caracal

It may not be just anything, but i would need more information. I don't know which total energy would be better. Bakhoum just treats the definite integral differently to get his total energy. Here is reference: https://arxiv.org/pdf/physics/0206061.pdf (page 6). But this is just mathematical trick. This does not prove that his suggestion is the right total energy. Work integral [math] E_{k} = \int_{0}^{s} F dx = \int_{0}^{p}v dp [/math] , is mathematically speaking always a difference between two things : [math] E_{k} = E(s) - E(0) [/math]. You can add same function to E(s) and E(0) and still get same [math] E_{k} [/math]. In other words, you don't know how much energy particle has at rest and how does the total energy vary between 0 and s. Bakhoum suggested that [math] E(s) = m \gamma v^2 = m c^2 (\gamma - 1/\gamma) [/math] and [math] E(0) = m c^2(1-1/\gamma) [/math]. I think Nuclear reactions support [math] E_{tot} = m \gamma c^2 [/math], but Bakhoum states that [math] E_{tot} = m \gamma v^2 [/math] reconcile some equations in relativistic quantum mechanics better. Bakhoum studied Dirac equation and Hydrogen atom, how different total energy change them. But Bakhoum also claims that nuclear fission would have different energy spectrum, and beta decay happens differently. I tend to think that he might be wrong there. I think the energy spectrum and beta decay could be explained well without this new total energy. I suggest two different kind of things to explain energies in nuclear reactions: 1. there are actually two different kind of total energies - energy that participates to relativistic QM and energy that participates to mass to energy conversions or 2. All nuclear reactions and decays just happen to involve high velocities + powerful interaction of the mass that is converting (to particles, kinetic energy or photons) , when [math] m \gamma v^2 [/math] will aprroach [math] m \gamma c^2 [/math]. For example in beta decay that seem to happen in rest - I listed these possibilities that could make option 2. possible - Heisenberg uncertainty principle, quark kinetic energies, potential energies and quark field interactions could make the reaction involve something that is accelerating from low velocities to high velocities - therefore mass to energy conversion look like to be form [math] \Delta E = \Delta mc^2 + \Delta E_{kin} + \Delta E_{binding} + etc [/math]. Actually equation [math] \Delta E = \delta mc^2 [/math] in nuclear reactions could be still right. That would mean that Mass is still the measure of the energy content of matter. But particles just have this different total energy [math] E = (\gamma - 1/\gamma ) mc^2 [/math] That would mean that the energy budget of protons mass or nucleus mass could be slightly different. There is that 'could' because my thinking is not working now. I don't know where de Broglie frequency is needed, if is it needed anywhere. If it is not needed, then different de Broglie frequency does not change any practical calculations. Looks like total energy of particle is used in relativistic quantum mechanics such as quantum field theory. I didn't manage to find information where it is needed exactly. But i understood that it is an assumption that is inserted to the theory. (It is assumed that total energy would be [math] m \gamma c^2 [/math].)

Reconciling this new total energy to quantum mechanical theories is what Bakhoum has done in his writings. I don't go to check them now. But what it seemed to be is that It just clears out some things in the theories of quantum mechanics. The simplest is that there is not anymore super luminal phase velocity. ... Yes i can see what kind of experiment that is. A sample of radioactive material contain an atom nuclei that undergo beta decays and that happens in rest. Energy,momentum,electric charge and spin are conserved. Proton has less mass than neutron and therefore it is energetically possible for neutron to decay to proton, electron and antineutrino. Down quark emits W- boson and changes to up quark. W- boson decays after some time into electron and antineutrino. In fact it does not matter whether the parent is in rest or in constant velocity - the situation would be same, only the sample would be rest in other inertial reference frame. That is because of Relativity principle. What i mean by this suggestion is high relative velocity between interacting particles + powerful interaction. But what happens at the moment when down quark changes to up quark and emit W- boson? That is a moment that takes about [math]N * 10^{-25}s [/math] or less since the lifetime of W boson is already [math] T = 3 * 10^{25}s [/math] And also what happens at the moment when W- boson decays into electron and antineutrino? I don't know. Particles just seem to appear or change to some other particles. There are three reasons why there might be high velocities present: - Quarks already have high kinetic energies - Potential energies - Heisenberg uncertainty principle Two last ones can temporarily give more kinetic energy to particles. This kinetic energy does not show up in the outcome of the reaction.

Looks like he appears in arxiv. I think he is not a bad writer. But he may not have 'proofreaders' or 'moderators' so there may be mistakes. The idea is very solid - just one change and one equation. [math] E_{tot} = m \gamma v^2 = mc^2(\gamma - 1/\gamma) [/math] and that's it. I repeat here a little.: The derivation of equation for kinetic energy [math] E_{kin} = mc^2(\gamma - 1) [/math] is done by looking how much work has to be done for particle in order it to gain velocity v. But after that it is interpreted this equation such that [math] E_{kin} = E_{tot} - E_{0} = m \gamma c^2 - mc^2 [/math] But actually what you have is [math] E_{kin} = (m \gamma c^2 + X ) - (mc^2 + X ) [/math] where this X can be anything - any function of v + any constant. Looks like there is room for alternative interpretation. That would be that [math] E_{tot} = m \gamma v^2 [/math] . This new energy term [math] E_{new} = mc^2(1-1/\gamma) [/math] appears in the equations of quantum mechanics. For example de Broglie frequency [math] f =\frac{m \gamma v^2}{h} [/math] which is now different. Most of Bakhoums writings deals with how quantum mechanics and i think QED changes when there is different total energy. But it seems that nothing practical comes out of it - equations just changes and he claims that they become better. I am not sure how it can be converted to other form of energies, but my own suggestion is that it can in particle decays and nuclear reactions but only when velocity of interacting particles approaches c. Lets go directly to the reactions of beta decay or annihilation. From vertexes like this, without knowing the details of the reaction, i can't say whether particles have high velocities at the moment of reaction itself or not - in the places where the vertices meet or separate. It is probably quantum tunneling what makes these reactions possible. But what happen during the quantum tunneling and during the reaction itself? Can particles have high velocities during quantum tunneling? (pictures) The mean Z and W bosons lifetime is [math] 3 * 10^{-25}s [/math] so i guess beta decay event when W-boson is ejected from proton is even less than this. The reaction itself where particles separate may therefore be very rapid and takes time about that much : [math] 10^{-25}s [/math] ? In order to estimate the event time of reaction: Radius of proton is [math] r = 0.87*10^{-15}m [/math]. If some particle is moving at light speed, it goes across proton in time T : [math] T = 2R/c = \frac{1.74*10^{-15}m}{2.998*10^{8} m/s} = 5.804*10^{-24} s [/math] In such very short time scales the Heisenberg uncertainty principle may also play bigger role: Particles may have for short period of time high kinetic energies such that [math] \Delta E \Delta T = \frac{h}{2 \pi} [/math] If [math] \Delta T = 10^{-25}s [/math] , then [math] \Delta E = \frac{6.582 * 10^{-16}eVs}{10^{-25}s} = 6.582 * 10^{9} eV = 6582 MeV [/math] Maybe particles could have high velocities during very short moment of tunneling and after that lose the velocity, possibly both due to Heisenberg uncertainty principle? In that case they could have also high velocities at the moment of reaction.

I answer here to all of the things you pointed out. There is actually one possibility more: Maybe there are two different kind of energies: Total energy [math] E_{tot} = m \gamma c^2 [/math] that is converted to other forms of energy in nuclear reactions and radioactive decays and "energy that is participating quantum mechanical behavior" [math] E_{qm} = m \gamma v^2 [/math] . It would be this later one that should be inserted to quantum mechanical theories instead of the total energy. Yes, rest energy was derived by Einstein but it is coming from interpretation of the formula of kinetic energy. You can add same term to both of the energies "rest energy" and "total energy" and have still same kinetic energy. citation: https://arxiv.org/abs/physics/0206061 "Fundamental disagreement of wave mechanics with relativity" page 4 about. This is the first of his papers. There he argues that some equations become nicer if you add different total energy. But by thinking that possible rule that "nuclear reactions happen only when some of the particles have high velocities", There are three reasons that comes to mind that could accelerate particle 1. Interactions: EM interaction, strong interaction and weak interaction 2. Quantum tunneling: It may be possible that particles have high velocities during quantum tunneling. 3. Heisenberg's uncertainty principle, the energy time form of it. [math] \Delta E \Delta t = h/2\pi [/math] It allows particle to have temporarily high amount of kinetic energy but that energy is taken back also very soon. Just before electron-positron annihilation the both particles are being accelerated towards each other. I don't know is this enough. E.Bakhoum also writes about this in the article "Dialogue on the principle of mass-energy equivalence"

Hi all. Life is short, and i have another interesting idea regarding special relativity. I want to write this thread independently from the other thread i have written earlier that is about pulsating dots. Don't bring that up here in this thread. There may not be any connection between these ideas. the reason is that whatever kind of dots there may be, they may be extremely small. I have found interesting articles that are written by E.Bakhoum starting from the year 2001 where he suggest different total energy for the particle. I want to bring it up here as i think it is an interesting idea. I don't know if it its true. I don't agree with everything and i have also own thoughts about the issue, so this thread is not just repeating what he has written. But i think it is better to introduce his texts here as a reference. https://arxiv.org/search/physics?searchtype=author&query=Bakhoum%2C+E+G Unfortunately, it looks like that most of the predictions from this new total energy are theoretical and does not have practical meaning. --- The usual comprehension of the subject of energies in special relativity is that the total energy, kinetic energy and momentum are (i don't know if latex works here) [math] E_{tot}= m \gamma c^2 [/math] [math] E_{kin} = m c^2 (\gamma-1) [/math] [math] p = m \gamma v [/math] [math] \gamma = \frac{1}{\sqrt{1-(v/c)^2}} [/math] ,where [math] \gamma [/math] is Lorentz factor and m is mass of the particle in rest. In this notation m is always 'mass in the rest' m=m0 . How this kinetic energy is derived? It is derived from the situation where particle starts to accelerate from rest into velocity v and determining how much work is required to do this. I don't go there. Where does this total energy is derived from? It is actually interpretation from the kinetic energy formula - which is coming from definite integral of work put to a particle that accelerates from the rest to velocity v. It looks like there is some kind of 'rest energy' when particle is at rest. But this definite integral is a difference of two things. You can add same thing to both of the components and still have same difference. It turns out that there is one thing that could be added to both of the components. First note that there is following important mathematical identity: [math] \gamma v^{2} = c^{2}( \gamma - 1/\gamma) [/math] E. Bakhoum suggest that there is such component in both of the terms that is [math] -mc^2(1/\gamma) [/math] . He suggest that the total energy would be different but kinetic energy and momentum would be same. [math] E_{tot} = m \gamma v^2 = m c^2 (\gamma - 1/\gamma )= pv [/math] [math] E_{kin} = m c^2(\gamma -1) [/math] [math] p = m \gamma v [/math] what this new total energy would mean? Particle does not have rest energy. Mass is converted to energy with ratio E=mc^2 only when particles collide in near light velocity. There is no 'relativistic mass' that is energy divided by c^2 , since the another energy component in the total energy besides kinetic energy - is not same but depends on the velocity: [math] E_{2} = mc^2(1-1/\gamma) [/math] instead of [math] E_{2} = mc^2 [/math] Now i ask question that comes clearly to my mind: How about radioactive decays? They are proven to follow formula [math] \Delta E = \Delta m c^2 [/math] ? I think the answer is that particles that participate on radioactive decay are accelerated before their interaction to near light velocity and decay does not happen otherwise. These accelerations may be 'hidden' under the phenomenon of quantum tunneling that is the most prominent quantum mechanical phenomenon in nuclear reactions and decays. Therefore i don't completely agree with E. Bakhoum. I think E. Bakhoum makes mistake when he assumes that particle decays and nuclear decays and reactions can also happen when interacting particles have non-relativistic velocities at the moment of the reaction. He thinks that that there are different kind of energy distributions. I think any reaction or decay can happen only when interacting particles approach the light velocity. If nuclear reactions and radioactive decays happen only when the participating particles are accelerated to high velocity, both total energies give same predictions for nuclear reactions and decays. With this rule added - i think the idea might work. You may not be aware that the total energy is inserted to theory of quantum mechanics from outside of the theory. If you insert this new total energy there, many things changes. For example: 1. phase velocity = group velocity = v 2. de Broglie frequency [math] f = \frac{E_{tot}}{h} = \frac{m \gamma v^{2}}{h} [/math] 3. Energy-momentum relation [math] E^{2} = (\frac{pv^{2}}{c})^{2} + (mv^{2})^{2} [/math] instead of [math] E^{2} = (pc)^{2} + (mc^{2})^{2} [/math] (which may be now unpractical equation) 4. Classical radius of electron [math] r_{e} = \frac{1}{4 \pi \varepsilon_0} \frac{e^{2}}{m_e \gamma v^{2}} [/math] The radius depend on velocity and is infinite at rest. So what does it mean, how is this interpreted then? Is this anymore useful property? I am not sure and i don't go into this here. All of these changes are coming from that you insert E_tot = muv^2 instead of E_tot = muc^2 during the derivation. Also some of the four-vectors changes, such as four-momentum: 5. four-momentum [math] P = (m \gamma v_{x} , m \gamma v_{y} , m \gamma v_{z}, i (\frac{v}{c})m \gamma v) [/math] (i am not sure about this) Also Lagrangian is now different. But i don't go into these. But i believe that everything is consistent. However most of these changes seem to be theoretical aspects without practical outcome. The rest here is my own thinking: ----------- An interesting physical quantity of mc^2/h ................................................................ What i think there might be new insight that the matter wavelength depends directly from the new total energy of the particle: [math] \lambda = \frac{h}{m \gamma v} = \frac{hv}{m \gamma v^2} = \frac{hv}{E_{tot}} [/math] [math] f = \frac{v}{\lambda} = \frac{m \gamma v^2}{h} = \frac{E_tot}{h} [/math] [math] f = \frac{m \gamma v^2}{h} = \frac{mc^2}{h}(\gamma - 1/\gamma) [/math] (note that phase velocity and group velocity are both v with this new total energy) Where is this frequency coming from? Could it be beating between [math] m \gamma c^2 / h [/math] and [math] mc^2(1/\gamma) [/math] ? That would mean that when particle is in rest, there is some kind of frequency that is related to its mass: [math] f_{rest} = \frac{mc^2}{h} [/math] This would be a kind of "rest frequency" of the particle - whatever it is. For some reason it divides to two components when particle start to move. It could be also a maximum of some frequency distribution that changes to distribution with two maximums when particle moves. This is just a schematic picture of how the situation would look like if you look frequency distributions. The spikes can be different kind of: There are two relativistic effects known to happen when particle moves: length contraction and time dilation. Maybe these components are related to these. This constant c^2/h has very high value: [math] \frac{c^2}{h} = 1.356 \cdot 10^{50} [\frac{Hz}{kg}] = 1.366 \cdot 10^{32} [\frac{Hz}{eV}] [/math] This frequency is very high even for lightweight particles such as neutrino which might have maximum mass of 0.120 eV/c^2. Therefore it is very difficult, if not impossible to observe. However the component mc2h(1/γ) could be observable since it goes down to zero when particle approaches light velocity. if [math] m = 0.120 eV/c^{2} [/math] => [math] f_{rest} = 0.120 eV * 1.366 * 10^{32} [Hz/eV] = 1.639 * 10^{31} Hz [/math] An effect of time dilation for solar neutrino on the frequency f_rest: [math] f'/f_{rest} = \sqrt(1-0.99999999995^2) = 10^{-5} => f' = 1.639 * 10^{26} Hz [/math] Wavelength of f': [math] \lambda' = c/f = \frac{2.998*10^{8} m/s}{1.639 * 10^{26} hz} = 1.829 * 10^{-18} m [/math] That is still small - about 1000 times smaller than the radius of proton. What does this frequency then cause? a second kind of interference pattern? I don't know. So if this thinking is right then there is some kind of frequency that is related to particle's mass and that divides to two components, and beating between these two components is somehow responsible for matter wave phenomenom. ----- this is all. I would say that there might be something interesting in this idea of different total energy. Whether it is really the total energy of a particle, i don't know.

Hi all. I came up with this idea that could explain the theory of special relativity and more over the Lorentz covariance and Lorenz transformation. This reminds of one commonly known special relativity "problem" or thought experiment that is about mirrors. A. Lets assume that some physical entity does not consist of rigid constituents, but dynamic sub entities what i call "pulsating dots" or in other words, expanding and contracting spheres. B. Lets assume also that they pulsate between single point and some sphere such a way that the velocity of the frontier is always light velocity c. It may be possible that assumption A and B can be more general and that i don't need to assume that the pulsating dots are spherical but more like areas that change their shape with frontier velocity c. I don't go into what kind of dynamical laws there are for such dots. C. Next lets assume that in that some physical entity consists of these dots and every single point or place in x,y,z coordinates always is inside of some pulsating dot at any given instant of time - if you just zoom in to this point long enough. The pulsating dot around this point can for example be 10^-200 m (zero point 200 decimals) in diameter when it is at its largest. Or even smaller. Eventually, if you just go to enough small length scales and look this point or place, you will find a pulsating dot that covers such point. (D. It may be possible that Pulsating dot can be inside another pulsating dot.) I could also imagine that when you continue zooming the environment of that point long enough, i observe pulsating dot that has center exactly at that point. But this may not be necessary. Now, what happens if this entity is moving at a constant velocity v to some direction if you still demand that the frontier of the pulsating dots still is c? The answer is that 1. The form of the maximum frontier of every dot become ellipsoid that is larger by Lorentz factor 1/sqrt(1-(v/c)^2) 2. The period of the pulsation of every dot becomes longer also by Lorentz factor. But this 2. looks similar general change as relativistic time dilation of moving object in the theory of special relativity. I now attempt to raise question whether i can explain the Lorentz covariance and Lorentz transform assuming that the structure of all physical entities is dynamical this way - they consist of pulsating dots, when you just zoom in long enough. The entities are not static but undergoing continuos changes with frontier velocity c when you zoom in just long enough. Note that while they are not static they can still be stationary, or at least some of the properties of them can be stationary. Note that the entity in the picture above may not be stationary in its overall shape. I don't go here into the theory of quantum mechanics or generally standard model. I end here. What do you think?

This is a recent study that has observed that supermassive black hole growth could be coupled with cosmic expansion. This study seems to have gained much attention and also critizism. news article: https://www.ralspace.stfc.ac.uk/Pages/first-evidence-black-holes-source-of-dark-energy.aspx "Scientists find first evidence that black holes are the source of dark energy" research paper: https://iopscience.iop.org/article/10.3847/2041-8213/acb704 "Observational Evidence for Cosmological Coupling of Black Holes and its Implications for an Astrophysical Source of Dark Energy" (Published 2023 February 15) Shrinking matter theory explains this cosmological coupling by that when matter shrinks, the shrinking depends on proper time and black holes do not shrink since they have infinite time dilation. Therefore black holes appear to grow. There are some mistakes in my previous post regarding derivation of Friedmann equation and modification to stellar dynamics. The task is i think quite simple. i just have to somehow contribute both stellar dynamics and scale factor time evolution that all distances appear to grow by $$ r_(t) = r_0 e^{k(t-t_0)} $$ where k is parameter. There is still a problem with this theory: Earth-sun distance should appear to grow but it is observed to do that only 15cm/year.

About Friedmann equation and distance expansion: I couldn't manage to write correct equations. But i decided to write anyway what i am thinking about. My calculations were wrong. i cant multiply a with apparent a to get something like $$a_{obs} * a_{apparent}*a_{real}$$ this is wrong. I should start with stating that because matter is shrinking, all distances appear to grow by $$ r(t) = r_0 e^{k(t-t_0)} $$ This is the only thing i add to cosmological model at first. K is here a parameter. Besides the ordinary expansion of space, all distances just 'magically' appear to grow like this. (well not all - stars and planets do not expand) Maybe the law for planetary motion when matter is shrinking is then: $$ r_x(t) = r*e^{k(t-t_0)} $$ $$ a(t) = -\frac{GM}{r_x^2}$$ => $$ a(t) = -\frac{GM}{r^2}*e^{-2k(t-t_0)}$$ (?) IF thinking increments the orbital elements of the planet should be continuously changing such that the radial vector increases but velocity vector remains same. For this reason, the planet should gradually migrate away from the sun. I don't know does the eccentricity increase if you start with circular orbit. I know i could do numerical calculations by adding following increments: 1. Planet travels along Keplerian orbit a small increment 2.The radius vector gets increment but the velocity remain same - the planet is now in different orbit that is slightly further from sum but it has still exactly same velocity. I could look that situation on other perspective: as seen by non-shrinking observer. In this perspective the gravitation field of sun becomes different and shrinks. In Newtonian inverse square law gravitation, the observed change in gravitation is a' = La , 0 < L < 1. But what is L(t)? I know that L(t') = 1/LL(t') = e^{k(t-t_0)} and that t' = integral 0->t L(t) dt. Second question is - How to derive Friedmann equation from this new equation of planetary motion? I know that for Newtonian gravitation the first Friedmann equation in flat space becomes (Newtonian derivation of FE): $$ (\frac{\dot a}{a})^2 = \frac{8\pi G}{3}*\rho $$ On the other hand i suggested that in empty space the apparent expansion of scale factor is $$ a(t)= e^{k(t-t_0)} $$ Which is similar than in cosmological constant-only universe Could i from this correspondence just add A*k^2 to RHS of Friedmann equation? $$ (\frac{\dot a}{a})^2 = \frac{8\pi G}{3}*(\rho + Ak^2) $$ where A is certain constant (that i didnt manage to find from internet)? In this case the A*k^2 would be substitute to cosmological constant. I would need some kind of reference to make correct planetary motion equation and Friedmann equation that i failed to find for now. --- About the cause of shrinking (this is more open speculation but i write here some thoughts) I don't know the mechanism or process what would make matter to shrink. But i can describe how it could do that. It could be that if something happens to spacetime, for example the flow of time was accelerating could be the cause. Space or matter could be coupled with or stuck with flow of time that way, that it experiences shrinking when time accelerates. But what makes time to accelerate, i don't know. Maybe empty space differentiates but matter for some reason stay intact and shrinks. Space-time could be entity that is very dynamic in very small scales - i think that is what is needed. It could be full of continuously occurring events in a small scale. But not knowing this i can still describe something about possibility on shrinking of matter. What kind of principles hold. I selected to keep Lorentz covariance, constant light velocity, Heisenberg uncertainty principle, Photon energy law E=hf, De Broglie equation lambda = h/p and constant Planck constant. (Newton didn't know what causes gravitation but he knew how it works. He wrote mathematical description for it - the Newtons law of gravitation F = G M1M2/r^2. Now we know that gravitation field is not a force field, it is curvature of space-time. Newton's gravitation law is weak field approximation. And there is gravitational time dilation what newton didn't know also present in weak gravitational field) --- About curves I think if this theory gives almost similar Friedmann equation and time evolution of scale factor, then i think all other curves behaves also close to same way than in Lamda-CDM model. It will also give almost similar answer to nucleosynthesis.

Thank you for those formulas and equations. I have to think about them for some time. (I hope the following math i represent does not disturb them.) I think this theory produces FLRW metric. But it is 'pseudo' metric or apparent metric, not real metric. (but only if all matter were shrinking similar way everywhere.) Actually the observed metric is combination of pseudo metric and real metric. But the Friedmann equation is slightly different. Since the metric is FLRW, solving the new Friedmann equation is i think quite simple. You just insert or substitute $$ a = a_{obs}/LL(t) = a_{obs}*e^{-k(t-t0)} $$ $$ \frac{da/dt}{a} = H = H_{obs} + k $$ to any kind of Friedmann equation or other equation or formula you have that describe expanding universe. k [1/s] is a parameter. I earlier guessed that k is a little higher than H0 but it could be something else as well. (But you have to assume that all matter is shrinking at equal rate in universe and the amount of black holes neutron stars and relativistic particles is negligible, and that there exist negligible amount of matter that belongs to different preferred scale - it would have different shrinking rate which causes that its gravitational influence and field changes relative to shrinking observer. All these matter would have different density components that depend differently from scale factor than ordinary matter) --- An example to solve Friedmann equation: I don't use here in this example Lamda-CDM what you did above - but old model where is no cosmological constant and i add shrinking of matter to it. Now i am considering situation where space is flat and there is ordinary expansion + gravitation + shrinking matter present in universe, but there is no cosmological constant - shrinking of matter is here a substitute for cosmological constant: $$ ds^2 = c^2dt^2 - a_{obs}^2(t)dr^2 $$ $$ a_{obs} = a_{real}a_{apparent} $$ $$ a_{apparent} = e^{k(t-t_0)} $$ $$ a_{obs0} = a_{real0} = a_{apparent0} = 1 $$ (Here i assume that apparent expansion that is coming from shrinking of matter is exponential) i mark: $$ a_{real} = a $$ in the following: $$ (\frac{\dot{a}}{a})^2 = (\frac{8\pi G}{3})(\frac{\rho_{m0}}{a^3} + \frac{\rho_{rad}}{a^4}) $$ First i get the ordinary Friedmann equation solution to solve a $$\frac{\dot{a} }{a} = H_0\sqrt{\Omega_{m0}(\frac{1}{a^3})+\Omega_{r0}(\frac{1}{a^4})}$$ $$\Rightarrow t = \frac{1}{H_0}\int_{0}^{a}\frac{xdx}{\sqrt{\Omega_{m0}x+\Omega_{r0}}} $$ (formula is taken from www.universeinproblems.com and modified by setting a/a_0 = a and a_0 = 1) Next step is just to insert to equation above the following: $$ a = \frac{a_{obs}}{e^{k(t-t_0)}} $$ With computer it might be quite easy to solve the inverse relation $ a_{obs}(t) $ numerically and then get other quantities and their curves (from 2nd friedmann eq and eq of state (?)) My guess is that all curves are just slightly different from benchmark model that has cosmological constant. (or are they?) --- About differences between this shrinking matter theory and expanding space theory: There are some significant differences in this shrinking matter theory. Some of them are in the range of observations. The differences of this shrinking matter theory from expanding space theory are: 1. Different kind of Friedmann equation - scale factor solution from 1FE is afterwards multiplied by LL-function that could be exponential LL = e^(k(t-t0)) , where k is parameter 2. Apparent expansion is velocity component - not a force - therefore it is not cancelled in solar system and milky way: R = R0*e^(k(t-t0)) 3. Black holes do not shrink 4. Neutron stars shrink less and old neutron stars start to have matter that has shrunk less than ordinary matter. 5. Relativistic matter shrink less 6. Gravitation of ordinary matter has very small delay effect by factor e^{kr/c} 7. There exist matter that prefers different length,time and energy scales. (I invented term 'scale difference' to describe the difference between two kind of matter that prefer different scales: There is matter that has scale difference) 8. Cosmological redshift - the apparent part - is now 'illusion' that comes from changes in shrinking observers meterstick,clock rate and energy units. However there may be ordinary expansion/contraction of universe present. 9. Universe can be contracting and at the same time it can look to be expanding in the viewpoint of shrinking observer. ---- 2,3,4,5 might be just barely in the range of observations 7 it may be possible to weight proton from meteorite sample - it is in range of measurements to see if it has more mass 7 may be difficult to observe from spectral lines since it is difficult to distinguish from doppler effect One problem with 2 : earth - sun distance should increase but it is observed that it does that only 10-15cm/year. ---

About nonlinearity: yes i know linear function is y=ax+b that is a straight line in xy-plane. This theory ,if there were only shrinking of matter present and universe is static, and assuming exponential apparent expansion, is similar than de sitter space and gives the age of CMB to be about 100 Byrs which is 7.5 times longer than current cosmological model gives. *** How i calculate the age of the universe in static + shrinking matter only universe: For age of CMB i solve following equation by assuming k = H0 = 0.0693 [1/Byrs] e^(H0(t'-t_now')) = 1/1100 => (t'-tnow')= Ln[1/1100]/(0.0693) = 100 Byrs (that is 7.54 times greater than in lamda-CDM model.) (I don't know can some observation verify that universe is 13.7 Byrs old, or is that extrapolation based on cosmological benchmark model. ) *** In that moment when universe is young i think some observables can go up but in this case they have gone long way. But they are nonlinear nevertheless. (I don't know which observations actually tells how old CMB and universe are or is it just extrapolation based on the benchmark model. I know that oldest known globular cluster stars are about 12Byrs old based on at least HR-diagram mainsequence cut-off.) If i add some amount of matter to this kind of space, it just pulls universe together and the observed apparent expansion is slower. (In principle gravitation could win the apparent expansion if matter is added to this universe more.) But if there were matter, radiation and 'ordinary expansion' but not cosmological constant and shrinking present in the universe the model gives somewhat similar age than benchmark model does. In this case the shrinking of matter would be substitute to cosmological constant. But the Friedmann equations are slightly different. Therefore i guess just in basis of the similarity with benchmark model - in this case all behavior of observables in early universe can go up when looking backward in time just the same way than in benchmark model. --- About the equations of change: I remind that L describes the factor of the shrinking - it is not directly scale factor (it is that only in the 'shrinking matter' + static universe). In order to get scale factor you need to solve friedmann equation. There is gravitation and may be ordinary expansion present in universe besides of the shrinking of the matter that also have effect on the time evolution of scale factor. LHS is dimensionless and L is dimensionless, also RHS is dimensionless. I think the correct way to write the equations with units marked in each quantity is: Equation for lengths : (s'[m])/(s [m]) = L Equation for duration of any event : (t')/(t) = L Equation for energies: (E'[J])/(E[J]) = 1/L Equation for masses: (m'[kg]/m[kg]) = 1/L and for example: Equation for forces: (F'[kg m/s^2] / F [kg m/s^2]) = 1/L^2 Equation for accelerations: a'[m/s^2]/a[m/s^2] = 1/L Equation for powers: (P'[J/s]/P[J/s]) = 1/L^2 m = meter s = second J = kg m^2/s^2 = Joule kg = kilogram --- About luminosity: You ask how does luminosity of distant star $$ L = F * 4\pi D_{L}^2 $$ appear to shrinking observer? -First In the viewpoint of shrinking observer, distant star has been bigger in past and has had 'expansion' by factor L>1 The light that was radiated by the star red shifted and the flux is time dilated which makes power go down like 1/L^2. 1) the power of that star is 1/L^2 times weaker Also since the meter unit of the observer has shrunk, the distance to that star is greater by factor L. 2) distance to that star is L times greater These two changes 1,2 makes luminosity of distant star to be 1/L^4 times weaker if it is observed by shrinking observer. Now if the universe was static, this 1/L would be exactly red shift: 1/L = proportional red shift ( in static universe, only shrinking of matter present in the universe ) --- About Temperature To get equation of change for temperature, i can look all the equations that connects energy,power,flux or wavelength to temperature. For example for black body radiation, Wien displacement law is lambda = b/T . I know that b must be universal constant and i get T'/T = b/(lambda'/lambda) = 1/L Or i can look Stefan-Boltzmann law flux = rho*T^4. Rho must be universal constant. Therefore i get (T'/T)^4 = flux'/flux = (power'/power)^2/(cross section area'/cross section area)^2 => T'/T = 1/L (Example: If sun would shrink by factor L=0.5, its mass gets 2 times greater, its temperature gets 2 times hotter and its flux becomes 16 times greater and total luminosity 4 times greater. However the nuclear reactions get only 2 times faster - all atom nuclei also change by certain ways: their interacting forces,radiating powers, masses, coulomb walls and binding energies changes also. But actually the star would behave as if light coming from it just have time dilation, redshift and that the star appears to been closer, in other words, the star just appears to behave as if the space was expanding and changing the picture and signal on the way)

About Rydberg emission lines: (This following may be unnecessary to write:) The Rydberg emission lines of hydrogen in distant universe should be similar than in expanding space theory. But now the light had red shift already when it was left from the matter in past. -atomic orbitals of matter have just been bigger in the past by factor (1+z), and the time needed for emission from exited state has been (1+z) times longer. The binding energy of electron would have been 1/(1+z) times less. But the velocity of light and velocity of electron have been exactly same. Planck constant has been exactly same also. it would be difficult to distinguish whether red shift is coming from expansion of space or from matter that has been bigger in the past. It is also difficult to distinguish whether red shift is coming from matter that is bigger or from Doppler effect. --- About non-linearity: You mean non-linearity = rapid spike up in the early universe in the redshift and temperature evolution curve? Both of the following are possible: 1 that there is ordinary expansion of space besides that gravity pulls matter together and matter shrink. 2 There is no ordinary expansion, only gravitation pulls universe back together and matter shrink. If 1 is true then i think the theory have almost similar non-linearity in very early universe than benchmark model. The difference is just slightly different Friedmann equations and their solutions. But if 2 is true i am not sure of that. In this case, non-lireaty can be achieved if the function LL(t') that might be something else than exponential, has been greater than exponential in early universe. But if not, then i think universe and CMB both should have been much older than about 13.7 Billion years. And that could have a problem or disagreement with current observations of early universe. I don't know which observations. If there were no gravitation present and 2 is true, the shrinking matter universe would be similar than de-sitter universe. (That is so if i assume that the factoral rate of apparent expansion is same all the time: dLL(t')/dt' = H_0 LL(t'). This assumption may be wrong.) In both cases 1 and 2 the universe would be asymptotically close to De Sitter universe in distant future. --- About equations of change: I think all equations of the change are correct, but i marked them in unclear manner. for example notation F'/F = 1/L^2 [kg m/s^2] means that when matter and its behavior changes, the ratio (new force/old force) is multiplied by factor 1/L^2 and what is in the bracket [] is just remainder that the dimension of force is kg m/s^2. The brackets is not a factor in equation. I could write the equation like this: new force = old force * 1/L^2 [remind that unit of the force is kg m/s^2] But it is a little more convenient to divide both sides by old force: new force/old force = 1/L^2 How do i calculate these equations? I start with these $t'/t = L$ $s'/s = L$ $m'/m = E'/E = 1/L$ I can end up to equation for force starting with: F'/F = (m'/m)(v'/v)/(t'/t) and i know already that m'/m = 1/L v'/v = 1 t'/t = L => F'/F = 1/L^2 and for example equation of change for power i start with P'/P = (E'/E)/(t'/t) and knowing E'/E = 1/L t'/t = L gives me that P'/P = 1/L^2 and equation of change for acceleration a'/a = (v'/v)/(t'/t) and knowing v'/v = 1 t'/t = 1 it gives => a'/a = 1/L ----- Actually (i forgot this statement earlier) if matter is shrinking continuously and the time is accelerating (duration of all events is all the time getting shorter) then equation t'/t = L holds only when t' and t are very short. The equation for elapsing time in this situation would be t(t') = definite integral(t1->t2) (1/L(t')) dt' (But who measures this kind of time t if all observers are shrinking?) ----- I earlier noted already that when i talk about apparent expansion of space observed by shrinking observer, i use function LL(t') that is not same as L. LL is a function that describes factoral change in photon wavelength or expansion of space if it was solely coming from the shrinking of matter - if they were observed by shrinking observer. LL is also expressed as function of the elapsing time of shrinking observer. I could mark this time as t'. LL(t') = e^[k(t'-t0')] LL(t0) = 1 If there were no gravity, a(t') would be just same as LL(t'). In this situation a(t') behaves exactly like in de sitter space. ---

Thank you for detailed answer! i am able to answer to 1,2,3 and maybe 4 but i don't understand 5. Actually the last paper considers black hole growth coupling with cosmic expansion and this shrinking matter theory also predicts that black holes should appear to grow when they are observed by shrinking observer. The following is quite lengthy and slightly repetitious. I think it is experimentally well established that: 1) If we do local observations, we see no change in standard model or gravitation over cosmological times. 2) If we do cosmological observations, we see -time dilation in signal -red shift in light -loss of photon momentum and energy -expansion of the space Also observations suggest that the velocity of light is constant, it does not accelerate. Light passes 299 800 km every second all the time. I basicly consider two things in this shrinking matter theory: A) I consider that the standard model and gravitation has a shift into smaller length scale as a whole. That would cause that fine structure constant and any of the coupling constants do not change over time if we measure things nearby us. B) And i consider that certain kind of changes in matter, standard model and gravitation, not only shrinking but multiple changes simultaneously, produces same cosmological observations that i mentioned above. But all these observations are now caused by that matter has been bigger and behaved differently in past - not by stretching the signal on the way. Therefore it only looks like that the space is expanding. these A) and B) gives that the effect of coupling constants remain the same except that the signal coming from distant place appear to have red shift and time dilation and space appear to expand. (and except that BH, neutron stars and relativistic matter behave differently) (It is possible that i did thinking error and The statement B) would not hold after all) Why these kind of things happens in matter? i don't know, but they might have some common cause. But what kind of changes are then happening in the behavior of matter? By demanding -that special relativity (for example factoral time dilation and factoral length contraction) holds -light velocity c = constant, -photon equation E = hf and -Heisenberg's uncertainty principle dp*dx = h/2pi with h being universal constant and demanding that all these three valid all the time and do not change, at least three changes should take place: 1)Matter shrinks (note that free photons do not shrink since they travel at light velocity) 2)time of the matter accelerates such a way that observation of light velocity stay constant 3)All energies, momentum and masses increase such a way that observation of photon coming from distant place appear to lose energy and momentum s'/s = L [m] all lengths t'/t = L - duration of all events in the behavior of matter E'/E = 1/L [J] energy of all events... p'/p = 1/L [kg m/s] momentum of any object or particle m'/m = 1/L [kg] mass of any object or particle ,where 0 < L < 1 is factor of the change (What do i mean by 'accelerating time of matter'? it is the shortening of duration of every event happening in matter, for example spin rate of the nucleus of deuteron and half life of carbon-14 and so on.) By doing dimensional analysis from the three statements 1),2),3) above, i also get following changes in dynamics of matter: f'/f = 1/L frequency [1/s] v'/v = 1 velocity [m/s] a'/a = 1/L acceleration [m/s^2] F'/F = 1/L^2 strength of local forces [kg m/s^2] P'/P = 1/L^2 power [J/s] T'/T = 1/L temperature [K] f'/f = 1/L^4 luminosity [W/m^2] where 0 < L < 1 (Note that any field like force field or potential would also shrink into smaller size) When this kind of change in matter happens it gives theory that passes so called Tolman test states that in expanding universe, surface brightness should be inverse proportional to the fourth power of cosmological redshift. This model gives this result. (except keeping in mind that compact objects shrink more slowly and black holes do not shrink at all) f = A/(1+z)^4 where A is some constant These kind of changes also produces Robertson Walker Metrics in flat space - which is now only 'apparent' metric, not real metric. ds^2 = c^2dt^2 - a(t)dr^2 There are important differences from expanding space theory. The three differences are that 1) expansion of distances is not a force - it is 'apparent growth' that can be seen as velocity vector. Also the Friedmann equations are different for the same reason. And 2) compact,relativistic matter and matter in strong gravitation field shrinks more slowly and black holes do not shrink. It is also now possible that 3) there exist matter that belong to different scale or have shrunk differently in the past. The time dependency of scale factor a(t) IS NOT LINEAR in this theory. IF cosmological principle holds, the observed change in distances if gravitys effect is reduced and photon wavelength are all the time same during same time interval of time of the shrinking observer. This would give exponential law: dL/dt = k*LL LL(t) = e^[k(t-t0)] = a(t) Note that L is now a function (of time of a shrinking observer t - not constant time t) and LL(t) > 1. it is not the same than L in the equations above. I used to mark this function as LL(t). But because gravity slightly pulls matter together, the real time evolution of scale factor is only nearly exponential, but it will eventually come close to exponential since the effect of gravity becomes all the time smaller. (this may be wrong pair of equations) 1)a(t) = a_f * e^[k(t-t0)] 2)((da_f/dt)/a_f)^2 = (3piG/8c^2)* rho(t) (Note also that the equation 2) does not need cosmological constant to get accelerating apparent expansion.) The rough linear approximation a(t) = 1 + H0(t-t0) i use only to get value for k that is close to H0 and to estimate how much distances should grow in solar system dr(t)/dt = (roughly) r(t0)*H0 This equation (that radius vector r increases or 'appear to increase' should be taken then account in numerical simulations. It does make escape velocity smaller like you said. --- I think nucleosynthesis and recombination are both possible in this shrinking matter theory and gives same results. Also inflation is possible - now inflation may be a period where matter shrinks very fast during short period of time. --- I hope this clarifies what i think of the possibility that certain kind of changes in matter, standard model and gravitation - simultaneously could give almost same cosmological observations than expanding space gives. .

I found one research article that is related to this topic. news article: https://www.hawaii.edu/news/2021/11/03/expansion-of-universe-black-hole-growth/ research article: https://arxiv.org/abs/2109.08146 "Cosmologically coupled compact objects: a single parameter model for LIGO--Virgo mass and redshift distributions" This study tests hypothesis that Black hole growth could be coupled with cosmic expansion. It does not say anything about shrinking matter theory. But it tries to have meaningful results for LIGO and Virgo data by hypothesing that black holes grow when space is expanding by following single parameter equation and then simulating stellar evolution: m(a)=m0(aai)k ,where a_i <= a and k is a parameter of the coupling. quote: "To investigate this hypothesis, the researchers simulated the birth, life, and death of millions of pairs of large stars. Any pairs where both stars died to form black holes were then linked to the size of the universe, starting at the time of their death. As the universe continued to grow, the masses of these black holes grew as they spiraled toward each other. The result was not only more massive black holes when they merged, but also many more mergers. When the researchers compared the LIGO--Virgo data to their predictions, they agreed reasonably well. “I have to say I didn't know what to think at first,'' said research co-author and University of Michigan Professor Gregory Tarlé. “It was a such a simple idea, I was surprised it worked so well." ... In the shrinking matter theory where black holes do not shrink, close this kind of dependency exists but the equation is not exactly this. it is in dust only -no lambda universe same with k=1, but in matter+radiation-no lambda universe it would be: m(a)=m0∗LL(a)=m0(aai)∗s(a) ,where k=1 and s(a) is a function that can be obtained by solving first Friedmann equation. s(a_i) is propably slightly more than 1, maybe by few percents - Shrinking of matter is more dominant than gravitation. By the way it means that universe is contracting, but because matter is shrinking even more, the shrinking observer sees that space appear to expand. I am not sure of this new kind of friedmann equation. It can be expressed best with two equations: atrue=a⋅exp(k(t−t0)) (a˙a)2=(3πG8)(ρm,0a3+ρr,0a4) ,where [math]a_{true}[/math] is 'true' scale factor. The exponential comes from cosmological principle: i assume that the observed redshift/time unit that is coming from shrinking of matter observed by shrinking observer is same at every moment. LL(t)=exp[1+k(t−t0)] K is here a parameter that is likely close to Hubble constant, but is slightly greater than that. There are lacking new terms in that equation: compact matter, relativistic matter and matter that belong to different scale. All of these shrink at different speed than ordinary matter. For example the density of black holes behaves almost like [math]\frac{\rho}{a^2}. Also small amount of matter becomes black holes and neutron stars or relativistic matter over time. Above i considered only flat universe. When matter shrinks, shrinking observer would measure that the curvature radius appears to increase - as all distances appear to grow. This would mean that since CMB the curvature radius has increased by factor of 1000 or was it 2000. What happens to Kerr black hole ergosphere? It has time dilation factor by few percents. Does it stay unchanged or does it shrink slightly? That goes to question what is actually shrinking when matter shrinks. Is it only matter that shrinks or can ergosphere of rotating BH shrink too? This is open question to me. ---

Hi all, I wanted to write new thread about this topic just because there is just too much text in the previous one. I also change the perspective and look directly to a important question: how cosmological observations constrains shrinking matter theories. I was thinking if this is repetition but i wasn't sure. If many of you think this is repetition maybe i then ask you to delete this thread. So Here in this topic i look what kind of observational constrains there are for shrinking matter theory. First why do i still bother thinking shrinking matter theory? --------- MRS Hawkings have published (2010) a result that quasars don't exhibit effects of time dilation. (https://arxiv.org/abs/1004.1824) This result is in fact the only good reason i still think shrinking matter model could be real. This result may be explained by that supermassive black holes do not shrink when ordinary matter shrinks. Hawking has proposed some other explanations at the end of the paper. I don't actually know how astronomers think of this study and do they think it is for example reliable. IF there is just one thing i ask you to discuss about it is this study. How to explain this kind of result? The rest could actually be something i have already discussed with other members in this forum. --------- I try to be as brief as possible. I don't concider CMD - lambda model in different coordinates - that would just be exactly the same theory than CMD lambda model but in different kind of coordinates where matter shrinks relative to the coordinates. There are two things what i concider that are different: 1)Shrinking depends on proper time (and of course the 'scale' of the matter) 2)Apparent expansion of space is only apparent and is observational illusion caused by shrinking of matter. Gravitation however does pull matter together On the scalability argument ---------- One main argument against shrinking matter theory was that standard model and gravitation should start to look very different if matter shrinks. However i think it is still possible that the whole standard model gravitation experiences just right kind of shift into smaller length scale when matter shrinks that it looks like in our perspective that nothing has changed in standard model or behaviour of matter. I don't however know what would be the underlying cause for this kind of shift. Another argument against SMT is that standard model should look different in distant object in cosmological distances that has been in the past. But actually the standard model can be different, if the behavior of matter looks exactly as if it has only changes that are equal to cosmological observations in expanding space model: Light signal coming from distant object has time dilation, red shift and has it looks like that matter has been closer together in the past. That observation of cosmic photons actually demands three things - not only length but also the time rate and all energies of matter should change in the shrinking matter. Only that way the cosmic photon appears to as as it has just to have lost some of its momentum, while the real explanation of this would be that the photon already in the moment of emission has these properties. This is because the energy, wavelength and frequency of photon follows equation E = hf ,where h is universal constant. Why shrinking matter theory is worth thinking? There are 3 reasons for that: 1)If the shrinking of matter depends on proper time, then black holes do not shrink and neutron stars shrink slower. This could suite together with MRS Hawkings result that quasars don't exhibit effects of time dilation. (https://arxiv.org/abs/1004.1824) This result is in fact the ONLY reason i still think shrinking matter model could be real. 2)The exiting part of the theory is that there should exist matter that has shrunk differently in the past and therefore has different 'preferred length scale'. This matter is in places where matter has have relativistic time dilation for long time. However the differences between matter in different places in solar system are very small, just barely observable. For example differences of proton or electron in meteorite samples. (This kind of matter could even be candidate for dark matter. Universe may for example contain two or more generations of matter that lies in different scales, for example protons that belongs to 200 times larger length scale than ordinary protons ) 3)There are two measurable modifications to stellar and interstellar/galactic dynamics a) - "Distance expansion" R1/R0 = 1 + k(t1-t0) = 1 + 6.93*10^-11 1/year [t1-t0] (about same as hubble constant) b) - delay effect correction to gravity g = -(1 + kc/r)(GM/r^2) (which is very small if the rate of shrinking is small) Distance expansion is not a force - therefore it is not cancelled by gravitation. "Distance expansion" is apparent growth of otherwise fixed distances if they are measured by shrinking observer. The delay effect comes from that matter has been different in the past - including its gravitation field. But this delay effect is very small if the shrinking of matter is slow. Observational problems ------- There is a Problem with "distance expansion" in solar system that could falsify shrinking matter theory: -Because of distance expansion, moon should receed from earth at least 2.6cm/year and earth should preceed from sun 10.4m/year. The observed values are 3.8cm/year and 10.4cm/year, later is 100 times too small and the former may be also too big but i dont know how much of this 3.8cm is coming from tidal decay. Is there a process that could cancel this 10.4m/year out? for example the influence of other planets? If there are no such processes, this would falsify shrinking matter theories. On the observational constrains for shrinking matter theory: -------------- What shrinking matter theory should do is the following -Standard model and gravitation is measured not to change over cosmological time when doing local measurements -Velocity of light is constant in empty space and space follows Lorentz covariance -When observing distant galaxies in the past the light signal should have 1)Redshift 2)Time dilation 3)The distant galaxy looks as if it has been closer in the past 4)Photons lose momentum Like i already wrote above, one main argument against shrinking matter theory is that standard model and gravitation should look very different if matter shrinks. However it is still possible that the whole standard model gravitation experiences just right kind of shift into smaller length scale when matter shrinks. Another argument against SMT is that standard model should look different in distant object in cosmological distances that has been in the past. But actually the standard model can be different, if the behavior of matter looks exactly as if it has only changes that are equal to 1)-4) and nothing else. 1),2) and c=constant requires that the time of the matter must accelerate by inverse of its shrinking factor 4) requires, if the law E = hf still holds and h is universal constant, that the energy and momentum meter we use, should shift that way that all energies, momentums and masses increases by same factor than the time accelerates. To put these into simple math: length unit: l_new = L * l_old time unit : t_new = L * t_old mass unit: m_new = (1/L) m_old Energy unit: E_new = (1/L) E_old Where L = 1 + k(t_new-t_old) , which is only linear approximation that fits to Hubble's law. K would be actually slightly greater than Hubble constant if you count that gravitation is sligthly pulling matter together. this k > Hubble Constant, propably by few percents?. For example the changes in photon that is emitted by same matter in the past and now are exactly same than effects in the picture in just looking or photon in cosmological distances in expanding space model. The change in all energies and masses is really DEMANDED by the observational fact that cosmic photon appears to lose momentum and energy. (Other possibility could be planck constant h is not constant in equation E = hf but i dont concider it here. I think it is an observed fact that h is universal constant also for cosmic photons.) About dynamics ------------ Next question is, how do dynamical properties of matter change? If newtons 2nd law is universal law, then F = ma for inner interactions => F_new/F_old = (1/L^2) This does not apply to cross interactions (interaction between two differently shrunk particles) The equation for cross interactions could be F_cross/F_old = 1/(L1*L2) (or is it?) Since power P = E/t => P_new/P_old = (1/L^2) Since acceleration is a = v/t => a_new/a_old = 1/L About Friedmann equations ----------------------------- What is the time dependency of scale factor? There are now two phenomena that has effect on scale factor: - Distance expansion - Gravitation And it is possible that there is no "ordinary expansion" of the space present. In empty space or space with small amount of dust there is only distance expansion. But how does distance expansion vary over cosmological time scales? A good guess could be that it is exponential in the viewpoint of shrinking observer: a(t) = exp(k(t-t0)) a(t0) = 1 This looks similar than dust + cosmological constant - universe What is the equation for a(t) if also gravitation is taken into account? I don't know... ---- End

I see. This is quite interesting. Space fountain and space tethers. After some search I found one interesting Science fiction blog text about space tethers "ToughSF - Space Tethers: Stringing up the Solar System "- http://toughsf.blogspot.com/2020/07/tethers-all-way.html And two science fiction texts about space fountain: "R.L.Forward indistinguishable from magic - chapter 4 ": - https://www.baen.com/Chapters/0671876864/0671876864___4.htm (this text is not reader friendly) ( is this copyrighted?) and "Orions arm universe project - encyclopaedia galactica - Technologies used to support large scale structures using momentum" - https://www.orionsarm.com/eg-article/47e1bb1fc898c (Which i have only read by glimpse this far.) I Give here yet illustration of two different kind of installations of this "free fall cable". Left one is a "Wünchenberg's chair" (this is not a real name). It is an installation of computer controlled chain loop, that either wobbles around or looks to stand still without moving while there is no any mechanical support in the system. It looks like a chair or table, that is why the name. Wünchenberg is just imaginary name of a mathematician. This kind of installation could well be a home decoration or a part of public statue or monument. Red-colored part of the chain is raising up, and the blue-colored part of the chain is falling down. Black line in the central axis is trajectory of the mass-center. The "mass-center" of the looped chain would follow upward free fall trajectory (drawn in black dotted line) if there were no air resistance. The tangential velocity of the chain is constant everywhere. If properly installed, it may be ? possible to achieve condition where the shape of the chain appear to stay still and only close look of the chain reveal that it is actually moving all the time. While not in this state, the chain will wobble around but stay inside some closed conical surface all the time. The conic central diameter may in fact not grow linearly upward. In principle the chain could consist of two chains that are moving into opposite directions. The system manages not to mess up only with constant computer control. The chain element joints contain actuators that can bend the angle between the elements on the fly and there is computer program that control the movement of these actuators. Right one is another kind of illustration where the chain mass center follows parabolic curve. It curls into helical shape like a old phone wire on the top of the curve to avoid gathering up when the free fall velocity slows down - this is also computer controlled mechanism.