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James A. Green May 2005, October 2013
Beyond Einstein's General Relativity
Dear Physicists, Editors, and Library Acquisitions Staff,
This is to submit some remarks, insights, and relevant Internet links concerning the National Geographic article Beyond the Big Bang: Einstein's Evolving Universe, by Marcia Bartusiak, which appeared in the May 2005 issue [6], and to suggest some alternative books. In the first place, there are good reasons for believing that the Special Theory of Relativity is everywhere good in local coordinates, but that classical General Relativity is wrong and needs to be replaced by unified quantum field theory [1,2]. According to Einstein's classical theory, if we consider two massive spheres on the y'-axis in coordinate system K' moving along the x-axis of a parallel coordinate system K, the two gravitating spheres no longer attract each other when the velocity v = c/2. At that speed, the grav-magnetic forces defined by the off-diagonal elements of the metric tensor balance the force of gravitational attraction. However, this prediction of the Einstein GR field equations predicts the existence then of a stationary frame in which two gravitating bodies do not attract [8]. This is contrary to the Principle of Relativity, according to which gravitation and all other natural forces are in synchronism with a local light-clock, so that a local observer cannot determine whether he is stationary or in motion.
How did this happen? Well, the Einstein field equations guarantee the local Lorenz covariance of the components of the metric tensor, NOT the local Lorentz covariance of the Force of Gravitation. That happened because the GR field equations were not properly chosen. In fact, what we did was this: we noted that the 4-divergence of the matter was zero, and that the result defined the conservation laws for matter and energy. Then we looked for a tensor derived from the Riemann curvature tensor that had an identically vanishing 4-divergence, the so-called Einstein tensor Eμ,ν = Rμ,ν - (1/2)gμ,νR, where the indices run over 0 through 3. If we set this tensor proportional to the matter tensor, we wind up with an interesting proposal for field equations that does get a lot of things right. However, detailed examination of its solutions together with its equation of motion produces the paradox that the Principle of Relativity is grossly violated for the local force of gravitation at v=c/2, as described above. Until we knew this, we did not quite understand what the consequences of the violation of local Lorentz covariance for the force of gravitation would be in General Relativity.
Einstein himself pointed directly at the problem complex here in his book The Meaning of Relativity [4], on or around page 102, where he shows that, to first order, Fgrav = m(Egrav + v x curl Agrav) + higher order terms. Of course, Bgrav = curl Agrav, where Agrav is derived from the off-diagonal elements of the metric tensor obtained as a solution of the Einstein field equations, which guarantee the local Lorenz covariance of the gμ,ν, but not of the gravitational forces described by the equation of geodesic motion. Incidentally, that Einstein's field equations predict geodesic motion minimizing the line element along a path of travel is usually stated a victory for his choice of field tensor equations

Eμ,ν = - κTμ,ν, or Rμ,ν - (1/2)gμ,νR = - κTμ,ν.

Instead of Einstein's GR field equations, we are looking at classical vector-boson field theory in local coordinates as the correct foundation for the unified field theory of forces, and have succeeded in deriving the 2nd-order effects of classical GR including the perihelion precession of Mercury, the curvature of light around the Sun, and the gravitational redshift on a different basis. The derivation of the field equations is more direct, proceeding from conservative hypotheses of special relativity and quantum mechanics. The principle of equivalence that Einstein introduced in 1911 [7] is retained, but the result is a space-time in which space is a flat vacuum and in which only gravitational time-dilation is as it was in classical GR. Superposition effects are required to obtain the observed perihelion precession and curvature of light around the Sun. In this view of the universe, the fundamental equations of physics are simple by comparison with classical GR, but we get the right symmetries and characteristics for both the weak and strong nuclear forces and the right local Lorenz covariance properties for both of the long-range forces of electromagnetism and gravitation. Indeed, locally their field equations have mirror-image form, so that all forces are all acting in synchronism within inertial systems.

I consider it quite unlikely that anything like the cosmological constant exists. When you have a feel for the derivation of fundamental laws of nature, you resist accretions to the fundamental forms, and discover how observed effects follow from the simplest assumptions. Let me explain how this can be. If we solve conservative field equations for gravitation for the case of an expanding universe of uniform density, we find that the Hubble constant has the form: H(t) = (2/3)/t meters/second per meter, where t is the age of the universe, assuming we are computing for the present matter-dominated era after the decoupling of radiation and matter. However, this does not yet account for observational time- delay, for the fact that when we look out in space a distance L meters, we are looking backwards in time by L/c seconds. Inserting this into our calculation to obtain the Hubble constant we expect to observe with terrestrial telescopes, we find H(t,L) = (2/3/(t - L/c). This has the consequence that the deeper we look into space, the larger the Hubble constant seems to be. I have plotted this function myself versus measurements of the Hubble constant presented by Micheal Rowan-Robinson, and I find the results compatible with his observations. That is, we do not need to introduce fantastic or strange "dark energies" to explain our observations, or weird new constants in the laws of the field of force, rather, we need to introduce rather obvious refinements that can be rigorously derived, such as observational time-delay L/c. Then we can deduce what we ought to measure and compare our measurements with observation. For instance, let H(t,L) = (2/3)/(t - L/c). Then the recession velocity of receeding galaxies equals the speed of light when c = H(L,t)L. Solving the equation for the Red Limit of visibility of receeding galaxies at this time, we find L = (3/5)ct. That is, we find whole, well-formed galaxies at the limits of observation rushing away at very close to light speed at the distance L = (3/5)ct, where t is the age of the universe in seconds. This shows up on some catalogs of the galaxies. Also, I have shown that the galaxies formed when the first bound atomic systems formed in the He++ to He+ phase transformation of the cooling Big Bang fireball, an event which caused a pressure jog. There are many other interesting details. For example, one starts the universe up by modeling the infall of dead galaxies from way out. Then one can obtain the subsequent oscillations of the system, which always preserves the total number of nucleons. I computed from an analysis of the inelastic properties of the Squeeze-Boom explosion that the period of the oscillating universe is about 238 billion years.
So, in the future, we are going to do experiments to test the unified quantum field theory of forces, which determines a different coupling strength for grav-magnetic forces than does General Relativity, which predicts that this force is 4 times as strong as it must be to preserve the Principle of Relativity in local coordinates.

Yours Truly,
James A. Green Jim Green's Home Page
Greenwood Research
[1] Green, James A., Gravitation and the Electroform Model: From General Relativity to Unified Field Theory, 11th edition, Greenwood Research.
[2] Green, James A., The Field Equations of the Electroform Model: From General Relativity to Unified Field Theory.
[3] Green, James A., The Red Limit: As Far as the Eye can See.
[4] Einstein, Albert, The Meaning of Relativity, 5th edition, Princeton Univ.
[5] Green, James A., Galaxy Formation, Greenwood Research.
[6] Bartusiak, Marcia, Beyond the Big Bang: Einstein's Evolving Universe, National Geographic, May 2005.
[7] Einstein, Albert, "On the Influence of Gravitation on the Propagation of Light", in The Principle of Relativity, Dover Publications, reprinted from the original 1911 article.
[8] Since all velocities v < c are observed in the expanding universe, frames of reference in which v = c/2 featuring co-moving material contents must exist in which, according to classical GR, the static gravitational field will be balanced by the grav-magnetic force due to the off-diagonal elements of the metric tensor for sources moving away from the viewer and normal to the expansion vector v = (c/2)(v/|v|). However, no one seriously believes that inertial frames of reference exist in the expanding universe in which the force of gravitation does not seem to be the same as it is in other inertial frames. Indeed, our solutions for the expanding universe themselves depend on the assumption that none of the inertial frames involved in the expanding universe has locally distinct properties, such as the property that gravitational forces are in balance for some material configurations with grav-magnetic forces and cease to act, as the Einstein GR field equations predict.
The location of the v = c/2 zone, neglecting observational time-delay, might be attempted from c/2 = [(2/3)/t]L, which yields L = (3/4)ct, where t is the age of the universe in seconds, but this is incorrect. If observational time-delay L/c is included, we find from c/2 = [(2/3)/(t - L/c)]L that L = (3/7)ct. Also, for the limit of observation when c = H(t,L)L, we find L = (3/5)ct, near which distance entire well-formed galaxies seem to fly away at velocities approaching the speed of light. (For the derivation, see The Red Limit: As Far as the Eye can See.)