Einstein's attempt to generalize the indefinite Minkowskian metric of general relativity was a pass in the wrong direction. The only thing it could treat was light trajectories in which increments in local time vanish. Rather, if we focus on velocity space and think of Newton's F=ma as Fdt=mdv, as a modification of velocity rather than a change in position, the metrics we obtain will be generalizations of the Riemann metric which will be appreciated as geodesic equations of Riemann geometry where the acceleration will have constant projection onto a plane that is tangent to a given surface.

We will focus our attention on potentials of inverse power laws of which the Kepler problem is an elementary example of. A key factor is the introduction of Levi-Civita's fictitious time ds=dt/r, which can be generalized to ds/dt=V(r), where V(r) is a potential belonging to the class of inverse power laws. For V(r)=1/r, we have the Newtonian potential, while for V(r)=1/r^2, the fictitious time is related to an angle variable, and implicit is the conservation of angular momentum. In fact, both potentials are related to angles. The former is related to the rate of change of the eccentric anomaly, E, since its rate of change is dE/dt=na/r, where n is the mean motion which is the constant angular velocity needed to complete one revolution in an elliptic orbit whose semi-major axis is a, and the true anomaly, v, whose rate of change is dv/dt=h/r^2, where h is the conserved angular momentum.

The two velocities are related to by Snell's law:

dE/dt/sin E=dv/dt/sin v,

from which we obtain the index of refraction as

eta=na/r d theta/dt,

the ratio of the constant angular velocity of an ellipse to that of a circle since the angular momentum, h=r^2 d theta/dt . When the index of refraction eta=1, there is no distortion, and the two media become a single medium.

In the Kepler problem, the equation of motion is |du/dt|=GM/r^2. This has a singularity at the origin where two bodies would collide. If we introduce the eccentric anomaly, we reduce the power by one giving (na)du/dE=GM/r. Then using the conservation of energy, GM/r-u^2/2=H, where H<0 is the total energy, we come out with

(na) du/dE=u^2/2-H.

Now inversion in a circle, w=u/|u|^2, which is a two-dimensional analog of three-dimensional stereographic projection, we arrive at

dE^2= 4(na)^2 dw^2/(1-2Hw^2)^2,

which, except for the scaling factor, is the form given by Riemann in his inaugural dissertation and published posthumously. For H<0, it is a Riemann metric for elliptic geometry, while for H>0, it becomes the metric of constant curvature for hyperbolic geometry.

We now come to the realization that the metric is identical to the geodesic equation with *constant* tangential acceleration. The acceleration of a curve a=r(x(t)) is

d^2a_k/dt^2=r_{k,ij}u_iu_j+r_{k,i}du_i/dt,

where u_i=dx_i/dt. To find the shape of a curve such that the acceleration has at most a constant (also zero) projection onto the tangent plane to a given surface, we multiply the above by r_{k,n}, where n=1,...M and obtain

du^m/dt+G^m_{ij} u^i u^j =A,

where A is the constant projection onto the plane tangent to the surface, and G^M_{ij} is the Christoffel symbol. The constant A plays the role of the total energy in the Keplerian example. Inverting the geodesic equation, we obtain an expression for dt^2, where t is a single time, and not a proper time. Commonly, it is referred to as an affine parameter so as to encompass both particle and light trajectories. Here we have no distinction between the two.

In summary, the expression for the indefinite metric of general relativity makes no sense interpreting it as a line element for some local time attached to particle motion. What is relevant is the equation of motion whose solution a generalized geodesic curve. The square of the equation of motion coincides with a Riemann metric in velocity space for a non-Euclidean geometry of constant curvature, corresponding to the constant projection of the acceleration onto the plane tangent to the given surface.

## Comments