Phys1201. Advanced Physics (version 1) Relativity Notes. Lecturer: Craig Savage.

Phys1201. Advanced Physics 2. 2008 (version 1) Relativity Notes Lecturer: Craig Savage. craig.savage@anu.edu.au. www.anu.edu.au/physics/savage These notes should be read before the relevant lectures, and studied in depth after them. Contents 1. Overview 2. Simulation 3. Light Clocks time dilation 4. Light Clocks length contraction and the relativity of simultaneity 5. The Lorentz Transformations 6. The Meaning of the Lorentz Transformations 7. The Twin Paradox 8. Four-dimensional Geometry 9. 4-vectors 10. Relativistic Optics 11. The relativity of simultaneity revisited Appendix Taylor expansion References 1. Overview One of the great themes in physics is unification: taking things that look different and discovering that they are really parts of a greater whole. Newton did this when he realised that apples fall and planets orbit the Sun for the same reason gravity. Maxwell and others realised that electricity, magnetism, and light were aspects of electromagnetism, summarised by Maxwell s equations. Summary Relativity unifies space and time into space-time. It is based on two postulates. Einstein and Minkowski made the next big unification when they realised that space and time were parts of a greater four-dimensional space-time. This unification is called the special theory of relativity and will be our major subject [1]. From it a number of other unifications followed, such as the unification of energy and mass expressed by the equation E = mc 2 [2]. Special relativity is based on two postulates out of which Einstein built the entire theory by logical deduction [1,3]. The relativity principle. The result of every experiment is independent of its speed. The constancy of the speed of light. Every measurement of the speed of light in a vacuum gives the same result: c = 3x10 8 m/s. The relativity principle is an extension of Galileo s relativity principle for mechanics to all of physics. However, the second postulate is radical, and seems crazier the more you think about it. It means you can never catch up with a light wave. It says that if PHYS1201 Relativity Notes, 2008. 1

you chase a pulse of light in a rocket, travelling at nearly the speed of light, you will measure its speed to be the same as if you had stayed at home. Crazy but true! The same year, 1905, Einstein also kick-started quantum mechanics by showing that light had a unified wave and particle nature [4]. In 1915 he continued unifying by showing that gravity was not a force, but rather a property of space-time. The physics of this is called the general theory of relativity. Fundamental physics has continued unifying since 1915. Important successes are: quantum mechanics (1925-30), which unified particles and fields. electro-weak theory (1970), which unified electromagnetism and the weak nuclear interaction (responsible for radioactive decay, and for nuclear fusion in the Sun) Currently, some speculative ideas, such as string theory, are being pursued with a view to unifying gravity with both the electro-weak interaction and the strong nuclear interaction (which holds nuclei together against the electrical repulsion of the protons). 2. Simulation Contents Computer simulations are used to conduct investigations that might otherwise be difficult or impossible. For example Summary modelling the causes of climate change, or the effects of changes The relativistic world cannot in economic policy. At ANU we have developed a simulation of be directly experienced, but it special relativistic physics called Real Time Relativity. You can be simulated. will do a lab using it, and need to get some practice before the lab. You can play with it on the computers in the tutorial room or download it and run it yourself, if you have a suitable computer [5]. 3. Light clocks time dilation Contents There are many different ways to develop the special theory of relativity. One is to look at certain crucial experiments and deduce it from them this is called the inductive approach. It is described in detail in a technical paper by H. Roberston [6]. Three experiments from which special relativity follows are: the Michelson-Morely, the Kennedy-Thorndike, and the Ives- Stilwell. The first two are optical interferometry experiments, while the last measures time dilation using the frequency of light emitted by fast moving hydrogen atoms. Summary The essential physics of relativity follows from analysing light clocks. Time dilation is the slowing of the ticks of moving clocks. We will take the deductive approach, as Einstein did [1]. This postulates certain truths and then uses logic to deduce consequences. The idea is that we do not doubt the postulates, unless we find a consequence that disagrees with a reliable experiment. The critical postulates are the two given in section 1 of these notes: the relativity principle and the constancy of the speed of light. We shall deduce special relativity starting from an analysis of light clocks. These are a conceptually simple kind of clock that no one has ever actually built. However they are beautifully adapted to deriving the consequences of the relativity postulates. PHYS1201 Relativity Notes, 2008. 2

A light clock uses the path of a pulse of light over a fixed distance as its unit of time. In relativity, and astronomy, it is convenient to measure distance in units of lightdistance, such as light-seconds or light-years. A light-second is the distance light travels in a second, 3 10 8 m, slightly less than the distance to the Moon. A lightnanosecond is 0.3 m. So a clock with a path length of a light-nanosecond would be about 30 cm long, or 15 cm if we use a round trip. A light clock consists of a light source and a light detector next to each other, with a mirror situated such that the emitted light is reflected back to the detector; see diagram. When the detector detects the light pulse it immediately triggers the source to emit another pulse. Each time a pulse is emitted the clock ticks off a nanosecond (for a 30 cm path). You can image a digital display counting off the ticks. The light clock works much the same way as any clock does it counts the number of some boringly repetitive physical phenomenon known to have a constant repeat time. In your watch it may be the mechanical vibration of a quartz crystal. The light clock has two advantages for us: 1) it is simple, so we can understand exactly what s going on, 2) its reliability is based directly on one of the postulates of relativity the constancy of the speed of light. We could use a more complicated clock for our analysis, but that would require a lot more work, and much more complicated arguments. What would a light clock look like if it were moving past us at close to the speed of light? mirror source & detector A light clock. Let s start by assuming that the light path is oriented perpendicular to the clock s motion, and that the length of the light path does not change we ll prove this later. Then the path taken by the light pulse is shown in the diagram. It is longer than for a stationary clock. Remember that we are postulating that every measurement of the speed of light in a vacuum gives the same result, so the speed of light has its usual value of c = 3 10 8 m/s. It therefore takes the light longer to travel the longer distance. Since the clock ticks are determined by the light pulses, a moving clock ticks slower than a stationary clock that is, ticks take longer. What does this mean? Are light clocks crazy clocks? A moving light clock. We can answer that by using the relativity principle. Attach your favourite clock to the top of a light clock. Perform the experiment of checking whether they stay synchronised they do, because both are good clocks. Now repeat the experiment with the clocks moving, and you moving along with them. The relativity principle, which we are postulating to be true, says that: the result of every experiment is independent of its speed. Hence we deduce that you will again find that the light clock and your favourite clock agree. If I m watching the clocks move past at close to the speed of light, because they agree for you they agree for me Why? Think about putting their digital readouts next to each other. Consequently, any moving clock will slow down in exactly the same way as the light clock does. If every clock slows down then time itself slows down for what is time but what clocks measure (for a physicist anyway). PHYS1201 Relativity Notes, 2008. 3

Time itself slows down! Go over the argument again. Where s the trick? there is none! If we assume the relativity principle and the constancy of the speed of light to be true then time slows down for moving objects. So if you don t like the conclusion you have to object to at least one of the postulates. Can we do experiments to check this? Yes. Countless experiments have verified this result, called time dilation. The GPS, or Global Positioning System, uses a set of satellites to determine the position of a receiver to within a few metres [7]. GPS receivers do this by precisely timing radio signals from satellites, and then determining distances by dividing by the speed of light. With distances to 4 satellites you can fix your position. Why 4 and not 3? See [7]. This system relies on precise timing, so the satellites contain atomic clocks clocks that count the oscillations of electrons in atoms. But the satellites are moving relative to the receiver at about 3.9 km/s (12 hour orbits). This is enough to produce a significant time dilation, and so the atomic clocks are purposely set to run at the wrong rate on Earth so that when they are launched into orbit they stay synchronised with Earth clocks! If this weren t done there would be a lot of lost cars, ships and planes. Hence time dilation is being tested every second of every day, and people s lives depend on it. Actually general relativity causes negative time dilation, which unfortunately does not cancel the positive time dilation due to the motion expect for 9,545 km radius orbits (GPS orbits are 20,200 km). The net result is that the atomic clocks have to be set to run slow on Earth, so that they stay synchronised in orbit. Here is an extract from Ashby s review of relativity in the GPS [7, section 5]: There is an interesting story about this frequency offset. At the time of launch of the NTS-2 satellite (23 June 1977), which contained the first Cesium atomic clock to be placed in orbit, it was recognized that orbiting clocks would require a relativistic correction, but there was uncertainty as to its magnitude as well as its sign. Indeed, there were some who doubted that relativistic effects were truths that would need to be incorporated! A frequency synthesizer was built into the satellite clock system so that after launch, if in fact the rate of the clock in its final orbit was that predicted by general relativity, then the synthesizer could be turned on, bringing the clock to the coordinate rate necessary for operation. After the Cesium clock was turned on in NTS-2, it was operated for about 20 days to measure its clock rate before turning on the synthesizer. The frequency measured during that interval was +442.5 parts in 10 12 compared to clocks on the ground, while general relativity predicted +446.5 parts in 10 12. The difference was well within the accuracy capabilities of the orbiting clock. This then gave about a 1% verification of the combined second-order Doppler (special relativistic time dilation) and gravitational frequency shift effects for a clock at 4.2 earth radii. In a fun 1971 experiment Joseph Hafele and Richard Keating flew four portable atomic clocks around the world twice in a commercial jet [8]. They verified time dilation to 10%. A more personal example is that time dilation is the reason you are now being bombarded by muons from cosmic rays. These are unstable elementary PHYS1201 Relativity Notes, 2008. 4

particles, a kind of heavy electron, that are produced about 50 km up in the atmosphere by collisions of cosmic protons and alpha particles with atmospheric nuclei. They last about 2.6 microseconds before decaying, which means at the speed of light they could travel 2.6 10-6 s 3 10 8 m/s = 780 m. They can only reach the ground, and pass through you, because of time dilation. Let s now do a quantitative analysis of the moving light clock. We ll refer to the diagram in which h is the height of the clock s light path, so that the time for a tick of a stationary clock, which we ll call a proper tick, is t p = 2h/c. Let t m be the time for the moving clock s tick and v be the clock s speed. The half distance travelled by the light to the mirror can be expressed in terms of the speed of light and the tick time, or by using Pythagoras theorem on the indicated triangles. Equating these allows us to find the tick time: ( ct m / 2) 2 = h 2 + ( vt m / 2) 2! t m 2 = h 2 c 2 / 4 " v 2 / 4 = 4h2 c 2 " v 2 = ( 2h / c)2 2 1" v 2 / c = t p 2 1 " v 2 / c 2. (3.1) ct m /2 h vt m ct m /2 A moving light clock.! t m = t p 1" v 2 / c 2 = # t p. This is the time dilation formula. We have introduced the gamma factor, or ( ) "1/2. Lorentz factor,! = 1 " v 2 / c 2 We can apply this to the 3.9 km/s speed of the GPS satellites. We find ( v / c) 2 = 3.9 / 3! 10 5 ( ) 2 = ( 1.3! 10 "5 ) 2 = 1.7! 10 "10. This is so small compared to 1 that if you try and calculate γ directly on a calculator you tend to get 1. To deal with this we Taylor expand the inverse square root (see the appendix to these notes), to ( ) "1/2 # 1+ 1 2 v2 / c 2 = 1 + 1.7 $ 10 "10. This means that seen from find:! = 1" v 2 / c 2 Earth the clock ticks are longer by 1.7 parts in 10 10, or that the clock runs slow by this amount, due to special relativistic time dilation. Let s revisit the assumption we made about lengths perpendicular to the motion not v changing. This is a consequence of the relativity principle. Consider an experiment in object which we pass an object through a gap in a solid sheet. The sheet and object are fired towards each other with equal and opposite velocities. Assume the object only just fits -v through, see diagram. Now move the whole experiment with velocity v. The velocity of the sheet is then zero and that of the object -2v. The object is still observed to pass sheet through the gap in the sheet. The object through the Why? hole experiment. Hence the moving object cannot get longer perpendicular to the motion. Repeat the experiment, but now moving it with velocity v. Now the object is at rest and the sheet moves with velocity 2v. The object passes through, so it cannot get shorter perpendicular to the motion. Hence we can deduce that moving objects neither shrink PHYS1201 Relativity Notes, 2008. 5

nor expand in directions perpendicular to their motion, and must therefore remain the same length. Stop and think about what we have just done shown that moving clocks run slow. How is this possible? It is only possible if we abandon the idea of absolute time the same time for everyone. What we have found is that relatively moving observers have different times. 4. Light clocks length contraction and the relativity of simultaneity Contents Thought experiments (sometimes called gedanken experiments ) with light clocks can reveal some of the other central physics of special relativity: length contraction, in which lengths parallel to the motion shrink, and the relativity of simultaneity, in which moving, spatially separated clocks lose synchronisation. To investigate these effects we consider a pair of light clocks, one oriented perpendicular to the motion as before, and one oriented parallel to the motion, see diagram. When at rest the two clocks stay synchronised. The principle of relativity implies that this is also the case when they are moving. Why? Summary Length contraction is the reduced length of moving objects along their direction of motion. The relativity of simultaneity is the disagreement between relatively moving observers about whether spatially separated clocks are synchronised. perpendicular v parallel We already know about the diagonal path the light in the perpendicular clock takes. In the parallel clock, the light has further to go to get to the mirror, as it runs away, and a shorter return distance as the detector catches up. Does the sum of these distances equal the distance along the perpendicular clock s diagonal path? No, it s greater, and that s why length contraction is implied by the relativity principle its needed to keep the clocks synchronised (see movies at [9]). Two light clocks. Let s work that out. We denote the length of the moving parallel clock s light path by h m, which we do not assume is equal to h. The time a light pulse takes to go from the emitter to the mirror is h m /(c-v), since c-v is the speed of the light relative to the moving mirror for an observer watching the clocks move with velocity v. Note that although the speed of light is always the same, its speed relative to something else, as observed by a third observer, is not: e.g. the observed relative speed of a red light pulse relative to a co-propagating green one is zero, while relative to a counter-propagating blue one it is 2c. Similarly, the time the light pulse takes to return from the mirror to the detector is h m /(c+v). For the clocks to stay synchronised, the sum of these times must equal the time for a tick of the perpendicular clock t m =! t p (Equation (3.1)): PHYS1201 Relativity Notes, 2008. 6

t m =! t p = t m = 2h / c 1" v 2 / c 2 and h m c " v + h m c + v = h m # 2c & $ % c 2 " v 2 ' ( = 2h m # 1 & c $ % 1" v 2 / c 2 ' ( (4.1) Equating the right hand sides: h m = h 1! v2 / c 2 1! v 2 / c 2 = h 1! v2 / c 2 = h / " (4.2) This is length contraction. The length of a moving object whose rest length, or proper length is h, is reduced by the inverse gamma factor! "1 : h m = h 1! v 2 / c 2. The relativity principle allows us to generalise from this case to all lengths parallel to the motion, just as we did for time in section 3. Stop and think: How does that argument go? There s something else interesting going on here: the times when the light pulses reflect from the mirrors of the two clocks are different, even though the times for each clocks round trip (tick) are the same. Why? The reflection time for the perpendicular clock is t m /2, while for the parallel clock we have already calculated it to be t 1/2 = h m /(c-v). Let s re-express this: t 1/2 = h m c! v = h 1! v2 / c 2 c! v = h ( c + v) 1! v2 / c 2 c 2! v 2 ( ) h / c + hv / c2 = = t / 2 + h v / c 2 p. 1! v 2 / c 2 1! v 2 / c 2 (4.3) The first term in the numerator of the last expression is the time for a half-tick of a stationary clock. The denominator provides the time dilation factor. If this was all, the reflections would occur simultaneously: t 1/2 =! t p / 2 = t m / 2 (false). However the second term in the numerator destroys simultaneity. It is the position of the parallel clock s mirror in its rest frame, h, times v/c 2. We will see this term again later when we study the Lorentz transformations. It is responsible for the relativity of simultaneity: spatially separated events which are simultaneous when at rest, are not simultaneous when they are moving. According to the result of Equation (4.3), the further apart they are (the bigger h is), the more out of sync they are. 5. The Lorentz transformations Contents A sophisticated view of space-time is that it is events related by geometry. An event is a point-happening: think of a firework exploding. It has a position and a time. We shall interpret these as the four-dimensional coordinates of the event. It s natural to PHYS1201 Relativity Notes, 2008. 7

generalise the idea of three-dimensional spatial coordinates to four-dimensions by including time. Summary The Lorentz transformations relate an event s time and space coordinates according to different observers. They incorporate all the physics of relativity: time dilation, length contraction, and the relativity of simultaneity. If you have ever used a grid reference on a map, or done a calculation using Cartesian coordinates, you will know that coordinates are really useful. Therefore, our goal in this section is to figure out the relationship between the coordinates of an event as measured by two relatively moving co-ordinate systems. They will be different because a point with fixed location in one system will have a changing location in the other, due to their relative movement. We consider only the case of inertial coordinate systems: ones with constant relative velocity. Inertial coordinate systems are referred to, more briefly, as inertial frames. Time Out. All this discussion of inertial frames and observers can seem quite abstract and confusing. To make it concrete think about an inertial frame as a smoothly cruising car or spaceship, and an observer as you in that vehicle. Changing frames means moving to a vehicle travelling with a different velocity. Figuring out the relationship between differnet observers coordinates can be quite a tedious business, if the coordinate systems are allowed to be oriented any-which-way, so we will focus on the essentials by considering coordinate systems S and S in standard configuration : that is, arranged so that there is an event at which the spatial coordinate origins coincide, x = x = y = y = z = z = 0, at time t = t = 0. Furthermore, we assume that S and S are oriented so that Two inertial frames. their coordinate axes are parallel, and we assume that the origin of S, x = y = z = 0, travels along the positive x axis at uniform speed v. This means that at time t, it is at x = vt. All these assumptions have their subtleties, but amount to simple and natural statements about the nature of space and time: primarily their homogeneity (here and now is not fundamentally different to any other place and time), and the isotropy of space (all directions are fundamentally the same). If you want to know all the details consult Rindler [10]. From the consideration of the object passing through a hole, at the end of section 3, we know that distances perpendicular to the motion do not change. Hence the y and z coordinates are the same in S and S : y = y and z = z. Newton goes on to say that time is absolute, t = t, and that x = x - vt. Let s assume that in relativity x remains independent of y and z and that it remains a linear function of x and t (see Rindler [10] for detailed justifications). Since x = vt implies x = 0, because of the way we have set up our coordinates, we must have x! = " ( x # vt), (5.1) PHYS1201 Relativity Notes, 2008. 8

where γ may depend on the speed v. This γ is the same one we have encountered before, as we are about to show. Now we can reverse the roles of S and S by considering S to be moving with speed v along the negative x axis of S. Then x = -vt implies x = 0, and because of the symmetry of the situation, ( ). (5.2) x =! x " + v t" Now we differ from Newton by using the postulate of the constancy of the speed of light. Light emitted by an event at the coincident coordinate origins satisfies both x = ct and x = ct. Substituting these in Equations (5.1) and (5.2) respectively gives c t! = " t c # v ( ) and ct =! t" ( c + v). (5.3) Multiplying these equations together, and dividing through by tt gives c 2 =! 2 ( c 2 " v 2 ) #! = 1 1" v 2 / c 2. (5.4) This is indeed the gamma factor, or Lorentz factor we met in section 3. The x coordinate in the S frame is therefore, according to Equation (5.1) x! = x " vt 1 " v 2 / c 2. (5.5) We can find the t coordinate in terms of x and t by substituting Equation (5.1) for x into Equation (5.2) ( ) =!! ( x # vt) + v t" x =! x " + v t" ( ) =! 2 x # v! 2 t + v! t" $ x( 1 #! 2 ) = #v! 2 t + v! t " $ t " =! t + x 1#! 2. v! (5.6) The last term can be simplified as follows: 1! " 2 v" = 1 # 1 & 1! v" $ % 1! v 2 / c 2 ' ( = 1 v 1!!v 2 / c 2 v2 / c 2 1! v 2 / c =!v / c 2 2 1! v 2 / c =!" v / 2 c2. (5.7) Then the expression in Equation (5.6) for t becomes t! = " t # x" v / c 2 = " ( t # xv / c 2 ) = t # xv / c2 1# v 2 / c 2. (5.8) We derived this expression, in a different form, at the end of section 4. There we interpreted the part containing t as causing time dilation and the part containing x as causing the relativity of simultaneity. PHYS1201 Relativity Notes, 2008. 9

We can now summarise the relativistically correct relationship between the coordinates of an event in coordinate systems S and S that are in standard configuration, called the Lorentz transformations : t! = t " xv / c2 1" v 2 / c, x! = 2 x " vt 1" v 2 / c, y! = y, z! = z. (Standard configuration) (5.9) 2 We can also express the S coordinates in terms of the S coordinates, in which case we usually refer to the inverse Lorentz transformations : t = t! + x! v / c2 1" v 2 / c, x = x! + v t! 2 1" v 2 / c, y = y!, z = z!. (inverse L.T.) (5.10) 2 These can be obtained by algebraic work, or more physically by swapping the S and S coordinates and changing the sign of v. Time Out. What do the Lorentz transformations mean? They mean that relatively moving observers have different position and time coordinates for an event. It s easy enough to understand why the time of an event has to be accounted for in calculating x " vt its position, as in the numerator of x! =, but it s harder to understand why 1" v 2 2 / c the position of an event has to be accounted for in calculating its time, as in the numerator of t! = t " xv / c2. It s because of some fundamentally new physics: the 1 " v 2 2 / c relativity of simultaneity, which we will discuss later. Let s look at some examples of how these are used. Example 1. What are the S coordinates of the event E with S coordinates t = 1s, x = 9 10 8 m, y = z = 1 m, when v = c 3 / 4? To use Equation (5.9) we first need the gamma factor. We chose the funny speed so that 1! = 1" 3 / 4 = 1 1 / 4 = 2. (5.11) Then Equation (5.9) gives ( ) = 2( 1 " 3 3 / 4 ) $ "3.2 s, ( ) = 2( 9 # 3 3 / 4 ) " 10 8 $ 12.8 " 10 8 m, t! = 2 1" 9 # 10 8 3 / 4 / 3 # 10 8 x! = 2 9 " 10 8 # 3 " 10 8 3 / 4 y! = 1 m, z! = 1 m. What does this mean? In S the event E occurs at an earlier time, and further from the origin, than in S. This is quite strange, as while in S the event E happens before the event of the coincidence of the S and S origins, event O, it happens after it in S. PHYS1201 Relativity Notes, 2008. 10

Hence the time ordering of events can be different in different inertial frames. Does this mean that causal order depends on the coordinate system? That is, could an event at O cause E in S, even though E precedes O in S? That would be really weird. Fortunately for the theory of cause and effect, the only events whose time ordering can reverse are those that are sufficiently far apart that not even light can travel between them. Such events are said to be space-like separated, and cannot be causally related, since nothing can travel faster than light. In fact, this is a major part of the reason we believe nothing can travel faster than light because if something could, then causality would be out the window. However there are other reasons too, including the fact that nothing has ever been observed to do so. Check that events E and O are space-like separated. Example 2. Two clocks at rest on the x axis in S are synchronised at t = 0. Clock 1 is at the origin x = 0, and clock 2 at x = 9 10 8 m. What times do the two synchronisation events correspond to in S, when v = c 3 / 4? Equation (5.9) gives t 1! = 2( 0 " 0) = 0, and t 2! = 2( 0 " 9 # 10 8 3 / 4 / 3 # 10 ) 8 = 2 ("3 3 / 4 ) $ "5.2 s. Since t 1! " t 2! the synchronisation didn t work in S! If the synchronisation involved setting both clocks to read zero, when clock 1 reads zero in S, clock 2 reads -5.2 s. This is an example of the relativity of simultaneity that we met in section 4. Let s now check that the Lorentz transformations contain the relativistic physics we have previously discovered using light clocks: time dilation, length contraction, and relativity of simultaneity. To do this we have to identify the events involved in measurements showing the particular effect. As an aside it s a general principle in relativity and quantum mechanics that deep understanding comes from focussing on measurable things. In relativity these are the times and positions of events. In quantum mechanics they are energies, momenta, positions, etc. of particles. Time dilation concerns the time for a tick of a clock that is at rest in S. Let event T 1 be the start of a tick and event T 2 the end of the tick. We might as well keep things simple by letting T 1 be the coordinate origin: x 1 =t 1 = 0. The clock is at rest in S, so T 2 has the same position, x 2 = 0, while its time is the proper tick time t p, t 2 = t p. The inverse Lorentz transformations (5.10) give the S time coordinates of T 1 and T 2 : t 1 = 0; t 2 =! t 2 " + x 2 " v / c 2 ( ) =! ( t p + 0) =! t p. This is time dilation: the moving clocks tick is longer than the proper tick by the gamma factor, which is always greater than 1. PHYS1201 Relativity Notes, 2008. 11

Length contraction concerns the result of measuring the length of a moving object. Let the object be at rest in S, with proper length L p. A length measurement in S finds the positions of both ends of the object at the same time it s crucial that the positions are found at the same time because the object is moving in S. For simplicity, let one of the measurement events, M 1, be at the coordinate origin, and the other at x 2 = L, t 2 = 0, so that L is the object s length in S. Using the Lorentz transformations (5.9): x 1! = 0, x 2! = " ( x 2 # vt 2 ) = " ( L # 0) = " L. x 2 is the object s length in S, in which it is at rest, x 2 = L p. Hence L p = γ L and L = γ 1 L p. This is length contraction: the moving object s length L is less than its proper length by the inverse gamma factor, which is always less than 1. The relativity of simultaneity concerns two events that are simultaneous in S, for example. Let one event be the coordinate origin, and the other be simultaneous, t 2 = 0, but at position x 2 = D. The times for these events in S are then, using the inverse Lorentz transformations (5.10): t 1 = 0, t 2 =! ( 0 + Dv / c 2 ) =! Dv / c 2. This is the relativity of simultaneity. The time difference between the events in S is proportional to their spatial separation in S. 6. The Meaning of the Lorentz transformations Contents The time and space coordinates used by different inertial observers in standard Summary configuration are related by the Lorentz The Lorentz transformations mix up space and time. transformations : t! = t " xv / c2 1" v 2 / c, x! = 2 x " vt 1" v 2 / c, y! = y, z! = z. (6.1) 2 In section 5 we applied these to specific events. Better understanding their significance is the next task. What do coordinates mean? Two inertial frames in standard configuration. They label or name events. Although they are not themselves space, time, or space-time, they help us to understand them. Consider the familiar two-dimensional Euclidean plane which has Cartesian x coordinates, x and y. These are uniquely defined once we fix an origin, an orientation of the x-axis, and the coordinate unit, e.g. metres. Other coordinates, x and y, may differ in any of these: origin, axis orientation, or x length unit. For example, if the dashed coordinate system is related to Two Cartesian the undashed one by a clockwise 45 degree rotation then (see diagram) coordinate systems. frames. PHYS1201 Relativity Notes, 2008. 12 y y

x! = 1 2 x " 1 2 y, y! = 1 2 x + 1 2 y. (6.2) In general, we can use geometry to get the coordinates of a point in one system from those in another. Because there are many possible Cartesian coordinate systems, there is no fundamental concept of x-ness or y-ness : the particular x and y coordinates a point has depend on choices we can make freely the location of the origin, the orientation of the x-axis, and the length unit. x-ness and y-ness are different things in different coordinate systems: one observer s x is another s y. All this is straightforward. But look again at the Lorentz transformations for x and t: t! = t " xv / c2 1" v 2 / c, x! = 2 x " vt 1" v 2 / c 2. (6.3) They are analogous to the transformations between x and y in the two-dimensional plane. Just as x and y get mixed up into x and y, x and t get mixed up into x and t. This analogy suggests that there is no fundamental concept of t-ness or x-ness : the particular t and x coordinates an event has depend on choices we can make arbitrarily. As well as the spatial choices that apply in the two-dimensional plane case, there is the choice of relative velocity. This means that t-ness depends on velocity. But t-ness is what we call time. This challenges how we usually think of time as something fundamental, something more than just a coordinate, and something profoundly different to space. But the Lorentz transformations suggest otherwise. Time and space are mixed by the Lorentz transformations: one observer s x is (part of) another s t. It was Hermann Minkowski who first fully got the significance of this. In an interesting twist, Minkowski was Einstein s maths teacher at university, and in 1899 described him as a lazy dog [11]. That was six years before Einstein s miraculous year of 1905 in which, with prodigious effort, he wrote four papers that changed the direction of physics, including his special relativity and E = mc 2 papers. In 1908 Minkowski summarised his new understanding of relativity as follows: The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. Here s a brief science fiction story to illustrate this point In a galactic cloud live gas-bags. They are 1.1 light years long and live for a year. They are born by condensing out of the cloud over a few days, age, and die by dissipating back into it. At birth they are dense and bright red. At death they are thin and grey. When humans first saw a gas-bag, they were flying along it in a space-ship travelling at 90% of light-speed. The humans mapped the gasbag at an instant of rocket frame time. They found that one end was red and the other gray. The head was dying while the feet were being born PHYS1201 Relativity Notes, 2008. 13

Who is a gas-bag at an instant in time? Is it its body at an instant in its own rest frame? Or at an instant of human time? Or is it something encompassing both? Exercise: verify that the humans observations agree with the Lorentz transformations. 7. The twin paradox Contents If a traveller goes on a relativistic round trip they come back younger than the people left behind on Earth this has been verified experimentally by taking atomic clocks on round trip aeroplane flights [8]. How can that be? According to time dilation the travellers observe Earth clocks to run slow, and hence less time should have elapsed on Earth. This incomplete argument is called the twin paradox. What do you think is missing from this argument? Summary Going fast keeps you young. If you don t include all the physics, you can get paradoxes. The time dilation argument ignores the effect of acceleration. It is easiest to understand why acceleration is important by considering the initial acceleration up to cruise speed. From the rocket frame the distance to be travelled decreases by the Lorentz contraction factor. Hence the journey is shorter than in the Earth frame, and takes less time. The travellers age less because they are going a distance shorter by the inverse Lorentz factor, at the same speed. The differential aging is only possible because different observers have different times there is no absolute time. But what about the slowed Earth clocks seen by the travellers? According to time dilation they re going slower than the travellers clocks, by the inverse Lorentz factor, and hence should read less elapsed time. What s missing from this argument is the effect of the relativity of simultaneity when the travellers change cruise frames as they turn around to come back. One might be tempted to ignore the turnaround because the travellers time involved can be as small a fraction of the total trip time as you like, in principle. However it cannot be ignored all the Earth clock s advance occurs during the turnaround. This surprising result is because of the way the Lorentz transformations mix space into time. Let s work out the details of this. To apply the Lorentz transformations we need three standard configuration frames: the Earth frame v = 0, denoted by a subscript E, the outgoing rocket cruise frame v = V relative to Earth, denoted by a subscript O, and the returning rocket cruise frame v = -V, denoted by a subscript R. Let s choose the space and time coordinate origins of the cruise frames to be coincident at the turnaround event. The position of Earth in each cruise frame at time zero is x E = - VT/2, where T is the travellers trip time. The inverse Lorentz transformations giving the time on Earth in terms of the cruise frame times immediately before and after turn around (t O = t R = 0 ) are PHYS1201 Relativity Notes, 2008. 14

t E, before = t + x V / O E c2 x = E V / c 2 1! V 2 / c 2 1! V 2 / c =!( T / 2)V 2 / c 2, 2 1! V 2 / c 2 t E, after = t R + x E (!V ) / c 2 =!x V / c 2 E 1! V 2 / c 2 1! V 2 / c = ( T / 2)V 2 / c 2 2 1! V 2 / c 2 (7.1) When the rocket changes from the frame with velocity V to that with V, the coordinate time on Earth jumps by t E, before! t E, after. If we add this jump to the dilated time of the cruise periods we find the total elapsed Earth time ( ) + TV 2 / c 2 T E = T 1! V 2 / c 2 + TV 2 / c 2 2 / c 2 1! V 2 / c 2 1! V 2 / c 2 = T 1! V 2 / c 2 (7.2) As expected, the Earth time exceeds the travellers time by the Lorentz factor. We can look at this another way using the space-time diagram. The Earth frame coordinates are x,t. The boxes represent synchronized Earth frame clocks, one set on t 2 Earth, and one set at the turnaround location. 2 The dashed x axis is the set of events simultaneous with t = 0 in S. The doubledashed x axis is the set of events simultaneous with t = 0 in S. Changing from the S frame to the S frame, the Earth clock which is simultaneous with the Earth frame clock reading zero at the turnaround location jumps from the one reading t = -1 to the one 1 0-1 -2 1 0-1 -2 reading t = +1. In this example Earth time jumps by 2 units, no matter how quick the Twin paradox space-time diagram. acceleration from S to S is. x x x What have we learnt? Firstly, that we have to include all the physics to understand a problem leave some physics out and you may get a paradox. In this case the relativity of simultaneity is required, as well as time dilation. Secondly, this problem emphasises that coordinates are just labels for events and don t necessarily correspond to anything physical. The jump in Earth time as the travellers turn around does not correspond to anything actually happening on Earth it s a change in which events the travellers regard as simultaneous, and therefore in how they label time on Earth. This example illustrates the conventional nature of time. You can choose to measure time using Earth time, S time, or S time they are all perfectly good, but they all differ as to which events are simultaneous. Which you choose is a matter of convention. When you state the time of an event you must also specify the inertial frame the time is defined in. If you forget to do this you may get paradoxes like the twin paradox. PHYS1201 Relativity Notes, 2008. 15

8. Four-dimensional geometry Contents We are now going to take a big step: exploring space-time as four-dimensional Summary geometry. It may be hard to get your head Geometry is about invariants. around at first, but work with it, because it Metrics are invariant generalised length measures. is a profoundly important aspect of reality. The interval is the metric in relativity. What do we mean by geometry? You probably think of various shapes, such as triangles and spheres, and their properties. But why are shapes interesting? It s because they stay the same when you move them (translation) or rotate them. And why is that? It turns out that a fundamental reason is that there is a measure of length or distance that is invariant under the transformations of translation and rotation. In terms of Cartesian coordinates the Pythagorean relation gives this length L, in ordinary three-dimensional space: L 2 = (!x) 2 + (!y) 2 + (!z) 2. (8.1) Here the Greek Delta Δ means difference. This equation is interpreted as: the square of the distance between two points whose coordinates differ by Δx, Δy, and Δz is given by Equation (8.1). Note that we tend to focus on the square of the length in relativity, for reasons we discuss later. The new thing is that it s possible to generalise the notion of distance by generalising Equation (8.1). This makes sense provided the new measure of distance is invariant under a physically significant set of transformations. For example, it wouldn t make sense to define a new craig-distance K by K 2 = (!x) 2 + (!y) 2 + 2 (!z) 2, (8.2) because it could change under a rotation about the x axis. For example, the two points x = y = z = 0 and x = y = 0, z = 1 have K 2 = 2; but after rotating the points by 90 degrees about the x axis, the second point becomes x = z = 0, y = 1, while the first point is unchanged, and so K 2 = 1. In fact, there s no point trying to come up with new distances in ordinary three-dimensional space nothing works. However if we go to four-dimensional space-time by combining space and time we find, in the words of Dr. Strangelove, that a new concept of distance is not only possible, but essential. Generalised measures of distance are called metrics. Our first thought for a distance in four-dimensional space-time might be the simplest generalisation of the Pythagorean relation Equation (8.1):!L 2 4 = (!x) 2 + (!y) 2 + (!z) 2 + ( c!t) 2. (wrong) (8.3) Note that we have multiplied time by the speed of light c to get units of distance that can be combined with the other distances in the expression. This is the opposite of dividing distances by the speed of light to get light-seconds as a unit of distance. PHYS1201 Relativity Notes, 2008. 16

Hence we may talk about light-metres of time, the time it takes light to travel a metre. Often physicists drop the light prefix and just talk about metres of time! For this new four-dimensional distance to be useful it should be the same in different inertial frames: i.e. invariant under the relativistic transformation of changing inertial frames. Let s see if it is by considering the events in Example 2 of section 5. In S the new distance between them is found from!l 2 4 = (!x) 2 + (!y) 2 + (!z) 2 + ( c!t) 2 = ( 9 " 10 8 ) 2 = 8.1" 10 17 m 2. (8.4) In S the Lorentz transformations give x! = 0 and x! = 2( 9 " 10 8 ) = 1.8 " 10 9 m. Evaluating! L4 2 in S, using the times from Example 2, t 1! = 0, t 2! = "5.2 s :!L 4! 2 = " x! ( ) 2 + (" y! ) 2 + (" z! ) 2 + ( c" t! ) 2 ( ) 2 + ( $5.2 # 3 # 10 8 ) 2 = 5.7 # 10 18 m 2. = 1.8 # 10 9 (8.5) Since (8.4) and (8.5) are not equal, L4! 2!!L 4 " 2, the Pythagorean generalisation isn t invariant between inertial frames. After a bit of playing around we can discover that changing the sign of the time term fixes the problem: L 2 4 = (!x) 2 + (!y) 2 + (!z) 2 " ( c!t) 2. (correct) (8.6) Let s check for Example 2. The distance in S doesn t change because Δt = 0,!L 2 4 =!x ( ) 2 + (!y) 2 + (!z) 2 " ( c!t) 2 = 8.1# 10 17 m 2. In S it gives L 4! 2 = " x! ( ) 2 + (" y! ) 2 + (" z! ) 2 # ( c" t! ) 2 ( ) 2 # (#5.2 $ 3 $ 10 8 ) 2 = 8.1$ 10 17 m 2. = 1.8 $ 10 9 (8.7) Wow! It s the same, L 2 4 = L 4! 2. We can prove that the new four-dimensional distance is the same in all inertial frames by using the Lorentz transformations, Equation (6.1). L 4! 2 = " x! ( ) 2 + (" y! ) 2 + (" z! ) 2 # ( c" t! ) 2 = ( $ ("x # v"t) ) 2 + ("y) 2 + "z = $ 2 "x 2 ( 1 # v 2 / c 2 ) # 2"x"t v # v ( ( )) 2 ( ) ( ) 2 # c$ "t # v"x / c 2 ( ( ) + "t 2 v 2 # c ) 2 + ("y) 2 + ("z) 2 = "x 2 # c 2 "t 2 + "y 2 + "z 2 = L 4 2 (8.8) It is conventional to change the sign. This is then often denoted by s 2 and is called the interval squared: PHYS1201 Relativity Notes, 2008. 17

s 2 = c 2!t 2 "!x 2 "!y 2 "!z 2. (8.9) s is called the interval. The set of all events together with the interval metric is what is meant by four-dimensional space-time geometry. Note that space-time is not just three-dimensional space plus one-dimensional time the new metric is central. Strangely, if s 2 < 0, s is imaginary. This corresponds to space-like separated events, because there is some inertial frame in which they occur simultaneously but at different positions. It turns out that this possibility doesn t really cause any problems, because we can just deal with the interval squared, s 2. When s 2 > 0, s/c is called the proper time, denoted by the Greek letter tau, τ. Events for which s 2 > 0 are said to be time-like separated, because there is some inertial frame in which they occur at different times but at the same position. Events for which s 2 = 0 are said to be light-like separated, since a light pulse can be present at both events. It s a strange idea that separate events can have zero separation but that s how things are in space-time. A useful representation of the geometry of space-time is the light cone. The light cone of an event E is all events that are light-like separated from it that is, all events that would be reached by a light pulse emitted by E, together with all events that would illuminate E if they emitted a light pulse. The former events form the future light cone of E, and the latter events form the past light cone of E. This is shown in the diagram, which has one spatial dimension suppressed, and units chosen so that light rays emitted by, or converging on, the marked event form the cone. Events inside the future light cone are those that can be reached from E by travelling slower than light they are said to be in the future of E. Events inside the backward light cone are those from which E could be reached by travelling slower than light they are said to be in the past of E. Light cone of an event. Events inside the light cone are said to be causally connected to E, even if they didn t actually effect E. Events outside the light cone are said to be elsewhere or causally disconnected from E. This is because E can neither influence or be influenced by those events. It is events elsewhere whose time order relative to E depends on the inertial frame (see discussion in section 5). Events in the future light cone are always in the future of E, in any frame. And events in the past light cone are always in the past of E, in any frame. Every event has its own light cone, so they are hard to draw as they overlap, but the diagram gives the idea. It has one spatial dimension suppressed, and time is along the vertical axis. The blue lines indicate the world lines of particles at rest. Light cones of different events. PHYS1201 Relativity Notes, 2008. 18

9. 4-vectors Contents Three-dimensional vectors are an aspect of three-dimensional Euclidean geometry that are very useful in Newtonian physics. Why? Summary Just as 3-vectors are important in Newtonian physics, 4-vectors are important in 4-dimensional relativistic physics. Useful 4-vectors are: 4-position, 4-velocity, 4-acceleration, 4-momentum, 4-wave-vector. 4-momentum conservation incorporates conservation of relativistic momentum and conservation of energy. One reason is that they have a length that is invariant under translation and rotation. There is also a scalar product between vectors that is an invariant. The scalar product is also called the dot product. length:! A =! A!! A = A x 2 + A y 2 + A z 2, dot product:! A!! B = A x B x + A y B y + A z B z =! A! B cos". (9.1) These quantities are the same no matter which coordinate system they are evaluated in. They are said to be coordinate independent. They may also be regarded as the simplest type of geometric object -- a number that is invariant under translation and rotation. When this aspect is being emphasised they are called scalars. In section 8 we derived an interval that is invariant under boosts between inertial frames. This allows us to regard the set of four coordinates, ct, x, y,z, as forming the four-dimensional space-time 4-position vector : R = ( ct, x, y,z) = ( R t, R x, R y, R z ) (9.2) 4-vectors will be denoted by a double underline. Note that all components have the same units. The first component, which we will refer to as the time component, has units of length because it is actually ct. Just as in Euclidean geometry we can generalise from this to define a general 4-vector as anything that has the same geometric properties as the position vector: A = ( A t, A x, A y, A z ) (9.3) Specifically, its components must transform between inertial frames in the same way as the components of the 4-position vector; i.e. they must obey the Lorentz transformations A t! = " ( A t # A x v / c), A x! = " ( A x # A t v / c), A y! = A y, A z! = A z (9.4) Note the difference to the Lorentz transformations for x and t because A t has the same units as A x, and hence we use the Lorentz transformations appropriate for x and ct. Later, we will introduce the 4-velocity and 4-acceleration, which are examples of 4- vectors. PHYS1201 Relativity Notes, 2008. 19

Analogously with 3D Euclidean geometry we can define a 4D length and scalar product for 4-vectors that are the same, no matter which inertial frame they are evaluated in: 4D length: A = A! A = A t 2 " A x 2 " A y 2 " A z 2, 4D scalar product: A! B = A t B t " A x B x " A y B y " A z B z. (9.5) The proof that these give the same numbers whichever inertial frame they are evaluated in is the same as the proof of the invariance of the interval in section 8, Eq.(8.8). Exercise. Prove that A! B = A t B t " A x B x " A y B y " A z B z = A t # B t # " A x # B x # " A y # B y # " A z # B z #, where the dashed frame components are related to the undashed ones by the usual Lorentz transformation, Eq.(6.1). From now on we will assume that the geometry of nature is that of four-dimensional space-time, and hence that fundamental physics should be formulated using 4-vectors just as Newtonian mechanics was formulated using Euclidean 3-vectors. This is a basic principle that has guided physics for the last hundred years, shaping our theories of fundamental particles, for example. The first task is to find some interesting 4-vectors. To do this we start from the one we already have, the 4-position R = ct, x, y, z ( ). In Newtonian physics we could differentiate the position 3-vector of a particle with respect to time to get a new vector the velocity. This works because in Newtonian physics time is a geometric scalar, independent of the coordinate system. The analogous geometric scalar in relativity is proper time. If we differentiate the 4- position with respect to proper time we get a new 4-vector, the 4-velocity: U = dr d! = dr d " #1 t ( ) = " d dt ( ) (9.6) $ ( ct, x, y,z) = " c, dx dt, dy dt, dz ' % & dt ( ) = " c,u x,u y,u z The u x,u y,u z are just the velocity components we are used to nothing special. Coordinate time t is related to proper time, here denoted τ, as discussed in section 3, Eq.(3.1): t =!" # " =! $1 t and! = 1 / 1$! u 2 / c 2. Why did we differentiate with respect to proper time, rather than coordinate time? Because proper time is a geometric invariant, but coordinate time is not. Coordinate time depends on a choice of inertial frame, and hence is conventional, having no significance beyond that choice. Proper time is defined by the object whose 4-velocity we are calculating. It is a property of the object, and therefore independent of any choice of inertial frame. Differentiating a 4-vector with respect to proper time therefore gives another 4-vector, another geometric object. PHYS1201 Relativity Notes, 2008. 20

According to our previous discussion the length of U should be a coordinate independent geometric invariant, and therefore potentially of physical significance: U = U!U = (" c) 2 # (" u x ) 2 # (" u y ) 2 # (" u z ) 2 = " c 2 # u! 2 = c (9.7) Well, the speed of light is a indeed fundamental invariant, but it s not very interesting. We can use the fact that the components of U transform according to the Lorentz transformation to determine how particle velocity transforms between inertial frames S and S. The transformation can t be as simple as subtracting the frame velocity v from the particle velocity u x, as in Newtonian mechanics, because this would imply particle velocities greater than the speed of light: e.g. if u x =!3c / 4 and v = c / 2, then Newton predicts u x! v =!5c / 4. Applying the Lorentz transformation Eq.(9.4) to the 4-velocity components gives! u" c =! v (! u c #! u u x v / c) =! v! u c( 1# u x v / c 2 ),! u" u x " =! v (! u u x #! u c( v / c) ) =! v! u ( u x # v),! u" u y " =! u u y,! u" u z " =! u u z. (9.8) Note that we were careful to use subscripts to distinguish between! u = 1 / 1" u! 2 / c 2 associated with the particle s velocity u!,! u " = 1 / 1# u! " 2 / c 2 associated with the particle s velocity u!! and! v = 1 / 1" v 2 / c 2 associated with the relative speed of the frames v. Dividing the second equation (9.8) by the first we get an expression for the x! component of the particle s velocity u! x c = u x " v ( ) # u! c 1" u x v / c 2 x = u x " v 1 " u x v / c 2 (9.9) Let s check this on the example above, u x =!3c / 4 and v = c / 2 : u x! = u x " v 1 " u x v / c = "3c / 4 " c / 2 2 1 " "3 / 8 ( ) = "5 / 4 11 / 8 c = " 10 11 c (9.10) This is less than light-speed and hence the formula is reasonable. Dividing each of the third and fourth equations of (9.8) by the first we obtain expressions for the y! and z! components of the particle s velocity, that are perpendicular to the relative frame velocity u y! = u y ( ), u! " v 1 # u x v / c 2 z = u z " v 1# u x v / c 2 ( ). (9.11) PHYS1201 Relativity Notes, 2008. 21

We can use these results to learn how velocities add up in relativity. In Newtonian physics if an object A has velocity v! relative to us, and another object B has velocity! u relative to A, then B has velocity v! + u! relative to us. This can t be right for relativistic velocities otherwise B might be predicted to have a speed faster than light. We can work out the correct way to combine velocities using Eqs.(9.9,9.11). Let the rest frame of A be S, with speed v relative to the S frame. Let the velocity of B through S be along the x axis with speed u x!. With a bit of algebra we can solve Eq.(9.9) for u x in terms of u x! : u x = u x! + v 1+ u x! v / c 2 (relativistic velocity addition) (9.12) The numerator is the familiar Newtonian velocity addition. The denominator is the relativistic correction. We can now differentiate the 4-velocity with respect to proper time to get a new 4- vector, the 4-acceleration: A = du d! = du d " #1 t ( ) = " d dt (" c," u x," u y," u z ) $ = " c d" dt, d" dt u x + " du x dt, d" dt u y + " du y dt, d" dt u z + " du z ' % & dt ( ) $ = " c d" dt, d"! u + " d! u ' $ % & dt dt ( ) = " c d" dt, d"! u + " a! ' % & dt ( ) (9. 13) where! a is the familiar 3-acceleration. The length of the 4-aceleration can be found by the trick of choosing a frame in which the components are simple. In the instantaneous rest frame of the object! u = 0 and A =! ( 0,! a! ) = ( 0, a! ) (instantaneous rest frame) (9.14) since γ =1, and d γ /dt = 0 since it is proportional to u (check!). Therefore A 2 =!! a p 2, with! ap the proper acceleration, the acceleration in the instantaneous rest frame. This is the acceleration felt by the object. It is physically important because it determines the physical effect of the acceleration on the object such as crushing. We can calculate the space-time scalar product of the 4-velocity and 4-acceleration in any inertial frame we like its always the same number. It s easiest to calculate in the instantaneous rest frame: A = ( 0, a! p ), U = ( c, 0! )! A "U = 0 (9.15) The 4-velocity and 4-acceleration are always relativistically orthogonal. PHYS1201 Relativity Notes, 2008. 22

Another useful 4-vector is the 4-momentum P. This is the product of a particle s rest mass m 0 and its 4-velocity: P = m 0 U = m 0! ( c, u! ) = m( c, u! ) (9.16) where we have defined the mass m = m 0!. The rest mass is the inertial mass of the particle in its rest frame, and is therefore a property of the particle itself. It is inertial frame independent, and a geometric scalar (rest mass also applies to systems of particles). The length of the 4-momentum is P = m 0 c. Why? The spatial part of the 4-momentum is called the relativistic momentum! p = m 0!! u [13]. For speeds small compared to light-speed, u / c! 1, it reduces to the Newtonian momentum. Since Newtonian momentum is conserved, we might hypothesise that relativistic momentum is also conserved. In fact, experiments confirm this. A useful mathematical theorem about 4-vectors is the zero component lemma [14]: if a particular component of a 4-vector is zero in all inertial frames then the 4- vector is the zero 4-vector. It is proved by contradiction: if any component of the vector were non-zero in some frame, there would be some relative frame velocity which Lorentz transforms it into the component that is assumed to be zero. Conservation of relativistic momentum means that the difference of the spatial components of the 4-momentum, at different times, is zero. And this is true in all frames. The zero component lemma therefore implies that the difference of the time components of the 4-momentum is also zero in all frames. So conservation of relativistic momentum gives us another conservation law for free, by the magic of 4- vector geometry. Looking at the 4-momentum time component in Eq.(9.16) we see that mc is conserved: or multiplying by c, that mc 2 is conserved. Since there aren t too may things that are conserved, and mc 2 has the units of energy, this is a strong hint that mc 2 is the relativistic energy of a particle: E = mc 2. This turns out to be true, and shows the power of 4-vectors in unifying parts of physics that we had previously thought were different. Here, momentum and energy. They turn out to be different components of the more fundamental 4-momentum. It is 4-momentum that is conserved. In a particular inertial frame, this means its components are conserved, and therefore that energy and momentum are conserved. 4-vectors are also needed to describe waves in space-time. A plane wave in 3- dimensional space is defined by the spatial planes on which the wave disturbance has constant phase, e.g. the wave peak. These phase-planes are separated by the wavelength λ. Let the normal vector to these planes be! n = n x,n y,n z ( ). If the plane is at a distance d from the origin, points on it satisfy! n!! r = d (sketch a diagram if this isn t clear to you). If the phase-planes propagate with speed w, the events of a phaseplane satisfy, assuming the plane passed the origin at t = 0, PHYS1201 Relativity Notes, 2008. 23

! n! r! = wt " wt # n!! r! = 0 " K! R = 0, where K = 1 % w $ c, n! ( % & ' ) * = f & ' f = w /! is the wave frequency, and K is called the 4-wave-vector. 1 c,! n ( w) *. (9.17) Although we ve used the 4-vector notation for K, we haven t actually proved that it is a 4-vector. This follows from K! R = 0 being true in all inertial frames. Since R is a 4-vector, and 0 is a scalar, K must be a 4-vector otherwise K! R = 0 would not be true in all frames. In 1923 de Broglie made the seminal quantum mechanical hypothesis that every particle has an associated wave, and every wave a particle. Specifically, he suggested that their 4-momentum and 4-wave-vector were related by P = hk! m( c, u! ) = hf The time component of this equation is! " 1 c, n % # $ w& ', h=6.6 ( 10-24 Js, Planck's constant (9.18) mc = hf / c! mc 2 = hf (9.19) Einstein had hypothesised in 1905 that the energy of a photon is E = hf [4]. De Broglie assumed that this relation applied to all particles, not just photons. This implies that the energy of a particle is E = mc 2. Einstein had already suggested this in his second 1905 relativity paper [2]. His argument there was based on the relativistic Doppler effect. 10. Relativistic optics Contents According to the previous section, Eq.(9.17), the 4-wave-vector for a light wave, with wave speed w = c is:!! 1 K = f c, n $ " # c % &. (10.1) It is convenient to define the wave 4-frequency by: F = Kc = f ( 1, n! ). (10.2) Recall that f is the wave frequency and! n the unit vector in the direction of propagation. The 4-vector Lorentz transformations, Eq.(9.4), of F relate the light wave frequency and propagation direction in different inertial frames: Summary If you were travelling near the speed of light things would look quite different because of: aberration, the Doppler effect, and the headlight effect. f! = " ( f # fn x v / c) = " f ( 1# n x v / c), (10.3) f! n x! = " ( fn x # fv / c) = " f ( n x # v / c), f! n y! = fn y, f! n z! = fn z PHYS1201 Relativity Notes, 2008. 24

The first equation is the relativistic Doppler effect. In terms of the photon model of light, discussed at the end of section 9, P = hk! ( E / c, p! ) = hf ( c 1, n! )! E = hf. (10.4) The photon energy is proportional to the wave frequency, and hence the Doppler effect changes the photon energy. The behaviour of the light propagation direction! n is known as relativistic aberration. It produces a small change in the observed direction of stars as the earth s orbital velocity changes, for example [5]. Of most interest to us are the transformations relevant to the Real Time Relativity simulation in the PHYS1201 lab. The rocket we fly around is the usual S frame and the world is the S frame. The inverse Lorentz transformations are: ( ) =! " ( f v / c) =! " f =! f " + f " n x" v / c fn x =! f " n x " + " fn y = f " n y ", fn z = f " n " f ( 1+ n x" v / c), f ( n x " + v / c),. (10.5) z As for Eq.(5.10) these follow by swapping dashed and undashed quantities and changing the sign of v in (10.3). Solving the Doppler equation for f! : S S y y f, θ θ x, x Aberration of incoming light.! n f,!! n ( ) #1 = f f! = " #1 f 1+ n x! v / c 1# v 2 / c 2 1+ n x! v / c. (10.6) Since n! is a unit vector, n x! is the cosine of the angle "! between the light ray and the x axis. Since we are interested in how things look, we are interested in rays coming in towards the observer. Then n x! = " cos #!, and f! = f 1 " v 2 / c 2 = Df, (10.7) 1" ( v / c)cos #! where this equation defines the Doppler factor D. The numerator may be interpreted as the effect of time dilation, while the denominator is due to wave-front compression or expansion. For waves incoming perpendicular to the relative motion, at "! = 90, the denominator is one and the observed frequency is less than the world frequency. This means that the time between wave crests, the period, is longer; which is exactly the effect of time dilation, if the emission of wave crests is regarded as a clock. Since the denominator depends on v/c, while the numerator depends on (v/c) 2, the denominator dominates the Doppler effect for low speeds. For light coming from in PHYS1201 Relativity Notes, 2008. 25

front, "! < 90 we have cos "! > 0, so the denominator is less than one, and the frequency is increased, or blue shifted. For light coming from behind, "! > 90 we have cos "! < 0, so the denominator is greater than one, and the frequency is decreased, or red shifted. Dividing the second equation of (10.3) by the first and using n x! = " cos #!, n x =! cos" we get: n x! = n x " v / c 1 " n x v / c # cos $! = cos$ + v / c 1 + ( v / c)cos$. (10.8) Dividing the third equation of (10.3) by the first we get the alternative form: n y! = n y " 1# n x v / c ( ) $ sin %! = sin% " 1 + ( v / c)cos% ( ). (10.9) Substituting these expressions into the trigonometric identity ( ) = sin! tan 1 2 "! " 1+ cos "! (10.10) we get a third form: tan( 1 2 "!) = = ( )( 1 + v / c) = tan( 1 ") 2 # ( 1+ v / c) = 1$ v2 / c 2 1+ v / c ( 1 + v / c) ( 1 $ v / c) 1+ v / c sin" # 1+ cos" tan( 1 " 2 ) = 1$ v / c 1+ v / c tan ( 1 " 2 ) tan( 1 " 2 ) (10.11) This form allows us to understand relativistic aberration using a construction due to Roger Penrose [16]. It is based on the idea that our instantaneous visual field may be regarded as painted on the inside of a sphere surrounding us called the view-sphere. Further, the view-sphere may be mapped to a plane by stereographic projection [17]. A stereographic projection of the northern hemisphere is shown. As you can see, θ /2 the stereographic projection has the property that circles on the sphere are P mapped to circles or lines on the map. A stereographic projection is made by choosing a point P on the sphere and projecting onto the tangent plane T at the opposite point. The projection is done by drawing a line from P through the O θ PHYS1201 Relativity Notes, 2008. 26 Stereographic projection. T Aberration construction based on a viewsphere of unit diameter. tan! 2

sphere to T. The point on the sphere intersected by the line is mapped onto the plane T, see diagram. In the diagram we have noted the geometric fact that the angle subtended by an arc of a circle at the circumference is half that subtended at the centre. Proof? Choosing the view-sphere to have unit diameter, the height of the stereographic projection point on the plane T is tan! / 2 ( ). This makes Eq.(10.11) useful: tan( 1 2 "!) = 1# v / c 1+ v / c tan ( 1 " 2 ). (10.12) If two observers, at rest in S and S, are coincident at the centre of the projection sphere, O, then the stereographic projections of their view-spheres differ only by the scaling factor 1! v / c / 1 + v / c. We can immediately deduce two things from this construction. Straight-line objects in S, such as rods, map to circular arcs on the view-sphere. These stereographically project to circular arcs or lines on the plane T. The aberration transformation Eq.(10.12) stretches or shrinks these uniformly so that they remain circular arcs or lines. Projecting the stretched plane T back onto the view-sphere gives the view-sphere of the relatively moving observer S. Because of the property of stereographic projection, S sees rods as a circular arcs or lines. This can be easily seen in the Real-time Relativity simulation, see image. Relativistic aberration. The other thing we can deduce is that spheres, which always have circular outlines (unlike circles which may have elliptical outlines), will continue to have a circular outline after aberration since stereographic projection maps circles to circles. This can also be seen in the Real-time Relativity simulation. However the details of the surface of the sphere are distorted by aberration. There is a third relativistic optics effect, in addition to the Doppler effect and aberration: the headlight effect. This is the increased intensity of light coming from objects we are moving towards. Here, intensity means power per unit solid angle. The opposite effect occurs for objects we are moving away from. There are three separate physical causes that combine to produce these intensity changes: the change in angular size of the emitting region, the Doppler change in energy of the photons, and the change in photon flux due to the combined effects of time dilation and the observer s motion. In terms of the Doppler factor in Eq.(10.7) these The headlight effect. PHYS1201 Relativity Notes, 2008. 27