Special relativity The minimal minimum John O’Neall 1. The principles of Special Relativity Special relativity (SR) is based on two principles: 1. The principle of relativity states that the laws of nature should be the same for all observers in inertial frames (defined below). 2. ll such observers, upon measurin" the speed of li"ht in a vacuum! will find the same result! c # 2$$!%$2!&'( km*sec. The first re+uirement is necessary in order for physics to be coherent. ,t means that observers in inertial systems use the same e+uations. Rather than "oin" on incessantly repeatin" -in an inertial reference frame.! let/s "et it done with once and for all by statin": SR considers only observers in inertial reference frames, those which move with constant (unaccelerated) velocity relative to one another. The first re+uirement may be stated di0erently: 1o experiment can measure the absolute velocity of an observer3 the results of any experiment performed by an observer do not depend on his speed relative to other observers who are not involved in the experiment.1 The second re+uirement is a result of rather astoundin" e2perimental results. There is no known reason for it! that/s 4ust how it is. 5ecause of the constant speed of li"ht! SR prevents us from considerin" space and time as bein" two separate thin"s and explains how they are related and lin)ed into a more "lobal entity! spacetime. 6ore on that in a moment. 2. The spacetime diagram and the Lorentz transform 7e use the notion of a four-vector in tensor notation:2 X # (t! 2! y! 8) :# ( 29! 21! 22! 2:) or . 7e also use standard relativist units, where c # 1! so that c # :x19( m*sec # 1 meanin" that time is measured such that 1 Schut8! 1. 2 6ore on that in my ;R overview. Special relativity << overview 1 2929<91<92 1 sec # :x19( m. ,n SR! we spea) of spacetime as the total of space and time! which while di0erent are no lon"er distinct. n event is somethin" which happens instantaneously at a specific value of time and space! i.e.! t and . n event may be simply a set of values of the four spacetime variables. n inertial observer is 4ust a spacetime coordinate system which records the coordinates of any event. =is space is >uclidean (in SR) ! distances between "iven points are independent of time and his cloc)s are synchronized and run to"ether. The essential "eometry of SR is shown in ?i"ure 1. ?or simplicity! we i"nore y and z. The black a2es show the x and t variables of observer 3 the blue! of observer ! whom we suppose is movin" with velocity v alon" the 2<direction. The path of the ori"in of as seen by is the blue line labeled t/. This is because the ori"in of is always at and so this path is his time axis. ?rom the dia"ram! the tan"ent of the upper an"le! ! is "iven by . ny li"ht wave on such a dia"ram must move with constant velocity! c # 1! and so is always represented by a &'@ line (since we chose c=1). ,ma"ine a li"ht wave in the system startin" at 2#9 and t # <a! so it will move upward at an an"le of &'°. Suppose there is a mirror to reflect it at the 2<axis where 2#a also (&'@! remember) so it will move upwards, always at &'° and reach 2#9 a"ain at t#Ba. Transferrin" this to the system of ?i"ure 1 with the rays still at &'° since the speed of li"ht is constant! will pick a point at 2/#a (not shown in the figure). Since we assume the particle started at 2#t#9! we have the two points needed to define a strai"ht line! the blue line makin" an e+ual an"le with the 2<axis of observer and which is the 2/ axis.: Figure 1: Geometry of SR – the spacetime diagram (after Collier) point C in space is therefore represented in /s reference frame by the intersections of the fine< dotted lines with the x and t a2es. They are obviously normal to t and 2. The coordinates of are : Schut8! D<(. Special relativity << overview 2 2929<91<92 a bit more complicated: Since lines parallel to 2/ are lines of constant t/! and vice versa! the upper dashed line intersects the t/ axis at the value t/ of C in the frame. Similarly! the lower dashed line intersects the 2/ axis at the value of 2/ for the point C. The fact that the speed of li"ht is constant for all observers leads (by means of several sorts of derivations) to the Lorentz transformations between the reference frames of the two observers: ! This is often expressed as a transformation matrix: (1)! where we use a bar instead of a prime to desi"nate the second system! and (2) 6ore precisely! this is a Eorentz boost! where one frame moves with a constant velocity relative to another. ,t may also di0er by spatial rotation. This point of view leads to the idea of the Eorentz transformation as a rotation includin" time! as explained in Section '. Then the transformation formula for a four<vector is . (:)! with and expressed as a column vectors. four<vector is a set of coordinates (displacements) which transform by the Eorentz transformation.& Fsin" the Eorentz transform! one can derive somewhat laboriously all the followin". The definedinterval between two events separated by the displacements (&) is an invariant +uantity across inertial systems. ,t can be expressed in terms of a metric tensor which is dia"onal and has ei"envalues (<1!1!1!1) as . space of this kind is called Minkowski space. The proper time is also an invariant: (5). ,t is clear from the definition that the proper time is the time measured by a clock at rest with & ,n ;R! a vector is a set of components and not a thin" which points from one point to another! since that would be impossible in a curved space. Special relativity << overview : 2929<91<92 respect to the particle or event! since then ! There exists in any frame a set of orthonormal basis vectors (6) so that a four<vector may be e2pressed as (%) where and is summed over. Such a four<vector is invariant in spacetime under Eorentz transforms, only its components are di0erent in di0erent frames. The e+uality can be used to show that the basis vectors transform as ((). Gomparin" this to shows that the transformation matrix for basis vectors is the inverse of that for vectors. They are not vectors, but one<forms! or dual vectors.' Figure 2: Light cones (by !" #ains$atsi via &ikimedia Commons() Ei"ht rays satisfy and so move at &'° an"les to the t<axis and 2<axis (and y<axis) on a spacetime dia"ram! extended to represent two spatial dimensions. ,nside the cone alon" the t<axis, ! so an event at the ori"in has time to influence any point within the cone without ' See the ;R overview. D https:**commons.wi)imedia.or"*wi)i*?ile:7orldHline.sv". Special relativity << overview & 2929<91<92 "oin" faster than the speed of life. The interior of the cone! for positive or ne"ative t! is therefore referred to as timelike. Iutside the cones, this is not possible! so these re"ions are called space- like.. The cone for tJ9 is the future relative to the ori"in! the cone for tK9! the past. 3. Four-velocity and four-momentum The four-velocity! ! of a particle may be defined as a vector tan"ent to its world line and one time unit lon" in its rest frame. Ibserver sees the ori"in of move alon" its world line in his frame. ,n the frame of ! this ori"in is at ! so its only non<zero component is alon" the basis vector . So the particle/s four<velocity is that vector in its inertial rest frame! . ?rom another point of view! it would be "ood to define four<velocity as a derivative! as in classical physics. 5ut instead of dt! an invariant +uantity! ! is used! so the components of the four<velocity are "iven by ($) which reduces to in the particle/s frame. Eook at the time component of . 5ut . Then the spatial components are ! where is the ordinary velocity. So . (19). 1ote that (11)! by the definition of . This is also an invariant. The four-momentum! ! is defined! similarly to classical mechanics, by ! (12) where m is the particle/s mass in its own frame. So (1:) and so is the particle/s ener"y and its three<momentum. lso! ! Special relativity << overview ' 2929<91<92 where refers to the three<momentum! and so . (1&) This formula is "ood even for massless particles li)e photons. 1ote that when v KK c! the ener"y ! the rest<mass ener"y plus the kinetic ener"y of classical mechanics. The relativistic Ea"ran"ian for a sin"le particle is and one can derive the =amiltonian by . Then it is simple to confirm the precedin" e+uations ,f particle p/ moves in the frame of p with velocity v alon" the 2<axis, and a particle is movin" with velocity W alon" the 2/<axis of p/ ! then its velocity in p/s frame is "iven by the Einstein law of composition of velocities: (1') which can never e2ceed the speed of li"ht (1! in this system).