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- 1 Milne’s Model of the Universe
- 1.1 The Three Cosmological Models
- 1.2 Philosophical Differences
- 1.3 Self-Consistent Assumptions of the Standard Cosmological Model
- 1.4 Milne Model
- 1.5 The Problems of Modern Cosmology
- 1.6 A Suggested Approach to Each Problem Using Milne's Model
- 1.6.1 The Flatness Problem
- 1.6.2 The Horizon Problem
- 1.6.3 The Dark Matter Problem
- 1.6.4 The Density Fluctuation Problem
- 1.6.5 The Inflation Problem
- 1.6.6 The Exotic Relics Problem
- 1.6.7 The Thermal State Problem
- 1.6.8 The Cosmological Constant Problem
- 1.6.9 The Singularity Problem
- 1.6.10 The Timescale Problem
- 1.7 Sources
- 1.8 See Also
Milne’s Model of the Universe
A comparison between the cosmology described by Edward Arthur Milne in "Relativity, Gravitation and World Structure" (RGWS) and the Standard Cosmological Model (SCM), each being possible descriptions of the universe, the former using Lorentz transformations (LT) and the latter using the Friedmann, Lemaitre, Robertson, Walker Metric (FLRW Metric).
(see Edward Arthur Milne)
The Three Cosmological Models
Overall, there are three distinct classifications for cosmological models which have been seriously considered. These are the steady-state model, the standard cosmological model, and the Milne model.
Briefly the three models can be described as follows:
The Milne Model: an infinite explosion that took place at a point in space at an instant in time, expanding into pre-existing space.
The Standard Cosmological Model (SCM): a static universe, where space is constantly expanding in a manner consistent with Einstein’s Field Equations.
The Steady-State Universe: (Hoyle?) A universe of infinite age and size where matter may be disappearing and reappearing over time.
This article focuses on the first two, though the third certainly deserves consideration as well.
The differences between Milne’s concept of space-time and Einstein’s concept of space-time appear to run much deeper than is commonly appreciated. For instance the Standard Cosmological Model maintains the big bang was not actually an explosion, but simply the expansion of space. This is completely at odds with the cosmology of Milne which is represented exactly by an explosion of matter into space.
The Scale Factor
The central question of the Standard Cosmological Model is to determine the scale factor of the universe, whether it be constant or varying, and thus determine the “shape” of space.
In contrast, Milne denies that space has a shape or a scale factor which can be measured or observed. In this respect, Milne and Einstein had an immense philosophical gulf between them.
Because of this difference, Milne's Model cannot be described in the context of the Friedmann (FLRW) metric which is an equation describing the scale of space.
Philosophical Conundrum, Is Space Possible Without Matter?
Einstein said that space without matter was impossible, while Milne’s model takes as a premise the existence of a Euclidian coordinate system pre-existing the Big Bang. This may be the root conflict between the two models.
A diagram of the Milne Model universe clearly shows white-space beyond the edge of a sphere, representing space *outside* the universe. This marks Milne’s model as being clearly distinct from any popular cosmology today.
An oft-made assumption is that space cannot exist without matter, and with this reasoning, the possibility of Milne’s model is eliminated, leaving the standard cosmological model as the next most likely possibility.
However, Milne argued, contrary to Einstein, that space, or at least a coordinate system, could exist before all matter, and beyond all matter. Milne claimed no difficulty in describing such a space, or conceptualizing such a space, and goes into some detail to describe it in RGWS.
The SCM holds as self-evident that space only exists between all things. By having a diagram of a model of the universe which explicitly represents the space beyond all matter of the universe (which he maintained was an infinite amount), he may have convinced other scientists of his time that he was crackers.
However, despite this difficulty of a well defined coordinate system beyond the infinite amount of matter in the universe, Milne remained convinced of the validity of his model.
This model is different from the Standard Model, and therefore analyzing the problems of current cosmology such as the Cosmic Background Radiation, Inflation, and Galactic Superstructure should yield some insight.
Edward Arthur Milne predicted the Cosmic Background Radiation through the use of this model, as can be seen clearly in the last sentence of the plate. "The particles near the boundary tend towards invisibility as seen by the central observer, and fade into a continuous background of finite intensity."
A Note on the Two Scale Factors (General vs. Special Relativity)
The concept of a scale factor based on the position of the observer is not the same as the concept of a scale based on the observer’s velocity.
Milne did not admit the existence of a scale factor based on the position of the observer, but he did accept the validity of the Lorentz Transformation equations which scale the distance between two events according to the observer's velocity.
A Note on Homogeneity
Sections 60-64 of Relativity, Gravitation and World Structure present a critique of the Standard Cosmological Model. Milne points out that in a “system homogeneous according to” … “the methods of current relativistic cosmology” … “observers O and O’ will not recognize, or analyze, the system open to their observations as homogeneous.”
Milne was not critical of the use of the definition of homogeneity used by cosmologists but cautioned strongly against its use as a cosmological principle. “This conventional homogeneity is only definite when the motion of the particles is first prescribed.” His alarmed but polite arguments continue throughout section 62, and in section 63, he states explicitly, that homogeneity has been shown to break down, then goes on to prove this in section 64.
However, his proof in section 64 uses relatively moving observers. By assuming the expansion of space instead of relative motion of nebulae, the standard cosmological model proves immune to this particular argument.
However, in section 60, he had already pointed out that a static universe is necessarily part of a favored reference frame, and fails to satisfy another equally valid cosmological principle.
There are at least two things that can save the Standard Cosmological Model from Milne's arguments. One is to assume a static universe, and thus a preferred reference frame, and describe "homogeneity" as the new Cosmological Principle.
The other is to maintain "No Preferred Reference Frame" as the Cosmological Principle, and use the stretching of space to describe a Standard Cosmological Model which is not static yet has no particles in relative motion. This is the aim of the FLRW metric.
Self-Consistent Assumptions of the Standard Cosmological Model
Here is a list of some of the properties associated with the Standard Cosmological Model.
1. The universe has roughly the same density throughout (if you make your scale coarse enough, say kilograms/cubic mega-parsec), though this density decreases over time due to the expansion of space, it is constant at any given time throughout all space.
2. The red-shift of distant galaxies is due to the expansion of space instead of the recession velocity of the galaxies. Of course there is Peculiar velocity which allows for small aberrations from this rule, which is necessary in order for galaxies to collide, or motion to appear at all. But the redshift which corresponds to Hubble flow is assumed to be due entirely to the expansion of space.
3. Because distant galaxies actually have no relative velocity, they are in the same reference frame, and thus no Lorentz Transformation is required to convert between the coordinates of the observers in these galaxies. Because no Lorentz Transformation is required (except possibly for Peculiar velocity), time can be considered to be a universal, or absolute, quantity.
4. Because no Lorentz Transformation is required, Special Relativity can be considered valid, though it's application becomes ambiguous in cosmology. Thus, Special Relativity is dismissed as being valid locally, but not globally.
5. The SCM assumes Einstein field equations; differential equations with boundary conditions. These equations and their solutions have been described in terms of differential equations of tensors (See Ricci tensor, Weyl tensor and Christoffel symbols).
- Roger Penrose (The Emperor's New Mind) has distinguished between the two types of tensors of General Relativity; one being primarily due to the universal curvature of space consistent with the Friedmann Metric, and the other being the effect caused by gravity, more consistent with the Schwartzchild metric.
6. The first solution to the Einstein Field Equations was published in a paper by Friedmann. ("Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes" (On the possibility of a world with constant negative curvature of space). In his solution, undetermined constants included the curvature of the scale of space which may be -1, +1, or 0, and another constant, called the cosmological constant, and the universe's density, which (Milne points out) is assumed a priori to be a constant over space, but nonconstant over time, which is reasonable, since this was how the question was posed by Einstein.
7. The specific boundary conditions necessary to determine the constants of the Friedmann solution should be observable in the map of the Cosmic microwave background radiation. Just as any single-valued function of one variable can be represented by a specific infinite series of sine waves called a Fourier series, or by an infinite series of polynomials called a Taylor Series, a single-valued function of two angle variables which represent the inner surface of a sphere can be represented by an infinite series of Spherical Bessel Functions. Determining the coefficients of an infinite series of terms of an orthonormal basis defined from the Einstein Field Equations is part of what astronomers are doing to establish the Boundary conditions of the Einstein Field Equations in order to find the undetermined constants.
To reiterate, the Standard Cosmological Model has successfully described a plausible possibility for a universe which does not makes the use of Special Relativity and the Lorentz Transformations ambiguous on the cosmological scale. So it accepts the legitimacy of Special Relativity, but only in applications at a local level. One benefit of all this is that absolute time becomes a possibility.
Much of the mathematics of General Relativity seeks to eliminate the ambiguity of applications of Special Relativity within the model, as Peculiar velocity of galaxies and nebulae require some consideration of the laws of relative motion.
The Standard Cosmological Model is self-consistent and might eventually be proven to be reality. If dark matter, dark energy, or any of the other conjectured necessary ingredients of the SCM are discovered, this would be strong evidence in favor of one version or another of the Standard Cosmological Model. These would be discoveries of profound importance—our ability to harness such resources might give us infinite mobility to explore our universe.
Milne developed another self-consistent model which should also be considered. It should be considered, at least as a matter of scientific history, if for no other reason than to consider its implications and identify its flaws if there are any.
This model does not predict dark matter or dark energy, and because of its compatibility with Special Relativity, it strictly disallows instantaneous travel between distant points without temporal difficulties. As such, this model, unfortunately, does not hold the same hope for future technologies as the Standard Cosmological Model.
Milne’s model describes
1. A literal explosion of matter from a point into pre-existing space. In any reference frame, space exists outside the finite region of the universe, and space pre-existed the big bang.
2. Space is considered to have no properties, so this model has no use for the concept of a scale factor, and thus the Friedmann metric does not apply. There is no positive, negative, or even zero curvature of space, and certainly no cosmological constant in Milne's model. Milne's Cosmology is not "Flat" in reference to the Friedmann Metric because it does not use the Friedmann Metric.
3. The mass of the universe is infinite. This assumption comes from consideration of the alternative of a finite universe. In the course of argument, Milne eliminates (to his satisfaction) all other possibilities as being either absurd, or not being compatible with the Principle of Relativity. Moreover, he finds the unique mathematical density function compatible (RGWS Chapter 5, Equation 9) with his assumptions of relative motion and no-preferred-reference-frame, and this density function is an explicit mathematical representation of an infinite amount of mass.
4. The universe began at a single instant of time and space. This is required to have no favored reference frame, and to have no materials popping into existence out at the edge of the universe. Milne devotes five pages to this argument in Chapter VII of RGWS, entitled "Creation" followed by a three page aside on the topic of "Creation and Deity." His arguments have been dismissed by other cosmologists as being primarily faith-based in nature. Source: "Cosmology and Controversy." Regardless of his verbal arguments, the density function which he describes (RGWS Chapter 5, Equation 9) is a unique solution, and it describes the matter in the form of a singularity at time t=0.
5. The red-shift of galaxies is caused almost entirely by recession velocity. (The slowing of time near gravitational sources also causes a redshift of light from massive objects, but this is a change in the speed of time--not the scale of space. i.e. the Ricci Tensor, not the Weyl Tensor)
6. Simultaneity, as perceived by relatively moving observers, is completely determined by using the Lorentz Transformation to convert between Euclidian coordinate systems of the different observers. Simultaneity in the view of a single observer O is determined by a region of constant t.
Milne also provides details (Relativity Gravitation and World Structure RGWS) on how to develop a "mixed" coordinate system where an event is associated with each observer and these events can be mapped into an infinite and homogeneous distribution. Milne takes pains to point out the flaws of doing this, but ironically, his construction was adopted by Robertson or Walker and has become the centerpiece of the Standard Cosmological Model.
7. As mentioned earlier, The density of the universe which corresponds to the cosmological principle of “no preferred reference frame” is a fully determined function. Equation (6) In Chapter 5, Section 89 of RGWS gives the VELOCITY DISTRIBUTION
f(u,v,w) du dv dw = (B du dv dw)/ (c^3 (1- (u^2 +v^2 +w^2)/c^2)^2
where u, v, and w are initial velocities. This can be described as the initial condition of the Milne Model, where all of the particles are located at point (0,0,0) at time 0. This is a fluid dynamics representation.
In the experience of an observer, the system assumes the SPATIAL DISTRIBUTION given in equation (9) of the same chapter.
8. The universe is an expanding sphere with infinite density on the outer surface, in the reference frame of any observer within it. The inner edge of this sphere is visible as the Surface of last scattering, or Cosmic background radiation, caused by a thick dense layer of of hydrogen atoms moving away from us at nearly the speed of light in their era of Recombination. He described this himself as "The total population of points is infinite. The particles near the boundary tend towards invisibility, as seen by the central observer, and fade into a continuous background of finite intensity."
To reiterate, Milne's model denies that space is scalable, maintains that General Relativity affects things locally (Ricci Tensor, and Schwartszchild Metric), and not globally (Weyl Tensor, and FLRW Metric), while Special Relativity affects the observers perception of the entire universe, not just a local region. This is opposite to the Standard Cosmological Model, except that the SCM only makes application of the LT's ambiguous, whereas Milne rejects the FLRW metric entirely.
Milne's Model gives a very specific unique solution compatible with Hubble's Law and the Principle of Relativity. It was rejected on account of the singularity, infinite mass, space beyond all matter, time before matter, lack of absolute time, and above all, Milne's three page aside about "Creation and deity."
Milne specifically addresses all of these topics, whereas most books on General Relativity simply dismiss them as being inconceivable. In the seven decades since RGWS, Milne's model has had few supporters, so significant experimental support for his model may have been overlooked.
The Problems of Modern Cosmology
John Baez has provided an extensive list of shortcomings of standard cosmology. http://www.damtp.cam.ac.uk/user/gr/public/bb_problems.html
These include the flatness problem, the horizon problem, the density fluctuation problem, the dark matter problem, the exotic relics problem, the thermal state problem, the cosmological constant problem, the singularity problem, and the timescale problem.
A Note on the Steady State Model
The thermal state problem and horizon problem can be considered to be distinct from the singularity problem in that they require different assumptions. The first two assume an initial nonzero size of the universe, whereas the singularity problem assumes an initial zero size.
It should be noted that a Lorentz Transformation of two non-touching simultaneous events can separate them to arbitrarily large distances in time and space. Thus if the Lorentz Transformation is admitted as valid on a cosmological scale, then assuming an initial finite size becomes identical to assuming a steady-state, with particles popping into existence at different times for different observers.
However, in general, the SCM does not admit the validity of applying LT's on a cosmological scale.
A Suggested Approach to Each Problem Using Milne's Model
Some of the shortcomings of SCM simply disappear with Milne’s model. Some require a little more explanation, and some of them switch from being questions of cosmology and become questions entirely about the very nature of matter itself.
The Flatness Problem
Assuming the SCM, observations of the universe indicate that the speed of the stretching of space is slowing asymptotically toward zero.
In Milne’s model, there is no stretching of space, so the question must be asked in a different way.
The relation between the distance of a nebula and its velocity in Milne’s model is simply
Distance = Velocity * Time.
The distance equals rate times time equation is familiar to algebra and physics students at all levels.
However, Milne did not preclude the possibility of acceleration after the beginning of time, so, in such cases where some acceleration occurred after the Big Bang.
Distance = Velocity * Time + Initial Distance,
Naturally, since it has been a very long time since the last such acceleration, the local objects are moving very slowly, and the objects which are moving quickly are so far away that we barely notice them. The universe, in Milne’s model shows no signs of slowing down whatsoever, and never would have been expected to. Nearby objects, of course, in a universe over 13.7 billion years old, had an initial velocity of d/13.7 billion so they are moving extremely slowly, and have mostly fallen into the earth, the sun, and the planets.
One other note, substituting the following into the Friedmann solution:
density of the universe = zero, and the cosmological constant = zero, curvature of the universe = -1,
after a little algebra, Friedmann's Solution becomes exactly Hubble's Law,
Velocity = H_0 * Distance
Which seems at first to be equivalent to Distance = Rate * Time, where Hubble's Constant is 1/Time. This seems to suggest that Milne's Cosmology is a model with zero density, zero cosmological constant, and a negative curvature. This is an incorrect assessment. Milne's model rejects the concept of a scale factor, and thus the Friedmann solution from the outset.
The Horizon Problem
The Horizon Problem is a question of why the cosmic background is so smooth. Implicit in the Horizon problem is the assumption that the initial size of the universe is nonzero. This would allow the possibility of objects which were noncausally connected. It would also allow the possibility that distant stars and galaxies would simply wink into existence as the light from them first reached us from the distance. This may be a common concept of the cause of the cosmic background.
Milne’s model assumes that the universe’s initial size was exactly zero, and therefore there are no objects in the universe that are not causally connected in some way. They were all touching at the moment of the big bang.
The Dark Matter Problem
Mundane Dark Matter vs. Mysterious Dark Matter
The Dark Matter Problem has been presented in two main ways. One is to question why the outer arms of spiral galaxies have the same velocity as the inner arms. It has been determined that 90% of the matter in these arms cannot be seen, and is thus “dark matter.” This is easily explained by mundane dark matter: meteors, planets, dust, and gas which does not glow and thus cannot be seen from here.
The other manner in which the Dark Matter Problem has been presented is as the question of what matter is present to warp the space in a manner consistent with the assumptions of the standard cosmological model and the observations of reality. A much larger amount of matter is necessary for this. One description of this matter is huge clouds of neutrinos, or some other neutral nonbaryonic particles. This nonbaryonic matter has been observed, but not in sufficient quantities. It is quite possible that we could miss large amounts of such matter even as it slips right through us.
The standard cosmological model requires large amounts of mundane dark matter and enormous amounts of mysterious dark matter in order to make the model consistent with observations.
The Milne model, on the other hand, requires large amounts of mundane dark matter (to explain the spiral arm motion), but does not require enormous amounts of mysterious dark matter. Stars and galaxies represent blemishes and concentrations among an otherwise smooth fluid of mundane dark matter.
The Density Fluctuation Problem
Why are there blemishes and concentrations of matter in the universe? “The perturbations which gravitationally collapsed to form galaxies must have been primordial in origin; from whence did they arise?”
Astronomers have noticed that the visible matter of the universe has organized itself into perturbations known as stars, galaxies, clusters, and super clusters. In fact, these super clusters seem to appear in great long swaths across the cosmos such as the Shapley Concentration, (AKA the Finger of God).
Many opinions are out there as to the nature of the cause of these perturbations. Some have suggested that they appear due to superstrings; monofilament remnant leftovers of the big bang, with the gravitational strength of black holes or neutron stars. Others have suggested that the phenomena are due to some kind of crystal defect in the homogeneity of the universe.
Another option is that a large mass passed through the region while it was still dense. Just as a fast moving boat creates whirlpools on the surface of a smooth pond, so did this large mass leave spiral galaxies forming in its wake. This idea is not consistent with an absolute time scale, or homogeneity, as it requires a body to come from a dense region into a less dense region, and remain in its primordial form as it travels through millions or billions of light years of space. This is consistent with Milne’s model, with it's hot outer, infinitely dense edge, from whence Brownian Motion could cause a primordial particle to accelerate inward into a region of lesser density.
The Inflation Problem
The challenge for proponents of the SCM is to explain inflation in a manner which does not invoke the Lorentz Transformations.
Milne’s model, on the other hand, does invoke the Lorentz Transformations, and thus provides a mechanism for inflation with no difficulty. The model inflates with respect to any observer who undergoes acceleration, in the manner described by putting an luminally expanding sphere through a Lorentz Transformation about an event which is not at the origin.
The first of these two images shows a two dimensional cross-section of the space of a large mass in its own reference frame. The second of the two animations shows the Minkowski Space-time diagram showing the path of this object, as well as its line of simultaneity at each moment, and the red dot which represents the stationary center of the universe from its perspective.
If a body were to be accelerated by Brownian Motion in the thermodynamics of the hot big bang, it would be moved to a new reference frame, where instead of seeing itself at the center (and least dense section) of very small universe, it would see itself at the very dense edge of a vast universe. The objects coming from the center of that universe would be approaching it at an alarming rate. If the mass of this body were large enough (which is highly likely if it came from the first fraction of a second... see The Singularity Problem, below), it would leave condensed matter and spiral galaxies in its wake.
The Exotic Relics Problem
“Phase transitions in the early universe inevitably give rise to topological defects, such as monopoles, and exotic particles. Why don't we see them today?”
Most of these phenomena would be blocked from view by the layer of newly formed Hydrogen atoms at the visible inner surface of the sphere of Milne’s Model. This is fairly well explained by the concept of “the surface of last scattering” although using the Milne model would give an even greater understanding of why this surface was so incredibly effective at blocking everything except for those photons unable to knock an electron out of the 1S orbital of a hydrogen atom.
The Thermal State Problem
Again, the question of why the universe began in a state of thermal equilibrium implicitly assumes the initial size of the universe was nonzero. This seems to be more of an argument against the Steady State model than the Standard Cosmological Model. And regardless, it doesn't apply to Milne's Model, because everything was in thermal contact at the first instant.
The Cosmological Constant Problem
Milne’s model does not admit the validity of Einstein’s Field Equations, the scale factor, or the Friedmann solutions. The cosmological constant problem in Milne's model is that anyone with a solid grasp of Milne's model well in mind will be unable to find any conceptual use for the Friedmann Solution, and no matter how strong his interest in the subject, he will be dismissed by some cosmologists as unteachable.
The Singularity Problem
This is an important problem but it is not compatible with the Thermal State Problem or the Horizon Problem, since both of them implicitly assume an initial nonzero size of the universe. Supporters of the Steady-State model do not need to worry about the singularity problem, but it is probably the most important problem of the Milne Model.
The Milne model requires the entire system to occupy a single point at first instant, as is defined by the Lorentz Invariant density equation of Milne's Model, and is this singularity is described in detail in his chapter about Creation. It remains a problem in his model, or at least a rich source of questions.
Specific Questions on the Singularity Problem
The Singularity Problem in Milne's Model might be essentially the same as the Singularity Problem of the Cosmological Model in regards to Quantum Mechanics: Since the distance between any two particles is zero at the instant of creation, this requires, by the Heisenberg Uncertainty Principle, (HUP) that either that Planck's Constant was zero during that event, or that the relative momentum between any two particles in the system be infinite (their relative velocity is the speed of light, or their mass is infinite), or Planck's constant is reduced to zero.
On the other hand, the explicit definition of initial conditions of Milne's Model provides a problem setup for statistical thermodynamics analysis of the system: Statistical thermodynamics and solid-state mechanics usually consider systems with an equipartition of momentum. A careful analysis will show that the Lorentz invariant distribution described by Milne is an equipartition of rapidity, which is a fundamentally different quantity than mass-independent momentum. (Though they are the same at low velocities). The laws of motion absent all forces would have predict an equipartition of momentum. This difference could lead to further insight.
The Timescale Problem
In the Standard Model, the Friedmann Metric affects both time and space in a global manner; regions of simultaneity are determined by 3-surfaces in spacetime. These surfaces are not flat; a favored surface shape is hyperboloid; hyperboloid surfaces of simultaneity allow an absolute universal time consistent with Hubble's Law.
In Milne's Model, Spacetime is Euclidian, and surfaces of simultaneity are flat. Gravity can affect time on a local level, and LT's rotate the flat surface of simultaneity.
Either way there is a timescale problem. Milne's timescale problem is that it's hard to understand Special Relativity. SCM's timescale problem is that no one knows what the shape of the 3-surfaces are, and it is hard to understand how gravity could affect things on a global level.
The following is a short list of books, websites, and articles consistent with Milne's Model. The Litmus tests of whether an idea is consistent or inconsistent with Milne's Model is to ask whether it attempts to apply Lorentz Transformations on a cosmological scale or whethery they primarily deal with the Friedmann metric.
Gravitation, Relativity and World Structure
Only Chapter's 1-7 and 16-Appendix are specifically referenced here. The content between chapters 8 and 15 are beyond the scope of this presentation.
The "Inflation-in-Milne-Model" animation presented is a non-calculus-based approach to the idea of motion in the Milne Model. In these chapters, Milne refers to particles traveling faster than the speed of light, which is ostensibly what the animation shows during periods of the instantaneous representation of the acceleration of the observer.
Milne takes a calculus based approach which treats uniform accelerations due to gravitational attraction instead of the instantaneous acceleration due to thermal collision. His approach is similar to that of Mike Fontenot (see CADO, below). Milne treats the change in the observed location of the particle with respect to the observer as an actual velocity, and so frequently describes "Faster than Light" motion of particles in chapters 8-15. This can be taken as the mechanism for cosmological inflation as presented by Milne, though this should be verified by better educated mathematicians and physicists.
This image shows an observer under non-instantaneous acceleration. It shows only events rather than the worldlines of objects, but it's not hard to imagine the animation with lines drawn between some of the events to represent worldlines of non-accelerating particles.
If they were, the intersection where these lines crossed the horizontal t=0 axis would move back and forth at rates higher than the speed of light. Distance between worldlines along a constant t is the definition of distance as used in Lorentz Transformation.) This might be called Faster Than Light Motion in some interpretation, though of course it is only faster than light in reference to the accelerating observer.
That is, there is no reference frame where any object is traveling faster than the speed of light, but the observer is not remaining in any single reference frame, so he can observe an object's position changing faster than the speed of light.
A much acclaimed title among laypersons, and much maligned among cosmology experts, Lewis Carroll Epstein’s 1976 work, Relativity Visualized, presents many of the same concepts presented by Milne. Having not sat through numerous arguments with Eddington, Russell, Einstein, etc, Epstein's work is much less argumentative.
Epstein's main goal was to explain the Theory of Relativity in a manner that was self-consistent and that *anyone* could understand. Even so, because of the difficulty of the two different coordinate systems presented, the Appendix should be skimmed before delving into the meat of the book. Epstein's use of a space-propertime diagram, distinct from a space-coordinate-time diagram is compatible with Milne's description of a mixed coordinate system.
The description and Illustration of the General Theory in Epstein's book is truly unique, and if the mathematics of the Schwarzschild Metric could be somehow illustrated by the nonmathematical models of Epstein, a great leap forward could be made in pedagogy.
Epstein views his own model as simply a self-consistent possibility, distinct from the concept of "expanding space," but has not made a careful critical analysis and comparison.
Website: Relativistic Flight through Stonehenge
The Relativistic Flight through Stonehenge describes what to expect visually from objects moving toward us near the speed of light.
The diagrams indicate that when the observer accelerates toward a stationary object, it appears further away. Extrapolation of this idea leads to the realization that this change in perceived distance is limited only by the acceleration of the observer.
This simultates how an object could be viewed to be moving "faster than the speed of light" and is consistent with Inflation of the Milne Model Universe.
CADO: Current Age of Distant Objects
Mike Fontenot had a research paper published on the topic of the Current Age of Distant Objects, which is consistent with the Milne Model, (and in particular the explanation for Inflation.) He determined that by accelerating toward a distant receding object, you increase its coordinate distance until you've matched pace with it, and cause its current coordinate age to change rapidly by accelerating toward or away from it.
RelLab is a software simulation program for Macintosh designed to simulate the Special Theory of Relativity. This program is capable of accurate applications of the Special Theory of Relativity at any scale, without use or mention of Friedmann’s scale factor, which makes it compatible for doing rudimentary simulations of Milne’s model.
JDoolin 20:27, 16 August 2006 (UTC)
http://world.std.com/~mmcirvin/milne.html ("Milne Cosmology, and Why I Keep Talking About It," by Matt McMirvin.")
http://www.phys-astro.sonoma.edu/BruceMedalists/Milne/MilneRefs.html Here is a list of further reading on Milne's life and work.