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":
Friedmann, "On the curvature of space" (1922) and "On the possibility of a world with constant negative curvature" (1924), chose a hyperbolic line element in which the spatial part was the hemisphere model equipped with a Riemann metric. There were two unknown parameters, R2 and M 2, that multiplied the space and time components of the metric, respectively, and were required to satisfy the Einstein field equations.
The Friedmann model of the universe forms the basis of the Standard Model. Yet, it is usually considered not as a pair of evolution equations, but, rather, a condition where the Hubble parameter and average density of the universe are determined empirically and the sum of all the densities, the mass density, the curvature term, and the vacuum energy density (cosmological term), add to unity.
The Euler-Lagrange equation in general relativity contains an additional term to the acceleration involving the Christoffel connection coefficients. In the weak field limit, Newton's law of gravity "pops" out of this term yielding Newton's second law for gravitational attraction. This supposedly supports the notion that non-Euclidean geometry with non-constant curvature is somehow related to gravitation.