Physics Unraveled: Accelerating Waves and the Mysteries of Time and Relativity

An artistic depiction of a wave encountering an exponentially curved spacetime. Credit: Matias Koivurova, University of Eastern Finland

Researchers have derived a new wave equation, linking wave mechanics with the general theory of relativity and the arrow of time, offering solutions to long-standing physics debates and introducing applications for novel materials.

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 to accelerating waves. The novel formalism has turned out to be an unexpectedly fertile ground for examining wave mechanics, with direct connections between accelerating waves, the general theory of relativity, as well as the arrow of time.

Light Interaction With Matter

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.

For example, when light is incident on an interface, the standard wave equation is satisfied on both sides. To analytically solve such a problem, one would first find what the wave looks like at either side of the interface, and then employ electromagnetic boundary conditions to link the two sides together. This is called a piecewise continuous solution.

However, at the boundary, the incident light must experience an acceleration. So far, this has not been accounted for.

“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.

By assuming that the speed of a wave can vary with time, the researchers were able to write down what they call an accelerating wave equation. While writing down the equation was simple, solving it was another matter.

“The solution didn’t seem to make any sense. Then it dawned on me that it behaves in ways that are reminiscent of relativistic effects,” Koivurova recounts.

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 realizing that. What followed was an investigation of the surprisingly far-reaching consequences of formalism.

No Hope for a Time Machine?

In a breakthrough result, the researchers showed that in terms of accelerating waves, there is a well-defined direction of time; a so-called ‘arrow of time.’ This is because the accelerating wave equation only allows solutions where time flows forward, but never backward.

“Usually, the direction of time comes from thermodynamics, where an increasing entropy shows which way time is moving,” Koivurova says.

However, if the flow of time were to reverse, then entropy would start to decrease until the system reached its lowest entropy state. Then entropy would be free to increase again.

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.

“Yet, we expect single particles to behave as if they have a fixed direction of time!” Koivurova says.

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.

Relativity Triumphs Over the Controversy

Another property of the framework is that it can be used to analytically model waves that are continuous everywhere, even across interfaces. This in turn has some important implications for the conservation of energy and momentum.

“There is this very famous debate in physics, which is called the Abraham–Minkowski controversy. The controversy is that when light enters a medium, what happens to its momentum? Minkowski said that the momentum increases, while Abraham insisted that it decreases,” Ornigotti explains.

Notably, there is experimental evidence supporting both sides.

“What we have shown, is that from the point of view of the wave, nothing happens to its momentum. In other words, the momentum of the wave is conserved,” Koivurova continues.

What allows the conservation of momentum are relativistic effects. “We found that we can ascribe a ‘proper time’ to the wave, which is entirely analogous to the proper time in the general theory of relativity,” Ornigotti says.

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 the length contraction that makes it seem like the momentum of the wave is not conserved inside a material medium.

Exotic Applications

The new approach is equivalent to the standard formulation in most problems, but it has an important extension: time-varying materials. Inside time-varying media light will experience sudden and uniform changes in the material properties. The waves inside such materials are not solutions to the standard wave equation.

This is where the accelerating wave equation comes into the picture. It allows the researchers to analytically model situations that were only numerically accessible before.

Such situations include an exotic hypothetical material called disordered photonic time crystal. Recent theoretical investigations have shown that a wave propagating inside the said material will slow down exponentially, while also increasing exponentially in energy.

“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.

The research has wide-reaching implications, from everyday optical effects to laboratory tests of the general theory of relativity, while giving an idea of why time has a preferred direction.

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

9 Commentson "Physics Unraveled: Accelerating Waves and the Mysteries of Time and Relativity"

1. Totally agree that time only moves in forward direction, but more so I think of time as expanding than moving growing in size of a comparative balloon shape.

• Torbjörn Larsson | October 27, 2023 at 7:31 pm | Reply

You seem to confuse time, something that can be measured by clocks, with the cosmological arrow of time – space is expanding (yesterday, now and) in the future.

2. Christopher Ducey | October 26, 2023 at 10:24 am | Reply

Could this phenomenon have any bearing on solving the conundrum cosmologists have measuring the expansion of the universe (“Hubble tension” problem)?

3. Torbjörn Larsson | October 27, 2023 at 7:29 pm | Reply

If time travel (TT) was possible, all problems would be equally easy (but they are not). A TT computer would always answer immediately.

As the article itself notes, there is no local arrow of time, all processer are reversible. That all radiative phenomena, including electromagnetism has a global arrow of time like in the article description, is seen in that waves radiate outwards. Like for entropy, it is much less likely that a convergent wave will happen since they are harder to make.

4. Fixed gravity for you. | October 28, 2023 at 3:24 am | Reply

Considering special relativity as being essentially about conserving light energy across different inertial reference frames (different velocities) in uniform gravity, a reason for supposing “c” is constant across regions with different values for small “g” (different gravitational accelerations) is critically lacking as velocities are obviously not accelerations. Not to put too fine a point on it, but gravitational energy shifts are all about shifts in small “g,” not shifts in time rates. Industrial time-rate gatekeepers will nonetheless insist on denying that a reference light source appearing frequency-shifted by gravity is a gravitational intensity indicator instead of an indicator of a time rate change, no doubt supposing they’re keeping their conception of humanity safer for it.

5. Fixed gravity for you. | October 28, 2023 at 4:02 am | Reply

When one adds gravity variations to special relativity problems, one ends up with foolish old paradoxes to be explained away unendingly to preserve foolish old egos essential to presentations of a valid unified “general relativity.” There’s always a preferred reference frame unless gravity is avoidable, but gravity is not really avoidable.

Anyway, “equal and opposite actions” in the news may only be defied by quantizing gravity properly, preserving causality despite any non-locality that may be involved. Light-speed and causality are de-coupled unless you have a thing for Wheeler-Feynman “absorber theory” gatekeeping nonsense. The concept of bent spacetime is of course not adequate for describing a universe with non-local effects.

I’d like to add something about motion assistance by sinuous rotation in a thixotropic medium, but that is probably the domain of condensed matter gatekeeping.

• Fixed gravity for you. | October 28, 2023 at 4:34 am | Reply

Quantizing gravity properly means gravitational information flows occur at the quantum level over any distance by exchange of radiated gravitational energy quanta, frankly. “Equal and opposite” gravitational interactions can be both local and non-local if matter can respond to gravity carrier flows retro-reflectively. This (focused, retroreflective) sort of response can also be classified as naturally “Hebbian.” Temperature is apparently critical in the case of non-local retro-reflections, regardless of context, which adds a facility for gravitational annealing-type “re-focusing” processes in cooled matter, to go along with Hebbian tendencies. More generally, Hebbian entanglement-building guided by cooled matter gravitational flows may define preferred conduits for electromagnetic energy.

• Fixed gravity for you. | October 28, 2023 at 5:05 am | Reply

As a consequence, cosmologically annealed retro-reflection assisted by cold hydrogen atoms (ionized, bound or unbound) aligning their preferred gravitational radiations, in terms of position, spin axis and spin direction, regardless of distance apart, apparently should play a major role in generating any dark matter filament effect. Just my opinion. A hotter universe may lack finer DM filament effects.

• Fixed gravity for you. | October 28, 2023 at 6:43 am | Reply

What I’ve left unmentioned is that stronger gravity implies faster light, not different meters or slower time or reverse causality, or instantaneous action at a distance. Entangled faster information can give an evolutionary edge to some trophic objects and may encourage “herding”-like behaviors.