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 , 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 . 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 , 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  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.
James A. Green Jim Green's Home Page
 Green, James A., Gravitation and the Electroform Model: From General Relativity to Unified Field Theory,
11th edition, Greenwood Research.
 Green, James A., The Field Equations of the Electroform Model:
From General Relativity to Unified Field Theory.
 Green, James A., The Red Limit: As Far as the Eye can See.
 Einstein, Albert, The Meaning of Relativity, 5th edition, Princeton Univ.
 Green, James A., Galaxy Formation, Greenwood Research.
 Bartusiak, Marcia, Beyond the Big Bang: Einstein's Evolving Universe, National Geographic, May 2005.
 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.
 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.)