I agree with you. In case of deflection of light, this angle vary from -pi/2 to +pi/2.

Dear Robert. I am not talking about delta theta at all. I am talking about psi. By including the factor Cos(psi) + Sin(psi) in Newton's law of gravitation I am able to calculate a very precise value of deflection of light around the Sun which I think is more accurate than that obtained in general relativity. NASA has looked at the feasibility study for carrying out precise measurement of deflection of light within a triangle formed between earth and two spacecrafts using laser. To my knowledge, this mission (LATOR) is not yet funded. When the results of such experiment is available, various competing theories can be tested for higher digit accuracy. This is the reason I was wondering why this fact went unnoticed for three hundred years. I agree with 't Hooft that Newton cannot be held accountable for this because relativity was not invented in his time and gravitational deflection of light did not become an issue till Einstein thought about it.

This is not an answer but a comment. I suspect the question was so ill posed that all who replied misunderstood. Anybody can call any angle "psi". Please give reference of that article so that maybe someone understands the question.

Sorry for the confusion. Since Robert has given the link to the reference it will be easy to see in two figures that psi is the angle between the radial vector and the velocity vector. If we draw a vector normal to the velocity vector then the angle between the radial vector and the normal vector is (pi/2 - psi). In Newton's theory gravitational potential is given by GM/r but in PR it is given by eqn.(4.3) which includes the factor Cos(psi) + Sin(psi). From eqn.(4.16) we see that gravitational acceleration is {GM/r^2](Cos(psi) + Sin(psi)) and gravitational mass is relativistic mass m. This equation introduces the curvature parameters. Deflection of light is computed by solving double integral eqn.(4.18) which does not have the parameter of eccentricity. Final result is given by eqn.(4.19) which is same as the lowest order term in general relativity. The higher order terms given by general relativity does not exist in PR. I do not think this double integral solution gives a plain hyperbolic trajectory.

Robert: Indeed-I was just saying that, *if* you started with the expression for the force,

(that contained a cos(psi)+sin(psi) factor), along with the definition of psi, *then* this would mean that the force doesn't depend only on the distance, but, also, on the orientation of the radial vector between the two masses. At this point it's not clear that there's a potential at all, if the curl (or rot? :) ) of the force field, as defined, doesn't vanish identically.  If the potential is defined as containing the cos(psi)+sin(psi) term, but not the force, then, again, the potential isn't a function only  of the distance, so angular momentum isn't conserved either.  In both cases the trajectories must be found anew (some components of the angular momentum *might* be conserved)-and they won't be, generically, conics.

And beyond the technical issues, the conceptual point is that, *if* this additional factor is introduced to describe the deflection of light, this is, simply, impossible, if a  scalar potential can be defined, because light cannot be deflected by a scalar potential. This doesn't, of course, mean that,  were such a potential absent, the deflection could be attributed to the modification, since if the orbits weren't correct, this isn't consistent with Newton's law.

Robert: You can *define* the force that way and put it on the right hand side of Newton's second law, of course-but then everything else (in particular shape of orbits and  conservation laws) must be proved. And one is starting with several of them manifestly absent. 

Indeed-because it's the RHS, that's being modified, namely the law of gravitation. 

If you look at  F = GM1*m2 / r^2, the multiplication of M1 * m2 can be viewed in

two extremes where M1 >>> m2 ( is vastly greater than ), and where

M1 = m2.  The first case works well with the Newton equation where

GM1/r^2 creates the force applied  on the vastly smaller mass m2.

The problems with the Newtonian method comes when M1 becomes

equal to or nearly equal to m2.   At that point the forces become equal

and vectorially addative.  this is actually the case for both conditions,

but when m2

Thank you for your input Michael. I have applied Newton's method when m1 = m2, and it seems to work pretty well. At the beginning of the universe, fundamental particles were produced in pairs of particle and anti-particles. Treating this as a two-body problem with m1=m2, I solved the quantum gravity problem as discussed in the article "Periodic quantum gravity and cosmology." I got some very interesting results.

@Robert. Your arguments are based on mistaken notion that, Neither the force nor the potential are continuous functions of the position and velocity of the object. The second example of apple in a rocket is also without any mathematical basis.

I think you have not clearly understood the statement following eqn.(4.25). "So any conversion of acceleration between radial and tangential directions is accompanied by the conversion factor (cosψ + sin ψ). This factor acts as a single scalar quantity and does not get split into normal and tangential vector components. It also needs to be understood that d2r/dt2 is a radial vector but dr/dt is not a radial vector which acts along the velocity vector v. Therefore factor (cosψ + sin ψ) does not play any role in this expression of velocity v = dr/dt which remains unaltered."

Your argument that I have defined F = ma, is also a false notion which can be clearly seen by looking at eqns.(4.25) and (3.8).

Your comment that the potential is velocity dependent (but not continuous where velocity is zero), is an incorrect conclusion.

@Stan. There is something wrong with your notion that the light cannot be deflected by a scalar potential. Gravitational attraction exists between energies (potential + kinetic) of two bodies and energy is a scalar. Rest of your arguments does not have any mathematical basis.

In Newtonian gravity,  *assuming* that the inertial mass is equal to the gravitational mass, one obtains an equation of motion that doesn't depend on the mass of the probe particle-so, seemingly,  could describe the gravitational attraction of massless particles to a massive particle. However, this is inconsistent if the probes are relativistic-which light surely is (and massless particles, in general, are). The reason is that this equation of motion  isn't invariant under reparametrization of the time parameter of the trajectory-which is equivalent to Lorentz invariance. It's valid for non-relativistic probes and singles out the proper time as a privileged parameter. It is, of course, possible to describe the trajectories of relativistic probes in general and of massless probes, in particular, by solving the constraints of reparametrization invariance for the particle's trajectory.

In the relativistic case it isn't the mass of the probe that couples to gravity, but the energy-momentum tensor.(Energy isn't a Lorentz scalar-it's a component of the energy-momentum 4-vector for a particle and a corresponding component of the energy-momentum tensor for a field theory.). Departures from the equivalence principle are expressed, in this case,  by the fact that fields other than the metric are necessary to describe the spacetime geometry-even in the absence of matter.

If gravity were described by a scalar field (which the scalar potential is) only-or in the approximation that it can be so described, i.e. in the Newtonian limit,   it could only couple to the trace of the energy-momentum tensor of the probe-which vanishes for the electromagnetic field, that describes light.(In fact it vanishes for any massless particle, expressing classical conformal invariance.)

That's one reason the deflection of light cannot be described  in the Newtonian approximation: in that approximation massless-thus relativistic-particles don't interact with the gravitational field, since no interaction term, consistent with the symmetries, can be written. 

In GR, the metric couples to the energy-momentum tensor, not, just, its trace and can give  rise to deflection for light. Therefore any deflection of light, by a gravitational field, *must* vanish in the Newtonian  (flat spacetime-another way to see this) limit, since it can only be a function of GM/(R c^2) and, in the limit of vanishing argument,  such a function must vanish-and does vanish.

It is, of course, possible, to express the motion of a particle in terms of the variables one wishes-so, in this particular case, to exchange the polar angle one habitually uses in polar coordinates for the angle psi. The expressions become correspondingly more elaborate-but this is a matter of taste. If the complexity of the resulting expressions leads to the use of  approximations, however, these latter must be recognized as such and shouldn't be taken for anything else-especially for this class of problems, where the answer *is* known exactly and in closed form in the variables (r(t), polar_angle(t)).

There isn't any *physical* effect that can be described in terms of (r(t),psi(t)) that cannot be described in terms of (r(t),polar_angle(t)).One should take care  to deal with what happens when the transformation becomes multiply-valued (or, even worse, infinitely-valued.)

It's, simply, a change of variables. The misunderstanding, relevant for the discussion at hand,  lies  in the assumption that equations that were deduced for massive, non-relativistic, particles, describe, also,  relativistic, even massless, particles. They don't.

@Robert. Regarding apple in a rocket, your first proposal of psi=pi/2 is illogical, because that angle is impossible without a large external force acting on the apple. In your second proposal, a slow spiraling trajectory is also not possible without an external force or atmospheric drag etc. In your third proposal you have mentioned about the rocket passing through a fixed point along a slow equiangular trajectory. This will give you a very small oscillating value of (pi/2 - psi), just like in case of a planet, and the effect will be so small that you will not even be able to measure it. For a rocket hovering at a fixed altitude, (pi/2 - psi) = 0, and therefore (Cos(psi) + Sin(psi)) = 1. This is a hypothetical circular orbit.

Regarding v=dr/dt v^, (dr v^) is resultant vector acting along the direction of the velocity vector. This is obtained by combining (dr/dt r^ + rdtheta/dt theta^). In case of a=d^2r/dt^2 (cos psi + sin psi), when you diferentiate (dr/dt r^ + rdtheta/dt theta^) second time, the tangential component of acceleration gets eliminated because r^2dtheta/dt is a constant h and the result is only the radial acceleration given by eqn.(4.29) and this acceleration is same as the Newtonian acceleration which is GM/r^2. So when the Newtonian acceleration is multiplied by Cos(psi) + Sin(psi), then same is the case with d^2r/dt^2. Therefore dv/dt is a radial vector. This is all I have quoted in my statement.  

On F=ma, eqn. 4.1 only deals with the gravitational potential energy Ep. The acceleration is not the standard Newtonian acceleration but includes the term Cos(psi) + Sin(psi). This additional term is due to the relative kinetic energy between the two bodies and if we look at this from Stam's point of view then the gravitational potential does not remain scalar any more. From my point of view, I see both potential as well as kinetic energy as a scalar quantity but in general relativity only the mass energy is treated as the energy and the momentum part is treated as non-scalar which is the kinetic energy part. Therefore there is nothing wrong with eqn.(4.1). It does not describe the particle energy.

@Stam. 

In general relativity, in case of energy momentun four vectors, only the mass energy is treated as energy and mometum is not treated as such. The momentum part is treated as non-scalar which is the kinetic energy part. But from my point of view, both are components of the total energy of the body or system and therefore scalar. if we look at the revised potential in PR from your point of view, then the gravitational potential does not remain scalar any more because of the additional term Cos(psi) + Sin(psi), which is associated with the velocity and the radial vectors. In PR also, deflection of light, by a gravitational field, does vanish in the Newtonian limit-and, for any scalar potential in this limit. In PR, flat Minkowski spacetime corresponds to n = 1. Newtonian gravity is recovered by putting n = 1 in eqn.(4.38) which results in eqn.(4.71). You can see that it is not possible to deflect the light for n = 1. The deflection of light in PR corresponds to another flat spacetime metric which is not the Minkowski flat spacetime. This flat spacetime is defined for n = 0. If we put n = 0 in eqn.(4.34), we get d^2s/dt^2 = 0. This is the condition introduced after eqn.(4.16) for computing the defelction of light. This is consistent with eqn.(4.15). If you replace tau by s and differentiate, you get the same result. But this is not the case in general relativity. If you replace tau by s in eqn.(4.12) and differentiate, you do not get zero. This is because of the small error introduced by way of the weak field approximation. This is the only way to introduce the Newtonian potential in general relativity, but this small error gets magnified in case of strong fields. Not only that, but the energy momentum tensor is also based on the weak field approximation. This is like introducing Newtonian potential into theory through a back door. Where as in PR it is introduced through a front door as can be seen in eqn.(5.1). Hence eqns.(5.5) to (5.11) are comparable to the energy momentum tensor in GR. However, use of these equations are restricted to describing general motion of the astronomical objects in the presence of matter. For problems such as deflection of light, gravitational redshift, perihelic precession of planets and orbital period derivative of a binary system a simple two body formalism provides much more accurate results. Not only that, at the begining of the universe, the particles were produced in pairs of particle and anti-particle. Such systems can also be best described as the two body systems as can be seen in the article "Periodic quantum gravity and cosmology". Therefore in PR, two body systems come first and the multi body systems come later. In general relativity it is opposite. This compromises the accuracy of the two body systems. In PR gravitational mass is equated to the relativistic mass which prevents any application of energy momentum tensor of general relativity in this theory.

General Relativity cannot be modified or replaced that easily, and anyway, what Mr Zaveri sees as shortcomings are not at all that. GR does the two body system as precisely as any many body system. See the Hulse Taylor double-pulsar. In standard Einstein gravity both energy and momentum (and stress) act as sources in a completely consistent way. What PR tries to do makes no sense to me at all. As RJ Low, I drop this discussion.

In PR the alleged invariant is invariant under Lorentz transformations that keep the asserted phase velocity constant, *not* the velocity of light in vacuum-so it's not surprising that everything else isn't as assumed. 

Of course why this would have anything to do with the rest of the discussion isn't clear. And I'm at a loss to understand the assertions in the last answer, so I'd rather stop here, too. 

Robert, Gerard and Stam. Thank you for your participation.

Analysis of Hulse Taylor double-pulsar was reevaluated in the article: Orbital period derivative of a binary system  http://arxiv.org/abs/0707.4544  Here I claim little higher accuracy than that obtained in general relativity.

Stam seems to have missed the point that for light, the phase velocity and the particle velocity are the same. This can be easily seen from the relation  V = c^2/v. Where V is the phase velocity and v the particle velocity. When you substitute particle velocity v = c for light, you get the phase velocity V = c. 

With respect to earlier discussions, particularly with Robert J. Low regarding his comment that -- if  v=dr/dt, then a=d^2r/dt^2 (cos psi + sin psi) cannot be correct (r,a and v are vectors here), -- I have prepared a detailed explanation presented in the attached article "Supplement to periodic relativity" which clearly shows why it is so.

Article Supplement to periodic relativity

Following article is now published in "Progress in Physics"

Zaveri V.H., Periodic relativity: deflection of light, acceleration, rotation curves. Progress in Physics, 2015, v.11(1), 43-49.

http://ptep-online.com/index_files/2015/PP-40-11.PDF