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MASTER THESIS

Barbora Bezdˇekov´ a

Electromagnetic Waves in Dispersive and Refractive Relativistic Systems

Institute of Theoretical Physics

Supervisor of the master thesis: prof. RNDr. Jiˇr´ı Biˇc´ak, DrSc., dr. h. c.

Study programme: Physics

Study branch: Theoretical Physics

Prague 2019

I declare that I carried out this master thesis independently, and only with the cited sources, literature and other professional sources.

I understand that my work relates to the rights and obligations under the Act No. 121/2000 Sb., the Copyright Act, as amended, in particular the fact that the Charles University has the right to conclude a license agreement on the use of this work as a school work pursuant to Section 60 subsection 1 of the Copyright Act.

In Prague 19.7.2019 Signature of the author

I would like to thank my supervisor prof. RNDr. Jiˇr´ı Biˇc´ak, DrSc., dr. h. c. for his patience in giving me a lot of advice, his frank interest in my academic work and also for his human approach during our whole cooperation.

I am grateful to doc. RNDr. Vladim´ır Balek, CSc. for helping me to success- fully reproduce and understand results based on his thesis.

My big thanks belongs to my family, especially to my parents for their per- manent support.

Last but not least, I would like to thank all the people who believed, more than me, in my capacity to complete my second thesis. ¹

1R´ada bych podˇekovala vedouc´ımu svoj´ı diplomov´e pr´ace prof. RNDr. Jiˇr´ımu Biˇc´akovi, DrSc., dr. h. c. za jeho trpˇeliv´e ud´ılen´ı rad, upˇr´ımn´y z´ajem o m˚uj profesn´ı r˚ust a za jeho osobn´ı pˇr´ıstup po celou dobu naˇs´ı spolupr´ace.

Dˇekuji doc. RNDr. Vladim´ıru Balekovi, CSc., ˇze mi pomohl ´uspˇeˇsnˇe zreprodukovat a pocho- pit v´ysledky vych´azej´ıc´ı z jeho diplomov´e pr´ace.

M˚uj velk´y d´ık patˇr´ı moj´ı rodinˇe, hlavnˇe m´ym rodiˇc˚um za jejich st´alou podporu.

Pˇredevˇs´ım bych chtˇela podˇekovat vˇsem lidem, kteˇr´ı vˇeˇrili tomu, ˇze dokonˇc´ım svoji druhou diplomovou pr´aci, mnohem v´ıc neˇz j´a.

Title: Electromagnetic Waves in Dispersive and Refractive Relativistic Systems Author: Barbora Bezdˇekov´a

Institute: Institute of Theoretical Physics

Supervisor: prof. RNDr. Jiˇr´ı Biˇc´ak, DrSc., dr. h. c., Institute of Theoretical Physics

Abstract: Study of light rays (photon world lines) in geometric optics limit plays a significant role in many astrophysical applications. Light rays are mainly studied in association with gravitational lensing. The majority of studies are mainly focused on light propagation in vacuum. If the refractive and dispersive medium characterised by refractive index n is considered, effects occurring due to the medium presence need to be taken into account, which significantly complicates the problem. In the present thesis, rays propagating through simple refractive and dispersive systems, such as plane differentially sheared or expanding medium, are studied. In order to simplify the problem, the Hamiltonian equations of motion are used. The ray trajectories in the vicinity of Kerr black hole as well as the accessible regions for the rays are also studied. Radial variation of the medium velocity is considered. Due to the recent increase of publications focused on the gravitational lensing in plasma, a detailed review summarizing the results obtained lately is included.

Keywords: electromagnetic radiation, geometric optics, dispersion, relativistic systems, special relativity, general relativity, gravitational field

N´azev pr´ace: Elektromagnetick´e vlny v disperzn´ıch a refraktivn´ıch relativistic- k´ych syst´emech

Autor: Barbora Bezdˇekov´a Ustav: ´´ Ustav teoretick´e fyziky

Vedouc´ı diplomov´e pr´ace: prof. RNDr. Jiˇr´ı Biˇc´ak, DrSc., dr. h. c., ´Ustav teoretick´e fyziky

Abstrakt: Studium paprsk˚u (svˇetoˇcar foton˚u) v limitˇe geometrick´e optiky hraje v´yznamnou roli v mnoha astrofyzik´aln´ıch aplikac´ıch a je pˇredmˇetem intenzivn´ıho v´yzkumu, pˇredevˇs´ım v r´amci tzv. gravitaˇcn´ıho ˇcoˇckov´an´ı. Proveden´e studie se vˇetˇsinou zab´yvaj´ı ˇs´ıˇren´ım svˇetla ve vakuu. V pˇr´ıpadˇe, ˇze je studov´an pr˚uchod svˇeteln´ych paprsk˚u refraktivn´ım a disperzn´ım prostˇred´ım charakterizovan´ym in- dexem lomu n, je tˇreba vz´ıt v potaz efekty, kter´e se v takov´em prostˇred´ı obje- vuj´ı, coˇz dan´y probl´em znaˇcnˇe komplikuje. Pr´ace se zab´yv´a studiem paprsk˚u pohybuj´ıc´ıch se ve zjednoduˇsen´ych refraktivn´ıch a disperzn´ıch syst´emech (napˇr.

rovinn´e vrstvy s rozd´ıln´ymi rychlostmi prostˇred´ı) za pouˇzit´ı Hamiltonov´ych po- hybov´ych rovnic. D´ale je studov´an pohyb paprsk˚u v okol´ı Kerrovy ˇcern´e d´ıry a zkoum´any jejich pˇr´ıstupov´e oblasti v pˇr´ıpadˇe radi´alnˇe se mˇen´ıc´ı rychlosti re- fraktivn´ıho prostˇred´ı obklopuj´ıc´ıho ˇcernou d´ıru. Vzhledem k ned´avn´emu zv´yˇsen´ı mnoˇzstv´ı publikac´ı zab´yvaj´ıc´ıch se zkoumanou problematikou byla tak´e seps´ana podrobn´a reˇserˇse shrnuj´ıc´ı nejv´yznamnˇejˇs´ı v posledn´ıch letech dosaˇzen´e v´ysledky.

Kl´ıˇcov´a slova: elektromagnetick´e z´aˇren´ı, geometrick´a optika, disperze, relativi- stick´e syst´emy, speci´aln´ı relativita, obecn´a relativita, gravitaˇcn´ı pole

Contents

List of Used Convention and Notation 2

Introduction - Light, Gravity, and Plasma 3

1 Relativistic Geometric Optics 5

2 Recent Results on Light Propagation in Plasma and Other Media

- A Review 9

2.1 Linearised Description of Light Propagation in Plasma . . . 9

2.2 Ray-tracing in Plasma Medium . . . 21

2.3 Studies of Black Hole Shadows Surrounded by Plasma . . . 24

3 Formalism of Optical Metric and Its Applications 31 3.1 Formalism of Optical Metric . . . 31

3.2 Application of the Formalism . . . 33

3.2.1 Plane Differentially Sheared Medium . . . 33

3.2.2 Expanding Medium . . . 36

3.3 Another Approach to Discussed Problems . . . 37

4 Radiation Transfer in a Refractive Medium 41 4.1 Plane Differentially Sheared Medium . . . 41

4.2 Rotating Differentially Sheared Medium . . . 43

4.3 Refractive Rotating Medium in Kerr Spacetime . . . 47

4.4 “Radially” Falling Medium in Kerr Spacetime . . . 54

Conclusion 56

Bibliography 58

List of Used Convention and Notation

α, β 0,1,2,3 i, j 1,2,3

g_αβ spacetime metric η_αβ flat spacetime metric

h_αβ small perturbation term of spacetime metric k_α wave vector

c speed of light in vacuum G gravitational constant m electron mass

m_p proton mass e electron charge

ϵ₀ permittivity of vacuum k_B Boltzmann constant ℏ reduced Planck constant

M mass of a gravitating object (i.e., the black hole)

J angular momentum

a= _cM^J angular momentum per mass (the Kerr parameter) R radius of the closest approach

r_g Schwarzschild gravitational radius γ Lorentz factor

n refractive index

ω photon frequency at infinity ω_p plasma frequency

ωpe plasma electron frequency ˆ

x unit vector in the xdirection

Unless otherwise stated, it holds that c=G= 1, η_αβ = diag(−1,1,1,1).

Properties of a weak gravitational field spacetime metric

In the case of a weak gravitational field approximation, spacetime metricg_αβ can be written as a sum of

a metric of a flat space ηαβ

and a small perturbation h_αβ; h_αβ →0 as xⁱ → ∞.

Thus, the metric is g_αβ =η_αβ+h_αβ or g^αβ =η^αβ−h^αβ such as g_αβg^βγ =δ^γ_α. It holds that η^αβ =η_αβ; h^αβ =η^αρη^βσh_ρσ.

Introduction - Light, Gravity, and Plasma

Electromagnetic waves provide a permanent source of knowledge about the Uni- verse and thus, their behaviour and detection represent the main interest in as- tronomy.

It is well-known that the influence of gravity on light propagation is one of the crucial gravitational effects, which has been studied for decades. Exactly one hundred years ago, sir Arthur Stanley Eddington together with astronomer Frank Watson Dyson undertook two expeditions to verify general relativity via bending of light rays in the vicinity of a gravitating object. Their observations of light deflection caused by Sun’s gravitational field during the total solar eclipse of 29 May 1919 on the island of Pr´ıncipe became one of the most famous evidence of general relativity, which made Albert Einstein indeed a first class celebrity.

Eddington observed precisely the effect predicted by Einstein’s theory – light coming from other stars and passing in the vicinity of the Sun was bent and thus, these stars seemed to be shifted [Dyson et al., 1920].

Moreover, it is also well-known that the light rays can be affected by medium through which they propagate. Not only gravity, but also plasma or other medium can cause deviations of light propagation. A light ray passing near a gravitating source (e.g., a black hole) which is surrounded by a disc of hot (>10³ K) ionised gas (so-called accretion disc) can be modified by this medium. Such medium is characterised by refractive indexn. Due to the dependence of wave frequency on n, the speed of light changes along the propagation path in this system. To be able to propagate in plasma, wave frequency ω has to be higher than the plasma frequencyω_p, which, in SI units, reads

ω_p =







√

N_je² ϵ0mj

.

Number density Nj as well as mass mj are given by the j-th plasma particle species. Typically, these are electrons and protons (ions) and plasma is regarded as a two-component medium. Considering the plasma quasineutrality, it holds thatNe=Ni. Thus, to be able to describe the light propagation through plasma, it is necessary to obtain a plasma density profile. Ifω ≫ω_p, the radiation passing through the medium exhibits nearly the same propagation characteristics as in the vacuum case.

It is worth mentioning that in many astrophysical applications the effects on wave propagation caused by the nature of refractive medium are negligible.

However, they start to be important when radio frequency waves are analysed or light penetration through dense atmospheres is studied. In such media, the refraction is significant and its effect has to be included into the description of the system.

Description of electromagnetic waves propagating in a refractive and disper- sive medium when gravity has to be taken into account is a quite tricky task. The equations accounting for both these factors are pretty complicated even when con- sidering solely electron plasma without magnetic field. However, it turns out that

this problem can be simplified when the geometric optics approach is used. This method was introduced in detail in the last chapter of the book Relativity: The General Theory by J. L. Synge [Synge, 1960]. It is based on the Hamiltonian description of rays propagating in a dispersive medium. In more physical and de- tailed way, the formalism was also studied and discussed by Biˇc´ak and Hadrava [1975]. The problem was more recently adapted by Bisnovatyi-Kogan and Tsupko [e.g., Bisnovatyi-Kogan and Tsupko, 2009, 2010, Tsupko and Bisnovatyi-Kogan, 2012] and others in order to find mainly the deflection angle of rays propagating in a refractive medium in vicinity of a gravitating object. Let us emphasise that in most of the mentioned works and in this entire thesis, the absorption of the waves by medium is assumed to be negligible and it is thus ignored.

Geometric optics approach allows us to deal with the problem in terms of rays. This terminology is possible due to the idea thatray represents a history of trajectory of a particle -“photon”. Photons are assumed to be particles with mass and charge equal zero. In fact, in geometric optics approximation it is possible to switch between these two terms (photon vs ray). Throughout the thesis, they are freely interchanged, with regard to the chosen approach. We hope that in spite of that, the description remains clear enough to deliver the message of the text in an understandable form.

The content of the thesis is organised as follows. The geometric optics ap- proach as set by Synge is described in Chapter 1. The review of recent studies of Bisnovatyi-Kogan and Tsupko and several other authors dealing with a propaga- tion of rays around a gravitating object in plasma is presented in Chapter 2. A comparison of description of ray propagation in differentially sheared media via geometric optics approach and Lorentz transformation of radiation phase speed is discussed in Chapter 3. In this chapter, also the optical metric approach is introduced. Chapter 4 summarises the results of the analysis of ray propagation in media defined in previous chapters. The regions accessible for rays are inves- tigated and the ray trajectories are constructed for differentially sheared media.

1. Relativistic Geometric Optics

Geometric optics approach was analysed in depth from the covariantly generalised Maxwell equations by Ehlers [1967]. The study gives the following restrictions under which the geometric optics is valid:

1. typical wavelengthλ_tis short in comparison with the variations of properties of the given medium (i.e., the index of refraction, the velocity),

2. the waves are locally monochromatic, i.e., the scales of variations of wave features (the amplitude, wavelength, polarization) are large comparing to λ_t,

3. the characteristic curvature radius of the spacetime is larger than λ_t, 4. over one typical wave period the medium varies negligibly.

The Hamiltonian description of the geometric optics in an isotropic dispersive medium was studied by Synge [1960]. In his book, it is shown that under the conditions mentioned above the waves can be considered as a hypersurface of a constant phaseϕ(x^α) = const. Each wave is associated with such surface. Thus, the wave vector kα, normal to the wave, is

k_α =∇ϕ= ∂ϕ

∂x^α. (1.1)

Figure 1.1: Illustration of the world lines of wave W detected by observer O.

These lines intersect at point A (adapted from Synge [1960]).

Speed of a single particle u^′ measured by observer O moving with the four- velocity U^µ is given by (see Fig. 1.1)

u^′2 = dξ_αdξ^α

ds² . (1.2)

The intersection of observer O with the wave W is denoted as A. This approach assumes there is an artificial particle moving by u^′ which is bounded with the

wave W and its world line lies on W. The infinitesimal element of the particle world line is denoted dx^α, whereas dξ^α is the infinitesimal vector orthogonal to O. As it can be seen in Fig. 1.1, it holds that

dξ^α =dx^α+U^αU_βdx^β as ds=−U_βdx^β. (1.3) Thus, in this case, the Pythagoras theorem takes the form

dξ_αdξ^α =dx_αdx^α+ (U_αdx^α)² (1.4) and the speed (1.2) can be written as

u^′2 = 1 + dx_αdx^α

(U_αdx^α)². (1.5)

Clearly,U_αU^α =−1 and dx_α =g_αβdx^β.

Since the wave speed is defined as the minimum speed u^′ of all particles accompanied by the wave, the infinitesimal element dx^α of the particle world line can be written as dx^α = νU^α+ℏk^α. As the element dx^α is bounded with k^α through k_αdx^α = 0, the minimalization of (1.5) can be performed by using Lagrange multiplier ν. Due to this condition, one gets

ν=−ℏk_αk^α

k_αU^α and dx_αdx^α =ν(ℏk_αU^α−ν) =−ℏ²k_αk^α

( k_αk^α (k_αU^α)² + 1

)

. (1.6) Finally, the phase speedu is

1

u² = 1 + kαk^α

(k_αU^α)². (1.7)

In the moving dispersive and refractive medium the formalism described above can be used, too. Let the four-velocity of the medium beU. Hence, the medium is characterised by the refractive indexn(k_αU^α, x^α), which is defined asn= 1/u.

Due to the definition of four-velocity U, frequency ω =−k0 and phase speed u are measured in the medium rest frame. The definition of frequency ω can be applied in the case that the Hamiltonian does not explicitly depend on the time coordinate. Indeed, HamiltonianHdefined in the case of a ray motion must satisfy the condition

H(x^α, k_α) = 1 2

[g^αβk_αk_β −(n²−1)(k_αU^α)^2]= 0. (1.8) The ray is a curve which extremalises the expression

∫ x2

x1

k_αdx^α. (1.9)

Note that in the nondispersive medium, i.e.,n does not depend onk_α, Hamil- tonian (1.8) can be rewritten by using so-called optical metric ˜g^αβ in the form

H(x^α, k_α) = 1

2˜g^αβk_αk_β, (1.10)

where

˜

g^αβ =g^αβ −(n²−1)U^αU^β (1.11) (for more details see further and Anderson and Spiegel [1975], which will be summarised later).

In fact, Hamiltonian (1.8) yields partial differential equation for phase ϕ.

Characteristic curves of the equation (1.8) are solutions of partial differential equations

dx^α

dλ = ∂H

∂k_α, (1.12a)

dk_α

dλ =−∂H

∂x^α, (1.12b)

whereλis a parameter. It was shown that in the vacuum case (i.e., whenn= 1), λ represents an affine parameter. The characteristic curves satisfying the set of the equations above are called optical rays. The equations determine not only the ray, but also the frequency vector at each point.

By using equation (1.12a), one gets ¹ dx^α

dλ =k^α−(n²−1)(k_βU^β)U^α−n ∂n

∂(k_αU^α)

(kβU^β)2U^α. (1.13) Formula (1.13) is valid in a dispersive isotropic medium and defines the ray tan- gential vector. It also shows that in general, the ray is not guided along the wave vector, i.e., kα is not tangential to the ray. However, in vacuum, when n= 1, the wave vector represents a tangential vector to the ray.

In order to save the concept of causality, the rays have to be either timelike or null. Thus, it must hold that

g_αβ∂H

∂k_α

∂H

∂k_β ≤0. (1.14)

Following the notation of Fig. 1.1, the speed of a ray relative to the medium can be obtained. Let us denote this speed by v. By using equations (1.3), (1.5), and (1.12a), the calculation ofv² gives

v² = 1 +g_αβ∂H

∂k_α

∂H

∂k_β

(

U_δ∂H

∂k_δ

)−2

. (1.15)

From (1.15) it can be seen that the restriction (1.14) is equivalent to the statement v ≤1.

When using the definition (1.8), one obtains g_αβ∂H

∂k_α

∂H

∂k_β =n²(k_αU^α)²

⎡

⎣1−

(

n+ ∂n

∂(k_δU^δ)k_αU^α

)2⎤

⎦, (1.16) U_δ∂H

∂k_δ =nk_δU^δ

(

n+ ∂n

∂(k_δU^δ)k_αU^α

)

.

1Balek [1976] states this equation wrong. In the second and third terms of equation /11.11/, velocityU^αis missed.

These expressions can be further simplified by writing n+ ∂n

∂(k_δU^δ)k_αU^α =n+ ∂n

∂ω_oω_o = ∂(nω_o)

∂ω_o , (1.17)

where it was denoted ω_o =k_αU^α. Frequency ω_o is the wave frequency measured by the observer O.

Finally, the speed v is simply v² = 1 +n²(k_αU^α)²

⎡

⎣1−

(∂(nω_o)

∂ω_o

)2⎤

⎦ [

nk_δU^δ

(∂(nω_o)

∂ω_o

)]−2

=

[∂(nω_o)

∂ω_o

]−2

. (1.18) This formula corresponds to the well-known definition of the group speed. Thus, the group velocity is equivalent to the ray speed defined in terms of the Hamil- tonian formalism. The restriction (1.14) represents the limitation of group speed which cannot be higher than c.

2. Recent Results on Light Propagation in Plasma and Other Media - A Review

Gravitational lensing is one of the crucial predictions of general relativity, which it has been verified by many observations [e.g., Falco, 2005]. This effect is usually characterised by a lensing angle. If a plasma surrounds a gravitating source, the problem tends to be more complicated due to the refraction and dispersion effects. During last few years, this has been dealt with in several papers. In this chapter, main results of these works are summarised and some interconnections are discussed.

2.1 Linearised Description of Light Propagation in Plasma

Bisnovatyi-Kogan and Tsupko [2009] demonstrated how to obtain the equation of light propagation in an inhomogeneous plasma in gravitational field using pertur- bation theory, i.e., by writing a general form of a spacetime metric g_αβ in terms of a flat metricη_αβ and small perturbations h_αβ.

If a static inhomogeneous plasma with the refraction index n =n(xⁱ, ω(xⁱ)), depending on the space location xⁱ and the photon frequency ω(xⁱ), is assumed, the electron number density N(xⁱ) is written as N(xⁱ) = N₀ +N₁(xⁱ), where N₀ =const and N₁, a small inhomogeneous term, is constrained by N₁(∞) = 0, N₁ ≪N₀. Electron plasma frequency ω_pe can be then written as ω_pe² =ω₀²+ω²₁, whereω²₀ =K_eN₀,ω₁² =K_eN₁(xⁱ),K_e = _mϵ^e2

0. Note that in CGS unitsK_e = ^4πe_m². In a general case, refractive index n is related to photon frequency ω(xⁱ) via

n² = 1− ω²_pe

ω²(xⁱ). (2.1)

Further, it is useful to define

n₀ =n(∞) =

√

1− ω₀²

ω². (2.2)

It holds that ω=ω(∞). Let us start with the variational principle δ

(∫

p_αdx^α

)

= 0, (2.3)

wherex^α,p_α denote spacetime and momentum coordinates, respectively, in order to find the photon trajectories in the gravitational field. Assuming the well-known relation between refraction index n, phase velocity u, and speed of light c, i.e., c=nu, we can define a scalar function W(xⁱ, p_i) as

W(xⁱ, p_i) = 1 2

[

g^αβp_αp_β−(n²−1)(p₀^√−g⁰⁰)²

]

= 0. (2.4)

The variational principle (2.3) combined with the restriction (2.4) results in a set of differential equations for the light trajectories

dx^α

dλ = ∂W

∂p_α, (2.5a)

dp_α

dλ =−∂W

∂x^α (2.5b)

Parameter λ changes along the light trajectories. The function W(xⁱ, p_i) can be further simplified as

W(xⁱ, p_i) = 1 2

[

g^αβp_αp_β−(n²−1)(p₀^√−g⁰⁰)²

]

= (2.6)

1 2

[g⁰⁰p₀p₀+g^ijp_ip_j −(n²−1)p²₀(−g⁰⁰)^]= 1

2

[g^ijp_ip_j −n²p²₀(−g⁰⁰)^],

which allows us to rewrite equations (2.5) in terms of spatial momentum compo- nentsp_i as

dxⁱ

dλ =g^ijp_j, dp_i dλ = 1

2

[

−g^jk_,ip_jp_k+⁽n²p²₀(−g⁰⁰)⁾

,i

]

. (2.7)

Further, it holds that 1

2

(n²p²₀(−g⁰⁰)⁾

,i = 1 2

(n²⁾

,ip²₀(−g⁰⁰) + 1

2n²⁽p²₀(−g⁰⁰)⁾

,i = 1

2

(

1− ω_pe² ω²(xⁱ)

)

,i

p²₀(−g⁰⁰) + 1

2n²p²₀⁽−g⁰⁰⁾

,i = 1

2

(

1− K_eN₀

ω²(xⁱ)− K_eN₁(xⁱ) ω²(xⁱ)

)

,i

p²₀(−g⁰⁰) + 1

2n²p²₀⁽−g⁰⁰⁾

,i = 1

2

(

1− K_eN₀

n²₀ω² − K_eN₁(xⁱ) n²₀ω²

)

,i

+1

2n²p²₀⁽−g⁰⁰⁾

,i=

−1 2

K_e ω² −ω₀²

∂N₁(xⁱ)

∂xⁱ +1

2p²₀⁽−g⁰⁰⁾

,i,

where the relation between time component p₀ and photon frequency ω(xⁱ) p₀^√−g⁰⁰=Cω(xⁱ) =−ω(xⁱ)

n₀ω (2.8)

was used. Note that the constant factor C was determined using the boundary conditions forp₀ and ω(xⁱ) at infinity. Finally, the set of equations (2.5) leads to

dxⁱ

dλ =g^ijp_j, (2.9a)

dp_i

dλ =−1

2g_,i^jkp_jp_k+ 1

2p²₀(−g⁰⁰)_,i− 1 2

K_e ω²−ω²₀

∂N₁(xⁱ)

∂xⁱ , (2.9b)

where ω denotes the photon frequency at infinity. Such equations can describe the light propagation in an inhomogeneous plasma including effects of dispersion in a gravitational field.

If a photon moves along thezdirection in a flat spacetime and thezcoordinate is taken as parameter λ, the components of the photon momentum are p_α = (−1/n0,0,0,1). Thus, in a weak gravitational field it holds that

1 2

[−g^jk_,i p_jp_k+ (−g⁰⁰)_,ip²₀^]= 1 2

[−g_,i³³p²₃+ (−g⁰⁰)_,ip²₀^]= 1

2

[

(h₃₃)_,i+ 1

n²₀(h₀₀)_,i

]

= 1 2

(

h₃₃+ ω² ω²−ω₀²h₀₀

)

,i

,

(2.10)

which allows to express equation (2.9b) in the form dp_i

dz = 1 2

(

h₃₃+ ω² ω² −ω₀²h₀₀

)

,i

− 1 2

K_e ω²−ω²₀

∂N₁(xⁱ)

∂xⁱ . (2.11) This formula describes the light propagation in a weak gravitational field and in an inhomogeneous plasma.

In the plane perpendicular to the light unperturbed trajectory, deflection angle αl is defined as

α_l = p_l(+∞)−p_l(−∞)

|p| ; |p|=^√p²₁+p²₂+p²₃; l= 1,2. (2.12) Thus, if the unperturbed light trajectory is aligned with thezaxis, the photon deflection angle formula reads

α_l = 1 2

∫ +∞

−∞

∂

∂x^l

(

h₃₃+ ω²

ω²−ω²₀h₀₀− K_e

ω²−ω₀²N₁(xⁱ)

)

dz. (2.13)

The first two terms arise due to an inhomogeneous gravitational field described byh₀₀and h₃₃, the third term appears due to the inhomogeneous plasma electron number densityN₁. Index l= 1,2 denotes x, y axes, respectively.

In the case of a spherically symmetric distribution of plasma density, for- mula (2.13) may be rewritten in terms of the impact parameter b. It holds that b = √

x²+y² and the radius vector r is defined as r = √

b²+z². If a homoge- neous dispersive medium is considered, i.e.,∂N₁(xⁱ)/∂x^l = 0, the deflection angle in terms of the impact parameter b can be rewritten as

α_b =

∫ +∞

0

∂

∂b

(

h₃₃+ 1

1−ω₀²/ω²h₀₀

)

dz. (2.14)

In order to obtaina particular form of the deflection angle, Bisnovatyi-Kogan and Tsupko [2009] calculated the deflection angle value for a photon moving in a homogeneous plasma with dispersion in the linearised spherically symmetric Schwarzschild metric

ds² =−

(

1− 2M r

)

dt²+

(

1− 2M r

)⁻¹

dr²+r²(dϑ²+ sin²ϑdφ²). (2.15)

If the Schwarzschild gravitational radius 2M is r_g (in standard units 2GM/c²), the perturbation components of metrich₀₀, h₃₃ can be written as

h₀₀ = rg

r, h₃₃= rg

r cos²θ = rg

r

(z r

)2

. (2.16)

After an (analytical) integration of these components, the formula for the deflec- tion angle becomes

α = r_g b

⎛

⎝1 + 1 1− ^ω_ω²⁰2

⎞

⎠. (2.17)

It is obvious that the deflection angle of rays, in contrast to purely gravitational effects, depends on the photon frequency. This formula is valid only if ω > ω0. If the photon frequency is low enough (i.e., close to ω₀), the image of the point source will be projected to a line or a ring. This is typically the case of radio frequencies. If ω → ∞, the deflection angle formula is 2rg/b, which corresponds to its vacuum value. Of course, in this limit n→1 as it follows from (2.2).

Bisnovatyi-Kogan and Tsupko [2009] also showed that in the case of a homo- geneous medium without dispersion (i.e.,n does not depend onω(x)) the photon trajectory remains the same as in vacuum, however, due to the presence of the medium, its propagation velocity decreases. The deflection angle is in this case defined as

αl = 1 2

∫ +∞

−∞

∂

∂x^l (h₃₃+h₀₀)dz. (2.18) The perturbation metric components h₀₀,h₃₃ are given by (2.16).

Non-linear effects in plasma and gravity were discussed by Bisnovatyi-Kogan and Tsupko [2010] and Tsupko and Bisnovatyi-Kogan [2012]. They showed that the magnification of the image is increased due to a propagation via a homo- geneous plasma. The larger the difference between the plasma densities at the source and observation regions is, the stronger the observed effect appears. The lower photon frequency, the larger amplification, which causes the original source spectrum to differ from the observed image spectrum (low-frequency part is more intense).

The deflection angle increases if further enhancement of the plasma inhomo- geneity is considered. In Bisnovatyi-Kogan and Tsupko [2010], a weakly inho- mogeneous plasma approach (i.e., the assumption N₁ ≪ N₀) is not assumed anymore; again, it holds ω_pe² =K_eN(xⁱ), where the refraction indexN(xⁱ) is not considered as a sum of a constant and a small inhomogeneous term. Using the same technique as described above, formula (2.13) can be rewritten in terms of the expansion in the plasma frequency, assuming condition ω²_pe/ω² ≪1. Hence, one obtains

α_b =−2r_g b − 1

2

∫ +∞

−∞

[ω_pe² ω²

r_gb r³ +K_e

ω² b r

dN(r) dr + ω²_pe

ω⁴ K_eb

r

dN(r) dr

]

dz. (2.19) The first term corresponds to a vacuum deflection, the second term is a correction due to the gravitational deflection caused by the plasma presence, the third term represents a non-relativistic effects given by the plasma inhomogeneity, and the

fourth term corrects the third one. Note that the second term occurs purely due to the plasma presence and affects the deflection angle value both in a homogeneous and in an inhomogeneous medium.

In Bisnovatyi-Kogan and Tsupko [2010], it is also derived how the spectra of the same source change due to the propagation in different media (charac- terised by the plasma density). Note that hereinafter, for the sake of consistent dimensionality of the problem,G, care not in geometrical units.

If a singular (with the centre at the origin) isothermal sphere is considered and its density distribution is given by

ρ(r) = σ_v²

2πGr², (2.20)

where σ²_v denotes a one-dimensional velocity dispersion, the number density of the plasma can be rewritten as

N(r) = ρ(r)

κm_p, (2.21)

proton mass is denoted by m_p and κ denotes a dark matter contribution coeffi- cient, which is assumed to beκ≈6. Thus, the correction term to the gravitational deflection due to the plasma presence and the correction caused by the plasma inhomogeneity (i.e., the second and the third terms in the equation (2.19)) are equal to

2 3c²

σ⁴_vK_e

κm_pGω²b², −1 4

σ_v²K_e

κm_pGω²b², (2.22)

respectively. If a non-singular isothermal gas sphere is concerned, i.e., instead of the singularity at r = 0, a finite core of radius r_c is taken into account with the density distribution given by

ρ(r) = σ_v²

2πG(r²+r_c²), (2.23)

it is possible to obtain the second and the third term of equation (2.19) in the forms

1 2c²

σ_v⁴K_eb

κm_pGω²r_c³, −1 4

σ_v²K_eb

κm_pGω²r_c³ (2.24)

in the case of r_c≫b. For r_c≪b, one gets 2

3c²

σ⁴_vK_e

κmpGω²b², −1 4

σ_v²K_e

κmpGω²b². (2.25)

In real situation, the absolute value of the ratio between these two terms (second term/third term) is≪1 (asσ_v < cand the gravitational radius of a core

< r_c), which means that the deflection effects caused by the non-uniform plasma distribution are much stronger than the gravitational plasma effects. Also note that the two terms have opposite effects.

Bisnovatyi-Kogan and Tsupko [2010] also discuss the deflection angle in the case of a black hole surrounded by an electron-proton plasma. The results ob- tained in the Newtonian approximation of the plasma density distribution show that the effects caused by the plasma presence become weaker as the plasma temperatureT increases. The ratio of the deflection angle terms discussed above is this time 2k_BT /m_pc², where k_B is the Boltzmann constant. The higher the temperature is, the more uniform the plasma becomes and the significance of ef- fects caused by the plasma inhomogeneity on the refraction thus decreases. This conclusion seems to be valid also in the case of completely relativistic description of the gas sphere.

In the case of a galaxy cluster, a singular isothermal sphere can be used to describe a distribution of the gravitating matter. This model was applied by Bisnovatyi-Kogan and Tsupko [2010] for calculating a plasma density distribution from the equation of hydrostatic equilibrium of a plasma,

k_BT m_pρ

dρ

dr =−2σ_v²

r , (2.26)

located in the gravitational field induced by the sphere. This approach allows to find an equation for plasma densityρ(r) with the boundary condition ρ(r₀) =ρ₀, which reads

ρ(r) =ρ₀

( r r₀

)−s

, s= 2m_pσ_v²

k_BT . (2.27)

Ifs ≪1, i.e., 2σ_v² ≪k_BT /m_p, the second and third terms of equation (2.19) are 2π

c²

σ_v²Keρ0

m_pω²

(r0

b

)s

, − π k_B

σ²_vKeρ0

T ω²

(r0

b

)s

, (2.28)

and their ratio is 2k_BT /mpc². This indicates that in the case of the relativistic plasma presence in a galaxy cluster, the plasma gravitational effects are larger than the plasma inhomogeneity effects. For example, in jets escaping from galac- tic nucleus, where the plasma is relativistic, the effects caused by its relativistic nature will be stronger than effects caused by its non-uniform distribution. Ad- ditionally, if the plasma distribution is not spherically symmetric, its density gradient direction can be opposite to the gravitational force direction (e.g., in case of rotation) and the terms of the deflection angle discussed above may have the same sign.

The same plasma density models as introduced above, i.e., singular isothermal sphere, non-singular isothermal gas sphere, and plasma in a galaxy cluster were also adapted by Benavides-Gallego et al. [2018]. Considering the case of a boosted Kerr black hole, they showed that the deflection angle formula (2.19) contains extra terms as follows

2J_r

Λb² sinχ, −2J_r Λω² sinχ

∫ ∞ 0

ω²_e

r³dz, 3b²Jr

Λω² sinχ

∫ ∞ 0

ω²_e

r⁵dz, (2.29) which result from the inertial frame dragging and the two latter depend on the plasma distribution model. Parameter Λ describes the boost of the metric. It holds that Λ = (coshγ+ sinhγcosθ)² and the boost velocity v is v = tanhγ =

sinhγ/coshγ, where γ denotes the Lorentz factor. Description of the remaining variables is discussed later on.

They showed that the influence of dragging and boost parameter Λ, and thus also the deflection angle formulas, change as a function of the plasma distribution model. Benavides-Gallego et al. [2018] also show that the deflection angle is larger in the case of a galaxy cluster compared to those obtained for the two models mentioned above.

All the results mentioned above were obtained assuming that the deflection angle (i.e., plasma and gravity effects) remains small, i.e., α ≪ 1. The exact photon deflection angle formula using Schwarzschild metric (2.15) was derived by Tsupko and Bisnovatyi-Kogan [2013]. The spherically symmetric distribution of plasma number density N(r) is assumed. It holds that ω_pe² =K_eN(r), where K_e is the same as defined above. If a static medium in a static gravitational field is considered, it holds

p₀^√−g⁰⁰=−ℏω(xⁱ), (2.30) whereℏis the reduced Planck constant. Using the variational principle (2.3), the condition (2.4) can be obtained, and by inserting (2.30), it can be rewritten into the form

W(xⁱ, p_i) = 1 2

[g^αβp_αp_β +ω_pe² ℏ²

]= 0. (2.31)

Using the equations of motion (2.5), the equations for space and momentum componentsxⁱ,p_i are

dxⁱ

dλ =g^ijp_j, (2.32a)

dp_i

dλ =−1

2g_,i^jkp_jp_k− 1 2ℏ²

(ω_pe^{2 )}

,i. (2.32b)

If a motion in the equatorial plane (θ = π/2) is assumed, the relevant spherical coordinates arer and φand the corresponding equations simplify to

dr

dλ =g^rrp_r, dφ

dλ =g^φφp_φ. (2.33)

Note that the main characteristics of the radiation transfer - not only in the vicinity of a gravitating object - can be described by the turning points of the investigated rays. At these points, it holds that

dr

dλ = 0. (2.34)

If the Schwarzschild metric (2.15) is considered, the equations (2.33) lead to dφ

dr = p_φ r²p_r

1

1−^2M_r . (2.35)

Further, using (2.31), one obtains p_r

(

1− 2M r

)

=±







√p²₀−

(

1− 2M r

)(

p²_φ

r² +ℏ²ω²_pe(r)

)

. (2.36)

The + sign corresponds to the photon motion in the direction of increasing r, whereas the− sign describes the opposite situation.

When the photon comes from infinity, reaches the closest approach R from the central object, and then returns back to infinity, the change of the angular coordinate φis

∆φ= 2

∫ ∞ R

p_φ r²

dr

√

p²₀−⁽1− ^2M_r ^{) (p}_r^2φ2 +ℏ²ω²_pe(r)⁾

. (2.37)

If the photon moves along a straight line, the change of the angular coordinate is

∆φ=π. Considering that, the deflection angle can be expressed as α= 2

∫ ∞ R

p_φ r²

dr

√

p²₀−⁽1−^2M_r ^{) (p}_r^2φ2 +ℏ²ω_pe² (r)⁾

−π. (2.38)

The point of the closest approach R represents a turning point, i.e., _dλ^dr⏐⏐_⏐

r=R = 0 and p_r|_r=R = 0. Combined with equation (2.36), this condition results in

p²₀ =

(

1− 2M R

)(

p²_φ

R² +ℏ²ω_pe² (R)

)

, (2.39)

wherep₀denotes the photon energy at infinity. It holds thatp₀ =−ℏω. Given the relationship between parameters R,p₀, p_φ mentioned above (equation (2.39)), it is possible to express one of them as the function of the remaining two. In Tsupko and Bisnovatyi-Kogan [2013], the variable p_φ is substituted in terms of R and p₀ as follows

p²_φ =R²p²₀

( 1

1− ^2M_R − ω²_pe(R) ω²

)

. (2.40)

When formula (2.40) is substituted into (2.38), the deflection angle of a photon coming from infinity, reaching the central object in the closest approach distance R, and then returning back to infinity is

α = 2

∫ ∞ R

dr

√r(r−2M)

√h²(r) h²(R) −1

−π, (2.41)

where

h(r) =r

√ r

r−2M − ω²_pe(r)

ω² . (2.42)

Another method how to obtain equation (2.41) was shown in the monograph by Perlick [2000]. He used the Hamilton’s equations of motion in the form

ˆ

g_ijx˙ⁱx˙^j = 1, (2.43)