# Light and the Luminiferous Ether

## An mathematical conception of light.

Prior to Einstein’s Special Theory of Relativity, the prevailing hypothesis in the scientific community was that light was the result of undulations in an all-permeating physical substance called the luminiferous ether. The most refined version of this hypothesis, known as the stationary ether hypothesis, was that proposed by H. A. Lorentz in the mid 1890’s. Lorentz claimed that the motion of matter did not affect the ether, so that it remained stationary even if material bodies moved through it. There are two main ramifications of this hypothesis. The first was dubbed by Einstein the principle of the constancy of the speed of light. According to Lorentz’s hypothesis, there is a unique frame of reference where light assumes a constant velocity c in all directions, namely the reference frame at rest with respect to the ether. In this frame the undulations of the ether (light) would spread with uniform speed in all directions, regardless of the velocity of the light source (because the ether is unaffected by the motion of the light-emitting body). The second ramification is that the stationary ether is incompatible with the principle of relativity, which asserts that the laws of physics are the same in all inertial (non-accelerating) reference frames. Surely a reference frame moving with respect to the ether would not observe the same laws of physics as a frame at rest with respect to the ether. An observer in the moving frame would measure a speed of light that differed from c by ± v, where v is the velocity of the moving frame with respect to the stationary ether.

Einstein contends against Lorentz’s hypothesis, and argues it is possible to reconcile the principle of the constancy of the speed of light with the principle of relativity, without the need for a luminiferous ether. I tend to agree with Einstein. It is entirely possible to forego the ether and retain both principles. By conceiving of light as an abstract mathematical byproduct of Maxwell’s equations and Einstein’s spacetime coordinates, we eliminate the need for a physical substance to account for the propagation of light. We can account for this on solely abstract terms, and, in doing so, reconcile the principle of relativity with the principle of the constancy of the speed of light.

First, let’s establish Einstein’s conception of reference frames and their relation with the speed of light. Einstein equates reference frames with 4-dimensional coordinate systems, comprised of 3 spatial dimensions and one temporal one. The spatial coordinates of a given point are determined in the conventional way: by counting how many unit rods (whose length is set arbitrarily) are needed to cover distance from the origin to the point’s orthogonal projection onto a given axis. This process determines the coordinates x, y and z. The temporal coordinate, t, has a more complicated formulation. This formulation is necessary, given the inexistence of absolute time. Without a universal clock against which to compare the passage of time, we must rely on something else to determine a consistent measure of duration. Einstein relies on the speed of light. His concept of time is as follows. Image two points in space A and B at a set distance d apart. He stipulates that light travels at the same speed from A to B and from B to A, a stipulation that follows intuitively from the symmetry of space and the principle of relativity.[1] Now consider two identically built clocks, one placed in A and the other placed in B. An interval of time, 𖣳∆t, is defined such that if a light ray emitted from A at time t (as measured by the clock in A), arrives in B at t + t (as measured by the clock in B), then a light ray emitted from B at time t (as measured by the clock in B), arrives in A at t + t (as measured by the clock in A) as well. In other words, time intervals in a coordinate system are defined such that the speed of light is the same in all directions. A more advanced, but perhaps more abstract, construction of the time coordinate would be one such that the spacetime interval of any point along a light worldline is always equal to 0.

The question remains open of how we can compare the coordinates of certain events between different coordinate systems, that is, between different reference frames. The Lorentz transforms help “translate” coordinates from one system to another. The important thing is that the spacetime interval is invariant under these transformations, which means the speed of light is the same in all directions in every inertial reference frame. That is, the Lorentz transformations and Einstein’s concept of coordinate systems guarantee that all inertial observers will measure the speed of light to be the same in every direction, regardless of their motion, so long as it is non-accelerating.

Nonetheless, something seems to be missing from Einstein’s construction. If there is no ether (and consequently no preferred frame), and the principle of the constancy of the speed of light holds, how is it that light propagates through space? We can answer this question by thinking abstractly about light. Light can be thought of as a wave in the electromagnetic (EM) field. Now, I am aware the reader might balk at this statement. It seems that the electromagnetic field is just another name for the ether, a pseudo-ether if you will. However, there is a fundamental difference between the two: the electromagnetic field is not a substance, it is not physical, it is an abstract mathematical construct.[2] It does the “bookkeeping”, as it were, for the laws of physics as they appertain to charged particles. Allow me to elaborate. The EM field is a vector field where each point in spacetime (whose coordinates are as defined by Einstein) has two associated vectors, one for the electric field and the other for the magnetic field. These vectors determine the force a unit charge particle would experience at that position while travelling at a specific velocity, given the configuration of all charged particles at that point in time. In a sense, the vectors simply keep track of the effects that a charged particle would experience at a given point in spacetime, but they aren’t anything physical per se. The EM field can be thought of as a set of “rules” that charged particles obey. At a fixed point in time the EM field describes what charged particles should do at that particular moment, in any given point in space.

Having this notion of a field, we can think more deeply about what light really is. Under a dynamic system of charged particles, Maxwell’s equations produce peculiar solutions. These solutions are oscillatory,[3] and propagate through the vector field at a constant velocity that is, not coincidentally, equal to the speed of light. What if light is these solutions? We can simply think of light as a field pattern (or a pattern of vectors if you prefer) that traverses the EM field. A wave-like change in the “rules” of behaviour of charged particles that propagates at a uniform speed c across space. Light, then, is a purely abstract concept. It is the mathematical, oscillatory reaction of the EM field to accelerating charges; the name we give to a changing set of “rules” for charged particles. In this manner, we can dispose of any ethereal substance upon which to ground the concept of light in physical terms, and simply conceive of it as an abstractly. When our eyes look up to a bright star, they aren’t perceiving a physical substance, but are rather recording a movement of charged particles in the retina according to an oscillating set of rules, whose oscillation came about by the movement of other charged particles in the distant star.[4]

But one puzzling question remains: if light is but an abstract wave-like change in a mathematical vector field, what does it mean when we talk about its velocity, and how do we guarantee its constancy? Well the velocity of light would correspond to a measure of the rate of change of the electromagnetic field vectors. The velocity of light keeps track of how quickly the aforementioned “rules” change, and specifically how quickly this change propagates through space. As it turns out, Maxwell’s equations are the same in all inertial reference frames (by the principle of relativity), which implies that the solutions to those equations propagate at c in all inertial frames. Moreover, the Lorentz transforms map these solutions unto other coordinate systems, without altering their rate of propagation. This fact is guaranteed by mathematical construction. Einstein’s spacetime is designed in such a way that the Lorentz transforms can be applied from any inertial frame to any other inertial frame, without affecting the speed of light. It is, thus, an analytic certainty that the principle of the constancy of the speed of light is satisfied along with the principle of relativity.

Einstein’s conciliation of the principle of relativity with the principle of the constancy of the speed of light, is a mathematical one. Thus, it is by considering light in its true abstract essence that we can do away with a physical ether.

#### Footnotes

[1] Space is the same in all directions. Therefore, there is no reason why any one direction should be phenomenologically different from the others. If there were one such direction the principle of relativity would be broken.

[2] This points merits some emphasis. Maxwell himself is known to have grappled with the difference between the ether and the electromagnetic field. He said: “But if the luminiferous aether and the electro-magnetic media occupy the same place, and transmit disturbances with the same velocity, what reason have we to distinguish the one from the other?” Maxwell’s fault was thinking of light as a physical disturbance in some physical medium, rather than an abstract ripple in a mathematical medium. The core difference between the ether and the EM field is that the former is physical and the latter abstract.

[3] I mean oscillatory in a mathematical sense: A travelling pattern in the vectors of the EM field that is decomposable into sines and cosines by the Fourier Transform.

[4] More formally, the charged particles in the retina are accelerating according to an oscillating vector field whose oscillation is the solution to Maxwell’s equations in the context of other accelerating charged particles in the distant star.

#### Bibliography

Einstein, A. (1911) Doc. 17: The Theory of Relativity. In The Collected Papers of Albert Einstein (Vol. 3). Princeton, New Jersey: Princeton University Press.

Maxwell, J. C., & Niven, W. D. (1965). The scientific papers of James Clerk Maxwell. New York: Dover Publications.