Lorentz Group

The mathematical form of the Kinematical Laws of Special Relativity , Maxwell's Field Equations in the theory of Electromagnetism , Dirac's Equation in the theory of the Electron , are each invariant under Lorentz transformations. Therefore the Lorentz group can be said to express a fundamental Symmetry of many of the known fundamental laws of nature. BASIC PROPERTIES The Lorentz group is a Subgroup of the Poincaré Group , the group of all Isometries of Minkowski Spacetime . The Lorentz transformations are precisely the isometries which leave the origin fixed. Thus, the Lorentz group is an Isotropy Subgroup of the isometry group of Minkowski spacetime. For this reason, the Lorentz group is sometimes called the ''homogeneous Lorentz group'' while the Poincaré group is sometimes called the ''inhomogeneous Lorentz group''. Lorentz transformations are examples of Linear Transformations ; general isometries of Minkowski spacetime are Affine Transformations . Mathematically, the Lorentz group may be described as the Generalized Orthogonal Group O(1,3), the Matrix Lie Group which preserves the Quadratic Form : $(t,x,y,z) \mapsto t^2-x^2-y^2-z^2$ on R⁴. This quadratic form is interpreted in physics as the Metric Tensor of Minkowski spacetime, so this definition is simply a restatement of the fact that Lorentz transformations are precisely the linear transformations which are also isometries of Minkowski spacetime. The Lorentz group is a 6-dimensional Noncompact Lie Group which is not Connected , and whose Connected Components are not Simply Connected . The Identity Component (i.e. the component containing the identity element) of the Lorentz group is often called the restricted Lorentz group and is denoted SO⁺(1,3). In pure mathematics, the restricted Lorentz group arises in another guise as the Möbius Group , which is the Symmetry Group of Conformal Geometry on the Riemann Sphere . This observation was taken by Roger Penrose as the starting point of Twistor Theory . It has a fascinating physical consequence for the appearance of the night sky as seen by an observer who is maneuvering at relativistic velocities relative to the "fixed stars", which is discussed below. The restricted Lorentz group arises in other ways in pure mathematics. For example, it arises as the ''point symmetry group'' of a certain Ordinary Differential Equation . This fact also has physical significance. Note: the Lorentz group also preserves the quadratic form $(t,x,y,z) \mapsto x^2+y^2+z^2-t^2$ and is therefore sometimes denoted O(3,1). A similar remark applies to its identity component and the subgroups introduced below. CONNECTED COMPONENTS Because it is a Lie Group , the Lorentz group O(1,3) is both a Group and a Smooth Manifold . As a manifold, it has four Connected Component s. Intuitively, this means that it consists of four topologically separated pieces. To see why, notice that a Lorentz transformation may or may not reverse the direction of time (or more precisely, transform a future-pointing Timelike Vector into a past-pointing one), reverse the Orientation of a Vierbein . Lorentz transformations which preserve the direction of time are called orthochronous. Those which preserve orientation are called '''proper''', and as linear transformations they have determinant +1. (The improper Lorentz transformations have determinant −1.) The subgroup of proper Lorentz transformations is denoted SO(1,3). The subgroup of orthochronous transformations is often denoted O⁺(1,3). The Identity Component of the Lorentz group is the set of all Lorentz transformations preserving ''both'' orientation and the direction of time. It is called the proper, orthochronous Lorentz group, or '''restricted Lorentz group''', and it is denoted by SO⁺(1, 3). It is a Normal Subgroup of the Lorentz group which is also six dimensional. ''Note'': Some authors refer to SO(1,3) or even O(1,3) when they actually mean SO⁺(1, 3). The Quotient Group O(1,3)/SO⁺(1,3) is isomorphic to the Klein Four-group . Every element in O(1,3) can be written as the semidirect product of a proper, orthochronous transformation and an element of the Discrete Group :{1, ''P'', ''T'', ''PT''} where ''P'' and ''T'' are the Space Inversion and Time Reversal operators: P T The four elements of this isomorphic copy of the Klein four-group label the four connected components of the Lorentz group. THE RESTRICTED LORENTZ GROUP As stated above, the restricted Lorentz group is the Identity Component of the Lorentz group. This means that it consists of all Lorentz transformations which can be connected to the identity by a Continuous curve lying in the group. The restricted Lorentz group is a connected Normal Subgroup of the full Lorentz group with the same dimension (in this case, 6 dimensions). The restricted Lorentz group is generated by ordinary Spatial Rotations and Lorentz Boost s (which can be thought of as hyperbolic rotations in a plane that includes a time-like direction). The set of all rotations forms a Lie Subgroup isomorphic to the ordinary Rotation Group SO(3). The set of all boosts, however, does ''not'' form a subgroup, since composing two boosts does not, in general, result in another boost. A boost in some direction, or a rotation about some axis, each generate a One-parameter Subgroup . An arbitrary rotation is specified by 3 Real Parameters , as is an arbitrary boost. Since every proper, orthochronous Lorentz transformation can be written as a product of a rotation and a boost, it takes 6 real numbers (parameters) to specify an arbitrary proper orthochronous Lorentz transformation. This is one way to understand why the restricted Lorentz group is six dimensional. (We'll study this in more detail in a later section on the Lie algebra of the Lorentz group.) To specify an arbitrary Lorentz transformation requires a further two bits of information, which pick out one of the four connected components. This pattern is typical of finite dimensional Lie groups. RELATION TO THE MöBIUS GROUP The restricted Lorentz group SO⁺(1, 3) is Isomorphic to the Möbius Group , which is, in turn, isomorphic to the Projective Special Linear Group PSL(2,C). It will be convenient to work at first with SL(2,C). This group consists of all two by two complex matrices with determinant one : $, \; ad - bc = 1$ We can write two by two Hermitian matrices in the form : $X = \left \begin{matrix} t+z & x-iy \ x+iy & t-z \end{matrix} ight$ This trick has the pleasant feature that : $\det \, X = t^2 - x^2 - y^2 - z^2$ Therefore, we have identified the space of Hermitian matrices (which is four dimensional, as ''real'' vector space) with Minkowski spacetime in such a way that the determinant of a Hermitian matrix is the squared length of the corresponding vector in Minkowski spacetime. But now SL(2,C) acts on the space of Hermitian matrices via is the Hermitian Transpose of $P$ , and this action preserves the determinant. Therefore, SL(2,C) acts on Minkowski spacetime by (linear) isometries. We have now constructed a Homomorphism of Lie groups from SL(2,C) Onto SO⁺(1,3), which we will call the '''spinor map'''. The Kernel of the spinor map is the two element subgroup ±''I''. Therefore, the quotient group PSL(2,C) is isomorphic to SO⁺(1,3). APPEARANCE OF THE NIGHT SKY This isomorphism has a very interesting physical interpretation. We can identify the complex number : $\xi = u+iv$ with a Null Vector in Minkowski space : $\left \begin{matrix} u^2+v^2+1 \ 2u \ -2v \ u^2+v^2-1 \end{matrix} ight$ or the Hermitian matrix : $N = 2\left \begin{matrix} u^2+v^2 & u+iv \ u-iv & 1 \end{matrix} ight$ The set of real scalar multiples of this null vector, which we can call a ''null line'' through the origin, represents a ''line of sight'' from an observer at a particular place and time (an arbitrary event which we can identify with the origin of Minkowski spacetime) to various distant objects, such as stars. But by Stereographic Projection , we can identify $\xi$ with a point on the Riemann Sphere . Putting it all together, we have identified the points of the Celestial Sphere with certain Hermitian matrices, and also with lines of sight. This implies that the Möbius transformations of the Riemann sphere precisely represent the way that Lorentz transformations change the appearance of the celestial sphere. For our purposes here, we can pretend that the "fixed stars" live in Minkowski spacetime. Then, the Earth is moving at a nonrelativistic velocity with respect to a typical astronomical object which might be visible at night. But, an observer who is moving at Relativistic velocity with respect to the Earth would see the appearance of the night sky (as modeled by points on the celestial sphere) transformed by a Möbius transformation. CONJUGACY CLASSES Because the restricted Lorentz group SO⁺(1, 3) is isomorphic to the Möbius group PSL(2,C), its Conjugacy Classes also fall into four classes: elliptic transformations hyperbolic transformations loxodromic transformations parabolic transformations (To be utterly pedantic, the identity element is in a fifth class, all by itself.) In the article on Möbius Transformation s, it is explained how this classification arises by considering the Fixed Point s of Möbius transformations in their action on the Riemann sphere, which corresponds here to Null Eigenspace s of restricted Lorentz transformations in their action on Minkowski spacetime. We will discuss a particularly simple example of each type, and in particular, the effect on the appearance of the night sky of the One-parameter Subgroup which it generates. At the end of the section we will briefly indicate how we can understand the effect of general Lorentz transformations on the appearance of the night sky in terms of these examples. A typical elliptic element of SL(2,C) is : $P_1 = \left \begin{matrix} \exp(i heta/2) & 0 \ 0 & \exp(-i heta/2) \end{matrix} ight$ and collecting terms, we find that our spinor map takes this to the (restricted) Lorentz transformation : $Q_1 = \left[ \begin{matrix} 1 & 0 & 0 & 0 \$ 0 & \cos( heta) & -\sin( heta) & 0 \ 0 & \sin( heta) & \cos( heta) & 0 \ 0 & 0 & 0 & 1 \end{matrix} ight] This transformation represents a rotation about the z axis. The one-parameter subgroup it generates is obtained by simply taking $heta$ to be a real variable instead of a constant. The corresponding continuous transformations of the celestial sphere (except for the identity) all share the same two fixed points, the North and South pole. They move all other points around latitude circles. In other words, this group yields a continuous counterclockwise rotation about the z axis as $heta$ increases. Notice the ''angle doubling''; this phenomenon is a ''characteristic feature of spinorial double coverings''. A typical hyperbolic element of SL(2,C) is : $P_2 = \left \begin{matrix} \exp(\beta/2) & 0 \ 0 & \exp(-\beta/2) \end{matrix} ight$ which also has fixed points $\xi = 0, \infty$ . Under stereographic projection from the Riemann sphere to the Euclidean plane, the effect of this Möbius transformation is a Dilation from the origin. Our homomorphism maps this to the Lorentz transformation : $Q_2 = \left[ \begin{matrix} \cosh(\beta) & 0 & 0 & \sinh(\beta) \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ \sinh(\beta) & 0 & 0 & \cosh(\beta) \end{matrix} ight]$ This transformation represents a boost along the z axis. The one-parameter subgroup it generates is obtained by simply taking $\beta$ to be a real variable instead of a constant. The corresponding continuous transformations of the celestial sphere (except for the identity) all share the same fixed points (the North and South poles), and they move all other points along Longitude s away from the South pole and toward the North pole. A typical loxodromic element of SL(2,C) is : $P_3 = P_2 P_1 = P_1 P_2 = \left[ \begin{matrix} \exp \left((\beta+i heta)/2 ight) & 0 \ 0 & \exp \left(-(\beta+i heta)/2 ight) \end{matrix} ight]$ which also has fixed points $\xi = 0, \infty$ . Our homomorphism maps this to the Lorentz transformation : $Q_3 = Q_2 Q_1 = Q_1 Q_2$ The one-parameter subgroup this generates is obtained by replacing $\beta + i heta$ with any real multiple of this complex constant. (If we let $\beta, heta$ vary independently, we obtain a ''two-dimensional'' abelian subgroup, consisting of simultaneous rotations about the z axis and boosts along the z axis; in contrast, the ''one-dimensional'' subgroup we are discussing here consists of those elements of this two-dimensional subgroup such that the rapidity of the boost and '''angle''' of the rotation have a ''fixed ratio''.) The corresponding continuous transformations of the celestial sphere (always excepting the identity) all share the same two fixed points (the North and South poles). They move all other points away from the South pole and toward the North pole (or vice versa), along a family of curves called '''loxodromes'''. Each loxodrome spirals infinitely often around each pole. A typical parabolic element of SL(2,C) is : $P_4 = \left \begin{matrix} 1 & \alpha \ 0 & 1 \end{matrix} ight$ which has the single fixed point $\xi=\infty$ on the Riemann sphere. Under stereographic projection, it appears as ordinary Translation along the Real Axis . Our homomorphism maps this to the matrix (representing a Lorentz transformation) : $Q_4 = \left[ \begin{matrix} 1+\alpha^2/2 & \alpha & 0 & -\alpha^2/2 \ \alpha & 1 & 0 & -\alpha \ 0 & 0 & 1 & 0 \ \alpha^2/2 & \alpha & 0 & 1-\alpha^2/2 \end{matrix} ight]$ This generates a one-parameter subgroup which is obtained by considering $\alpha$ to be a real variable rather than a constant. The corresponding continuous transformations of the celestial sphere move points along a family of circles which are all tangent at the North pole to a certain Great Circle . All points other than the North pole itself move along these circles. (Except, of course, for the identity transformation.) Parabolic Lorentz transformations are often called null rotations. Since these are likely to be the least familiar of the four types of nonidentity Lorentz transformations (elliptic, hyperbolic, loxodromic, parabolic), we will show how to determine the effect of our example of a parabolic Lorentz transformation on Minkowski spacetime, leaving the other examples as exercises for the reader. From the matrix given above we can read off the transformation : $\left \begin{matrix} t \ x \ y \ z \end{matrix} ight ightarrow \left \begin{matrix} t \ x \ y \ z \end{matrix} ight + \alpha \; \left \begin{matrix} x \ t-z \ 0 \ x \end{matrix} ight + rac{\alpha^2}{2} \; \left \begin{matrix} t-z \ 0 \ 0 \ t-z \end{matrix} ight$ Differentiating this transformation with respect to the group parameter $\alpha$ and evaluate at $\alpha=0$ , we read off the corresponding vector field (first order linear partial differential operator) : $x \, \left( \partial_t + \partial_z ight) + (t-z) \, \partial_x$ Apply this to an undetermined function $f(t,x,y,z)$ . The solution of the resulting first order linear partial differential equation can be expressed in the form : $f(t,x,y,z) = F(y, \, t-z , \, t^2-x^2-z^2)$ where $F$ is an ''arbitrary'' smooth function. The arguments on the right hand side now give three ''rational invariants'' describing how points (events) move under our parabolic transformation: : $y = c_1, \; t-z = c_2, \; t^2-x^2-z^2 = c_3$ (The reader can verify that these quantities standing on the left hand sides are invariant under our transformation.) Choosing real values for the constants standing on the right hand sides gives three conditions, and thus defines a ''curve'' in Minkowski spacetime. This curve is one of the flowlines of our transformation. We see from the form of the rational invariants that these flowlines (or orbits) have a very simple description: suppressing the inessential coordinate y, we see that each orbit is the intersection of a ''null plane'' $t = z+c_2$ with a ''hyperboloid'' $t^2-x^2-z^2 = c_3$ . In particular, the reader may wish to sketch the case $c_3 = 0$ , in which the hyperboloid degenerates to a light cone; then orbits are parabolas lying in null planes just mentioned.
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	:<math> Rac{d\xi}{d Heta} { Heta	0} = -rac{1+\xi^2}{2} </math>
	{{cite Book Author	Carmeli, Moshe

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Lorentz Group