Lensing by Schwarzschild Black Holes
Annika van den Brink
July 8, 2020
Studentnumber 12014877
Supervisor dr. J´acome Armas
Second Examiner dr. Alejandra Castro Anich
Course Bachelor Project Physics & Astronomy
Size 15 EC
Conducted between 30-03-2020 and 08-07-2020 Version Final version
Submission Date 08-07-2020
Faculteit der Natuurwetenschappen, Wiskunde en Informatica UvA & Faculties Faculteit der B`etawetenschappen VU
Abstracts
Popular (Dutch)
Volgens Newton werkte zwaartekracht doordat objecten krachten op elkaar uitoefenen, en deze theorie was lange tijd de standaard binnen de natuurkunde. Echter, Einstein had daar een ander idee over. In zijn relativiteitstheorie werkt zwaartekracht juist doordat de ruimtetijd gekromd is. Maar wat is ruimtetijd, en hoe is het gekromd? Ruimtetijd is, zoals de naam doet vermoeden, de combinatie van ruimte en tijd. Einstein zag in dat we deze twee niet los kunnen denken, maar dat ze samengevoegd moeten worden. Zware objecten kunnen deze ruimtetijd dan ’buigen’, zoals bijvoorbeeld een bal op een trampolinedoek dat ook doet. En wanneer we bijvoorbeeld een knikker over de trampoline laten rollen, zullen we zien dat deze wordt afgebogen rond de zware bal. Dit gebeurt volgens Einstein dus ook in de ruimtetijd; zware objecten kunnen de ruimtetijd krommen en lichtere deeltjes en licht kunnen daardoor worden afgebogen. Dit gebeurt ook met een zwart gat en licht wat daardoor wordt afgebogen. In dit project is onderzocht hoe het licht precies wordt afgebogen rond zo’n zwart gat. Rond een zwart gat bevindt zich een zogenaamde accretieschijf, die bijvoorbeeld planeten bevat. Deze schijf zendt licht uit, wat uiteindelijk ons bereikt. Ook andere lichtstralen bereiken ons. Gebaseerd op hoeveel energie een lichtstraal heeft, blijkt het dat sommige van deze lichtstralen in het zwarte gat vallen, andere er langsop vliegen, en weer andere een aantal keer om het zwarte gat cirkelen waarna ze weer ontsnappen. Al deze lichtstralen die ons uiteindelijk bereiken, bepalen hoe wij het zwarte gat zien. Het blijkt dat belangrijkste dat dit bepaalt de straling van de accretieschijf is. Deze lichtstralen hebben het grootste effect op hoe wij uiteindelijk het zwarte gat zien.
Scientific
The main focus of this thesis was to understand the bending of light rays, lensing, around a Schwarzschild black hole. In order to do so, first of all the behaviour of light rays in spacetime had to be studied. In this thesis, we studied general relativity, where gravitational effects arise because of the curvature of four-dimensional spacetime [1]. This curvature of spacetime affects the paths light rays will follow through spacetime. To understand how exactly, one has to study geodesics. Geodesics tell how light rays will move through a curved spacetime, based on the principle that the shortest path will be taken. In this thesis, Schwarzschild black holes were studied, for which the curvature of spacetime is defined by the so-called Schwarzschild geometry. Using this geometry, the geodesic equation was solved to yield results predicting and explaining light’s behaviour around Schwarzschild black holes. One important feature is the derivation of a bound circular orbit at r = 3GM . Around this radius, the so-called photon ring occurs, resulting from light rays orbiting around the black hole close to this bound circular orbit before they escape to infinity [2]. How these and other light rays around the black hole will eventually be seen by a distant observer depends on a few things, among which orientation and the properties of the emission near the black hole. In this thesis, we had a look at emission from a geometrically and optically thin disk, observed from a face-on orientation. We learned that the characteristic features of the observed image by a distant observer were dominated by those properties of the emission near the black hole. The example treated provides a good starting point. However, it also shows an idealised case and leaves room for a lot of further research on this topic. Most importantly in expanding the research to Kerr black holes, since Schwarzschild black holes are only theoretical entities. But also the emission around the black hole, and the orientation are idealised scenarios.
Contents
1 Introduction 3
2 Theory 4
2.1 Introduction to general relativity . . . 4 2.2 Geodesics . . . 5 2.3 The Schwarzschild solution . . . 7 3 Black hole shadows, photon rings, and lensing rings 10 3.1 Photon rings and lensing rings . . . 10 3.2 Negligible photon rings . . . 12 3.3 Optically and geometrically thin disk emission . . . 14
4 Discussion 15
1
Introduction
Figure 1 shows the first picture ever made of a black hole, and it was only taken last year [3]. It shows the black hole located in the galaxy called M87. A lot of research has built up to the moment this picture was taken, and a lot can again be learned from it. And apart from that, it is an impressive achievement in itself that this picture has been made. But what is the physics behind it? Why does this black hole look the way it does? What has affected its appearance and could it have looked entirely different? All of those are questions that will be investigated and answered in this thesis.
An important topic that has to be addressed first is establishing a definition of a black hole. This will later be treated more extensively, now we just touch on the subject by saying that black holes are regions of spacetime where gravity is so strong that nothing can escape from it [4]. In order to understand this definition, and as such to understand black holes, general relativity has to be studied. So, this thesis will start by giving a short introduction to general relativity. This way, we will gain an understanding of how gravity works in Einstein’s theory, and how matter behaves in spacetime as a result of this. Understanding the way matter behaves in spacetime and around objects makes it possible for one to calculate and predict the paths matter takes through spacetime.
The most important thing for understanding the picture in Figure 1 is to understand how light moves through and behaves in spacetime. The study of this is called geodesics. Geodesics are based on the principle that light (or matter generally speaking) takes the shortest path between points through spacetime. This way, the path of light can be predicted in any so-called curved spacetime. The curvature of spacetime is the starting point of general relativity, and since we can know the specific curved spacetime in the vicinity of a black hole, the path of light around it can be predicted and explained, from which it can be explained why Figure 1 looks the way it does.
In this thesis the goal is to understand how light rays bend around the horizon of a black hole, which is called lensing. The focus will be on Schwarzschild black holes, a special case of black holes that do not rotate and are not charged. In order to do so, first of all an introduction to general relativity will be provided. After this the geodesic equation will be derived and explained, after which the equation will be solved for the spacetime geometry around a Schwarzschild black hole. The next step is tracing back light rays from a distant observer to look at different regimes of light rays bending around a black hole. The concepts of a direct image, lensing ring and photon ring will be introduced here, and will be used to explain the way a black hole eventually looks to a distant observer.
2
Theory
2.1
Introduction to general relativity
In Newtonian gravity, gravitational phenomena arise from forces and fields on and between objects. This Newtonian gravitational interaction between objects is spontaneous. However, in special relativity, simul-taneity is not the same in all inertial frames, leading to the result that the Newtonian law of gravity can be true in one frame, but false in another. This is inconsistent with the principle of relativity: ’Identical experiments carried out in different inertial frames give identical results.’ [1]. This led Einstein to his theory of general relativity, where gravitational phenomena arise from the curvature of four-dimensional spacetime [1]. The key idea behind this theory is the equality of gravitational and inertial mass. This idea also gives rise to the Einstein Equivalence Principle: ’In small enough regions of spacetime, the laws of physics reduce to those of special relativity; it is impossible to detect the existence of a gravitational field by means of local experiments.’ [5], or in other words ’there is no experiment that can distinguish a uniform acceleration from a uniform gravitational field’ [1].
In general relativity, a spacetime geometry is summarised by a line element giving the spacetime distance between any two nearby points. So, when describing a general geometry, a system of four coordinates is used to label the points, written as xµ, where summation over the index µ is implied. The line element, ds2,
between nearby points separated by coordinate intervals dxµ is then given by [1]:
ds2= gµνdxµdxν, (1)
where gµν is called the metric. The time component of the metric tensor can be thought of as the equivalent
of the Newtonian gravitational potential, but the entire metric has a lot more additional uses [5]. The metric is a symmetric, position-dependent matrix. As an example, we could take a look at the Minkowski metric for flat spacetime, conventionally called ηµν:
gµν = ηµν = −1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 , (2)
leading to the line element in flat spacetime (using natural units, where c = 1, which will be used throughout the thesis):
ds2= −dt2+ dx2+ dy2+ dz2. (3) As stated, instead of arising from forces and fields, gravitational phenomena arise from the curvature of spacetime in general relativity. To quantify this, Einstein proposed his Einstein field equations:
Rµν−
1
2Rgµν+ Λgµν = 8πGTµν. (4) In this equation, a lot of variables play a role. First of all Rµν, which is the Ricci curvature tensor. This
tensor is a contraction of the Riemann tensor, which is a quantification of the curvature of four-dimensional spacetime. The Ricci tensor can be considered as a measure of the extent to which the geometry of a certain metric differs from that of Euclidian or pseudo-Euclidian spacetime [5]. R is called the Ricci scalar, and is the trace of the Ricci tensor. Λ is the cosmological constant, representing the energy density of spacetime. G is Newton’s gravitational constant, and finally Tµν is the stress-energy tensor. This tensor is the generalization
of mass density [5].
What does this equation have to do with the curvature of spacetime? It relates the curvature of spacetime to the presence of energy-momentum. The left-hand side is purely geometrical, whereas the right-hand side only contains terms dealing with momentum and energy. Thus, if we were to know the right-hand side of the equation, theoretically, we could derive gµν. This tensor then comes to play a role in the line element,
Equation 1, which will play an important role in the next topic that will be discussed; the movements of test particles and light rays through curved spacetime.
2.2
Geodesics
The fundamental principle behind the movements of test particles and light rays through curved spacetime is the following: ’The world line of a free test particle between two timelike separated points extremizes the proper time between them.’ [1]. These world lines are called geodesics, and the equations of motion that determine them are derived from the geodesic equation, which will be derived in this section. In this section, first of all the geodesic equation for massive particles will be derived, from which a natural generalization can be done as to arrive at the geodesic equation for massless particles, or light rays. The latter is also called null geodesics.
For the first step in deriving the geodesic equation for massive particles, let us have a look at the equations of motion in flat spacetime. Geodesics represent the shortest path between two points, which simply is a straight line in this case. The curve describing the movement of particles in spacetime is called a world line, and for massive particles this is a so-called timelike world line. Timelike world lines are just any world line for particles moving with a speed less than the speed of light. So, for massive particles the world line can be specified parametrically by xµ(τ ). Here the variable τ has been introduced, which is called the proper time, which is the time that would be measured by a clock carried along the world line [1]. So this variable τ measures a spacetime distance along the world line. Now the principle stated at the start of this paragraph comes into play, implying for the equations of motion in flat spacetime the following equation:
d2xµ
dτ2 = 0. (5)
Now, we’d like to generalize this to a general geodesic equation, also applicable to curved spacetime. As a starting point, we take Equation 1. Again, particles moving on this world line can be specified parametrically by xµ(τ ). Combining this with Equation 1, we know that:
ds2= gµνdxµ(τ )dxν(τ ). (6)
Rewriting Equation 6 to an equation including dτ2 and writing out the derivatives yields: ds2= gµν dxµ dτ dxν dτ dτ 2 = gµνx˙µx˙νdτ2. (7)
Now we want to look at the action S defined as: S = Z τ2 τ1 Ldτ = −m Z τ2 τ1 dτ, (8)
where L is the Lagrangian and m is the rest mass. The Lagrangian is a powerful tool in situations that require extremizing a certain value, exactly what we’d like to do. As told earlier, it is the case that we are looking at timelike events. For these events, ds2< 0, and τ yields the distance on the curve. It is therefore convenient to introduce dτ2≡ −ds2. We now want to look at the variation of the action, δS, which has to
be equal to 0 to find the minimum path. Using Equation 7 yields:
δS = −mδ Z τ2 τ1 p −ds2= −mδ Z τ2 τ1 p−gµνx˙µx˙νdτ = −m Z τ2 τ1 δp−gµνx˙µx˙νdτ = 0. (9)
Now, writing out Equation 9 further, and working out the variation inside the integral, yields:
δS = −m Z τ2 τ1 δ(−gµνx˙µx˙ν) 2p−gµνx˙µx˙ν dτ = −m 2 Z τ2 τ1 δ(−gµνx˙µx˙ν)dτ = m 2 Z τ2 τ1 δ(gµνx˙µx˙ν)dτ = m 2 Z τ2 τ1 (δgµνx˙µx˙ν+ gµνδ ˙xµx˙ν+ gµνx˙µδ ˙xν)dτ, (10)
where we have used that the derivative and the δ commute. Since µ and ν are dummy indices, the last two terms in the integral in Equation 10 are essentially the same. For the first term, the relation δgµν= ∂αgµνδxα
can be used, yielding:
δS = m 2 Z τ2 τ1 ((∂αgµν)δxαx˙µx˙ν+ 2gµνδ ˙xµx˙ν) dτ = m 2 Z τ2 τ1 (∂αgµν)δxαx˙µx˙ν+ d dτ (2gµνδx µx˙ν) − 2 ˙g µνδxµx˙ν− 2gµνδxµx¨ν dτ, (11) where we have rewritten the second term of the integral. The second term of Equation 11 will give us 0, so this term vanishes. This is because, when evaluating the integral of this term over dτ , the end points τ1 and
τ2 won’t vary. Whatever path the particle follows through spacetime, those end points won’t vary, so when
we substitute them after the integral has been calculated, the variation will yield 0. The other terms can be rewritten, and some labels can be swapped as to yield a result that will prove useful for finding the geodesic equation: δS = m 2 Z τ2 τ1 (∂αgµν)δxαx˙µx˙ν− 2(∂βgµν) ˙xβδxµx˙ν− 2gµνδxµx¨ν dτ =m 2 Z τ2 τ1 ((∂µgαν)δxµx˙αx˙ν− (∂αgνµ) ˙xαδxµx˙ν− (∂νgµα) ˙xνδxµx˙α− 2gµνδxµx¨ν) dτ. (12)
Now the terms in the integral can be collected as to provide a convenient form to derive the geodesic equation from: δS = m Z τ2 τ1 1 2(∂µgαν− ∂αgνµ− ∂νgµα) ˙x νδxµx˙α− g µνδxµx¨ν dτ = m Z τ2 τ1 1 2(∂µgαν− ∂αgνµ− ∂νgµα) ˙x νx˙α− g µνx¨ν δxµdτ = 0, (13)
where it has been stated again, as was done in Equation 9, that δS has to be 0 in order to find the minimum path. This has to be the case for any value of δxµ, not just the case when δxµ equals 0. So this means from
Equation 13 that the whole term inside the brackets has to be 0, meaning: 1
2(∂µgαν− ∂αgνµ− ∂νgµα) ˙x
νx˙α
− gµνx¨ν= 0. (14)
The first term in Equation 14 resembles the Christoffel symbols, an array of numbers describing a metric connection, ’which gives us a way of relating vectors in the tangent spaces of nearby points’ [5]. The Christoffel symbols are formally given by:
Γµν= 1 2g
λσ(∂
µgνσ+ ∂νgσµ− ∂σgµν). (15)
Equation 14 can be rewritten into a form containing the Christoffel symbols, at which point the geodesic equation will be reached:
gµνx¨ν+ 1 2(−∂µgαν+ ∂αgνµ+ ∂νgµα) ˙x νx˙α= 0 gµγgµνx¨ν+ gµγ 2 (−∂µgαν+ ∂αgνµ+ ∂νgµα) ˙x ν ˙ xα= 0 δνγx¨ν+g µγ 2 (−∂µgαν+ ∂αgνµ+ ∂νgµα) ˙x νx˙α= 0 ¨ xγ+g µγ 2 (∂αgνµ+ ∂νgµα− ∂µgαν) ˙x νx˙α= 0. (16)
From Equation 16, the Christoffel symbols from Equation 15 can be recognised and combining them will yield the definitive version of the geodesic equation:
d2xγ dτ2 = −Γ γ να dxν dτ dxα dτ . (17)
Equation 17 works for massive particles, however, it does not work for massless particles, so for light. This has to do with the world line light moves on. As told before, massive particles all move on timelike world lines, light however moves on a null world line. For null world lines, it is the case that the line element ds2 = 0, so it does not have proper time. Deriving a geodesic equation for light would go similarly. For
massless particles, it is also possible to define the action, work out the variation of the action and require that it is equal to 0. We could also make a natural generalization, yielding the same result as working out the action for massless particles [1]:
d2xγ dλ2 = −Γ γ να dxν dλ dxα dλ . (18)
The only change in Equation 18 with regards to Equation 17 is the parameter λ replacing the proper time τ . This affine parameter λ does not measure a spacetime distance, as τ did, but it is a parameter chosen such that Equation 18 takes the form of the geodesic equation [1]. Null geodesics, Equation 18, will be the geodesics we are interested in when studying how light rays bend around a Schwarzschild black hole.
2.3
The Schwarzschild solution
As stated, the derived null geodesics in the previous section describe the motion of light in curved spacetime. In this thesis, where the path of light around a black hole will be researched, Equation 18 is the equation needed. But before this question will be answered, the black hole has to be clarified. Because how can we understand a geometry, and in our case the Schwarzschild geometry, as a black hole? In the introduction, it was stated that black holes are regions of spacetime where gravity is so strong that nothing can escape from it [4]. More precisely, this is because ’the surfaces defining black holes are null surfaces with this one-way property: you can fall through one but you can never get back out’ [1], where a null surface is a surface generated by light rays. In four dimensions and for Einstein-Maxwell theory, a black hole is described entirely by only three parameters: mass M, spin J and charge Q, which is known as the no-hair theorem [6]. A spinless and chargeless black hole is called a Schwarzschild black hole and a black hole with mass and spin is called a Kerr black hole. In this thesis, the focus will be on Schwarzschild black holes. The spacetime around such a black hole is expected to be spherically symmetric, and at infinity it will resemble flat spacetime. This geometry is called the Schwarzschild geometry, and the line element summarizing it is given by (in natural units and listed in order (t, r, θ, φ) as will be the case in this thesis) [1]:
ds2= − 1 − 2GM r dt2+ 1 −2GM r −1 dr2+ r2 dθ2+ sin2θdφ2 = − 1 − Rs r dt2+ 1 −Rs r −1 dr2+ r2 dθ2+ sin2θdφ2 , (19) where we have defined the Schwarzschild radius Rs= 2GM . This implies for the metric:
gµν = − 1 −Rs r 0 0 0 0 − 1 − Rs r −1 0 0 0 0 r2 0 0 0 0 r2sin2θ . (20)
Knowing the metric for the Schwarzschild geometry, this can be used to study the geodesics of light rays in such a geometry, and thus around a Schwarzschild black hole. Since the metric in Equation 20 is time-independent and spherically symmetric, the laws of conservation of energy and angular momentum hold. These two conserved quantities will turn out to play an important role in the geodesics in the Schwarzschild geometry. To use this quantitatively, first the concept of a Killing vector has to be introduced: ’a Killing
vector is a general way of characterizing symmetry in any coordinate system’ [1]. So the Schwarzschild metric being time-independent means that there is a Killing vector ξ, such that ξµ = (1, 0, 0, 0) [1]. And, parallel to this, the Schwarzschild metric being spherically symmetric means that there is a Killing vector η, such that ηµ = (0, 0, 0, 1). This Killing vector η tells us that angular momentum is invariant under spatial rotations. Angular momentum has one magnitude component, and two components for its direction, so since angular momentum is conserved, this means our particle or light ray moves in a plane. This means that for a single particle we can choose uθ= 0 and θ =π
2. Using the aforementioned two Killing vectors, we can now
respectively define the conserved energy per unit rest mass E and the conserved angular momentum per unit rest mass L [1]: E = −ξ · u = 1 −2GM r dt dτ, (21) and L = η · u = r2sin2θdφ dτ, (22)
where u is the four-velocity of the test particle, defined by uα = dxdτα. Furthermore, the geodesic equation implies that the quantity = u · u is constant along the path:
= u · u = gµνuµuν. (23)
Equation 23 can be written out using the Schwarzschild metric from Equation 20, and by using the earlier mentioned condition for θ. This can be executed for massive particles as well as light rays, and in the following both will be done. In the former case, Equation 23 will yield -1, whereas in the latter case it will yield 0. So, first of all for the case of massive particles, this results in:
− 1 − 2GM r dt dτ 2 + 1 −2GM r −1 dr dτ 2 + r2 dφ dτ 2 = −1. (24)
Now Equations 21 and 22 come into play, by substituting them into Equation 24:
− 1 − 2GM r −1 E2+ 1 − 2GM r −1 dr dτ 2 +L 2 r2 = −1. (25)
This can be further rewritten into the final form: E2− 1 2 = 1 2 dr dτ 2 +1 2 1 −2GM r 1 + L 2 r2 − 1 . (26)
Now the constant E can be defined:
E = 1 2 dr dτ 2 + Vef f(r), (27)
where E = E22−1 and we have defined the effective potential for radial motion of particles:
Vef f(r) = 1 2 1 − 2GM r 1 + L 2 r2 − 1 = −GM r + L2 2r2 − GM L2 r3 . (28)
What can be seen from Equation 28 is that orbits in a Schwarzschild metric can be treated in the same way as in Newtonian mechanics. The only difference with a Newtonian central potential is the last additional term in Equation 28. This additional term has a negligible contribution for large values of r, but becomes increasingly more important as r decreases. From Equation 28 two stable extremal orbits can be found: a
local minimum and a local maximum stable orbit. These two orbits can be found by solving dVef f
dr = 0, which
gives us:
GM r2c− L2rc+ 3GM L2= 0, (29)
where rcis the critical radius, representing the radii where circular orbits occur. There is one local minimum
and one local maximum for a certain value of L and solving Equation 29 for rc leads to the following equation
for those: rc= L2 2GM 1 ± s 1 − 12 M L2 2 . (30)
From Equation 30, it can be seen that the innermost stable circular orbit occurs at L =√12GM , for which:
rc= 6GM. (31)
Now the geodesic equation for light will be solved. In this case, Equation 23 yields 0, so writing this out leads to the following equation:
− 1 − 2GM r dt dλ 2 + 1 −2GM r −1 dr dλ 2 + r2 dφ dλ 2 = 0. (32)
Again, making use of the conserved quantities in Equations 21 and 22 (where every τ has been replaced by a λ) leads to the following:
− 1 −2GM r −1 E2+ 1 − 2GM r −1 dr dλ 2 +L 2 r2 = 0 (33) Multiplying this by 1− 2GM r
L2 leads to our final form for null geodesics:
E2 L2 = 1 L2 dr dλ 2 + 1 r2 1 −2GM r 1 b2 = 1 L2 dr dλ 2 + Wef f(r), (34)
where we have defined a new parameter b ≡ EL. Since Equation 34 has the form of an energy formula, the effective potential for light rays could also be concluded from it, namely:
Wef f(r) = 1 r2 1 −2GM r . (35)
The parameter b can also be called the impact parameter, and will be encountered again later in this thesis. It also has an important reason as to why it was introduced, which has to do with the affine parameter λ in null geodesics. Suppose we would multiply λ by some constant, then it would still be as good an affine parameter as it was before. This is because the value of will still be 0, and the geodesic equation, Equation 18, also won’t change. However, both the individual values of L and E do change. ’Therefore, only the ratio
L
E has physical significance and determines the properties of light ray orbits.’ [1].
As was executed for massive particles, for light rays there also exists a bound circular orbit, which can be found by taking the derivative of Equation 35 with respect to r and setting this equal to 0. This yields for rc for light rays:
rc= 3GM. (36)
This is an unstable circular orbit, and only massless particles can stay in this orbit. ’Photons can orbit forever in a circle at this radius, but any perturbation will cause it to fly away either to r = 0 or r = ∞’ [5]. It is interesting to now have a look at a more qualitative view of the effective potential for radial motion
of massless particles. This can be seen in Figure 2 [1]. This figure shows three kinds of light ray orbits in the Schwarzschild geometry. Three cases with different orbits corresponding to different values of b are distinguished. The left shows the effective potential plotted against GMr , this way showing its relationship to
1
b2. The right shows the shapes of the orbits in the x, y−plane. As you can see, at r = 2GM , the potential is
always zero. This is because the event horizon of the black hole is at that radius, so inside this radius is the black hole. For light rays coming from infinity, so coming from a large value of r, there is always a barrier. But a sufficiently energetic photon will go over this barrier and be dragged down to the center, which can be seen in the bottom case. Note that sufficiently energetic means in comparison to its angular momentum. If the photon is not sufficiently energetic, then the orbit will have a turning point and again escape to r = ∞, as is shown in the second case. The top case shows the case at the top of the barrier, where the previously derived bound circular orbit can be found [5].
It has now been shown that, in a Schwarzschild metric, there are two conserved quantities: energy and angular momentum, and using this makes it possible to calculate circular orbits in such a metric, and in our case, around a Schwarzschild black hole. We know that the stable circular orbits for freely falling test particles are given by Equation 30, with the innermost stable orbit being given by Equation 31. For photons, there is one, unstable, circular orbit, given by Equation 36. This value for rc will prove to be mostly important in
the rest of this thesis, where we will look at the paths of light around a Schwarzschild black hole.
3
Black hole shadows, photon rings, and lensing rings
3.1
Photon rings and lensing rings
It is now interesting to look at applications of the previously developed and explained theory. We will look at how light bends near a Schwarzschild black hole, and what will be seen as a result of it by a distant observer. This will all be done by studying the paper ”Black Hole Shadows, Photon Rings, and Lensing Rings” by S.E. Gralla, D.E. Holz and R.M. Wald [2]. An important parameter in this examination is the aforementioned impact parameter b, which is given by:
b = L E = r q 1 −2GMr , (37)
where L and E are the parameters from respectively Equation 22 and 21 [7] [8]. From Equation 36, it followed that the bound orbits of light occur at rc = 3GM . From this, it follows that the critical curve is
b = 3√3GM ≈ 5.2GM .
A few definitions are now first to be made clear. First of all, a black hole shadow. ’The term ”black hole shadow” has come to represent the interior of the critical curve. The model problem where this region corresponds to some kind of ”shadow” is when the black hole is illuminated by a distant, uniform, isotropically emitting spherical screen surrounding the black hole (and the observer is far away from the black hole, but within the radius of the screen.’ [2]. Secondly, the photon ring: ’[t]he photon ring is a region of enhanced brightness near the critical curve that arises if optically thin matter emits from the region where unstable bound photon orbits exist. The light rays that comprise the photon ring can orbit many times through the emission region and thereby ”pick up” extra brightness.’ [2]. A distinction has now to be made between a photon ring and a lensing ring. ’We define the lensing ring to consist of light rays that intersect the plane of the disk twice outside the horizon, and we define the photon ring to consist of light rays that intersect three or more times.’ [2]. This last difference can be best explained by looking at Figure 3. The figure on the left shows the fractional number of orbits n = 2πφ, where φ is the total change in (orbit plane) azimuthal angle outside the horizon. The dashed line represents an approximation, where the thick line represents the exact solution. The colours correspond to n < 0.75 for the black line representing direct trajectories; 0.75 < n < 1.25 for the gold line representing lensed trajectories and n > 1.25 for the red line representing photon ring trajectories. In the right figure, it is shown what this means eventually for the photon paths. Here, the solid disk represents the black hole, and the dashed line the photon orbit at r = 3GM . The distant observer is placed at the right of this figure. The colours represent again direct, lensed and photon ring trajectories, and the spacing in impact parameter is 1/10, 1/100 and 1/1000 respectively. These three distinct kinds of light rays can, along with the impact parameter, be placed in one overview as follows [2]:
Figure 2: Three kinds of light ray orbits in the Schwarzschild geometry; three cases with different orbits corresponding to different values of b. The left shows the effective potential plotted against GMr , this way showing its relationship to b12. The right shows the shapes of the orbits in the x, y−plane. From the top
Figure 3: The left figure shows the fractional number of orbits n = 2πφ, where φ is the total change in (orbit plane) azimuthal angle outside the horizon. The dashed line represents an approximation, where the thick line represents the exact solution. The colours correspond to n < 0.75, the black line representing direct trajectories; 0.75 < n < 1.25, the gold line representing lensed trajectories and n > 1.25, the red line representing photon ring trajectories. The right figure shows a selection of associated photon trajectories, treating r, φ as Euclidean polar coordinates. The solid disk represents the black hole, and the dashed line the photon orbit at r = 3GM . The spacing in impact parameter is 1/10, 1/100 and 1/1000 in the direct, lensed, and photon ring bands, respectively [2].
1. Direct: n < 3/4 b/M /∈ (5.02, 6.17) 2. Lensed: 3/4 < n < 5/4 b/M ∈ (5.02, 5.19) or (5.23, 6.17) 3. Photon ring: n > 5/4 b/M ∈ (5.19, 5.23)
3.2
Negligible photon rings
In the following, three different cases of emission near a black hole will be treated. But they all have one thing in common: the effect of the photon ring on the total observed intensity is negligible. This is because of the slow, logarithmic increase of the path length with the distance from the critical impact parameter b [2]. This relation will now be derived, by following the paper [2]. We start off by recalling that null geodesics satisfy: dr ds = ± q Vef f(r) Vef f = 1 − b2 r2 1 −2GM r , (38)
where r is the Schwarzschild coordinate. Equation 38 can be derived be rewriting the effective potential for the radial motion of light rays from Equation 35. The photon orbit is known to be at rc = 3GM , with
b = 3√3GM ≈ 5.2GM . We will now look at geodesics slightly away from rc, which will give photon rings.
rc= 3GM (1 + δr) b = 3
√
3GM (1 + δb). (39) Equation 39 can be plugged in Equation 38, and expanding this leads to an approximation for Vef f:
Vef f ≈ 3δr2− 2δb. (40)
Whenever Vef f from Equation 40 equals 0, so-called turning points occur, where the geodesics change
di-rection. These turning points occur for δr = ±q2
3δb. When δb < 0, this means the geodesic leads into the
black hole, so there are no turning points. For δb > 0 only the outermost turning point is of interest, so the turning point can be concluded to be:
δrturn=
r 2
3δb. (41)
We now want to look at a small region near the photon orbit defined by
δR < δr < δR, (42)
for some positive δR 1, and we ask how much distance is being covered in this region. From Equation 38, it is known that ds = √dr
Vef f
. Solving this will yield the path length ∆s. This can be done for both cases; as well for δb < 0 as δb > 0. First of all, δb < 0:
∆s = 3GM Z δR δR dδr √ 3δr2− 2δb =√3GM log 3 −2δb δR + r δR2−2 3δb !2 . (43)
Secondly, δb > 0, where we use Equation 41 as a boundary:
∆s = 2 · 3GM Z δR δrturn dδr √ 3δr2− 2δb =√3GM log 3 2δb δR + r δR2−2 3δb !2 . (44)
Lastly, in the case of δb → 0, it can easily be seen that the path length diverges. This can also be understood physically, since δb → 0 means the photons will start orbiting at rc = 3GM , where they will stay forever,
since this is a bound orbit. Now one last approximation can be done for looking at distances close to the turning point, which are distances of our interest. Mathematically, this means we will look at the case where |δb| 3
2δR
2. This value can be plugged in both Equation 43 and 44, yielding:
∆s ≈√3GM log 6δR
2
|δb|
. (45)
Lastly, the time ∆t and angle ∆φ covered when circling around rc = 3GM can be estimated. From the
definitions of E and b we know that:
dt ds= 1 1 −2GM r ≈ 3, (46) and dφ dt = b r2 ≈ 1 √ 3GM. (47)
Using Equation 45 yields now for ∆t and ∆φ: ∆t ≈ 3√3∆φ (48) ∆φ ≈ log 6δR 2 |δb| . (49)
3.3
Optically and geometrically thin disk emission
The image of the black hole that a distant observer will see is dependent on a lot of parameters, among which orientation and the kind of emission near the black hole. In this thesis, it will be assumed that the orientation is face-on. With regards to the emission, the paper ”Black Hole Shadows, Photon Rings, and Lensing Rings” distinguishes three distinct cases: a backlit black hole, optically and geometrically thin disk emission, and geometrically thick emission. In this thesis, the focus will be on the second case, where the emission originates from optically and geometrically thin disk emission whose specific intensity Iν depends
only on the radial coordinate. So, taking the disk to lie in the equatorial plane, the emitted intensity can be defined as Iem
ν = I(r), where ν is the emission frequency in a static frame. The observed intensity at a
frequency ν0 can be specified, using the fact that Iν
ν3 is conserved along a ray as follows:
Iνobs0 = g3I(r), (50)
where g is defined as g =q1 −2GM
r . So the total intensity integrated scales then as:
Iobs= g4I(r). (51)
’If a light ray followed backward from the observer intersects the disk, it will pick up brightness from the disk emission. But if the light ray has an impact b such that, in the notation of Figure 3, we have n > 3/4, then the light ray will bend around the black hole and hit the opposite side of the disk from the back. It will therefore pick up additional brightness from this second passage through the disk.’ [2]. This is the case for every extra hit with the disk, so that in the end the total observed intensity is the summed total of the intensities from each intersection with the disk:
Iobs(b) =X
m
g4I|r=rm(b), (52)
where rm(b) is the radial coordinate of the mth intersection with the disk plane outside the horizon [2].
The functions rm(b)(m = 1, 2, 3, ...) will be referred to as transfer functions, which directly show where on
the disk a light ray of impact parameter b will hit. The first transfer function (m = 1) corresponds to the direct image of the accretion disk, corresponding to the black lines in Figure 3. The second transfer function (m = 2) corresponds to the lensing ring, corresponding to the yellow lines in Figure 3. Finally, the third transfer function (m = 3) corresponds to the photon ring, corresponding to the red lines in Figure 3. The photon ring, however, has a negligible contribution to the total flux, as will later be seen in Figure 4, and has been derived mathematically in the previous section. This mathematical derivation can be even more elaborated, however [2]. First of all, the edges of the mthimage are defined as b±m, so that for the mthimage
we have b ∈ (b−m, b+m), where m = 1, 2, 3+ corresponds to the direct emission, lensing ring and photon ring
respectively. From Equation 49, it can be derived that:
∆b ∼ Ce−φ, (53)
where ∆b is defined as |b − bc|, and C is some constant. Equation 53 applies in the case where b → b±c, thus
the width ∆bm= b+m− b−mis exponentially suppressed:
∆bm≈ e−π∆bm−1, (54)
for m → ∞ [2]. So this means that, the more intercepts with the disk, the brighter a light ray becomes. However, every time another intersection occurs, the number of light rays for which this happens, is expo-nentially suppressed. So the intensity may be higher for light rays intersecting the disk more times, but since
the number of light rays for which this is the case gets low really fast, the total contribution to the flux will get low really fast too. This is the reason why the photon ring, for which m ≥ 3, has a negligible contribution to the total observed intensity, as will be seen when Figure 4 will now be discussed.
Figure 4 shows the observational appearance of a geometrically and optically thin disk of emission near a Schwarzschild black hole, viewed from a face-on orientation. As explained, only the images for which m = 1, 2 or 3 are included in this plot. The emitted and observed intensities Iem and Iobsare normalized to
the maximum value I0 of the emitted intensity outside the horizon. The left column shows three different
emission profiles of the disk near the black hole. The middle column shows the resulting emission profiles that will be seen by a distant observer, and the right column shows the resulting image of that emission profile [2]. Let us now have a look at the three different cases. First of all, the upper case. In the first case, the emission profile is sharply peaked near r = 6GM . The observed emission profile of the direct emission looks a lot like this emission profile, however, the peak occurs at around r = 7GM . This is because of gravitational lensing, which is an effect in general relativity that bends the light away from the source as it travels to a distant observer [9]. As one can see, the lensing ring shows apart from the direct emission, at its radius near b ∼ 5.5GM . It is a very small ring compared to the direct emission, so it can be seen the lensing ring already has a small contribution to the total flux. The photon ring however, has a negligible contribution as has been derived. At first sight, the photon ring is not visible, but when zoomed in the ring can be spotted near r = 5.2GM . This can be seen in Figure 5, where the photon ring can be seen inside the lensing ring.
Secondly, we will have a look at the middle case. This shows an emission profile peaking at the bound circular photon orbit at r = 3GM . The observed emission profile shows that the direct image has been moved a little bit to a higher radius again, but more importantly, gravitational redshift has noticeably decreased the observed flux. This is the effect that wavelengths of photons will be increased when deeper in a gravitational well. Furthermore, an important difference with the first case is that the lensing ring and photon ring are now superimposed on the direct image. In the middle column, this can be seen as the spikes on the direct emission profile. This plot also shows that the effect of the lensing ring on the total intensity is, again, small and that of the photon ring negligible. In the image that will eventually be seen by a distant observer, the lensing ring can be seen as the bright ring. The direct emission is represented by the surrounding observed light. The photon ring can again not be seen. Lastly, the bottom row shows an emission profile that starts with the highest intensity at the black hole horizon at r = 2GM and decreases slowly all the way to r = 6GM . The observed emission profile and image look a lot like those of the second case. The lensing ring and photon ring are again superimposed on the direct image, and gravitational lensing and gravitational redshift have a noticeable contribution to the observed emission profile. In the observed image, the bright ring is the lensing ring, and the light that can be seen surrounding it originates from the direct emission. The photon ring has again a negligible contribution.
Two key points can be learned from these plots [2]. First of all, the emission is dominated by the direct emission. In every image the direct emission has the greatest contribution to the total flux observed, contrasted with that of the lensing ring and the, negligible, photon ring. Secondly, the size of the dark central area is very much dependent on the emission model. When the emission stops at some inner edge, as is the case for the upper two rows, the radius of the main dark hole is the apparent position of those edges. For the first case, this would be at around r ∼ 7GM and for the second case at somewhat smaller than r ∼ 4GM . When the emission extends to the horizon, the radius of the main dark hole is the apparent position of the horizon. This is the case in the bottom row, where the radius of the black hole seems to be r ∼ 3GM . The black holes in these three cases all have the same size, however, the size we observe them to have differs vastly depending on the emission profile. The black hole shadow, corresponding to the critical curve of b ≈ 5.2GM plays no role in determining the size of the main dark area [2].
4
Discussion
In this thesis it has been examined what makes that we see a black hole the way we see it. By what circumstances are light rays affected as to create for example the picture in Figure 1. One example was thoroughly examined: a Schwarzschild black hole with optically thin and geometrically thin disk emission viewed from a face-on orientation. This example provides a good starting point when applying the derived behaviour of light in spacetime to real world situations. However, it is also a rather limited example, and this accounts for all three aspects of it.
Figure 4: Observational appearance of a geometrically and optically thin disk of emission near a Schwarzschild black hole, viewed from a face-on orientation. Only the first, second and third intersections with the disk plane are included in this plot. The emitted and observed intensities Iem and Iobsare normalized to the maximum
value I0 of the emitted intensity outside the horizon. The left column shows three different emission profiles
of the disk near the black hole. The middle column shows the resulting emission profiles that will be seen by a distant observer, and the right column shows the resulting image of that emission profile. Two key points can be learned from these plots. First of all, the emission is dominated by the direct emission. In every image the direct emission has the greatest contribution to the total flux observed, contrasted with the lensing ring and the, negligible, photon ring. Secondly, the size of the dark central area is very much dependent on the emission model. When the emission stops at some inner edge, as is the case for the upper two rows, the radius of the main dark hole is the apparent position of those edges. When the emission extends to the horizon, the radius of the main dark hole is the apparent position of the horizon. The black hole shadow, corresponding to the critical curve of b ≈ 5.2GM plays no role in determining the size of the main dark area [2].
Figure 5: A close up view of the upper right plot from Figure 4. This shows the image of the black hole as seen by a distant observer in the case of an emission profile peaking at r = 6GM . The direct image is clearly visible at the outer edge of the image, and the lensing ring can be seen within the direct image. The photon ring can be spotted within the lensing ring, when one zooms in, which has been done here. As can be seen, the lensing ring has a small contribution to the total flux as compared to the direct image, and the photon ring even has a negligible contribution.
First of all, in this thesis, we only had a look at a Schwarzschild black hole. However, all astrophysical objects rotate, and this also accounts for black holes. The Schwarzschild black hole is just a theoretical concept and highly unlikely to be found in nature. A rotating black hole, which is, as already mentioned, called a Kerr black hole, on the other hand, will be found in nature. It would therefore be really interesting to investigate Kerr black holes in a future research in the same way as was done regarding Schwarzschild black holes in this thesis. To do this, one would have to solve the geodesic equation for the spacetime surrounding a Kerr black hole, which is described by the Kerr metric. Would one also want to take charge into account, the Kerr-Newman metric would be needed. Using one of those metrics, the research can be carried out similarly as was done in this thesis. However, apart from this exact solution, we could also have a more qualitative look at the case of a Kerr black hole. This way we can touch on the subject and find resemblances as well as differences between the Schwarzschild and the Kerr case. As the paper ”Black Hole Shadows, Photon Rings, and Lensing Rings” has done exactly this, we will follow them in giving a qualitative overview of the Kerr black hole. Their (and our) main conclusions for a Schwarzschild black hole were, first of all, that the photon ring has a negligible contribution to the total observed flux and, secondly, that there is a visible lensing ring in the case of optically thin emission [2]. The negligible photon ring contribution followed from the fact that the path length diverged, which has been derived in this thesis. In the Kerr case, they derive that ’the affine path length diverges at most logarithmically at any point near the critical curve of the Kerr black hole. This shows that successive images are at least exponentially demagnified, making the photon ring negligible.’ [2]. So, for a Kerr black hole, the photon ring will also have a negligible contribution to the observed flux. For the second main conclusion, the face-on orientation also comes into play. Would we consider an emission disk lying in the plane orthogonal to the spin axis, and would we still view this face-on, the propagation of the photons isn’t qualitatively affected in the radial direction, keeping the typical properties of the lensing ring similar to the Schwarzschild case [2]. However, would we not view the black hole face-on, but nearly edge-on or something in between, the photons would behave rather differently in the Kerr case than in the Schwarzschild case [2].
However, although this is a likely situation, there could also be other kinds of emission disks. The paper discusses a backlit black hole, which is of no physical interest, but it also considers optically thin, but geometrically thick disk emission [2]. And not only could this factor vary for real world black holes, also the considered emission profiles in Figure 4 are highly idealised, and so are the eventual intensity profiles as seen by a distant observer in this figure.
Lastly, this topic has already come across in this section, but viewing the black hole from a face-on orientation is also an idealised scenario. In the case of an inclination, we could still define the photon ring and the lensing in the same way as before, however, the lensing ring would not take the shape of a ring as it does in a face-on orientation (neither does the photon ring, but its effect is negligible in any case) [2]. This is because the backside image of the far side of the disk would make a large contribution, as compared to the direct image. The brightness along the ring will also vary due to Doppler-shifts [2]. If the inclination were to be nearly face-on, the results would be qualitatively the same, but in a nearly edge-on case, differences would be visible [2]. An inclined orientation is also a more realistic case and therefore worth investigating in the future.
5
Conclusion
This thesis aimed to understand the bending of light rays, lensing, around a Schwarzschild black hole, as to understand why black holes look the way they do to distant observers. A key concept in understanding this is geodesics. Geodesics tell how light rays will move through a curved spacetime, based on the principle that the shortest path will be taken. The use of ’curved spacetime’ implies that we are looking at general relativity, where gravitational effects arise because of the curvature of four-dimensional spacetime [1]. The geodesic equation for light reads: d2xγ
dλ2 = −Γ γ ναdx ν dλ dxα
dλ, where λ is an affine parameter parametrising the null
world line. From this, the effective potential for radial motion of massless particles can be derived, being Wef f(r) =r12 1 −
2GM
r . From this equation, a lot about the movements of light rays can be predicted and
explained. One important feature is the derivation of a bound circular orbit at r = 3GM . From this, the existence of lensing rings and photon rings can be explained, where the lensing ring is defined as light rays that intersect the plane of the disk twice outside the horizon, and the photon ring as light rays that intersect three or more times [2]. By using this same source, in this thesis we had a look at three simplified cases of emission from a geometrically and optically thin accretion disk surrounding the Schwarzschild black hole, as observed by a distant observer from a face-on orientation. From this, two key points could be learned: firstly, that the emission is dominated by the direct emission, and secondly, that the size of the dark central area is very much dependent on the emission model [2].
All in all, this thesis provided a good starting point at examining how light behaves around black holes and explaining why we see black holes the way we see them. However, the example treated shows an idealised case and leaves room for a lot of further research on this topic. First of all, the research should be expanded to Kerr black holes, since Schwarzschild black holes are only theoretical entities. Secondly, the case of optically and geometrically thin disk emission is just one possibility, and also the three emission profiles in Figure 4 are highly idealised. A face-on orientation is the last factor treating black holes in a simplified way. Would a distant observer have, for example, a nearly edge-on orientation, this would change the results because of several effects.
References
[1] J. B. Hartle, Gravity: An Introduction to Einstein’s General Relativity, ch. 3, 6 - 9. Pearson Education Limited, 2014.
[2] S. E. Gralla, D. E. Holz, et al., “Black Hole Shadows, Photon Rings, and Lensing Rings,” Physical Review D, vol. 100, no. 2, 2019.
[3] K. Akiyama, A. Alberdi, et al., “First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,” The Astrophysical Journal Letters, vol. 875, 2019.
[5] S. Carroll, Spacetime and Geometry: An Introduction to General Relativity, ch. 2 - 5. Addison Wesley, 2004.
[6] J. D. Bekenstein, “Novel ”No-Scalar-Hair” Theorem for Black Holes,” Physical Review D, vol. 51, pp. 6608–6611, 1995.
[7] J.-P. Luminet, “Image of a Spherical Black Hole with Thin Accretion Disk,” Astronomy and Astrophysics, vol. 75, pp. 228–235, 1979.
[8] D. E. Holz and J. A. Wheeler, “Retro-MACHOs: π in the Sky?,” Astrophysical Journal, vol. 578, no. 1, pp. 330–334, 2002.
