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Localization of Energy Implies the Absence of Gravitational Induction

The localization of energy is encapsuled in Poisson's equation. Moreover, it requires the gravitational field to be static. The existence of tidal forces requires the electric component of the Weyl tensor to be symmetric. This implies that there is no Faraday induction in gravitation. And since all fields are static, there is not time rate of change of Maxwell's displacement current in Ampere's equation.

The localization of energy is encapsuled in Poisson's equation. Moreover, it requires the gravitational field to be static. The existence of tidal forces requires the electric component of the Weyl tensor to be symmetric. This implies that there is no Faraday induction in gravitation. And since the gravitational field is static, there is no time rate of change of the displacement current. Hence, there are no gravitational waves.

Any lack of symmetry of the gravitational force tensor would lead to a non-localization of energy. Hence, even if GW waves could be salvaged through the antisymmetric components, there could never be a localization of energy that would be the source of the propagation of these waves. Even without these revelations, it should have seem strange that GW plane waves are OK, but spherical waves are not. The latter would necessarily indicate a source for such waves as dropping a pebble in pond and observing the spherical waves grow from the center.

So, bivectors and a time dependent Maxwell tensor are compatible with energy flow (Poynting vector) and energy localization, but no tidal forces. Gravity, on the other hand, is compatible with a description through tidal forces, whose field can be represented by field line when the divergence of the tidal force vector vanishes. This would be analogous to a static Coulomb field. But, in EM, the electric field is a sum of the gradient of the Coulomb field and the time rate of change of the vector potential. Herein lies the dichotomy: the electric field has one foot in a static Coulomb universe, and another food in a time dependent motion of charges. Circular currents lead to magnetic fields and their interaction to phenomena of induction. Gravity has no charge so inductive phenomena are ruled out and Newtonian tidal forces hold sway.

It is claimed that whereas there is no gravitational analog to Faraday's law on account of the fact that the electric tensor is symmetric, there is an analog to Ampere's equation without the displacement current [Costa & Herdeiro, "A gravito-electromagnetic analogy based on tidal tensors"). This, however, is inaccurate. It is only necessary to consider the homogeneous Friedmann universe. The non-zero time-time, and space-space, Ricci components are:



respectively, where r is a function of time only, the primes indicating the derivative with respect to time. It's the second equation that shows that the Ricci tensor is devoid of physical meaning.

Ampere's law, in the absence of the displacement current is


Analogously, Newton's current is

2r'^2-rr''= V_N^2

which comes from his expression for the radius of curvature. Note that in both cases, the current results from the difference between individual accelerations and interacting currents. Not so with the space-space component of the Ricci tensor, R_r^r. The individual terms have no meaning, only their differences. In Newton's case accelerating particles with decrease Newton's force, while mutual interaction through their motion will increase it.

The generalized Newton's law is


Whereas, Weber's force, which is synthesis of Coulomb's and Ampere's is


where c_W is Weber's constant, greater than the speed of light by the square root of 2. Moreover, Newton's force equation can be satisfied by F_N r^2=constant, under constant angular momentum, or F_N=constant under constant angular velocity.

Since R_0^0 is a tidal force, it cannot be presented as the sum of the density and 3 times the hydrostatic pressure. The pressure should always appears as a surface term, and not in the Poisson equation. So, too, R_r^r is not Ampere's expression for the velocity, and this explains why Einstein's equations do not make any sense. The interaction term (r'/r)^2 cannot be identified as Hubble's parameter. It has no meaning of a recessional velocity. Again, only differences between individual acceleration and motion of pairs of particles have physical meaning. Therefore, the conclusions obtained from the Friedmann model of a homogeneous universe are without physical significance. Unfortunately, it is as O'Rahilly so poignantly put it: Once a method has become fashionable in physics, reasons will be found for justifying it" even if that means twisting the laws of physics and logic.

We have shown [Seeing Gravity, p. 236] that Ampere's law is a geodesic equation, and at constant angular velocity, one particle will trace out a parabola with respect to the other. The Riemann tensor which determines geodesic deviation is R^i_{0l0}. For weak static fields, this consists of 4 terms, 3 of which contain derivatives with respect to time and only one is the second mixed spatial derivatives of the g_{00} component of the metric.. Since g_00 is proportional to the Newtonian potential, R^i_0l0 is the second derivative of this potential with respect to space coordinates x_i and x_l. The tidal force, F^i, is R^i_0l0 multiplied by the displacement x^l. When referred to a tidal ellipsoid, the divergence of F^i vanishes. and the field can be represented as field lines, just like in the case of charges. This is the condition for tidal equilibrium which has been confused with Einstein's condition of "emptiness." Nothing is empty that produces tidal forces!

In summary, the localization of energy prohibits potentials from becoming time dependent resulting in an antisymmetric electric tensor (the covariant derivative of the electric field). EM specifically includes a non-potential, time-dependent term in the expression for the electric field which is responsible for inductive phenomena: the curl of the electric field and the time rate of change of the magnetic field, and the curl of the magnetic field which gives rises to the time rate of change of Maxwell's displacement current.

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