Kusse B.R., Westwig E.A. Mathematical Physics: Applied Mathematics for Scientists and Engineers
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TENSORS IN NON-ORTHOGONAL COORDINATE
SYSTEMS
g,
Figure
14.8
The Two-Dimensional
Lorentz Coordinate Systems
1’
___
1
-
X
Figure
14.9
Lorentz
Transformations
of
Events
NON-ORTHONORMAL
COORDINATE
SYSTEMS
587
Figure
14.10
Motion
at
a
Velocity
Less
Than
the
Speed
of
Light
determine the primed velocity from:
(14.121)
This
velocity can be calculated directly from the transformation equations as
(14.122)
As
can
be seen from
this
expression and from the figure, as
v
--+
co,
v’
-+
co.
Another classic consequence of special relativity is immediately evident from the
graphical representation of coordinates in Figure
14.9.
Two events which occur at
the same time in the unprimed frame, occur at different times in the primed frame.
Imagine two events with the same value of
cot,
but different values of
x.
They would
occur on a horizontal line in Figure 14.9.
In
the primed system, not only will they
have different values of
x’,
but they also
will
have different values of
cot’.
The representation of Figure 14.8 for describing
the
transformations of special
relativity does have problems, however. First, consider the magnitude of the basis
vectors of the primed system. These covariant
basis
vectors are specified in terms of
the
unprimed basis vectors by Equation 14.120. Specifically for
g;
:
g;
=
Yo
gl
+
YOPO
g2.
(14.123)
588
TENSORS IN NON-ORTHOGONAL COORDINATE
SYSTEMS
The magnitude squared of
this
basis vector
is
obtained by
an
inner product of the
basis vector with itself
lgj(’
=
g;
*
g;.
=
(Yo
gl
+
%Po
g2).
(Yo
gl
+
%Po
gz)
=
2<1
+
P3,
(14.124)
where
the
last step follows because the unprimed basis vectors have unit magnitude.
As
long
as
the primed system is moving with respect to the unprimed system, that is
if
vo
#
0,
Equation 14.124 says that
g{
and, by a similar argument
gi,
are not unit
vectors.
This
in itself is not that serious a problem, because
our
formalism does easily
handle nonunit
basis
vectors. It does, however, complicate
the
geometrical picture
of
Figure 14.8, because you cannot simply axis-parallel-project a vector onto a basis
vector to obtain a vector component. You must also divide by the magnitude
of
the
corresponding basis vector.
A
more serious problem arises when you consider
the
inner product in
this
system.
In particular, let’s look at
i
-
f,
the inner product of the position vector with itself.
This
quantity
is
a scalar and should
be
invariant
to
transformations.
In
the unprimed
frame,
(1
4.125)
-
r
=
x
gl
+
cot
g2,
so
we expect the quantity
f
*
T;
to
be
given by the expression
x2
+
(cot)’.
(14.126)
But we already determined in Equation 14.117 that special relativity requires the
quantity
x2
-
(cot)’
to be invariant. It is not possible for both quantities to be
invariant. The problem lies in
our
assumption about the geometry of the space. The
value of
T;
.
i=
is not given by Equation 14.126!
This
is
discussed in
the
next section.
The
Metric
of
Special
Relativity
Both the problem with the magnitudes of the
primed covariant basis vectors and the invariance
of
the magnitude
of
the position
vector involve some assumptions about how to take an inner product.
In
both cases,
we took the inner product
of
a
vector with itself in the unprimed coordinate system
assuming an
Euclidian
metric
of
the form
In effect,
this
was the metric used
in
the calculation of Equation 14.126:
(
14.127)
(14.128)
589
NON-ORTHONORMAL COORDINATE SYSTEMS
With a little reflection, it can
be
seen that the invariant quantity of Equation 14.1
17
can be generated if we use an alternate metric:
Mi,+
['
0
-1
"1
.
(
14.129)
When
this
metric is used in the unprimed system, Equation 14.126
is
modified to
=
x2
-
(cot)?
(14.130)
How does ths new metric affect the orthogonality and the magnitudes of the basis
vectors? Using the metric
of
Equation
14.129
to
form
the inner product
of
gl
with
itself gives
=
1.
(14.131)
The
g1
basis vector
is
still a unit vector. The inner product between
gl
and
g2
is
=
0.
(
14.132)
The covariant basis
vectors
of
the unprimed frame are still orthogonal. But the
magnitude of
g2
is
=
-1.
(14.133)
This says that, while
/g2
I
=
1,
g2
itself must be imaginary! Very odd.
equations given by Equation
14.120,
we can write
What about the basis vectors
of
the primed frame? Using the transformation
2;
=
Yo
gl
+
YoPo
g2
g;
=
YoPo
gl
+
Yo
g2.
Using the unprimed metric, the magnitude of
gi
is
determined by
(
1
4.1
34)
(14.135)
=
1. (14.136)
590
TENSORS IN NON-ORTHOGONAL COORDINATE SYSTEMS
So
gi
has
now become a unit vector. The inner product between
gi
and
gi
is
=
0.
(14.137)
Now
the
primed, covariant basis vectors of the skewed system are
also
orthogonal!
Finally, the magnitude
of
gi
is determined by
=
-1. (14.138)
The
g;
basis vector is an imaginary
unit
vector,
similar
to
gz.
The inner products in the primed system could have also been performed using
Mij,
the metric
of
the primed system. Since the metric is a tensor, the primed metric
is related to the unprimed metric
in
the normal manner:
M!.
=
gm
g".
M
(14.139)
1J
1
J
mn.
In
matrix notation, the prjmed metric is given by
[M']
=
[
YOPO
'Yo
?f']
['
0
-1
]
[Go
Yt]
=['
0
-1
01
(
14.140)
Notice the form
of
the metric
is
identical in the primed and unprimed systems. With
this metric, the inner product between the covariant primed basis vectors can
be
determined:
=
1,
=
0,
and
=
-1.
(14.141)
(14.142)
(14.143)
These are exactly the same results obtained previously, using the unprimed metric.
NON-ORTHONORMAL COORDINATE SYSTEMS
591
The fact that one of the basis vectors is imaginary seems like a very strange thing.
But there is nothing mathematically inconsistent with it. The distance along the the
axis is now simply measured in terms
of
“imaginary” meter sticks instead of “real”
ones. The new metric, while not very intuitive, solves our problems with the simple
picture in Figure 14.9. The basis vectors are now
unit
vectors and the invariance of
x2
-
is built into the inner product operation.
This
weird geometry is often
called Minkowski space, while the new metric is called the Minkowski metric.
We actually can devise an equivalent coordinate system which does not use the
imaginary time axes. In this construction, we write
the
position vector as
(14.144)
-
r
=
x
21
+
ic,t
Q,
where the basis vectors
61
and
22
are now real, orthonormal, unit vectors. The same is
true in the primed system. In this case inner products are performed with the “normal”
Euclidian metric. This is equivalent to
our
earlier construction, using the imaginary
basis vector. since all we have really done here is pull out a factor of
i
from the basis
vector and put it into the coordinate. The problem with this view is that you lose the
convenient representation of Figure 14.9. There is no way to interlace the two sets
of axes, represent the same event in both systems as
a
single point, and visualize the
transformation with axis-parallel projections. The best we can do is draw two distinct
coordinate systems, as shown in Figure 14.1 1.
Extension to Three Dimensions
dimensions. The position vector, in the unprimed and primed systems, becomes
These ideas can easily be extended to
three
spatial
-
r
=
xgl
+
y
g2
+
zg3
+cot
&I
=
x’
g;
+
y’
g;
+
z‘
g;
+
cot‘
g.
The Minkowski metric expands to
[MI
=
,
ic,t
I
0
001
0’
000-1
E:
:I
(14.145)
(14.146)
ic$
Figure
14.11
A
Second Picture
of
the
Coordinate Systems
for
Special Relativity
592
TENSORS IN NON-ORTHOGONAL COORDINATE
SYSTEMS
The Lorentz transformation between the
two
systems becomes
Yo
0
0
--YoPo
(14.147)
10
-YoPo
0
0
yo
In
this
expression, the relative motion between the coordinate systems is along the
x
direction. The appropriate transformation for an arbitrary dii-ection of relative motion
can
be
obtained by a rotational transformation of
this
matrix.
The position vector,
as
written in Equation
14.145,
is usually called the position
“four-vector.” There
are
many other
important
four-vectors, some
of
which are de-
veloped
in
the exercises at the end
of
this
chapter. One of their more useful properties
is
that the inner product between any
two
four-vectors is invariant to coordinate
transformation.
EXERCISES
FOR
CHAPTER
14
1.
Consider the primed and unprimed coordinate systems shown below:
X2
2
2‘
--f
I
X’
Let the two systems have a common origin. Each system has a set of covariant
basis vectors
g~,
g2
and
g:,
gi.
(a)
Find
the expressions for
x1
and
x2
in terms
of
x“
and
x’~
and the inverse
(b)
Identify the covariant basis vectors
in
the primed system by drawing their
(c)
The vector
v,
expressed in the unprimed system, is given by
relations.
magnitudes and directions.
-
v
=
2g,
+
g2.
What are the contravariant and covariant components of
this
vector in the
primed system?
(d)
Using its contravariant and covariant components, determine the magnitude
of the vector by forming the inner product
of
the vector with itself. Show
EXERCISES
593
that the same result is obtained if just the contravariant components are used
with the appropriate metric.
2.
Consider the unprimed, two-dimensional Cartesian system and a primed, skewed
system, as shown in the figure below:
2
2‘
(a)
Determine the set
of
equations that relate the unprimed coordinates to the
(b)
Determine the elements
of
the transformation matrices
ti,
and
gij.
(c)
Determine the elements of the
Mij
metric in the primed system, using the
elements of the transformation matrix
gij
and Equation
14.92.
(d)
Show that the elements of the metric in part (c) could
also
have been obtained
by using
Mil
=
g{
.
g;.
3.
We have shown that the metric is a tensor. It can be expressed using purely
contravariant components
as
M’J,
or
with purely covariant components as
Mi;,
primed
coordinates.
4.
5.
so
that
- -
M
=
Milg’g2
=
Milg1g2.
(a)
Evaluate the product
MzkMk1.
(b)
Use the result for part (a) to simplify the following expressions:
i.
ViMiJMlk.
ii.
VkM”M,k.
Show that the cross product
A
X
B
expressed in a skewed system can
be
written
as
A’B1&k€ilk
and determine
v.
Starting with Equation
14.92
and taking the unprimed system to be orthononnal,
show that
M,!,
=
(s)
(s)
.
594
TENSORS IN NON-ORTHOGONAL COORDINATE SYSTEMS
The definition of the metric for the primed system
is
Mij
=
gi .g$,
which depends
only on the properties of the primed system. The above equation, however, seem
to also imply that the primed metric depends on the unprimed system. Explain
this
apparent contradiction.
6.
In
this
problem, the ideas of contravariance and covariance will be used to analyze
a two-dimensional polar system. Designate a two-dimensional, Cartesian system
with coordinates
(x,
y)
+
(x', x2)
as
the unprimed system. Let the polar system
be primed, with coordinates
(p,
6)
4
(,I1,
.I2).
The relationships between the
two sets of coordinates are
XI1
=
Jm
(a)
Obtain the elements of the metric for the primed polar system,
Mij
using the
(b)
Show for the metric you
obtained
in
part
(b)
that
expression
Mij
=
glidj
where
g',
=
dx[/dx/'.
Z
6;
*
e;,
where the
$:
are the traditional
polar
basis vectors
(gP,
6%)
defined in Chap-
ter
3,
(c)
Using contravariantkovariant transformation techniques, obtain the con-
travariant
(@',
ge)
and covariant
($,
ge)
basis vectors of the polar system
in terms
of
the Cartesian basis vectors
(Q,
$).
Compare these contravariant
and covariant polar basis vectors to the traditional polar basis vectors
(&,
66).
Pick a point in the first quadrant and plot
Q,
$,
eP,
&,
gp,
and
be.
(d)
The displacement vector can be expressed using the covariant polar basis
vectors
as
dI;
=
dx'g,!
=
dpgp
+
dogo.
so
that the contravariant components
of
the displacement vector in the polar
system are simply
dp
and
do.
What are the covariant components of the
displacement vector in the polar system?
(e)
Using the contravariant and covariant components of the displacement vector
in the polar system, form the inner product
dT
.
dF.
7.
Show
that
if
a set
of
primed coordinates
(x,
ct)
are related to
a
set
of
unprimed
coordinates
(x',
ct')
by a Lorentz transformation, then
x2
-
(c2)2
=
XI2
-
(dy.
8.
This
problem illustrates the differences between Euclidian geometry and
Min-
kowski geometry. First for the
Minkowski
geometry:
EXERCISES
595
(a)
Draw
an
orthonormal set of axes for an (x,ct)-system and plot the point
(b)
On the same plot, draw the
x’
and
ct’
axes of a primed system, moving at
(c)
Transform the coordinates of the point
in
part (a) into the primed system and
Now for Euclidian geometry:
(a)
Start with a (x1,x2)-system that is orthonormal. Make a labeled sketch of
the axes of this system and plot the point
x1
=
2cm,
x2
=
3cm.
(b)
Determine the
g’,
and the
tij
transformation matrices that relate the
(xi,
x2)-
system to a skewed primed system, where the
x’l
and
xI2
axes lie in the same
position
as
the
x’
and
ct’
axes of the above Minkowski system.
(c)
Transform the point in part (a) into the primed coordinate system and indicate
the values of
(x”,
xI2)
on the axes
of
your drawing.
(d)
Notice and comment on the differences in scaling of the
x’
and ct’ axes of
the Minkowski system and the scaling of the primed axes of the Euclidian
system.
9.
Consider a wave propagating along the x-axis with the a space-time dependence
x
=
2cm,
ct
=
3cm.
velocity
v
=
.58c with respect to the unprimed system.
indicate the values of
(x’,
ct’) on the axes of your drawing.
of
The time period of the wave is
T
=
2a/o, and the spatial wavelength
is
A
=
2a/k.
(a)
Assume that
o
and
k
are positive real numbers.
Is
the wave traveling in the
positive or negative x-direction?
(b)
Now observe this wave from a coordinate system moving along the
x-
direction, with a positive velocity
v
=
pc
with respect to the original
system. Should the frequency of the wave in the moving system be larger
or
smaller than
o?
(c)
Call the moving frame the primed frame.
In
this system, the space-time
dependence of the wave becomes
&k’x’
+oft’)
The primed quantities can
be
obtained from the unprimed quantities by using
the transformation techniques of special relativity. First, it is necessary to
construct a pair of “four-vectors” (actually, since we have only one spatial
dimension in this problem, they
are
“two-vectors”). The terms in the expo-
nential brackets can be expressed as the inner product between two of these
vectors,
R
and
K.
R
is
the position two-vector,
_-
-
R
=
$1
+
c&,