** ****DISPROOF OF EINSTEIN’S PRINCIPLE OF EQUIVALENCE**** **

**DISPROOF OF EINSTEIN’S PRINCIPLE OF EQUIVALENCE**

**Yamin Choudhury**

The Aga Khan School

Dhaka, Bangladesh

June 18, 2010

A flaw is found in the very the principle of equivalence as formulated by

Albert Einstein and a new deduction of observed results is undertaken without the

use of the principle. The non equivalence of gravitational and inertial masses

readily leads to the deduction of bending of light by twice the amount when going

past a star compared to the classical deduction and the correct precession of perihelion

of Mercury's orbit around the Sun. Further, the gravitational red shift formula of General Relativity is believed to be wrong and a correct one, presumably, is deduced

which not only gives the correct time adjustments for GPS systems but also provides an

explanation for the apparent runaway acceleration of the universe without recourse to

any 'dark energy' concepts.

The principle of equivalence was formulated by Einstein on the presumption that from inside of a closed compartment there was no way to tell the difference as to whether the compartment was accelerating in space or was stationary in a gravitational field that would produce the same acceleration in the opposite direction in a free falling mass in it from inside of the compartment without looking outside of the compartment. Or, in terms of forces, that a spring balance having a mass hanging on it would show the same stretch if the compartment was accelerating upwards or if the compartment was stationary on a planet with a gravity, that would cause same acceleration downwards in a freely falling mass, pulling it downwards.

The fact that till to date general relativity could not be unified with either electromagnetism nor with quantum mechanics demands, I believe, a critical re-thinking into its very foundations with the hope of discovering on one hand flaw or flaws in it and on the other hand an explanation as to why some observations are reported to be its confirmations.

Raising the presumption of equivalence principle to the status of a postulate Einstein concluded that light traveling horizontally across the compartment would bend downwards under gravity since accelerating the compartment upwards in space would cause the light ray to do so and under the equivalence gravity would have to do the same. The flaw or error in the principle is right here at the start!

The thought model used by Einstein uses a ray of light entering the compartment from outside in a horizontal direction (perpendicular to the direction of gravity force or to the direction of acceleration) through a tiny hole in the wall of the compartment, say for example through the left wall, and moves on to hit the right wall at some point on it. (This is already equivalent to looking at outside from inside the compartment! It is already in violation of the principle, Einstein's derivation is like a sleight of hand, after all how does one look at the outside but by allowing light to come from outside. Then how can one not tell the difference between gravity and accelerating frames if one is looking outside?). Since the compartment is accelerating upwards and the speed of light though large is finite the right wall would move upwards in that time to some extent. The path of light would have had been a straight line sloping downwards if the compartment had been moving up at a constant speed. Since it is accelerating upwards the speed of the compartment keeps on increasing upwards causing light to continually bend or slope down more and more, in a downward arc relative to the inside of the compartment, until it hits the right wall. So the path traced out would be a downward curve. Einstein concluded light would trace out a similar curve under gravity with the compartment stationary in it since he assumed impossibility of detecting the difference between gravity and pure acceleration.

What, I think, Einstein failed to see was that the ray would bend downwards not only inside the compartment space as it was traveling across it but also at the very entry point in the left wall! Whatever the direction of a ray entering through the hole in the wall was would keep on entering with greater and greater downward slope, as it is accelerating upwards, and thereafter would bend further down through the space in the compartment before hitting the right wall. The result would be that the light spot on the right wall would not be stationary! The spot formed on the wall would keep on moving downwards and that too with increasing speeds. (That is how the information of 'outside' is carried by light into the 'inside'). Whereas, under pure gravity with the compartment stationary in it the spot formed on the wall would be stationary. The principle is in principle wrong! The diagram below depicts what is being asserted.

Even if we did not have the compartment but just looked at light with the observer under pure acceleration free from gravity the light would bend more and more 'downwards' because of the increasing speed of the observer 'upwards' but that would cause the slope or gradient of the ray path to continually change at every point of the space in the neighborhood of the observer not only for an instant but continually throughout time so long the rays are on and the observer keeps on moving under acceleration causing the path to continually keep on bending and displacing downwards. Therefore under acceleration the curvature of the path is time dependent whereas under pure static gravity it would be time independent at any given point in space, that is, the bending would be constant in time, or static, under pure gravity alone. So whether we look at light from inside of a compartment accelerating or under pure acceleration without resorting to a compartment there is no equivalence with static gravity. It is this time dependence of curvature under acceleration that got mixed up in Einstein's formulation with gravity due to the erroneous identification of acceleration with gravity that led him to commit his 'greatest blunder' in inventing the cosmological constant to get rid of the time dependence, which then he regretted upon Hubble's discovery of cosmic expansion. The constant is again being used but is arbitrary and does not follow logically from theory.

To avoid looking outside the compartment one could consider firing the light horizontally from within the compartment from,say, the left wall to the right wall horizontally and then claim equal bending of a light beam under pure acceleration or under an equivalent static gravity. But this would not still establish the equivalence, let us see how. Since the energy E of light is given by *E = mc ^{2}* where

*m*is the effective mass of light and

*c*its speed while for an equivalent classical particle (

*by an equivalent classical particle we mean a particle not subject to special relativistic laws*) of same mass and momentum

*mc*the energy E is given by

*E = (1/2) mc*. Hence, even if light crossed across a lift sitting stationary in a gravity field and fired horizontally from within the lift , to avoid having to look outside of the compartment, it would reveal the difference between pure acceleration and gravity, for light would bend down by double the amount a classical particle would since it would have to fall down through twice the height of what a classical particle would do to gain twice the energy,compared to a classical particle for the same change in momentum traveling at the same initial speed. This is because E = pc for light the change of E would be c times the change of its momentum p, but E=1/2 pc for a classical particle at speed c, hence, its change of energy E would be 1/2 c times of its change of momentum p. Since change of momentum would be same for both due to same force of gravity acting on both for same time, as they go across the distance in same time light would have to fall through twice the distance the particle of same mass would fall through to gain twice the energy change! Light would bend down by twice the amount an equivalent classical particle would do.

^{2}The bending would be same for both light and particles if the lift alone were purely accelerating upwards at a rate equal to that for free falling under the gravity force. It is easy to see this just consider a ray of light aimed from the left horizontally at a body on the other side as the light is fired the lift also starts to accelerate upwards the light is going to hit the body since the acceleration of the lift through space can have no effect on the motions of the light and the body. Thus the light will fall relative to the lift by the same amount as the body will. But this will be half the distance light would fall under a gravity producing the same size of free fall acceleration in a body. Or consider viewing rays of light traveling a straight distance aimed at and hitting a body, both in an inertial frame, by an observer who is in a frame accelerating in some direction, 'up', the rays of light would keep hitting the body for frame acceleration could not affect the event of the light hitting the body. Hence, both light rays and the body would be seen to accelerate equally downwards.

If one considers the case of inertial
frames it is easy to see now that light and a particle would go along
straight paths in deep space far away from forces of any kind. But in a
frame falling freely under gravity though the particles moving in it would follow
straight paths at non-relativistic speeds inside the lift light would
not! Since light bends, or falls in this case, by twice the amount by
which a classical particle would fall in the same time and which is the
amount by which the lift falls down through space its path would be bent
downwards or towards the gravity source relative to the inside of the
lift. This destroys the equivalence of pure acceleration and
gravitational field in space and also destroys the equivalence of
inertial frames and frames free-falling under gravity. We are talking
about both the strong and weak principle of equivalence as being wrong. Pure acceleration
and gravity are just not equivalent. They are equivalent only in
classical mechanics without consideration of special relativity or the
electromagnetic nature of light in it. The equivalence is approximately valid for very low (non-relativistic) velocities only.

Other considerations making equivalence principle impossible.

Throughout we have assumed uniform acceleration possible, which only
gravity can produce at distances far away from the source of gravity. We assumed
so only to take on Einstein in terms of his own formulation of the
equivalence principle. In practice to produce uniform acceleration in a
frame of reference by other than gravity is impossible as
because the compartment has to be either pushed up or pulled up and the
signal from one vertical end, floor or the roof, to the other end will
take time due to light speed being the limit for travel of any signal and cause a relative speed
between floor and the roof to arise. If the compartment is being
pulled up then the roof will keep moving up and away faster than the
floor and the total height of the compartment will keep getting larger.
While if it is being pushed up the floor will keep moving up faster than
the roof and the total height will keep on getting shorter. For example
let us take a lift of height *h* that is going to be accelerated up at a
rate *a* by being pulled up. The signal for the pull would take a time *t =
h/c* to reach the floor from the roof and pull it up with an
acceleration *a*. In that time the roof will have acquired an upwards
speed v relative to the floor given by *v = ah/c* thus the roof will be
going away with a speed ever increasing with the increasing of height h
that takes place. Thus a mass hanging from the roof of the compartment
by a spring will be accelerated up by the pull of the roof on the
spring, therefore the spring will keep elongating under acceleration
while it will remain of constant length under static gravity. The
opposite happens when the floor is pushed up with an acceleration a,
then the roof keeps coming down with speeds decreasing with decreasing
heights. The equivalence of gravity and pure acceleration cannot be formulated.

The relative speed, say, u between floor and roof leads to a further
non-equivalence of gravity and pure accelerations. Let a beam of light
be fired from the floor towards the roof where an ideal reflector
reflects it back to the floor and let the compartment be accelerated by being pulled up from the roof. Say, the speed of the floor moving up at
the time of firing the beam of light up be *v *then the roof would be at slightly larger velocity v+u since it would have sensed the pulling up signal earlier. When the light arrives at the roof the roof would have increased speed * *to
v+2u (as light and the pulling signal both travel at same speeds) and the floor would reach speed v+u at that time. Upon reflection
at the roof light would undergo a Doppler red shift proportional 2u + 2u
= 4u (reflection causing double Doppler shifting) and when it reaches back to the floor the floor would be at a further increased
speed of v + 2u, as it would pick up a further speed u upwards in the time light
travels down from roof to floor. Viewed from the floor light would undergo Doppler blue
shift proportional to 2u, this would cause a net red shift in the light
viewed proportional to 4u - 2u = 2u. Thus light would show a net red
shift upon being sent up and reflected back under pure acceleration. The opposite would happen if the lift were to be accelerated by being pushed from underneath, that is,would undergo net blue shift.
Under gravity in a frame at rest or in uniform velocity relative to a
planet the light sent up would undergo a certain amount of red shift on
reaching the roof due to a loss of kinetic energy and then undergo an
equal amount of blue shift on reflecting back to the floor, due to gaining back the kinetic energy, producing
zero net Doppler shift. By this method Einstein's passenger in the 'lift
experiment' could have determined whether they were accelerating in
space or were sitting stationary on Earth.

We cannot have equivalence under the conditions. That is why a charged particle sitting stationary inside the compartment would be able to sense pure acceleration by radiating and sense only pure gravity by not radiating.Again if the charge is free falling in an accelerating frame there is no radiation but free falling under gravity would cause it to radiate, that is how matter falling into a black-hole on ripping apart into free charges radiate intensely. And, as explained earlier, accelerating frames would cause the stretched spring to keep on stretching further in time whereas under pure gravity alone the stretch would remain constant. **It is because charges can sense the difference between pure acceleration and gravitational field that electromagnetism cannot be reconciled or unified with general relativity based on the equivalence of the two.**

Physicists had used Newtonian dynamics for
light in computing the bending of light by the Sun and had gotten the
wrong measure in spite of Maxwell having had laid down the correct
formula of energy-momentum relation for light many decades before. When Einstein first formulated his general
relativistic formula for the bending using warping of space it was not
correct. Thanks to WW1 his formula could not be tested immediately and
found wrong and he had had time to correct it. What did he correct? I
suspect (I do not know) he
simply replaced E = ^{1}/_{2} mc^{2} by E = mc^{2} for light somewhere in the
formulation and as for all the tensor formulations they were
redundant since his equations become solvable only when space is ‘flat’
which is another way of saying curvatures of spaces are not there. The
curvatures are just a lot of confusion screening the simple truth: E =
pc for pure energy. Simply using E = mc^{2}, and interpreting it correctly, instead of E = ^{1}/_{2} mc^{2} in Newtons equations for
hyperbolic motion of a body going past a heavy mass pulling it
by gravity gives the correct bending for light. Gravitational lensing and GPS settings are
possible to be explained by using only the equation E = pc or its
equivalent E = mc^{2} for light
and E^{2}
= m_{0}^{2}c^{4} + (mvc)^{2} for particles with rest masses along with Newton's gravity properly.

The equation E = mc^{2} is a statement that the inertial mass of light is different from its gravitational mass.

Double bending of
light under gravity is a disproof of the equivalence principle since
light
would not bend as much under pure acceleration as it would under
equivalent gravity. By equivalent
gravity is meant the gravity that would produce the same free fall
acceleration in a mass released from rest in it. Light shows it has a
difference in its gravitational and inertial mass and by special
relativity any mass when moving would have different gravitational and
inertial masses. In the case of light we have that the inertial mass of light as computed from its change of
momentum under a force or collision is always less than its mass as computed from its change of gravitational potential energy or the change in its kinetic
energy. For instance if light went out vertically from the surface of a
gravitating body its energy would change by c times its change of
momentum instead of 1/2 c times which means that we use 2V
instead of V in the potential energy term in the equation to be derived shortly as but the cognition of this fact. The equivalence of the
inertial and the gravitational masses does not exist except for masses
at rest and never for light. The assumed equivalence has led, I believe, GR to produce a wrong formula for gravitational red shifts which has caused the wrong interpretation of accelerating expansion of the universe.There is no need to invent the concept of 'dark energy' to explain the apparent
runaway acceleration of the cosmos. It seems that GR is really invalid and that all its correct predictions must be deducible from special relativity and Newton’s
gravity, all the so called verification of GR have so far have been for extremely small gravity and accelerations. We
shall try to deduce some of the results below.

Classical bending of hyperbolic paths of particles past the Sun.

(One familiar with classical mechanics may skip this section)

In the conventional
method of classical mechanics taking the energy E of light to be ½ mc^{2}
gives c^{2} to be equal to 2E/m the equation of motion of a particle
moving past a gravitating body in a hyperbolic orbit is formulated by writing E =
½ (mv_{r}^{2 }+mv_{θ}^{2})+V in polar
coordinates. Here v_{r} and v_{θ} are the radial and angular
components of the velocity v of the particle as measured from the origin of the
coordinate system. The expression is rearranged to give

v_{r}^{2 }= (2/m)∙(E – V - mv_{θ}^{2}/2)
(1)

taking the root on both sides of the eqn. we get:

v_{r} = dr/dt = √(2/m)∙√(E – V - mv_{θ}^{2}/2) (2)

giving dt
= dr / √(2/m)∙√(E – V - mv_{θ}^{2}/2) (3)

now angular momentum L = mv_{θ}r using which we have

mv_{θ}^{2}/2 = L^{2 }/ (mr^{2}) (4)

and eqn (3) becomes

dt = dr /
√(2/m)∙√(E – V- L^{2 }/ 2mr^{2}) (5)

L = mv_{θ}r
= mr^{2} dθ/dt since v_{θ} = r dθ/dt and we have dθ = Ldt/mr^{2}
which gives

dθ = L dr /(mr^{2})√((2/m)∙(E – V- L^{2 }/
2mr^{2}))

or dθ = dr /(mr^{2}/L)√((2/m)∙(E – V- L^{2
}/ 2mr^{2})) which
simplifies to

dθ
= dr /(r^{2})∙√(2mE/L^{2}
– 2mV/L^{2}- 1^{ }/ r^{2}) (6)

setting u = 1/r and V = ─ k/r converts the equation to

dθ
= ─ du / √ (2mE/L^{2} + 2mku/L^{2}-
u^{2}) (7)

integrating both sides gives the solution

θ = θ’
─ arc cos ((L^{2}u/mk─1) / (√(1+2EL^{2}/mk)) (8)

simplifying leads to

u = 1/r = (mk/L^{2})∙(1
+ e∙cos(θ ─ θ’)) (9)

here e = √(1+2EL^{2}/mk). We have 1/r = GMm/L^{2} (1 + e cosΦ) , (10)

where e = (1 + (2E/m)(L/GM))^{1/2} and is very
large with respect to our Sun for velocity of particle equal to speed of light,
here L is the angular momentum of the particle, r is its distance from the Sun
or the star center, M is the mass of the Sun or star, Φ is the angle from one
end of the path at infinity to the other end at infinity, or the total sweep,
about the radius R_{o} the minimum distance of approach, and its
maximum would be equal to π if there was no deflection.

The deflection angle δΦ
is found by assuming small angle of change and then setting 1/r = 0 for r = ∞
and using small angle approximations to get the conditions. First we get the
nearest distance R_{o} of the particle from Sun center by setting cosΦ
=1 and thus making 1/r a maximum at r = R_{o} and get 1/R_{o} =
(GMm/L^{2})(1 + e). Rearranged the equation gives GMm/L^{2} =
1/(R_{o}(1+e)). Then setting r → ∞ or 1/r → 0 we get:

1/∞ = 0 = GMm/L^{2} (1 + e cos(Φ/2))

or, 0 = GMm/L^{2 }+ (GMm/L^{2})
e cos(Φ/2)

or using the relation
GMm/L^{2} = 1/(R_{o}(1+e)) in the second term we can simplify
it to

0 = GMm/L^{2} +(1/(R_{o}(1+e))) e cos(Φ/2) = (e/(1+e)). cos(Φ/2)
/R_{o}) + GMm/L^{2}

The ratio e/(1+e) is practically unity since e is extremely
large and the relation in a gravitational field being independent of the mass
of the particle just considering for a unit mass leads to no loss of
generality. For a unit mass L = R_{o}c and GMm/L^{2} becomes
equal to GM/(R_{o}c)^{2} the above equation simplifies

to: 0 = (1/R_{o}) cos
(Φ/2) + GM/(R_{o}c)^{2} ,

where Φ now is equal to π + δΦ and R_{o} is the
nearest distance of approach, here fixed at the radius of the Sun or the star,
and for which Φ = 0. The condition simplifies to:

-cos (π/2 +δΦ/2) = sin (δΦ/2) ≈ δΦ/2 = GM/(R_{o}c^{2}) whence
δΦ = 2GM / (R_{o}c^{2})

This is the incorrect formula for bending as worked out by
using Newton’s gravity and Newton’s mechanics. What is to be understood
here is that the value of c^{2 }as computed classically from c^{2}
= 2E/m is double that for light. Light is subject to its own rule of having c^{2}
= E/m, hence, for light the value of c^{2} computed by classical
mechanics should be replaced by c^{2}/2 in the formula for bending as
G, M and R_{o} are independent of whether it is light or classical
particles and we get:

δΦ/2 = GM/(R_{o}c^{2 }/ 2)
= 2GM / R_{o}c^{2} or δΦ = 4GM / (R_{o}c^{2})

This formula is the correct form. The error was in applying Newton’s mechanics along
with his laws of gravity upon light. It is only Newton’s gravity that was under investigation
and not his laws of mechanics which was already understood to be not applying
to light since special relativity and/or electromagnetism was already known. We can get the formula by applying the correction right in the beginning in the formulation of the equation.

The bending of light under gravity.

In the case of light one has to understand that the energy E of light in terms of its mass and speed is given by E = mc^{2} and not by E = ½ mc^{2}. Light cannot change its magnitude of speed but it can change its components perpendicular to its direction of motion by any magnitude less than the speed c. It can pick up any small speed from zero to v << c perpendicular to its path and thus will be subject to the usual rules of classical mechanics along those directions. To accelerate to v from zero it will pick up kinetic energy ½mv^{2} along the direction since speed in increasing from zero to v averages v/2 speed unlike light's instantaneous constant velocity c along its direction of motion.Thus for the small changes that light velocity will undergo in the case of passing by the Sun the kinetic energy part of the energy expression ½ (mv_{r}^{2 }+mv_{θ}^{2}) remains the same. However the energy equation E = mc^{2 }= pc (p is the momentum) for light demands that light accelerates perpendicular to its motion at a rate such that it picks up twice the energy gain an equivalent classical particle would gain under same momentum change under the same force. The only way light can do so is to present half its gravitational mass as inertial mass and accelerate at twice the rate of the equivalent classical particle to acquire twice the speed of the classical particle for then it would gain twice the energy with same gain in momentum. Light has half the inertia for the gravitational mass it has when falling transverse to its path of motion! That is special relativity, a point that has never been emphasized properly. A lesser mass subject to same force as a greater mass will undergo same change of momentum in same time but gain greater energy in the same time. And light has to do just that to satisfy the equation: E = mc^{2}. In formulating the equation of motion the kinetic energy part of the equation remains the same as is for the classical particle but to cater for the extra gain of energy by virtue of reduced inertia, in this case half inertia, the potential energy lost by light for equivalent changes in momentum has to be double that for the loss of potential energy by the equivalent classical particle. Thus the equation E =
½ (mv_{r}^{2 }+mv_{θ}^{2}) + V for total energy of the light in its motion has to be replaced by E =
½ (mv_{r}^{2 }+mv_{θ}^{2}) + 2V thereafter following standard procedures, after appropriate rearrangements, the equation (3) for time element dt becomes:

dt
= dr / √(2/m)∙√(E – 2V - mv_{θ}^{2}/2) and that makes dθ to be given by

dθ
= dr /(r^{2})∙√(2mE/L^{2} + 4mk/L^{2}- 1^{ }/ r^{2})

simplifying using u =1/r and integrating like before leads to

u = 1/r = (2mk/L^{2})∙(1
+ e∙cos(θ ─ θ’)), here e = √(1+EL^{2}/2mk).

We have: 1/r = 2GMm/L^{2} (1 + e cosΦ) or for a unit mass, without loss of generality,1/r = 2GM/L^{2} (1 + e cosΦ), which is the form that gives, by the same procedure as applied for the classical particle, the correct bending angle given by δΦ = 4GM / (R_{o}c^{2}). Though the value of e is changed but its value is so large that the ratio e/(1+e) remains virtually unity and does not alter the bending angle δΦ = 4GM / (R_{o}c^{2}) noticeably. Thus all one has to
do is multiply the effective potential energy change of light by two when subject to force acting on it since its effective inertia is half that of masses at rest.That is what produces the double bending of light. And also gives the formula for gravitational lensing effect from pure special relativistic considerations alone. If the error in calculating the bending of light using the relation *E = ½ mc ^{2 }*instead of using

*E = mc*had not been made at the start the correct bending of light would have had been found and the next thing to occur naturally in the minds of the physicists, I believe, would have had been the noticing of the difference of inertial masses of light and classical particles and physicists would have had thought differently.

^{2 } Precession of the perihelion of planet Mercury around the Sun.

So far we have
considered the special case of light only. Now we shall generalize the method
to include the case of relativistic particle at any speed *v*. Total energy of any particle represents total mass and defines the quantity for gravitational attraction, hence, the bending under gravity is directly proportional to the total energy. Momentum on the other hand represents resistance to bending and makes the bending inversely proportional to it. For deducing the formula we take our cue from the previous case of light beam bending under gravity. We saw that bodies moving with some speed would fall under gravity by amounts different from what a classical particle would do due to relativistic reasons forcing greater increase of energy for a given increase of momentum. We find that for classical or non-relativistic particles change of energy *E* (due to falling under gravity in this case) per unit momentum *p* is *1/2* *m**v ^{2}* /

*mv*=

*1/2 v*while for light it is

*mc*/

^{2}*mc*=

*c*which is twice what it would be for a classical particle at speed

*c*.

We saw that a body would increase in kinetic energy due to falling under gravity. The relativistic form of kinetic energy increase for a given momentum is more than what a corresponding classical particle's kinetic energy would increase by for same momentum. To do that the relativistic particle would have to fall by a greater height and would fall by greater and greater height for greater and greater kinetic energy. Greater height of fall amounts to greater bending of the particle's horizontal path past the massive body. So to find the increase of bending under gravity due to the reduced inertia due to relativistic conditions we take the ratio of *E / p* of
kinetic energy *E* and the momentum *p* as a measure of increase of bending or the reduction of
inertia of a moving mass at speed *v *under gravitational attraction, going horizontally or orthogonal to the direction of the gravity field. For purely classical particles or in
this case of particles at speeds *v*
very much smaller than light speed *c*
the factor or coefficient multiplying the potential energy term *V* in the expression for total energy is
unity. For light the factor or coefficient is two to cater for the double
increase of energy compared to the increase of kinetic energy in classical
particle for the same increase of momentum in both. It is obvious that for a
particle at any velocity *v*, small or
large, the factor has to be a value between one and two. The factor should
reduce to unity for *v* small compared
to *c* and reduce to two for *v = c*. To deduce the general form of the
factor for a particle at any speed *v*
we have to consider the ratio *E / p*
which we now do.

The kinetic energy *E* of a mass *m _{o}* at speed

*v*is given by

*E = mc*where

^{2 }─ m_{o}c^{2},*m*is the rest mass of the body and

_{o}*m*its rest mass energy and

_{o}c^{2}*m*its total or relativistic mass at speed

*v*and

*mc*its total energy at speed v, the expression simplifies further, by using the relation

^{2}*m = γ m*from special relativity, to

_{o}*E = (γ─1)*

*m*where

_{o}c^{2}*γ = (1 ─ v*. The momentum

^{2}/c^{2})^{─1/2}*p*

*= mv*becomes

*p = γ m*which makes the ratio

_{0 }v*E/p*to be:

*E / p = (γ─1)* *m _{o}c^{2 }/ γ m_{0 }v = [( γ ─1) / γ]∙(c^{2}/v)
= (1 ─ 1/γ)∙(c^{2}/v) *(Z1)

*γ* = ∞ for *v = c *and* *the ratio *E / p *becomes* [(1 ─ 1/∞] ∙ (c ^{2}/c)
= c *which is the value of

*E / p*for a particle at light speed like light itself and not

*c/2*as would have been the case for a non-relativistic particle. For

*v ≈ 0*or for

*v*very much smaller than

*c*the last part of expression Z becomes

*(0/0) c*as

^{2}*γ*=1 for

*v = 0*and becomes indeterminate and so has to be expanded in binomial terms for the limiting ratio. We have

*(1 ─ 1/γ) = 1─ (1 ─ v*which on expanding becomes

^{2}/c^{2})^{1/2}*1─ (1 ─ v*, ignoring terms of higher powers of (

^{2}/2c^{2})*v/c*) than the second, and we get

*E / p = ( v*the correct value of

^{2}/2c^{2})∙(c^{2}/v) = v / 2*E / p*for a classical particle or for a particle moving slow compared to light speed.This clearly shows how the bending of light is twice that of an equivalent particle, not subject to relativistic mass increase with speed, traveling at light velocity.

In the energy
equation *E =m(dr/dt) ^{2 }+ V + l^{2}/mr^{2}*
the potential energy gain term

*V*has to be multiplied by a coefficient to take into account the different inertial masses at different speeds falling through different heights in a gravitational field causing different changes in

*V*. For light it has to be multiplied by two and for a classical particle it has to be multiplied by unity only. We get the bending factor for

*V*as a ratio of bending at any velocity

*v*to that for purely classical non-relativistic bending by dividing (Z1) by

*v/2*the factor for classical bending

*.*The coefficient becomes

*2*∙

*[(1 ─ 1/γ)∙(c*, which then gives two for light and one for a classical particle. If we substitute

^{2}/v^{2})]*v = c*Lorentz factor

*γ*becomes infinity and we get the factor is equal to

*2*∙

*[(1 ─ 1/*

*∞*)∙(c

^{2}/c

^{2})] = 2. If we let

*v*be very small, say

*v ≈ 0*then

*2 *∙ *[(1 ─
1/γ)∙(c ^{2}/v^{2})]= 2∙[1 ─ (1 ─ v^{2}/c^{2})^{1/2})∙(c^{2}/v^{2}) *

which becomes: *2∙[1 ─ 1]/0 = 0/0 *an indeterminate.

Hence, we must expand the indeterminate term in small values of * v. *And then we get the factor is equal to

*2∙[1
─ (1 ─ v ^{2}/(2c^{2}))]∙(c^{2}/v^{2}) = 2∙[v^{2}/(2c^{2})]∙(c^{2}/v^{2})
= 1, *

we have ignored powers greater than the squares in the expansion.For a relativistic particle at speed *v* the potential term is

* 2∙[(1
─ 1/γ)∙(c ^{2}/v^{2})]*

*∙V*

and the expression for energy *E* of the particle becomes upon *V* being multiplied by the factor

* E = m*∙*(dr/dt) ^{2 }+ 2 *∙

*[(1 ─ 1/γ)*∙

*(c*∙

^{2}/v^{2})]*V + l*(Z2)

^{2}/mr^{2 } The equation for
orbit around a star or the Sun has to be solved using the above equation
inverted to give *r *as a function of *θ* the angle the particle is at with
respect to the radius of nearest approach to the Sun or the star by the
particle. For very small ratio of *v/c*
the term *2 *∙ *[(1 ─ 1/γ)* ∙* (c ^{2}/v^{2})]*
could be considered practically a constant throughout the orbit as the change in velocity of Mercury in going around the Sun is very slight from its average speed due the small eccentricity of the orbit.

The term *2 *∙ *[(1
─ 1/γ)* ∙* (c ^{2}/v^{2})]*
may be considered the ‘bending factor’ or the 'whirling factor'

*b*(v) under relativistic conditions and is what is probably needed to formulate a 'Lagrangian' for a relativistic particle in a central force field like gravity, replacing

*V*by

*b(v)V*and is what gives through special relativity everything that GR predicts. The factor being a measure of bending under gravity is an inverse measure of inertia and a factor of two is an inertia that is half of classical normal while a factor of one is for inertia equal to normal classical. It gives the value unity for no extra bending above the classical. For light the factor is two and gives the rate of bending of light as twice that of an equivalent classical particle and we have seen that it gives the correct bending for hyperbolic orbit of light going past the Sun. Obviously, for particles moving at speed v, the bending will be more than for an equivalent classical particle and but will be lesser than that for light due to their reduced relative inertia.

*b*= 1.000000078. Hence, the planet in completing a full turn around the Sun undergoes a turn of 1.000000078 x 360

^{o}= 360.0000281

^{o}. Or the increase of angle is 0.0000281

^{o}per revolution or upon multiplication by 3600 (the number of arc seconds per degree) gives an increase in angle of 0.10125” (arc seconds) per revolution. The planet revolves around the Sun once every 88 days or 365/88 = 4.147 revolutions per year or 414.7 revolutions per century. That makes the change of angle of the planet’s velocity, compared to the classical path, to be 0.10125” x 414.7 = 41.988375” (arc seconds) per century which is the observed precession of the perihelion of Mercury! We have used the average speed to make the calculations simple, if we solved it using the ‘bending factor’ as a variable all along then we would get a more accurate result, the actual value being 42.9 arc seconds per century. We have deduced the two most famous deductions of general relativity, the correct bending of light which is also the cause of gravitational lensing effect and the precession of perihelion of planet Mercury, using only special relativity, Newton's gravity and some straightforward common sense only. The 'bending factor' is based on the non-equivalence of gravitational and the inertial masses of moving bodies and is, in fact, a measure of the very non-equivalence. The fact that the bending factor gives the precession so readily is itself an indication that our contention that by special relativity the gravitational and the inertial masses of bodies are not equivalent when the masses are moving is correct!

GPS time settings and gravitational red-shifts.

Since light is pure
kinetic energy (KE) having a mass *m* given by *m* = *E/c ^{2}* where

*E*is its energy and

*c*the speed of light not only does it lose KE but also its mass in coming out of a gravitational well in doing so. If light climbed up from the surface of a planet of mass

*M*and radius

*R*to a height at a distance

_{o}*R*from the center of the planet then it would lose KE equal to the gain of potential energy (PE) against gravity. As mass keeps decreasing continually with decrease of KE we have to integrate along the path to get the final KE of light at distance

*R*from the center.

For a small increase
of height *dr* the loss in KE which is equal to gain in PE is given by ─ *dE =(GME/(c ^{2}r^{2}))dr,*, where,

*G*is the universal constant of Newton’s law of gravitation and

*E = mc*for light. The total loss ─Δ

^{2}*E*of the KE of light as it moves up from position

*R*to position

_{o}*R*is found by integrating from to

*R*to

_{o}*R*. To integrate we first separate the variables by rearranging the equation into the form (I apologize for not being able to use all types of mathematical symbols for which reason I have used words in some places)

─*dE / E
=((GM/c ^{2}r^{2}))∙dr*, then

integration
of ─*dE / E (from **R _{o} *to

*R*) = (GM/c

^{2})∙integration of (dr/r

^{2})

*(from*

*(11)*

*R*to_{o}*R)*this then gives after multiplying throughout by (─1)

ln (*E/E _{o}) *= (

*GM/c*(12)

^{2})∙(1/R─1/R_{o})taking the exponential of both sides of equation (12) we have

*E/E _{o} = exp[(GM/c^{2})*∙

*(1/R─1/R*

_{o})] =*exp[*

*∙*

*─*(GM/c^{2})*(1/R*(13)

_{o}─1/R)] or *E = E _{o}(*

*exp[*

*∙*

*─*(GM/c^{2})*(1/R*(13A)

_{o}─1/R)])for the case of planet Earth the exponent *(GM/c ^{2})*∙

*(1/R*is extremely small and to a first order we may ignore powers higher than the first in the expansion of the exponential and we get, by making the exponent positive by reversing the order of the reciprocal radii within the bracket

_{o}─1/R)

_{ }

_{} *E* = *E _{o}* (1+

*(GM/c*) (14)

^{2})∙(1/R─1/R_{o}) The total change of kinetic energy
*ΔE* is then given by

*E─E _{o}*
=

*E*

_{o}*((GM/c*) (14A)

^{2})∙(1/R─1/R_{o})) since* E* = hf =* h/T* we get
after substituting it in (14)

*T _{o}* =

*T*(1+

*(GM/c*(15)

^{2})∙(1/R─1/R_{o})) or Δ*T* = *T-T _{o}* =

*T(GM/c*(15A)

^{2})∙(1/R_{o}─1/R ) Where *T *and *T _{o}*
are the time periods at the final and the initial parts of the light ray path.

For the GPS satellite systems *R* is about 4 *R _{o}* (3

*R*above Earth surface), then using

_{o}*R*= 6.4 x 10

_{o}^{6}m and mass of Earth as 6 x 10

^{24}kg and the standard value of the constant

*G*we get a slowing down of clocks at Earth surface levels with respect to clocks at orbit heights to being near about 45,000 ns per day due to gravity which is the correct order of slowing down of clocks as used by the GPS systems (see the web) and which are being erroneously attributed to general relativistic reasons. Not only is the relative speeding up of Earth clocks by about 7000 ns per day due to the relative speed (14,000 kph) of the satellite with respect to Earth surface found by SR, as is thought to be the only thing SR does, the slowing down of the clock speeds at Earth levels relative to the GPS satellites due to gravity too is found by the same SR. This leads to a net slowing down of clocks on Earth surface relative to GPS satellites to be 38,000 ns. General relativity is solvable only when there is no curvature involved or when there is practically no gravity nor any accelerations. The equivalence principle is valid only trivially as the equivalence of zero gravity and zero acceleration! With zero acceleration or zero curvature at all times there is no time dependence in the curvature as like in the case of curvature defined by gravity and qualitatively (and quantitatively as well) the two become equivalent.

**Black holes**For a mass m_{0}
leaving the surface of a spherical body of mass M and radius R_{0} at a
velocity v (vertical to the surface of M) its total energy is γm_{0}c^{2} where c is the speed of light in
vacuum and γ = 1 /
√(1 – v^{2}/c^{2}) (Lorentz factor) . As the mass rises the loss of energy for an infinitesimal rise
of magnitude dR is given by:

-d(γm_{0}c^{2}) = (GMm_{0}γ)(dR/R^{2})

cancelling m_{0}
on both sides dividing throughout by c^{2} we have:

dγ / γ = - (GM.dR/c^{2} R^{2})

integrating left
side from γ = γ to γ = 1 (for m_{0} to come
to rest) and integrating the right side from R_{0} to R we get:

ln(1) - ln(γ) = (GM.dR/c^{2} ) . (1/R - 1/R_{0})

or -ln(γ) = ln(1/γ) = (GM.dR/c^{2} ) . (1/R - 1/R_{0})

or 1/γ = exp((GM.dR/c^{2} ) . (1/R
- 1/R_{0}))

or γ = exp(-(GM.dR/c^{2} ) . (1/R
- 1/R_{0})) = exp((GM.dR/c^{2} ) . (1/R_{0} - 1/R))

squaring both sides this gives :

c^{2}/(c^{2}-v^{2})
= exp((2GM.dR/c^{2} ) . (1/R_{0} - 1/R))

for escape speed we set R = ∞ or 1/R = 0 then after rearranging we get:

v^{2}
= c^{2} (1 - exp(-2GM/R_{0}c^{2}))

this shows that for sufficiently high speed still less than the speed of light any mass can escape from the gravity of a body of any mass no matter how large short of infinity and of radius greater than zero. There is no black hole!

If we take the small limit of mass M in the above formula and expand the exponential to the second term only we get:

v^{2} =
c^{2} (1 - (1 + (-2GM/R_{0}c^{2}))

or v^{2}
= c^{2} (2GM/R_{0}c^{2}) = 2GM/R_{0}

and that is the correct formula for Newtonian escape speeds! The deduction of our formula for escape speeds is consistent. General relativity is always being found to be correct for only the small value limits where special relativity and Newton's gravity alone are sufficient. In the next section we see a possible application of our formula under large value conditions wherein general relativity has so far been unable to give a satisfactory answer.

Apparent accelerations in cosmic expansion.

Speeds of stars and galaxies and, thus, their accelerations and retardation are found in astronomy by means of Doppler shifts which happens not only due to speeds of recession only but also due to loss of energy of light in coming out of the gravitational fields of the stars themselves. At higher speeds not only is Doppler effect larger due to the speed but the relativistic mass of the star also becomes larger causing greater gravitational red-shifts. The overall effect is due to reduced acceleration leaving the stars with greater speeds causing greater red-shifts which is further multiplied by a exponential of γ times decrease due to increase of relativistic or gravitational mass of star due to its relative speeds. Not only that, at nova the radius of the star is considerably lesser than its radius as a star and can cause increased gravitational red-shifts in the radiations emanating from its exploding surface. This can lead observations to imply exponential increase in accelerations of cosmic recession speeds. Let us try to see this:

By equation (13A) we
have that the energy *E* of light at a distance *R* from a star of mass
*M _{o}* and radius

*R*after emitting from it is given by

_{o}*E = E*), where

_{o }exp( -GM_{o}/R_{o}c^{2}*E*is the energy of light at the surface of the star during emission. In terms of wavelengths energy

_{o}*E*is given by

*E = hc/λ*where

*λ*is the observed wavelength and

*c*is the speed of light. This makes the measure of wavelength at a distance R from the star to be given by:

*λ = λ _{o} exp(GM_{o}/R_{o}c^{2}) *(A)

where the star is
traveling at a speed *v* relative to the observer, the relativistic mass
of the star is γ*M _{o}* where γ is the Lorentz factor and

*λ*is the wavelength of the emitted light at the surface of the star. Therefore the exponential part of (A) gets multiplied by γ due to relativistic increase of mass of the star and we have:

_{o}
*λ = λ _{o} exp(γ GM_{o}/R_{o}c^{2})* (B)

*R*, say, to

_{o}*1/n*times

*R*and this causes the equation (B) to become:

_{o}* λ = λ _{o}
exp(nγ GM_{o}/R_{o}c^{2}) * (C)

The Doppler shift
due to velocity *v* of the star is given by special relativity to be:

*λ'* = (*√(c+v)/√(c-v)) ∙ λ _{o}*,
which upon gravitational corrections becomes

*λ'* = (*√(c+v)/√(c-v)). **λ _{o }*

*exp(nγ GM*

_{o}/R_{o}c^{2})*(D)*

The term *nγ *in the exponential*
*makes the Doppler effect extremely large for even a not very large recession speed. On top of it the nova must be starting at an even lesser radius than the original normal star causing an even greater red-shift in the light emitted. I
believe the increased gravitational red-shifts due to reduced radius of stars
at supernova is not being taken into account and is leading to the idea of runaway
accelerations in the expansion rates of the universe. However, even without taking reduced radius at nova since the temperatures would be exorbitantly large the wavelengths may be much shorter having a canceling effect on net Doppler shift produced, the exponential nature may still offset the cancelling effect to some extent) the factor *exp*(*γ) *alone can cause larger and larger factor of shift with greater and greater recession speeds. Some values of Doppler shift factor are calculated and shown below: for a star of mass *M _{o}* and radius

*R*

_{o} Let *exp(GM _{o}/R_{o}c^{2}) = Z *for some star of mass

*M*and radius

_{o}*R*

_{o}*, it's value is greater than 1.00 since*

_{}*GM*is greater than zero, and further it is very large compared to standard red shift factor.

_{o}/R_{o}c^{2} For recession speeds at 50% the radius of the universe γ = 1.09 and the Doppler shift factor, relative to a stationary of similar mass and radius at the given distance, is Z^{1.15} ,

For recession at 75% radius the shift factor is Z^{1.51},

For recession at 90% radius the factor is Z^{2.29},

For recession at 95% radius the factor is Z^{3.20},

For a very massive star typical of supernovas visible very far away the factor* exp(GM _{o}/R_{o}c^{2})* is already very large compared to the standard Doppler shift factor and then when it is raised by the above powers, due to recession speeds, it gives extremely large shift factors with increasing distances of the stars.

There is no need to invent dark energy to explain the apparent ever increasing acceleration of the cosmos. The reasons for increased red-shifts are:

(i) Extra gravitational red-shifts (not deducible from GR formula since it is less sensitive to increase in mass) take

place due to comparable increase in relativistic mass of the star due to it's recession speed.

(ii) And further exponentially greater gravitational red-shifts take place due to decreased radius of star at nova

causing novas very far away to appear excessively redder.The redshift formula of GR is insensitive to decrease

of emitting star's radius unless small enough to be near Schwarzchild radius.

(iii) Due to the intense Doppler shifts causing reduced frequencies of radiation emitted the apparent brightness of

the nova will also be fainter than expected as intensity of waves are reduced proportionately to the square of

frequencies.

GR does not seem to have any explanation for the observed fact, I believe this is due to not having the idea or the definition of vertical emission of light from a gravitating body in its very mathematical essence and for taking light to have an effective rest mass every now and then in its formulation. Its formula for light escape speeds are purely in terms of Newtonian particles of constant non zero rest mass! The Schwarzchild radius is like as that calculated from the Newtonian expression for escape speed by simply putting *c *to be the limiting speed in that expression and assuming light of mass *m *as having kinetic energy equal to *m**c ^{2}*. To apply it to light is to assume light is like a classical particle having a constant rest mass which then loses all its kinetic energy upon trying to rise out of a 'black-hole'.

Almost all the verification of GR have been in either small velocity range or small gravity ranges where the error in its predictions are hidden or obscured by small approximations. For instance the red shift formula of GR

*z = [1/√(1─GM/Rc ^{2})] ─ 1*

gives the same value for red shift for small values of gravity as the special relativistic formula we have deduced.* *However, the GR formula does not give as much red shifts for moderate reductions in *R *as our formula does* *and, hence, cannot explain the 'runaway' accelerations of cosmic recessional speeds. When experiments to validate GR red shift formulas such as Mossbauer effects using gamma rays over a distance of some tens of meters are cited I fail to understand how it is not noticed that the range belongs to Newton's physics. In spite of being extremely accurate the Mossbauer effects are measured over very small distances under small and constant gravity forces. Newton's physics is sufficient and that is what is used : decrease df in frequency is proportional to gain in potential energy mgh where m is the gravitational mass of light g is acceleration due to Earth's gravity and h the height risen under the constant gravity. The mass change of light under the circumstances is of the order *m∙(gh/c ^{2})* which is negligible for the values of gravity field strength and the heights of movements involved thus the mass of light remains practically constant and simple Newton's physics is enough. The experiments are really a praise for the extreme accuracy of Mossbauer effect based experiments rather than as verification of GR predictions, GR is simply not being tested in such experiments. And so do most of the other verification of GR red shifts verified, if it concerns light then mostly low gravity and if it concerns particle then mostly low velocities.

There seems to be evidence that the galaxies in which the apparently very far away supernovas are taking place are themselves calculable to be nearer than the supernovas! How can that be? This necessitates correcting the GR red-shift formula which is what is causing the novas to seem to be farther away than they are by associating extremely large red-shifts with extreme distances.

A strange kind of presumption about bending of light is prevailing in the modern world of today. The presumption is that general relativity announced to the world that gravity bends light. That gravity bends light was assumed ages ago before the advent of relativity only nobody knew how to calculate it. The calculations needed *E = mc ^{2}* or the equivalent form for relativistic particles along with Newton's gravity but when the electromagnetic or relativistic form was found and known people miscalculated using

*E*

*= (1/2) mc*instead of the relation

^{2}*E = mc*for the energy of light going past the Sun and got a wrong result. Thereafter formulating GR warping space and time, which also had to be corrected before giving correct results (see history) the c

^{2}*orrect bending was got after smoothing out the warps, a result that follows logically from Newton's gravity upon correctly interpreting and using the correct formula for the energy of light.*

A more appropriate experiment to compare GR and SR views would be in the arena of extremely far supernovas and the analysis of their red shifts. The reported runaway accelerations with distance from Earth of cosmic recessional speeds could be analyzed using each of the two ways separately. The formula we have developed for red shifts maybe found to be closer to the observed values.

Conclusions.

Gravitational and inertial masses are equivalent only for masses at rest and are approximately equivalent for masses at speeds very low compared to the speed of light, that is for classical mechanics only. When masses move at relativistic speeds the inertial masses become different from their gravitational masses.The consistency of GR is no proof of its validity as a scientific theory, it may be incomplete. Classical mechanics is a mathematically consistent theory but it is not valid in the modern sense any more at atomic levels. A theory has to be not only consistent but also has to be complete or real in some sense to be valid. A consistent theory has to relate to the physical reality out there to be valid. The consistent results of GR are at low values of gravity strengths, its inconsistency or incompleteness, we contend, will be evident at high values of gravity field strengths. The theory has to be wrong essentially otherwise it would have had been unified with quantum theory by now, after all almost a hundred years have elapsed since the stating of the theory.

I think (this is pure speculation as I am not qualified enough to make the statements) the essential kinematic nature of GR caused the physicists to lose intuitive or dynamical grasp of the phenomena and then in the complicated mess of warped space-times it became impossible to keep track of things, and thus end up taking even light to have an effective rest mass every now and then, especially so as the very principle of equivalence the core of GR is erroneous to start with. Just imagine the equivalence principle assumes light cannot tell the difference between purely accelerating frames and frames stationary in a gravitational field and using that finds that light bends doubly under gravity than under pure accelerations, an utter self contradiction. An implication of the lack of intuitive grasp is in the fact that physicists have thought of things like the 'worm holes' and even of such things as bending of time to go backwards in an intense gravity field for which Stephen Hawking apologized to the science-fiction readers and writers some years back after it was found not to be a consistent picture.Lack of intuitive grasp leads to erroneous modeling. Further the very concept of energy is made rather abstruse in GR for gaining what ?

We are not asserting that gravity does not warp space, it does because slowing down of time in signals leaving the surface of a massive body due to loss of kinetic energy does imply some sort of compression of space along paths coming out radially from the body (longitudinal compression). As the speed of light must remain constant when measured at any point in space it must have traveled a longer distance to come out of the gravity field or potential well to account for the delay. What is being asserted is that the GR measure of the compression leading to infinite curvature at the surface of a 'black hole' is erroneous. We found that time slowing down is exponential and leaves it not equal to infinite at the surface.

Our derivation has achieved the following:

(i) Explained the double bending of light when going past a massive body

(ii) Explained the exact precession of perihelion of Mercury in its orbit around the Sun

(iii) Gives a plausible explanation of apparent runaway acceleration of the outer universe without

resorting to 'dark energy'

(iv) Gives a plausible explanation of expansion of the universe at the very beginning of coming into

existence: As at the beginning all particles had relativistic energies or speeds both particles and radiation

could overcome gravity and move out from the 'dot' state and make expansion possible. As for there being

more matter than anti-matter in the universe involves quantum mechanics as well and is a different

problem altogether.

(v) As for the gravitational waves necessary to be radiated to allow for falling in of orbiting masses due to increased curving

caused by reduction of inertial mass compared to its gravitational mass, or the non equivalence, one has to define a

gravitational form of energy waves much like the earlier discovered electromagnetic form of wave energy.