˙
x3 = 0, (3.32c)
[ ˙x0(0)]2 =n2{[ ˙x2(0)]2+ [ ˙x1(0)]2}=n2[ ˙x2(0)]2 1
sin2θ0. (3.32d) Inserting the obtained relations into equation (3.31a) gives
x˙0 = V(1−n2) ˙x1+nx˙2(0)(1−V2)sin1θ
0
1−n2V2 = V(1−n2) + (1−V2)nsintanθθ
0
1−n2V2 x˙1. (3.33) By using equation (3.24), one gets the final result 3
tan2θ = γ2sin2θ0
1−γ4(1−n2V2) sin2θ0. (3.34) The ray oriented parallel to the flow keeps its direction. Ifn >1, the photons tend to move in the same direction as the expansion. In this case, it must hold γ4(1−n2V2) < 1. As V increases, i.e., V → 1, γ → ∞, then θ → 0. In the opposite case, i.e., when n < 1, the photons tend to turn away from the flow direction. Assuming n is constant, the square of expansion speed V2 is limited by the critical value when tan2θ→ ∞, i.e., γ4(1−n2V2) sin2θ0 →1.
where µ = cosα. Wave frequency ωext is measured by the external observer.
According to Lerche [1974a], variables ˙x1, ˙x2 actually represent the components of local group velocity⃗vg measured by the external observer defined as
⃗vg = ˆx1 ∂ω
∂kx1 + ˆx2 ∂ω
∂kx2, (3.36)
where ˆx1 and ˆx2 denote the unit vectors in the corresponding directions and kx1, kx2 are individual wave vector components. Due to the validity of (1.12), variables
˙
x1, ˙x2 correspond to those defined by Anderson and Spiegel [1975]. Thus, the slope (3.26) is
tanθ= sinα(u−µ∂u∂µ)
(uµ+ (1−µ2)∂u∂µ). (3.37) Let us assume there is an observer situated in a medium rest frame at height y who measures the wave with frequency ωA, wave vector kA, phase speed u0 at angle α0 and thus, it holds µ0 = cosα0. In order to express wave frequency ωext
and wave vector kext measured by the external observer in terms of ωA, u0, and α0, the velocity addition formulae give
ωext=γωA
(
1 + µ0V u0
)
, (3.38)
kext =γωA µ
(µ0 u0 +V
)
. (3.39)
Medium velocityV =V(y) is measured by the external inertial observer. Further, by using the Snell’s law one gets
kextsinα=kAsinα0. (3.40) Wave vector kext can be expressed by using (3.39) and (3.40) as follows
kext=
[
γ2ωA2
(µ0 u0 +V
)2
+k2A(1−µ20)2
]12
(3.41) and phase speedu measured by the external observer is
u= ωext kext
= u0+V µ0
[(µ0+u0V)2+ (1−µ20)γ−2]12
, (3.42)
where formula for medium rest frame phase speed u0 =ωA/kA was used. When considering expressions (3.38) and (3.39), the relationu=ωext/kext leads to
µ
u = µ0+u0V
u0+µ0V . (3.43)
Regarding the fact that u0 = 1/n at heighty (asc= 1), formulae (3.42) and (3.43) allow to omit variables measured by the observer in a medium rest frame and the relation
u
(
1− V2 n2
)
=µV
(
1− 1 n2
)
+ (nγ)−1
[
1− V2
n2(1−µ2)−V2µ2
]1
2
(3.44)
is obtained. Differentiation of (3.44) leads to
∂u
∂µ = V(n2−1)[nγQ(µ)−V µ]
nγQ(µ)(n2 −V2) , (3.45) where
Q(µ) =
[
1− V2
n2(1−µ2)−V2µ2
]12
. (3.46)
In order to describe the angle between the x axis and the ray seen by the observer set aty= 0 (as followed by Anderson and Spiegel [1975]), in this case it holds that α =π/2 →µ= 0 and thus, by using equations (3.37), (3.44), (3.45), and (3.46) one gets
tanθ =
(1− Vn22
)1
2
nγV (1− n12
). (3.47)
This result is identical with formula (3.27) derived above.
Further, formula (3.30) was obtained by Lerche [1974c] by using the same formalism as was introduced in Lerche [1974a]. This time, a nearly cold plasma is considered, i.e., the medium is dispersive and wave frequencyωAis related with wave vectorkA via
ωA2 =ω2p
(
1− 5 6vT2
)
+k2A
(
1 + 1 3vT2
)
=ω2∗+c2∗k2A, (3.48) where ω∗ and c∗ are introduced by this relation. The plasma is considered to be nearly cold, i.e., thermal velocityvT is taken into account only up toO(v2T). Note that as units withc= 1 are used, the dimensionless form of the defined condition isO(vc2T2); the terms includingvT would actually consist of vcT.
In the case of the nearly cold plasma, refractive index n measured by the observer in the medium rest frame is modified in comparison with the previous case, since formula (3.48) holds and thus
n2 = kA2 ωA2 = 1
c2∗
(
1− ω∗2 ωA2
)
. (3.49)
The phase speed measured by the observer in the medium rest frame is thus given by
u0 =c∗
(
1− ω∗2 ωA2
)−1
2
(3.50) As was discussed above, the variables measured by the external observer can be expressed in terms of those measured in the medium rest frame. However, the velocity addition formulae can be applied also in the opposite way in order to find the expressions forωA and kA as function of ωext,u, and µ, which are
ωA=γωext
(
1− µV u
)
, (3.51)
kA=γωext µ
(µ u −V
)
. (3.52)
Along with (3.40), equations (3.51), (3.52) give the expression for the phase speed measured in the medium rest frame in the form
u0 = ωA
kA = u−V µ
[(µ−uV)2 + (1−µ2)γ−2]
1 2
. (3.53)
Finally, when using (3.50), (3.51), and (3.53) one gets that the phase speed of a ray propagating in a plane differentially sheared nearly cold plasma measured by the external inertial observer is
u
(
1−c2∗V2− ω2∗ γ2ω2ext
)
=µV(1−c2∗) +c∗γ−1
[
1−c2∗V2− ω∗2
γ2ω2ext (3.54) +µ2V2(c2∗ −1)
(
1− ω∗2 c2∗ωext2
)]1
2
.
This formula is the generalisation of expression (3.44) obtained for a nearly cold plasma characterised by thermal velocity vT. The differentiation of (3.54) leads to
∂u
∂µ = V(1−c2∗)[Q∗(µ) +γ−1c∗µV (1−c2ω∗2
∗ωext2
)]
Q∗(µ)[1−c2∗V2−γ2ωω2∗2ext
] , (3.55)
where
Q∗(µ) =
[
1−c2∗V2− ω∗2
γ2ωext2 +µ2V2(c2∗−1)
(
1− ω∗2 c2∗ωext2
)]12
. (3.56)
Considering the same conditions as in the previous case (i.e., µ remains zero at any given heighty, if aty = 0, it holds µ= 0, as it can be seen from (3.45) and (3.55)), the slope (3.37) is given by
tanθ = c∗
[1−c2∗V2− γ2ωω2∗2ext
]12
γV(1−c2∗) . (3.57)
This formula is consistent with the expression (3.30) as was obtained by Anderson and Spiegel [1975]. This can be seen when considering a cold plasma dispersion relation
ωp2
ωA2 = 1−n2. (3.58)
4. Radiation Transfer in a Refractive Medium
In this chapter, we shall extend the study of the radiation transfer as was de-veloped by Balek [1976]. The plane differentially sheared medium as well as the rotating differentially sheared medium as introduced in Chapter 3 are considered.
These are studied further. Quantitative description of the introduced systems is performed.
In the second part of the chapter, the ray accessible regions in rotating and falling refractive media situated in the Kerr spacetime are discussed.
4.1 Plane Differentially Sheared Medium
Following the notation used by Lerche [1974a], let us describe a plane differentially sheared medium moving by velocity V⃗ (this is measured by an external inertial observer). The medium is refractive, but nondispersive, it is realised by several infinitesimally thin layers dy, and moving along the x axis (see Fig. 3.1). Thus, the velocity components are defined as
Vx =V(y)≡V, Vy =Vz = 0. (4.1) Light rays propagating in the system introduced above can be described by the Hamiltonian
H= 1 2
[−ω2 +kx2+ky2+kz2−γ2(n2−1)(ω−kxV)2]. (4.2) This relation was obtained from the general form of Hamiltonian (1.8) by using the specific properties of the medium. As in the previous chapters, it holds that ω=−k0 andγ = 1/
√
1−V⃗2. In the medium local rest frames, n=const. Since Hamiltonian (4.2) does not depend on coordinatesx, z, equation of motion (1.12b) implies that kx, kz are constant along a given ray. Thus, due to the simplicity (following Lerche [1974a]), the rays propagating in the (x, y)-plane are assumed, i.e., kz = 0.
Under these restrictions, frequency ω is given by ω(kx, ky, y) = 1
n2 −V2
[(n2−1)V kx+γ−1√kx2n2(1−V2) +ky2(n2−V2)]. (4.3) This formula was obtained by using (4.2) when setting H = 0. The sign of the second term in equation (4.3) is given by the requirement that time coordinatet increases along affine parameter λ: it must hold that dλdt >0.
In order to visualise the ray trajectories obeying the conditions given above, we constructed Fig. 4.1. It shows the ray paths in a plane differentially sheared moving medium as was introduced by Lerche [1974a] characterised by different re-fractive indices. The ray paths were obtained using the Hamilton equation (1.12a) describing the evolution of photons along the ray path:
dx
dλ = ∂H
∂kx =kx+γ2(n2−1)(ω−kxV)V, dy
dλ = ∂H
∂ky =ky. (4.4)
Notice thatky(in comparison withkx) is a tangential vector to the ray. Therefore, the ray motion can be described by the equation
dy
dx = ky
kx+γ2(n2−1)(ω−kxV)V . (4.5) This equation corresponds to the formula for the angle between the x axis and group velocity ⃗vg obtained by Lerche [1974a] and described in Chapter 3.
Equation (3.37) can be derived from expression (4.5) when considering that
u= ω
√k2x+ky2; µ= cosα= kx
√k2x+ky2. (4.6) In order to simplify the problem, let kx be equal to zero. Because of Hamilto-nian equation (1.12b), it stays so along the whole ray path. Hence, equation (4.3) takes the form
ω= ky γ√
n2−V2 (4.7)
and equation of motion (4.5) becomes dy
dx = (n2−V2)12
γV(n2−1). (4.8)
It can be seen that whenV →1, thusγ → ∞and dydx →0. Hence, the ray motion is indeed limited by the speed of the medium.
This result corresponds to expressions (3.27) and (3.47) obtained by Anderson and Spiegel [1975] and Lerche [1974a], respectively. Therefore, the Hamiltonian formalism represents the third way how to describe the given problem as was discussed in Chapter 3.
Fig. 4.1 shows the ray paths obtained by using equation (4.5) in the case of different refractive indices. The dotted lines show the medium layers of different velocity V(y). The linear dependence of V on y is assumed. The dashed line show the ray path for the case when n = 1. From equation (4.4) it can be seen that if n = 1, component x does not change along the ray path (characterised by parameter λ), since kx is constant along the ray (due to equation (1.12b)).
Hence, the ray moves directly in the direction ofy axis.
The rays with n ̸= 1 evolve as it is depicted in Fig. 4.1. The closer the refractive index value is to 1, the closer the ray is to the limiting case n = 1.
The values of refractive indices were chosen from 0.3 up to 0.95 in the case when n <1, while in the case n >1, they range from 1.05 to 2.
If n <1, the rays can propagate until they reach the height y=L1 at which V(L1) = n (see equation (4.8)). The direction of ray propagation is opposite to the direction of rays propagating whenn >1. This means the light propagates in the opposite direction to the motion of the medium. Thus, the larger refractive index (i.e., the closer to one) is assumed, the greater height can be reached by a ray. When reaching heightL1, rays cannot penetrate higher and start propagating upstream. This can be seen in Fig. 4.1.
If n >1, the rays move along with the increasing V till they reach a critical height y = L2 at which the medium speed becomes V(L2) = 1. Approaching the critical height, the rays cannot penetrate further and they are aligned in the same direction as the medium.
Figure 4.1: The ray trajectories in a plane differentially sheared medium obeying the equation (4.8) in the case of different refractive indices (solid curves). The vertical dashed line corresponds to the case when n = 1. The medium is char-acterised by the velocity V = ˆxV(y). The individual velocity layers are outlined by the dotted lines. The refractive index values were chosen from 0.3 up to 0.95 in the case when n < 1 and between 1.05 and 2 for the case n > 1. The rays evolving closer to x= 0 correspond to the cases when n ≈1.