AMA331

SCHOOL OF MATHEMATICS AND STATISTICS Autumn Semester 2007-08

INTRODUCTION TO RELATIVITY 2 Hours

Marks will be awarded for your best FOUR answers.

AMA331 1 Turn Over

AMA331

1 Define what is meant by an inertial frame. (2 marks)

Two inertial frames, R : (ct, x) and R : (ct, x) are related by the Lorentz transformation

(ct

x

)= γ(u)

1 −u

c−uc

1

(ct

x

),

where

γ(u) =

(1− u

2

c2

)− 12

.

Verify thatc2t2 − x2 = c2t2 − x2. (6 marks)

Show that a light ray passing through the origin of the inertial frame R and travelling withspeed c along the x-axis of the inertial frame R (in the direction of increasing x) alsotravels with speed c in the inertial frame R. (2 marks)

Now consider a second light ray passing through the point x = D at time t = 0 in theframe R, and travelling with speed c along the x-axis in the direction of increasing x .

Write down the path of the second light ray in the frame R and show that the path ofthis second light ray in the frame R is

γ(u)

(ct − ut − uD

c−ut + ct +D

).

(4 marks)

At time T in the frame R, show that the second light ray is at the point

x = cT + γ(u)D(1 +u

c

).

(6 marks)

Hence show that, at time T in the frame R, the light rays are a distance L apart, where

L = D

√c + u

c − u .

(5 marks)

AMA331 2 Continued

AMA331

2 Write down the (4× 4) matrix which is the metric tensor g, and show that g2 = I,the (4× 4) identity matrix. (3 marks)

Two inertial frames R : (ct, x, y , z) and R : (ct, x , y , z) are related by the transformation

ct

x

y

z

= L

ct

x

y

z

, (∗)

where L is a (4× 4)-matrix with constant entries.

Write down the conditions on the matrix L for the transformation (∗) to be a properorthochronous Lorentz transformation. (3 marks)

Suppose that the matrices L and M represent proper, orthochronous Lorentz transforma-tions. Show that the matrix LM also represents a proper Lorentz transformation.

(5 marks)

Suppose that the matrices L and M represent proper, orthochronous Lorentz transforma-tions and that the matrix M has the form

M =

1 0 0 0

0 M11 M12 M

13

0 M21 M22 M

23

0 M31 M32 M

33

.

Show that the matrices LM and ML represent orthochronous Lorentz transformations.(4 marks)

Suppose that the matrix L is symmetric, so that LT = L, and that L represents a proper,orthochronous Lorentz transformation.

Show that the matrix L has inverse

N = gLg

and that the matrix N also represents a proper, orthochronous Lorentz transformation.(10 marks)

AMA331 3 Turn Over

AMA331

3 Define what is meant by the displacement vector between two space-time eventsE1 = (ct1, x1) and E2 = (ct2, x2) in an inertial frame R. (2 marks)

Two space-time events, E1 and E2, have space-time co-ordinates (ct, x) given by (0, 0)and (1, 2) respectively, in some inertial frame R : (ct, x).

Find the displacement vector X between the two events in the inertial frame R and showthat the two events are space-like separated. (3 marks)

The inertial frame R :(ct, x

)is related to R by the Lorentz transformation

(ct

x

)= γ(u)

1 −u

c−uc

1

(ct

x

),

where

γ(u) =

(1− u

2

c2

)− 12

.

Find the displacement vector X between the two events in the frame R. (3 marks)

Find the speed u of the inertial frame R with respect to R if:

(i) The events E1 and E2 are simultaneous in R; (2 marks)

(ii) The event E2 precedes the event E1 by a time 1/c . (5 marks)

Let R :(ct, x

)be the inertial frame in which the two events are simultaneous, and

R′ : (ct ′, x ′) be the inertial frame in which the event E2 precedes the event E1 by a time1/c .

Find the spatial separation of the two events in the frames R and R′. (5 marks)

Draw a clear space-time diagram showing the axes of the three inertial frames R, R andR′, and the position of the two events E1 and E2. (5 marks)

AMA331 4 Continued

AMA331

4 In an inertial frame R, an observer has worldline X(t) = (ct, x, 0, 0).

Define the quantities proper time, four-velocity and four-acceleration for this observer.(4 marks)

Show that the four-velocity of the observer takes the form

V = γ(v) (c, v , 0, 0) ,

where v is the speed of the observer in the inertial frame R : (ct, x, y , z). (3 marks)

The inertial frame R is related to a second inertial frame R : (ct, x , y , z) by the Lorentztransformation (

ct

x

)= γ(u)

1 −u

c−uc

1

(ct

x

),

with y = y and z = z , where

γ(u) =

(1− u

2

c2

)− 12

.

Show that, along the path of the observer,

dt

dt= γ(u)

(1− uvc2

).

(3 marks)

Find the components of the four-velocity in the inertial frame R, and show that, if v isthe speed of the observer in the inertial frame R, then

v =v − u1− uv

c2

.

(6 marks)

Deduce that there is a value of u for which the observer is at rest in the frame R.(2 marks)

Let a =dv

dtand a =

dv

d tbe the accelerations of the observer in the inertial frames R and

R respectively. Show that

a = a(1− uvc2

)−3γ(u)−3.

(5 marks)

Deduce that, if a 6= 0, there is no value of u for which the observer is not accelerating inthe inertial frame R. (2 marks)

AMA331 5 Turn Over

AMA331

5 Define the rest mass and four-momentum of a particle. (3 marks)

A particle of rest mass m has three-velocity v in an inertial frame R.

Write down expressions for the energy E and three-momentum p of the particle in theinertial frame R. (2 marks)

Two protons, each of mass m, are moving with speed v in opposite directions along thex-axis of an inertial frame R. The two protons collide and two new particles are formed,each with rest mass 10m.

Assuming the new particles are formed at rest in R, show that γ(v) = 10 and hence findthe energy of each proton before the collision in terms of m and c only. (6 marks)

Suppose now that the protons each have the same energy before they collide. However,before the collision one proton is moving along the x-axis, while the other is moving alongthe y -axis in R. On collision the protons form a single new particle. For the remainder ofthis question, you may use the value γ(v) = 10 calculated above.

Show that the rest mass of the new particle is√202m and find, in terms of m and c only,

its energy and the magnitude of its three-momentum in the frame R. (14 marks)

End of Question Paper

AMA331 6