In this approach, nonsensical meta-phenomena like
time dilation and length contraction, along with other
SR's badly-resolved paradoxes, will be avoided.

We can invoke a strictly 3-d flat manifold plus time, described
by Galilean relativity equipped with a specific rule for Doppler
shifts, where addition of velocities remains purely as Euclidean
vectorial sum:

1. Define the frequency Doppler shift z as,

f' = f Exp(-v/c),
z = (f' - f) / f = Exp(-v/c) - 1,
v/c = - ln(z + 1)

2. Consider a thought experiment similar to those carried
out by Friedrich Hasenöhrl, as described in his papers
of 1904 and 1905.

Let two photons of equal frequency f, with combined
energy E_0 = 2hf, reflect back and forth on two mirrors
at rest, one in front the other, such that each reflection
occurs simultanously on each opposite mirror. Now, let the
mirrors move with relative speed v. This would mean
that a photon is seen as Doppler blue-shifted by a mirror
and the other photon is seen as red-shifted. So, according
to the definition of Doppler shift, provided above, we get a
difference between momenta as

p = (hf/c)(Exp(v/c) - Exp(-v/c)),
p = (2hf/c) sinh(v/c),
and since the combined energy is E_0 = 2hf, we get
p = (E_0/c) sinh(v/c)

To first order approximation, if v -- 0, then sinh(v/c) -- v/c,
so we would get p -- (E_0/c) (v/c) -- E_0 v/c^2 .
Also to first order approximation, the momentum of a particle
with mass m moving at small v -- 0, tends to p -- mv . Thus,
we get

E_0 v/c^2 = mv,
E_0 /c^2 = m,
E_0 = mc^2,
which is the famous mass-energy equivalence
equation.

Since this equivalence must be true for any particle at rest
with mass m, it implies that the momentum at speed v,
deduced above, p = (E_0 /c) sinh(v/c), must be expressed
in function of m, v and c, as

p = mc sinh(v/c).

Now, total energy E of a particle can be expressed as

E = mc^2 + E_k,
where E_k is kinetic energy, defined as
E_k = mc^2 (cosh(v/c) - 1),

thus, we get

E = mc^2 + mc^2 (cosh(v/c) - 1)
E = mc^2 cosh(v/c)

This latter equation means,

since
cosh(v/c) = sqrt(1 + sinh^2 (v/c)),
then
E = mc^2 sqrt(1 + sinh^2 (v/c))
so,
E^2 = m^2c^4 (1 + sinh^2 (v/c))
E^2 = m^2c^4 + m^2c^4 sinh^2 (v/c),

and as we have already attained above that
p = mc sinh(v/c), it yields

E^2 = (mc^2)^2 + (pc)^2,

which is the famous energy-momentum relation.

Summarizing:
We can see that E = mc^2 and E^2 = (mc^2)^2 + (pc)^2
are correct formulas, but, under Galilean relativity equipped
with the above mentioned Doppler shift rule, we can express
the momentum as

p = mc sinh(v/c),
and the kinetic energy as
E_k = mc^2 (cosh(v/c) - 1),

So, total energy E can be also expressed as

E = mc^2 cosh(v/c).

Discussion:
What's the difference between this approach and the well-known
relativistic approach, described in SR? Firstly, both approaches
are experimentally undistinguishable. Secondly, we can see that
the lorentz factor, gamma = 1/sqrt(1 - v^2/c^2) is replaced by the
factor, for some cases,

gamma' = (c/v) sinh(v/c),
which means that if v -- 0, then sinh(v/c) -- v/c,
so it would be gamma' -- (c/v)(v/c) = 1.

Therefore, gamma' = (c/v) sinh(v/c) can be used to deduce the
momentum

p = mv gamma' = mc sinh(v/c),

and total energy

E^2 = m^2c^4 + m^2c^4 sinh^2 (v/c),
E^2 = m^2c^4 + m^2c^4 (v/c)^2 gamma'^2 ,
E = mc^2 sqrt(1 + ( (v/c) gamma' )^2) ,

which in the limit gamma'= 1, it yields

E = mc^2 sqrt(1 + v^2/c^2),
with first order approximation as
E = mc^2 + mv^2/2,
and second order approximation as
E = mc^2 + mv^2/2 - mv^4/8c^2.

Under this approach, we can easily afford gravitation issues.
For example, if we are wanting to enhance Newtonian gravitation,
then the speed c should be replaced by the speed of gravity c_g,
so an orbiting test body with mass m, around a massive body,
would be endowed with momentum

p = m c_g sinh(v/c_g),

and angular momentum L would be then

L = m r c_g sinh(v/c_g),

where r is the radial distance to the center of
the massive body.

If we compare this angular momentum L with Newtonian
L_n = m r v , we see L_n is not preserved when relativistic
effects are taken into account due to strong gravitational
fields, whereas L must be preserved along any orbit.

Let's examine deeply what a momentum p = mc sinh(v/c)
means. Newtonian force F is defined as the rate of change
of momentum p. So, we get F = dp/dt = m d(c sinh(v/c))/dt.
Therefore, d(c sinh(v/c))/dt must be an acceletation,

a = d(c sinh(v/c))/dt = c cosh(v/c) d(v/c)/dt =
= c cosh(v/c) (1/c)d(v)/dt =
= cosh(v/c) d(v)/dt =
= cosh(v/c) a_n,
where a = a_n is acceleration for the case c = oo,
or for the case v -- 0.

IOW, a_n is Newtonian acceleration. For Newtonian gravity,
it is a_n = GM/r^2, thus

a = cosh(v/c) GM/r^2.

But now the question is, what is that ratio v/c? Is that c the
speed of light?, or is it the speed of gravity considered as
superluminal? If we can answer this latter question, then we
can answer what the ratio v/c is. If c is defined as the speed
of gravity, regarded as a finite superluminal speed, then v must
be a real tangential speed of the orbiting body. If c is defined
as the speed of light, then v must be an apparent speed deduced
through observed Doppler and ranging values. In any case,
the force of gravity would be

F = cosh(v/c) GMm/r^2.

Since experimental astronomical tests are performed by means
of light observations and measurements of Doppler and ranging,
we should choose c as the speed of light, so v would be the
apparent deduced speed.

We can now modified the Newtonian equation of the orbit of
a test particle,

r = (h^2/GM)/(1 + e cos(phi - phi_0)),
where
h is angular momentum per unit mass,
e is orbital eccentricity,
r and phi are polar coordinates wrt
center of the massive body with mass M,

making h = r c sinh(v/c), it yields

r = ( r c sinh(v/c))^2 /GM)/(1 + e cos(phi - phi_0)),
GM / c^2 = (r sinh^2 (v/c)) / (1 + e cos(phi - phi_0)).


Regards

Under this novel approach, we can calculate, for instance,
a escape velocity:

Kinetic energy of an orbiting test particle is

E_k = mc^2 (cosh(v/c) - 1)

And potential energy is

E_p = cosh(v/c)GMm/r

Now, we equate E_k = E_p, and it yields

mc^2 (cosh(v/c) - 1) = cosh(v/c) GMm/r,
c^2 (cosh(v/c) - 1) = cosh(v/c) GM/r,
cosh(v/c) = 1 / ( 1 - GM/rc^2),
v = c arccosh(1 / (1 - GM/rc^2))

A first-order approximation is v_e = sqrt(2GM/r),
which is the well-known Newtonian escape velocity.

Let's compute, for example, the complete escape
velocity (all higher order considered), for the mean
position of the Earth in solar system:

GM = 132,712,440,018 Km^3/s^2, is Sun's gravitational parameter,
R_e = 149597870.691 Km, is mean Earth-Sun distance, and
c = 299792.458 km/s, is the speed of light in vacuum,

Then

v = c arccosh(1/(1 - GM/R_ec^2)),
v = 42.1219 Km/s

On Jun 7, 2:21 am, Albert****o wrote:


1. Define the frequency Doppler shift z as,

f' = f Exp(-v/c),

No reason to define it in such an idiotic way.

On Jun 7, 2:56 pm, Dono wrote:
On Jun 7, 2:21 am, Albert****o wrote:



1. Define the frequency Doppler shift z as,

f' = f Exp(-v/c),

No reason to define it in such an idiotic way.

There are reasons that you will never be capable
of understanding, troll.

On Jun 7, 6:01 pm, Albertito wrote:
On Jun 7, 1:38 pm, Albertito wrote:



Under this novel approach, we can calculate, for instance,
a escape velocity:

Kinetic energy of an orbiting test particle is

E_k = mc^2 (cosh(v/c) - 1)

And potential energy is

E_p = cosh(v/c)GMm/r

Now, we equate E_k = E_p, and it yields

mc^2 (cosh(v/c) - 1) = cosh(v/c) GMm/r,
c^2 (cosh(v/c) - 1) = cosh(v/c) GM/r,
cosh(v/c) = 1 / ( 1 - GM/rc^2),
v = c arccosh(1 / (1 - GM/rc^2))

A first-order approximation is v_e = sqrt(2GM/r),
which is the well-known Newtonian escape velocity.

Let's compute, for example, the complete escape
velocity (all higher order considered), for the mean
position of the Earth in solar system:

GM = 132,712,440,018 Km^3/s^2, is Sun's gravitational parameter,
R_e = 149597870.691 Km, is mean Earth-Sun distance, and
c = 299792.458 km/s, is the speed of light in vacuum,

Then

v = c arccosh(1/(1 - GM/R_ec^2)),
v = 42.1219 Km/s

Vis viva equation (orbital velocity):
Under this novel approach, let m be the mass of the
test body. Total energy of the test body in its orbit is the
sum of its kinetic and potential energies,

E = mc^2 (cosh(v/c) - 1) - cosh(v/c) GMm/r

For orbits that are circular or elliptical, the total energy is
also given by

E = - cosh(v/c) GMm/2a
where
a is semi-major orbital axis

Now, equating

- cosh(v/c) GMm/2a = mc^2 (cosh(v/c) - 1) - cosh(v/c) GMm/r,

and solving for v, it yields

v = c arccosh(1 - GM/rc^2 + GM/2ac^2).

For a circular orbit, it is a = r, so

v = c arccosh(1 - GM/2rc^2 ),
is the velocity of orbiting body

Let's compute, for example, the orbital velocity
(all higher orders considered), for the mean circular
orbit of the Earth in solar system:

GM = 132712440018 Km^3/s^2, is Sun's gravitational parameter,
R_e = 149597870.691 Km, is mean Earth-Sun distance, and
c = 299792.458 km/s, is the speed of light in vacuum,

v = c arccosh(1 - GM/2R_ec^2 ),
v = 29.7847 Km/s

Sorry, there is a typo, I meant equating

- cosh(v/c) GMm/2a = mc^2 (cosh(v/c) - 1) - cosh(v/c) GMm/r,

and solving for v, it yields

v = c arccosh(1/(1 - GM/rc^2 + GM/2ac^2)).

For a circular orbit, it is a = r, so

v = c arccosh(1/(1 - GM/2rc^2)),
is the velocity of orbiting body

Let's compute, for example, the orbital velocity
(all higher orders considered), for the mean circular
orbit of the Earth in solar system:

GM = 132712440018 Km^3/s^2, is Sun's gravitational parameter,
R_e = 149597870.691 Km, is mean Earth-Sun distance, and
c = 299792.458 km/s, is the speed of light in vacuum,

v = c arccosh(1/(1 - GM/2E_ec^2)),
v = 29.7847 Km/s