[27] **viXra:1502.0246 [pdf]**
*submitted on 2015-02-28 06:08:52*

**Authors:** Brett A Collins

**Comments:** 6 Pages.

In this paper I postulate the existence of a second time dimension, making five dimensions, three space dimensions and two time dimensions. I will postulate some basic properties based on a smoking gun and then use these basic properties to derive the time dilation equations of "Special Relativity", which helps defne additional properties of the second time dimension. The conclusion being that the Universe has five dimensions but that we only perceive four dimensions. Further more I will demonstrate that Newton's second law of motion still holds if you ignore the time contribution from the second time dimension and additionally as a result of this paper we will understand a little bit more about the very nature of time. I believe this to be quiet a significant paper.

**Category:** Relativity and Cosmology

[26] **viXra:1502.0235 [pdf]**
*submitted on 2015-02-26 12:34:18*

**Authors:** Juergen Altenbrunn

**Comments:** 16 Pages. needs Theory of relativity without singularities

The cosmologic red shift is an effect of general relativity. It is possible to calculate cosmologic red shift of light, passing through a transparent continuum with mass. It is necessary to consider the mass variation caused caused from variation of potential energy demonstrated in [3] for this calculation. Result of this calculation is also an intensified increasing of cosmologic red shift at very long distances. Accelerated expansion of our universe or dark energy is not needed for this calculation, only pure theory of general relativity. It is possible to calculate the average density of our universe, if you know Hubble-constant. Calculated average density is 1.76*10^-26kg/m^3 by using a Hubble-constant of 21km/s per million light years.

**Category:** Relativity and Cosmology

[25] **viXra:1502.0234 [pdf]**
*submitted on 2015-02-26 15:00:00*

**Authors:** Friedwardt Winterberg

**Comments:** 19 Pages.

An attempt is made to explain the Allais effect, the anomalous behavior of pendulums during a total solar eclipse, by Poisson diffraction of gravitational waves into the lunar shadow, with the waves emitted through large mass motion in the center of the sun by a thermonuclear fusion reaction driven magnetohydrodynamic dynamo. Thermomagnetic currents in the tachocline shield the strong magnetic field in the solar core.

**Category:** Relativity and Cosmology

[24] **viXra:1502.0233 [pdf]**
*submitted on 2015-02-26 12:04:28*

**Authors:** Juergen Altenbrunn

**Comments:** 28 Pages.

Singularities are avoidable in theory of relativity by consideration of mass variation. Mass varies not only by changing of kinetic energy but also by changing of the potential energy of the mass. This variation of the mass follows the relation E = m * c^2. The mass will be lighter, it decreases in the same way as potential energy of this mass decreases. This will be very clearly by consideration of distances in the same range like Schwarzschild radius. Standstill mass is the potential energy against infinity.
The dissemination of the gravitational field varies near the mass also by consideration of mass variation. Also red shift varies near a heavy dense mass by consideration of mass variation. The event horizon disappears. Physics are possible inside of Schwarzschild radius.

**Category:** Relativity and Cosmology

[23] **viXra:1502.0218 [pdf]**
*submitted on 2015-02-24 07:58:41*

**Authors:** Hasmukh K. Tank

**Comments:** Four-page letter

Astronomical observations of cosmological red-shift are currently interpreted in terms of ‘expansion of universe’ and ‘accelerated-expansion of the universe’. Whereas a straight forward derivation presented here suggests that: it is not the universe, which is expanding, rather it is the photon which is decelerating, at the rate of H0 c; here H0 is Hubble’s constant, and c is the speed of light. (As soon as a new mechanism for the ‘cosmological red-shift’ is proposed, it is argued that: tired-light interpretations of cosmological red-shift’ are not compatible with the observations of ‘time-dilation of super-novae light-curves’. So in a paper titled: “Wave-theoretical insight into the Relativistic ‘length-contraction’ and ‘time-dilation of super-novae light-curves’” [1] it has been already shown that any mechanism which can cause ‘cosmological red-shift’ will also cause ‘time-dilation of super-novae light-curves’. Therefore, we now need not remain confined to the Big Bang Cosmology, to understand the universe.) Since, as per this new interpretation of the ‘cosmological red-shift’, the ‘reduction-in-energy of the photon’ (h Δν) goes on reducing with every subsequent unit distance traveled by the photon; so the Red-shift-Distance-Curve automatically becomes non-linear, which has been creating an impression of ‘accelerated-expansion of the universe’. As a supportive evidence for this proposal, it is shown that even the space-probes Pioneer-10, Pioneer-11, Galileo and Ulysses too did decelerate at the same rate H0 c [2]. And to counter-balance this deceleration, the accelerations at the boundaries of: the nucleus-of-atom, the globular-clusters, the spiral-galaxies and the galactic-clusters too, are of the same order as H0 c. This coincidence was numerically found by Sivaram [3]; and is explained here, in terms of equilibrium between self-gravitational-acceleration of structures and the acceleration of the un-manifest-energy contained in space.

**Category:** Relativity and Cosmology

[22] **viXra:1502.0213 [pdf]**
*submitted on 2015-02-23 12:12:17*

**Authors:** sangwha Yi

**Comments:** 5 Pages. Thank you for reading

In the general relativity theory, if we use Einstein’s gravity field equation, we can discover the General Schwarzschild solution. We obtain new solution in the General Schwarzschild solution.

**Category:** Relativity and Cosmology

[21] **viXra:1502.0209 [pdf]**
*submitted on 2015-02-22 20:37:09*

**Authors:** George R. Briggs

**Comments:** 1 Page.

There are 4 particles Intrinsically (mc^2) more energetic than the copious tetraquark (4.430 GEV): The top quark (173 GEV), the Z particle (91.2 GEV), and the two W particles (80.4 GEV each). These are all much more energetic than the tetraquark and should be considered as a class of particle by themselves and as a group signaled by the copious tetraquark. The group is partly responsible for the E8-symmetry breaking at the time of the Big Bang.

**Category:** Relativity and Cosmology

[20] **viXra:1502.0184 [pdf]**
*submitted on 2015-02-21 05:23:36*

**Authors:** Lubomir Vlcek

**Comments:** 39 Pages. Through the work of Max Planck, Albert Einstein, Louis de Broglie, Arthur Compton, Niels Bohr, and many others, current scientific theory holds that all particles also have a wave nature (and vice versa).[1] This phenomenon has been verified

Einstein's theory of relativity can not explain ...
1. Movement principles of the fast-spinning pulsars,
2. Nuclear Fusion ,
3. Wave - Particle Duality as Kinetic Energy Against and In Direction of Motion
4. the 4th Maxwell's equation,
5. Lorentz equals without the help of Space-Time,
6.Confinement of quarks
7. Great Table of Elementary Particles
8. Spectral line Hα
9. Neutrino Oscillations
10. Form of the interference field must be non-linear.
11.Form of Intensity of the Moving Charge Electric Field must be asymmetrical.
12.Kinetic energy of a charge moving at the velocity of v has two different values: Kinetic energy against direction of motion as wave
Tkin ad = mc2[ln |1+v/c|- (v/c)/(1+v/c)]
Kinetic energy in direction of motion as particle
Tkin id = mc2[ln|1-v/c|+ (v/c)/(1-v/c)]
13. Yukawa potential

**Category:** Relativity and Cosmology

[19] **viXra:1502.0155 [pdf]**
*replaced on 2015-02-18 10:16:50*

**Authors:** Wenceslao Segura González

**Comments:** 15 Pages. Spanish

In 1943 Erwin Schrödinger began a series of publications on Unitary Field Theory, with which wanted to unify the gravitational, electromagnetic and mesonics fields on a geometric basis. In this article, which is a continuation of [36] and [37], we analyze the third of the purely affine theories Schrödinger (Schrödinger-III), where he used a connection asymmetric capable of accommodating the unification of the three fields

**Category:** Relativity and Cosmology

[18] **viXra:1502.0151 [pdf]**
*replaced on 2018-10-23 09:11:16*

**Authors:** Sjaak Uitterdijk

**Comments:** 2 Pages. In [v3] the name of the reference has been replaced

This article shows, from several points of view, why the equation E=mc2 must be an untenable equation.

**Category:** Relativity and Cosmology

[17] **viXra:1502.0148 [pdf]**
*replaced on 2015-02-18 04:02:37*

**Authors:** Radwan M. Kassir

**Comments:** 6 pages - [v2] minor editing

In this paper, the conclusive origin of the Special Relativity (SR) mathematical conflicts identified in the author’s prior related works will be presented and linked to these various conflicts. It is shown that for inertial reference frames in relative motion, the constancy of the speed of light postulate inherently results in time transformations independent of the spatial coordinates. The time of events occurring along the longitudinal coordinate axis is transformed with a different scaling factor than that of events occurring along the transverse coordinate axes. The incorporated x coordinate (considering that the relative motion is in the X-direction) in the Lorentz transformation time equation is the result of the SR assumption forcing the transformation to take a linear form as a function of the time and x coordinate. The SR derived Lorentz transformation time equation takes the assumed form by the aid of the imposed constancy of the speed of light equation, resolving itself as x=ct and x’=ct’ to supply the x and x’ terms to the assumed time transformation and its inverse, respectively. The resulting time transformation equation (and its inverse) dependence on the x (and x’) coordinate is therefore fake, since x (x’) in the transformation is nothing but ct (ct’) making the transformation time-dependent only, yet contradicting the time transformation for events in the transverse Y-Z plane.

**Category:** Relativity and Cosmology

[16] **viXra:1502.0107 [pdf]**
*replaced on 2015-09-18 17:30:15*

**Authors:** D. G. Taylor

**Comments:** 14 Pages.

The escape velocity equation shows mathematical parallels between General Gravitational Relativity & Special Relativity Time distortions. Like the light speed limit that SR puts to Real velocity, GR puts a parallel limit to escape velocity. Time Distortion would mean all Bosons are slowed. There is a mathematic argument that General Relativity Graviton slowdown leads to that escape velocity limit – the fundamental escape velocity from any body will never be greater than light speed. The principal equation introduced in this paper is a rephrased GR Time equation. The Escape velocity equation |vesc=(2GM/r)^½| can also be as written |vesc^2=(2GM/r)|.
So |1 – 2GM/rc^2| can be re-expressed as |1 – vesc^2/c^2|. It is reasoned the expression will never have a zero value, only a Graviton||Boson slowdown. Time passage would never cease; only approach cessation. Time slowdown predicted by Relativistic distortion is confirmed by muon decay. The different phrasing of the Gravitational Force equation |F=GMm/r^2| would mean a limitless gravitational force. While Bosons compression would be unlimited, any matter could escape after formation absorbing sufficient kinetic energy from slowing Photons||Bosons. This reasoning is illustrated by calculating Classic Relativity interpretations for distortions for Sagittarius A [SA] body at the center of the Milky Way.

**Category:** Relativity and Cosmology

[15] **viXra:1502.0106 [pdf]**
*replaced on 2017-07-02 06:04:36*

**Authors:** Yong Bao

**Comments:** 7 Pages.

The power product relationship between physical quantities (PQs) is determined by the generalized relational expression (GRE). Selecting two, three and four PQs in the GRE, we find the corresponding general formula. Assuming that the exponents of the physical constants in the formula are zero or fitted number, we obtain many famous equations without factors, such as Einstein's mass-energy relation, Schwarzschild black hole (SBH) event horizon temperature formula, and holographic dark energy (HDE) equation model, Casimir effect equation, Planck blackbody radiation formula, Stefan-Boltzmann's law, Einstein field equation, Newton's law of attraction, Schrodinger equation, Coulomb's law, Newton's second law, Clapeyron equation, superconducting thin film power law, two formulas of critical temperature Formula LSCO, centrifugal force formula, etc. Some new relationships are given, including the relationship between the square of total energy and the energy density of HDE, the one between the square of energy and the density of SBH, the one between the sixth-order radius of SBH and the energy density, the one between the pressure and the entropy density of the center of SBH, and so on. We show that GRE can determine the power product relationship between two, three and four PQs, which is useful and significant.

**Category:** Relativity and Cosmology

[14] **viXra:1502.0104 [pdf]**
*replaced on 2015-02-17 02:44:27*

**Authors:** Hasmukh K. Tank

**Comments:** Six-page letter

Accepting Einstein’s General Relativity Theory, that the changes in gravitational field can propagate at the speed of light, it is proposed here that: before an electron in an atom emits a photon, the energy (h f0) of the photon was a part of total energy of the atom; contributing to establishing the gravitational-field around the atom. As soon an electron in that atom emits a photon of energy h f0 , and the photon starts moving away from the atom, the gravitational-field around the atom partly reduces, proportional to the photon’s energy h f0 , and this wave of ‘reduced gravitational field’ propagates radially-outwards at the speed of light. And a part of energy of the photon gets spent in “filling” the ‘gravitational potential-well’ produced by its energy, when it was a part of energy of the atom. From the derivation presented here we find that the energy spent by the photon to “fill” the ‘gravitational potential-well’, during its inter-galactic journey manifests as the ‘cosmological red-shift’.

**Category:** Relativity and Cosmology

[13] **viXra:1502.0101 [pdf]**
*replaced on 2017-07-02 06:10:30*

**Authors:** Yong Bao

**Comments:** 6 Pages.

We propose a generalized relational expression (GRE), which unifies all uncertainty relations (URs) with dimensions through dimensional analysis. Here we find the general expression of UR, where the product of two or n dimensional non-commutative physical quantities (PQs) is equivalent to the power product of the reduced Planck constant h, the gravitational constant G, the speed of light in vacuum c, the Boltzmann's constant k and the base charge e, and find out that any dimensional PQ has a corresponding Planck scale. Since the dimensions are the same, many PQs have the same Planck scale. All Planck scales fall into two categories, one for the basic Planck scale and the derived Planck scale, and the other for the Femi-Planck scale, the Bose-Planck scale, and other-Planck scales. The corresponding Planck scale for any PQ is proven to be equal to the power product of Planck length, Planck time, Planck mass, Planck temperature, and base charge (or Planck charge). The GRE is found that the power product of the non-commutative PQs is found to be equal to the corresponding Planck scale. Applying GRE we also find the Big Bang UR on the temperature and volume of the Big Bang, and the Schwarzschild Black Hole (SBH) UR on the mass and volume of the SBH. These URs suggest no singularities in the Big Bang and SBH considering quantum effects. We show that GRE is generalized, interesting and significant.

**Category:** Relativity and Cosmology

[12] **viXra:1502.0099 [pdf]**
*submitted on 2015-02-13 09:29:09*

**Authors:** Hasmukh K. Tank

**Comments:** Four-page letter

Accepting Einstein’s theory, that the changes in gravitational field can propagate at the speed of light, it is proposed here that: before an electron in an atom emits a photon, the energy (h f0) of the photon was a part of total energy of the atom; contributing to establishing the gravitational-field around the atom. As soon an electron in that atom emits a photon of energy h f0 , and the photon starts moving away from the atom, the gravitational-field around the atom partly reduces, proportional to the photon’s energy h f0 , and this wave of ‘reduced gravitational field’ propagates radially outwards at the speed of light. And a part of energy of the photon gets spent in “filling” the ‘gravitational potential-well’ produced by its energy when it was a part of energy of the atom. From the Gauss’s law for gravity we find that the energy spent by the photon to “fill” the ‘gravitational potential-well’ during its inter galactic journey manifests as the ‘cosmological red-shift’.

**Category:** Relativity and Cosmology

[11] **viXra:1502.0092 [pdf]**
*submitted on 2015-02-12 09:26:57*

**Authors:** Valentin Danci

**Comments:** 54 Pages.

The Neo-classical Theory of Relativity uses concepts of Classical Mechanics and Classical Electromagnetism to describe the relativity of inertial motion better than it is described in the Special Theory of Relativity conceived by Albert Einstein in 1905.

**Category:** Relativity and Cosmology

[10] **viXra:1502.0089 [pdf]**
*replaced on 2015-02-14 14:27:30*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 11 Pages. In portuguese

We transform the Schwarzschild metric in spherical coordinates to rectangular, confirming that the conclusion obtained by Einstein for rulers disposed perpendicular to a gravitational field remains unchanged when using the exact solution of Schwarzschild, obtained under the conditions of static field in vacuum and with spherical symmetry. Beyond this transformation also talked again about the speed of light in radial and perpendicular directions measures in the Earth's surface, we recall tests for the verification of the principle of the constancy of the speed of light and calculate the acceleration of light and bodies in generally, in a radial static gravitational field.

**Category:** Relativity and Cosmology

[9] **viXra:1502.0075 [pdf]**
*submitted on 2015-02-10 12:07:24*

**Authors:** Mohammad Bahrami, Mehdi Zareie, Davood Hashemian

**Comments:** 13 pages, 3 figures and 1 table

We found out that force in nature cannot be infinite and by deriving Lorentz factor of force, we were able to obtain the relativistic form of the gravitation law and the second law of Newton. These relativistic equations are compatible with special relativity. We also propose a procedure for solving the dark matter problem in which the validity of these equations can be tested.

**Category:** Relativity and Cosmology

[8] **viXra:1502.0070 [pdf]**
*submitted on 2015-02-09 21:00:19*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 3 Pages.

Calculate the acceleration of light and bodies in general under the influence of a static gravitational field with spherical symmetry, in the vacuum and in a radial movement.

**Category:** Relativity and Cosmology

[7] **viXra:1502.0067 [pdf]**
*submitted on 2015-02-09 09:40:00*

**Authors:** Paul R. Gerber

**Comments:** 2 Pages.

The consistency and, hence, validity of the theory of General Relativity is questioned.

**Category:** Relativity and Cosmology

[6] **viXra:1502.0058 [pdf]**
*replaced on 2015-02-07 08:11:46*

**Authors:** Hasmukh K. Tank

**Comments:** Two-page letter

Inverse square-law is followed by electric charges, gravity, and light. Light has shown even an unexpected property, of double-slit-interference of single photons. So it is not unreasonable to expect inverse-square-law to be followed by single photons. Assuming that the single photon is emitted either from the surface of an electron, or a globular-cluster, or a galaxy, the derivation presented here suggests that even a single photon seems to follow the inverse square-law.

**Category:** Relativity and Cosmology

[5] **viXra:1502.0052 [pdf]**
*submitted on 2015-02-06 23:33:41*

**Authors:** Rodney Bartlett

**Comments:** 7 Pages.

This was adapted by me from my entry in the Foundational Questions Institute’s (FQXi’s) 2015 contest, “Trick or Truth: the Mysterious Connection Between Physics and Mathematics”. My entry is “Explaining The Mysterious Connection Between Physics and Mathematics By Reconciling the Unified Field and Anthropic Principle” (http://fqxi.org/community/forum/topic/2312). I’ve tried to make the article below understandable without referring to the FQXi entry, though it may well be recommended to view FQXi at times for more information. The article below touches on the following -
1) 3 POSITIVES OF SPACE-TIME, BLACK HOLES AND GRAVITY BALANCED BY A NEGATIVE HIGHER DIMENSION,
2) PARTLY REVISED GRAVITY REQUIRES THE IDEA OF NEWTON AND KEPLER THAT THE MOON CAUSES THE TIDES, TO BE JOINED WITH GALILEO’S PARTLY CORRECT IDEA THAT THE EARTH’S MOVEMENTS SLOSH ITS WATER
3) MICROWAVE BACKGROUND WITHOUT A BIG BANG, AND ALTERNATIVE TO INFLATION,
4) SUPERPOSED ELECTROMAGNETISM AND GRAVITY, QUANTUM SUPERPOSED PHOTON AND GRAVITON,
5) COSMIC EXPANSION VIA BARYON ACOUSTIC OSCILLATIONS,
6) RELATING MASS GENERATION BY PHOTON-GRAVITON INTERACTION TO BLACK HOLES, THE SOLAR NEBULA AND THE HIGGS BOSON,
7) HOLOGRAPHIC UNIVERSE, 5TH-DIMENSIONAL HYPERSPACE, AND DARK ENERGY

**Category:** Relativity and Cosmology

[4] **viXra:1502.0044 [pdf]**
*submitted on 2015-02-05 14:16:23*

**Authors:** Martin Erik Horn

**Comments:** 12 Pages.

The Dirac operator is the general relativistic generalization of Minkowski’s Lorentz operator and describes the geometric derivative in general relativity. But while the Dirac operator is discussed in length in books about Geometric Algebra, the discussion of the Lorentz operator usually is avoided. This leads to a huge didactical gap, as a discussion of the geometric derivative in special relativity becomes impossible without reciprocal frames.
These didactical problems are a consequence of three shocks special relativity suffered in its youth.

**Category:** Relativity and Cosmology

[3] **viXra:1502.0027 [pdf]**
*replaced on 2015-05-03 14:20:25*

**Authors:** Vladimir S. Netchitailo

**Comments:** 42 pages, 2 tables, 5 figures, 134 references

This paper provides an overview of the 4-D World – Universe Model (WUM). The World – Universe Model unifies and simplifies existing Cosmological models and results into a single coherent picture, and proceeds to compare the origin, evolution, structure, ultimate fate, and parameters of the World to the Big Bang Model.
WUM explains the experimental data accumulated in the field of Cosmology and Astroparticle Physics over the last decades: the size and age of the World; critical energy density and the gravitational parameter; temperatures of the cosmic microwave background radiation and the peak of the far-infrared radiation; observed expansion of the World and cosmological redshift; gamma-ray background and macrostructure of the World. Additionally, the Model makes predictions pertaining to masses of dark matter particles, photons, axions, and neutrinos; proposes new types of particle interactions (Super Weak and Extremely Weak) and the fundamental physical parameters of the World; explains “Pioneer Anomaly” and the rise of the solar luminosity steadily from about 70% of its current value during the last 4.6 Byr.
The Model proposes to introduce a new fundamental parameter Q in the CODATA internationally recommended values for calculating time dependent parameters of the World.

**Category:** Relativity and Cosmology

[2] **viXra:1502.0016 [pdf]**
*replaced on 2018-01-24 03:29:23*

**Authors:** Henok Tadesse

**Comments:** 6 Pages.

This paper reveals a mysterious law governing the Doppler effect of light. A new exponential term is introduced: eV/c . Einstein's thought experiment ( chasing a beam of light ) is very compelling and can be considered as self-evident. It is an immediate consequence of the non-existence of light carrying medium ( ether). The speed of light should be constant c independent not only of source velocity, but also of observer velocity. However, Einstein made a serious mistake when he tried to explain this. He misinterpreted the constancy of the speed of light as the relativity of space and time. This is an extraordinary, farfetched claim. Therefore, relativistic Doppler effect formula is based on a wrong foundation. Moreover, all existing theories of light ( special relativity, ether theory, emission theory) fail to explain Doppler effect for source- observer relative velocities greater than the speed of light. Although special relativity imposes that no ( relative ) velocity is greater than the speed of light, it is easy to obtain relative velocities greater than the speed of light without violating the universal speed limit. This can be done by an arrangement consisting of light source, observer and a mirror. Suppose that a light source and an observer are at rest relative to each other and located close to each other. Light emitted by the source goes to a distant mirror and is reflected back to the observer, while the mirror is moving towards the source and observer with velocity close to the speed of light. In this case, the velocity of the mirror image of the source relative to the observer is equal to 2c, for which the relativistic Doppler effect formula will be undefined because β = V/c will be greater than 1, resulting in a square root of a negative number. Theoretically, it is possible to get unlimited relative velocity between the mirror image of the source and the observer by using multiple moving mirrors. Therefore, a new theory of the real law governing the Doppler effect of light is required. Einstein's thought experiment can be understood as the constancy of phase velocity, not group velocity. The phase velocity of light is always constant c irrespective of source and observer motion. The group velocity of light behaves in a more conventional way: it is independent of source velocity, but dependent on observer velocity. The constancy of the phase velocity requires, unconventionally, not only change in frequency, but also change in wavelength for a moving observer. Even though this is also counter intuitive, it is not as farfetched as 'length contraction and time dilation'. Therefore, λ . f = λ’. f ’ = c . This theory invokes another problem: then what is the law governing the Doppler effect of light. The new law of Doppler effect of light is proposed as: f ‘ = f . e V/ c , λ’ = λ . e -V/ c , where V is positive for source and observer approaching each other. This theory explains the Ives-Stilwell experiment.

**Category:** Relativity and Cosmology

[1] **viXra:1502.0004 [pdf]**
*submitted on 2015-02-01 09:10:45*

**Authors:** Pierre Réal Gosselin

**Comments:** Français, 7 pages, site web: http://phrenocarpe.org/zhp/index.html , English at: http://vixra.org/abs/1501.0255

Nous faisons un retour sur l’expression de la loi de Hubble, réduisons Ho à trois constantes fondamentales et définissons la surface de Hubble
σH.
We review Hubble’s law formulation, we reduce Ho to a combination of three fundamental constants and define the Hubble surface σH .

**Category:** Relativity and Cosmology