@Vikram i'm not so sure inertial mass and rest mass mean the same thing.Inertial mass is usually defined using momentum.As such it is only equal to rest mass when v=0.In non -relativistic mechanics we can picture gravity using only the rest masses of the gravitationally interacting bodies.In relativistic mechanics we have to consider the contribution of the four momentum to the curvature of spacetime,leading to gravity.The equivalence principle implies that these quantities are in fact the same,excluding rest mass.

Thank you for your response Carringtone. I quote below the original writing of Einstein.

A little reflection will show that the law of the equality of the inertial and gravitational mass is equivalent to the assertion that the acceleration imparted to a body by a gravitational field is independent of the nature of the body. For Newton's equation of motion in a gravitational field, written out in full, it is:

(Inertial mass) x (Acceleration) = (Intensity of the gravitational field) x (Gravitational mass).

It is only when there is numerical equality between the inertial and gravitational mass that the acceleration is independent of the nature of the body.

— Albert Einstein

In Newton's equation of motion the gravitational mass ia always the rest mass because in his time there was no consideration for relativity. For acceleration to be independent of the nature of the body, the inertial mass also has to be numerically equal to the rest mass. In order to introduce relativity, the coordinate system is fixed on the gravitational mass. So again the gravitational mass remains the rest mass in its own rest frame, and it is equal to the inertial mass. The relativity is now introduced into the acceleration part of the equation by introducing four vectors, Riemannian geometry, geodesics, weak field approximation etc. The track of the coordinate system fixed on the gravitational mass is determined by the parallel displacement of the vector acting on the gravitational mass. The track produced is the geodesic which is assumed to be time like in case of the gravitational force.

GENERAL RELATIVITY has been very successful in predicting,

1) the gravitational redshift of light

2) the gravitational deflection of light

3) perihelic precession of planets

4) orbital period derivative of the binary stars

but it is unsuccessful in giving the quantum theory of gravity.

All the experiments that have been conducted for testing the principal of equivalence have been conducted in the rest frame of the gravitational mass. This includes Newton's Pendulum, Eotvos experiment, Torsion Balance, vacuum free fall, shuttle experiments etc. In these experiments gravitational mass may have some negligible velocities.

PERIODIC RELATIVITY:

I propose that in order to get the quantum theory of gravity, we start all over from Newton's equations of motion. We abondon the geometrical approach for introducing relativity because space-time does not have a physical existence. So we keep the acceleration part of the equation as it was originally proposed and we fix our coordinate system on the source of the gravitational field. Now the gravitational mass will always be in motion in this coordinate system. Because of this motion it will also have some kinetic energy. From general relativity we have learned that gravitational force acts on both, mass as well as energy. Newton's equation of motion does not account for this, because in weak field and low velocity, it hardly makes any difference. In quantum gravity we encounter strong field and high velocities. So here I propose that the gravitational mass in Newton's theory should be relativistic mass. Photon does not have a rest mass, but only relativistic mass equivalent to its kinetic energy. Introduction of relativistic mass now allows to convert Newton's equation of motion into a Schrodinger like wave equation for a two-body system of pair of particle and anti-particle. Complete treatment is given in "Periodic quantum gravity and cosmology." In

addition, the theory is also successful in predicting,

1) the gravitational redshift of light

2) the gravitational deflection of light

3) perihelic precession of planets

4) orbital period derivative of the binary stars

Note that (three-dimensional) F=ma does not hold in relativity (neither in special nor in general relativity, and neither for the invariant mass, not for the "relativistic mass", aka energy; note that in general the acceleration does not even go in the direction of the force; you can easily test that by calculating F=dp/dt and a=dv/dt for a particle that is accelerated in a direction neither parallel nor perpendicular to its velocity). Therefore the concept of "equivalence between inertial and gravitational mass" does not work too well in that context. Instead it is replaced by the general equivalence principle which states that free fall is locally indistinguishable from lack of gravitation.

Note also that in general relativity, the source of gravitation is not mass, but energy and momentum.

Thank you Christopher. General equivalence principle can probably be verified for larger objects but not for fundamental particles. Vector quantities such as velocity, acceleration and force are dependent on space-time which do not have a physical existence, and are plagued by Heisenberg's uncertainty principle. Energy is the most fundamental parameter of the universe and it is a scalar. All vectors and masses are reducible to energy. If you have three or four forces acting in different directions, it will be difficult to get the resultant force but you can compute individual energies for those forces and make a simple arithmatic sum and get the total energy. Similarly it can be shown that the source of gravity in general relativity is just the energy. Gravitational potentials due to different masses are transformed in terms of energy density with the help of Poission's equation. Similarly rate of change of momentum involves rate of change of the mass. In relativity ratio of relative mass to proper mass is same as ratio of relative time to proper time. This is again same as ratio of proper de Broglie frequency to relative frequency. Now if differentiate the standard relativistic expression for the particle energy with respect to time you get force made up of two components. The first part is (relativistic mass x acceleration) and the second part is related to the rate of change of particle frequency which I call de Broglie force. This can be done for fundamental particles but it is not possible to do so for astronomical bodies.Therefore invariance between the force and energy gets lost. With this analysis we can see that source of gravitation arising from momentum is same as saying that the source of gravitation arises from energy. Therefore in ultimate analysis there is only one source of gravitation and that is energy.

Can I ask, showing total ignorance, 'Do we need a quantum theory of gravity?

If we take say the Cartan n-forms view of the world, there are three layers

1. Differential Rules - here quantum effects in one form or the other are paramount.

2. Integral Rules - here wave packets are paramount but I suspect that quantum rules may also apply.

3.Generalised Curvilinear Coordinates - where the impact of the previous two layers have modified the shape of the space-time coordinates. Changes will be made by various operations at the differential level; I suspect due to application of the Cartan equivalent of 'curl'. These changes will propagate at the speed of light, along with the rearrangement of the differential wave fields.

4. The Gravitational forces are then a function of the shape of space time. Space-time provides the equivalent of a gravitational potential, and I cant see why it needs to have particulate properties.

Why do we need gravitons?

Forgive me, but I have found several properties where they seem to have been quantised, just because that is what one does. If I have been too simplistic, please knock me down kindly.

All these properties refer to spacetimes that are solutions of the Einstein equations. If we wish to compute transition probabilities from one spacetime to another, then we need to ``sum over all possible spacetimes'', as intermediate configurations. How to do this isn't known. And just one reason why such a calculation is necessary is that, as a black hole radiates (which, as Hawking found, generically, it does) the spacetime it describes changes. The radiation *is* composed of quanta-and to compute their effect on the spacetime (its ``backreaction'') is an open problem, that requires dealing with spacetimes that are not solutions of the Einstein equations-particularly when the size of the black hole decreases and there isn't any known mechanism that would stabilize its size-so when this becomes small enough that quantum effects become important, then, surely, the description in terms of classical physics must change. How, it is not, yet, known for this case.

Another example: the evolution of the very early Universe. While it is possible to describe inflation (or equivalent scenaria) in terms of an effective potential, the origin of the different mechnisms requires control of situations where gravitational effects and quantum effects are of comparable importance.

Thanks for the answer. My question was a bit outside the scope of the topic, but I really wanted exactly the level of information you gave me.

You're welcome. Regarding the question posed here: the answer is No. The reason is that, first of all, the ``relativistic mass'' isn't Lorentz invariant, so it isn't a useful quantity. Next that, in curved spacetime, the only way to define mass in an invariant way is at the boundary of the spacetime (or infinity, assuming the spacetime is asymptotically flat). Then, if the spacetime in question is described exclusively by its metric, the interaction of matter with gravity is given by the invariant product of the metric and the energy-momentum tensor-which leads to the equality of the inertial mass, that enters in the response of matter to force, and the gravitational mass, that enters in the geodesic equation.

If the spacetime is not exclusively described by the metric however (e.g. in scalar-tensor theories), then the inertial mass, isn't equal to the gravitational mass, since there's the contribution of the scalar.

@Stam. If we try to replace gravitational mass with relativistic mass in general relativity, then that will not be possible, I agree. Because in GR we are locked into fixing the coordinate system on the gravitational mass. Therefore geodesics are always described in terms of proper time which is same as proper mass.

I am talking about completely throwing out the general relativity with its space-time and metrics, geodesics and tensors and its quick fix versions called scalar-tensor theories. I am talking about starting from scratch. From Newton's equations and special relativity. I also disagree with your comment that relativistic mass isn't Lorentz invariant. How this is done is already shown in detail in the article "Periodic relativity: basic framework of the theory." I want to point out a little correction here. What is described as the true force in this article is same as the Lorentz force and what is described as the de Broglie force is part of the Lorentz force. This de Broglie component makes the relativistic mass Lorentz invariant. This expression for the true force is just the modified version of dp/dt.

Another point: If one starts from Newton's equations and special relativity (so here this would *mean* Newton's law of gravitation), then, as Einstein found, the end result will be general relativity-if the equivalence principle is assumed-or a theory that includes additional fields, beyond the metric tensor, that parametrize the non-equivalence of inertial and gravitational mass-since spacetime isn't determined just by the metric tensor. The Brans-Dicke scalar-tensor theories are one version, supergravity, as first noticed by Scherk, another.

@Stam. If one starts from Newton's equations and special relativity (so here this would *mean* Newton's law of gravitation), and if equivalence of gravitational mass and the relativistic mass is assumed then the end result will be periodic relativity and periodic quantum gravity and cosmology which is able to generate the entire table of standard model particles from a single formula. And it unifies gravity with other fundamental forces. This is what Brans-Dicke scalar-tensor theories cannot do.

The gravitational mass is equal to the intertial static mass, where the motion speed tends to 0. The described "mass increase" as L.OKUN in his paper affirms, is just an energy increase, caused by the space-time local warpage .

The energy needed to further increases of speed (LORENTZ) is stored in the same volume occupied by the mass. That energy goes into the warped space-time, which can represent over 99% of the volume. The relativistic mass is an "improper term" and cannot be equal to the gravitational mass. Planets don't have a different gravitational attraction because they move at some km/s around the Sun. The gravitational field of such masses don't increase, the inertia does. This process occurs because the speed of light in warped space time decreases (clock rate gets longer). The body which has a relative lower light speed compared to the surroundings is able to store more energy than any another body of the same type moving at lower speed. Any energy contribution this way will have a lower and lower effect on the further speed increase of the body. The SPACE-TIME defends itself from speed increments of bodies (which are responsible for increaisng the Einstein Stress Tensor), by lowering the local speed of light which is the speed of any energy exchange.

Dear Stefano. Your point of view is presentable within the context of general relativity but it has no chance in a theory which does not recognize the existence of space-time.

Dear Vikram, which theory are you talking about then??

@Stefano

Periodic relativity: basic framework of the theory and

Periodic quantum gravity and cosmology

The space-time is central, I don't consider such theories

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