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Einstein's Attempt to Find a Gravitational Analog to EM Induction was Stillborn

Faraday defines an "electrotonic" state as a wire traversed by an electric current placed in a magnetic field. As long as the magnetic field remains stationary there is no effect; "it is when it is changing that it is operative." The quote is taken from JJ Thomson's Yale lectures on electricity and motion that he delivered in 1905.

To Thomson, the particle is immersed in an ether, and its Faraday tubes drag the ether along when it is in motion. This causes an increase in the mass of the particle proportional to the magnetic energy. In Thomson's words, " the whole of the mass of any body is just the mass of the ether surrounding the body which is carried along by the Faraday tubes associated with the mass." The greater the speed, the greater is the contribution of the charge to the mass of the particle. This contribution to the mass vanishes when the particle is at rest.

Considering a shell of finite thickness, d, The pulse of the same thickness is produced by the slowing down of the particle which "is the seat of intense electric and magnetic forces which diminish inversely as the distance from the charged particle, whereas the forces before the particle was slowed down diminished inversely as the square of the distance." If the particle were made to vibrate periodic electromagnetic waves would be propagated outside the body.

Einstein thought that there was a gravitational analog of this when he wrote "Is there a gravitational effect which is analogous to electrodynamic induction?" Probably he was fully congizant of the shaky ground he was one since it was published in a medical journal. With a vague notion of why Mach's principle should apply to the increase in mass of a particle due to the presence of all the other particles in the universe, Einstein reasoned that a closed physical system should have its mass increased by a factor E/c^2 due to its energy content. It is imperative to observe that this is a completely static effect where a shell K(M) of mass M is surrounding a mass, m, at the center P(m). If R is the radial distance from P to any point on K, the mass at the center should be increased by

m' = m(1+GM/Rc^2). (1)

We are now told that the particle at P moves with velocity q, and that the energy of the material point is

mc/{1-q^2/c^2}^1/2 or (m/2) q^2/c

"in the first approximation. However, this looks more like a momentum than an energy.

The fudging starts when he says that "[i]n order to obtain the kinetic energy in the customary unit, one has to multiply this expression by the constant c_0 which is equal to the velocity of light at infinity...Thus, in the customary units the kinetic energy L is

L=(m/2)q^2(c_0/c). (2)

But, this is not the non-relativistic kinetic energy of a particle of mass m! The distinction made between c_0 and c would pave the way for the introduction of an index of refraction, like that for an Eaton-type lens,


where a is a contact, were the ratio (c_0/c) in L were squared. But Einstein chose the risker path of writing the equation of motion as F=-c dc/dx in the x-direction, linearizing it and setting it equation to dV/dx, where V is the Newtonian gravitational potential. In this way he cam out with (1), as desired.

If Einstein had resorted to using an index of refraction, he would have found that since the frequency doesn't change as light passes into a denser medium, its speed, c, will decrease causing a shortening of the wavelength. Then from de Broglie's relation he would have found that the momentum increases since the latter is inversely proportional to the wavelength. This is in flagrant contradiction with mechanics where an increase in velocity leads to an increase in momentum with mass remaining constant.

The velocity has entered by the contorted definition of kinetic energy, (2). There is no reason to believe that the mass, m, would be dependent on the velocity, q; and, in fact, it isn't. The only way then is for the dependency to arise in the ratio c_0/c, and this would imply a medium with a different index of refraction than that at infinity.

No mention is made of what the velocity, q, is: whether it is measured with respect to an inertial system, i.e. the external force, or whether it is the velocity of m with respect to the rotating body axes. The distinction is crucial since the velocity and acceleration in the rotating frame vanish in an inertial one.

Motion is crucial when it comes to processes like induction in which Einstein is trying to draw a parallelism between electromagnetic induction and gravitational induction. The addition of E/c^2 to the mass due to the presence of a shell of mass M does not show "that the presece of the inertial shell K increases the inertial mass of the material point P inside the shell." In some vague way, Einstein is trying to draw on Mach's principle (which he coined in 1918) that the inertia of a body depends on the distribution of heavenly bodies. That's undoubtedly why he turns his attention to determine the relation between the two forces that are necessary to put the two masses in motion.

Einstein calls induction the following: He wants to determine the force exterted (induced) on m by the spherical shell K in order that it leaves the mass at rest in P. "This force has the same sign as the acceleration, in contrast to the corresponding interaction between equivalent electrical masses." However, this is not induction.

Inductive effects, as explained by the grand master, J J Thompson, consist in a moving charge of electricity will carry with it its own magnetic field. The magnetic field has energy and this contributes to the kinetic energy outside of the sphere of the particle. Thus, part of the mass of a sphere is due to its charge. If the charge remains stationary, as in Einstein's case without the charge, there will be no effect.

J J was worried about the violation of Newton's third law. The particle will gain additional energy, which he reasoned was compensated by the energy loss by the field. In the gravitational case, however, the field energy is negative, and there was nothing to take from. Negative energy was the primary reason why Maxwell gave up on a theory of gravitation that would parallel his theory of electromagnetism. Whereas the electromagnetic field has positive energy, the gravitational field has negative energy.

Whereas Einstein had to artificially add E/c^2 to the original mass, J J showed that the magnetic energy is an additional kinetic energy, which identifies the extra mass as due to the charge of the particle. And it was even conjectured that maybe all the mass is electrostatic in origin.

The term J J began with was the magnetic component of the Lorentz force. This component is normal both to the velocity of the particle and its radial distance to an observer. It has the same form as a Coriolis force; the force is normal to the direction of motion and the velocity of the particle. So what is claimed is that the Lorentz force

e dv/dt = e (E + v X B), (3)

has the mechanical analog

m dv/dt = m (g + v X w), (4)

where the charge, e, is replaced by the mass, m, and the gravitational acceleration stands in for the electric field, and the magnetic component of the Lorentz force is replaced by the Coriolic force where the angular velocity, w, stands in for the magnetic field.

Now, the centrifugal force is related to the magnetic energy according to

w x (w x r_s)= w(w.r_s)-w^2 r_s =-(eB/m)^2 r_s

because (w.r_s)=0. And, although (3) and (4) have similar forms, the velocity in (3) is with respect to an inertial frame, i.e., to an external force, while the v in (4) refers to the axes of a rotating frame. The gravitational force, mg, is considered as an internal force, rather than an external force.

The Coriolis force depends on a rotating coordinate system like the rotation of the earth. The torque depends on the nonorthogonality between the radial vector referred to the rotating axes and the angular velocity. In inertial systems, the radial vector, referred to a system, is always orthogonal to the angular velocity, where the angular velocity w=eB/m.

A moving mass does not create a gravitomagnetic field like a moving charge creates a magnetic field. The latter live in inertial frames, the former do not. The Coriolis force does not depend on the motion of the source. A magnetic field can be made to disappear by the relative motion of the source.

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