pictures of rapidly moving distant objects are undistorted
in
46 AN INTRODUCTION TO THE SPECIAL THEORY
OF RELATIVITY
shape. But the picture taken in the laboratory from one
direction
will be the same as obtained in the proper frame, but from
a
different v
and .  Mesons," Phys. Rev. 88, 179 (1952).
D. H. Frisch and J. H. Smith, "Measurement of
Relativistic Time Dilation
Using p. Mesons," Am. J. Phys. 31, 342 (1963).
D. H. Frisch, J. H. Smith, and F. L. Friedman, Time
Dilation (a film),
Educational Services
need in this book for the more general equations of
transformation
which could be derived from Eqs. 29.1.
In the Galilean transformation the acceleration of a
particle
is the same in all inertial frames. This is not true at high
velocities
where the Lore
event takes place in the time interval during which the
shell
passes, information about the event is swept along by the
shell
and will be recorded by the camera.
In spite of inferences that might be drawn incorrectly
from a
casual inspection of the Lorent
circular, the moon was bounded by circles, the planets
were circular,
and the stars seemed to describe circular paths. On the
surface of the earth, and for horizontal motions, a state of
rest
seemed the natural state, for objects came to rest when
left by
Math 260
Assignment #4
Spring 2011
webpage: http:/www.metu.edu.tr/bcanan
1. Evaluate the following determinants:
1
0
0
1
1 2
0
1 1 1
1
(a) 4 1 2
(b)
1
2
0 1 0
0
1
4 1 1
0
a b
2. If det
= 2, calculate:
c d
2
2 0
3c
a
+
c
1
(a) det c + 1 1 2a
Math 260
Assignment #3
Spring 2011
In order to receive full marks, students must indicate the various steps taken.
1. Find the inverse of
1 2 3
(a) A = 2 5 3
1 0 8
1 0
2. Let A = 2 4
3 1
the following matrices (if possible)?
1 6
4
(b) A = 2 4 1
1 2
5
5
2
Math 260
Assignment #1
Spring 2011
1. Solve the following matrix equation for a, b, c and d.
a b
c 3d d
=
c d
2a + d a + b
3 0
1 5 2
4
1
1
4
2
2. Consider the matrices A = 1 2 , B =
, C =
, D = 1 0 1
0
2
3 1 5
1 1
3 2 4
6 1 3
and E = 1 1 3 . Compute th
MATH 260 BASIC LINEAR ALGEBRA
Answers to Problem Set 2 (Systems of linear equations)
1. (i) System is inconsistent, since [
] [
2. (i)
(ii) Consider as a parameter. System is consistent if
[ ]
[ ]
[

]
[
(ii) Solution is unique,
]
.
]
Then, solution is u
cube passes the origin by a distant observer located along
the y axis.
At what angle with respect to V would a photographer in
the proper
frame have to be located to take the same picture? [127]
(29.1 b)
(29.1a)
or
48 AN INTRODUCTION TO THE SPECIAL
THE
dx'
Since we have set ourselves the limitation that V'a: = dt' =
0, the
preceding equation becomes
dt = y dt'. (29.3)
Dividing each of Eqs. 29.2 by 29.3, we obtain the
transformation
equations for the acceleration as
a = dUx = ldU'., = 'V3a' (29.4a)
direction at 0.98c in the laboratory frame. Find (c) the
acceleration as
measured in the laboratory, and (d) the angle the
acceleration makes
with the x axis in the laboratory frame. [(a) 5 cm/sec2 (b)
37 (c) 0.038
cm/sec2 (d) 81.4]
CHAPTER BIBLIOGRAPHY
R
() =7T/2, so that from Eq. 27.3 cos ()' = p and ()' is
greater than
7T12, as shown.
28.1 A small object is in the form of a sphere of radius R
in its proper
frame. Determine the shape of the particle (a) as measured
(observed)
by a set of observers in
flashes of a moving clock, as the rod clock, we ask
laboraTHE
LORENTZ TRANSFORMATION 45
tory observers to note whether they were coincident with
the moving
clock at the time it flashed, and if so what was the reading
on their own clock at the time the mov
laboratory frame, and the events he notes may have taken
ages
to reach him. A telescope camera photographing distant
stars
during a solar eclipse may have its shutter open a very
short time.
The light which enters the shutter may come from distant
stars,
By seeing, or by photographing, we mean something else.
The
difference is most clear in connection with photography.
When
we take a photograph, there is only one lens. When we
see, there
is only one observer. The difference between observing
and
seeing is
must divide Eqs. 29.2 by an appropriate equation relating
time
differentials in the two systems. From the Lorentz
transformation
equations (21.3) we have
t = y(t'  Vx' / c2),
and, taking differentials,
dt = y(dt'  V dx' / c2),
dt = y dt' (1 _!:' dX
The analysis we call Newtonian mechanics is based on a
similar
conceptual structure. Motions are analyzed in terms of
forces.
But the concept of a natural motion is somewhat more
abstract.
Newton's first law expresses the motion a body will
describe if
le
and are decelerated, radiating light. Particles accelerated
in highenergy accelerators move in curved paths, being
centripetally
accelerated in this deflection; they thereby radiate
electromagnetic
energy. In both cases this radiation is predominantly
in
results from aberration, and is known as the headlight
effect.
Suppose a source of radiation radiates energy uniformly
in all
directions in its proper frame (the primed frame), as
shown in
Fig. 27.2. The source moves in the +x direction with
respect to
T
a =0.51 cm/sec2]
29.2 A particle experiences an acceleration of a' = (3 I",.
+4 11/')
cm/sec2 in a coordinate frame in which it is
instantaneously at rest.
Find (a) the magnitude of the acceleration, and (b) the
direction it
makes with the x' axis. The p
MATH 260 BASIC LINEAR ALGEBRA
Answers to Problem set 3 (Determinants)
1
1. (Ex. 3, p. 117) Given that
a
1 1
a
(i)
x2
2
b 4 , calculate the following determinants
2 3b
(ii)
(iii)
(iv)
x
x
1
b 4x
ax
2
3b
2
2
1
2a
x
2x
a bx
1 bx
b 4x
2
4
2b 16
1
1
b
ax
ax
2
MATH 260 BASIC LINEAR ALGEBRA
Answers to Problem set 6
I. Linear transformations, matrix representation of linear transformations
1. Which of the following are linear transformations?
(i) : 3 2 given by (1 , 2 , 3 ) = (1 + 2 , 2 + 3 )
(ii) : 2 2 given by
MATH 260 BASIC LINEAR ALGEBRA
Problem set 4 (Vector Spaces)
1. Textbook (C.Ko) p.145, No 6
2. Textbook (C.Ko) p.145146, No 11, 12
3. Textbook (C.Ko) p.146, No 13 (b)
4. Textbook (C.Ko) p.146, No 14
5. Find the conditions on , , , under which (, , , ) (3,
MATH 260 BASIC LINEAR ALGEBRA
Problem set 6
I. Linear transformations, matrix representation of linear transformations
1. Which of the following are linear transformations?
(i) : 3 2 given by (1 , 2 , 3 ) = (1 + 2 , 2 + 3 )
(ii) : 2 2 given by (1 , 2 ) =
, B .1 I J L; , wl 1 .I F
W , E J, 9% %t x 5. s
, t: 1".e P a a. e
w a rIT L
1? e mm? +9 W .L P w u. I
, Iv. T mix, I» .I t Jm &u . ?&1 c
L W 6 cm, 7 1\ no. rlu a M 0 1M v5 W u
w Hi ta. \ « JIIIIJ '9 lo Idmmlwl +71 , I) c a
m f 1m61w whose ,_ ®L m V