Electromagnetic and gravitational radiation in all dimensions: A classical field theory treatment

How long does a light bulb shine in odd dimensional flat spacetimes, according to a distant observer? This question is nontrivial because electromagnetic and gravitational waves, despite being comprised of massless particles, can develop tails; they travel inside the light cone. To this end, I attempt to close a gap in the literature by first deriving, strictly within classical field theory, the real-time electromagnetic dipole and gravitational quadrupole energy and angular momentum radiation formulas in all relevant dimensions. The even-dimensional case, where massless signals travel strictly on the null cone, depends on the time derivatives of the dipoles and quadrupoles solely at retarded time, whereas the odd-dimensional ones involve an integral over their retarded histories. Despite the propagation of light inside the null cone, however, I argue that a monochromatic light bulb of some intrinsic duration in odd dimensions remains approximately the same apparent duration to a distant detector, though the tail effect does produce a phase shift and adds to the signal several transitory nonoscillatory inverse square roots in time. Analogous remarks apply to a distant gravitational wave detector hearing from a finite duration quasiperiodic quadrupole source.

Figure 1

Causal structure of signals in odd versus even dimensional Minkowski spacetime. The Dirac delta function of Eq. (1), when employed in Eq. (6), tells us the signal received by the observer can only come from her past light cone. This is the dashed segment, representing the intersection between the observer’s backward light cone and the world tube of the source responsible for the waves. In contrast, the step function of Eq. (3) when employed in Eq. (6) says the signal detected by the observer comes from within her null cone (gray region). This means the signal she receives at a given moment is the superposition of the waves emitted from the source’s world tube lying within this gray region.

Figure 2

Top panel: The time-dependent part ($\mathcal{E}[{\omega }_{\ell }(t-r)]$) of the leading order electromagnetic radiation field, for a “light bulb” of duration ${\omega }_{\ell }T=100$, as a function of retarded time ${\omega }_{\ell }(t-r)$. The bold line is the exact time dependence in Eqs. (110) and (111); whereas the dashed line is their asymptotic expansion in Eqs. (114) and (115). We see that the gradual downward shift of the sinusoidal waveform is due to the $1/\sqrt{{\omega }_{\ell }(t-r)}$ in Eq. (114); whereas according to Eq. (115), after the active duration the signal decays to zero rather quickly and without any oscillations. Note that the energy flux $\mathrm{d}E/(\mathrm{d}t\mathrm{d}\mathrm{\Omega })$ of the “light bulb” in Eq. (103) is proportional to the square of the bold line of this plot [cf. Eq. (109)]. Bottom left panel: Spacetime configuration of the observer receiving signals from the source when her retarded time $t-r$ lies within its active duration (dashed segment). This corresponds to the $0\le {\omega }_{\ell }(t-r)\le 100$ region of the plot in the top panel. Bottom right panel: Spacetime configuration of the observer receiving signals from the source after her retarded time $t-r$ has passed its active duration (dashed segment). This corresponds to the ${\omega }_{\ell }(t-r)>100$ region of the plot in the top panel.