2.7 P h o t o n s

No four-velocity

Photons move on null lines, so, for a photon path,

Therefore dτ is zero and Eq. (2.31) shows that the four-velocity cannot be defined. Another way of saying the same thing is to note that there is no frame in which light is at rest (the second postulate of SR), so there is no MCRF for a photon. Thus, no e0 in any frame will be tangent to a photon’s world line.

Note carefully that it is still possible to find vectors tangent to a photon’s path (which, being a straight line, has the same tangent everywhere): dx is one. The problem is finding a tangent of unit magnitude, since they all have vanishing magnitude.

Four-momentum

The four-momentum of a particle is not a unit vector. Instead, it is a vector where the components in some frame give the particle energy and momentum relative to that frame. If a photon carries energy E in some frame, then in that frame p0 = E. If it moves in the x direction, then py = pz = 0, and in order for the four-momentum to be parallel to its world line (hence be null) we must have px = E. This ensures that

where ν is its frequency and h is Planck’s constant, h = 6.6256 × 10−34 J s.

This relation and the Lorentz transformation of the four-momentum immediately give us the Doppler-shift formula for photons. Suppose, for instance, that in frame O a photon

This is generalized in Exer. 25, § 2.9.

Any particle whose four-momentum is null must have rest mass zero, and conversely. The only known zero rest-mass particle is the photon. Neutrinos are very light, but not massless. (Sometimes the ‘graviton’ is added to this list, since gravitational waves also travel at the speed of light, as we shall see later. But ‘photon’ and ‘graviton’ are concepts that come from quantum mechanics, and there is as yet no satisfactory quantized theory of gravity, so that ‘graviton’ is not really a well-defined notion yet.) The idea that only particles with zero rest mass can travel at the speed of light is reinforced by the fact that no particle of finite rest mass can be accelerated to the speed of light, since then its energy would be infinite. Put another way, a particle traveling at the speed of light (in, say, the x direction) has p1/p0 = 1, while a particle of rest mass m moving in the x direction has, from the equation p · p = −m2, p1/p0 = [1 − m2/(p0)2]1/2, which is always less than one, no matter how much energy the particle is given. Although it may seem to get close to the speed of light, there is an important distinction: the particle with m =0 always has an MCRF, a Lorentz frame in which it is at rest, the velocity v of which is p1/p0 relative to the old frame. A photon has no rest frame.

2.8 F u r t h e r re a d i n g

We have only scratched the surface of relativistic kinematics and particle dynamics. These are particularly important in particle physics, which in turn provides the most stringent tests of SR. See Hagedorn (1963) or Wiedemann (2007).

2.9 E xe rc i s e s

1 Given the numbers {A0 = 5, A1 = 0, A2 = −1, A3 = −6}, {B0 = 0, B1 = −2, B2 = 4,

B3 = 0}, {C00 = 1, C01 = 0, C02 = 2, C03 = 3, C30 = −1, C10 = 5, C11 = −2, C12 =

−2, C13 = 0, C21 = 5, C22 = 2, C23 = −2, C20 = 4, C31 = −1, C32 = −3, C33 = 0}, find:

(a) Aα Bα ; (b) Aα Cαβ for all β; (c) Aγ Cγ σ for all σ ; (d) Aν Cμν for all μ; (e) Aα Bβ for all α, β; (f) AiBi; (g) AjBk for all j, k.

2Identify the free and dummy indices in the following equations and change them into equivalent expressions with different indices. How many different equations does each

expression represent?

(a) Aα Bα = 5; (b) Aμ¯ = μ¯ ν Aν ; (c) TαμλAμCλγ = Dγ α ; (d) Rμν − 12 gμν R = Gμν .

3Prove Eq. (2.5).

6In the t − x spacetime diagram of O, draw the basis vectors e0 and e1. Draw the corresponding basis vectors of the frame O¯ that moves with speed 0.6 in the positive x direction relative to O. Draw the corresponding basis vectors of O a frame that moves with speed 0.6 in the positive x direction relative to O¯ .

7(a) Verify Eq. (2.10) for all α, β.

(b)Prove Eq. (2.11) from Eq. (2.9).

8(a) Prove that the zero vector (0, 0, 0, 0) has these same components in all reference frames.

(b)Use (a) to prove that if two vectors have equal components in one frame, they have equal components in all frames.

9Prove, by writing out all the terms, that

Since the order of summation doesn’t matter, we are justified in using the Einstein

(a)Write down the matrix of ν μ¯ (−v).

(b)Find Aα¯ for all α¯ .

(c)Verify Eq. (2.18) by performing the indicated sum for all values of ν and α.

(d)Write down the Lorentz transformation matrix from O¯ to O, justifying each entry.

(e)Use (d) to find Aβ from Aα¯ . How is this related to Eq. (2.18)?

(f)Verify, in the same manner as (c), that

for v and v , as given in (c), and show that the result does not equal that of (c). Interpret this physically.

14 The following matrix gives a Lorentz transformation from O to O¯ :

(a)What is the velocity (speed and direction) of O¯ relative to O?

(b)What is the inverse matrix to the given one?

O →

(c) Find the components in of a vector A (1, 2, 0, 0).

O¯

15(a) Compute the four-velocity components in O of a particle whose speed in O is v in the positive x direction, by using the Lorentz transformation from the rest frame of the particle.

(b)Generalize this result to find the four-velocity components when the particle has arbitrary velocity v, with |v| < 1.

(c)Use your result in (b) to express v in terms of the components {Uα }.

(d)Find the three-velocity v of a particle whose four-velocity components are (2, 1, 1, 1).

16Derive the Einstein velocity-addition formula by performing a Lorentz transformation with velocity v on the four-velocity of a particle whose speed in the original frame

was W.

18(a) Show that the sum of any two orthogonal spacelike vectors is spacelike.

(b)Show that a timelike vector and a null vector cannot be orthogonal.

19A body is said to be uniformly accelerated if its acceleration four-vector a has constant spatial direction and magnitude, say a · a = α2 0.

(a)Show that this implies that a always has the same components in the body’s MCRF, and that these components are what one would call ‘acceleration’ in Galilean terms. (This would be the physical situation for a rocket whose engine always gave the same acceleration.)

(b)Suppose a body is uniformly accelerated with α = 10 m s−2 (about the acceleration of gravity on Earth). If the body starts from rest, find its speed after time t. (Be

sure to use the correct units.) How far has it traveled in this time? How long does it take to reach v = 0.999?

(c)Find the elapsed proper time for the body in (b), as a function of t. (Integrate dτ

along its world line.) How much proper time has elapsed by the time its speed is v = 0.999? How much would a person accelerated as in (b) age on a trip from

Earth to the center of our Galaxy, a distance of about 2 × 1020 m? 20 The world line of a particle is described by the equations

in some inertial frame. Describe the motion and compute the components of the particle’s four-velocity and four-acceleration.

21 The world line of a particle is described by the parametric equations in some Lorentz frame

where λ is the parameter and a is a constant. Describe the motion and compute the particle’s four-velocity and acceleration components. Show that λ is proper time along the world line and that the acceleration is uniform. Interpret a.

22(a) Find the energy, rest mass, and three-velocity v of a particle whose four-momentum has the components (4, 1, 1, 0) kg.

(b)The collision of two particles of four-momenta

results in the destruction of the two particles and the production of three new ones, two of which have four-momenta

Find the four-momentum, energy, rest mass, and three-velocity of the third particle produced. Find the CM frame’s three-velocity.

23A particle of rest mass m has three-velocity v. Find its energy correct to terms of

order |v|4. At what speed |v| does the absolute value of 0(|v|4) term equal 12 of the kinetic-energy term 12 m|v|2?

24Prove that conservation of four-momentum forbids a reaction in which an electron and positron annihilate and produce a single photon (γ -ray). Prove that the production of two photons is not forbidden.

25(a) Let frame O¯ move with speed v in the x-direction relative to O. Let a photon have

frequency ν in O and move at an angle θ with respect to O’s x axis. Show that its frequency in O¯ is

(b)Even when the motion of the photon is perpendicular to the x axis (θ = π/2) there is a frequency shift. This is called the transverse Doppler shift, and arises because

of the time dilation. At what angle θ does the photon have to move so that there is no Doppler shift between O and O¯ ?

(c)Use Eqs. (2.35) and (2.38) to calculate Eq. (2.42).

26 Calculate the energy that is required to accelerate a particle of rest mass m =0 from speed v to speed v + δv (δv v), to first order in δv. Show that it would take an infinite amount of energy to accelerate the particle to the speed of light.

27 Two identical bodies of mass 10 kg are at rest at the same temperature. One of them is heated by the addition of 100 J of heat. Both are then subjected to the same force.

moving at speed v = 0.6 in the positive x direction relative to O, with its spatial axes

→

30 The four-velocity of a rocket ship is U O (2, 1, 1, 1). It encounters a high-velocity

energy of the cosmic ray as measured by the rocket ship’s passengers, using each of the two following methods.

(a) Find the Lorentz transformations from O to the MCRF of the rocket ship, and use

it to transform the components of P.

(b)Use Eq. (2.35).

(c)Which method is quicker? Why?

31A photon of frequency ν is reflected without change of frequency from a mirror, with an angle of incidence θ . Calculate the momentum transferred to the mirror. What momentum would be transferred if the photon were absorbed rather than reflected?

32Let a particle of charge e and rest mass m, initially at rest in the laboratory, scatter a photon of initial frequency νi. This is called Compton scattering. Suppose the scattered photon comes off at an angle θ from the incident direction. Use conservation of four-momentum to deduce that the photon’s final frequency νf is given by