Electromagnetic Waves on Curved Spaces versus Gravitational Waves on Flat Spaces

Einstein developed a set of field equations in order to determine the metric coefficients in what would be otherwise the hyperbolic metric of special relativity. At least one metric, so derived, called the Schwarzschild metric can be written as a Hamilton-Jacobi equation (Landau and Lifshitz, Classical Theory of Fields, p. 306 4th edition). The Hamilton-Jacobi equation is equivalent to the eikonal equation of geometrical optics.

Derivation of Laplace's Equation for the Speed of Gravity from Newtonian Dynamics

Derivations of Laplace's formula for the change in period of the moon, that yielded a formula for the speed of gravity, has been derived by a number of authors. The key point is to consider the tangential force acting on the moon due to aberration due to the finite speed at which (supposedly) gravity propagates. To the central force, $\mu/a^2$, where $\mu$ is the gravitational parameter containing the mass of the earth, $GM$, and $a$ the semi-major axis, the aberration term $v/v_G$ is appended where $v_G$ is the speed at which gravity supposedly propagates.

Time Dilation and Space Contraction from Keplerian Orbits

We have discussed repulsive gravity within the context of the conformal equivalence between the squares of Hookean ellipses and the orbits in a gravitational field. In particular attractive oscillators corresponds to attractive gravity while a negative Hookean law is equivalent to repulsive gravity. Here, we show that oscillators and gravitational forces lead to non-Euclidean geometries. This has been discussed before by Milnor, "On the geometry of the Kepler problem" in Not. Am Math Soc. 90 (1983) 353-365.

Projections from the Sphere and Pseudosphere and their Indices of Refraction

Based on the eikonal equation of geometrical optics and the Legendre transform, Luneburg was able to associate an elliptical orbit created by a Coulomb field with the line element of a sphere. This meant that to any sphere of a given radius, $r_0$, there belongs a conjugate sphere of radius $r_1=1/r_0$, which represents a perfect undistorted optical image of the original sphere. Its image, however, is inverted, and its magnification is the ratio of the radii, $r_1/r_0$. The line element, $$ds^2=4\frac{dx^2+dy^2}{(1+r^2)^2},$$ is that of a sphere of radius $r$.

Twisted Tales from General Relativity: The Deflection of Light

Supposedly one of the greatest predictions of twentieth century physics, was Einstein's prediction of the deflection of light from a massive star like the sun. His pre-relativistic calculation gave half the value that his post-relativistic calculation gave, and this was heralded as a great triumph for general relativity. Yet, there are so many loop holes in the calculation that it's amazing that no one has ever questioned it.

Gravitational Waves: From Schwarzschild to Michelson

We know very well how a static gravitational field influences the propagation of light from Schwarzschild's solution of Einstein's field equations. But, do we know how light behaves in a passing gravitational wave? It is argued that gravitational waves are "ripples in spacetime", and their effect is to stretch and contract it so that spacetime is a new aether and not a vacuum in which light (and gravitational waves) propagate.

Weber versus Schwarzschild: GR Much Ado About Nothing

In our last blog we proposed a metric, $$-c^2d\tau^2=-rc^2dt^2+\frac{1}{2r}\left(dr^2+r^2d\varphi^2\right),$$ in the plane, $\vartheta=\pi/2$, for the Weber force, $$F_W=\mbox{const.}\times\left(\ddot{r}-\frac{\dot{r}^2}{2r^2}+\frac{c^2H^2}{r^2}\right),$$ whose vanishing coincides with the Euler-Lagrange equations, or, equivalently, the geodesics.

The Metric for the Weber Force

We have seen, in the last blog, that there is a rather strict analogy between the Schwarzschild metric and Gerber's potential, both of which give the same expression to the perihelion shift of Mercury to order $1/c^2$. It has long been known that to this order the force derived from Gerber's potential is the same as the Weber force. There is a slight difference, however.