Late-time behavior of massive Dirac fields in a Schwarzschild background

The late-time tail behavior of massive Dirac fields is investigated in the Schwarzschild black-hole geometry and the result is compared with that of the massive scalar fields. It is shown that in the intermediate late times there are three kinds of differences between the massive Dirac and scalar fields: (i) The asymptotic behavior of massive Dirac fields is dominated by a decaying tail without any oscillation, but the massive scalar field by a oscillatory inverse power-law decaying tail; (ii) the dumping exponent for the massive Dirac field depends not only on the multiple number of the wave mode but also on the mass of the Dirac field, while that for the massive scalar field depends on the multiple number only; and (iii) the decay of the massive Dirac field is slower than that of the massive scalar field.

Figure 1

The figure describes $\ln |G^C(r_{*},r_{*}^{\prime },t)|$ versus $t$ for different $k$ with the same mass ($\mu =0.01$), which shows that the dumping exponent depends on the multiple number of the wave mode.

Figure 2

The figure describes $\ln |G^C(r_{*},r_{*}^{\prime },t)|$ versus $t$ for different mass $\mu$ with the same multiple number ($k=1$), which indicates that the dumping exponent depends on the mass $\mu$ of the Dirac fields.

Figure 3

The above four figures describe $\ln |G^C(r_{*},r_{*}^{\prime },t)|$ versus $t$ for $\mu =0.02$, and the below four for $\mu =0.01$. The lines (a) represent the result of the Green function of the massive Dirac field with different $k$. For comparing, we also show the corresponding result of the scalar field with lines (b). It is shown that the late-time behavior of massive Dirac fields is dominated by a decaying tail without any oscillation, and the decay of the massive Dirac field is slower than that of the massive scalar field.