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 = mc2 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) mc2. 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.

  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 mc2 by E = mc2 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 = mc2, and interpreting it correctly, instead of E = 1/2 mc2 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 = mc2 for light  and E2 = m02c4 + (mvc)2  for particles with rest masses along with Newton's gravity properly.

 The equation E = mc2 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 mc2 gives c2 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 (mvr2 +mvθ2)+V in polar coordinates. Here vr and vθ 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

                                                                         vr2 = (2/m)∙(E – V - mvθ2/2)                                              (1)

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

                                                                          vr = dr/dt = √(2/m)∙√(E – V - mvθ2/2)                               (2)

                     giving                                            dt = dr / √(2/m)∙√(E – V - mvθ2/2)                                    (3)

now angular momentum L = mvθr using which we have                

                                                                         mvθ2/2 = L2 / (mr2)                                                            (4)

and eqn (3) becomes

                                                                        dt = dr / √(2/m)∙√(E – V- L2 / 2mr2)                                   (5)

L = mvθr = mr2 dθ/dt since vθ = r dθ/dt and we have dθ = Ldt/mr2 which gives

                                                                       dθ =  L dr /(mr2)√((2/m)∙(E – V- L2 / 2mr2))     

                                                     or               dθ =  dr /(mr2/L)√((2/m)∙(E – V- L2 / 2mr2)) which simplifies to

                                                                       dθ =  dr /(r2)∙√(2mE/L2 – 2mV/L2- 1 / r2)                           (6) 

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

                                                                       dθ =  ─ du / √ (2mE/L2 + 2mku/L2- u2)                              (7)

integrating both sides gives the solution

                                                                       θ = θ’ ─ arc cos ((L2u/mk─1) / (√(1+2EL2/mk))                (8)

simplifying leads to

                                                                       u = 1/r = (mk/L2)∙(1 + e∙cos(θ ─ θ’))                                 (9)

here e = √(1+2EL2/mk). We have                  1/r = GMm/L2 (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 Ro 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 Ro of the particle from Sun center by setting cosΦ =1 and thus making 1/r a maximum at r = Ro and get 1/Ro = (GMm/L2)(1 + e). Rearranged the equation gives GMm/L2 = 1/(Ro(1+e)). Then setting r → ∞ or 1/r → 0 we get:

                                                                        1/∞ = 0 = GMm/L2 (1 + e cos(Φ/2))

                                    or,                                0 = GMm/L2 + (GMm/L2) e cos(Φ/2)

 or using the relation GMm/L2 = 1/(Ro(1+e)) in the second term we can simplify it to

                                           0 = GMm/L2 +(1/(Ro(1+e))) e cos(Φ/2) = (e/(1+e)). cos(Φ/2) /Ro) + GMm/L2

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 = Roc and GMm/L2 becomes equal to GM/(Roc)2 the above equation simplifies

                                                        to:            0 = (1/Ro) cos (Φ/2) + GM/(Roc)2 ,

where Φ now is equal to π + δΦ and Ro 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/(Roc2) whence δΦ = 2GM / (Roc2)

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 c2 as computed classically from c2 = 2E/m is double that for light. Light is subject to its own rule of having c2 = E/m, hence, for light the value of c2 computed by classical mechanics should be replaced by c2/2 in the formula for bending as G, M and Ro are independent of whether it is light or classical particles and we get:

                                                    δΦ/2 = GM/(Roc2 / 2) = 2GM / Roc2 or δΦ = 4GM / (Roc2)

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 = mc2 and not by E = ½ mc2. 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 ½mv2 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 (mvr2 +mvθ2) remains the same. However the energy equation E = mc2 = 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 = mc2. 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 (mvr2 +mvθ2) + V for total energy of the light in its motion has to be replaced by E = ½ (mvr2 +mvθ2) + 2V thereafter following standard procedures, after appropriate rearrangements, the equation (3) for time element dt becomes:

                                                         dt = dr / √(2/m)∙√(E – 2V - mvθ2/2)  and that makes dθ to be given by

                                                        dθ =  dr /(r2)∙√(2mE/L2 + 4mk/L2- 1 / r2)        

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

                                                         u = 1/r = (2mk/L2)∙(1 + e∙cos(θ ─ θ’)), here e = √(1+EL2/2mk).

  We have: 1/r = 2GMm/L2 (1 + e cosΦ) or for a unit mass, without loss of generality,1/r = 2GM/L2 (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 / (Roc2). 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 / (Roc2) 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 mcinstead of using E = mc2 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.

 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 mv2 /mv = 1/2 v while for light it is mc2 / 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 mo at speed v is given by E = mc2 ─ moc2, where mo is the rest mass of the body and moc2 its rest mass energy and m its total or relativistic mass at speed v and mc2 its total energy at speed v, the expression simplifies further, by using the relation m = γ mo from special relativity, to E = (γ─1) moc2 where γ = (1 ─ v2/c2)─1/2. The momentum p = mv becomes p = γ m0 v which makes the ratio E/p to be:                           

                                           E / p = (γ─1) moc2 / γ m0 v = [( γ ─1) / γ]∙(c2/v) = (1 ─ 1/γ)∙(c2/v)                                               (Z1)

γ = ∞ for v = c and the ratio E / p becomes [(1 ─ 1/∞] ∙ (c2/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) c2 as 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 ─ v2/c2)1/2 which on expanding becomes 1─ (1 ─  v2/2c2), ignoring terms of higher powers of (v/c) than the second, and we get E / p = ( v2/2c2)∙(c2/v) = v / 2 the correct value of 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 + l2/mr2 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)∙(c2/v2)], which then gives two for light and one for a classical particle. If we substitute v = c  Lorentz factor γ  becomes infinity and we get the factor is equal to 2 [(1 ─ 1/)∙(c2/c2)] = 2. If we let v be very small, say v ≈ 0  then 

                                                 2 [(1 ─ 1/γ)∙(c2/v2)]= 2∙[1 ─ (1 ─ v2/c2)1/2)∙(c2/v2)

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 ─ v2/(2c2))]∙(c2/v2) = 2∙[v2/(2c2)]∙(c2/v2) = 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)∙(c2/v2)]∙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) (c2/v2)] V + l2/mr2                                                           (Z2)

  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) (c2/v2)] 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) (c2/v2)] 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.

  If we apply the formula to determine the precession of perihelion of Mercury around the Sun then, as we mentioned above, we can try for a quick solution by taking the average velocity of the planet and consider the factor a virtual constant as velocity does not change much in going around the Sun. The average speed of Mercury is 48 km/s and that make the factor b = 1.000000078. Hence, the planet in completing a full turn around the Sun undergoes a turn of 1.000000078 x 360o = 360.0000281o. Or the increase of angle is 0.0000281o 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.

  What is being passed off as correct predictions of general relativity are but SR and Newton's gravity.  Because light has mass it can lose and gain kinetic energy under gravity which leads to change in its frequencies that explain the GPS settings and gravitational red shifts. No warping of space and time outside of the Lorenz transformations are needed.

  Since light is pure kinetic energy (KE) having a mass m given by m = E/c2 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 Ro to a height at a distance 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/(c2r2))dr,, where, G is the universal constant of Newton’s law of gravitation and E = mc2 for light. The total loss ─ΔE of the KE of light as it moves up from position Ro to position R is found by integrating from to Ro to 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/c2r2))∙dr, then

                            integration of  dE / E (from Ro to R) = (GM/c2)∙integration of (dr/r2) (from Ro to R)                         (11)

                                                           this then gives after multiplying throughout by (─1)

                                                                ln (E/Eo) = (GM/c2)∙(1/R─1/Ro)                                                                          (12)

 taking the exponential of both sides of equation (12) we have

                                                   E/Eo = exp[(GM/c2)(1/R─1/Ro)] =  exp[ (GM/c2)(1/Ro─1/R)]                                   (13)

 or                                                           E = Eo( exp[(GM/c2)(1/Ro─1/R)])                                                                  (13A)

for the case of planet Earth the exponent (GM/c2)(1/Ro─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 

                                                              E = Eo (1+ (GM/c2)∙(1/R─1/Ro))                                                                            (14)

 The total change of kinetic energy ΔE is then given by

                                                              E─Eo = Eo ((GM/c2)∙(1/R─1/Ro)))                                                                        (14A)

 since E = hf = h/T we get after substituting it in (14) 

                                                             To = T (1+(GM/c2)∙(1/R─1/Ro))                                                                              (15)

 or                                                         ΔT = T-To = T(GM/c2)∙(1/Ro─1/R )                                                                        (15A)

 Where T and To 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 Ro (3 Ro above Earth surface), then using Ro = 6.4 x 106 m and mass of Earth as 6 x 1024 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 m0 leaving the surface of a spherical body of mass M and radius R0 at a velocity v (vertical to the surface of M) its total energy is γm0c2 where c is the speed of light in vacuum and γ1 / √(1 – v2/c2) (Lorentz factor) . As the mass rises the loss of energy for an infinitesimal rise of magnitude dR is given by:

                                                                 -d(γm0c2) = (GMm0γ)(dR/R2)

cancelling m0 on both sides dividing throughout by c2 we have:

                                                                  dγ / γ = - (GM.dR/c2 R2)

integrating left side from γ = γ to γ = 1 (for m0 to come to rest) and integrating the right side from R0 to R we get:

                                                      ln(1) - ln(γ) = (GM.dR/c2 ) . (1/R - 1/R0)

                                          or       -ln(γ) = ln(1/γ) = (GM.dR/c2 ) . (1/R - 1/R0)

                                         or        1/γ = exp((GM.dR/c2 ) . (1/R - 1/R0))

                                         or         γ = exp(-(GM.dR/c2 ) . (1/R - 1/R0)) = exp((GM.dR/c2 ) . (1/R0 - 1/R))

squaring both sides this gives :

                                                     c2/(c2-v2) = exp((2GM.dR/c2 ) . (1/R0 - 1/R))

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

                                                    v2 = c2 (1 - exp(-2GM/R0c2))

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:

                                                    v2 = c2 (1 - (1 + (-2GM/R0c2)) 

                                         or       v2 = c2 (2GM/R0c2) = 2GM/R0

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 Mo and radius Ro after emitting from it is given by E = Eo exp( -GMo/Roc2), where Eo is the energy of light at the surface of the star during emission. In terms of wavelengths energy 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 = λo exp(GMo/Roc2)                                                                                      (A)

where the star is traveling at a speed v relative to the observer, the relativistic mass of the star is γMo where γ is the Lorentz factor and λo 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:

                                                                         λ = λo exp(γ GMo/Roc2)                                                                                    (B)

  At nova the star compresses to a fraction of Ro, say, to 1/n times Ro and this causes the equation (B) to become:

                                                               λ = λo exp(nγ GMo/Roc2                                                                                  (C)

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

                                                       λ' = (√(c+v)/√(c-v)) ∙ λo, which upon gravitational corrections becomes

                                                       λ' = (√(c+v)/√(c-v)). λo exp(nγ GMo/Roc2)                                                                         (D)

   The term 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 Mo and radius Ro

    Let exp(GMo/Roc2) = Z for some star of mass Mo and radius Ro, it's value is greater than 1.00 since  GMo/Roc2 is greater than zero, and further it is very large compared to standard red shift factor.

  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 Z1.15 ,

 For recession at 75% radius the shift factor is Z1.51,

  For recession at 90% radius the factor is Z2.29, 

  For recession at 95% radius the factor is Z3.20,

  For a very massive star typical of supernovas visible very far away the factor exp(GMo/Roc2) 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.

   Thus novas extremely far away will show very much larger red shift than expected. Even if the reduced radius effect were not very large the compound effect would would cause exponential increase of Doppler shifts and decrease of associated brightness.

  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 mc2. 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/Rc2)] ─ 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/c2) 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 = mc2 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) mc2 instead of the relation E = mc2 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 correct 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.

 

 


 



 

 
 

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