Anisotropic Coordinates

It is not so easy to free oneself from the idea that coordinates must have a direct metrical significance.

                                                                           Einstein, 1949

In flat spacetime it’s usually most convenient to describe phenomena in terms of an inertial coordinate system, defined as a system of space and time coordinates in which the equations of Newtonian mechanics (in homogeneous and isotropic form) hold good quasi-statically. In any such system of coordinates the speed of light in vacuum equals c in all directions, i.e., the speed of light is isotropic. As is well known, inertial coordinate systems are related by Lorentz transformations. Since the speed of light has the same value in all directions in terms of such coordinates, it follows that the round-trip speed of light is the same around any closed path in terms of these coordinates. However, the converse is not true, because there exist coordinate systems in which the round trip speed of light is the same around any closed path, and yet the speed of light is anisotropic, i.e., it varies depending on the direction.

Before describing these anisotropic coordinate systems, it’s worth pointing out a common misconception. It’s often said that, although the one-way speed of light is conventional, the two-way speed of light is empirically unambiguous. This is not true. Consider, for example, a system S of inertial coordinates, and another system S’ of coordinates related to S by a Galilean transformation. Neither the one-way nor the two-way speed of light is invariant in terms of S’. Indeed the prediction of Michelson of a second-order anisotropy in the two-way speed of light was based on the assumption that inertial coordinate systems are related by Galilean transformations. The fact that no anisotropy in the speed of light was detected by the inertial measures implicit in Michelson’s interferometer does not imply that light speed is isotropic in terms of S’, but that S’ is not an inertial coordinate system.

As discussed in the note on Round Trips and One-Way Speeds, the necessary and sufficient condition for the speed of light around any closed path to be invariant, equal to a scaled value of 1, is that the speed must be of the form

where k is a constant between -1 and +1. Let S denote a system of inertial coordinates t,x,y, and suppose a pulse of light is emitted from the origin at time t=0. At time t_p=1 the locus of positions of the light pulse (or wave crest) is x_p=cos(θ), y_p=sin(θ). Now consider another coordinate system Sʹ, related to S by

where A and B are (as yet) unknown constants. In terms of these coordinates the speed of the pulse corresponding to any given value of θ is

We can’t yet compare this with equation (1) because the parameter θ in this expression is the angle in S, whereas we need to express cʹ in terms of the corresponding angle θʹ in Sʹ, taking aberration into account. To do this, we note that the spatial hypotenuse H of the ray from the origin to x_pʹ,y_pʹ is given by H² = (x_pʹ)² + (y_pʹ)², so the cosine of θʹ is

Solving this for cos(θ), we have

Letting α denote Btan(θʹ), this can be written as

In terms of the squared parameter q² = 1 + α²(1−v²) we have

Substituting this into equation (2), we get

Reverting back to the a parameter, this can be written as

Multiplying through the numerator and denominator by cos(q’), we get

For a Galilean transformation we have A = B = 1, and the directional speed of light in terms of Sʹ is

Naturally this agrees with the result that can be derived immediately from the condition

which signifies that the expanding circular wave centered on a point moving with speed –v along the x axis. Solving this equation for c’ gives

which is easily shown to equal to the previous expression. Since this describes a circular locus, instead of an ellipse as required by equation (1), the round-trip speed of light is not invariant in terms of these coordinates, differing in the second order.

On the other hand, if we set A = (1−v²)^1/2 and B = 1/A in equation (4), the directional speed of light in terms of Sʹ is

and therefore the round-trip speed of light around any closed path in terms of these coordinates is invariant, even though the speed of light is anisotropic in the first order. In this case the aberration effect is

This corresponds to the transformation from S to Sʹ given by

These are not inertial coordinates (even though they are un-accelerated), because the equations of Newtonian mechanics are not even quasi-statically valid in terms of these coordinates. In particular, the “third law” of equal action and reaction is violated. This is due to the fact that mechanical inertia is not isotropic in terms of these coordinates, nor is the speed of light. However, if we define another system of coordinates Sʺ by a simple time skew

the transformation from S to Sʺ is

In terms of these coordinates, the speed of light is invariant, both one-way and round-trip, and mechanical inertia is isotropic. Both S and Sʺ are inertial coordinate systems. It follows that we can transform from an inertial coordinate system to an anisotropic coordinate system, in which the one-way speed of light is direction-dependent in the first order but the round-trip speed of light is invariant for any closed path, by a simple time skew, i.e., tʹ = tʺ + vxʺ, xʹ = xʺ, yʹ = yʺ. We may call these Lorentzian coordinates, because they are scaled with Lorentz’s time dilation and length contraction factors, but their temporal foliation is skewed relative to the inertial foliation.

The figure below shows the directional speed of light in terms of inertial coordinates, Galilean coordinates, and Lorentzian coordinates.

As explained above, the round-trip speed of light is invariant for the inertial coordinates and for the Lorentzian coordinates, but not for the Galilean coordinates. The one-way speed of light is invariant only for the inertial coordinates, i.e., the coordinates in terms of which mechanical inertia is isotropic.

Return to MathPages Main Menu