Semiclassical bremsstrahlung from a charge radially falling into a Schwarzschild black hole

A semiclassical investigation of the electromagnetic radiation emitted by a charged particle in a radially freely falling motion in Schwarzschild spacetime is carried out. We use quantum field theory at tree level to obtain the one-particle-emission amplitudes. We analyze and compare the energy spectrum and total energy released, which are calculated from these amplitudes, for particles with varying initial positions and for particles originating from infinity with varying kinetic energy. We also compare the results with those due to a falling charged “string” extended in the radial direction.

Figure 1

The zero-frequency limit of the partial energy spectrum, ${\mathcal{E}}^{\mathrm{up};0\ell }$, as a function of $r_0$ for some choices of $\ell$.

Figure 2

The partial and total energy spectra, given by Eqs. (42) and (44), respectively, with $r_0=3M$ and $v_0=0$, as a function of $M\omega$. We consider the first 60 multipoles.

Figure 3

The partial energy spectrum ${\mathcal{E}}^{(\ell )}\equiv 10^{\frac{3\ell }{|\ell -2|}}(M/q{)}^2{\mathcal{E}}^{\mathrm{in};\omega \ell }$ depicted as a function of $r_0/M$ and $M\omega$, with $v_0=0$, for $\ell =1$ (left) and $\ell =5$ (right). The (gray) facegrids mark the positions of $M{\omega }_{\ell }^{\mathrm{qnf}}$. The (purple) disks mark the points of the global maximum of each function in the given range, at $r_0/M\approx 5.4504$, 2.7598 and $M\omega \approx 0.1924$, 1.0619, respectively.

Figure 4

The partial and total energy spectra, given by Eqs. (42) and (44), respectively, with $r_0=\infty$ and $v_0=0$, as a function of $M\omega$. We consider the first 9 multipoles.

Figure 5

The log-linear-scaled plot of the percentage of initial energy of the particle radiated to infinity, depicted as a function of $r_0/M$, with $v_0=0$. The vertical (gray) line marks the BH horizon, and the horizontal (gray) line marks the value of the quantity in the limit $r_0\to \infty$.

Figure 6

The partial and total energy spectra, given by Eqs. (42) and (44), respectively, with $r_0=\infty$ and $v_0=0.99$, as a function of $M\omega$.

Figure 7

The total energy spectrum, given by Eq. (44), for $r_0\to \infty$, with some choices of $v_0$, as indicated.

Figure 8

The partial energy spectrum ${\mathcal{E}}^{\mathrm{up};\omega \ell }$ is shown as a function of $M\omega$ (left) and $M\omega /\ell$ (right), with $r_0=\infty$ and $v_0=0$.

Figure 9

The partial energy spectrum ${\mathcal{E}}^{\mathrm{up};\omega \ell }$ for $\ell =1$ (top) and $\ell =10$ (bottom), with finite $r_0$ and $v_0=0$.

Figure 10

The partial energy spectrum ${\mathcal{E}}^{\mathrm{up};\omega \ell }$ with $\ell =1$ and $\ell =3$, as a function of $M\omega$, for $r_0=\infty$ and some choices of $v_0>0$.

Figure 11

Top: the radial positions of the inner and outer ends of the string as a function of the time $t$. The difference $r(t-\mathrm{\Delta }t)-r(t)$ is also plotted. Bottom: the process of the string falling radially into the BH is represented for four different values of the parameter $t$, with $r_0=15M$ and $v_0=0$.

Figure 12

The function ${\zeta }_N(\omega )$, given in Eq. (60), for some choices of $N$ and $\mathrm{\Delta }t=2M$.

Figure 13

The relative multipole contribution to the absorbed energy for a charged string released from $r_0=\infty$, as the percentage in the sum up to $\ell =27$, denoted here by ${\mathcal{E}}^{\mathrm{up}}$, as a function of the multipole number $\ell$. The vertical gridlines indicate the values of $\ell$.

Figure 14

The absorbed energy for a charged string released from $r_0=\infty$, as a function of $\mathrm{\Delta }t$. The $\ell$-sum in Eq. (45) is truncated at ${\ell }_{\max }$. Different choices of ${\ell }_{\max }$ are considered.

Figure 15

The total energy spectrum for a charged string projected toward the BH from $r_0=3M$ with $v_0=0$ and some choices of $\mathrm{\Delta }t$, as indicated.