Two massive spheres slowly drift apart in empty space. From the standpoint of ordinary mechanics, nothing special is happening. But Einstein's equations say otherwise: accelerating masses stir spacetime itself, and from them spread gravitational waves — ripples that stretch and squeeze everything in their path.
The authors took the simplest case: two identical point masses that do not attract each other but simply recede along a prescribed trajectory. They applied the standard Einstein formula for gravitational radiation power but rewrote the motion in so-called imaginary time. This mathematical trick, as if the clock were ticking at right angles to its usual course, turned the trajectory into a closed loop. And then the emission spectrum took on a familiar shape: the power at each frequency turned out to be proportional to ω³/(e²πcω/κ−1).
Remarkably, this spectrum requires neither thermal equilibrium, nor an event horizon, nor random jolts. Everything emerges from the pure geometry of motion. The total energy and the total number of gravitons in this radiation are finite — the calculations do not blow up to infinity, as often happens in quantum problems.
This result is a bridge between the world of large bodies and the quantum world. It hints that the Planck spectrum may arise far more often than previously thought, and not only in thermal processes. Two spheres, flying apart in the void, sing a gravitational song, and its notes exactly echo the thermal noise of distant stars.
Planck's formula for thermal radiation was the first in the history of physics to feature the constant ℎ. Now the same mathematical structure resurfaces in gravitational waves from simple mechanical motion.
In Carl Sagan's novel Contact, the alien signal contained a Planckian distribution, as a hint of its artificial origin. Now it turns out that such a spectrum can also be produced by natural gravitational radiation.