Synchrotron radiation in strongly coupled conformal field theories

Using gauge/gravity duality, we compute the energy density and angular distribution of the power radiated by a quark undergoing circular motion in strongly coupled $\mathcal{N}=4$ supersymmetric Yang-Mills theory. We compare the strong coupling results to those at weak coupling, finding them to be very similar. In both regimes, the angular distribution of the radiated power is in fact similar to that of synchrotron radiation produced by an electron in circular motion in classical electrodynamics: the quark emits radiation in a narrow beam along its velocity vector with a characteristic opening angle $\alpha \sim 1/\gamma$. To an observer far away from the quark, the emitted radiation appears as a short periodic burst, just like the light from a lighthouse does to a ship at sea. Our strong coupling results are valid for any strongly coupled conformal field theory with a dual classical gravity description.

Figure 1

A cartoon of the gravitational description of synchrotron radiation at strong coupling. The arena where the gravitational dynamics takes place is the five-dimensional geometry of ${\mathrm{AdS}}_5$. A string resides in the geometry and an endpoint of the string is attached to the four-dimensional boundary of ${\mathrm{AdS}}_5$, which is where the dual quantum field theory lives. The trajectory of the endpoint of the string corresponds to the trajectory of the dual quark. Demanding that the endpoint rotates results in the string rotating and coiling around on itself as it extends in the ${\mathrm{AdS}}_5$ radial direction. The presence of the string in turn perturbs the geometry and the near-boundary perturbation in the geometry induces a four-dimensional stress tensor on the boundary. The induced stress has the interpretation as the expectation value of the stress tensor in the dual quantum field theory.

Figure 2

A sketch of the solutions (2.8). As the quark moves along its circular trajectory, it emits radiation in a narrow cone of angular width $\alpha \sim 1/\gamma$ in the direction of its velocity vector. The diagram is a snapshot at the time when the quark is at the top of the circle. The thick solid curve shows where the radiation emitted at earlier times is located at the time of the snapshot. The width $\Delta$ of the spiral scales like $\Delta \sim 1/{\gamma }^3$, as explained in Fig. 3.

Figure 3

A close-up illustration of the emission of radiation at 2 times $t_1$ and $t_2$. The radiation is emitted in the direction of the quark’s velocity vector, within a cone of angular width $\alpha$. An observer at the point $p$ is illuminated by a pulse of radiation of duration $\Delta t\sim R_0\alpha$ and of spatial thickness $\Delta$. The leading edge of the pulse observed at $p$ is emitted at $t_1$ and the trailing edge observed at $p$ is emitted at $t_2$. At time $t_2$ the radiation emitted at $t_1$, denoted by the solid curve directly above the point $t_2$, has traveled a distance $R_0\alpha /v$ toward $p$. The chordal distance between the two emission points is $R_0\alpha$ in the $\alpha \to 0$ limit. The width $\Delta$ is therefore $\Delta =R_0\alpha (1/v-1)$.

Figure 4

Left: a cutaway plot of $r^2\mathcal{E}/P$ for $v=1/2$. Right: a cutaway plot of $r^2\mathcal{E}/P$ for $v=3/4$. In both plots the quark is at $x=R_0$, $y=0$ at the time shown and its trajectory lies in the plane $z=0$. The cutaways coincide with the planes $z=0$, $\phi =0$ and $\phi =7\pi /5$. At both velocities the energy radiated by the quark is concentrated along a spiral structure which propagates radially outwards at the speed of light. The spiral is localized about $\theta =\pi /2$ with a characteristic width $\delta \theta \sim 1/\gamma$. As $v\to 1$ the radial thickness $\Delta$ of the spirals rapidly decreases like $\Delta \sim 1/{\gamma }^3$.

Figure 5

Plot of $r^2\mathcal{E}/P$ at $\theta =\pi /2$ and $\phi =5\pi /4$ at $t=0$ as a function of $r$ for $v=1/2$. The plot illustrates the fact that the pulses of radiated energy do not broaden as they propagate outward. This implies that they do not broaden in azimuthal angle, either. Strongly coupled synchrotron radiation does not isotropize.

Figure 6

The energy density at $\theta =\pi /2$ and $\phi =5\pi /4$ for quark velocity $v=1/2$, as in Fig. 5. $r\approx 33R_0$ corresponds to the location of a spiral in the energy density. Directly ahead of and directly behind the spiral, the energy density is slightly negative. To see how slightly, compare the vertical scale here with that in Fig. 5.

Figure 7

$r^2\mathcal{E}/P$ for $v=1/2$ in the $z=0$ plane. The color code is the same as in Fig. 4, with zero energy density blue and maximal energy density red. The spiral curve marked with dots is (2.12), namely, the place where the spiral of synchrotron radiation would be in electrodynamics or in weakly coupled $\mathcal{N}=4$ SYM theory. We see that the spiral of radiation in the strongly coupled gauged theory with a gravitational dual is at the same location. This indicates that, as at weak coupling, strongly coupled synchrotron radiation is beamed in the direction of the motion of the quark. For reference, the solid curve is (2.14), namely, the place where the synchrotron radiation would be if the quark were emitting a beam of radiation perpendicular to its direction of motion.

Figure 8

The normalized time-averaged angular distribution of power at both weak and strong coupling at $v=0.9$. At both weak and strong coupling, the time-averaged angular distribution of power is localized about $\theta =\pi /2$ with a characteristic width $\delta \theta \sim 1/\gamma$. The (slight) difference in the shapes of the two angular distributions is as if the ratio of power radiated in vectors and scalars is $2:1$ in the strongly coupled radiation, while it is $4:1$ in the weakly coupled case.