Researchers at Tampere University and the University of Eastern Finland have reached a milestone in a study where they derived a new kind of wave equation, which applies for accelerating waves.
The novel formalism has turned out to be an unexpectedly fertile ground for examining wave mechanics, with direct connections between accelerating waves, general theory of relativity, as well as the arrow of time.Whenever light interacts with matter, light appears to slow down. This is not a new observation and standard wave mechanics can describe most of these daily phenomena.
"Basically, I found a very neat way to derive the standard wave equation in 1+1 dimensions. The only assumption I needed was that the speed of the wave is constant. Then I thought to myself: what if it's not always constant? This turned out to be a really good question," says Assistant Professor Matias Koivurova from the University of Eastern Finland.
Working together with the Theoretical Optics and Photonics group, led by Associate Professor Marco Ornigotti from Tampere University, the researchers finally made progress. To obtain solutions that behave as expected, they needed a constant reference speed -- the vacuum speed of light. According to Koivurova, everything started to make sense after realising that. What followed was investigation of the surprisingly far-reaching consequences of the formalism.
This is the difference between 'macroscopic' and 'microscopic' arrows of time: while entropy defines the direction of time for large systems unambiguously, nothing fixes the direction of time for single particles.Since the accelerating wave equation can be derived from geometrical considerations, it is general, accounting for all wave behavior in the world. This in turn means that the fixed direction of time is also a rather general property of nature.
Since the wave experiences a time that is different from the laboratory time, the researchers found that accelerating waves also experience time dilation and length contraction. Koivurova notes that it is precisely length contraction that makes it seem like the momentum of the wave is not conserved inside a material medium.The new approach is equivalent to the standard formulation in most problems, but it has an important extension: time-varying materials.
"Our formalism shows that the observed change in the energy of the pulse is due to a curved space-time the pulse experiences. In such cases, energy conservation is locally violated," Ornigotti says.
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