Milne model

# Milne model

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The Milne model was a special relativistic
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

cosmological
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...

model proposed by Edward Arthur Milne in 1935. It is mathematically equivalent to a special case of the FLRW model in the limit of zero energy density
Energy density
Energy density is a term used for the amount of energy stored in a given system or region of space per unit volume. Often only the useful or extractable energy is quantified, which is to say that chemically inaccessible energy such as rest mass energy is ignored...

(in other words, an empty universe), and it obeys the cosmological principle
Cosmological Principle
In modern physical cosmology, the cosmological principle is the working assumption that observers on Earth do not occupy an unusual or privileged location within the universe as a whole, judged as observers of the physical phenomena produced by uniform and universal laws of physics...

. The Milne model is also similar to Rindler space, a simple re-parameterization of flat Minkowski Space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...

.

Since it features both zero energy density and maximally negative spatial curvature, the Milne model is contradicted by cosmological observations
Observational cosmology
Observational cosmology is the study of the structure, the evolution and the origin of the universe through observation, using instruments such as telescopes and cosmic ray detectors.-Early observations:...

. Astronomers actually observe the universe's density parameter to be consistent with unity and its curvature to be consistent with flatness.

## Milne metric

The Milne universe is a special case of a more general Friedmann–Lemaître–Robertson–Walker model (FLRW). The Milne solution can be obtained from the more generic FLRW model by demanding that the energy density, pressure and cosmological constant all equal zero and the spatial curvature is negative. From these assumptions and the Friedmann equations it follows that the scale factor must depend on time coordinate linearly.
Setting the spatial curvature and speed of light to unity the metric for a Milne universe can be expressed with hyperspherical coordinates as:

where
is the metric for a two-sphere and
is the curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

-corrected radial component for negatively curved space that varies between 0 and .

The empty space that the Milne model describes can be identified with the inside of a light cone of an event in Minkowski space by a change of coordinates.

Milne developed this model independent of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

but with awareness of special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

. As he initially described it, the model has no expansion of space, so all of the redshift (except that caused by peculiar velocities) is explained by a recessional velocity
Recessional velocity
Recessional Velocity is a term used to describe the rate at which an object is moving away, typically from Earth.-Application to Cosmology:This term is generally only used in reference to distant Galaxies. The most common reason for the use of this term is Hubble's Law, which states that the...

associated with the hypothetical "explosion". However, the mathematical equivalence of the zero energy density
Energy density
Energy density is a term used for the amount of energy stored in a given system or region of space per unit volume. Often only the useful or extractable energy is quantified, which is to say that chemically inaccessible energy such as rest mass energy is ignored...

() version of the FLRW metric to Milne's model implies that a full general relativistic treatment using Milne's assumptions would result in an increasing scale factor and associated metric expansion of space
Metric expansion of space
The metric expansion of space is the increase of distance between distant parts of the universe with time. It is an intrinsic expansion—that is, it is defined by the relative separation of parts of the universe and not by motion "outward" into preexisting space...

with the unique feature of a linearly increasing scale factor for all time since the deceleration parameter is uniquely zero for such a model.

## Incompatibility with observation

Even though the Milne model as a special case of a Friedmann-Robertson-Walker universe is a solution to General relativity, the assumption of zero energy content limits its use as a realistic description of the universe. Besides lacking the capability of describing matter Milne's universe is also incompatible with certain cosmological observations
Observational cosmology
Observational cosmology is the study of the structure, the evolution and the origin of the universe through observation, using instruments such as telescopes and cosmic ray detectors.-Early observations:...

. In particular it makes no prediction of the cosmic microwave background radiation
Cosmic microwave background radiation
In cosmology, cosmic microwave background radiation is thermal radiation filling the observable universe almost uniformly....

nor the abundance of light elements
Big Bang nucleosynthesis
In physical cosmology, Big Bang nucleosynthesis refers to the production of nuclei other than those of H-1 during the early phases of the universe...

which are hallmark pieces of evidence that cosmologists agree support Big Bang cosmology over alternatives.

## Milne's density function

Milne proposed that the universe's density changes in time because of an initial outward explosion of matter. Milne's model assumes an inhomogeneous density function which is Lorentz Invariant (around the event t=x=y=z=0). When rendered graphically Milne's density distribution shows a three-dimensional spherical Lobachevskian pattern with outer edges moving outward at the speed of light. Every inertial body perceives itself to be at the center of the explosion of matter (see observable universe
Observable universe
In Big Bang cosmology, the observable universe consists of the galaxies and other matter that we can in principle observe from Earth in the present day, because light from those objects has had time to reach us since the beginning of the cosmological expansion...

), and sees the universe as homogeneous and isotropic in the sense of the cosmological principle
Cosmological Principle
In modern physical cosmology, the cosmological principle is the working assumption that observers on Earth do not occupy an unusual or privileged location within the universe as a whole, judged as observers of the physical phenomena produced by uniform and universal laws of physics...

.

Unless the universe modeled has zero density, Milne's proposal does not follow the predictions of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

for the curvature of space caused by global matter distribution, as seen in, for example statistics associated with large-scale structure.

## Differences between Milne model and other models

In order to explain the existence of matter in the universe, Milne proposed a physical explosion
Explosion
An explosion is a rapid increase in volume and release of energy in an extreme manner, usually with the generation of high temperatures and the release of gases. An explosion creates a shock wave. If the shock wave is a supersonic detonation, then the source of the blast is called a "high explosive"...

of matter which would not affect the universe's geometry. This is in contrast to the metric expansion of space
Metric expansion of space
The metric expansion of space is the increase of distance between distant parts of the universe with time. It is an intrinsic expansion—that is, it is defined by the relative separation of parts of the universe and not by motion "outward" into preexisting space...

that is the hallmark feature of many of the more famous cosmological models including the Big Bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...

and Steady State
Steady State theory
In cosmology, the Steady State theory is a model developed in 1948 by Fred Hoyle, Thomas Gold, Hermann Bondi and others as an alternative to the Big Bang theory...

models. Milne's universe shares a superficial similarity to Einstein's static universe
Static universe
A static universe, also referred to as a "stationary" or "Einstein" universe, is a model in which space is neither expanding nor contracting. Albert Einstein proposed such a model as his preferred cosmology in 1917...

in that the metric
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

of space
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

is not time-dependent. Unlike Einstein's initial cosmology, Milne's proposal directly contradicts the Einstein equations for cosmological scales. Special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

becomes a global property of Milne's universe while general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

is confined to a local property. The reverse is true for standard cosmological models, and most scientists and mathematicians agree that the latter is self-consistent while the former is mathematically impossible.

Edward Arthur Milne
Arthur Milne
Edward Arthur Milne FRS was a British astrophysicist and mathematician.- Biography :Milne was born in Hull, Yorkshire, England...

predicted a kind of event horizon
Event horizon
In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case...

through the use of this model: "The particles near the boundary tend towards invisibility as seen by the central observer, and fade into a continuous background of finite intensity." The horizon arises naturally from length contraction
Length contraction
In physics, length contraction – according to Hendrik Lorentz – is the physical phenomenon of a decrease in length detected by an observer of objects that travel at any non-zero velocity relative to that observer...

seen in special relativity which is a consequence of the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

upper bound for physical objects. In Milne's universe, the velocities of objects approach this upper bound while the distance to these objects approaches the speed of light multiplied by the time since the event of the initial explosion of material. Beyond this distance, objects do not lie in the observable part of the Milne universe.

At the time Milne proposed his model, observations of the universe did not appear to be in a homogeneous form. This, to Milne, was a deficiency inherent in the competing cosmological models which relied on the cosmological principle
Cosmological Principle
In modern physical cosmology, the cosmological principle is the working assumption that observers on Earth do not occupy an unusual or privileged location within the universe as a whole, judged as observers of the physical phenomena produced by uniform and universal laws of physics...

that demanded a homogeneous universe. “This conventional homogeneity is only definite when the motion of the particles is first prescribed.” With present observations of the homogeneity of the universe on the largest scales seen in the cosmic microwave background and in the so-called "End of Greatness", questions about the homogeneity of the universe have been settled in the minds of most observational cosmologists
Observational cosmology
Observational cosmology is the study of the structure, the evolution and the origin of the universe through observation, using instruments such as telescopes and cosmic ray detectors.-Early observations:...

.