Passage of radiation through wormholes of arbitrary shape

We study quasinormal modes and scattering properties via calculation of the $S$ matrix for scalar and electromagnetic fields propagating in the background of spherically symmetric and axially symmetric traversable Lorentzian wormholes of a generic shape. Such wormholes are described by the general Morris-Thorne ansatz. The properties of quasinormal ringing and scattering are shown to be determined by the behavior of the wormhole’s shape function $b(r)$ and shift factor $\Phi (r)$ near the throat. In particular, wormholes with the shape function $b(r)$, such that $b^{\prime }(r)\approx 1$, have very long-lived quasinormal modes in the spectrum. We have proved that the axially symmetric traversable Lorentzian wormholes, unlike black holes and other compact rotating objects, do not allow for superradiance. As a by-product we have shown that the 6th order WKB formula used for scattering problems of black or wormholes gives quite high accuracy and thus can be used for quite accurate calculations of the Hawking radiation processes around various black holes.

Figure 1

Wormhole shapes $z(r)$; $b(r)=b_0^{1-q}{r}^q$, $\Phi =0$.

Figure 2

The effective potential for the scalar field $\ell =0$ as a function of the tortoise coordinate; $b(r)=b_0$, $\Phi =0$.

Figure 3

Real (right panel) and imaginary (left panel) parts of quasinormal modes for scalar field for $\ell =0$ (red, bottom), $\ell =1$ (blue), and $\ell =2$ (green, top); $b(r)=b_0^{1-q}{r}^q$, $\Phi =0$.

Figure 4

Real (right panel) and imaginary (left panel) parts of quasinormal modes for the Maxwell field for $\ell =1$ (red, bottom), $\ell =2$ (blue), and $\ell =3$ (green, top); $b(r)=b_0^{1-q}{r}^q$, $\Phi =0$.

Figure 5

Real (right panel) and imaginary (left panel) parts of quasinormal modes for the Maxwell field for $\ell =1$; $b(r)=b_0^{1-q}{r}^q$, $\Phi =1/r^p$.

Figure 6

Real (right panel) and imaginary (left panel) parts of quasinormal modes for scalar field for $\ell =1$; $b(r)=b_0^{1-q}{r}^q$, $\Phi =1/r^p$.

Figure 7

Transmission coefficients for scalar (left panels) and electromagnetic (right panels) fields $\ell =1$ (top panels) and $\ell =2$ (bottom panels), $\Phi =0$, $b(r)=b_0^{1-q}{r}^q$, $q=-1$ (violet), $q=-1/2$ (cyan), $q=0$ (blue), $q=1/2$ (green), $q=3/4$ (yellow), $q=7/8$ (red), and $q=15/16$ (magenta). As $q\to 1$ the transmission coefficient approaches the theta function $\theta ({\omega }^2{b}_0^2-\ell (\ell +1))$.

Figure 8

Transmission coefficient for scalar (left panels) and electromagnetic (right panels) fields $\ell =1$ (top panels) and $\ell =2$ (bottom panels), $\Phi =\frac{b_0}{r}$, $b(r)=b_0^{1-q}{r}^q$, $q=-1$ (violet), $q=-1/2$ (cyan), $q=0$ (blue), $q=1/2$ (green), $q=3/4$ (yellow), $q=7/8$ (red), $q=15/16$ (magenta), and $q=31/32$ (orange). As $q\to 1$ the transmission coefficient approaches the theta function $\theta ({\omega }^2{b}_0^2{e}^{-2\Phi (b_0)}-\ell (\ell +1))$.

Figure 9

Transmission coefficient for scalar (left panel) and electromagnetic (right panel) fields $\ell =1$, $b(r)=b_0$, $\Phi =(\frac{b_0}{r}{)}^p$: $p=-1/2$ (magenta), $p=-1/4$ (red), $p=0$ (green), $p=1/4$ (blue), and $p=1/2$ (cyan). The more sloping line corresponds to the lower value of $p$.

Figure 10

On the left figure we show the gray body factors for the scalar field $\ell =1$ in the Schwarzschild background found with the help of the WKB formula (red line) and the accurate values found by the shooting method (blue line). The right figure shows the difference between the WKB formula and the accurate results.

Figure 11

Energy emission rate of the Schwarzschild black hole for the scalar field found with the help of the WKB formula (red, bottom) and the accurate values obtained by the shooting method (blue, top).