Planckian Gravitons from an Imaginary-Time Clock

Abstract

We present a simple derivation of the exact Planck spectrum of the quadrupole radiation from point masses moving apart nonrelativistically, essentially an analog for gravitational radiation. The standard Einstein quadrupole radiation formula gives emitted power proportional to the square of the third derivative of $x(t)^2$. In our moving-mass picture, imaginary-time periodicity appears as a product-log trajectory of a quadrupole source. In the frequency domain, the power becomes proportional to the Planck distribution, $ω^3/(e^{2πcω/κ}-1)$. The resulting Planckian graviton energy spectrum has finite total energy and finite graviton number. The emitted spectrum is purely kinematic in origin: no equilibrium, horizon, or stochastic source is assumed.