Lucious Posted January 19, 2015 Posted January 19, 2015 Assuming a trivial sphere model of the universe where there is a unique center (so long as there is equally a unique center in all parts of the interior of the sphere) then there is using Newtons law a gravitational equilibrium with all other gravitating systems [latex]G \equiv \frac{Rc^2}{m^{N}}[/latex] of all [latex]N-states[/latex] of the various systems of mass [latex]m^{N}[/latex] We can take from any center of origion, [latex]x=1[/latex] for simplicity and is effected by how everything else is moving (These would be physical subsystems in set algebra where the universe is a set [latex]\mathcal{S} \in \mathcal{U}[/latex] where [latex]\mathcal{S}[/latex] are the systems, or subsystems and [latex]\mathcal{U}[/latex] is the universe). When [latex]x=1[/latex] its distance from the center at [latex]x=0[/latex] is by definition the scale factor [latex]a[/latex]. It's kinetic energy [latex]\frac{1}{2}\dot{a}^2[/latex] of outward motion is using a standard equation of cosmology is [latex]\frac{1}{2}\dot{a}^2 - \frac{4 \pi}{3} a^3 \rho G[/latex] for our sphere model. We know it is in an equilibrium so we can replace [latex]G[/latex] with [latex]\frac{1}{2}\dot{a}^2 - \frac{4 \pi}{3} a^3 \rho \frac{Rc^2}{m^{N}}[/latex] Now divide by the distance which is just [latex]a[/latex] [latex]\frac{1}{2}\dot{a}^2 - \frac{4 \pi}{3} \frac{a^3 \rho (\frac{Rc^2}{m^{N}})}{a}[/latex] Using a quotation from Susskind, this equation under the form of Newton is independant of time. In other words its a timeless model - but it does describe change. It's also constant which we will denote as [latex]\frac{1}{2}\dot{a}^2 - \frac{4 \pi}{3} a^2 \rho \frac{Rc^2}{m^{N}} = \pm k[/latex] We can further mutipy it by 2 to remove the half [latex]\dot{a}^2 - \frac{8 \pi}{3} a^2 \rho \frac{Rc^2}{m^{N}} = \pm k[/latex] To obtain the Hubble constant, we further divide it by [latex]a^2[/latex] [latex](\frac{\dot{a}}{a})^2 = \frac{8 \pi}{3} \rho \frac{Rc^2}{m^{N}} - \frac{k}{a^2}[/latex] Which leads to the well-known F-R equation of cosmology except in the condition of explicit gravitational equilibrium. It almost seems absurd to think of a universe which is in a gravitational equilibrium when we observe inequality of gravitational forces between different bodies of masses. So what could any good come from thinking in this completely strange way? We must remember, no location inside the sphere can be unique, meaning that every point inside of space is the center of the universe. Because of this complication of fact according to modern interpretation, the equilibrium exists as a network of non-trivial points inside the universe. This may explain the almost uniform observations of what has come to be known as the ''flatness problem.'' In each point in space, there is an [latex]x = 0[/latex] with a scale factor [latex]a[/latex] to a point [latex]x=1[/latex] in some three dimensional hypersphere - but in the model of there also being a location [latex](x', x'', x'''... x^{N} = 0)[/latex] points in space that are all metrically connected to every scale factor [latex](a',a'',a'''... a^N)[/latex] separating everything object of mass [latex]m^N[/latex] in space: You could say that when you calculate all the forces it conserves to zero. In other words, every action and negative reaction which has happened in space holistically comes to a big zero - a form of conservation in its own right. So it is possible to have a Robertson-Friedman equation for a gravitational equilibrium between the masses of interactions, so long as we don't care when they happen. Perhaps the relativity of simultaneity backs this up - perhaps no one can truly agree when events actually happen, so there is an obscurity when you think about the interactions in the first.
Rolando Posted January 19, 2015 Posted January 19, 2015 Could you please explain what you mean by m^N to those who are not familiar with set algebra?
Lucious Posted January 19, 2015 Author Posted January 19, 2015 (edited) Could you please explain what you mean by m^N to those who are not familiar with set algebra? Hi! That isn't a set exactly, it is a notation used for any mass to an N-power, or it can be a sum as shown [latex]\sum_{i}^{N} m_i[/latex] - The [latex]N[/latex] just represents any integer which may count the systems inside the universe, this is where set theory comes into it. All the systems inside the universe can be simply denoted as [latex]\mathcal{S}[/latex] where it is the ''set of subsystems.'' This is given ''in'' a system, denoted by [latex]\in \mathcal{U}[/latex] where the last symbol represents the whole, or holistic Newtonian system in which all subsystems gravitationally interact. So the question of when you add all causal dynamics in the universe should equally conserve the overall equilibrium, even if local distortions don't agree with this example. Just as we when we look in all directions of the universe, there is no center. To resolve this, all locations in the universe denoted [latex]\mathcal{U}[/latex] are the center and equally the origin of big bang. Or at least, this is what most cosmologists interpret it as. You could actually do more complex math using causal subsystems, but Fotini Markopoulou has written most of the work concerning what the universe looks like from within. (edited) Just to note, the fact the FR-equation is derived independent of time may reflect its more complex quantized form using the General Relativistic equations which lead to the famous Wheeler de witt equation, which has been interpreted to mean the universe is also time independent. It's obtained by making quantized path integrals on the Einstein field equations which yield a very simple, timeless Schrodinger equation [latex]\psi|H> = 0[/latex] As you can see it is a simplified version of the timeless wavefunction, with no time derivative on the RHS. This may have significance with the timeless derivation of FR-equation of cosmological evolution scales. [latex](\frac{\dot{a}}{a})^2 = \frac{8 \pi}{3} \rho \frac{Rc^2}{m^{N}} - \frac{k}{a^2}[/latex] Let's put everything to the RHS [latex]\frac{8 \pi}{3} \rho \frac{Rc^2}{m^{N}} - \frac{k}{a^2} = (\frac{\dot{a}}{a})^2[/latex] We introduce a universal wavefunction which is dependent on only the generalized coordinate [latex]q[/latex] [latex](\frac{8 \pi}{3} \rho \frac{Rc^2}{m^{N}} - \frac{k}{a^2})\psi(q) = (\frac{\dot{a}}{a})^2 \psi(q)[/latex] Keeping this equation in mind, it is totally time independent and if we really wanted to make an evolution in the equation, all we would really need to do is use the variational principle using another state vector which dictate a variational change in the generalized position [latex]\delta q[/latex] (though a small one). A spacially isortopic metric in the line element can be given as [latex]ds^2 = -A(dx^{2}{1} + dx^{2}{2} + dx^{2}{3}) + Bdx^{2}_{4}[/latex] ref 1. (We will come back to this in a moment) For the small quantities we took in [latex]\delta q[/latex] the displacement of the positions of [all] systems in the universe - again, retaining the Newtonian holistic connection between all gravitating bodies. As the scale factor increases according to the equations so must a gravitational redshift between all systems inside the universe. This can be calculated by taking a new metric I derived a while back as an expression form [latex](1 - 2\frac{Gm}{mc^2} \frac{M}{r} + \frac{GQ^2}{c^4 R^2})^2[/latex] If the overall charge of the universe is zero, then the metric charge will also be conserved [latex]Q=0[/latex] reducing it to the more simpler [latex](1 - 2\frac{Gm}{mc^2} \frac{M}{r})[/latex] Now going back to the line element we have [latex]ds^2 = -A(dx^{2}{1} + dx^{2}{2} + dx^{2}{3}) + Bdx^{2}_{4}[/latex] Einstein showed that this equation represents a definition of inertia [latex]\frac{MA}{\sqrt{B}}[/latex] He wanted this inertia to tend to zero [latex]\frac{MA}{\sqrt{B}} \rightarrow 0[/latex] at a spacial infinity. What I noticed is that this couldn't be true under the accepted Newtonian interaction of inertial particles which are continuously connected by the force of gravity. This meant to me at least, that [latex]\frac{MA}{\sqrt{B}} \rightarrow 0[/latex] couldn't be plausible in the classical sense if Newtons first principle of gravitation is wrong. I doubt many could dispute this. ref. 1 http://www.platonia.com/barbour_modern_cosmology.pdf I'll come back to this soon to finish it. Later I am going to show how Einsteins definition of inertia was wrong, but Newtonian physics had it within tangential relationship to the Machian Universe which Einstein based at least 70% of his work upon. The contradiction was that that Einstein wanted to have inertia dissipate to zero at spatial infinity, included to my objection above, I don't believe that infinity is even a real mathematical object of physical significance. I believe that singularities are in fact a direct non-negotiable breakdown of the physics at hand. Before I go into the depth of that math, I have some stuff I want you to read: It is work by myself explaining Machian physics which may be interesting to people here - but on top of this, I show how General Relativity can allow a holistic model in Machian terms according to Einstein's very own principles. This may take about several reads to understand; The problems are numerous and in this essay, I must tackle these issues efficiently. First of all, we need to define what time is. Newton, the father of classical physics defined time as: Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time ...'' And concerning absolute space, he said Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies: and which is vulgarly taken for immovable space ... Absolute motion is the translation of a body from one absolute place into another: and relative motion, the translation from one relative place into another. Thinking time flows equably without any reference to anything external, means that time flows relative to the human being. This ''flow of time'' has been in physics literature for a very long time and the conclusions are startling. And though there is nothing external which we can say is in any reference to time, means that any mathematical we use in our equations, assume immediately without any evidence that there is an external time present without the observer. We of course, do not believe in any absolute space or time in the context Newton implied them. In Newton's picture, the universe was a pure vacuum with no quantum activity. In the quantum picture, time doesn't even have a flow. In relativity, there is no true ''flow to time'' either, because fundamental to it is the relativity of simultaneity which argues all events actually happen side by side... time itself was an illusion. Now, this external, intrinsic flow to the observer, which seems to extend from past to future... isn't actually the way modern scientists think about time at all [1] - If time really existed in the quantum sense it would instead be sharp beginnings and ends, fleeting moments of existence which are not continuous. This is the quantum picture of time. The relativity of space says it is continuous. Minkowski (Einstein's teacher) extended his relativity principle to say space-time is continuous also as a fourth dimension of space. (1) The quantum picture of time at first differed with the relativity of time. But sooner or later, the mistakes would show.... Minkowski space was probably based on faulty principles including the idea that time is continuous or discrete... it turns out that time probably doesn't exist at all (this still leaves open the question of whether space is fundamental, maybe something I will tackle in a future paper) (2). Minkowski when he extended Einstein's idea's made time as a fundamental property of the universe. When Paul Dirac was asked about he thought, he said he was ''inclined to disagree to think that four dimensional unity was fundamental.'' Einstein had a bit of a schizophrenic nature on time concerning ''some'' comments over his career, but I think it's clear Einstein never considered time fundamental within physics. He was definitely more than aware that his very own GR which is hailed to this day through experimentation was essentially timeless. After Minkowski had published his theory on four dimensional space, Einstein was known later in his years to comment that the mathematicians ''butchered his theory'' as he knew it. Before Minkowski's idea shot off, Einstein's theory was still basic in how they were interpreted and Machian relativity almost vanished from the minds of physicists altogether; though, not many know, but Einstein was heavily influenced by the idea's of Mach, one of the founding fathers of relativity. Actually Machian theory was more truer in the general relativistic sense Einstein intended his theory than the four-dimensional case proposed by his teacher Minkowski. To be relativistic, you need to talk about each point in space; but points in space are not really physical, only interactions are. This happens because General Relativity is manifestly Covariant which makes sure that the laws of physics remain the same in every coordinate frame. This allows physicists to use diffeomorphism invariance, a beautiful mathematical consequence in which the universe isn't desccribed by a cosmological ... instead it arises as symmetry of the motion of the theory. In other words, it's measuring change using diffeomorphisms to describe ''time without time.'' Machian relativity doesn't use time either to describe the evolution of systems, instead as he famously put it ''we arrive at the abstraction of time, from the changes of things.'' Interestingly, Leibniz also made a similar argument, believing that space in itself made no sense, unless speaking about locations. Time in itself made no sense, unless inferred from the relative movement of bodies... But Mach's idea's where already bubbling, he wanted to advocate a holistic relativistic model ''When, accordingly, we say that a body preserves unchanged its direction and velocity in space, our assertion is nothing more or less than an abbreviated reference to the entire universe.'' Gravitation and Inertia, p. 387 On this relativistic ''space-scape'' I call it, changes of physical systems in this universe are our definition of time, without it, if systems did not change, there would be no way to define time at all. Here is one way to articulate the problem, if change is the true definition of time and change happens in space then time isn't space itself, it's a measure of disorder in space. The closest thing we have to the definition of time as we understand it, is entropy and this thermodynamic law gives rise the cosmological arrow of time as it is known using popular science buzzwords. (Later I will provide a ''toy'' model towards a theory of everything using entropic gravity, the entropy part being our definition of the measure of change and gravity being described from the entropic laws). To believe there is an order to things which gives rise to a ''direction'' in time is a very wrong picture to adopt within relativity because of simultaneity. The relativity of simultaneity forbids us from absolutely saying every event in the universe can ever be agreed upon. Because of this, it argues that events may as well happen side-by-side. Time appears between moving observers which dilates when thing happen in different frame of references; in a funny sense, it is time dilation that gives rise to time [anime *]. Also, there is no place in the universe we can point to and say ''it all happened there.'' In fact, according to modern theory, the big happened at every point throughout space. So where is this linear view of time, with a definite past to some definite future? It's completely analogous to applying concepts like ''up'' and ''down'' in the universe because there are none. So equally, there is no ''true'' arrow in space for the evolution of things either. There is only a measure of change in a given instant. Not only is Newtonian time linear, but he implies it is external flowing to us. To this day, many many years later, no physicist has ever shown evidence of this. Time is not an observable and so cannot be ''measured'' in the quantum sense; it doesn't have any corresponding Hermitian matrix. One can ''measure'' time in relativity, but it isn't a true quantum measurement, nor should any scientist really believe a clock measures time... it measures a mechanical change and we measure this by the displacements of it's hands. There is also no-non trivial operator for time in physics. So as you can see, so far the definition of time is tricky, but Newtonian time is most likely incorrect thinking that external time exists. There are no problems with understanding a subjective time existing... in fact there are internal gene regulators in the brain which act a theory to why humans have a sense of time at all, many animals have these internal circadian rhythms; so we have a perfectly reasonable biological explanation for the subjective experience of time. The sense Newton gave his flow of time without relation to anything external, probably because there is no way you can physically relate something external with time (the measurement of time is ill-defined in relativity whereas in quantum theory, time isn't an observable). Time is likely emergent, induced from the presence of matter fields; this would mean time isn't fundamental after all and there are many great reasons to think this might be the case we will get into later. With all the definition nonsense out of the way, let's just get to some hard facts about the model of timelessness. Why is it a problem? How does time fall out of physics? ref. (1) -Recent experiments have shown space-time still to be continuous to great degree by measuring the time taken of distant photons. This was to measure how ''grainy'' space time was. (2) - Markopoulou has already offered a theory in which time is fundamental but space isn't. She argues that the quantum theory of gravity will essentially be spaceless. I have already surmised myself that maybe not only time is not fundamental but maybe space isn't either: this could provide a reason to entanglement - systems continue to be ''connected'' because separation is a macroscopic illusion and isn't fundamental for quantum systems allowing intrinsic non-locality. Perhaps the final theory will not only be timeless then, but also spaceless. The trick is to able to describe ''time without time,'' just as General Relativity does. [anime *] http://en.wikipedia.org/wiki/File:Re..._Animation.gif As you can see in this on-line animation, events A, B and C all occur at different times depending on the motion of the observer. Because no one can agree on when an event happens, time appears to be an illusion where the past and future are simply products of the human mind. Our distinction of the past and future is called the psychological arrow of time; there are biological gene regulators which play the role for our perception of time and the different speeds at which perception happens. Note the latter phenomenon is biological, while simultaneity is a relativistic phenomenon. The Time Problem of Physics It is true in fact, that time falls out of physics in several different ways, in this work we will investigate a few of these approaches, one of the most famous is the Wheeler de Witt equation. You see, in the 1960's Wheeler came to de Witt for help constructing a theory of quantum gravity. To do this, de Witt quantized the General Relativistic equations to find what has become familiar as we have shown it before [latex]\psi|H> = 0[/latex] cont. next page Such a simple equation, it is in fact the timeless Schrodinger equation. What he found was that if you apply quantum mechanics to the universe, you find out it doesn't have a fundamental clock! This was an interesting equation for some other reasons as well: Apparently, this ''field of gravity'' was unlike any of the other quantum fields they dealt with in quantum theory. It was manifestly real, the field wasn't a complex one. Fields in quantum mechanics are inherently complex, so this was an unusual feature of quantum theory. But moreover, it showed that gravity wasn't what it seemed. Later I will propose a way in which scientists might look forward to: This will require a better understanding of gravity (in which there is no universal agreement with any theory as thus far) - we will see later though, the problem might be it is not a quantum field any way. Before we close off this small section of this work, I want to explain not to be deceived by the idea that time hasn't been used in relativity, of course, special relativity incorporates it without any troubles. The unification of GR to SR might find a deep philosophical breakdown of space with time. Already, motion in GR is described by no time parameter while in special relativity, moving observers experience time. It turns out there is a solution: the problem lies in the use of ''observer.'' In quantum field theory, an observer can be a particle, doesn't need to be a conscious moving observer measuring the time it take for light to reach one place to another (this is misnomer), you are actually measuring your own time, not the photons. Instead, particles can act as observers and when you apply these principles in a cosmological approach, you do find time disappearing - at least our ability to describe it. We will find out later why this is. The Model Towards the Fully Relativistic Space-Scape When we think about Minkowski space, it treats one of it's four space dimensions as an imaginary dimension to account for time. A popular way to think about space and time, is that they form a single system that stretches perhaps... forever. In many ways, it is a bit like a landscape. Time and space in this unified picture make a landscape for all the events inside the universe. However, we are arguing of course there is no time... we argue time is a measure of change [in] space, it isn't space itself. Treating it as a dimension of space is treated as faulty in this work. Change is what defines time, if there was no change, there would be no way to describe time. Change then will be described in a 'space-scape' which was just a fancy way of saying ''arena'' and the arena of space is where statistical averages (particles) interact and change with respect to each other - configuration space is the arena in which the physicist works. This fully space theory of relativity, is probably more relativistic if you can imagine such a thing, than even Minkowski space time. Minkowski space time assumes there is an external time coordinate, whereas this space theory correlates strongly to Machian Relativity which describes ''time without time''. Around Newtons time, he knew there where three different ways you could describe motion in the universe, 1. With an asbolute space-time background 2. With boundary conditions at spatial infinity 3. By Machian relativity It was considered that 3. was the hardest of the approaches which said motion was relative and holistic (in the sense that motion is driven in a causal manner by all other systems in the universe). Minkowski relativity slices up in space and time using Lorentz boosts and you can go from coordinate frame to another. Machian relativity is quite different indeed, it says that any point is relative to every other point in the universe. Now... points in space themselves are not physical according to modern relativity, only interactions are. It is believed that the laws of physics should be the same in every coordinate frame, similar to the Lorentz boost allowing you to go to one coordinate frame to another, diffeomorphism invariance allows you to shuffle freely between coordinates in GR. The motion then in GR arises as a symmetry of the theory, you don't actually have a time parameter. This is important, because while special relativity describes moving observers and uses a notion of local time, General Relativity actually acts a lot more like Machian Relativity at times. The technical difference between the boost and diffeomorphism, is that the boost preserves a spacetime interval naturally and Diffeomorphisms shuffle coordinates without any reference to time, again, the motion appears as a symmetry of the theory. The addition General Relativity has done with the motion is to be able to say points themselves are not physical, which means Machian Relativity needs to account for this. In Machian relativity, how can points be relative to each other but points are not physical in themselves only their interactions are? While Lorentz transformations preserved the assumption the rules of physics should be the same in every coordinate frame, it is General Covariance that allows General Relativity to assume the laws of physics are the same in every coordinate frame. As it turns out, Einstein solved this problem concerning how to give meaning back to coordinates in space. Basically you get physical points back when you consider the entire world line and their interaction. (Already, you need to think ''holistically'' about a systems history, than just a point, might we find you also need to think of the universe holistically as well?) There are other valid reasons to think Machian relativity has to describe the world, as we saw earlier, one of the popular timeless problems is when you quantize GR you get back the Wheeler de Witt equation, which is really just the time-independent Schrodinger equation. Moreover, the probabilities are for complete three-geometries and the values of matter fields on them which involves the complete configuration of the universe: not for values of some metric on some underlying manifold [1]. This means already, we are getting a sense of Machian relativity when we have to think holistically about the universe as a whole. Indeed, it wasn't long until Einstein realized the metric tensor could describe such a system. Now we can see how points in Machian Relativity can be physical; we have to formulate the theory not only with relative positions in Machian Relativity but it also must make world lines relative, to give positions a physical meaning. To say Machian Relativity not only makes the points but world lines physical, is akin to General Relativity dragging matter with the metric under Diffeomorphisms. So how do we write this new definition of Machian Relativity? I believe it is an important realization that position and world lines need to be included into Machian relativity. This isn't about ''replacing'' current General Relativity, it's doubtful any theory will better it for a long time to come. Instead, we want to incorporate a timeless understanding using Machian relativity (which in many ways already satisfies many contentions GR makes), again, this is largely due to Einstein being heavily influenced by Machian relativity. Points as it turns out, can be physical in Machian space just so long as you include their world lines and their interactions. And so here is the model suggested, a worldline holistic ensemble of relative moving systems in which perhaps order doesn't actually matter according to the relativity of simultaneity. Now to the math! To complete what was being spoken about before we came onto all this stuff about Machian theory. Ok... so if we actually can deal with relative positions then what about all our mathematical tools that use time to describe, say action... how do you understand any of that without time? Well, actually, we have quite number of the tools required to describe timelessness already. We already have a timeless Schrodinger equation. There is also a timeless action using only generalized coordinates [latex]\int \mathbf{p} \cdot d\mathbf{q}[/latex] This is also related to momentum [latex]\int mv\ ds = \int \mathbf{p} \cdot d\mathbf{q}[/latex] With there being a timeless action, there is also naturally a timeless path integral that exists in quantum mechanics [1]. The Newtonian mechanics of a universe of N-gravitating particles in Euclidean space is given quite beautifully by Julian Barbour. These particles will have a mass [latex]m_i[/latex] where [latex](i = 1,2... N)[/latex] and [latex]q = (r_i)[/latex] then a metric can be written [latex]< dq | dq > = \sum_i m_i dr_i \cdot dr_i[/latex] The metric is flat and too simple to yield non-trivial dynamics which can be obtained by the conformal factor which Barbour derives an action [latex]I = \int [<q_{\lambda}|q_{\lambda}> V(q)]^{\frac{1}{2}} d\lambda[/latex] You can go on from here to derive the Euler Langrange equations once you define a kinetic energy term [latex]T = \sum_i m_i \frac{dr_i}{d \lambda} \cdot \frac{dr_i}{d \lambda} = <q_{\lambda} | q_{\lambda}>[/latex] It should be explained, that the equation doesn't explicitely depend on [latex]\lambda[/latex] which usually plays the role of time. In fact, as Julian Barbour says, it is much more illuminating to think of it in it's timeless form, in which our kinetic energy absorbs that with a new term [latex]T^{*} = \sum_i m_i dr_i \cdot dr_i[/latex] Note, that [latex]r[/latex] has been playing the role our generalized coordinates, we will now convert to normal convention. The kinetic energy is related to the action with a [latex]\lambda[/latex]-derivative [latex]2T = \mathbf{p} \cdot d\dot{\mathbf{q}}[/latex] We can also absorb that [latex]\lambda[/latex] term again to simply write the timeless action [latex]2T^{*} = \mathbf{p} \cdot d\mathbf{q}[/latex] As explained, Julian Barbour has indeed wrote out a very nice framework for the theory. And it encorporates these holistic, relativistic idea's Mach wanted to think of the universe. Mach was no doubt influenced by the Newtonian idea that all matter in the universe effected all other systems. Mach believed inertia itself was a property induced by all the systems in the heavens, as he once said: ''You are standing in a field looking at the stars. Your arms are resting freely at your side, and you see that the distant stars are not moving. Now start spinning. The stars are whirling around you and your arms are pulled away from your body. Why should your arms be pulled away when the stars are whirling? Why should they be dangling freely when the stars don't move?'' You can feel a bit of this theory in Machian Relativity as well when he argues all systems are relative to each other in the universe, their positions are relative to all other systems. At least in my work, that may only be true if you consider their world lines and their interactions with it. But this subject of inertia brings me to a last discussion which will take us to Julian Barbour's idea which are largely responsible for my interest on the subject. Relativity says there is a problem, if motion is indeed relative then how do you locally define inertia? Well, actually, Julian Barbour has presented many idea's over the years that might help pave a way to an understanding of how to solve the time problem. He has dedicated work to show that Machian Relativity deals with changes of systems where positions are only relative to each other. He defines it by taking a generic solution of the Newtonian many body problem [2] of celestial planets [3]. Instead of giving time to systems described by their various configurations, you can simply give a sequence of the events. He omits the information (he says which is characterized by six numbers) which describes the position and orientation of the system, he does this by specifying the semi-metric [latex]r_{ij}[/latex] - and finally, he omits the scale information (he calls one number) contained in the metric (separation). He says this is best done by normalizing them by the square root of the moment of inertia [latex]I = \sum _{i} M_i \mathbf{x}_{i}^{2} = \frac{1}{M} \sum_{i <j} m_i m_j r^{2}_{ij}, M = \sum_{i} M_i[/latex] ''The resulting information'' he says, ''can be plotted as a curve on phase space.'' But perhaps more importantly he seems to have solved this problem about how one defined inertia within the theory; it keeps all the valuable information that the original Machian theory was based upon and that involving relative positions. In prof. Susskinds lecture he says the scale factor doesn't depend on the position [math]x[/math], but this would also be in disagreement with Newtonian N-body gravitating systems holistically-connected. If this is the case, then the scale factor very much depends on the position of all systems in the universe ref 1. He also claims the Hubble constant doesn't change with position but changes with time. Assuming he is wrong about the dependency of position, then he is in disagreement with a very workable Machian theory involving change and position including those all-important worldines. The hubble constant in the context could very well include all systems, in opposition to Susskinds claim that the hubble constant IS a function of time but NOT a function of which galaxy we are talking about. This may have something to do with only considering localized events rather than considering a holistic picture within General Relativity. more later [EDITED] REF. - http://www.academia.edu/4228885/Is_the_universe_in_equilibrium Edited January 19, 2015 by Lucious
Rolando Posted January 19, 2015 Posted January 19, 2015 (edited) That isn't a set exactly, it is a notation used for any mass to an N-power, or it can be a sum as shown [latex]\sum_{i}^{N} m_i[/latex] - The [latex]N[/latex] just represents any integer which may count the systems inside the universe, this is where set theory comes into it. OK, then m^N means Σm. Assuming a trivial sphere model of the universe where there is a unique center (so long as there is equally a unique center in all parts of the interior of the sphere) then there is using Newtons law a gravitational equilibrium with all other gravitating systems [latex]G \equiv \frac{Rc^2}{m^{N}}[/latex] of all [latex]N-states[/latex] of the various systems of mass [latex]m^{N}[/latex] We can take from any center of origion, [latex]x=1[/latex] for simplicity and is effected by how everything else is moving Could you also describe this "gravitational equilibrium with all other gravitating systems". What is it that is meant to be in equilibrium? Why is there no factor of 2 in the denominator of your equation (which I could imagine to be there)? Some parts of your text are hard to read, such as the last line the quotation. I don't get it, and so I am unable to follow your reasoning, although the equations that follow in your first post are intelligible to me. At one place, you multiply just one side of an equation by two, which is [insert adequate English word here]. But I find the question you raise, whether the universe could be in gravitational equilibrium, quite interesting. Edited January 19, 2015 by Rolando
Lucious Posted January 19, 2015 Author Posted January 19, 2015 (edited) OK, then m^N means Σm. Could you also describe this "gravitational equilibrium with all other gravitating systems". What is it that is meant to be in equilibrium? Why is there no factor of 2 in the denominator of your equation (which I could imagine)? Funnily enough, I ommited the factor of 2 and well seen! The reason why was because it was a constant and could be configured under the N-power. The correct original equation which would have been given is [latex]G \equiv \frac{Rc^2}{2m}[/latex] But beware, this is only for a two particle Einstein universe example. Instead, we want to talk about the sum of all the systems. I'm glad you mentioned this because for the condition of a simplified two-particle universe [latex]G \equiv \frac{Rc^2}{2m}[/latex] The gravitational equilibrium is missing an extra term, the Torsion, which would make it part of the Full Poncaire Group [latex]\Omega^2\ r = \frac{Gm}{R^2}[/latex] where [latex]\Omega[/latex] is the Torsion field. The Torsion field may also be quantized [latex]\Omega r^2 m = \hbar[/latex] Edited January 19, 2015 by Lucious
swansont Posted January 20, 2015 Posted January 20, 2015 ! Moderator Note Large sections of your text are identical to what's found here http://www.thestudentroom.co.uk/showthread.php?t=2653476 Plagiarism/copyright is something we take very seriously. Did you copy this material, or are you graviphoton, the author?
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