Embed Size (px)
Citation preview
Tensor Theory
Introduction and definitions
In n-dimensional space Vn (called a "manifold" in mathematics), points are specified by assigning values to a set of n continuous real variables x
1,x2.....xn called the coordinates.
In many cases these will run from -∞ to +∞, but the range of some or all of these can be finite.
Examples: In Euclidean space in three dimensions, we can use cartesian coordinates x, y and z, each of which runs from -∞ to +∞. For a two dimensional Euclidean plane, Cartesians may again be employed, or we can use plane polar coordinates r, whose ranges are 0 to ∞ and 0 to 2 respectively.
Coordinate transformations. The coordinates of points in the manifold may be assigned in a number of different ways. If we select two different sets of coordinates, x
1,x2.....xn and
x 1, x 2, ..... x n, there will obviously be a connection between them of the form
x r fr(x1,x2....xn) r = 1, 2........n. (1)
where the f's are assumed here to be well behaved functions. Another way of expressing the same relationship is
x r x r(x1,x2....xn) r = 1, 2........n. (2)
where x r(x1,x2....xn) denotes the n functions fr(x1,x2....xn), r = 1, 2......n.
Recall that if a variable z is a function of two variables x and y, i.e. z = f (x, y), then the connection between the differentials dx, dy and dz is
dzf
xdx f
ydy. (3)
Extending this to several variables therefore, for each one of the new coordinates we have
1
2
d xr
∂ xr
∂xss1
n∑ dxs
3
. r=1, 2........n. (4)
The transformation of the differentials of the coordinates is therefore linear and homogeneous, which is not necessarily the case for the transformation of the coordinates themselves.
Range and Summation Conventions. Equations such as (4) may be simplified by the use of two conventions:
Range Convention: When a suffix is unrepeated in a term, it is understood to take all values in the range 1, 2, 3.....n.
Summation Convention: When a suffix is repeated in a term, summation with respect to that suffix is understood, the range of summation being 1, 2, 3.....n.
With these two conventions applying, equation (4) may be written as
4
5
d xr
∂ xr
∂xsdxs
6
. (5)Note that a repeated suffix is a "dummy" suffix, and can be replaced by any convenient alternative. For example, equation (5) could have been written as
7
8
d xr
∂ xr
∂xmdxm
9
. (6)where the summation with respect to s has been replaced by the summation with respect to m.
10
Contravariant vectors and tensors. Consider two neighbouring points P and Q in the manifold whose coordinates are xr and xr + dxr respectively. The vector
11
12
is then described by the quantities dxr which are the components of the vector in this coordinate system. In the dashed coordinates, the vector
13
r P Q
14
is described by the components
15
d xr16
which are related to dxr by equation (5), the differential coefficients being evaluated at P.
17
The infinitesimal displacement represented by dxr or
18
d xr19
is an example of a contravariant vector.
Defn. A set of n quantities T r associated with a point P are said to be the components of a contravariant vector if they transform, on change of coordinates, according to the equation
20
21
Tr
∂ xr
∂xsTs
22
. (7)
where the partial derivatives are evaluated at the point P. (Note that there is no requirement that the components of a contravariant tensor should be infinitesimal.)
Defn. A set of n 2 quantities T rs associated with a point P are said to be the components of a contravariant tensor of the second order if they transform, on change of coordinates, according to the equation
23
24
Trs
∂ xr
∂xm∂ xs
∂xnTmn
25
. (8)
Obviously the definition can be extended to tensors of higher order. A contravariant vector is the same as a contravariant tensor of first order.
Defn. A contravariant tensor of zero order transforms, on change of coordinates, according to the equation
26
27
TT28
, (9)
i.e. it is an invariant whose value is independent of the coordinate system used.
Covariant vectors and tensors. Let be an invariant function of the coordinates, i.e. its value may depend on position P in the manifold but is independent of the coordinate system used. Then the partial derivatives of transform according to
29
30
∂φ
∂ xr∂φ
∂xs∂xs
∂ xr31
(10)
Here the transformation is similar to equation (7) except that the partial derivative involving the two sets of coordinates is the other way up. The partial derivatives of an invariant function provide an example of the components of a covariant vector.
32
Defn. A set of n quantities
33
associated with a point P are said to be the components of a covariant vector if they transform, on change of coordinates, according to the equation
34
35
Tr
∂xs
∂ xrTs
36
. (11)
By convention, suffices indicating contravariant character are placed as superscripts, and those indicating covariant character as subscripts. Hence the reason for writing the coordinates as xr. (Note however that it is only the differentials of the coordinates, not the
coordinates themselves, that always have tensor character. The latter may be tensors, but this is not always the case.)
Extending the definition as before, a covariant tensor of the second order is defined by the transformation
37
38
Trs
∂xm
∂ xr∂xn
∂ xsTmn
39
(12)
and similarly for higher orders.
40
Mixed tensors. These are tensors with at least one covariant suffix and one contravariant suffix. An example is the third order tensor
41
Tstr
42
which transforms according to
43
44
Tstr
∂ xr
∂xm∂xn
∂ xs∂x
∂ xtTnm
45
(13)
Another example is the Kronecker delta defined by
46
47
dsr 1, rs48
49
0, r≠s50
(14)
51
It is a tensor of the type indicated because (a) in an expression such as
52
Bpq..mn..dm
t53
, which involves summation with respect to m, there is only one non-zero contribution from
54
the Kronecker delta, that for which m = t, and so
55
Bpq..mn..dm
t B..tn..
56
; (b) the coordinates in any coordinate system are necessarily independent of each other, so
57
that
58
∂xr
∂xsds
r
59
and
60
∂ xr
∂ xs ds
r61
; so these two properties taken together imply that
62
63
dsr
∂ xr
∂xm∂xn
∂ xsdnm64
. (15)
65
Notes. 1. The importance of tensors is that if a tensor equation is true in one set of coordinates it is also true in any other coordinates. e.g. if
66
Tmn0
67
(which, since m and n are unrepeated, implies that the equation is true for all m and
68
n, not just for some particular choice of these suffices), then
69
Trs0
70
also, from the transformation law. This illustrates the fact that any tensor equation is covariant, which means that it has the same form in all coordinate systems.
2. A tensor may be defined at a single point P within the manifold, or along a curve, or throughout a subspace, or throughout the manifold itself. In the latter cases we speak of a tensor field.
Tensor algebra
71
Addition of tensors. Two tensors of the same type may be added together to give another tensor of the same type, e.g. if
72
Astr
73
and
74
Bstr
75
are tensors of the type indicated, then we can define
76
77
Cstr Ast
r Bstr
78
. (16)
79
It is easy to show that the quantities
80
Cstr
81
form the components of a tensor.
82
Symmetric and antisymmetric tensors.
83
Ars
84
is a symmetric contravariant tensor if
85
ArsAsr
86
and antisymmetric if
87
Ars−Asr88
. Similarly for covariant tensors. Symmetry properties are conserved under transformation of
89
coordinates, e.g. if
90
ArsAsr91
, then
92
93
Amn
∂ xm
∂xr∂ xn
∂xsArs
∂ xm
∂xr∂ xn
∂xsAsr Anm
94
. (17)
95
Note however that for a mixed tensor, a relation such as
96
Ars As
r 97
does not transform to give the equivalent relation in the dashed coordinates. The concept of symmetry (with respect to a pair of suffices which are eithe
Any covariant or contravariant tensor of second order may be expressed as the sum of a symmetric tensor and an antisymmetric tensor, e.g.
98
99
Ars
12(ArsAsr)
12(Ars−Asr)
100
. (18)
Multiplication of tensors. In the addition of tensors we are restricted to tensors of a single type, with the same suffices (though they need not occur in the same order). In the multiplication of tensors there is no such restriction. The only condition is that we never multiply two components with the same suffix at the same level in each. (This would imply summation with respect to the repeated suffix, but the resulting object would not have tensor character - see later.)
101
To multiply two tensors e.g.
102
Ars
103
and
104
Bnm
105
we simply write
106
107
Crsnm ArsBn
m108
. (19)
109
It follows immediately from their transformation properties that the quantities
110
Crsnm
111
form a tensor of the type indicated. This tensor, in which the symbols for the suffices are all
112
different, is called the outer product of
113
Ars
114
and
115
Bnm
116
.
117
Contraction of tensors. Given a tensor
118
Tnpm
119
, then
120
121
Tnm
∂ xm
∂xr∂xs
∂ xn∂xt
∂ xTstr122
. (20)
Hence replacing n by m (and therefore implying summation with respect to m)
123
124
Tmm
∂ xm
∂xr∂xs
∂ xm∂xt
∂ xTstr125
126
∂xs
∂xr∂xt
∂ xTstr127
128
dr
s ∂xt
∂ xTstr129
130
∂xt
∂ xTsts131
(21)
132
so we see that
133
Tmpm
134
behaves like a tensor
135
Ap136
. The upshot is that contraction of a tensor (i.e. writing the same letter as a subscript and a superscript) reduces the
137
Note that contraction can only be applied successfully to suffices at different levels. We may of course construct, starting with a tensor
138
Aqrs
p
139
say, a new set of quantities
140
Aqrr
p141
; but these do not have tensor character (as one can easily check) so are of little interest.
142
Having constructed the outer product
143
Crsnm ArsBn
m 144
in the example above, we can form the corresponding inner products
145
Cmsnm AmsBn
m 146
and
147
Crmnm ArmBn
m148
. Each of these forms a covariant tensor of second order.
Tests for tensor character. The direct way of testing whether a set of quantities form the components of a tensor is to see whether they obey the appropriate tensor transformation law when the coordinates are changed. There is also an indirect method however, two examples of which will now be given:
149
Theorem 1. Let
150
X r
151
be the components of an arbitrary contravariant vector. Let
152
Ar153
be another set of quantities. If
154
ArXr
155
is an invariant, then
156
Ar157
form the components of a covariant vector.
158
Proof: Since
159
X r
160
is a tensor, it obeys the tensor transformation law. Invariance of
161
ArXr
162
means that
163
164
ArX
r As Xs As∂ xs
∂xrX r165
(22)
166
and so
167
(Ar− As
∂ xs
∂xr)X r0
168
. (23)
169
Hence, since
170
X r
171
is an arbitrary tensor,
172
173
Ar
∂ xs
∂xrAs
174
. QED (24)
As an extension of this theorem, it is easy to show that any set of functions of the coordinates, whose inner product with an arbitrary covariant or contravariant vector is a
175
tensor, are themselves the components of a tensor. For example, if
176
ArsXs
177
is a tensor
178
Br179
, then
180
Ars
181
is a second order contravariant tensor.
182
Theorem 2. If
183
arsXrXs
184
is invariant,
185
X r186
being an arbitrary contravariant vector and
187
ars
188
being symmetric in all coordinate systems, then
189
ars
190
are the components of a covariant tensor of second order.
191
Proof: From our assumption about the invariance of
192
arsXrXs
193
,
194
195
amnXmXn ars Xr Xs196
197
198
ars
∂ xr
∂xm∂ xs
∂xnX mX n199
(25)
200
Hence
201
bmnX
mXn ≡(amn − ars∂ xr
∂xm∂ xs
∂xn)XmX n 0
202
. (26)
203
Since
204
Xm
205
is arbitrary and the total coefficient of
206
XmXn
207
is
208
bmnbnm209
, we deduce that
210
bmnbnm 0211
, i.e.
212
213
amn anm ars
∂ xr
∂xm∂ xs
∂xn ars
∂ xr
∂xn∂ xs
∂xm214
215
( ars asr)
∂ xr
∂xm∂ xs
∂xn216
(27)
217
on interchanging the summation variables r and s in the second term. But
218
amnanm
219
in all coordinate systems, hence
220
221
amn ars
∂ xr
∂xm∂ xs
∂xn222
. QED (28)
The metric tensor
223
The Euclidean space. Consider first the familiar Euclidean space in three dimensions, i.e. a space in which one can define Cartesian coordinates x, y and z so that the distance
224
dl
225
between two neighbouring points
226
x,y,z
227
and
228
x dx, y dy, z dz
229
is given by
230
231
dl 2(dx)2 (dy)2 (dz)2232
. (29)
233
If we choose any other coordinates
234
x1,x2,x3
235
to identify points in this space, the original coordinates will be functions of these new coordinates, and their differentials will be linear combinati
236
237
dl 2amn dxm dxn238
(30)
239
where the
240
amn
241
will be functions of
242
xm243
. (For example in spherical polar coordinates
244
x1r, x2 , x3 φ 245
we have
246
a111, a22 r2 , a33 r
2sin2 247
and all other a's are zero.)
248
We now show that
249
amn
250
is a covariant tensor of second order. The proof goes as follows:
251
(a)
252
amn
253
may be taken to be symmetric since each
254
apq
255
occurs only in the combination
256
apqa
257
on the RHS of (30).
258
(b)
259
dl 2amn dxm dxn 260
is invariant, since the distance between two points does not depend on the coordinates used to evaluate it.
261
(c) By keeping one point fixed and letting the second point vary in the neighbourhood of the first,
262
dxr
263
may be considered an arbitrary contravariant tensor.
264
Hence, using the theorem above,
265
amn
266
is a covariant tensor of second order. It is called the metric tensor for the Euclidean 3-space. A similar tensor obviously exists in the case of a two dimensional Euclidean space.
Riemannian space. A manifold is said to be Riemannian if there exists within it a covariant tensor of the second order which is symmetric. This tensor is called the metric tensor and
267
normally denoted by
268
gmn269
. Its significance is that it can be used to define the analogue of "distance" between points, and the lengths of vecto
270
Defn. The interval ds between the neighbouring points
271
xr
272
and
273
xr dxr274
is given by
275
276
ds2gmn dxm dxn277
. (31)
278
This is of course invariant. In the familiar Euclidean space where
279
gmn
280
is just the
281
amn
282
above,
283
ds2dl2 ≥0284
, being zero only when the two points coincide. In other cases however, e.g. in spacetime in
285
relativity theory,
286
ds2
287
may take on negative values, so that
288
ds
289
itself is not necessarily real. If ds = 0 for
290
dxr
291
not all zero, the displacement
292
dxr
293
is called a null displacement. Note that there is no requirement that ds should necessarily have the physical dimensions of length.
294
The conjugate metric tensor. From the covariant metric tensor
295
gmn
296
we can construct a contravariant tensor
297
gmn
298
defined by
299
300
gmngnpd
m301
. (32)
302
To show that
303
gmn
304
is a tensor, we note that, for any contravariant vector
305
Vp306
,
307
gmngnpV
pdmV V m
308
. This means that the inner product of
309
gmn
310
with the arbitrary covariant vector
311
gnpVp
312
is a tensor,
313
Vm314
, and so we deduce that
315
gmn
316
is indeed a tensor of the type indicated. It is said to be conjugate to
317
gmn318
. It is easily shown that when the metric tensor is diagonal, i.e. when
319
gmn0, m ≠n320
, the conjugate tensor is also diagonal, with each diagonal element satisfying
321
gnn1/ gnn322
.
323
The following theorem can be proved, but will just be quoted here: if g is the determinant of the matrix
324
gmn
325
(i.e. choosing to write the components of the tensor
326
gmn
327
in the form of a matrix array), then
328
329
gmn ∂
∂xrgmn
∂∂xr
lng330
. (33)
331
Raising and lowering suffices. Given a tensor
332
T rsm
333
, we may form another tensor
334
Tmrs
335
defined by
336
337
TnrsgnmT rsm
338
(34)
339
Note that
340
gmnTnrsg
mn gntT rst dt
m T rst T rs
m341
. (35)
342
The tensor
343
Tnrs
344
may therefore be regarded as possessing a special relationship with the original tensor
345
T rsm
346
in that either of them may be found from the other by the operation of forming the inner product of the f irst with the metric tensor or its conjugate. For this reason, the same symbol is used (T in this instance), and we describe the above processes by saying that in (34) we have "lowered the suffix m", and that in (35) we have "raised the suffix n". The process of raising or lowering suffices can be extended to cover all the indices of a tensor. For example we
347
can raise one or both of the suffices in the tensor
348
Tmn
349
, generating the corresponding tensors
350
T nm
351
,
352
Tnm
353
and
354
Tmn
355
. Notice the distinction between the two forms of the mixed tensor, effected by leaving appropriate gaps in the set of indices. When the tensor is symmetric however this distinction
356
disappears and we simply write either of these as
357
Tnm
358
.
Cartesian tensors
359
Flat space. A space or manifold is said to be flat if it is possible to find a coordinate system for which the metric tensor
360
gmn
361
is diagonal, with all diagonal elements equal to ± 1, otherwise the space is said to be curved.
The familiar Euclidean space in two or three dimensions is obviously flat, the diagonal elements then being all equal to + 1. We normally assume that the ordinary three dimensional space which we inhabit is flat, likewise in the special theory of relativity that the 4-dimensional "spacetime" is flat. In the general theory of relativity however this assumption must be abandoned, and we have to deal with the consequences of spacetime being curved.
It should not be assumed however that curved spaces never arise in elementary physics or mathematics. Take for instance the surface of a sphere, where it is natural to identify position
362
on the surface by spatial coordinates
363
( , φ)
364
; these are the second and third members of the set of three spherical polar coordinates
365
(r, , φ)366
, the first one having been set equal to a constant, viz. the radius of the sphere. The expression for the line element on the surface of a sphere is
367
368
dl 2a2(d 2 sin2 dφ2)369
(36)
370
where a is the radius of the sphere. No coordinate transformation can be found from
371
( , φ)
372
to new coordinates
373
(x1, x2) 374
such that the line element can be re-expressed in the form
375
376
dl 2(dx1)2 ±(dx2)2377
(37)
and so the space is by definition curved. Of course in this case the result is in accordance with our everyday notions regarding curvature. Geometry in a curved space is intrinsically different from that for flat spaces, e.g. parallel lines do eventually meet, and the sum of the angles in a triangle is not 180o.
Homogeneous coordinates. These are coordinates for which the metric tensor is diagonal with all diagonal elements taking the values +1. The metric expression is then
378
379
ds2(dx1)2 (dx2)2 (dx3)2 .... ..380
(38)
Clearly such coordinates can exist only if the space in question is flat. If this condition is satisfied, it must always be possible to find a set of homogeneous coordinates, since any minus signs in an expression for the metric can be transformed away by re-defining coordinates (albeit with imaginary values) with appropriate factors of i inserted.
Cartesian coordinates in the Euclidean plane or the Euclidean 3- space are obviously homogeneous.
381
Orthogonal transformations. These are linear transformations between two sets of homogeneous coordinates,
382
xm
383
and
384
xm
385
of the form
386
387
xm An
m xn Am388
(39)
389
where the coefficients
390
Anm
391
and
392
Am
393
are constants. Since the set
394
xm
395
are homogeneous,
396
397
ds2d xm d xm398
. (40)
399
But, from (39),
400
d xm Anm dxn401
(41)
402
and so
403
ds2Anm dxnA
m dx404
. (42)
405
But the coordinates
406
xm
407
are also homogeneous, and so the RHS of (42) is required to be equal to
408
dxpdxp409
. Hence
410
411
AnmAp
mdxndx412
(43)which requires
413
414
Anm Ap
m 1415
, n = p = 0, otherwise (44)
Cartesian tensors. If we are dealing with a flat space, homogeneous coordinates are an obvious preferred choice since they facilitate geometrical calculations. Any change of coordinates will normally involve orthogonal transformation equations satisfying equation (39). It is convenient therefore to define Cartesian tensors as quantities which transform according to the usual tensor transformation equations when the coordinates undergo an orthogonal transformation, i.e. as we pass from one set of homogeneous coordinates to another.
Note carefully that orthogonal transformation equations are a subset of all possible transformation equations. Therefore "Cartesian tensors" will not in general obey the tensor laws when subjected to an arbitrary coordinate transformation. On the other hand any (unrestricted) tensor automatically satisfies the definition of being a Cartesian tensor, since the conditions for the latter are a subset of the conditions for the former. We therefore have the seemingly paradoxical statement that "all tensors are Cartesian tensors, but not all Cartesian tensors are tensors".
Consider now the inverse transformation equations for an orthogonal transformation. Starting from (39) in the slightly modified form
416
417
xm A
m x Am418
, (45)
419
we have
420
Anm xm An
mAm x An
mAm421
(46)
422
423
xn An
m Am424
(47)using (44). So the inverse equations are
425
426
xnAn
m xm An427
(48)
428
where
429
An −AnmAm430
. (49)
The whole point of this analysis is now revealed: from equations (39) and (48) we see that
431
432
∂ xm
∂xnAn
m433
,
434
∂xn
∂ xmAn
m435
. (50)
The two differential coefficients involved in these equations are therefore equal; but we see, looking back at equations (7) and (11), that it was the presumed difference between them which was the whole basis of the distinction between covariant and contravariant tensors. Therefore if we restrict ourselves to Cartesian tensors, the distinction between covariant and contravariant tensors disappears, and there is no reason to continue to differentiate between indices used as superscripts and those used as subscripts. For convenience, subscripts are almost invariably the preferred choice in practice.
For example, in solid state physics we may require to calculate the electrical conductivity of a metallic crystal. In an isotropic medium such as a polycrystalline material the conductivity
436
equation
437
ji sEi
438
relates the components of the current density j to the components of the electric field E, with
439
the conductivity
440
s
441
taken to be constant. But in a single crystal the general relationship would be expressed as
442
ji siφEφ
443
where
444
siφ
445
is the conductivity tensor and the usual summation convention applies. In most textbooks on such topics the underlying assumption that the crystal or other system under consideration is e bedded in a flat space is taken for granted, an
N C McGill
446
