<i>CMB</i>—A Geometric, Lorentz Invariant Model in Non-Expanding Lobachevskian Universe with a Black Body Spectral Distribution Function

In the present paper, based on Lobachevskian (hyperbolic) static geometry, we present (as an alternative to the existing Big Bang model of CMB) a geometric model of CMB in a Lobachevskian static universe as a homogeneous space of horospheres. It is shown that from the point of view of physics, a horosphere is an electromagnetic wavefront in Lobachevskian space. The presented model of CMB is an Lorentz invariant object, possesses observable properties of isotropy and homogeneity for all observers scattered across the Lobachevskian universe, and has a black body spectrum. The Lorentz invariance of CMB implies a mathematical equation for cosmological redshift for all z. The global picture of CMB, described solely in terms of the Lorentz group—SL(2C), is an infinite union of double sided quotient spaces (double fibration of the Lorentz group) taken over all parabolic stabilizers P&sub;SL(2C). The local picture of CMB (as seen by us from Earth) is a Grassmannian space of an infinite union all horospheres containing origin o&isin;L3, equivalent to a projective plane RP2. The space of electromagnetic wavefronts has a natural identification with the boundary at infinity (an absolute) of Lobachevskian universe. In this way, it is possible to regard the CMB as a reference at infinity (an absolute reference) and consequently to define an absolute motion and absolute rest with respect to CMB, viewed as an infinitely remote reference.

Creation of a Non-Expanding, Non-Singular Universe

A theory of gravitation in flat space-time is applied to homogeneous, isotropic cosmological models. There are non-singular cosmological models. A natural interpretation is a non-expanding universe. The redshift is an intrinsic effect and not a Doppler effect. The universe contains only energy in the beginning, i.e. no matter exists. In the course of time matter and radiation are created from energy where the whole energy is conserved. Matter increases with time but a certain time after the beginning of the universe the creation of matter is finished and the universe appears like a static one. A modified Hubble law is considered which may explain the high redshifts of objects in the universe without the assumption of dark energy.

Misconceptions of Universe Expansion, Accelerated Universe Expansion, and Their Sources. Virtual Reality of Inflationary Cosmology

In this work, we present our theory and principles of the mathematical foundations of Lobachevskian (hyperbolic) astrophysics and cosmology which follow from a mathematical interpretation of experimental data in a Lobachevskian non-expanding Universe. Several new scientific formulas of practical significance for astrophysics, astronomy, and cosmology are presented. A new method of calculating (from experimental data) the curvature of a Lobachevskian Universe is given, resulting in an estimated curvature-K on the order of 10&minus;52 m&minus;2. Our model also estimates the radius of the non-expanding Lobachevskian Universe in a Poincare ball model as approximately 14.9 bly. A rigorous theoretical explanation in terms of the fixed Lobachevskian geometry of a non-expanding Universe is provided for experimental data acquired in the Supernova Project, showing an excellent agreement between experimental data and our theoretical formulas. We present new geometric equations relating brightness dimming and redshift, and employ them to fully explain the erroneous reasoning and erroneous conclusions of Perlmutter, Schmidt, Riess and the 2011 Nobel Prize Committee regarding “accelerated expansion” of the Universe. We demonstrate that experimental data acquired in deep space astrophysics when interpreted in terms of Euclidean geometry will result in illusions of space expansion: an illusion of “linear space expansion”—Hubble, and an illusion of “accelerated (non-linear) space expansion”—Perlmutter, Schmidt, Riess.

Comparing Gravitation in Flat Space-Time with General Relativity

General relativity (GR) and gravitation in flat space-time (GFST) are covariant theories to describe gravitation. The metric of GR is given by the form of proper-time and the metric of GFST is the flat space-time form different from that of proper-time. GR has as source the matter tensor and the Einstein tensor describes the gravitational field whereas the source of GFST is the total energy-momentum including gravitation and the field is described by a non-linear differential operator of order two in divergence form. The results of the two theories agree for weak gravitational fields to the order of measurable accuracy. It is well-known that homogeneous, isotropic, cosmological models of GR start from a point singularity of the universe, the so called big bang. The density of matter is infinite. Therefore, our observable universe implies an expansion of space, in particular an inflationary expansion in the beginning. This is the presently most accepted model of the universe although doubts exist because infinities don’t exist in physics. GFST starts in the beginning from a homogeneous, isotropic universe with uniformly distributed energy and no matter. In the course of time, matter is created out of energy where the total energy is conserved. There is no singularity. The space is flat and the space may be non-expanding.

The Hubble Constant Problem and the Solution by Gravitation in Flat Space-Time

General Relativity implies an expanding Universe from a singularity, the so-called Big Bang. The rate of expansion is the Hubble constant. There are two major ways of measuring the expansion of the Universe: through the cosmic distance ladder and through looking at the signals originated from the beginning of the Universe. These two methods give quite different results for the Hubble constant. Hence, the Universe doesn’t expand. The solution to this problem is the theory of gravitation in flat space-time where space isn’t expanding. All the results of gravitation for weak fields of this theory agree with those of General Relativity to measurable accuracy whereas at the beginning of the Universe the results of both theories are quite different, i.e. no singularity by gravitation in flat space-time and non-expanding universe, and a Big Bang (singularity) by General Relativity.

Gravitation as Geometry or as Field

General relativity (GR) and gravitation in flat space-time (GFST) are covariant theories to describe gravitation. The metric of GR is given by the form of proper-time and the metric of GFST is a flat space-time form different from that of proper-time. The source of GR is the matter tensor and the Einstein tensor describes the gravitational field. The source of GFST is the total energymomentum including gravitation. The field is described by a non-linear differential operator of order two in divergence form. The results of the two theories agree for weak gravitational fields to the order of measurable accuracy. It is well-known that homogeneous, isotropic, cosmological models of GR start from a point singularity of the universe, the so called big bang. The density of matter is infinite. Therefore, our observable big universe implies an expansion of space, in particular an inflationary expansion in the beginning. Doubts are stated because infinities don’t exist in physics. An explanation to the present, controversial discussion of expanding accelerating or non-accelerating universe as well as non-expanding universe is given. GFST starts in the beginning from a homogeneous, isotropic universe with uniformly distributed energy and no matter. In the course of time matter is created out of energy where the total energy is conserved. There is no singularity, i.e. no big bang. The space is flat and non-expanding.

Gravitational Energy and No Big Bang Starts the Universe

Gravitation in flat space-time is described as field and studied in several articles. In addition to the flat space-time metric a quadratic form formally similar to that of general relativity defines the proper-time. The field equations for the gravitational field are non-linear differential equations of second order in divergence form and have as source the total energy-momentum tensor (inclusive that of gravitation). The total energy-momentum is conserved. It implies the equations of motion for matter in this field. The application of the theory gives for weak fields to measurable accuracy the same results as general relativity. The results of cosmological models are quite different from those of general relativity. The beginning of the universe starts from uniformly distributed gravitational energy without matter and radiation which is generated in the course of time. The solution is given in the pseudo-Euclidean metric, i.e. space is flat and non-expanding. There are non-singular solutions, i.e. no big bang. The redshift is a gravitational effect and not a Doppler effect. Gravitation is explained as field with attractive property and the condensed gravitational field converts to matter, radiation, etc. in the universe whereas the total energy is conserved. There is no contraction and no expansion of the universe.

Cosmology with Bounce by Flat Space-Time Theory of Gravitation and a New Interpretation

General relativity predicts a singularity in the beginning of the universe being called big bang. Recent developments in loop quantum cosmology avoid the singularity and the big bang is replaced by a big bounce. A classical theory of gravitation in flat space-time also avoids the singularity under natural conditions on the density parameters. The universe contracts to a positive minimum and then it expands during all times. It is not symmetric with regard to its minimum implying a finite age measured with proper time of the universe. The space of the universe is flat and the total energy is conserved. Under the assumption that the sum of the density parameters is a little bit bigger than one the universe is very hot in early times. Later on, the cosmological model agrees with the one of general relativity. A new interpretation of a non-expanding universe may be given by virtue of flat space-time theory of gravitation.

F_(p) as a Discrete Metric Space

We define a metric that makes the algebraic closure of a finite field F_(p) into a UDBG(uniformly discrete with bounded geometry)metric space.This metric stems from algebraic properties of F_(p).From this perspective,for F_(p)we explore common research themes in metric spaces,reveal how peculiar properties naturally arise,and present it as a new type of example for certain well-studied questions.