This was to be my first post on your Gravitations forum:
The subject would be COVARIANCE
I’d like to discuss covariance. I have attached
a picture of the identity of 1/(1-x), what I will
call the “implicit covariant factor.” The right
hand side of the identity I will call the “explicit
covariant factor.” Never mind, for now, how the
probabilities and Shannon Entropy packets are
derived, they only introduce concepts that the
reader may not be familiar with, yet. It is left
as an exercise to show that 1/(1-x) has as a second
family member, [1/(1-x^2)]^0.5, the covariant Lorentz
factor when (v/c) is substituted for x, where the
first factor is the Pythagorean way to add vectors.
That is, the whole right hand side is how to add
vectors covariantly; the Lorentz way.
Now for the intuition; since the first factor on the
right hand side of the identity is (1+x), then I
suggest that the whole right hand side is how to
add scalars covariantly; the proposed way.
So you might ask? “Covariant inertia explains the
correct way to view simultaneous, synchronous, and
coincidental events…what would your so-called
covariant addition give us a correct view of?”
Umm, maybe a correct view of where gravity waves
eminate from… maybe from the covariant part,
and conserving the gravitational mass addition as
a simple (conserved) addition with the substitution
of GM/R/(c^2) for x; where the first power of M is
an un-creatable, un-annihilatable, constant of the
universe, the gravitational mass. And the possibility
that the covariant part is a “captive wave,” capable
of being radiated.
Intuition or fantasy? Why or why-not? Don’t ask me,
I just like the thought of something like “covariant
addition,” and can not help it. Lorentz and Einstein
taught us why covariance is necessary to fix the
Lorentz transform matrix by multiplying by the Lorentz
my feeling is that the addition of gravitational masses
is also covariant. Rule-me-out, but please don’t throw
me onto the brier-patch, said brother-rabbit to brother-
And if that isn’t enough, that the covariance is a first
principle that demands that the speed of light and gravity
be finite and equivalent, a reversal of a “common-sense”
notion of causality; and that covariance is the property
that accounts for chemical electromagnetic bonds on the
inertial side of the equation and nuclear electromagnetic
bonds on the gravitational side of the equation.
the role of mass, as used in physics, is widely discussed in
One statement was “the expression m0(1 – v2/c2)−1/2 is best suited for THE mass of a moving body”.
It is argued that the relativistic mass formula holds for all particles, including those moving at the speed of light, while the formula only applies to a slower than light particle (a particle with a nonzero rest mass) with ϒ = (1-v^2/c^2)^-1/2.
However, the relation E = m(rel)∙c^2 (Einstein) is generally invalid; firstly because moving particles always radiate and secondly, it is not distinguished between the gravitational mass mg and the inertial mass mi. In general, the energy E of moving particle is not conserved and the mg and mi are different by composed particles. Only by the elementary particles, e, p, P and E, we can be sure that mg = mi because they are not composed of other particles. The statement that E = m(rel)∙c^2 holds for all particles is a physically invalid claim.
Supposing all particles are composed of the elementary particles e, p, P and E the is the gravitational mass
mg = |(NP – NE)mP + (Np-Ne)me|
and the inertial mass
mi = (NP + NE) mP + (Np + Ne) me – E(bound)/c^2.
Both masses are greater or equal zero and are expressed with the numbers of elementary particles, with the elementary masses mP and me and in the inertial mass also the bound energy E(bound) enters.
COVARIANCE is connected to the covariant equation of motion of particles, which carrying two kinds of fields, however, neither the positions, nor the velocities of particles are exactly known. This uncertainty is a principle problem at the motion of particles. In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In physics, a basis is sometimes thought of as a set of reference axes. However, as a “set of reference axes” cannot be connected to “inertial frame of references” because the observers cannot define constant velocities with real physical particles. In the Minkowski space a basis with four unit vectors e0, e1, e2 , e3 can be introduced, and also an invariant distance between two points. However, it is dangerous to define covariance with the use of Lorentz factor ϒ = (1-v^2/c^2)^-1/2. In the equation of motion of particles all appearing term are covariant terms.