Möbius Transformations

Möbius Transformations Bachelor Thesis Author: Luis González-Miret Zaragoza Supervisor: Dr. Juan Carlos García Vázquez Faculty of Physics University of Seville 2019 Abstract In this thesis the Möbius transformations are studied along with two of its applications. Firstly, Möbius transformations are characterized as transformations of the extended complex plane to itself M:C∞7→ C∞. Some of their geometric properties, as preservation of generalized circles and angles are studied. Their fixed points are then reviewed, and with the help of the concept of cross-ratio, a way to find Möbius transformations with specific behaviour is found. Lastly, the set of all Möbius transformations Mis studied as a group isomorphic to PSL(2,C), and a classification into four categories is made. In the second part, their role in the Poincaré half-plane model of hyperbolic geometry is studied. Such model is briefly introduced, defining the concepts of hyperbolic length and distance in the half-plane, as well as lines and geodesics. Using the geometric properties of Möbius transformations, the lines of the model are derived. Lastly, the connection of the Möbius group and the Lorentz transformations is reviewed. The foundamentals of special relativity are briefly introduced. Then, the isomorphism between the Möbius group and a subset of the Lorentz gorup, the restricted Lorentz group, is fully reviewed. This will allow the study of transformations of the celestial sphere using Möbius transformations. I A todos los que me han aguantado estos cuatro años. Litany Against Fear I must not fear. Fear is the mind-killer. Fear is the little-death that brings total obliteration. I will face my fear. I will permit it to pass over me and through me. And when it has gone past I will turn the inner eye to see its path. Where the fear has gone there will be nothing. Only I will remain. Frank Herbert Dune Luis González-Miret Zaragoza IV Contents 0 Introduction 1 1 Meeting Möbius 3 1.1 Definition ............................... 3 1.2 TheRiemannsphere ......................... 3 1.3 Basicresults.............................. 7 1.4 Fixed points and the cross-ratio . . . . . . . . . . . . . . . . . . . 10 1.5 Möbius transformations as matrices . . . . . . . . . . . . . . . . . 14 1.6 Classification ............................. 17 1.6.1 Hyperbolic transformations . . . . . . . . . . . . . . . . . 17 1.6.2 Elliptic transformations . . . . . . . . . . . . . . . . . . . 18 1.6.3 Loxodromic transformations . . . . . . . . . . . . . . . . . 18 1.6.4 Parabolic transformations. . . . . . . . . . . . . . . . . . . 19 2 Poincaré half-plane 23 2.1 Brief introduction to hyperbolic geometry . . . . . . . . . . . . . 23 2.2 The Poincaré half-plane model . . . . . . . . . . . . . . . . . . . . 23 2.3 Geodesics and hyperbolic lines . . . . . . . . . . . . . . . . . . . . 25 2.4 Isometries of the Half-plane . . . . . . . . . . . . . . . . . . . . . 26 2.5 Derivationoflines........................... 29 3 Möbius as Lorentz 31 3.1 Framework............................... 31 3.2 Minkowsky space-time and Lorentz transformations . . . . . . . . 32 3.3 Lightcones .............................. 34 3.4 Classification of Lorentz transformations . . . . . . . . . . . . . . 34 3.5 Null directions and the celestial sphere . . . . . . . . . . . . . . . 37 3.6 Equivalence of Möbius and Lorentz transformations . . . . . . . . 38 3.7 Transforming the celestial sphere . . . . . . . . . . . . . . . . . . 43 V Chapter 0 Introduction Möbius transformations are a set of transformations of the extended complex plane into itself M:C∞7→ C∞with a series of beautiful properties and deep connections to both non-euclidean geometry and physics. These properties are in stark contrast with their simple appearance. The porpoise of this thesis is to give a general overview of the Möbius transformations, as well as a review of two of their applications.The thesis, then, is naturally divided in three sections: the first one covering their general properties and the other two reviewing their application in the Poincaré half-plane model of hyperbolic geometry and in the special theory of relativity. The first part is mostly based on the third chapter of Visual complex analysis [5], a magnificent manual covering extensively the Möbius transformation. In this section we define Möbius transformations acting on the extended complex plane, which is introduced through the concept of the Riemann sphere, adding another point, the infinity ∞, to the complex plane. After this, Möbius transformations are proven to leave angles and circles invariant, properties that will be key in the next sections. Studying them as a group, we will prove the isomorphism between them and PSL(2,C)and then we will classify them into four distinct categories. In the second section we find our first example of the applicability of the Möbius transformations. Here we review the basics of the Poincaré half-plane model of hyperbolic geometry and the key role the Möbius transformations in the derivation of its lines. For this part, we have mainly used Geometría Hiperbólica I. Movimientos rígidos y recetas hiperbólicas [1], an article appearing in an Argentinian publication which covers precisely the topic. The Poincaré half-plane in this thesis is defined as the set of complex numbers with positive real part. The particular definitions of length and distance of the hyperbolic half-plane model 1 CHAPTER 0. INTRODUCTION Luis González-Miret Zaragoza allows us to define lines that are neither parallel nor secant. For their derivations we will use a specific subset of the Möbius transformations that preserve the halfplane Last, but no least, the third part covers the unexpected link between the Möbius transformations and the special theory of relativity. The link between the Lorentz transformations, which relate different inertial systems of reference in this theory, and the Möbius transformations was studied by Roger Penrose in his book Spinors and space-time[8], co-authored by Wolfgang Ridler. This is the main reference we have used for this chapter. Here, after we introduce the basics of special relativity, we derive the isomorphism between both the Möbius and restricted Lorentz group, consisting the first on the set of all Möbius transformations and the second on a certain subset of the Lorentz transformations. This connection will allow us to divide the restricted Lorentz group into the four distinct categories we found in the first part. 2 Luis González-Miret Zaragoza CHAPTER 1. MEETING MÖBIUS the same equation. For w=R(z), we have Azz+Bz+Bz+C= 0 ⇒Aw−β αw−β α+Bw−β α+Bw−β α+C= 0 ⇒A ααww +B αw+B αw+C−Bβ α+Bβ α=A0ww +B0w+B0w+C0= 0 with A0=A/(αα),B0=B/α and C0=C−Bβ/α −Bβ/α. For w=I(z), then A1 ww +B1 w+B1 w+C= 0 ⇒A+Bw +Bw +Cww = 0 ⇒A0ww +B0w+B0w+C0= 0 with A0=C,B0=Band C0=A. This result is key to understand the geometrical significance of Möbius transformations. First, as the stereographical projection also maps circles to circles, then Möbius transformations preserve circles in the Riemann sphere. Actually, if Möbius transformations are conformal, then they also preserve angles in the Riemann sphere. In fact, we are going to prove that they are: Theorem 1.3.4. Möbius transformations are conformal. Proof. Let γ: (a1, c1)7→ C∞,: (a2, c2)7→ C∞be curves so that γ(b1) = (b2) with a1< b1< c1and a2< b2< c2in the extended complex plane. Möbius transformations are conformal if they preserve the quotient γ0(b1) 0(b2). Again, we are going to prove this for an inversion and a rotation-expansion-traslation. For M[γ(t1)] = αγ(t1) + β, then M0[γ(t1)] = αγ(t1), so M0[γ(b1)] M0[(b2)] =αγ0(b1) α0(b2)) =γ0(b1) 0(b2) For I(z) = 1 z, then I0(γ(t1) = −γ0(t1) γ2(t1), so I[γ(b1)] I0[(b2)] =−γ0(b1)2(b2) −0(b2)γ2(b1 =γ0(b1)) 0(b2)) 9 CHAPTER 1. MEETING MÖBIUS Luis González-Miret Zaragoza These basic propierties are going to be useful to apply Möbius transformations in different fields, as we will do in following sections. Now we are going to find a way to characterize each transformation by how they transform a set of three points. 1.4 Fixed points and the cross-ratio Definition 1.4.1. Let z, q, r, s ∈C∞. The cross ratio of these numbers, expressed as [z, q, r, s], is the quantity [z, q, r, s] :=                          z−q z−s r−s r−q,ifz, q, r, s 6=∞; r−s r−q,ifz=∞;q, r, s 6=∞; r−s z−s,ifq=∞;z, r, s 6=∞; z−q z−s,ifr=∞;z, q, s 6=∞; z−q r−q,ifs=∞;z, q, r 6=∞; (1.5) This quantity will come handy to find Möbius transformations sending a set of points to another set of points in C∞. First, we will take notice of the next propierty. Proposition 1.4.1. The Möbius transformation M(z) = [z, q, r, s] maps qto 0,sto ∞and rto 1 Proof. Immediate Remark. It is important to note that this Möbius transformation is, in general, not normalized, so we would need to multiply all coefficients by a normalization constant. Another interesting result is: Proposition 1.4.2. Möbius transformations leave cross ratios invariant, i.e [z, q, r, s]=[M(z), M(q), M(r), M(s)] Proof. This can be proven by simple algebra. 10 Luis González-Miret Zaragoza CHAPTER 1. MEETING MÖBIUS Let’s now consider the points that a Möbius transformation leaves invariant, i.e., its fixed points. Theorem 1.4.1. A Möbius transformation M(z) = az +b cz +dthat is not the identity has at most two fixed points, which are ξ±=(a−d)±p(a+d)2−4 2c Proof. Fix points fullfil M(ξ) = ξ. Then ξ=aξ +b cξ +d⇒cξ2+ (d−a)ξ−b= 0 ⇒ξ±=(a−d)±p(a−d)2−4cd 2c=(a−d)±√a2+d2−2ad + 4cd 2c =(a−d)±√a2+d2+ 2ad −4ad + 4cd 2c=(a−d)±p(a+d)2−4 2c From this, we can establish a way to fully characterize a Möbius transformation Theorem 1.4.2. There is exactly one Möbius transformation M(z) = wtransforming three determined points into three other determined points, i.e. for z1, z2, z3, w1, w2, w3∈C∞then M(z1) = w1, M(z2) = w2, M(z3) = w3. Such transformation is given by solving [w, w1, w2, w3] = [z, z1, z2, z3] with w=M(z) Proof. Let’s consider the Möbius transformations f(z) = [z, z1, z2, z3]and g(w) = [w, w1, w2, w3]. As f(z1) = g(w1) = 0,f(z2) = g(w2)=1and f(z3) = g(w3) = ∞, then the transformation M(zi) = g−1◦f(zi) = wifor i= 1,2,3. To check its uniqueness, let’s suppose there are two Möbius transformations M1(z) and M2(z)so that M1(zi) = M2(zi) = wi. Then the transformation M(z) = M−1 1◦M2(z)has z1, z2, z3as three fixed points. As a non-identity transformation has at most two fixed points, then Mis the identity and M1=M2. This property will allow us to find specific transformations that will send points to other points at our will. 11 CHAPTER 1. MEETING MÖBIUS Luis González-Miret Zaragoza Lemma 1.4.3. A Möbius transformation with a fixed point at ∞must have the form M(z) = Az +B, A, B ∈C, A 6= 0 Proof. As stated in theorem 1.4.1, ∞is a fixed point if c= 0 for M(z) = az +b cz +d. On the other hand, if ∞is a fixed point, then M(∞) = a/c =∞, so c= 0. Actually, more restrictive statements can be made. Lemma 1.4.4. If a Möbius transformation has a fixed point at 0 and another at ∞then it has the form M(z) = Az, A ∈C, A 6= 0. If its only fixed point is ∞, then it is a translation M(z) = z+B Proof. According to lemma 1.4, if ∞is a fixed point, then M(z) = Az+B, A, B ∈ C, A 6= 0. If 0is a fixed point, then M(0) = B= 0. Then, M(z) = Az If ∞is the only fixed point and A6= 1, then z=B/(1 −A)would be a fixed point to, so A= 1. Let’s now consider the cross ratio [z, ξ+, ξ−,∞], being ξ±the two fixed points of a Möbius transformation M(z)with two distinct fixed points none of them being ∞. According to proposition 1.4.2, applying M(z), we have: [z, ξ+, ξ−,∞] =[w, ξ+, ξ−, a/c]⇒z−ξ+ z−ξ−=a/c −ξ− a/c −ξ+ z−ξ+ z−ξ− ⇒z−ξ+ z−ξ−=ξ−−a/c ξ+−a/c z−ξ+ z−ξ− (1.6) with m= (ξ−−a/c)/(ξ+−a/c)being the multiplier. This way of writing a Möbius transformation is called the normal form. This way of expressing a Möbius transformation is particularly visual, as successive applications of the same Möbius transformations can be seen as multiplying the ratio (z−ξ+)/(z−ξ−)by m that many times. In fact, we will see that we can understand the fixed points as "sources" or "sinks", as multiple applications of M(z)bring closer or take further the other points to them. The cross ratio also allows us to determine whether four points lie in a circle: 12 Luis González-Miret Zaragoza CHAPTER 1. MEETING MÖBIUS Theorem 1.4.5. Let z, q, r, s ∈C∞. All four points lie in a circle Cif and only if =[z, q, r, s]=0 Proof. As stated in proposition 1.4.1, the Möbius transformation M(z) = [z, q, r, s] sends qto 0,rto 1and sto ∞. According to theorem 1.3.3, Möbius transformations preserve generalised circles. As three points determine a circle in C∞, then Msends the circle Cthat contains q, r, s to the real axis. Consequently, if z∈C, it is mapped to the real axis, so =[z, q, r, s]=0. On the other hand, if =[z, q, r, s]=0, that means that =M(z)=0, with M(z)=[z, q, r, s]. As Möbius transformations preserve circles, and the circle Ccontaining q, r, s is mapped to the real axis, then =[z, q, r, s]=0only if zis in such circle. Let O={q, r, s}with q, r, s ∈Cbe a set of three ordered points.Then we can assign a positive orientation to O, which consists on going over a circle from qto spassing through r. This divides the extended complex plane into three regions: •The circle containing such points C. •The region to the left defined as L(C) = {z∈C∞| =[z1, q, r, s]>0} •The region left to the right. R(C) = {z∈C∞| =[z1, q, r, s]<0} Regions to the left and to the right can easily be seen to be simply connected. This property allows us to determine an important geometric result. Theorem 1.4.6. Möbius transformations preserve orientation of circles, i.e. map left regions to left regions and right regions to right regions. Proof. It can be proven that Möbius transformations map connected regions of C∞into connected regions. Möbius transformations leave cross ratios invariant, so left regions are mapped to left regions and right regions to right regions. 13 CHAPTER 1. MEETING MÖBIUS Luis González-Miret Zaragoza 1.5 Möbius transformations as matrices Möbius transformations have deeper connections to complex algebra as it may seem. To start, we are going to prove that the set of all Möbius transformations has group structure and then we will proceed to identify it with a very important group of linear algebra. Theorem 1.5.1. The set of all Möbius transformations M=M(z) = az +b cz +d|ac −bd = 1 is a group under function composition. Such group is known as the Möbius group. Proof. All group properties must be fulfilled. Closure For M1= (a1z+b1)/(c1z+d1)and M2= (a2z+b2)/(c2z+d2), M2◦M1=(a2a1+b2c1)z+a2b1+b2d1 (c2a1+d2c1)z+c2b1+d2d1∈ M which can easily be proven to be normalized. Associativity As function composition is associative, it is fulfilled. Identity element The Möbius transformations I(z) = zacts as the identity. Inverse We proved in thorem 1.3.2 that there exist an inverse for every Möbius transformation. This vision of the Möbius transformations allow us to formulate the next map. Definition 1.5.1. The map Γ : SL(2,C)7→ M acts on 2×2 with determinant 1 matrices as a b c d!7→ az +b cz +d 14 Luis González-Miret Zaragoza CHAPTER 1. MEETING MÖBIUS Remark. It is easy to check that, as the matrices have determinant one, these Möbius transformations are normalized. We could also define the map Γ0: GL(2,C)7→ M, where we would get not-normalized Möbius transformations. Through this map, we are going to prove that we can work equivalently with Möbius transformations and members of SL(2,C) Proposition 1.5.1. The map Γis a surjective homeomorphism Proof. To do this, first we prove that the dentity matrix corresponds to the Möbius transformation. In fact: 1 0 0 1!7→ 1z 1=z Then the product of two matrices corresponds to the composition of the corresponding Möbius transformatios a1b1 c1d1! a2b2 c2d2!= a2a1+b2c1a2b1+b2d1 c2a1+d2c1c2b1+d2d1!7→ (a2a1+b2c1)z+a2b1+b2d1 (c2a1+d2c1)z+c2b1+d2d1 And, lastly, that the inverse matrix is maped to the inverse Möbius transformation −d b c−a!7→ −dz +b cz −a. The fact that the homeomorpism is surjetive comes from the fact that exactly 2 matrix are mapped to the same Möbius transformation ± a b c d!7→ az +b cz +d.(1.7) Equation (1.7) allows us to determine a isomorphism of the Möbius transformations with a more restricted group. Theorem 1.5.2. The Möbius group and the projective special linear group, PSL(2,C) = SL(2,C)/±Iare isomorphic, i.e. M∼ =PSL(2,C) Proof. As every member in PSL(2,C)corresponds to ±A,A∈SL(2,C), then every member of M corresponds to a member of PSL(2,C)and vice-versa. 15 CHAPTER 1. MEETING MÖBIUS Luis González-Miret Zaragoza As working with this group is a bit less intuitive, for the next discussions we are going to use the homeomorphism with SL(2,C). Actually, there is a way to characterize the action of Möbius transformations through the action of 2-column vectors. Definition 1.5.2. The homogeneous coordinates of z∈C∞are is defined as a pair (δ, η)∈C2such that z=δ/η. Actually, λ(δ, η), λ ∈C2determine the same complex number. To solve this, there exist the concept of normalized homogeneous coordinates, where λis chosen so that δ2+η2= 1. However, we are not going to use them, as it does not provide any advantage in this context Proposition 1.5.2. The action of PSL(2,C)on C2is equivalent to the action of Mon C Proof. If we consider a member of C2as complex homogeneus coordinates of z∈C, then a b c d! δ η!= aδ +bη cδ +dη!7→ aδ +bη cδ +dη =aδ/η +b cδη +d=az +b cz +d=M(z) From this, we can establish another way of seeing fixed points. Lemma 1.5.3. Fix points of a Möbius transformation M(z) = az +b cz +dare given by the eigenvectors of the matrix A= a b c d! Proof. The eigenvectors of Aare the vectors v= (δ1, δ2)T, δ1, δ2∈Csuch as Av =λv,for some λ∈C. That corresponds to M(z) = M(δ1/δ2) = λδ1/(λδ2) = z To find this eigenvectors, we have first to solve the characteristic equation |A−λ1|= 0. The eigenvalues will have a very relevant geometric role that we will discuss in next sections. Also, diagonalizing this matrices will help us classify Möbius transformations according to their effect on the Riemann sphere. 16 Luis González-Miret Zaragoza CHAPTER 1. MEETING MÖBIUS 1.6 Classification As we have previously stated, Möbius transformations have at most two fixed points. Let F(z)be a Möbius transformations such F(z) := z−ξ+ z−ξ−,(1.8) where ξ+and ξ−are fix points of a Möbius transformation M(z)with two distinct and finite fixed points. This transformation is, in general, not normalized. It is easy to see that F(z)maps ξ+to 0and ξ−to ∞. Now, let’s consider the transformation: ˜ M(z) := F{M[F−1(z)]}=F◦M◦F−1(z).(1.9) It can be seen that in this new transformation, the fixed points are 0 and ∞. So as we stated in lemma 1.4.4, ˜ M(z)must have the form: ˜ M(z) = mz, where m=ρeiφ is the multiplier that we derived in previous sections. We can see this as from (1.9) we can get to the normal form of the Möbius transform F◦M=˜ M◦F⇒w−ξ+ w−ξ−=mz−ξ+ z−ξ− where M(z) = w. From this fact we can now classify Möbius Transformations according to the form of m. 1.6.1 Hyperbolic transformations Hyperbolic transformations are characterized by m=ρ6= 1 If ρ > 1, applying ˜ Min the Riemann sphere will cause flow lines to go perpendicular to its latitudes from Sto N, whereas for ρ < 1they flow from Nto S. For M(z),ξ−will act as a sink, whereas ξ+will be a source, as shown in figure 1.4 17 CHAPTER 1. MEETING MÖBIUS Luis González-Miret Zaragoza Figure 1.4: Hyperbolic transformation [5] 1.6.2 Elliptic transformations Elliptic transformations fulfill m=eiφ. In the Riemann sphere, ˜ Mequates to a rotation along the z-axis. Applying M(z) will result on points describing elliptic orbits around ξ−and ξ+, without being neither of them either a source nor a sink, as shown on figure 1.5 Figure 1.5: Elliptic transformation [5] 1.6.3 Loxodromic transformations Loxodromic are a combination of the last two, where m=ρeiφ. In the Riemann sphere, applying ˜ Mmake points in spiral trajectories form Sto N. As is the case with elliptic transformations, applying M(z)will result on ξ− 18 Luis González-Miret Zaragoza CHAPTER 2. POINCARÉ HALF-PLANE •ρ(z1, z2) + ρ(z2, z1)≥ρ(z1, z3),∀z1, z2∈ H. The first three conditions are trivially derived from the previous definition. The fourth one is a bit less straightforward: the distance 4)ρ(z1, z2) + ρ(z2, z3)is the shortest distance between z1and z3traversing z2, which is going to be equal or greater than ρ(z1, z3). These are the basic ingredients of the model. We will proceed now to find its geodesics and lines, which will be defined in the next section. For this, the Möbius transformations will prove essential. 2.3 Geodesics and hyperbolic lines Definition 2.3.1. Ageodesic is defined as a curve γ: (a, b)7→ H that provides the path of integration used to compute the distance between each of its points. An hyperbolic line arc is the image of a geodesic [γ]. A hyperbolic line is a hyperbolic line arc which is not contained in any other hyperbolic line arc. We will now provide the simplest geodesic in the model, which will be crucial to derive the rest. Proposition 2.3.1. Let z1=ai, z2=bi with a, b > 0. The hyperbolic distance between them is ρ(z1, z2) = |log(b/a)| Proof. Let’s asume a < b (the case b < a is analogous).The hyperbolic length of a curve γ: [t1, t2]7→ H such us γ(t1) = ai,γ(t2) = bi is bounded as l(γ) = Zt2 t1p(γ0 1(t))2+ (γ0 2(t))2 γ2(t)dt ≥Zt2 t1 |γ0 2(t)| γ2(t)dt ≥Zt2 t1 γ0 2(t) γ2(t)dt = log γ2(t2) γ2(t1) (2.4) If we use the curve β(t) = it with a≤t≤b, then we get the length l(β) = Zb a dt t= log b a.(2.5) which coincides with the smallest value of the length of a curve joining the points ai and bt. This is the definition of the hyperbolic distance, so ρ(z1, z2) = log(b/a) As the points chosen on the imaginary axis were arbitrary, it follows inmediately that the curve β(t) = it with 0< t < ∞is a geodesic. It remains to be proven whether it is also an hyperbolic line. With this in mind, a sufficient condition for a curve to be a hyperbolic line is derived. 25 CHAPTER 2. POINCARÉ HALF-PLANE Luis González-Miret Zaragoza Proposition 2.3.2. Let γ: (a, b)7→ H be a geodesic such as for a point c∈(a, b) the curves γ|(a,c): (a, c)7→ H and γ|(c,b): (c, b)7→ H have infinite length. Then γ: (a, b)7→ H is an hyperbolic line. Proof. Reductio ad absurdum. Let σ: (a0, b0)7→ H be a geodesic such as [γ]⊂[σ]. Let’s assume ∃σ(t0) = z0/∈[γ]. Let z1be a point contained in both [σ]and [γ] such as γ(c) = σ(t1) = z1. Let assume t0< t1(for t0> t1the procedure would be analogous). As the restricted curve σ|(t0,t1)is a geodesic, its length l(σk(t0,t1)) must be finite, which would imply l(γ|(a,c))<∞, as it is contained in the other curve. This contradicts the initial hypothesis. With this result we can establish the next fundamental result Theorem 2.3.1. The image [β]of the curve β(t) = it, 0< t < ∞(2.6) is an hyperbolic line. Proof. We already proved that the length curve σ(t) = it for a<t<bis l(σ) = log(b/a). For an arbitrary point z1=ct with c∈(0,∞)the lengths of the curves σ(t)|(0,c)and σ(t)|(c,∞)can easily be seen to be infinite. In the next sections we will use this hyperbolic line to find all the others in the model. 2.4 Isometries of the Half-plane For now, we know the distance between points lying in a particular line of our model. We are interested now to find a way to move any pair of points outside this particular line into it in such a way that the distance between them does not vary. We establish now which are the transformations that will allow us to leave the distance between points invariant. Definition 2.4.1. An isometry in His a bijective transformation T:H 7→ H that preserves distances i.e. ρ(T(z1), T(z2)) = ρ(z1, z2) It will be proven that some subset of the Möbius transformations are isometries of H. 26 Luis González-Miret Zaragoza CHAPTER 2. POINCARÉ HALF-PLANE Definition 2.4.2. We define Tas the set consisting on the normalized Möbius transformations with real coefficients T={T(z) = az +b cz +d|a, b, c, d ∈R, ab −cd = 1} It can be easily proven that Tis a subgroup of Munder composition. Lemma 2.4.1. A Möbius transformation belonging to Tpreserves H. Proof. Let z=x+iy ∈ H. Appliying a transformation M(z)we get T(z) = az +b cz +d=ac|z|2+bcz +adz +bdz |cz +d|2. with real and imaginary part <[T(z)] = |z|2+bc(|z|2+ 1) |cz +d|2, =[T(z)] = (ac −dc)=z |cz +d|2==z |cz +d|2>0 So G(z)∈ H. It is also worth noting that with this same proof it can be seen that transformations of real coefficients with ab −cd > 0also preserve the H. These also form a subgroup under composition. It will be proved now that these are, in fact, isometries of the model, although not the only ones for H. Proposition 2.4.1. The transformations T∈ T preserve the hyperbolic length Proof. As it was shown in theorem 1.3.1, Möbius transformations can be expressed as compositions of two translations, an a rotation-expansion and a complex inversion. If we prove lengths to be constant under this transformations, then they are also for a general transformation T. As the coefficients of Tare real, the rotation-expansion becomes a simple expansion.For expansions and rotations deducing this from the formula (2.2) is trivial. For an inversion this is a bit more tricky. Computing from M(γ(t)) = 1/γ(t)with γ(t) = γ1(t) + iγ2(t)we obtain <[M(γ(t))] = γ1(t) |γ(t)|2=[M(γ(t))] = −γ2(t) |γ(t)|2 [<[(M(γ(t))]0=γ0 1(t)|γ(t)|2−2γ1(t)[γ1(t)γ1(t)0+γ2(t)γ0 2(t)] |γ(t)|2 27 CHAPTER 2. POINCARÉ HALF-PLANE Luis González-Miret Zaragoza [=[(M(γ(t))]0=γ0 2(t)|γ(t)|2−2γ2(t)[γ1(t)γ1(t)0+γ2(t)γ0 2(t)] |γ(t)|2 [<[(M(γ(t))0]2+ [=[(M(γ(t))0]2=[γ0 1(t)]2+ [γ0 2(t)]2 |γ(t)|4. Applying this in (2.2), we can see that the length is preserved. One can intuitively see (as it can also be proven) that for every pair of points in Hthere is exactly one semicircle or line both orthogonal to the real axis that contain them. This, and the fact that Möbius transformations transform generalized circles into generalized circles are used now to derive the next result. Proposition 2.4.2. Let z1, z2∈ H. There exist a Möbius transformation belonging to Tsuch as M(z1) = ai, M(z2) = bi with a, b > 0. Proof. If z1and z2lie in a line orthogonal to the real axis, then such transformation is G(z) = z−α, being αthe euclidean distance between the line and the imaginary axis. If the points lie on the a circle orthogonal to the real axis, then the transformation is more complex. First, the transformation G1(z) = z−α, being αthe distance between the complex axis and the left intersection between the circle and the real axis, as applied. This transformation moves the semicircle so its left extreme touches the origin. Then, the transformation G2=−1/z is applied, turning the semicircle into a line orthogonal to the real axis at x= 1/d =β, being dits diameter. The sign minus is added so the transformation preserves H. Finally, the transformation G3=z−βfinally takes the points to the real axis. Then, the transformation we are looking for is G(z) = G1◦G2◦G3=1 z−α−β For both cases, we have only used Möbius transformations with real numbers, all of them preserving the real axis, so the condition ab−cd > 0must be fulfilled. To get the corresponding transformation belonging to T, we just have to normalize the coefficients. This is the last ingredient we needed to get to establish Tas a set of isometries of the model. Theorem 2.4.2. The elements of Tare isometries of the Poincaré half-plane model. Proof. The elements of Tpreserve lengths and for every pair of points there exist a transformation that places them in the imaginary axis, where we know the distance between two points, so distances are preserved From this fact it is possible to derive a general formula for the distance between each pair of points. 28 Luis González-Miret Zaragoza CHAPTER 2. POINCARÉ HALF-PLANE Figure 2.2: Caption 2.5 Derivation of lines With this, and using the circle preserving property of the Möbius transformations we can establish all the lines of the model. As Tare isometries of the plane, a geodesic αis such if and only if it can be written as P◦αwith P∈G. The same can be said of the hyperbolic lines. Then, the next theorem is derived. Figure 2.3: Paralel lines Theorem 2.5.1. Hyperbolic lines are euclidean lines and semicircles perpendicular to the real axis. Arcs of hyperbolic lines are arcs of circles and segments of these hyperbolic lines and geodesics the curves they are the image of. Proof. Möbius transformation are conformal and every hyperbolic line [β]can be expressed as [β] = T◦[α]for every other α. We already know that the complex axis is an hyperbolic line, which is perpendicular with the real axis at x= 0 and for ∞. This can be easily seen if we represent both axis over the Riemann sphere. 29 CHAPTER 2. POINCARÉ HALF-PLANE Luis González-Miret Zaragoza As Möbius transformations transform lines into lines or circles, any other line will be a semicircle or a line perpendicular to the real axis. This notion of line will allow us to define parallel, secant and perpendicular lines in such a way that Euclid’s fifth postulate is not fulfilled. Definition 2.5.1. Two lines are parallel if they intersect on the real axis. Figure 2.4: Secant lines Definition 2.5.2. Two lines are secant if they intersect in any point other than the real axis. Figure 2.5: Caption Definition 2.5.3. Two lines are perpendicular if the intersect with a straight angle. Figure 2.6: Not secants nor parallel lines One can easily realize, as shown in figure (2.6) that this definition allows the existence of lines that are neither parallel nor secant, which contradicts Euclid’s fifth postulate. It can be proven that the model fulfills the other four postulates opening, making it valid for hyperbolic geometry. 30 Chapter 3 Möbius as Lorentz Within the framework of special relativity, events are represented as points in the four-dimensional Minkowsky space-time. Lorentz transformations substitute Galilean transformations as the coordinate transformations between inertial frames of reference. The equivalence between the Möbius group acting on C∞ and the proper orthochronous Lorentz group on world vectors is studied. This relation will be used to characterize how celestial spheres change with Lorentz transformations. As the focus of this work is not the exhaustive study of the Lorentz group but its relation with the Möbius transformations, only a quick review of its properties and subgroups is presented here. More information about this can be found on the extensive existing literature about special relativity. 3.1 Framework Special relativity was developed in the early XX century mainly by Albert Einstein. Its initial goal was to find a formalism where the equations of electromagnetism are invariant, as they are not under Galilean transformations. It is built from two postulates: •The laws of Physics are invariant in all inertial systems. •The speed of light in vacuum cis a invariant for all inertial observers. These postulates imply that time and space are not absolute anymore. A series of new equivalencies and conservation laws can be deduced from them. The most important, which is going to be key for the next results, is that the quantity known as space time interval,i.e. (∆s)2= (c∆t)2−(∆x)2−(∆y)2−(∆z)2(3.1) 31 CHAPTER 3. MÖBIUS AS LORENTZ Luis González-Miret Zaragoza is the same for all inertial observers, being ∆tthe difference of time and ∆x, ∆y, ∆xdifferences of distances in a Cartesian coordinate system between two events measured by an observer O. From now on, we are going to use natural units, so c= 1, which will simplify all expressions. 3.2 Minkowsky space-time and Lorentz transformations In special relativity, events are represented as elements of a four-dimensional vector space, with one temporal coordinate and three spatial ones. Definition 3.2.1. Minkowski space is defined as the vector space M:= {X= (t, x, y, z)∈R4} alongside the inner product (X1,X2) = t1t2−x1x2−y1y2−z1z2,X1,X2∈M. Vectors in M are known as world vectors. Each world vector represents an event in space time. This vectors are equivalent to the position vector in Newtonian mechanics. The introduced inner product allows the definition of the quantity X2= (X,X), which coincides with the definition of space-time interval (3.1). This quantity is known as the Lorentz norm1 The previously introduced inner product can be also expressed as: (X1,X2) = gµυxµ 1xυ 2(3.2) where gµυ is the metric tensor, with g00 = 1, g11 =g22 =g33 =−1and gµυ = 0 for µ6=υ.Here, x0=t, and we are using Einstein’s summation convention, were repeated indices imply summation from 0 to 3, for example, aibi= 3 X i=0 aibi. The metric tensor can be expressed as a 4×4 matrix g=      1 0 0 0 0−1 0 0 0 0 −1 0 0 0 0 −1       (3.3) 1Although we call it norm, it is not one, as it can have negative values. 32 Luis González-Miret Zaragoza CHAPTER 3. MÖBIUS AS LORENTZ The special relativity postulates allow us to define a set of transformations that preserve the space-time interval. Definition 3.2.2. Lorentz transformations are linear and real transformations Λ:R47→ R4acating on world vectors as: ˜xµ= Λµυxυ that preserve the Lorentz norm, i.e. (˜ X,˜ X)=(X,X) Lorentz transformations can also be seen as 4×4 matrices acting on world vectors       ˜ t ˜x ˜y ˜z       =Λ      t x y z       (3.4) All objects that transform with Lorentz transformation as world vectors are known as four-vectors. It can be shown that every Lorentz transformation fulfill a certain condition Proposition 3.2.1. For every Lorentz transformation Λ, g=Λ|gΛ (3.5) Proof. A lorentz transformation fulfils ˜ X2=X2. Using Einstein’s notation: ˜ X2=gµυ ˜xυ˜xµ=gµυΛυαxαΛµ βxβ X2=gαβxβxα ˜ X2=X2⇒gµυΛυαxαΛµ βxβ=gαβxβxα⇒gµυΛυαΛµ β=gαβ ⇒g=Λ|gΛ From this, we can go further and generalize the idea: Definition 3.2.3. The Lorentz group Lis defined as L={Λ|g=Λ|gΛ}. It can be proven that matrices fulfilling this condition conform a group under matrix multiplication. This group also contains time reversal (t, x, y, z)7→ (−t, x, y, z) and space parity (t, x, y, z)7→ (t, −x, −y, −z)between others. 33 CHAPTER 3. MÖBIUS AS LORENTZ Luis González-Miret Zaragoza 3.3 Light cones Vectors in Minkowski space-time can be classified into three categories according to their Lorentz norm and their time coordinate: Definition 3.3.1. A world vector Xis called time-like if X2>0,null or light-like if X2= 0 and space-like if X2<0. Definition 3.3.2. Vectors in Minkowski space-time are future-pointing if t > 0 and past-pointing if t < 0. In Minkowski space, null vectors conform two three dimensional cones with their vertex in the origins as shown in figure (3.1): Future-pointing light-like vectors lie inside the future cone, whereas past-pointing light-like vectors lie inside the past cone. Figure 3.1: Light cones. Here, only two spatial and one temporal dimensions are represented. Minkowski space-time is four-dimensional.[3] 3.4 Classification of Lorentz transformations The Lorentz group can be classified into four disjoint subgroups. To introduce this classification, let’s review some characteristics of their members. Proposition 3.4.1. Let Λ∈ L.Then its determinant is either 1 or -1 34 Luis González-Miret Zaragoza CHAPTER 3. MÖBIUS AS LORENTZ If we find an unitary matrix for rotations along every spatial axis, then we would be able to find a matrix corresponding to any rotation, as an arbitrary rotation can be expressed composition of rotation along the three axis. A rotation along the z axis of S+, and thus the Riemann sphere, has already been derived. It corresponds to the elliptic hyperbolic transformation ˜ ζ=eiφζ. This can be expressed with the two matrices Rz=± eiφ/20 0e−iφ/2! as every Möbius transformation corresponds to two matrices, one the opposite of the other. Now, we need to find a transformation for rotations along the Yand Xaxis. The matrices R± y=± cos θ/2−sin θ/2 sin θ/2 cos θ/2! are unitary and leave the quantity ξη −ηξ invariant, thus, according to equation 3.10, Yis preserved, which corresponds to a rotation on the y-axis. Computing the expression 3.13, it can be shown that it is a rotation of an angle θ. The same procedure can be applied to verify that the matrices R± x=± cos χ/2isin χ/2 isin χ/2 cos χ/2! corresponds to a rotation of an angle χalong the x axis.All of this allow us to state the next result. Proposition 3.6.4. Every spatial rotation corresponds to an unitary matrix in SL(2,C). Proof. As we have found unitary matrices corresponding to rotations along the three axis, every other rotation must be the product of these matrices. As the product of unitaries matrices are unitary matrices, every rotation corresponds to an unitary matrix. As we said, we have to still proof that a boost corresponds to a matrix in SL(2,C). A boost in the z axis corresponds to how coordinates change for an 41 CHAPTER 3. MÖBIUS AS LORENTZ Luis González-Miret Zaragoza observer traveling at a constant velocity v. Its explicit form is ˜ T= (1 −v2)−1/2(T+vZ),˜ X=X, ˜ Y=Y, ˜ Z= (1 −v2)−1/2(Z+vT)(3.14) This transformation can also be written as ˜ T+˜ Z=w(T+Z),˜ T−˜ Z=w−1(T−Z),˜ X=X, ˜ Y=Y(3.15) with w=1 + v 1−v1/2 .(3.16) Looking at the matrix 3.10, it can be seen that this corresponds to the application of the matrix B±=± w1/20 0w−1/2!,(3.17) which, in turn, corresponds to the Möbius transformation ˜ ζ=wζ (3.18) which lead us to the conclusion Proposition 3.6.5. Every Lorentz boost corresponds to two hermitian matrices in PSL(2,C)and every hermitian matrix in PSL(2,C)to a Lorentz Boost . Proof. As we stablished before, we can get a Lorentz boost in any direction by a combination of rotations and boosts in the z axis as we stated before. A boost in the z direction is unequivocally determined by the two matrices (3.17), which is hermitian. As the product U−1BU is hermitian if Bis hermitian and Uunitary, then every boost corresponds to two hermitian matrices. Also, every hermitian matrix Hcan be diagonalized by applying an unitary transform Aas A−1HA, which can be equated to (3.17), as determinant and hermiticity must be preserved by unitary transformations. We have proven all the relations we wanted to proof. As there is a one to one correspondence between the restricted Lorentz group and the Möbius group, then it follows: Theorem 3.6.1. The restricted Lorentz group and the Möbius group are isomorphic, M∼ =L↑ +. This total correspondence allows us to identify every Lorentz transformation with each of the different types of Möbius transformations. 42 Luis González-Miret Zaragoza CHAPTER 3. MÖBIUS AS LORENTZ •Hyperbolic transformations to boosts in any direction •Elliptic transformations correspond to rotations along any axis •Loxodromic transformations correspond to compositions of boosts and rotations in the same axis. •Parabolic transformations correspond to a composition of boosts and rotations such as only one point is left invariant in the celestial-sphere. 3.7 Transforming the celestial sphere We got a one to one correspondence with the Möbius Group and the restricted Lorentz. This will allow us to study how cellestial spheres changes for observers going through rotations and/or Lorentz Boosts. Let’s supose we are on a space-ship and from its windows we can see distant stars. We can see each of the stars as points in the cellestial sphere. Then, the algorithm we must follow is the following •Apply the anti-sky mapping Σto the cellestial sphere to go to S+ •Using the stereographic formulae to assign a complex point to each point in S+, i.e. mapping S+to the Riemann sphere. •Applying the corresponding Möbius transformations to the Riemann sphere. •Apply the inverse stereographic map and the inverse anti-sky mapping to get to the transformed S− The process followed is shown in figure 3.4, where we apply this process for a boost in the z-direction. This simple treatment only covers how the position of light sources would change in the sky. Other effects, as red shift, are of course not included in this treatment. 43 CHAPTER 3. MÖBIUS AS LORENTZ Luis González-Miret Zaragoza Figure 3.4: Boost as seen by an spaceship. Figure heavily modified taken from [8] 44 Bibliography [1] JC Boggino and RJ Miatello. “Geometrıa Hiperbólica I. Movimientos rıgidos y recetas hiperbólicas”. In: Revista de Educación Matemática 3.1 (). [2] Guillem Cobos Zara. “The Lorentz Group”. In: (2015). [3] Wikimedia Commons. SVG version of Image:Worldline.png. 2007. url: https://commons.wikimedia.org/wiki/File:World_line.svg. [4] NOEMÍ PATRICIA KISBYE. “El plano de Poincaré”. In: Facultad de Matemática Astronomıa y Fısica (2009). [5] Tristan Needham. Visual complex analysis. Oxford University Press, 1998. [6] Blagoje Oblak. “From the Lorentz group to the celestial sphere”. In: arXiv preprint arXiv:1508.00920 (2015). [7] John Olsen. “The geometry of möbius transformations”. In: Rochester: University of Rochester (2010). [8] Roger Penrose and Wolfgang Rindler. Spinors and space-time: Volume 1, Two-Spinor Calculus and Relativistic Fields. Vol. 1. Cambridge University Press, 1984. [9] Marıa Teresita Carrión Rebellato and Rafael Potrie. “TESELACIONES EN EL PLANO HIPERBÓLICO”. In: (). [10] Jakob Schwichtenberg. Physics from Symmetry. Springer, 2015. 45