Unveiling the Complexities of Physics: Accelerating Waves in the Context of Time and Relativity

A visual illustration of a wave traversing through a spacetime that bends exponentially. Attribution: Matias Koivurova, University of Eastern Finland

Scientists have formulated a groundbreaking wave equation that marries wave mechanics with general relativity and the concept of time’s arrow. This development offers resolutions to enduring debates in physics and paves the way for applications in innovative materials.

Researchers from Tampere University and the University of Eastern Finland have achieved a significant milestone by developing a new type of wave equation specifically designed for accelerating waves. This novel theoretical framework has proven to be an immensely productive area for exploring wave mechanics. It establishes direct links between accelerating waves, the general theory of relativity, and the unidirectional flow of time.

The Interplay Between Light and Matter

The phenomenon of light slowing down when it interacts with matter is not new. Conventional wave mechanics can adequately describe most of these occurrences in our daily lives.

For instance, when light strikes an interface, the conventional wave equation is valid on both sides. To analytically solve this type of problem, the wave form at each side of the interface is first determined, followed by the application of electromagnetic boundary conditions to connect the two sides. This approach is termed as a piecewise continuous solution.

Yet, it is important to note that light undergoes acceleration at the boundary, a factor that has not previously been considered.

Assistant Professor Matias Koivurova of the University of Eastern Finland elaborated, “I successfully derived the standard wave equation in 1+1 dimensions based solely on the assumption of constant wave speed. Then it occurred to me to question what would happen if the speed were variable. This proved to be a highly pertinent question.”

Upon positing that wave speed could change over time, the researchers formulated what they describe as an accelerating wave equation. Though the formulation was straightforward, solving the equation was far more complex.

Koivurova further explains, “The initial solutions appeared nonsensical, until it became clear that they exhibited characteristics similar to relativistic effects.”

In collaboration with the Theoretical Optics and Photonics group led by Associate Professor Marco Ornigotti at Tampere University, the team made significant headway. To produce logical solutions, a constant reference speed was required—the vacuum speed of light. According to Koivurova, clarity emerged upon this realization, leading to a comprehensive exploration of the far-reaching implications of this theoretical framework.

On the Impossibility of Time Machines

A groundbreaking finding of the research is that accelerating waves reveal a definite ‘arrow of time’—that is, time only flows in one direction and never backwards.

“Usually, thermodynamics provides the arrow of time through increasing entropy,” Koivurova states.

However, if time were to reverse, entropy would begin to decrease until reaching its minimum state, after which it would increase once again.

This delineates the difference between macroscopic and microscopic perspectives on the flow of time: while entropy determines the direction of time for large systems, the same cannot be said for individual particles.

“Yet, we expect individual particles to operate as if they possess a fixed arrow of time,” adds Koivurova.

Given that the accelerating wave equation can be derived from geometrical considerations, it is universally applicable, thereby implying that the unidirectional flow of time is a general property of the universe.

Resolving Controversies Through Relativity

Another salient feature of this theoretical framework is its ability to analytically model continuous waves, even across interfaces, thus having significant implications for the conservation of energy and momentum.

Associate Professor Marco Ornigotti discusses a contentious issue known as the Abraham–Minkowski debate, which centers on what happens to the momentum of light as it enters a medium. While there is experimental evidence supporting both increased and decreased momentum, Koivurova clarifies, “Our framework shows that the wave’s momentum remains unchanged, effectively being conserved.”

This conservation is attributed to relativistic effects. “We discovered that a ‘proper time’ can be assigned to the wave, analogous to the concept in the general theory of relativity,” Ornigotti elucidates.

Because the wave undergoes a different experience of time compared to laboratory time, accelerating waves are subject to time dilation and length contraction. Koivurova notes that it is precisely this length contraction that creates the illusion of non-conservation of momentum within a material medium.

Applications Beyond the Ordinary

While the new formulation aligns with the standard approach in most cases, it offers a crucial extension to time-varying materials. In such media, light undergoes abrupt and uniform shifts in material properties. Traditional wave equations fail to account for these dynamics.

The accelerating wave equation provides a means to analytically model scenarios that were previously only possible through numerical methods. Such scenarios include a theoretical material known as disordered photonic time crystal, in which a wave would experience an exponential slowdown while simultaneously gaining energy.

“The change in energy is due to the curved space-time experienced by the pulse. In these circumstances, the principle of energy conservation is locally breached,” states Ornigotti.

The research has profound implications, ranging from commonplace optical effects to experimental tests of general relativity, and it provides insights into why time appears to have a directional preference.

Reference: “Time-varying media, relativity, and the arrow of time” by Matias Koivurova, Charles W. Robson, and Marco Ornigotti, published on 19 October 2023 in Optica.
DOI: 10.1364/OPTICA.494630

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• Original Article: “Time-varying media, relativity, and the arrow of time”