Newton could tells us, but Einstein can't. According to his "general" theory of relativity, all particles follow what are known as 'geodesics'. These are paths of the shortest distance between any two points. In flat space, these paths are straight lines, but in non-euclidean spaces, these are curves where a point moving along such a segment moves at constant velocity with zero acceleration. How can Newton's apple be reconciled with such behavior?

The old adage that mass tells how space curves and curved space tells how mass moves is vacuous. Place a mass on rubber sheet. It will make an indent in the sheet, but, not thanks to space (time?), but rather because the pull of gravity.

We need a force; geometry in the form of a rubber sheet, or whatever, will not do.

Just take a look at the Schwarzschild solution: there is a central mass but the solution was solved using Einstein's condition of emptiness. But, how can it be empty if there is a mass present? O'Neill tells us point blankly not to model the mass as if it is not there.

Geodesics are restricted to convex parts of a surface. Wrap an apple in a rubber sheet. At the stem of the apple there is an indention. The rubber sheet cross the pit without touching it. Geodesics are not allowed to do this and must take land, and not air, transportation.

Geodesics have been incorporated in general relativity through a variational principle. They are curves with zero tangent acceleration. Consider a parametric curve,

$$\alpha(t)=s(x(t)),$$

where $t$ is time. The velocity is defined as

$$\dot{\alpha}_j=s_{j,i}\dot{x}_i,$$

where the dot denotes the derivative with respect to time and the comma denotes the derivative with respect to that component. The acceleration is the second derivative

$$\ddot{\alpha}_k=s_{k,ij}\dot{x}_i\dot{x}_j+r_{k,i}\ddot{x}_i.$$

The condition for zero acceleration is that it have zero projection on the plane tangent to the curve, i.e.,

$$\dot{\alpha}_kr_{k,j}=0,$$

for $j=1,\ldots$. This can be written in a more eloquent form by introducing the metric tensor, $g_{ij}(x)=r_{k,i}r_{k,j}$, so that the condition for constant velocity becomes

$$\ddot{x}^{k}+\Gamma^{k}_{ij}\dot{x}_i\dot{x}_j=0,$$

where $\Gamma^k_{ij}$ is the Christoffel symbol of the second kind. Details can be found in any text on Riemann geometry so we can dispense with the details.

The vanishing of the acceleration is the condition for a geodesic. The surface on which the curve is traced appears only in the metric tensor $g$ and its first derivative (through the $\Gamma_{ij}^k$). As such it can tell us nothing about the curvature of the surface which requires second derivatives in the $g_{ij}$. Since the Christoffel symbols, and their products, enter into Einstein's tensor of gravity, which is not a real tensor but a *pseudo* one, it can be eliminated through a coordinate transformation---which is hardly a desirable property for gravity to have!

In the special case where $g$ is constant, the equation of a geodesic reduces to $\ddot{x}_i=0$, or a *straight* line.

If apples don't fall like geodesics why should they be important to us? Consider a light beam traversing a material with an index of refraction $n$, which we suppose is equal to the a single space variable $y$, the height. It is sufficient to consider the behavior of the light beam in the plane $x,y$. The equations of a geodesic take the form

$$ \ddot{x}+2\frac{n^{\prime}}{n}\dot{x}\dot{y}=0,$$

and

$$\ddot{y}+\frac{n^{\prime}}{n}\left(\dot{y}^2-\dot{x}^2\right)=0.$$

By simple manipulations, these equations can be combined and brought into the form

$$\frac{1}{n^2}\left\{\left(\frac{dy}{dx}\right)^2+1\right\}=\mbox{const},$$

independent of time. Setting $dx/dy=\tan\vartheta$, where $\vartheta$ is the angle from the normal vector to the plane separating the two media of different indices of refraction, ($n$ is the relative index of refraction), and the beam, the above expression reduces to Snell's law

$$n\sin\vartheta=\mbox{const}.$$

As an example take the unevenly heat plane which is a physical model of the Poincare half-plane. Temperatures reduce to zero along the $x$-axis and increase along the $y$ axis. The index of refraction follows a similar trend,

$$n(y)=\frac{1}{y}.$$

Snell's law,

$$\sin\vartheta=\frac{y}{r},$$

becomes the equation of a circle of radius $r$. Geodesics, therefore, in the Poincare half-plane model, are semi-circles, and their asymptotes are straight lines with $\vartheta=0$, or equivalently, $x=\mbox{const}.$ for $r=\infty$.

In the last blog, we considered a source of electromagnetic radiation propagating along an axis with $\vec{v}=\mbox{const}.$, and considered the distances $s'$ and $s$ to a fixed observer, from a past position, $s'=c(t-t')$, to a present position $s(t)$.

The law of sines,

$$\frac{\sin(\omega'-\omega)}{\beta s'}=\frac{\sin\omega}{s'}=\frac{\sin\omega'}{s},$$

is nothing other than a statement of Snell's law, where $\beta=v/c$ is the relative velocity. In particular, Snell's law,

$$\frac{\sin(\omega'-\omega)}{\beta s'}=\mbox{const}.,$$

gives an index of refraction,

$$n=\frac{1}{\beta s'},$$

just like in the half-plane model for constant speed. There are none other than geodesics of our model.

This has often been used as a model that intends to show that the electric force always points in the direction of the present position, and that somehow a miraculous cancellation of all first order terms occurs, according to Carlip. He is right in claiming that all terms in the Lienard-Wiechert expression

$$\vec{E}/e=\frac{(\vec{s}'-\vec{\beta}s')(1-\beta^2)}{(s'-\vec{\beta}\cdot\vec{s}')^3}+\frac{\vec{s}'(\wedge(\vec{s}'-\vec{\beta}s')\wedge\vec{a}'/c)}{(s'-\vec{\beta}\cdot\vec{s}')^3},$$

where all quantities are calculated from the past position. Carlip (arXiv:gr-qc/9909087v2, like other authors, neglects the second term, which is the radiative term but explicitly involves such a term in his rationalization of why the cancellation takes place. That $\vec{s}'-\vec{\beta}s'=\vec{s}(t)$ was reason enough for claiming that the electric field (or the velocity component) points to the present position.

Carlip introduces the caveat that the velocity field component "does not point toward the 'instantaneous' position of the source, but only toward its position extrapolated from this retarded data." The vector $\vec{s}(t)$, however, is the vector from the observer to the present position of the source. He continues, " if a source abruptly stops moving at a point, a test particle at [that] position will continue to accelerate toward the extrapolated position of the source until the time it takes for a signal to propagate from [that] position to [the present] at the speed of light."

Confusion abounds here insofar as Carlip has excluded the acceleration component of the electric field, where $\vec{a}'$ is its acceleration at the retarded time, $t'$. Moreover, reference to Maxwell's theory treating an accelerating source is also not justified. Maxwell's equations lead to an undamped wave propagating at speed $c$, far away from sources $\varrho$, and their current, $J$. But, there is no reference to those sources accelerating at $\vec{a}'$.

All reference to aberration refers to the acceleration field, not the velocity field, in the expression for the electric field. Sadly, the observer will see nothing since the only visible acceleration is perpendicular to the line of sight, and this contributes to the radiated electric field. The invisible component of the acceleration parallel to the line of sight does not radiate so our observer will see nothing. What you see is what you get! And aberration occurs for the tangent acceleration so that the answer to Carlip's question "Is the Cancellation a Miracle?" is a flat *no, since aberration does not enter into the velocity field component of the electric field*.

Perhaps, Carlip was led astray by Poincare's observation that "any Lorentz-invariant model of gravitation necessarily requires additional velocity-dependent interactions, which can provide 'a more or less perfect compensation' for the effects of aberration $\ldots$ Poincare did not actually demonstrate that the cancellation of terms of order $v/c$ is necessary, but he showed that aberration terms can be naturally excluded without doing violence to the theory."

A "Lorentz-invariant model of gravity with light-speed propagation" is nonsense since you need accelerating mass to produce gravity waves, if conventional wisdom is correct. The claim that "deviations from Newtonian gravity are at most of order $v^2/c^2$" is also inaccurate since aberrations terms enter in odd powers of the relative velocity $v/c$. It would, therefore, appear much more profitable to accept Carlip's alternative "in which gravity propagated instantaneously, but, as in electromagnetism, only at the expense of 'deunifying' the field equations and treating gravity and gravitational radiation as independent phenomena."

And then there is the Achilles' heel of the putative observation of gravitational waves based on comparing spectra with that obtained from numerical relativity. As Carlip rightly admits "there is no preferred time-slicing in general relativity, and thus no unique definition of an 'instantaneous' direction." Numerical relativity depends crucially on this "time-slicing" and hence does not know whether it is coming or going!