularity when that data was released not to say those models aren't still around today but they have significantly lost favor. It is incredibly difficult to justify a seemingly homogeneous and isotropic when you involve either a WH or a BH. Not when you consider that Schwartzchild BH's are incredibly unlikely. Thanks to conservation of angular momentum anytime a BH feeds, it gains rotational energy. The accretion jets themselves are a result of that rotation being applied. Well a rotating body in space imparts its rotation through tidal locking on other nearby particles causing them to orbit. this then imparts more particles to join in. This then also affects both redshift and temperature. A simplistic formula for the thermodynamics Pv=nRT. As the density increases so does the pressure and temperature. So how does one maintain a homogeneous and isotropic universe inside or outside a massive BH or WH...? I've seem attempts some are seemingly plausible right up until you study how a BH feeds. It isn't continuous.... Its intermittent as material becomes available. Ths changes both rotation speed, as well as the amount of available mass. Hence should also change temperature and redshift. See where I am coming from, we see literally no evidence that can remotely be identifiable to support a BH type universe model. Instead we see plenty of evidence that makes it further and further unlikely.

Who said anything about spinning, a Schwartzchild BH though doesn't exist that we have ever found know of any? Either way the directional components to gravity itself in a BH is anistropic itself which doesn't match observations. One day you might actually listen to what I have stated in your previous threads on this topic and try to actually understand the mathematics behind your claims instead of regurgitating the same poor claims based on older attempts prior to WMAP data.

So your two page theory has lots of claims but zero math which means it has no predictability or testability. Hence it isn't a theory to begin with but rather a descriptive of your understanding. Not much to go on. Eather isn't supported by modern physics so that part you have right.

A universe that is expanding outward via a Schwartzchild will not be homogeneous or isotropic. Expanding outward is unto itself a preferred direction. Observational evidence does not support this. MOND isn't currently well accepted as it cannot account for the same range of observational evidence that LCDM can. Mainly it cannot account for early large scale structure formation. It was this problem that led MOND to include a dark matter variation later on which defeated the purpose of MOND in the first place.

Beyond our observable portion, its all conjecture. If one were to take the ever so slight curvature term and halt expansion. The math supports a possibility that a light ray would take 880 Billion years to arrive back at the origin. However that is only based on the ever so slight curvature term. There is no affect whether the universe is finite or infinite in terms of expansion or how our Observable universe evolves. Expansion is best described as a reduction in density, as the volume increases. Its unknown how small the entire universe could feasibly start at in the origin however no finite quantity can become infinite. So if the universe is infinite now then it will always be infinite in the past.

Think of it this way, Our Observable universe. Key note our observable portion of the universe originated at a point smaller than an atom. However this is just the portion that makes up our Observable portion of the universe today. Now here is the trick that smaller than an atom portion now makes up the entirety of our observable universe today. We reside within that portion and everywhere we look we see within that portion. So that original smaller than an atom is our entire observable universe today. The only centre is the center of our observable portion but how much bigger the rest of the universe is we do not know. We can only measure our observable universe and never beyond it. However we are certain there is a much larger universe we can never see or measure. We can also tell it must have similar conditions in mass density to our own observable portion due to the way a homogeneous and isotropic expansion occurs. homogeneous no preferred location ie a centre. isotropic no preferred direction. An explosion has a specific dynamic in that it radiates from a centre outward. this is inhomogeneous and anistropic. However measurements on how galaxies expand from one another shows no preferred direction to expansion rather it is simply a density decrease in all directions equally. Now center of mass. In a homogeneous and isotropic volume where the mass density is uniform, one can choose any arbitrary location. Then measure around that point and measure the same amount of mass surrounding every location. No location will have a higher gravitational potential than any other location, so one can arbitrarily choose any location as a center of mass as there is no difference between any other location in potential strength. Ie there is no discernable center of mass which would be a location of higher potential.

When you have a homogenous and isotropic mass density distribution, for any application of a CoM one can arbitrarily choose any coordinate as an effective centre for applications of shell theorem. Any coordinate for those purposes under math will be valid. So no there is no centre.

I will be honest TeVeS never really gained much strength when it was popular. I agree that the GW waves, place an even greater unlikelyhood of TeVeS being viable. GR is still the most successful model to observation in my opinion.

Oh my you really need to look into locality vs global interactions. You also need to establish a coordinate system and establish the range of interactions into time slices. The conditions outside of causality has no influence on a particle. When it comes to how a particle moves through a field this is incredibly important. You haven't got anywhere near the mathematics to account for causality regions not even close to the required mathematics for that. https://www.researchgate.net/publication/263468413_Is_the_State_of_Low_Energy_Stable_Negative_Energy_Dark_Energy_and_Dark_Matter 13 and 14 in this is wrong as well. Einstein field equations applies to EACH INDIVIDUAL FIELD. When those fields overlap then you determine what the function value is for each coordinate. Thus generating a new field. You can't just add stress tensors to the field equation as each field will have its own resulting Ricci curvature, scalar and metric signature relations. A tensor such as the stress tensor involves vector symmetry relations which you are not taking into account. The formulas you have been using are involved specifically to the Newton approximation under the correspondence principle. You need to work directly with the Einstein field equations and not the Newton approximations. Do so for each field and define each individually. You haven't got any time dependancies associated with causality. You need to establish local vs global influences using regional causality to a given coordinate. The entirety of a field does not influence a particle in the same infinitesimal time slice. Both the field and the particle is limited in the range of interaction to a given event by the locality of causality defined by c. You need to address this extremely important issue under a coordinate basis using 4D spacetime. However the single most biggest mistake your making is not realizing that the Cosmological Principle of homogeneous and isotropy for the FLRW metric only applies on size scales of 100 Mpc or greater. The mathematics of the FRW metric are not applicable for size scales of a galaxy. That is an anistropic local condition, that the FRW metric is not suitable to describe. The 100 Mpc is a suitable size scale where the mass distributions into locale LSS regions becomes effectively washed out. The [latex] 8\pi G[/latex] is a correspondence involving the shell theorem and Newton approximation of a UNIFORM matter distribution within a given volume. ( Not that anything I've seen from you can produce the worldline of light involved in the FLRW metric....specifically the ds^2 worldline involving the curvature terms. [latex]d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{r^2}d\Omega]^2[/latex] [latex]S\kappa(r)= \begin{cases} R sinr/R &k=+10\\ r &k=0\\ R sinr/R &(k=-1) \end {cases}[/latex]

There is a sign convention with [latex]F=-\frac{GMm}{r^2}[/latex] and subsequenty [latex] U=-\frac{GMm}{r} [/latex] The negative sign in this case indicates gravity as being a downward force towards the centre of mass. Many books however typically drop the minus sign but the convention includes it.

What in the world are you doing with these unit vectors I,j ? Lord only knows what your doing in that box considering you didn't even start with the correct force law. How in the world is the above expression suppose to define total potential energy it makes absolutely no sense. You really need to study how potential energy and kinetic energy is defined http://web.mit.edu/8.01t/www/materials/modules/chapter14.pdf You can't tell me the above works even ignoring the negative mass terms. [latex] a\times b=(a_yb_z-a_zb_y)i+(a_zb_x-a_xb_z)j+(a_xb_y-a_yb_x)k[/latex] the notation of the determinant is, this is a cross product [latex]\begin{bmatrix}i&j&k\\a_x&a_y&a_z\\b_z&b_y&b_z\end{bmatrix}[/latex] lets say you have a vector representing 10 Newtons of force at 30 degrees. [latex] F_y=(10N)sin(30)=5N[/latex], [latex]F_x=(10N)cos(30)=8.7 N[/latex] then [latex]\vec{F}=(8.7N)\hat{I}+(5N)\hat{j}[/latex] see the rules for dot and cross products https://en.wikipedia.org/wiki/Dot_product https://en.wikipedia.org/wiki/Cross_product this equation is just plain wrong no matter how you look at it. You obviously don't know how mass is derived if you have this expression. [latex]E_t=\underbrace{(+E)-(-E)}_{pair creation}=\underbrace{\sum +m_{+c^2}}_{positive mass}+\underbrace{\sum-m_{-c^2}}_{negative mass}+\underbrace{\sum K_{+m}+\sum K_{-m}}_{kinetic}+\underbrace{\sum U=0}_{potential}[/latex] How does this possibly describe the conservation of energy when you have the terms that describe mass being added to the existing mass ? Your adding the potential and kinetic energy twice for total mass even ignoring the negative mass terms. The more I read your article the more errors I find...

The average energy density for the Cosmological constant which is constant despite the volume, is [latex]7.0 *10^{-10}[/latex] joules per cubic meter. This is a positive energy density, (all energy density values are positive....) explain how this gives negative mass ? Its a fairly simple question. Though would be incredibly difficult to answer. ( you would need to rewrite the energy momentum tensor for example). Provide a proper mathematical answer to how negative mass can arise from a positive energy density. NO I am not talking negative effective mass. that is easy using lattice gauge. We are talking about an invariant/rest mass. I wouldn't be surprised if other use my call sign. Though I haven't encountered any others doing so on science based forums. I often encounter others in various video games. Mordred is the evil son of King Arthur so its probably a common used call sign by those familiar with the King Arthur legends. So as you insist the above equations for the constant or proportionality [latex]\kappa=-\frac{8\pi G}{c^4}[/latex] then show how you can take a positive energy density and get a negative mass term. You simply cannot add minus signs arbitrarily where you choose when you involve tensors. Such as those of the Einstein field equations. How do you get the energy momentum stress tensor involved in deriving the constant of proportionality to give a negative mass value from a positive energy density. start with [latex] G^{\alpha\gamma}=\kappa T^{\alpha\gamma}[/latex] The only you can get the below is to suddenly have a negative gravitational constant which is now adding yet another speculation argument to make your model work.

That really is a cop out response to my question. So lets clarify show in mathematical detail how your model can keep the cosmological constant with an expanding volume. I already read your article I don't need you to repeat the details contained within it. What I would like to see is you perform empirical tests that you can predict the expansion rates as a function of redshift comparing the scale factor today to the scale factor then. Then predict the galaxy rotation curves on a graph generated by your model. Not a collection of other works that you think support your model when they have nothing to do with your model. There is a lot of maybe's and ifs in your last post. Her is the problem you fail to recognize BOTH dark matter and dark energy have completely different dynamics, influences and equations of state. Yet you falsely claim to solve both by simply conjecturing a negative mass but have still not addressed an extremely important question. I repeat..... How can you have a positive energy density and have a negative mass ? This is in itself an impossibility. Now keep in mind I haven't even begun to start challenging your model. I am still waiting for a proper answer to the last question....

The topic is a bit difficult to grasp, with pop media's poor coverage being no help. It is often tricky for some to understand why the universe has accelerating expansion. Yet separate the laws of inertia from that descriptive. For some the expression the space is expanding between galaxies isn't sufficient. The difficulty however often lies with first understanding that in order to qualify as an acceleration one must apply a directional force. The OP has already stated he can accept this, one problem solved. The next hurdle is trying to explain how expansion works with the thermodynamic laws and the matter, radiation and cosmological terms. Unfortunately this step is a bit more difficult to grasp. I recommend reading an article developed by someone who visited forums much like this one who later received his PH.D in Philosophies of Cosmology. His target audience is forum members so he kept the article as low math as possible. Pay close attention to the difference between separation distance and those laws of inertia I mentioned. http://tangentspace.info/docs/horizon.pdf :Inflation and the Cosmological Horizon by Brian Powell lets start here then move into the fluid equations.

The volume between galaxy clusters increase this gives us the illusion of motion called recessive velocity via Hubbles law [latex] V={recessive}-H_0D[/latex] however this is what is called an apparent not a true velocity. In essence everything is in motion, gravity tries to contract matter etc but does not always succeed. Now lets look closer at everything is in motion. In expansion there is no inherent direction to that motion it has no net directional component. So atoms do act upon atoms etc, but not in any particular direction. Much like Brownian motion of an ideal gas. How this works is the evolution of the average density of matter. As more matter collapses to form galaxies the average density of matter on the global scale decreases, as this density decreases so does the average gravitational potential. This allows that Brownian type motion greater freedom to expand. The key formula is the critical density formula. This is the point where an expanding universe will stop expanding and then undergoes collapse. (Prior to discovery of the cosmological constant.). Indeed radiation and matter is involved in expansion the cosmological constant describes why this expansion is in essence accelerating rather than slowing down as originally predicted. The FRW metric was rewritten to include lambda. Here is the thing expansion has very little energy per cubic metre via calculations with the critical density formula this equates to roughly [latex] 7.0*10^{-10} [/latex] joules per cubic metre. So locally to an LSS gravity easily overpowers this miniscule influence. Hence Galaxies themselves are unaffected by expansion. Be careful with the balloon or raisin analogy, it primary use is to describe how a homogeneous and isotropic expansion does not change any angles between dots or raisins and without any inherent directional average velocity.

Ok lets try this angle. expansion or contraction of our observable universe involves the distribution of matter, radiation and the cosmological constant. It is in essence a thermodynamic process involving these three factors. Each of these contributors has an equation of state that relates the pressure vs the contributors kinetic energy with the contributors density. For example matter has a low kinetic energy term so its pressure contribution is zero. These contributors evolve over time in that the average density decreases as the volume increases, except for the cosmological constant. Heuristically one can think of pressure as performing the work for expansion however this isn't precisely accurate. However expansion requires no new forces to cause expansion the equations of state for the three categories mentioned determine the contributors to the FRW fluid equation. Here is the equations of state. https://en.wikipedia.org/wiki/Equation_of_state_(cosmology) notice the term [latex] w=\frac{P}{\rho}[/latex] this shows the correlation between the mean density and the pressure relations. Now why do we say space between galaxies expand rather than galaxies are moving. This is a consequence of the Cosmological principle. The universe mass density is a homogeneous and isotropic distribution as well as expansion is a homogeneous and isotropic process. The distribution and dynamics has no preferred direction nor a centre or location. Combined this expresses a uniform distribution. Now in order for inertia change to occur ie acceleration force must be applied. However the distribution around galaxies is uniform, so the force due to pressure surrounding galaxies are also uniform. There is no net force in any direction as the pressure is uniform surrounding the galaxies. Hence those galaxies gain no momentum due to expansion. Instead the volume between the galaxies expand without causing an inertia change to the galaxies themselves. Here is a more complete explanation http://www.astronomy.ohio-state.edu/~dhw/A5682/notes4.pdf

excellent post Marcus Those rules work far better than what you have presented to us. You should study those rules to see where your errors are. Perhaps then you might have a chance presenting something that makes sense under physics instead of the wishy washy haphazard presentation you provided where you have to change every rule under physics. If that is your only response to all the issues we have raised then this thread is going nowhere nor accomplishing anything. In the interest of adding something useful to this thread and to go along with Marcus excellent post who has already covered the stress tensor and curvature aspects. There is under GR and SR a particular relation. The inertial mass and gravitational mass are equivalent. So lets examine this. [latex] m_i=m_g[/latex] Ok so lets use Newtons laws of force for a central potential system for gravity. [latex]F=\frac{GMm}{r^2}[/latex] then set up Newtons second law. [latex]F=\frac{GMm}{r^2}=ma[/latex] so the mass of the object factors out leaving the gravitatonal field potential. [latex]a=\frac{GM}{r^2}[/latex]. What an interesting conversation piece this makes if one things about it. If the mass or composition of the object does not matter to freefall motion, then there is no way to shield gravity. Unlike the electromagnetic force which definitely relies upon the charge of the particle. This is what GR realizes, not only is there no way to shield gravity but it also couples to everything...unlike other fields which only interact with specific particles. Now due to this freefall consequence we can set up a series of geodesics that describe the motion of all particles simply by looking at its momentum compared to c (which is a constant). when the particles velocity equals c the geodesic becomes null (timelike). This means that the gravitational field is a consequence of spacetime curvature not a field into an of itself. One where the absence of gravity is the absence of curvature... Food for thought considering I used a central potential force equation under Newton...to reflect where GR starts to step in. ie adding the time dimension to understand how curvature works. It s a heuristic view I so far as we have not examined parallel transport of two force vectors and the loss of parallel transport defined by tidal forces for a central potential system. (simple to describe take a circle and from the centre look at the degrees, the distance between these two vectors decrease as one approaches CoM. However just to be more complete the geodesic (striaghtline for flat no curvature) is [latex]\frac{d^2x^\mu}{d\tau}^2=0[/latex] were [latex]x^\mu[/latex] has coordinates [latex]( x^0,x^1,X^2,x^3)=(ct,x,y,z)[/latex] the ct gives time unit dimenionality of length where all coordinates have the same unit of length. (See Natural unit). the curved spacetime geodesic is [latex]\frac{d^2x^\alpha}{d\tau^2}+\Gamma^{\alpha}_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}=0[/latex] [latex]G_{\mu\nu}=kT^{\mu\nu}[/latex] can be thought of as gravity=curvature= energy via e=mc^2 where [latex]G_{\mu\nu}[/latex] is the Einstein tensor, k is some proportionality constant [latex]T_{\mu\nu}[/latex] energy momentum tensor.

Ok Dhimoktritis, a lot of people have a problem accepting the Higgs field, however I should point out that the mathematics predicting the Higg's was out long before the discovery and confirmation of the Higg's boson. However lets address the electron first. You feel that the electron is comprised of sub particles yet all experimental evidence shows the electron has no substructure. It is fundamental in so far as it is not made up of even smaller particles. The proton is but the electron isn't. Your charge neto and mass neto I'm sorry does not make any sense. Particularly since all particles are field excitations.

lol hadn't got that far yet

However lets look at the first equation shall we [latex] E=D*m*G,[/latex] well the consequence of this equation setting G at some constant is that as you increase the distance the energy increases. So where does your conservation apply ?

So in order for your model of gravity being a consequence of the laws of inertia, to work you must reinvent those laws to begin with ? Got it. You have your work really cut out for you proving those laws are invalid. Better get cranking out the mathematics including inner and outer products under differential geometry. (Not to mention every physics model out there, as they all apply kinematics lol. So here is the definition of the very word inertia. "Inertia is the resistance of any physical object to any change in its state of motion. This includes changes to the object's speed, direction, or state of rest. Inertia is also defined as the tendency of objects to keep moving in a straight line at a constant velocity." https://en.wikipedia.org/wiki/Inertia You obviously have a completely different definition involved in your article via requiring orbital motion around some central potential. So you then change the meaning of acceleration. Yet your paper only has two formulas that neither one applies force to calculate work done on a central potential system. however I am curious what equation you used to plot your image for path of equilibrium as a plot of the equation you have on wolframalpha is significantly different. (I You can plug values into your variables if you like... won't get the same curve as you have in your article.