User:Al'Beroya/Cosmological General Relativity

An activist nominated the full CGR article for deletion. The result was no consensus. The activist deleted it anyway. You can read the discussion here.

Current CGR link (after deletion) points to Moshe Carmeli. The history of edits can be found here.

There were some good points made about sourcing in the discussion. Originally, I tried to stick very closely to sources from the two authors most closely involved with developing the theory, so that it would be as true as possible to the core of the theory. However, the consensus seems to be that I should include as many viewpoints from as many sources as possible, which is a fair critique. My intent is to rewrite and re-source the article based on input in the discussion (at least from the non-biased crowd). Below is my original submission.

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Cosmological General Relativity (CGR) is a cosmological theory developed by Israeli theoretical physicist Moshe Carmeli that provides an alternative view of the fundamental framework of the universe in contrast to the standard cosmological model or Big Bang Theory. It extends Einstein’s theories of general relativity and special relativity from a four-dimensional space-time to a five-dimensional space-velocity. Physicist John Hartnett and others have extended the theory, and examined its implications. While the theory is neither widely known nor accepted, it has mathematical elegance, predictive and explanatory power equal to or greater than the theory of general relativity, and if correct, revolutionary implications.

Key Features
The theory has a number of significant differences from the standard cosmological model, also known as the Lambda Cold Dark Matter model, which is based in turn on the FLRW (Friedmann–Lemaître–Robertson–Walker) exact solution to the Einstein field equations. Some key features of the model include:
 * Assumes the Hubble law as a fundamental property of the universe.
 * Establishes the cosmic time $$\tau$$ as a fundamental quantity, in the same manner that $$c$$ is a fundamental quantity that represents the speed of light. It is equal to the Hubble time (the inverse of the Hubble constant) in the absence of gravity, and is distinct from ordinary local time (proper time).
 * Predicts the now well-known cosmic inflation of the universe, in three separate phases: decelerating expansion, constant expansion, and accelerating expansion, with an approximate length of each.
 * Provides an alternative formula for radioactive decay based on cosmic time instead of local time, which reduces to the standard formula for short local times.
 * Eliminates dark matter and dark energy. Instead, these are treated as a fundamental property inherent in the 5-D space-velocity fabric of the universe.
 * Establishes the theoretical underpinnings of the Tully-Fisher Relation that describes the velocity of stars in spiral galaxies as a function of their radial distance from the galactic center.
 * Establishes the theoretical underpinnings for the MOND (Modified Newtonian Dynamics) theory which attempts to explain the galaxy rotation problem that led to the theory of dark matter in galactic halos.
 * Alters the relation of redshift for distant galaxies to their age and distance.
 * Addresses the temperature of the universe indicated by the cosmic microwave background.
 * Attempts to reconcile the well-known problem of incompatible discrepancies between the age of the universe derived from redshift measurements versus its age as determined from stellar evolution and nucleosynthesis.
 * Proposes an approach to reconcile quantum theories with gravitational theories, towards a true quantum theory of gravity.
 * Allows compatibility with M-theory, string theory and the related brane world theory. Carmeli proposed a 5-D version of brane world theory based on CGR.

Carmelian Cosmological Relativity is a serious attempt to deal with many of the most important and challenging unsolved problems in cosmology and astrophysics. Nevertheless, it is not a widely accepted theory. This is likely due to two reasons. First, it presents a serious challenge to the established standard Lambda-Cold Dark Matter (&Lambda;-CDM) Big Bang theory. Secondly, Hartnett has adopted it as the basis for much of his work on a creationist view of the universe, since CGR provides a credible solution to the starlight problem present in Young Earth Creationist cosmologies. Hartnett describes this as applying "creationist boundary conditions" to Carmeli's theory, namely that the universe has a center and an edge. Creationist origin theories are generally controversial in the scientific community, particularly in Western culture.

Authorship
Moshe Carmeli was the Albert Einstein Professor of Theoretical Physics, Ben Gurion University (BGU), Beer Sheva, Israel and President of the Israel Physical Society. He received his Doctor of Science from the Technion-Israel Institute of Technology in 1964. He became the first full Professor at BGU's new Department of Physics. He did significant theoretical work in the fields of cosmology, astrophysics, general and special relativity, gauge theory, and mathematical physics, authoring 4 books, co-authoring 4 others, and publishing 128 refereed research papers in various journals and forums. Other physicists that have collaborated with Carmeli on various papers related to CGR include Silvia Behar (BGU), John Hartnett, Tanya Kuzmenko (BGU), Shimon Malin (Colgate University), and Firmin Oliveira (Joint Astronomy Centre, Hilo, HI). Moshe Carmeli died in 2007.

John Hartnett is a Research Professor with the Frequency Standards and Metrology research group for the School of Physics at the University of Western Australia (UWA), leading the development of the world's best ultra-stable cryogenic sapphire oscillator clock for applications in very large baseline interferometry radio astronomy. He received his PhD with distinction in Physics from UWA, where he is currently a tenured professor. He is a book author, has authored over 200 refereed research papers, and dozens of other articles and publications, and holds a patent for temperature compensated oscillators. Other physicists that have collaborated with Hartnett on various papers related to CGR include Moshe Carmeli, Koichi Hirano (Tokyo University of Science), Firmin Oliveira (Joint Astronomy Centre, Hilo, HI), and Michael Tobar (UWA).

Cosmological Special Relativity
Like Einstein, in order to derive his general theory, Carmeli first had to develop a theory of Cosmological Special Relativity (CSR). In this theory, Carmeli assumed zero matter, and therefore zero gravity in the universe. However, once he extended CSR to the general theory (CGR), he reintroduced matter and gravity as necessary preconditions.

Five-Dimensional Space-Velocity
As we noted in the introduction, the most important feature of the Carmelian cosmology is that it extends Einstein's 4-dimensional space-time to a 5-dimensional Riemannian space-velocity framework, with the 5th dimension of cosmic time incorporated into the double element of space-velocity. The timelike coordinates in Einstein's field equations become velocitylike in Carmeli's theory. Furthermore, the physical meanings of the energy-momentum tensor and coupling constant change. In CGR velocity primarily refers to the velocity of the expansion of space, and not usually the classical meaning of an object's speed and direction or the time derivative of distance. And as space and time are relative in Einstein's relativity, space-velocity is relative in Carmeli's cosmological relativity.

Basic Principles
In CGR, Carmeli posited several key principles:
 * 1) The Hubble Law is a fundamental property of the universe.
 * 2) The Hubble time is constant at all cosmic times for a given redshift.
 * 3) The principles of physics are the same at all cosmic times.
 * 4) The universe is stress free when the matter density of the universe is equal to its critical density.
 * 5) The universe is in a state of expansion at all cosmic times.

Cosmological (Copernican) Principle


In order to make the standard Big Bang theory work, scientists follow the Copernican principle, which assumes that the universe is homogeneous, with matter distributed approximately evenly throughout the universe when measured on the largest scales. Under this assumption, the universe has no center, and the Big Bang happened everywhere at the same time.

However, Cosmological General Relativity does not require this assumption, allowing for the additional possibility of a universe with a center. (It should be noted that this is not the same as saying that planet earth is at the center of the universe; it merely means that it is possible that the galaxies are distributed about a center point.) The theory of CGR does not require the earth to be at any special place, although it allows the possibility for it to be near the center. Some astrophysicists, including Halton Arp and Hartnett, have argued that the universe does in fact have a center, and that we are near it, cosmologically speaking. Hartnett points to recent large-scale redshift surveys of the universe that appear to indicate that the Milky Way galaxy may be relatively close to the center (within a few million light years). These surveys include the 2002 2dF Galaxy Redshift Survey and the more recent Sloan Digital Sky Survey, and arguably show concentric circles of galaxy groups and filaments with the Milky Way near the center. Arp showed that there is a statistically significant grouping of galaxy redshifts at regular intervals.

Shape of the Universe
CGR assumes that the universe is spherically symmetric with the observer at the center. This means that the universe is isotropic (i.e., approximately the same in all directions) from the observer's perspective. Furthermore, he assumes that the universe is infinite, open, flat in 3-D space, curved in 5-D space-velocity, and infinitely expanding.

However, by removing the need for the assumption of the Copernican principle, additional configurations are possible. With the possibility of a center of the universe, concepts as to whether or not the universe is bounded (i.e., whether or not it has an edge), have new meaning. Hartnett assumes a bounded, finite universe with a center.

Hubble's Law


Hubble's Law relates the distance of a galaxy to the observed change in the frequency of its light as it recedes away from us. This change in frequency, equivalent to a Doppler shift is known as redshift, and is normally interpreted as an indication of the expansion of space that carries distant galaxies away from us. Hubble’s law states that the recessional velocity of a galaxy as indicated by its redshift, is equal to the product of the Hubble constant times its distance.
 * $$h = {v \over r}~\approx~70.4~km/sec/Mpc$$
 * where $$v$$ is the recessional velocity, $$r$$ is the radial distance, and $$h$$ is Hubble’s constant. The value given for $$h$$ is the combined value from measurements from WMAP and other missions, using the &Lambda;-CDM model.

Carmeli notes that "The Hubble Law is assumed in CGR as a fundamental law. CGR, in essence, extends Hubble’s law so as to incorporate gravitation in it; it is actually a distribution theory that relates distances and velocities between galaxies."

As such, in CGR, Hubble's constant is no longer truly constant. It is dependent on gravitation, matter density, and redshift. Thus, the corrected form of Hubble's constant is now:


 * $$H_0~=~h \left [ 1 - { (1-\Omega_m)z^2 \over 6}\right ]~=~h \left [ 1 - { (1-\Omega_m)r^2 \over 6c^2 \tau^2}\right ]$$
 * where $$H_0$$ is the corrected Hubble constant (in the lowest approximation), $$\Omega_m$$ is the density parameter (explained below), and $$z$$ is the redshift.

Now the expanded Hubble constant, better described as the Hubble parameter, tells us the expansion rate of the universe at a given cosmic time. Furthermore, CGR predicts that the Hubble expansion is speed limited, meaning that the cosmological expansion is governed by the Relativistic Doppler effect.

Cosmic Time Defined
Perhaps the second most important element of the Carmelian theory, is the distinction between local time (or proper time in scientific language) and cosmic time. Simply put, time on cosmic scales runs differently than time on local scales. Although "cosmic time" is used in the Friedmann–Lemaître–Robertson–Walker (FLRW) solutions of Einstein's field equations, Carmeli defines the term differently and uses different assumptions (e.g., the aforementioned Copernican principle).

In CGR, the cosmic time $$\tau$$ is defined as the Hubble time (inverse of the Hubble parameter) in the limit of zero or negligible gravity. It is approximately equal to the age of the universe in the standard cosmological model. It is measured from the present epoch backwards towards the beginning of the universe. In practice in CGR, the cosmic time $$\tau$$ is typically calculated at a redshift of z=1, which gives it a slightly smaller value than the Hubble time, which is based on a redshift of z=0.

In standard theories, cosmic time is considered absolute, but in CGR it is considered to be relative. The effect is similar to the time dilation experienced in Einsteinian general relativity for objects moving at a significant fraction of the speed of light, with the speed of light as an absolute maximum. In CGR, the Hubble time in the limit of zero gravity becomes a maximum time allowed by nature. Just as Einstein showed that time dilation relative to a stationary observer can be dramatic at great speeds, Carmeli and Hartnett showed that time dilation can also be dramatic for large cosmic times relative to an observer at the present cosmic time.

Again, in Einsteinian relativity, all observers in all frames of reference measure the same speed of light. In like fashion, in CGR, all observers measure the same Hubble time, no matter at what cosmic time it is measured. This application of cosmic time in a 5-dimensional space-velocity framework has dramatic implications for the universe.

Radioactive Decay & Dating
Radioactive decay is a natural process where certain elements emit high energy particles, causing a transformation from one nuclide to another. Understanding this process is foundational to radiometric dating. Perhaps the most well-known example of this kind of dating is the technique of radiocarbon dating, where a material's age is determine by the ratio of Carbon-14 ($$C_{14}$$) nuclides to Carbon-12 ($$C_{12}$$) nuclides present in a sample of material whose age we wish to know. By calculating how many $$C_{14}$$ atoms in a sample have decayed to $$C_{12}$$ atoms, and using certain key assumptions, we can determine the age of the material. For individual atoms, the decay process is unpredictable according to quantum mechanics; however, for large quantities of atoms, the rate is predictable according to an exponential rate given by the standard formula:


 * $${dN \over dt}={1 \over T}N$$


 * where $$T$$ is the half-life of the element, and $$N$$ is the number of nuclides.

This is also expressed in equivalent forms as follows:


 * $$-{dN \over N} = \lambda dt$$


 * $$N(t) = N_0 e^{\left(-t/T\right)}$$


 * where $$\lambda$$ is the decay constant for the specific element, $$N_0$$ is the beginning number of nuclides, and $$t$$ is the time interval. The $$N(t)$$ term on the left hand side of the equation is a key term in the age equation in radiometric dating.

However, in the Carmelian framework, the time component that determines the half-life of the element now includes the cosmic time as well as the local time. Carmeli's new formula is given by:


 * $$N(t) = N_0 e^{ \left [ -t \alpha (t)/T \right ] }$$


 * where $$ \alpha (t) = {\frac {1 -t^2/3 \tau ^2}{1 -T^2/3 \tau ^2}} \ge 1$$; $$(t \le T)$$

The result is that for small cosmic times, Carmeli's formula reduces to the standard formula. But for large cosmic time periods, there is a significant deviation from the usual result. Using Carmeli's formula, it becomes clear that the elapsed local time is much shorter, meaning that the material decays faster than traditionally calculated. In short, the cosmological relativity of time shortens the radioactive decay period for the local observer.

Carmeli contends that this revised formula presents a possible resolution for a well known problem in astrophysics. As we mentioned before, we can estimate the age of the universe from redshift measurements. The light from the stars also shows emission lines and absorption lines that tell us their chemical composition. We can then also estimate the age of the universe by applying radioactive decay measurements to the elements that compose the stars, together with an understanding of stellar evolution. These two methods, especially when applied to distant galaxies, yield incompatible results; the oldest stars according to stellar evolution theory are much older than they should be when compared to redshift measurements. Carmeli contended that his revised formula for radioactive decay, applied to stellar evolution theory (together with more accurate measurements than were available at the time his 1999 paper was published), could resolve the apparent conflict. It also implies a younger age for our solar system and planet.

Age of the Universe
The age of our universe is generally accepted to be about 13.8 billion years. Carmeli agrees, but in like fashion to Einstein, asks the question: by which clocks? In Carmeli's cosmology, in contrast with standard cosmology, cosmic time is not absolute but instead relative. Thus, the lengths of days at the beginning of the universe as measured in cosmic time, are enormously different from our present and local 24-hour day.

Carmeli's formula for the length of day number $$n$$ is:


 * $$T_n = {\tau \over n + (n-1) - n(n-1) / \tau}

$$


 * where $$T_n$$ is the length of the $$n$$th day in cosmic time units, and $$\tau$$ is the Hubble time.

Carmeli reduces this to $$T_n = {\tau \over 2n-1}$$ for the early universe by neglecting the last term in the denominator in the first approximation (which is approximately zero when $$n$$ is much less than $$\tau$$). Thus, the length of the first few days of the universe in cosmic time is given by:


 * $$T_1 = \tau, T_2 = {\tau \over 3}, T_3 = {\tau \over 5}, T_4 = {\tau \over 7}, T_5 = {\tau \over 9}, T_6 = {\tau \over 11}, ...

$$

To add cosmic times, Carmeli uses the following:


 * $$T_{1+2} = {T_1 + T_2 \over 1 + T_1T_2 / t^2}

$$

This means that adding cosmic times in CGR is analogous to adding velocities in Einsteinian relativity (substituting $$T$$ for $$V$$ and $$t$$ for $$c$$). Thus, the elapsed time from day 1 to day 2 is not $$\tau+{\tau \over 3}~=~{4\tau \over 3}$$, but instead $${\tau + \tau/3 \over 1 + \tau^2/3\tau^2}~=~{4\tau/3 \over 4/3}~=~\tau$$.

It follows that the total elapsed time since the beginning of the universe, will always be the Hubble time $$\tau$$, no matter when it is measured. This is analogous to how an observer at any speed will always measure the same speed of light, no matter how fast they are moving.

Thus, in CGR, the age of the universe is determined by the Hubble time, corrected for gravitation. density and redshift.

Density Parameter
After developing the generalized solution of the 5-D field equations, Carmeli investigated the curvature of space implied by his results. The field equations include the energy-momentum tensor, which depend on the matter density of the universe. Rather than using the traditional average mass density in the Einsteinian cosmology, Carmeli used the effective density of the universe, given by:


 * $$\rho_{eff}=\rho-\rho_c$$


 * where $$\rho$$ is the average mass density, and $$\rho_c$$ is the critical mass density defined by:


 * $$\rho_c = \frac {3}{8\pi G\tau^2}$$


 * where G is the gravitation constant. This works out to $$1.194E-29 g/cm^3$$, which is roughly a few hydrogen atoms per cubic centimeter.

Carmeli uses the density parameter, calculated from the above terms, and represented by $$\Omega_m$$, to determine the ratio of average mass density to the critical mass density ($$\rho/\rho_c$$). This ratio represents a slight positive pressure in CGR (in contrast with a slight negative pressure in the standard theory), which determines how the universe expands.

However, Carmeli's calculation differs from the standard procedure. As previously noted, the cosmic time parameter $$\tau$$ is the inverse of the Hubble parameter, in the limit of zero gravity. Carmeli also noticed a redshift-dependence on the measurement of the Hubble parameter (explained below), and applies that to the value of cosmic time $$\tau$$ used in the above formula. With these adjustments, Carmeli determines the density parameter to be $$\Omega_m=0.245$$, which is equivalent to $$0.32$$ in the standard theory.

Flat Universe
Using values from CGR, Carmeli shows that the universe is within approximately 0.9% of being flat, which is consistent with the WMAP probe measurements of the cosmic microwave background, as well as results from measurements of distant supernovae. Thus, the 3-dimensional space component of the universe is Euclidean.

Three Phase Expansion
Carmeli was then able to show that the universe was expanding, in three different stages, corresponding to the value of $$\Omega_m$$.


 * 1) For the early universe when matter was compressed more than the critical density, $$\Omega_m$$ is greater than one, and the universe experiences a decelerating expansion.
 * 2) For an instant, $$\Omega_m$$ is equal to one, and the universe coasts with a constant expansion.
 * 3) As the matter stretches apart, the density drops below the critical value, and $$\Omega_m$$ is less than one.  In this final stage, the universe experiences an accelerating expansion.

Based on estimates of the universe's matter density, yielding $$\Omega_m$$ = 0.245, Carmeli estimated that the transition to the final phase occurred approximately 8.5 billion years ago. This also means that the universe will expand forever, eliminating any possibility of a big crunch or big bounce. Carmeli made his prediction about two years before supernovae observations revealed an accelerating universe to the scientific community at large.

Cosmological Redshift


The cosmological redshift is one of the three types of redshift:
 * Doppler redshift due to the Doppler effect (including relativistic effects)
 * Gravitational redshift due to relativistic effects in a gravitational field
 * Cosmological redshift due to the expansion of space

According to the standard formula, redshift is given by:


 * $$z = \frac{\lambda_o}{\lambda_e}-1 = \sqrt{\frac{1+v/c}{1-v/c}}-1 \approx \frac{v}{c} \ .$$


 * where $$z$$ is the redshift, $$\lambda_o$$ is the observed wavelength, $$\lambda_e$$ is the emitted wavelength, $$v$$ is the velocity, and $$c$$ is the speed of light.

However, in CGR, we must account for the fact that the expansion of space is occurring in 5-D space-velocity, and thus we must include the effect of the cosmic time parameter. Carmeli adjusts the formula as follows:


 * $$z = \frac{\lambda_o}{\lambda_e}-1 = \sqrt{1 + \frac{(1- \Omega_m)r^2}{c^2 \tau^2}}-1$$


 * where $$r$$ is the radial distance, and $$\Omega_m$$ is the previously discussed critical mass density.

In the lowest approximation, when the radial distance is much less than $$c\tau$$, then:


 * $$z \approx \frac{(1- \Omega_m)r^2}{2c^2 \tau^2}$$

The result, as Carmeli puts it, is that "redshift is less red" in CGR.

Cosmological Constant
The cosmological constant, represented by $$\Lambda$$, is a term used in Einstein's equations for general relativity to account for the expansion of the universe. It describes the density of the vacuum energy of space, and is sometimes viewed as a possible form of dark energy. According to George Gamow, Einstein later recanted, calling it his "biggest blunder". Although it is debatable whether or not Einstein actually used those exact words, he clearly abandoned and rejected the cosmological constant.

CGR does not have a cosmological constant. Instead, the value and effect is inherent in the 5-D space-velocity geometry that describes the universe. But Carmeli was able to show the equivalent value for Einsteinian 4-D relativity predicted by CGR, is given by:


 * $$\Lambda={3 \over \tau^2}=1.934E-35~sec^-1$$

This indicates a positive pressure, and confirms the accelerating expansion of the universe. Furthermore it is very close to the value needed to make the standard cosmological model match current observations.

Post-Newtonian Gravitation
One of the major problems of modern astrophysics is that Newton's law of universal gravitation doesn't always live up to it's 'universal' moniker. The law works well on local scales, such as in our solar system, or for satellites in orbit around the earth. However, on scales where gravitation is weak, such as for lightweight objects in interstellar space, or stars in the outer reaches of galaxies, or in intergalactic space, then the law breaks down, leading to the "missing mass problem". It is exactly this problem that led to the theory of dark matter and energy.

CGR introduces a new regime of gravitation for weak acceleration where a new law dominates. For traditional solar-system scales, Newtonian gravitation dominates, and the effects of Carmelian gravitation are negligible. However, for scales where gravitation is weak, most notably in the outer arms of spiral galaxies, the Carmelian gravitation term dominates, and the Newtonian force becomes negligible.

Newtonian Gravitation


According to Newton's law of universal gravitation, published in 1687, the force between two bodies is proportional to the product of their masses, and inversely proportional to the square of the distance between them. Mathematically, this is given by:


 * $$F=G{m_1m_2 \over r^2}$$


 * where $$F$$ is force, $$G$$ is the gravitational constant, $$m_1$$ and $$m_2$$ are the two masses, and $$r$$ is the radial distance between them.

A few decades earlier, between 1609 and 1619, Johannes Kepler solved the problem of the motion of planets, introducing elliptical orbits as the solution. Together with Newton's law of gravitation (with relativistic corrections), we can very accurately predict the motion of planets around the sun, and satellites around the earth. But as we move to larger scales, such as predicting the motion of stars around the disc of a spiral galaxy, Newton's law begins to break down.

Newtonian Problem: Spiral Galaxy Rotation Curves


In order to study the motion of stars around spiral galaxies, astrophysicists apply the Doppler effect to the light received from those distant stars. The redshift or blueshift of the star's light tells us the motion relative to us. Using the cosmic distance ladder, we can determine the distance to the star. Repeated measurements over a period of years tell us the angular speed or proper motion. Using a little trigonometry, we can then determine the tangential velocity. Plotting these velocities gives us what is known as a galaxy rotation curve.

According to Newton's law, the velocities of stars should decrease in an inverse-square relationship along a smooth curve as you move farther away from the center of mass of the galaxy. However, this is not what we observe. Instead, we see the velocities initially drop more or less in accordance with Newton's law, but as the stars reach a regime where the gravity from the central mass of the galaxy is very weak, then their velocities briefly increase, and flatten out with an approximately steady velocity all the way to the edge of the galaxy.

Attempted Solution: Tully-Fisher Relation
In 1977, astronomers R. Brent Tully and J. Richard Fisher tried to explain the relationship of the velocities in the rotation curve by relating them to the intrinsic luminosity of the galaxy. The intrinsic luminosity is one of the ways astronomers estimate the mass of galaxies, and its accuracy depends on accurately knowing the distance to the measured galaxy.

Tully and Fisher determined that the luminosity was proportional to the galaxy's rotational velocity (at the edges) raised to a power between approximately 3 and 4, given by:


 * $$L \propto v_{rot}^\eta$$


 * where $$L$$ is the luminosity, $$v_{rot}$$ is the rotational velocity of the outer stars in the galaxy, and $$\eta$$ is a value between 3 and 4, depending on the brightness (divided into bands) of the galaxies being studied; generally speaking, the relationship is taken to be the fourth power of the velocity.

However, the Tully-Fisher relation has no theoretical basis; it merely describes an approximate relationship which generally holds, but which must be adjusted depending on the brightness, mass, and distance of the galaxy being observed. Direct application of the model results in inconsistencies and discrepancies that the model does not explain. The traditional explanation requires invoking the existence of cold dark matter.

Attempted Solution: Modified Newtonian Dynamics (MOND)
In 1983 Mordechai Milgrom took a slightly different approach to explain the rotation curve problem, proposing that in the regimes where gravitational acceleration is very weak, that Newton's law no longer holds, and acceleration is no longer linearly proportional to gravitational force. MOND proposes a similar relationship to that developed by Tully and Fisher, but with a different basis. Mathematically, the relation is:


 * $$v^4=GMa_0$$


 * where $$v$$ is the rotational velocity, $$G$$ is the gravitational constant, $$M$$ is the mass of the galaxy, and $$a_0$$ is a constant of acceleration..

First, Milgrom picks the fourth power of velocity, consistent with the range of values picked by Tully and Fisher, but instead of relating it indirectly to mass via luminosity, he relates it directly to the mass of the galaxy. To get the relationship to work, he introduces the acceleration constant $$a_0$$, whose value is calculated empirically from observations; $$a_0$$ is approximately equal to $$10^{-10} m/s^2$$.

About this constant, Milgrom said : "... It is roughly the acceleration that will take an object from rest to the speed of light in the lifetime of the universe. It is also of the order of the recently discovered acceleration of the universe." Milgrom's constant is only within a factor of 5.8 of his description; however, as we shall see, he was tantalizingly close to what Carmeli discovered.

CGR Solution: Carmelian Gravitation Regime
Carmeli's approach is similar to Milgrom's, but in five dimensions, and with an underlying basis for Milgrom's acceleration constant. Carmeli again takes the fourth power of the velocity, and like Milgrom, relates it to the gravitational constant and the mass of the galaxy. However, Carmeli goes a step further than Milgrom in his treatment of the constant of acceleration.

Carmeli defines $$a_0$$ in terms of the speed of light, and the cosmic time (from the Hubble parameter) as follows:


 * $$a_0~=~c/\tau~=~cH_0$$


 * where $$c$$ is the speed of light, $$\tau$$ is the cosmic time, and $$H_0$$ is the Hubble parameter.

In Carmeli's theory, when gravity weakens to the point that it only produces accelerations on the order of the critical acceleration $$a_0$$ (or $${2 \over 3}a_0$$ in Hartnett's adjusted version), then the force exerted by gravity no longer follows Newton's inverse square law. Instead, the force is proportional only to the simple inverse, which dramatically influences motion in weak gravity regimes.

Carmeli presents a revised version of Newton's equation for gravity as follows:


 * $$\nabla^2\phi(x)=4\pi k\rho(x)$$


 * where $$\phi(x)$$ is the gravitational potential, $$k=G{\tau^2 \over c^2}$$, $$\rho(x)$$ is the mass density, and $$\nabla$$ is the Laplace operator.

The primary difference, is that in Newton's equation, $$k~=~G$$, without Carmeli's $${\tau^2 \over c^2}$$ term. However, Carmeli points out that unlike in the classical Newtonian equations which deal in the motion in 3-dimensional space, in CGR the motion of particles is in the sense of the Hubble distribution; it applies on a cosmological scale in 5-D space-velocity with a different physical meaning. In CGR, unlike the standard assumption, stars in galaxies are not immune to the Hubble expansion; their motion is affected by it.

Like some other theories, CGR also predicts that gravity propagates in vacuum as a wave at the speed of light. This is a post-Newtonian concept first predicted by Einstein's theory of general relativity. Thus CGR confirms Einstein's predictions, but with a more general form of the field equations than that presented in traditional general relativity. Traditional theories of gravitation predict gravitational waves dependent only on space and time; Carmeli's theory of gravitation in CGR includes gravitational waves propagating through space-velocity (including ordinary 4-D spacetime), but also including a redshift dependence relative to the source of the gravitational wave.

Resolution of the Rotation Curve Problem
Carmeli's short-form equivalent to Milgrom's formula for the velocity relationship is:


 * $$v_c^4\propto GM{c \over \tau}$$

Hartnett, using later observational data for the rotation curves of a representative selection of galaxies, defines a constant of proportionality ($$2/3$$), making the above relation into an equality, as follows:


 * $$v^4={2 \over 3}GMa_0$$

However, Hartnett uses Carmeli's definition of the acceleration constant, $$a_0$$. Using a combined value for the Hubble parameter of $$70.4 km/sec/Mpc, a_0~=~6.8E-10 m/sec^2$$, and in the limit of low acceleration, Hartnett's formula is consistent with Milgrom's results.

Finally, Carmeli presents a post-Newtonian equation for the velocity of stars circling a galaxy:


 * $$v_c^4=\left ( \frac{GM}{r} \right )^2~+~2GMa_0~+~a_0^2r^2$$

On the right side of this equation, the first term is the Newtonian component, the second is the Tully-Fisher/Milgrom component, and the third term is purely Carmelian. (However, it should be noted that the middle term does not include Hartnett's proportionality adjustment.)

The end result is that for normal gravitational regimes, such as those inside a solar system, the Newtonian gravitational term dominates. As distances increase and gravity weakens, the post-Newtonian terms begin to dominate, and velocities become proportional to the simple inverse of distance. In Milgrom's MOND theory, at great distances there is an arbitrary attractive force. But in Carmeli's CGR, there tends to be a repulsive force, which provides a mechanism for the aforementioned expansion of space on a cosmological scale.

Now using Carmeli's formula, it can be seen (as in Figure x ) that observations of the velocity of stars in the outer discs of spiral galaxies very closely match the theoretical predictions of CGR. Not only do they match without any adjustments or banding, they provide a better match than any competing theory to date.

Approach to a Quantum Theory of Gravity
One of the most important challenges in physics is the development of a theory of everything. Although there are a few theories which show promise towards that ultimate unification of all the great theories, none have yet achieved a complete description of all the forces of the universe. However, it is generally understood that a quantum theory of gravity is a necessary component of the ultimate theory, unifying quantum theory and the theories of gravity based on Einstein's general relativity.

General relativity has been very successful at describing the behavior of the universe on very large scales, while quantum mechanics has been very successful at describing the behavior of the universe on very small scales. However, attempts at combining the two theories tend to result in unresolvable problems, or formulations that have no meaning when applied to the physical world. Gravity, which has thus far been described well by Newton's laws and general relativity, cannot be described well in terms of quantum mechanics. Furthermore, the current state of technology in particle accelerators cannot recreate the incredibly high energies required to study the quantum effects of gravity. Quantum gravitational effects may not even be observable except near the Planck length, the theoretical smallest possible length, which is many orders of magnitude smaller than what we can currently measure. It is theorized that at this scale, and at the highest energies, is where the fundamental forces, including gravity, will unite.

Quantum gravity is typically treated with an effective field theory, which come with high energy limitations. Carmeli proposed an approach using gauge theory, which is a specific type of field theory. The gauge theory allows the mathematical description of gravity's behavior in a field which represents all points and scales in the spacetime or space-velocity of the universe. Specifically, he proposes using the SL(2,C) gauge theory of gravitation as a way to describe the behavior of gravity in the regime of length, time, and energy where gravity can be quantized. His approach attempts to show that "There is a cosmological phase which is generically prior to metric spacetime, and it should be possible to quantize gravity at this phase."

Mathematically, Carmeli's SL(2,C) gauge theory is a group theoretical formulation of the Newman-Penrose formalism, equivalent to general relativity. But the gauge theory can be applied to additional geometries (specifically affinely connected manifolds and general differentiable manifolds). The resulting mathematical functions can then be transformed into systems of non-linear partial differential equations. The generalized solution representing the quantization of gravity described in the equations can then, in principle, be developed. However, Carmeli notes that such a generalized solution is "mathematically challenging," in much the same way that it took years to find even the first few solutions to Einstein's field equations from general relativity; addtional exact solutions are still being discovered to this day. However, Carmeli died before developing a complete theory of quantum gravitation in the framework of CGR's 5-D space-velocity.

Einsteinian vs. Carmelian Relativity
One of the strongest features of 5-D Cosmological General Relativity is that in all equivalent cases, it reduces to Einsteinian 4-D General Relativity when the 5th dimension of space-velocity (the velocity of the expansion of space) is reduced to zero. It is when the 5th dimension is included that effects on cosmological scales become important. In the table below, Einstein's theory (and its derivatives in the Friedmann–Lemaître–Robertson–Walker standard model, and successor Lambda Cold Dark Matter standard model) is presented on the left, while the corresponding Carmelian version is presented on the right.

Galaxy Rotation Curves


As discussed above, galaxy rotation curves provide an excellent test of Carmelian gravitation in CGR. Hartnett has tested Carmeli's predictions against the curves from several galaxies, including:
 * The Milky Way Galaxy (type Sb barred spiral galaxy)
 * Galaxy IC 342 (type Sc spiral in the constellation Camelopardalis)
 * Galaxy NGC 598, also known as Messier 33 (M33), the Triangulum Galaxy, or the Pinwheel Galaxy (type Sc/SA(s)cd II-III spiral in the constellation Triangulum)
 * Galaxy NGC 1097 (type SBb I-II barred spiral in the constellation Fornax)
 * Galaxy NGC 2590 (type Sb spiral in the constellation Hydra)
 * Galaxy NGC 2841 (type Sb I spiral in the constellation Ursa Major)
 * Galaxy NGC 3198 (type SBc/Sc II barred spiral in the constellation Ursa Major)
 * Galaxy NGC 6503 (type Sc III dwarf spiral galaxy in the constellation Draco)

In all cases, the measurements of the velocities of stars and tracer gases in the outer discs of the spiral galaxies provide a nearly exact match to the velocities predicted by Carmelian gravitation.

Type 1a Supernovae


A Type 1a supernova is a special type of supernova that has a very specific brightness profile when it explodes, called a light curve. The light curve is (arguably) almost identical in all supernovae of this type. Because of this regularity, astronomers can use this as a standard candle to measure distances to the most distant galaxies, hundreds of millions, or even billions of light-years away. Normally, the light from distant galaxies is very faint and redshifted; other than Type 1a supernovae and the D–&sigma; relation (a type of standard ruler), redshift is our only available measure for the farthest distances.

However, supernovae briefly outshine their entire host galaxy, allowing for much better observation. If we are lucky enough to catch a Type 1a supernova, then we can independently measure the distance to the galaxy as a check on the distance derived from redshift.

The trick is that the distance we calculate from redshift is dependent on the model of the universe used, and we must use different formulas for different ranges of distances. How well the supernova-derived distances match the redshift-derived distances varies significantly depending on the cosmological model used. A correct cosmological model should yield a good match between the two different types of measurement.

Hartnett compared Carmeli and Behar's model for redshift distance measurement against the Type 1a supernovae measurements from the High-Z Supernova Search Team of Riess et al,, the Supernova Legacy Survey of Astier et al, and other high redshift data published in the Astrophysical Journal in 2003-2004. As mentioned previously, Carmeli and Behar predicted a redshift-distance relationship limited by the relativistic Doppler effect on the expansion of the universe. This is given by:


 * $${r \over c\tau}={sinh \left( \varsigma \sqrt{1-\Omega_m (1+z)^3} \right ) \over \sqrt{1-\Omega_m (1+z)^3}}$$
 * where $$\varsigma={v \over c}={(1+z)^2-1 \over (1+z)^2+1}$$.

Using the theory of CGR, Hartnett was able to show an excellent fit for supernovae data with the speed-limited version of the redshift distance relationship. He further shows that the fit does not require any dark matter when considered in Carmeli's 5-D space-velocity.

Hubble Parameter Measurements


An important difference between CGR and standard cosmology is that CGR recognizes that the Hubble parameter $$H_0$$ is dependent on redshift. In this theory, the Hubble parameter is only a constant in vaccuum when there is no gravity. However, our universe has matter, gravity, curvature, and expansion. As previously mentioned, Carmeli's new calculation to adjust the Hubble parameter explicitly includes the density of the universe and the redshift. CGR predicts smaller values for $$H_0$$ as distances increase.

Early estimates for Hubble's parameter ranged as low as 50 and as high as 90. In 1958, Allan Sandage posited the value to be about 75, although it took decades for his estimate to be validated. In 2001, the Hubble Space Telescope provided our first look into deep space, and the first accurate mesurement of the Hubble parameter at 72&plusmn;8 km/sec/Mpc. Many subsequent space missions have provided additional measurements of the Hubble parameter, such as the Chandra X-Ray Observatory, Spitzer space telescope, Wilkinson Microwave Anistropy Probe (WMAP), and the Planck mission. These probes, and upcoming missions such as the James Webb space telescope, allow us to look ever farther into space (and further back in time).

The overall trend shown by these measurements is that the measured value tends to decrease as we measure it at greater distances and higher redshift. The data so far is consistent with CGR's predictions, and appears to confirm the theory.

The Pioneer Anomaly


The deep space probes Pioneer 10 and Pioneer 11 were launched from Cape Canaveral in 1972 and 1973, respectively. Pioneer 10 became the first probe to transit the asteroid belt, visit the planet Jupiter, and leave the solar system. Pioneer 11 was the first probe to visit Saturn and the second to visit the asteroid belt and Jupiter. As the Pioneers moved beyond the orbits of Saturn and Uranus, astronomers at NASA's Jet Propulsion Laboratory (JPL) noticed that Pioneer 10 and 11 were moving just the tiniest bit too slowly. Both spacecraft experienced a tiny slowdown of approximately 8.74 x 10-10 meters/sec2, meaning that each year the spacecraft were about 400km short of where they were predicted to be.

Carmeli applied the principles of CGR to an explanation of the anomaly. By using cosmological transformations, and relating the maximum possible velocity (the speed of light, c) to the CGR's maximum time $$\tau$$, Carmeli calculated the minimum allowable acceleration $$a_{min}$$. Using a value for the Hubble parameter of 70 km/sec/Mpc, Carmeli finds that this minimal acceleration is given by:


 * $$a_{min} = {c \over \tau} \approx 7.6E-10~meters/sec^2$$

Carmeli argues that the effect is due to the Hubble expansion, and that because the spacecraft is carried along with space as it expands, it is also affected by an additional acceleration; in this case, it is towards the sun. His minimal acceleration corresponds very closely with the observed anomaly. This could be viewed as related to the clock acceleration explanation.

However, in 2012, seven years after Carmeli published his attempted explanation, Slava Turyshev et al published an explanation from thermal recoil force that was previously unaccounted for. This is now the generally accepted explanation.

On the other hand, however, there is enough uncertainty that it is possible that the thermal recoil explanation does not explain 100% of the anomaly (by some accounts it is only 10-20% of the effect). There may be cosmological/expansion effects or new physics, such as Carmelian gravitation/CGR at work.

Unfortunately, the last signal received from the Pioneers was in 2003, so we cannot continue to monitor and the anomaly. Most other probes do not lend themselves well to similar studies, due their design and flight profile. However, some have suggested dedicated deep space probes to study the phenomena in the regime of weak gravitation at extreme distances from the sun. Such probes could shed light on non-Newtonian gravitational regimes, including CGR.

Implications

 * Dark Matter and Energy: Dark matter and energy do not exist; their purported effects are features of the 5-D fabric of space.  The most significant features of the standard model of the universe must be abandoned, in the same way that it had to be admitted that the planet Vulcan did not exist, after the precession of the orbit of Mercury was explained by Einstein's General Relativity.
 * Cosmological Constant: The cosmological constant is inherent in the 5-D curvature of space.
 * Hubble law: The Hubble Law is a fundamental property of the universe.
 * Cosmic time: The cosmic time is a fundamental quantity and universal constant.
 * Temperature: The temperature of the universe is relative to the cosmic time.
 * Density: The density of the universe is relative to the cosmic time.
 * Expansion of the Universe: The universe is now in a state of accelerating expansion, which will last forever.
 * Fate of the Universe: The universe will not end in a Big Crunch or Big Bounce.  The Big Freeze or Big Rip are still possible.
 * Redshift: "Redshift is less red" in CGR; thus, distant galaxies are closer than previously estimated.
 * Radiometric dating: Ages based on radioactive decay must be adjusted for cosmic time. The older the material, the greater the error in age.  Thus, the age of the earth, based on radiometric dating of rocks, must be re-evaluated; the true age in local time is shorter.
 * Newtonian gravitation: Newton's law of gravitation does not correctly describe motion where the acceleration of gravity is extremely weak, typically at galactic and intergalactic distances. Gravitational energy dissipates into heat at these distances.  New formulas must be used.