A useful form of the quotient rule for three-dimensional

Using arguments similar to those of Section 15.2, prove that Newton's first and third laws are invariant under the Galilean transformation.

Consider a classical inelastic collision of the form A -I- B D. (For example, this could be a collision such as Na + Cl > Na+ + Cl in which two neutral atoms exchange an electron and become oppositely charged ions.) Show that the law of conservation of classical momentum is invariant under the Galiliean transformation if and only if total mass is conserved as is certainly true in classical mechanics. (We shall find in relativity that the classical definition of momentum has to be modified and that total mass is not conserved.)

A low-flying earth satellite travels at about 8000 m/s. What is the factor y for this speed? As observed from the ground, by how much would a clock traveling at this speed differ from a ground-based clock after one hour (as measured by the latter)? What is the percent difference?

What is the factor y for a speed of 0.99c? As observed from the ground, by how much would a clock traveling at this speed differ from a ground-based clock after one hour (one hour as measured by the latter, that is)?

A space explorer A sets off at a steady 0.95c to a distant star. After exploring the star for a short time, he returns at the same speed and gets home after a total absence of 80 years (as measured by earth-bound observers). How long do A's clocks say that he was gone, and by how much has he aged as compared to his twin B who stayed behind on earth? [Note: This is the famous "twin paradox." It is fairly easy to get the right answer by judicious insertion of a factor of y in the right place, but to understand it, you need to recognize that it involves three inertial frames: the earth- bound frame 5, the frame 8' of the outbound rocket, and the frame 8" of the returning rocket. Write down the time dilation formula for the two halves of the journey and then add. Notice that the experiment is not symmetrical between the two twins: A stays at rest in the single inertial frame 5, but B occupies at least two different frames. This is what allows the result to be unsymmetrical.]

When he returns his Hertz rent-a-rocket after one week's cruising in the galaxy, Spock is shocked to be billed for three weeks' rental. Assuming that he traveled straight out and then straight back, always at the same speed, how fast was he traveling? (See note to Problem 15.5.)

The muons created by cosmic rays in the upper atmosphere rain down more-or-less uniformly on the earth's surface, although some of them decay on the way down, with a half-life of about 1.5 ,u,s (measured in their rest frame). A muon detector is carried in a balloon to an altitude of 2000 m, and in the course of an hour detects 650 muons traveling at 0.99c toward the earth. If an identical detector remains at sea level, how many muons should it register in one hour? Calculate the answer taking account of the relativistic time dilation and also classically. (Remember that after n half-lives, 2' of the original particles survive.) Needless to say, the relativistic answer agrees with experiment.

The pion (n- or 7r) is an unstable particle that decays with a proper half-life of 1.8 x 10-8 s. (This is the half-life measured in the pion's rest frame.) (a) What is the pion's half-life measured in a frame S where it is traveling at 0.8c? (b) If 32,000 pions are created at the same place, all traveling at this same speed, how many will remain after they have traveled down an evacuated pipe of length d = 36 m? Remember that after n half-lives, 2' of the original particles survive. (c) What would the answer have been if you had ignored time dilation? (Naturally it is the answer (b) that agrees with experiment.)

One way to set up the system of synchronized clocks in a frame 8, as described at the beginning of Section 15.4, would be for the chief observer to summon all her helpers to the origin 0 and synchronize their clocks there, and then have them travel to their assigned positions very slowly. Prove this claim as follows: Suppose a certain observer is assigned to a position P at a distance d from the origin. If he travels at constant speed V, when he reaches P how much will his clock differ from the chief's clock at 0? Show that this difference approaches 0 as V 0.

Time dilation implies that when a clock moves relative to a frame \(\mathcal{S}\), careful measurements made by observers in \(\mathcal{S}\) will find that the clock is running slow. This is not at all the same thing as saying that a single observer in 8 will see the clock running slow, and this latter statement is not always true. To understand this, remember that what we see is determined by the light as it arrives at our eyes.Consider an observer standing close beside the x axis as a clock approaches her with speed V along the axis. As the clock moves from position A to B, it will register a time \(\Delta t_{\mathrm{o}}\), but as measured by the observer's helpers, the time between the two events ("clock at A" and "clock at B") is \(\Delta t=\gamma \Delta t_{o}\) . However, since B is closer to the observer than A is, the light from the clock at B will reach the observer in a shorter time than will the light from A. Therefore, the time \(\Delta t_{\mathrm{see}}\) between the observer's seeing the clock at A and seeing it at B is less that \(\Delta t\). (a) Prove that \(\Delta t_{\mathrm{see}}=\Delta t(1-\beta)=\Delta t_{0} \sqrt{\frac{1-\beta}{1+\beta}}\) (which is less than \(\Delta t_{o}\)). Prove both equalities. (b) What time will the observer see once the clock has passed her and is moving away? The moral of this problem is that you must be careful how you state or think about time dilation. It's fine to say "Moving clocks are observed, or measured, to run slow," but it is definitely wrong to say "Moving clocks are seen to run slow."

As a meter stick rushes past me (with velocity v parallel to the stick), I measure its length to be 80 cm. What is v?

Consider the experiment of Problem 15.8 from the point of view of the pions' rest frame. What is the half-life of the pions in this frame? In part (b), how long is the pipe as "seen" by the pions and how long does it take to pass the pions? How many pions remain at the end of this time? Compare with the answer to Problem 15.8 and describe how the two different arguments led to the same result.

a) A meter stick is at rest in frame So, which is traveling with speed V = 0.8c in the standard configuration relative to frame S. (a) The stick lies in the xoyo plane and makes an angle Bo = 60 with the xo axis (as measured in So). What is its length 1 as measured in 8, and what is its angle B with the x axis? [Hint: It may help to think of the stick as the hypotenuse of a 30-60-90 triangle of plywood.] (b) What is 1 if 19 = 60? What is 80 in this case?

Like time dilation, length contraction cannot be seen directly by a single observer. To explain this claim, imagine a rod of proper length lo moving along the x axis of frame S and an observer standing away from the x axis and to the right of the whole rod. Careful measurements of the rod's length at any one instant in frame S would, of course, give the result / = /o/ y . (a) Explain clearly why the light which reaches the observer's eye at any one time must have left the two ends A and B of the rod at different times. (b) Show that the observer would see (and a camera would record) a length more than 1. [It helps to imagine that the x axis is marked with a graduated scale.] (c) Show that if the observer is standing close beside the track, he will see a length that is actually more than lo; that is, the length contraction is distorted into an expansion.

Solve the Lorentz transformation equations (15.20) to give x, y, z, t in terms of x', y', z', t'. Verify that you get the inverse Lorentz tranformation (15.21). Observe that you could have found the same result by interchanging primed and unprimed variables and changing V to V.

Consider two events that occur at positions r1 and r2 and times tl and t2. Let Ar = r2 r1 and At = t2 t1. Write down the Lorentz transformation for r1 and t1, and likewise for r2 and t2, and deduce the transformation for Ar and At. Notice that differences Ar and At transform in exactly the same way as r and t. This important property follows from the linearity of the Lorentz transformation.

Consider two events that occur simultaneously at t = 0 in frame 3, both on the x axis at x = 0 and x = a. (a) Find the times of the two events as measured in a frame S' traveling in the positive direction along the x axis with speed V. (b) Do the same for a second frame 3" traveling at speed V but in the negative direction along the x axis. Comment on the time ordering of the two events as seen in the three different frames. This startling result is discussed further in Section 15.10.

Use the inverse Lorentz transformation (15.21) to rederive the time-dilation formula (15.8). [Hint: Consider again the thought experiment of Figure 15.3, with the flash and the beep that occur at the same positions as seen in frame 3'.]

A traveler in a rocket of proper length 2d sets up a coordinate system S' with its origin 0' anchored at the exact middle of the rocket and the x' axis along the rocket's length. At t' = 0 she ignites a flashbulb at 0'. (a) Write down the coordinates xF, tF and x/3' , tB for the arrival of the light at the front and back of the rocket. (b) Now consider the same experiment as observed from a frame S relative to which the rocket is traveling with speed V (with S and 3' in the standard configuration). Use the inverse Lorentz transformation to find the coordinates xF, tF and xB, tB for the arrival of the two signals. Explain clearly in words why the two arrivals are simultaneous in 3' but not in S. This phenomenon is called the relativity of simultaneity.

Newton's first law can be stated: If an object is isolated (subject to no forces), then it moves with constant velocity. We know that this is invariant under the Galilean transformation. Prove that it is also invariant under the Lorentz transformation. [Assume that it is true in an inertial frame 3, and use the relativistic velocity-addition formula to show that it is also true in any other 3'.]

A rocket traveling at speed 4c relative to frame S shoots forward bullets traveling at speed ic relative to the rocket. What is the speed of the bullets relative to 3?

A rocket is traveling at speed 0.9c along the x axis of frame 3. It shoots a bullet whose velocity v' (measured in the rocket's rest frame 3') is 0.9c along the y' axis of 3'. What is the bullet's velocity (magnitude and direction) as measured in 5?

As seen in frame 3, two rockets are approaching one another along the x axis traveling with equal and opposite velocities of 0.9c. What is the velocity of the rocket on the right as measured by observers in the one on the left? [This and the previous two problems illustrate the general result that in relativity the "sum" of two velocities that are less than c is always less than c. See Problem 15.43.]

A robber's getaway vehicle, which can travel at an impressive 0.8c, is pursued by a cop, whose vehicle can travel at a mere 0.4c. Realizing that he cannot catch up with the robber, the cop tries to shoot him with bullets that travel at 0.5c (relative to the cop). Can the cop's bullets hit the robber?

A rocket is traveling at speed V along the x axis of frame 8. It emits a signal (for example, a pulse of light) that travels with speed c along the y' axis of the rocket's rest frame 8'. What is the speed of the signal as measured in 8?

Two objects A and B are approaching one another, traveling in opposite directions along the x axis of frame S with speeds vA and vB. At time t = 0, they are at positions x = 0 and x = d. Write down their positions for an arbitrary time t and show that they meet at time t = d/(vA vB). Notice that this implies that the relative velocity of the two objects is VA vB as measured in the frame 8 in which they are both moving. This may seem surprising at first thought, since we can clearly choose values of VA and vB for which this relative velocity is larger than c.32

Frame 8' travels at speed V1 along the x axis of frame 8 (in the standard configuration). Frame 8" travels at speed V2 along the x' axis of frame 8' (also in the standard configuration). By applying the standard Lorentz transformation twice find the coordinates x", y", z", t" of any event in terms of x, y, z, t. Show that this transformation is in fact the standard Lorentz transformation with velocity V given by the relativistic "sum" of V1 and V2.

The relativistic velocity-addition formula is the answer to the following question: If u is the velocity of an inertial observer B relative to an observer A, and v is the velocity of C relative to B, what is the velocity w of C relative to A? Let us denote the answer by w = "u v." In classical physics, this is just the ordinary vector sum of u and v; in relativity, it is given by the inverse of the velocity addition formulas (15.26) and (15.27) (at least for the case that u points along the x axis). Taking u = (u, 0, 0) and v = (0, v, 0), write down the components of "u + v" and also of "v + u." [Be careful to distinguish between the y factors yu and y, pertaining to u and v.] Show that "u + v" A "v + u," but that the two vectors have equal magnitudes and differ only by a rotation about the z axis. This rotation is sometimes called the Wigner rotation and is the cause of the so-called Thomas precession, which has an important effect on the fine structure of atomic energy levels.

(a) Find the 3 x 3 matrix R(0) that rotates three-dimensional space about the x3 axis, so that el rotates through angle 0 toward e2. (b) Show that [R(0)]2 = R(20), and interpret this result.

The "angle" 0 introduced in connection with Equation (15.40) has several useful properties. For any speed v < c (with corresponding factors ,8 and y) we can define 0 so that y = cosh 4). Defined in this way, 0 is called the rapidity corresponding to v. Prove that sinh 4 = py and that tanh = 13.

Here is a handy property of the rapidity introduced in Problem 15.30: Suppose that observer B has rapidity 01 as measured by A and that C has rapidity 4)2 as measured by B (with both velocities along the x axis). That is, the speed of B relative to A has Su = tanh Oland so on. Prove that the rapidity of C as measured by A is just 4) = 41 + 02

In Section 15.8, I claimed that the 4 x 4 matrix AR corresponding to a pure rotation has the block form (15.44). Verify this claim by writing out the separate components of the equation x' = ARx and showing that the spatial part (x1, x2, x3) is rotated, while x4 is unchanged.

(a) By exchanging x1 and x2, write down the Lorentz transformation for a boost of velocity V along the x2 axis and the corresponding 4 x 4 matrix AB2. (b) Write down the 4 x 4 matrices AR+ and AR_ that represent rotations of the x1x2 plane through +7/2, with the angle of rotation measured counterclockwise. (c) Verify that AB2 = AR_ AB lAR, where AB1 is the standard boost along the x1

Let AB(0) denote the 4 x 4 matrix that gives a pure boost in the direction that makes an angle 0 with the x1 axis in the x1x2 plane. Explain why this can be found as AB(0) = AR (-0)AB(0)AR(0), where AR(0) denotes the matrix that rotates the x1x2 plane through angle 0 and AB(0) is the standard boost along the x1 axis. Use this result to find AB(0) and check your result by finding the motion of the spatial origin of the frame S as observed in 8'.

Prove the following useful result, called the zero-component theorem: Let q be a four-vector, and suppose that one component of q is found to be zero in all inertial frames. (For example, q4 = 0 in all frames.) Then all four components of q are zero in all frames.

We have seen that the scalar product x x of any four-vector x with itself is invariant under Lorentz transformations. Use the invariance of x x to prove that the scalar product x y of any two four-vectors x and y is likewise invariant.

Verify directly that x' y' = x y for any two four-vectors x and y, where x' and y' are related to x and y by the standard Lorentz boost along the x1 axis.

As an observer moves through space with position x(t), the four-vector (x(t), ct) traces a path through spacetime called the observer's world line. Consider two events that occur at points P and Q in spacetime. Show that if, as measured by the observer, the two events occur at the same time t, then the line joining P and Q is orthogonal to the observer's world line at the time t; that is, (xp xQ) dx = 0, where dx joins two neighboring points on the world line at times t and t + dt.

Suppose that a point P in spacetime with coordinates x = (x, x4) lies inside the backward light cone as seen in frame S. This means that x - x < 0 and x4 < 0 at least in frame S. Prove that these two conditions are satisfied in all frames. Since this means that all observers agree that t < 0, this justifies calling the inside of the backward light cone the absolute past.

Show that the statement that a point x in spacetime lies on the forward light cone is Lorentz invariant.

In the proposition on page 627, it is obvious that at least one of the three statements has to be true. In the proof given there, I showed that if statement (1) is true, then so are statements (2) and (3). To complete the proof, show that (2) implies (1) and (3). [Strictly speaking you should also check that (3) implies (1) or (2), but this is so similar to the argument already given that you needn't bother.]

Prove that if x is time-like and x y = 0, then y is space-like.

(a) Show that if a body has speed v < c in one inertial frame, then v < c in all frames. [Hint: Consider the displacement four-vector dx = (dx, c dt), where dx is the three-dimensional displacement in a short time dt.] (b) Show similarly that if a signal (such as a pulse of light) has speed c in one frame, its speed is c in all frames.

(a) Show that if q is time-like, there is a frame 8' in which it has the form q' (0, 0, 0, (4). (b) Show that if q is forward time-like in one frame 8, then it is forward time-like in all inertial frames.

The quotient rule derived at the start of Section 15.11 is only one of several similar quotient rules. Here is another. Suppose that k and x are both known to be four-vectors and that in every inertial frame k is a multiple of x. That is, k = Ax in frame 8, and k' = A'x' in frame 8', and so on. Then the factor A (the "quotient" of k and x) is in fact a four-scalar with the same value in all frames, A = A'. Prove this quotient rule.

(a) Show that in the case that the source is approaching the observer head on, the Doppler formula (15.64) can be rewritten as w = w0V (1 + 13)1(1- p). (b) What is the corresponding result for the case that the source is moving directly away from the observer?

Consider the tale of the physicist who is ticketed for running a red light and argues that because he was approaching the intersection, the red light was Doppler shifted and appeared green. How fast would he have to have been going? (Area ;--'=, 650 nm and Agree 530 nm.)

The factor y in the Doppler formula (15.64), which can be ascribed to time dilation, means that even when 0 = 90 there is a Doppler shift. (In classical physics there is no Doppler shift when 6 = 90 and the source has zero velocity in the direction of the observer.) This transverse Doppler shift is therefore a test of time dilation, and has yielded some very accurate tests of the theory. However, except when the source is moving very close to the speed of light, the transverse shift is quite small. (a) If V = 0.2c, what is the percentage shift when 6 = 90? (b) Compare this with the shift when the source approaches the observer head-on.

Show that the four-velocity of any object has invariant length squared u u = c2

For any two objects a and b, show that the scalar product of their four-velocities is ua ub = C2Y (vrei), where y (v) denotes the usual y factor, y (v) = 1/V1 v2/c2, and vrei denotes the speed of a in the rest frame of b or vice versa.

(a) For the collision shown in Figure 15.11, verify that all four components of the total four-momentum pa + Pb [with the individual momenta defined relativistically as in (15.68)] are conserved in the frame S of part (a). (b) In two lines or less, prove that total four-momentum is conserved in the frame 8' of part (b). [This problem does not, of course, prove that the law of conservation of four-momentum is generally true, but it does at least show that the law is consistent with the collision of Figure 15.11.]

(a) Suppose that the total three-momentum P = E p of an isolated system is conserved in all inertial frames. Show that if this is true (which it is), then the fourth component P4 of the total four-momentum P = (P, P4) has to be conserved as well. (b) Using the zero-component theorem of Problem 15.35, you can prove the following stronger result very quickly: If any one component of the total four- momentum P is conserved in all frames, then all four components are conserved.

For any two objects a and b, show that Pa Pb = maEb = mbEa = mambc2Y (vrel) where ma is the mass of a, and Eb is the energy of b in a 's rest frame, and vice versa, and vro is the speed of a in the rest frame of b (or vice versa).

(a) Using the correct relativistic velocity-addition formulas make a table showing the four velocities as seen in frame S' of the collision of Figure 15.11(b) in terms of the initial velocity of a in S. [Give the latter some simple name, such as va 0).] (b) Add a column showing the total classical momentum may', mbvb before and after the collision, and show that the y component of the classical momentum is not conserved in 8'.

Since the four-velocity u = y (v, c) is a four-vector its transformation properties are simple. Write down the standard Lorentz boost for all four components of u. Use these to deduce the relativistic velocity- addition formula for v.

When oxygen combines with hydrogen in the reaction (15.80) about 5 eV of energy is released (that is, the kinetic energy of the two final molecules is 5 eV more than that of the initial three molecules). (a) By how much does the total rest mass of the molecules change? (b) What is the fractional change in total mass? (c) If one were to form 10 grams of water by this reaction, what would be the total change in mass?

When a radioactive nucleus of astatine 215 decays at rest, the whole atom is torn into two in the reaction 215At 211,, 4 151 + He. The masses of the three atoms are (in order) 214.9986, 210.9873, and 4.0026, all in atomic mass units. (1 atomic mass unit = 1.66 x 10-27 kg = 931.5 MeV/c2.) What is the total kinetic energy of the two outcoming atoms, in joules and in MeV?

(a) What is a particle's speed if its kinetic energy T is equal to its rest energy? (b) What if its energy E is equal to n times its rest energy?

If one defines a variable mass mvar = ym, then the relativistic momentum p = ymv becomes mvary which looks more like the classical definition. Show, however, that the relativistic kinetic energy is not equal to imvarv2.

A particle of mass ma decays at rest into two identical particles each of mass mb. Use conservation of momentum and energy to find the speed of the outgoing particles.

A particle of mass 3 MeV/c2 has momentum 4 MeV/c. What are its energy (in MeV) and speed (in units of c)?

A particle of mass 12 MeV/c2 has a kinetic energy of 1 MeV. What are its momentum (in MeV/c) and its speed (in units of c)?

(a) What is a mass of 1 MeV/c2 in kilograms? (b) What is a momentum of 1 MeV/c in kgm/s?

As measured in the inertial frame 8, a proton has four-momentum p. Also as measured in 8, an observer at rest in a frame 8' has four-velocity u. Show that the proton's energy, as measured by this observer, is u p

The relativistic kinetic energy of a particle is T = (y 1)m c2. Use the binomial series to express T as a series in powers of /3 = v/c. (a) Verify that the first term is just the nonrelativistic kinetic energy, and show that to lowest order in /3 the difference between the relativistic and nonrelativistic kinetic energies is 3,84mc2/8. (b) Use this result to find the maximum speed at which the nonrelativistic value is within 1% of the correct relativistic value

In nonrelativistic mechanics, the energy contains an arbitrary additive constant no physics is changed by the replacement E E+ constant. Show that this is not the case in relativistic mechanics. [Hint: Remember that the four-momentum p is supposed to tranform like a four-vector.]

Two balls of equal masses (in each) approach one another head-on with equal but opposite velocities of magnitude 0.8c. Their collision is perfectly inelastic, so they stick together and form a single body of mass M. What is the velocity of the final body and what is its mass M?

Consider the elastic, head-on collision of Example 15.10, in which two particles (masses ma and mb) approach one another traveling along the x axis, collide, and emerge traveling along the same axis. In the CM frame (by its definition) pian = pibn. Use conservation of momentum and energy to prove that pafin = pia; that is, the momentum of particle a (and likewise b) just reverses itself in the CM frame.

(a) Show that the four-momentum of any material particle (m > 0) is forward time-like. (b) Show that the sum of any two forward time-like vectors is itself forward time-like, and hence that the sum of any number of forward time-like vectors is itself forward time-like.

(a) Use the results of Problem 15.69 to prove that for any number of material particles there exists a CM frame, that is, a frame in which the total three-momentum is zero. (b) Relative to an arbitrary frame 8, show that the velocity of the CM frame is given by /3 = E pc/ E E.

One way to create exotic heavy particles is to arrange a collision between two lighter particles ad-b-- >ded-+g where d is the heavy particle of interest and e,- , g are other possible particles produced in the reaction. (A good example of such a process is the production of the particle in the process e+ e+ in which there are no other particles e,- , g.) (a) Assuming that and is much heavier that any of the other particles, show that the minimum (or threshold) energy to produce this reaction in the CM frame is Ecm ti mdc2. (b) Show that the threshold energy to produce the same reaction in the lab frame, where the particle b is initially at rest, is Elab md2c2/2mb. (c) Calculate these two energies for the process e+ e- > , with me ', 0.5 MeV/c2 and m* P-', 3100 MeV/c2. Your answers should explain why particle physicists go to the trouble and expense of building colliding-beam experiments.

A mad physicist claims to have observed the decay of a particle of mass M into two identical particles of mass m, with M < 2m. In response to the objections that this violates conservation of energy, he replies that if M was traveling fast enough it could easily have energy greater than 2mc2 and hence could decay into the two particles of mass m. Show that he is wrong. [He has forgotten that both energy and momentum are conserved. You can analyse this problem in terms of these two conservation laws, but it is much simpler to go to the rest frame of M.]

Consider the head-on elastic collision of Example 15.10, in which the final velocity vb of particle b is given by (15.95). (a) Show that, in the special case that the masses are equal (ma = mb), vb = va, the initial velocity of particle a. Show that in this case the final velocity of a is zero. [This result for equal-mass collisions is well known in classical mechanics; you have now shown that it extends to relativity.] (b) Show that in the nonrelativistic limit (15.95) reduces to vb = 2vama/ (ma mb). By doing the necessary nonrelativistic calculations, show that this agrees with the nonrelativistic answer for elastic head-on collisions.

A particle a traveling along the positive x axis of frame S with speed 0.5c decays into two identical particles, a b b, both of which continue to travel on the x axis. (a) Given that ma = 2.5mb, find the speed of either b particle in the rest frame of particle a. (b) By making the necessary transformation on the result of part (a), find the velocities of the two b particles in the original frame S.

A particle of unknown mass M decays into two particles of known masses ma = 0.5 GeV/c2 and mb = 1.0 GeV/c2, whose momenta are measured to be pa = 2.0 GeV/c along the x2 axis and pb = 1.5 GeV/c along the x1 axis. (1 GeV = 109 eV.) Find the unknown mass M and its speed.

Particle a is pursuing particle b along the x1 axis of a frame S. The two masses are ma and mb and the speeds are I), and vb (with I), > vb). When a catches up with b they collide and coalesce to form a single particle of mass m and speed v. Show that M2 = mQ Mb2 Ll anl bY (V a))/ (V b)(1 V al) b I C2) and find v.

Consider the elastic head-on collision of Example 15.10, in which particle a collides with a stationary particle b. Assuming that ma A mb, show that the final kinetic energy of particle a satisfies Tafin < (ma mo2c2/2mb. [Hint: Look at the CM frame where you can show that the four-vector pin pbin is time-like, so that (pafin pbin)2 < O.] (b) The result of part (a) implies that if Tain is large, almost all this incoming energy is lost to b. This is quite different from the nonrelativistic situation. Prove that in nonrelativistic mechanics the proportion of kinetic energy retained by a is fixed, independent of Tr. Specifically, Tafin = Tanana mb)2/(ma mo2.

Consider the elastic collision shown in Figure 15.17. In the lab frame 8, particle b is initially at rest; particle a enters with four-momentum pa and scatters through an angle 8; particle b recoils at an angle 'ilr. In the CM frame 8', the two particles approach and emerge with equal and opposite momenta, and particle a scatters through an angle 0'. (a) Show that the velocity of the CM frame relative to the lab frame is V = pac2/ (Ea mbc2). (b) By transforming the final momentum of a back from the CM to the lab frame, show that sin 0' tan 6 = (15.153) yv (cos 0' V/vt) where va` is the speed of a in the CM frame. (c) Show that in the limit that all speeds are much smaller than c, this result agrees with the nonrelativistic result (14.53) (where X = ma /mb). (d) Specialize now to the case that ma = mb. Show that, in this case, V/ vcc = 1, and find a formula like (15.153) for tan Vr Figure 15.17 Problem 15.78. (e) Show that the angle between the two outgoing momenta is given by tan(6 = 2/(6yv sin 0'). Show that in the limit that V <.< c, you recover the well-known nonrelativistic result that 9 + = 90

Consider an object of mass m (which you may assume is constant), acted on by a force F. From the definition (15.100) prove that F = yma + (F v)v/c2, where a = dv/dt is the object's acceleration. Notice that it is certainly not true in relativity that F = ma. Nor is it true that F = mvara, where mvar is the variable mass mvar = Ym, except in the special case that F happens to be perpendicular to v. In general, F and a are not even in the same direction.

A particle of mass m and charge q moves in a uniform, constant magnetic field B. Show that if v is perpendicular to B, the particle moves in a circle of radius r =11)1031. (15.154) [This result agrees with the nonrelativistic result (2.81), except that p is now the relativistic momentum p = ymv.]

An electron (mass 0.5 MeV/c2) moves with speed 0.7c in a circular path in a magnetic field of 0.02 teslas. Using the relativistic result (15.154) of Problem 15.80, find the radius of the electron's orbit. What would your answer have been if you used the classical definition of momentum? [Needless to say, the relativistic result is confirmed by experiment, and this gave some of the first evidence of the correctness of relativistic mechanics.33]

(a) Verify the result (15.106) for the position of a particle moving in a uniform electric field, by integrating the expression (15.105). (b) When t is small, the particle should be moving slowly and (15.106) should agree with the nonrelativistic result x = -lat2. Verify that it does. (c) Show that when t is large, x = F(ct + const) and explain this result.

Starting from the definition (15.100) of the force F on an object, prove that the transformation of the components of F as we pass from a frame S to a second frame 8', traveling at speed V in the standard configuration relative to 8, is F f3F 71c F'= F2 F3 , = 1 1 - pd vc 2 - ' y(1 - Pvilc) Y(1 Pvilc) (15.155) where 3 fi (V) and y = y ( V) relate to the relative speed of the two frames and v is the velocity of the object as measured in S.

A mass m is thrown from the origin at t = 0 with initial three-momentum po in the y direction. If it is subject to a constant force F, in the x direction, find its velocity v as a function of t, and by integrating v find its trajectory. Check that in the nonrelativistic limit the trajectory is the expected parabola.

We have seen that there are processes in which the mass of an object varies with time. (a) Starting from (15.85), prove that dmIdto= u K1c2, where to is the object's proper time, u is its four-velocity, and K is the four-force on the object. (b) This means that the necessary and sufficient condition that a force doesn't change an object's mass is that u K = 0. It is an experimental fact that if a charged particle is at rest in an electromagnetic field (even instantaneously) then dE/dt = 0. Use this to argue that electromagnetic forces do not cause a particle's mass to change.

The neutral pion rt-0 is an unstable particle (mass m = 135 MeV/c2) that can decay into two photons, 7r y + y. (a) If the pion is at rest, what is the energy of each photon? (b) Suppose instead that the pion is traveling along the x axis and that the photons are observed also traveling along the x axis, one forward and one backward. If the first photon has three times the energy of the second, what was the pion's original speed v?

A neutral pion (Problem 15.86) is traveling with speed v when it decays into two photons, which are seen to emerge at equal angles B on either side of the original velocity. Show that v = c cos 8.

Two particles a and b with masses ma = 0 and mb > 0 approach one another. Prove that they have a CM frame (that is, a frame in which their total three-momentum is zero). [Hint: As you should explain, this is equivalent to showing that the sum of two four-vectors, one of which is forward light-like and one forward time-like, is itself forward time-like.]

Show that any two zero-mass particles have a CM frame, provided their three-momenta are not parallel. [Hint: As you should explain, this is equivalent to showing that the sum of two forward light- like vectors is forward time-like, unless the spatial parts are parallel.]

The first positrons to be observed were created in electronpositron pairs by high-energy cosmic-ray photons in the upper atmosphere. (a) Show that an isolated photon cannot convert to an electronpositron pair in the process y e+ + e. [Show that this process inevitably violates conservation of four-momentum.] (b) What actually occurs is that a photon collides with a stationary nucleus with the result y + nucleus --> e+ + e + nucleus. Convince yourself that the formula (15.98) can be used to find the minimum energy for a photon to induce this reaction. [The derivation of (15.98) assumed that the incident particle had m > 0.] Show that, provided the mass of the nucleus is much greater than that of the electron, the minimum photon energy to induce this reaction is approximately 2mec2. [This is exactly the energy one would have calculated for the process y >- e+ e and shows that the role of the nucleus is just as a "catalyst" that can absorb some three-momentum.]

An excited state X* of an atom at rest drops to its ground state X by emitting a photon. In atomic physics it is usual to assume that the energy Ey of the photon is equal to the difference in energies of the two atomic states, AE = (M* M)c2, where M and M* are the rest masses of the ground and excited states of the atom. This cannot be exactly true, since the recoiling atom X must carry away some of the energy DE. Show that in fact Ey = A E[1 A E/(2M*c2)]. Given that AE is of order a few eV, while the lightest atom has M of order 1 GeV/c2, discuss the validity of the assumption that E=

A positive pion decays at rest into a muon and neutrino, n- ,u+ + v. The masses involved are m, = 140 MeV/c2, m = 106 MeV/c2, and m, = 0. (There is now convincing evidence that my is not exactly zero, but it is small enough that you can take it to be zero for this problem.) Show that the speed of the outgoing muon has /3 = (m,2 mL2)/(mn2 mA2). Evaluate this numerically. Do the same for the much rarer decay mode 7+ e+ v, (me = 0.5 MeV/c2).

Consider a head-on elastic collision between a high-energy electron (energy Eo and speed i5oc) and a photon of energy Ey0. Show that the final energy Ey of the photon is E E 1 + 2 + (1 13(0E01 E yo [Hint: Use (15.123).] Show that Ey < E0, but that if /3,, 1, then E y I E0 1; that is, a very high- energy electron loses almost all its energy to the photon in a head-on collision. What fraction of its original energy would the electron retain if Ee, ti 10 TeV and the photon was in the visible range, Eyo 3 eV? (Remember that the mass of the electron is about 0.5 MeV/c2; 1 TeV = 1012 eV.)

Prove that for any two matrices A and B, where A has as many columns as B has rows, the transpose of AB satisfies (AB) = 13' A.

By making suitable choices for the n-dimensional vectors a and b, show that if aCb = abb for any choices of a and b (where C and D are n x n matrices), then C = D.

Prove that if T and a are respectively a four-tensor and a four-vector, then b = Ta = T Ga is a four- vector; that is, it transforms according to the rule b' = Ab.

(a) A tensor T is said to be symmetric if TA, = Tvii. Prove that if T is symmetric in one inertial frame, then it is symmetric in all inertial frames. (b) T is antisymmetric if Tii, Prove that if T is antisymmetric in one inertial frame, then it is antisymmetric in all inertial frames. (An example of the latter property is the electromagnetic field tensor, which is antisymmetric in all frames.)

(a) Use the invariance of the scalar product a b = aGb to prove that the 4 x 4 Lorentz transformation matrices A must satisfy the condition (15.136), AGA = G. (b) Verify that the standard Lorentz boost (15.43) does satisfy this condition.

A useful form of the quotient rule for three-dimensional vectors is this: Suppose that a and b are known to be three-vectors and suppose that for every orthogonal set of axes there is a 3 x 3 matrix T with the property that b = Ta for every choice of a, then T is a tensor. (a) Prove this. (b) State and prove the corresponding rule for four-vectors and four-tensors.

(a) The statement that V is a vector operator means that if 0 (x) is any scalar, then the three components of V0 = (80/8x1, 80/ax2, 80/ax3) transform according to the three-vector transformation law (15.126). Prove this last statement. [Hint: Remember the chain rule, that 80/8x, = Ej(axyaxi)ao/axi.i.i (b) Prove that in four-dimensional spacetime, if 0 is any four-scalar, the quantity0 4) defined with the components ao ao ,_ao) I=14) ax2' ax3 ax4l (15.156) (note well the minus sign on the fourth component) is a four-vector. This result is crucial in writing down Maxwell's equations for the electromagnetic field.

(a) Prove that E B and E2 c2B2 are both invariant under any Lorentz tranformation. [Use the transformation equations (15.146) to prove the required results for the standard boost and then explain why if either quantity is invariant under the standard boost then it is invariant under any Lorentz transformation.] Use these results to prove the following two propositions: (b) If E and B are perpendicular in frame 3, then they are perpendicular in any other frame 8', and (c) if E > cB in a frame 3, then there cannot exist a frame in which E = 0.

(a) Starting from the transformation equations (15.146) for the standard boost along the x1 axis, find the corresponding boost along the x3 axis. (b) Write down the inverse of this transformation and then verify the results (15.151) and (15.152) for the fields of a moving line charge.

(a) Using the transformation equations (15.146), show that if E = 0 in frame 3, then E' = v x B' in 3'. (b) Similarly, show that if B = 0 in frame 3, then B' = v x E'/c2.

We defined the electromagnetic field tensor by the equation K = q3 u qTGu, where K is the four-force on a charge q and u is its four-velocity. (a) Starting from the Lorentz force (15.139) write down the four components of K [as in (15.141)]. (b) Use these to find the matrix TG and show that the tensor a' has the form claimed in (15.143).

Since T is a four-tensor it has to transform according to the rule (15.145), = AajC. Using the form (15.143) for a' and the standard Lorentz boost for A, find the matrix 9' and verify the transformation equations (15.146) for the electromagnetic fields.

Derive the Lorentz-force law from Coulomb's law as follows: (a) If a charge q is at rest in frame 3', then Coulomb's law tells us that the force on q is F' = qE'. Use the inverse of the force transformation (15.155) in Problem 15.83 to write down the force F as seen in S. (Answer in terms of E' for now.) (b) Now use the field transformation (15.146) to rewrite your answer in terms of E and B and show that F = q(E v x B).

It is a result well known in classical electromagnetism that one can introduce a three-scalar potential 0 and a three-vector potential A such that the fields E and B can be written as E = Wp aA and B=V x A. In relativity these potentials are combined to form a single four-potential A = (A, 0/c). Prove that = q4A quAi, where q is the four-dimensional gradient operator defined in (15.156) of Problem 15.100. (If we accept that A really is a four-vector, this gives an alternative proof that is a four tensor.)

Consider an electric charge distribution, with charge density Q, moving with velocity v relative to a frame S. (a) Show that p= ypo, where Q0 is the charge density in the rest frame. (Notice that v can vary with position, so different parts of the distribution will have different rest frames, but that's all right.) (b) The three-current density is defined as J = Qv. Show that the four-current density, defined as J = (J, cp), is a four-vector. (c) It is a well-known result in electromagnetism that conservation of charge implies the so-called equation of continuity, V J .90/3t = 0 (where V J = 8J, /axi is the so- called divergence of J). Show that this condition is equivalent to the manifestly invariant condition q J = 0, where q is the four-dimensional gradient defined in (15.156) of Problem 15.100.

Two equal charges q are moving side-by-side in the positive x direction in frame S. The distance between them is r and their speed is v. Find the force on either charge due to the other in two ways: (a) Find the force in their rest frame 3' and transform back to 3, using the force transformation (15.155) of Problem 15.83. Note that the force in S is less than in the rest frame. (b) Find the electric and magnetic fields in S' and thence in 3, using the field transformation (15.146). Use these fields (in 3) to write down the Lorentz force on either charge in 3. Note that in S there is an attractive electric force and a repulsive magnetic force. As )3 1 they become nearly equal and their resultant approaches zero.

A charge q is moving with constant speed v along the x axis of frame 3 with position uti. (a) Write down the electric and magnetic fields in the charge's rest frame 3'. (b) Use the inverse of the field transformation (15.146) to write down the electric field in the original frame S. [In the first instance, you will find E in terms of the primed variables x', y', z', t', but you can use the standard Lorentz transformation to eliminate them in favor of x, y, z, t.] Show that the field at position r and time t is E kq(1 $2) R (1 S2 sine 6)3/2 R2 (15.158) where R = r vti is the vector pointing from the charge's position to the point of observation r, and 8 is the angle between R and the x axis. (c) Sketch the behavior of the field strength as a function of for fixed R, and make a sketch of the electric field lines at one fixed time t.

Two of Maxwell's four equations read V x B = accoJ and V E= 1 Q C^ atco (15.159) where J and Q are the current and charge densities that gave rise to the fields. Show that these two equations can be written as the single four-vector equation q = go", where q is the four-dimensional gradient operator introduced in Problem 15.100, J is the four-current (J, cQ), and the scalar product q J'