When light interacts with matter, it appears to slow down—a phenomenon that has been long observed. In standard wave mechanics, this can be explained by employing electromagnetic boundary conditions to link the two sides of an interface. However, this approach fails to account for the acceleration experienced by incident light at the boundary. Assistant Professor Matias Koivurova from the University of Eastern Finland sought to address this issue by exploring the possibility of a time-varying wave speed.
By assuming that the speed of a wave can vary with time, Koivurova and his colleagues developed what they call an accelerating wave equation. While the equation itself was straightforward to write down, solving it proved to be a challenge. Nevertheless, they persisted and eventually discovered that the behavior of the solution resembled relativistic effects.
Working alongside the Theoretical Optics and Photonics group led by Associate Professor Marco Ornigotti from Tampere University, the researchers began to unravel the far-reaching consequences of their findings. The study, titled “Time-varying media, relativity, and the arrow of time,” reveals a well-defined direction of time associated with accelerating waves.
Traditionally, the direction of time is explained by thermodynamics and the concept of increasing entropy. However, Koivurova proposes that the accelerating wave equation also puts forth a fixed direction of time. The solutions derived from this equation only allow time to flow forward, never backward. This distinction between the “macroscopic” and “microscopic” arrows of time arises from the fact that entropy determines the direction of time for large systems, while individual particles lack a fixed direction.
What makes this discovery particularly significant is that the accelerating wave equation accounts for all wave behavior, making the fixed direction of time a general property of nature. Furthermore, this framework enables analytical modeling of continuous waves across interfaces, presenting implications for the conservation of energy and momentum.
A long-standing debate in physics revolves around the conservation of momentum for light entering a medium. While there is experimental evidence supporting both sides of the argument, the researchers’ work suggests that, from the perspective of the wave itself, nothing happens to its momentum upon entering a medium.
The conservation of momentum is possible due to the relativistic effects present in the accelerating wave equation. The researchers found that they could attribute a “proper time” to the wave, similar to the proper time in the general theory of relativity. This alternative formulation reconciles the diverging views of Abraham and Minkowski regarding momentum changes in light.
Koivurova emphasizes that length contraction plays a crucial role in the conservation of momentum. Inside a material medium, the contraction of the wave’s length gives the illusion of momentum not being conserved. However, this new approach demonstrates that momentum is indeed conserved.
The accelerating wave equation not only provides insights into the conservation of momentum but also enables analytical modeling of waves within time-varying media. In such materials, light encounters sudden and uniform changes in their properties that cannot be described by the standard wave equation.
Using the accelerating wave equation, the researchers can now investigate situations that were previously only accessible through numerical simulations. One example is the study of disordered photonic time crystals, a hypothetical material in which waves exponentially slow down and increase in energy. The formalism developed by Koivurova and Ornigotti shows that this energy change is a consequence of the curved space-time experienced by the pulse. Energy conservation is locally violated in such cases.
The implications of this research reach far and wide, from the understanding of everyday optical effects to laboratory tests of the general theory of relativity. By unraveling the connection between accelerating waves, relativity, and the arrow of time, the researchers have shed new light on the fundamental nature of time itself.
While this study presents groundbreaking insights, further exploration is warranted to fully grasp the intricacies of accelerating waves and their implications. The concept of a fixed direction of time for individual particles, as proposed by Koivurova, presents exciting avenues for future research. By exploring how single particles behave within the framework of the accelerating wave equation, scientists may uncover new dimensions of the arrow of time and its influence on the behavior of matter and light.