This is the title taken from a little known (perhaps intentionally!) of a 1912 article of Einstein to show that the mass of a system is modified by the presence of other matter by an amount $GMm/Rc_0^2$, where $m$ is within a hollow sphere of mass $M$ of radius $R$, and $c_0$ is the speed of light at infinity. This is the amount of increase in mass independently of whether we are considering $m$ or $M$. Instead of the electrodynamic repulsion between two circuits, embodies in Ampere's law, Einstein always comes out with an attraction between the masses.
Einstein made a big story out of Ricci's tensor and its dubious relation to a matter stress tensor. In the very few analytical models that are solvable, the force does not correspond to a linear combination of sectional curvatures. But general relativity does not define forces, but it does define pressures, and the pressure is the force per unit area.
If we agree that the sectional curvatures of a surface have something to do with the forces that perhaps create them, then it is manifestly obvious that the average curvature afforded by the Ricci tensor has nothing to do with the forces themselves. For it would be equivalent to the condition of mechanical equilibrium whereby the sum of the forces vanish in a state of mechanical equilibrium.
In our last blog, we showed that Weber's force could be written as an affine linear combination of two sectional curvatures: radial and tangential curvatures.
Friedmann considered an isotropic and homogeneous world modelled as three-dimensional generalization of the hyperbolic Poincare' half-plane model. Through a fractional transform, known as a Mobius transform, which preserves angles, but not distances, the half-plane model can be transformed into a disc model of the hyperbolic plane. The resulting metric is known as the Robertson-Walker metric which has a constant spatial negative curvature, $k$, of the Poincare' model.
Rather than being supportive of Einstein's "General" theory of relativity, Friedmann's solution of the Einstein's equations makes it untenable for it introduces a non-constant spatial curvature, and it contains an energy density even though the sum of the curvatures in any given direction vanishes.
Any time there is a variation in the gravitational force, tidal forces are produced that can affect the mutual attraction of masses. Depending upon direction, tidal forces can be both attractive and repulsive, but the magnitude is always $m/r^3$. Moreover, they can give rises to tidal waves, and because of the Lorentzian form of the metric they can be described as gravitational waves in the linearized approximation to Einstein's equation.
The Ricci tensor is a collection of Gaussian curvatures, or, more precisely their average in one direction. What is the physical significance of adding Gaussian curvatures, and claiming---as Einstein did---that the vanishing of their average is a condition of "emptiness".
The analogy between electrodynamics and gravitational radiation is well-known, and documented, e.g.,
Consider the Schwarzschild solution for illustrative purposes. The Ricci scalar vanishes for spacetime, implying a 'compensation' of time and space components of the tidal forces (-2+1+1=0).
Proper time, or "local" time as Lorentz referred to it, and coordinate time are related by a change in the sign of the constant velocity in the Lorentz transform. So what the observer "sees" at either position should not change the physics (apart from currents, magnetic fields or whether an electric charge radiates or not).
A pillar of special relativity is that uniform motion is undetectable as far as physical laws are concerned (Poincare'). This was generalized by Einstein in general relativity to his principle of "universal covariance":