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, x 2 .....x n 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, x 2 .....x n and
x 1 , x 2 , ..... x n , there will obviously be a connection between them of the form
x r  f r (x 1 , x 2....x n )
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 (x 1 , x 2....x n )
r = 1, 2........n. (2)
r 1 2
n
r 1 2
n
where x  (x ,x ....x ) denotes the n functions f (x ,x ....x ) , 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
dz 
f
f
dx 
dy .
x
y
(3)
Extending this to several variables therefore, for each one of the new coordinates we have
n  xr
r
dx    s dx s .
s 1 x
1
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
r
r x  s
dx  
dx .
(5)
x s
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
r
r  x
m
dx   m dx .
(6)
x
where the summation with respect to s has been replaced by the summation with respect to m.
Contravariant vectors and tensors. Consider two neighbouring points P and Q in the
manifold whose coordinates are xr and xr + dxr respectively. The vector P Q
is then described by the quantities dxr which are the components of the vector in this
coordinate system. In the dashed coordinates, the vector P Q is described by the components
dx r which are related to dxr by equation (5), the differential coefficients being evaluated at
r
P. The infinitesimal displacement represented by dxr or dx  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
r
r  x s
T  
T .
 xs
(7)
2
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
r
s
rs x   x mn
T   m n T
.
 x x
(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
T T ,
(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
  xs

 xr  xs  xr

(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.
Defn. A set of n quantities T r 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
T 
r
x s
T .
 xr s
(11)
3
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
 xm x n
T 

(12)
rs x r x s T mn
and similarly for higher orders.
Mixed tensors. These are tensors with at least one covariant suffix and one contravariant
r
suffix. An example is the third order tensor T st which transforms according to
p
x r x n x m
r
T 
st   xm x s  xt Tnp
(13)
Another example is the Kronecker delta defined by
 sr  1, r  s
 0, r  s
(14)
mn.. t
It is a tensor of the type indicated because (a) in an expression such as Bpq..  m , which
involves summation with respect to m, there is only one non-zero contribution from the
mn.. t
tn..
Kronecker delta, that for which m = t, and so Bpq..  m  B pq.. ; (b) the coordinates in any
 xr
r
coordinate system are necessarily independent of each other, so that s   s and
x
r
 x
r
 s ; so these two properties taken together imply that
s
 x
sr 
 xr  xn m
 .
x m  xs n
(15)
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 T mn  0 (which, since m
4
and n are unrepeated, implies that the equation is true for all m and n, not just for
 0 also, from the transformation
some particular choice of these suffices), then T 
rs
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
Addition of tensors. Two tensors of the same type may be added together to give another
r
r
tensor of the same type, e.g. if Ast and Bst are tensors of the type indicated, then we can
define
Cr  Ar  Br .
(16)
st
st
st
r
It is easy to show that the quantities Cst form the components of a tensor.
rs
Symmetric and antisymmetric tensors. A is a symmetric contravariant tensor if
Ars  Asr and antisymmetric if Ars  Asr . Similarly for covariant tensors. Symmetry
rs
sr
properties are conserved under transformation of coordinates, e.g. if A  A , then
Amn 
x m  xn rs  xm x n sr
A 
A  Anm .
r
s
r
s
 x x
x  x
(17)
s
r
Note however that for a mixed tensor, a relation such as Ar  As 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 either both subscripts or both superscripts) can obviously be
extended to tensors of higher order.
Any covariant or contravariant tensor of second order may be expressed as the sum of a
symmetric tensor and an antisymmetric tensor, e.g.
Ars 
1 rs
1
(A  Asr )  (Ars  A sr ) .
2
2
5
(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.)
m
To multiply two tensors e.g. Ars and Bn we simply write
m
Crsn
 ArsBnm .
(19)
m
It follows immediately from their transformation properties that the quantities Crsn form a
tensor of the type indicated. This tensor, in which the symbols for the suffices are all
m
different, is called the outer product of Ars and Bn .
m
Contraction of tensors. Given a tensor T np , then
m
s
t
m x   x  x r
Tnp
 
p Tst .
 xr x n x 
(20)
Hence replacing n by m (and therefore implying summation with respect to m)
m
s
t
m x  x  x r
T 

mp
p T st
 xr x m x 
x s x t r
 r
p T st
 x  x
t
s x
r
 r
p Tst
x 

m
 xt s
T
 xp st
(21)
so we see that T mp behaves like a tensor Ap . The upshot is that contraction of a tensor (i.e.
writing the same letter as a subscript and a superscript) reduces the order of the tensor by 2
and yields a tensor whose type is indicated by the remaining suffices.
Note that contraction can only be applied successfully to suffices at different levels. We may
p
p
of course construct, starting with a tensor Aqrs say, a new set of quantities Aqrr ; but these
do not have tensor character (as one can easily check) so are of little interest.
6
m
m
Having constructed the outer product Crsn  ArsBn in the example above, we can form the
m
m
m
m
corresponding inner products Cmsn  AmsBn and Crmn  ArmBn . 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:
r
Theorem 1. Let X be the components of an arbitrary contravariant vector. Let Ar be another
r
set of quantities. If ArX is an invariant, then Ar form the components of a covariant vector.
r
r
Proof: Since X is a tensor, it obeys the tensor transformation law. Invariance of ArX
means that
 xs r
X
 xr
s
ArX r  A
sX  As
and so
xs r
(Ar  As r )X  0 .
x
(22)
(23)
r
Hence, since X is an arbitrary tensor,
x s
Ar 
As.
 xr
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
rs
r
tensor, are themselves the components of a tensor. For example, if A Xs is a tensor B , then
Ars is a second order contravariant tensor.
r s
r
Theorem 2. If arsX X is invariant, X being an arbitrary contravariant vector and ars being
symmetric in all coordinate systems, then ars are the components of a covariant tensor of
second order.
r s
Proof: From our assumption about the invariance of arsX X ,
amnX mX n  ars
 X r Xs
x r x s m n
 ars
 m
X X
 x x n
7
(25)
bmnX mXn  (a mn  a
rs
Hence
x r x s m n
)X X  0 .
x m x n
(26)
m
m n
Since X is arbitrary and the total coefficient of X X is bmn  bnm , we deduce that
bmn  bnm  0 , i.e.
 xr  xs
x r  xs

a

rs
 xm  xn
x n x m
x r x s
 (ars
  a
)
sr
x m x n
amn  anm  ars

(27)
on interchanging the summation variables r and s in the second term. But amn  anm in all
coordinate systems, hence
amn  a
rs
 xr  xs
.
x m  xn
QED
(28)
The metric tensor
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 d
between two neighbouring points x, y, z and x  dx, y  dy, z  dz is given by
d 2  (dx)2 (dy)2  (dz)2 .
1
2
(29)
3
If we choose any other coordinates x , x ,x to identify points in this space, the original
coordinates will be functions of these new coordinates, and their differentials will be linear
combinations of the differentials of the new coordinates. Thus in terms of the latter
coordinates,
d 2  amndxm dxn
(30)
m
where the amn will be functions of x . (For example in spherical polar coordinates
x 1  r, x 2   , x3   we have a11  1, a 22  r 2 , a 33  r 2 sin2  and all other a's are zero.)
We now show that amn is a covariant tensor of second order. The proof goes as follows:
(a) amn may be taken to be symmetric since each apq occurs only in the combination
apq  a qp on the RHS of (30).
8
2
m n
(b) d  amndx dx is invariant, since the distance between two points does not depend
on the coordinates used to evaluate it.
(c) By keeping one point fixed and letting the second point vary in the neighbourhood of the
r
first, dx may be considered an arbitrary contravariant tensor.
Hence, using the theorem above, amn 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
normally denoted by gmn . Its significance is that it can be used to define the analogue of
"distance" between points, and the lengths of vectors. We will assume that all manifolds that
we will be dealing with from now on are Riemannian.
r
r
r
Defn. The interval ds between the neighbouring points x and x  dx is given by
ds2  gmn dx mdx n .
(31)
This is of course invariant. In the familiar Euclidean space where gmn is just the amn above,
ds2  d 2  0 , being zero only when the two points coincide. In other cases however, e.g. in
2
spacetime in relativity theory, ds may take on negative values, so that ds itself is not
r
r
necessarily real. If ds = 0 for dx not all zero, the displacement dx is called a null
displacement. Note that there is no requirement that ds should necessarily have the physical
dimensions of length.
The conjugate metric tensor. From the covariant metric tensor gmn we can construct a
mn
contravariant tensor g defined by
g mngnp   pm .
(32)
mn
p
To show that g is a tensor, we note that, for any contravariant vector V ,
g mngnpV p   pmV p  V m . This means that the inner product of g mn with the arbitrary
p
mn
m
covariant vector gnpV is a tensor, V , and so we deduce that g is indeed a tensor of
the type indicated. It is said to be conjugate to gmn . It is easily shown that when the metric
9
tensor is diagonal, i.e. when gmn  0, m  n , the conjugate tensor is also diagonal, with each
nn
diagonal element satisfying g  1/ gnn .
The following theorem can be proved, but will just be quoted here: if g is the determinant of
the matrix gmn (i.e. choosing to write the components of the tensor gmn in the form of a
matrix array), then
g mn


gmn  r ln g .
r
x
x
(33)
m
Raising and lowering suffices. Given a tensor T rs , we may form another tensor T mrs
defined by
Note that
T nrs  gnmT m
rs
g mnTnrs  g mngntTtrs  tmTtrs  Tm
rs .
(34)
(35)
The tensor T nrs may therefore be regarded as possessing a special relationship with the
m
original tensor T rs in that either of them may be found from the other by the operation of
forming the inner product of the first 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 can raise one or both of the suffices in the tensor T mn , generating
m
m
mn
the corresponding tensors T n , T n and T . 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 disappears and we simply write either of
m
these as T n .
Cartesian tensors
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 gmn 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
10
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
on the surface by spatial coordinates ( ,  ) ; these are the second and third members of the
set of three spherical polar coordinates (r,  ,  ) , 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
(36)
d 2  a 2 (d 2  sin2  d 2 )
where a is the radius of the sphere. No coordinate transformation can be found from ( ,  )
to new coordinates (x1 , x 2 ) such that the line element can be re-expressed in the form
d 2  (dx1)2  (dx 2 )2
(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
ds2  (dx1 )2  (dx2 ) 2  (dx 3 )2  ......
(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.
Orthogonal transformations. These are linear transformations between two sets of
homogeneous coordinates, x m and x m of the form
x m  Anm xn  Am
11
(39)
where the coefficients Anm and Am are constants. Since the set x m are homogeneous,
But, from (39),
and so
ds2  dx mdx m .
(40)
dx m  Anm dxn
(41)
n m
p
ds2  Am
n dx Ap dx .
(42)
But the coordinates x m are also homogeneous, and so the RHS of (42) is required to be
equal to dx p dx p . Hence
n
p
AnmAm
(43)
p dx  dx
which requires
Anm Apm  1 , 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
p
m
x m  Am
p x A ,
12
(45)
we have
p
m m
Anmxm  AnmAm
p x  An A
(46)
m
 x n  Am
nA
(47)
using (44). So the inverse equations are
m
n
x n  Am
n x   A
where
An  AnmAm .
(48)
(49)
The whole point of this analysis is now revealed: from equations (39) and (48) we see that
 xm
 xn
m
 An ,
 Anm .
n
m
x
 x
(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
equation ji   Ei relates the components of the current density j to the components of the
electric field E, with the conductivity  taken to be constant. But in a single crystal the
general relationship would be expressed as ji   ijE j where  ij 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 embedded in a
flat space is taken for granted, and Cartesian tensors are automatically implied by the choice
of a Cartesian coordinate system.
N C McGill
13