Physics Lorentz Transformations March Lorentz Transformations This handout summarizes the lecture discussion of Lorentz transforma tions Introduction When two observers are moving relative to each other we use Lorentz trans formations to relate their observations Four vectors Lorentz transformations op erate on fourvectors more generally ten sors with fourvector indices An example of a fourvector is the space time co ordinate formed from the time co ordinate t and the threedimensional vector space co ordinate x We multiply the time by the velocity of light ct x x We can lab el the comp onents of the fourvector as x with ranging from 0 to The sup erscript is conventional Then the time comp onent is x ct Another common fourvector is the fourmomentum E p pc 0 The comp onent p is the energy which must include the rest energy The fourmomentum of a particle of mass m at rest is just 2 mc p Lorentz transformation The Lorentz transformation is the central feature of sp ecial relativity that was adopted in order to account for the remarkable observation that the sp eed of a light ray measured by an observer was apparently the same regardless of how fast the observer was moving 0 Supp ose observers in frames S and S are moving with a velocity v relative to each other To b e more precise lets align the spatial co ordinate 0 systems for each observer so S is moving along the z axis of observer S and 0 0 the x and y axes are parallel with the x and y axes Supp ose also that b oth observers set their clo cks so that when their origins are on top of each other their clo cks read also Then if an observer in frame S sees an event at the space time co ordinate 0 0 x and an observer S sees the same event at spacetime co ordinate x the co ordinates are related by the Lorentz transformation 00 0 3 x x x 10 1 x x 20 2 x x 30 3 0 x x x We have introduced v c p 2 In terms of t x y and z the Lorentz transformation is 0 ct ct z 0 x x 0 y y 0 z z ct As we mentioned this transformation assures that if b oth observers see the same light ray and measure its sp eed they b oth get the same result namely c A simple way to check this is to supp ose that the light ray leaves the origin at time t Later on the observer S notices that it has reached 0 the spacetime co ordinate x while observer S notices that it has reached 0 x The relationship b etween these observations is given by the Lorentz transformation It is easy to check the Lorentz invariance prop erty 0 2 1 2 2 2 3 2 00 2 10 2 20 2 30 2 x x x x x x x x or 2 2 0 2 0 2 ct r ct r p 2 2 2 where we have introduced the distance to the origin r x y z for p 0 0 0 2 0 2 0 2 y z for observer S For a light ray observer S and r x moving with velocity c we must have r ct So b oth sides of the equation ab ove must b e zero More gnerally Both observers always get the same 0 0 value for any Lorentz invariant quantity So we conclude that also r ct and b oth observers agree that the sp eed is c If the relative sp eed v is much less than the sp eed of light we have and 0 ct ct 0 x x 0 y y 0 z z v t which is the Galilean relativity that we learned in our rst course in classical mechanics Time dilation and LorentzFitzerald contrac tion As an application of a spacetime Lorentz transformation consider how time intervals are recorded by dierent observers As a sp ecic application supp ose a particle is placed at rest at the origin of observer S at time 0 0 t t It decays after a p erio d of time Observer S also sees the decay According to hisher clo cks what is the elapsed time 0 The answer is easily found from the relation ct ct z which we apply to the decay event Notice that to use this relation we have to sp ecify z Since the decaying particle is still at the S origin we have z and 0 t So fast moving particles app ear to have longer lifetimes according to the Lorentz factor This is the p eculiar phenomenon of time dilation As a second application consider how distance measures are recorded by dierent observers Supp ose that observer S places a meter stick along 0 the z axis and observer S measures its length What will the result b e As always we answer by considering the relationship b etween a pair of 0 events In this case the events are S measuring the two ends of the meter 0 stick which she must do at the same instant t Otherwise she would have to correct for the motion of the meter stick We can use the Lorentz transformation equations given ab ove without mo dication if we supp ose 0 0 0 that S makes the measurement at t when the S and S origins coincide 0 Both observers agree that the left end of the meter stick is at z z 0 when t t S would say the right end of the meter stick is always 0 0 at z and S records it as z What seems p eculiar is that they will disagree ab out the simultaneity of the events of measuring b oth ends If we 0 0 lo ok at the relation ct ct z we see that t requires ct z so only for an event at z or for for vanishingly low velocities v c ! do 0 b oth observers agree on the time So when observer S measures the right 0 end of the meter stick at time t observer S says that measurement happ ens at time ct z with z meter But from the fourth relation 0 0 z ct observer S must b e doing this measurement at p osition z 0 2 0 z z z z So to S the meter stick is short by a factor This is the LorentzFitzgerald contraction Note that b oth observers always agree on lengths p erp endicular to their relative velocity The contraction is only along the direction of relative motion Lorentz transformations and other fourvectors The fourmomentum transforms just like the spacetime co ordinate under a Lorentz transformation 00 0 3 p p p 10 1 p p 20 2 p p 30 3 0 p p p so 0 E E p c z 0 p c p c x x 0 c p c p y y 0 p c p c E z z In fact a fourvector is dened to b e any quantity that transforms in this way Velocity transformation Lets see how velocities transform under a Lorentz transformation We con sider a particle moving along the z axis with z w t according to observer 0 S He says it has velocity w Observer S moving with velocity v relative 0 0 0 to S says it has velocity w z t We want to nd it Now we are dealing with three velocities the velocities of the particle as seen by the observers and the relative velocity of the observers For simplicity lets do this in only one dimension so supp ose all velocities are along the z axis Then a Lorentz transformation relates the p osition of the particle as seen by b oth observers 0 z z ct w v t 0 ct ct z c v w ct We divide the equations to get w v 0 w 2 v w c 0 In the low velocity limit we recover the familiar Galilean result w w v 0 But at the other extreme if w c we get w c so all observers agree on the same velocity for a particle moving at the sp eed of light Lorentz invariants We have already seen that the quantity 2 0 2 1 2 2 2 3 2 2 2 x x x x x ct x is invariant under a Lorentz transformation By the same token the quantity 2 0 2 1 2 2 2 3 2 2 2 2 2 4 p p p p p E p c m c is also invariant since it transforms the same way This result is fortunate since then all observers agree on the same mass This relativistic relationship b etween energy momentum and mass is sometimes called the mass shell condition More general invariants can b e constructed from any pair of fourvectors For example if we have two fourmomenta k and p the combination 0 0 1 1 2 2 3 3 p k p k p k p k p k is also invariant Notice that this expression denes an inner pro duct rather like a dot pro duct but the signs metric are dierent from the familiar Cartesian dot pro duct The familiar dot pro duct of threedimensional vec tors is similarly invariant under rotations Photon fourmomentum A photon has zero mass If it has frequency then its wavelength is c and if it is moving in the z direction its energy and momentum are given by E B C B C p B C A pc where E h and p h E c This result is consistent with the statement that 2 2 2 p E cp meaning the photon has zero mass Relativistic kinematics Kinetic Energy A moving particle has kinetic energy In Newtonian mechanics we thought of kinetic energy as b eing the primary quantity and then tacked on the rest mass when we b egan talking ab out lowvelocity nuclear physics In sp ecial relativity we treat the total energy as the primary quantity given p 2 4 2 2 m c p c and the kinetic energy K is then derived from it by E using 2 K E mc It is a useful exercise to check that the relativistic massenergymomentum relation ab ove turns into the standard nonrelativistic result for sp eeds much less than the sp eed of light q 2 2 2 4 2 2 K m c p c mc p m Twobo dy collisions Consider a twobo dy reaction of the type ! Let the fourmomenta of the particles b e p p p and p resp ectively 1 2 3 4 Notice we are now using subscripts to lab el the particle Each of these terms represents a full fourvector Let the masses similarly b e m m 1 2 m and m Energy and momentum must b e conserved The conservation 3 4 condition can b e written concisely in terms of fourvectors as p p p p 1 2 3 4 Supp ose that particle is at rest and particle arrives along the p ositive z axis Then the initial fourmomenta are 2 E m c 2 B B C C B B C C p p B B C C 1 2 A A pc 4 2 2 2 2 c so there is really only one Of course we must also have E p c m 1 kinematic parameter for the incoming particle Let