Advanced EM - Master in Physics Special Relativity: first steps We have seen two different approaches towards special relativity: Experimental approach, which shows the existence and validity of the Lorentz invariant LI. Theoretical postulate of the invariance of the speed of light (A. Einstein) They can be shown to be equivalent, in the sense that one is the consequence of the other. To prove it, let us…. Start with the invariance of “c”; and examine the case of two successive events in the same space point, separated by a time interval 2 τ, in the Inertial Reference Frame IRF’. We want to know how they appear in another IRF. Well, instead of imagining a system in IRF’ consisting simply of a clock, we can imagine a system made of a pulsed lamp sending out a light pulse toward a mirror which sends it back to the source, where a fast light detector is located. 2 τ is the time between light pulse emission and detection. The regular repetition of this process, with period τ, is a clock. Now let us imagine we want to measure the coordinates of the two events: emission and reception of light E and R from another Inertial Reference Frame which moves wrt IRF’ with uniform velocity V; and set the axis lamp-mirror in the experiment perpendicular to the relative velocity V. We want to calculate the quantity for the pair of events E and R both in IRF’ and in IRF. In IRF’, where the setup is at rest, this quantity is simply (2c τ ) 2

Advanced EM - Master in Physics In the reference frame IRF, where the setup is seen to be moving with speed v, the system is seen as in the figure: Pulsed lamp Mirror Light detector τcτc 2vt The quantity we are investigating for being an invariant is Ev. 1Ev. 2 That is, requiring that the light moves with velocity “ c ” both in IRF and in IRF’ causes the invariance of that quantity which is then called the Lorentz invariant LI. The equality has in fact been demonstrated for any value of the velocity v. The inverse demonstration – from the Lorentz invariant to the speed of light in any reference system – can be obtained just reversing the argument.

Advanced EM - Master in Physics The dilatation of times Let us now use the Lorentz invariant (which implies the invariance of the speed of light, and thus removes the inconsistency with the EofM) to evaluate how the time between two events changes when we change the reference frame IRF. Let a rocket leave from Earth direction a distant star, at a distance a million light-years (a modern example of a distance measured in units of time). Let the rocket velocity (in units of “ c ”) be 0.99; In the Earth IRF the rocket will take /0.99 = years to cover that distance. The LI will take the value (“ c ” in this case is obviously =1) of E10 (light-years)2. A clock on the rocket will indicate, upon arrival to the star, a time which will be the square root of the LI, i.e years. Or, more general, L being the distance and β the rocket velocity measured in the earth frame, in the Earth IRF the LI is : LI = (t 2 – L 2 ) = [t 2 – (βt) 2 ]= t 2 (1- β 2 ). Since a clock on the rocket will have run no distance in its own IRF, the time it will show upon arrival is the square root of the LI : To put numbers, if the rocket going to the star had had a γ of 10 6 a clock on it would have indicated 1 year of proper time upon arrival. This is the dilatation of times. The time between two events is the shortest in the IRF in which the two events happen in the same place. That IRF is called the rest frame, because an object in that frame will not have moved between the two events; and that time is called the proper time between those events. Times measured in all other IRFs in movement wrt the rest one are larger than the proper time by a factor of γ.

Advanced EM - Master in Physics So, the proper time is the minimal time between two events. As long as, of course, the LI is positive. Because, a characteristics of that odd 4-dimension space which we call the space-time, introduced by that odd minus sign in the definition of the LI, is that it can be either positive or negative. If a couple of events has a positive LI the pair is said to be “ time-like ”. If the LI is negative, the pair is said to be “space-like” instead: the two objects can not be related by a “travel”. In such case the LI will not be the square of the proper time because there is no system in which the two events happen at the same place. There will be, instead, an IRF in which the two events are simultaneous. In that IRF the distance will be minimal. The property of being “time-like” or “ space-like” belongs to pairs of events. If applied to a single event, it is really to two events, the other being the origin of the IRF. Needless to say, there is no Lorentz transformation which can bring a spacelike event pair to be timelike in a new IRF – and vice versa. Positive and negative LI So far we have deduced various properties of the new space without mentioning how the 4 coordinates of an event transform in the change of the IRF. Finding the proper transformation of coordinates, similar in principle to what happens with rotations in three-dimensional space, is not difficult, given the condition of the invariance of the LI. The other condition that will help us in finding the proper transformation is the request that it be linear. The meaning of such request is that the relative coordinates of a pair of events do not depend on where they are: if the transformation had, p.ex. a quadratic term in the “x” coordinate, then X A ’ – X’ B = K (X A 2 – X B 2 ) would be dependent on where is the origin of the IRF. i.e. on a translation of axis..

Advanced EM - Master in Physics The Lorentz transformation Events immersed in spacetime are the physical world. And all the Physics is contained in the events and in their relative Lorentz Invariants. However, to describe, associate those events and write the physical laws that describe the relations between events we need numbers that define positions, times and other quantities of events. To define those numbers and give them meaning is the role of the IRFs. These numbers – the coordinates (4 coordinates, three for space and one for time) – define the position of an event in spacetime. Precisely in the same way as for rotations in 3-dimensional space, in a change of reference system a linear relation exists between old and new coordinates of the same event. Such linear relation in the case of spacetime is bound by the condition of invariance of the LI, and is called the Lorentz transformation. Once the condition of linearity is satisfied, the conditions of invariance of coordinates orthogonal to the direction of the relative motion AND of Lorentz invariance are sufficient to determine all the coefficients. The equations for the transformation from IRF’, with three space axes parallel to those of IRF and moving wrt IRF in the direction of positive X (at time t=0, x=0 also t’, x’ are zero) are: and y x z Y’ X’ Z’ IRF IRF’ v

Advanced EM - Master in Physics The x_coordinate (in IRF) of the IRF’ origin is The invariance of the origin of IRF’ at time t’ writes: Let us use the same units for time and space ( c = 1 ) Therefore, when x’=0 then t=γt’. We have found coefficient D. Then we immediately find F: from Motion of IRF’ origin (x’=0) We have replaced t with γt’ ; and we have found coefficient F: we will substitute D and F. We have now to find B and E. We will do it by imposing the invariance of LI Then:

Advanced EM - Master in Physics And the Lorentz transformation reads: It is now easy to calculate what the law of “composition of the velocities” is –when the velocities are parallel (velocities in units of “ c ”). Where dx’ and dt’ refer to the moving coordinates of an object in IRF’: obviously dx’/dt’ is the object velocity in IRF’. We then divide both num. and denomin. by dt’: And obtain the “law of composition of the velocities” when, as in this example, they are both parallel. Note that this velocity is always lower than 1. And, in case the relative velocity of the two IRFs is very low (i.e. non-relativistic, or <<c ), it tends to (Galilean relativity). In case the two velocities (that of an object wrt IRF’ and that of IRF’ wrt IRF) are not parallel, a different law applies: Let v y ’=(dy’/dt’ ) be the y-component of the velocity of the object in IRF’, then: The velocity in IRF will of course also have a component along “x”, which can be as well obtained from the previous formula (composition of parallel velocities). If v’ is directed along the Y axis, then v y =v’ v / γ

Advanced EM - Master in Physics We have seen how easy is finding the law of composition of velocities when using the Lorentz transformation. But we have aid before that the Physics lays in the network of Lorentz Invariants between all the events in the system, not in their coordinates. Which is to say that we could derive the law of composition of velocities directly by asserting the invariance of the speed of light. Let the system we discuss be a school bus, full of fairly undisciplined students. The bus is cruising at speed β. Suddenly a student in the back row shoots a gun towards the bus front. How do the two velocities – the bus β and the bullet velocity v – add up? The system as described so far does not allow the comparison with the speed of light: let us introduce a light pulse travelling forward from the back row, in the same direction as the bus, and starting exactly at the same time as the bullet. The light goes faster than a bullet, so the light reaches the front driver’s mirror before the bullet does, is reflected back, and eventually, next to a student sat at a given row does meet the bullet still travelling forward. L is the bus length, and f is the fraction of Lat where the bullet meets the light. It takes the light a time t f to reach the mirror, where t f is given by and a time t b to reach from the mirror to the bullet again For arriving to the light the bullet will have run a distance which, moving members around, becomes ( v is the bullet speed in the chosen IRF): Replacing the values just found for t f and t b, we obtain:

Advanced EM - Master in Physics We repeat the same argument in IRF’, where the bus is at rest, and find Now, f=f’: that the bullet and light meet at a certain point in space, x=fL is a Physics fact; bullet, and light meet near row 4, and that happens both in IRF and in IRF’. So we can write Which will reduce to just as found, much more easily, with the Lorentz transformation.

Advanced EM - Master in Physics We have already seen, in the discussion of the train paradox, that in the change from an IRF at rest to an IRF in movement wrt a rigid object (a train?) the rigid body seems to be shorter (in the direction of the relative motion between the 2 frames). And, in the case of the train, we have seen, very qualitatively, that the problem seems to be that for the Observer External OE the lightnings seem to have struck the train simultaneously while the Observer Passenger on the train the lightning on the train head struck first. Let us try to evaluate the coordinates of the events in the IRF of OE and then do a Lorentz transformation to the frame IRF’ to understand what OP sees. And…. Let us call L the train length (in IRF’, its own reference frame) and l is the distance measured by OE between the burns on the rail.

Advanced EM - Master in Physics Let us calculate the coordinates (time and space) of the various events as seen by observer OE (the observer outside the train) in the IRF frame. Event coordinates [t,x] as seen in IRF (OE) Coordinates as seen in IRF’ (OP)

Advanced EM - Master in Physics Coordinates as seen in IRF’ (OP) Let’s write again the last table REMARKS: in IRF the train’s length is “L”, in IRF’ it is l. They are related by: The contraction of lengths seen in moving objects is confirmed; and quantified. While in IRF the flashes are simultaneous at t=0, in IRF’ they happen at t’=βL/2 at the tail and t’=-βL/2 at the front. OP sees the flash from the train tail arrive a time δt’=βL later than the flash from the front

Advanced EM - Master in Physics

Advanced EM - Master in Physics