Contra-variant Four-vector. The line-element is defined by the four
components _dx__{ν}, whose transformation law is expressed by the
equation
(5) $$ dx'_{\sigma} = \sum_{\nu} \frac{\partial x'_{\sigma}}{\partial
x_{\nu}} dx_{\nu} $$
The _dx′__{σ}’_s_ are expressed as linear and homogeneous function of
_dx__{ν}’_s_; we can look upon the differentials of the co-ordinates as
the components of a tensor, which we designate specially as a
contravariant Four-vector. Everything which is defined by Four
quantities A^{σ}, with reference to a co-ordinate system, and transforms
according to the same law,
(5a)
$$ A^{\sigma} = \sum_{\nu} \frac{\partial x'_{\sigma}}{\partial x_{\nu}}
A^{\nu} $$
we may call a contra-variant Four-vector. From (5. a), it follows at
once that the sums (A^{σ} ± B^{σ}) are also components of a four-vector,
when A^{σ} and B^{σ} are so; corresponding relations hold also for all
systems afterwards introduced as “tensors” (Rule of addition and
subtraction of Tensors).
_Co-variant Four-vector._
We call four quantities A_{ν} as the components of a covariant
four-vector, when for any choice of the contra-variant four vector B^{ν}
(6) ∑_{ν} A_{ν} B^{ν} = _Invariant_. From this definition follows the
law of transformation of the co-variant four-vectors. If we substitute
in the right hand side of the equation
∑_{σ} A′_{σ} B^{σ′} = ∑_{ν} A_{ν} B^{ν}.
the expressions
$$ \sum_{\sigma} \frac{\partial x_{\nu}}{\partial x_{\sigma'}}
B^{\sigma'} $$
for B^{ν} following from the inversion of the equation (5a) we get
$$ \sum_{\sigma} B^{\sigma'} \sum_{\nu} \frac{\partial x_{\nu}}{\partial
x_{\sigma'}} A_{\nu} = \sum_{\sigma} B^{\sigma'} A'_{\sigma} $$
As in the above equation B^{σ′} are independent of one another and
perfectly arbitrary, it follows that the transformation law is:—
$$ A'_{\sigma} = \sum \frac{\partial x_{\nu}}{\partial x_{\sigma'}}
A_{\nu} $$
_Remarks on the simplification of the mode of writing the expressions._
A glance at the equations of this paragraph will show that the indices
which appear twice within the sign of summation [for example ν in (5)]
are those over which the summation is to be made and that only over the
indices which appear twice. It is therefore possible, without loss of
clearness, to leave off the summation sign; so that we introduce the
rule: wherever the index in any term of an expression appears twice, it
is to be summed over all of them except when it is not expressedly said
to the contrary.
The difference between the co-variant and the contra-variant four-vector
lies in the transformation laws [(7) and (5)]. Both the quantities are
tensors according to the above general remarks; in it lies its
significance. In accordance with Ricci and Levi-civita, the
contravariants and co-variants are designated by the over and under
indices.
§ 6. Tensors of the second and higher ranks.
Contravariant tensor:—If we now calculate all the 16 products A^{μν} of
the components A^{μ} B^{ν}, of two contravariant four-vectors
(8) A^{μν} = A^{μ}B^{ν}
A^{μν}, will according to (8) and (5 a) satisfy the following
transformation law.
(9)
$$ A^{\sigma \tau'} = \frac{\partial x'_{\sigma}}{\partial x_{\mu}}
\frac{\partial x'_{\tau}}{\partial x_{\nu}} A^{\mu \nu} $$
We call a thing which, with reference to any reference system is defined
by 16 quantities and fulfils the transformation relation (9), a
contravariant tensor of the second rank. Not every such tensor can be
built from two four-vectors, (according to 8). But it is easy to show
that any 16 quantities A^{μν}, can be represented as the sum of
A^{μ}B^{ν} of properly chosen four pairs of four-vectors. From it, we
can prove in the simplest way all laws which hold true for the tensor of
the second rank defined through (9), by proving it only for the special
tensor of the type (8).
_Contravariant Tensor of any rank_:—It is clear that corresponding to
(8) and (9), we can define contravariant tensors of the 3rd and higher
ranks, with 4³, etc. components. Thus it is clear from (8) and (9) that
in this sense, we can look upon contravariant four-vectors, as
contravariant tensors of the first rank.
_Co-variant tensor._
If on the other hand, we take the 16 products A_{μν} of the components
of two co-variant four-vectors A_{μ} and B_{ν},
(10) A_{μν} = A_{μ} B_{ν}.
for them holds the transformation law
(11)
$$ A^{\sigma \tau'} = \frac{\partial x'_{\mu}}{\partial x_{\sigma'}}
\frac{\partial x'_{\nu}}{\partial x_{\tau'}} A^{\mu \nu} $$
By means of these transformation laws, the co-variant tensor of the
second rank is defined. All re-marks which we have already made
concerning the contravariant tensors, hold also for co-variant tensors.
_Remark_:—
It is convenient to treat the scalar Invariant either as a contravariant
or a co-variant tensor of zero rank.
_Mixed tensor._ We can also define a tensor of the second rank of the
type
(12) A_{μ}^{ν} = A_{μ}B^{ν}
which is co-variant with reference to μ and contravariant with reference
to ν. Its transformation law is
(13)
$$ A^{\tau'}_{\sigma} = \frac{\partial x_{\tau'}}{\partial x_{\beta}}
\frac{\partial \alpha}{\partial x_{\sigma'}} A^{\beta}_{\alpha} $$
Naturally there are mixed tensors with any number of co-variant indices,
and with any number of contra-variant indices. The co-variant and
contra-variant tensors can be looked upon as special cases of mixed
tensors.
_Symmetrical tensors_:—
A contravariant or a co-variant tensor of the second or higher rank is
called symmetrical when any two components obtained by the mutual
interchange of two indices are equal. The tensor A^{μν} or A_{μν} is
symmetrical, when we have for any combination of indices
(14) A^{μν} = A^{νμ}
or
(14a) A_{μν} = A_{νμ}.
It must be proved that a symmetry so defined is a property independent
of the system of reference. It follows in fact from (9) remembering (14)
$$ A^{\sigma \tau'} = \frac{\partial x_{\sigma'}}{\partial x_{\mu}}
\frac{\partial x'_{\tau}}{\partial x_{\nu}} A^{\mu \nu} = \frac{\partial
x_{\sigma'}}{\partial x_{\mu}} \frac{\partial x_{\tau'}}{\partial
x_{\nu}} A^{\nu \mu} = A^{\tau \sigma'} $$
_Antisymmetrical tensor._
A contravariant or co-variant tensor of the 2nd, 3rd or 4th rank is
called _antisymmetrical_ when the two components got by mutually
interchanging any two indices are equal and opposite. The tensor or
A^{μν} or A_{μν} is thus antisymmetrical when we have
(15) A^{μν} = -A^{νμ}
or
(15a) A_{μν} = -A_{νμ}.
Of the 16 components A^{μν}, the four components A^{μμ} vanish, the rest
are equal and opposite in pairs; so that there are only 6 numerically
different components present (Six-vector).
Thus we also see that the antisymmetrical tensor A^{μνσ} (3rd rank) has
only 4 components numerically different, and the antisymmetrical tensor
A^{μνστ} only one. Symmetrical tensors of ranks higher than the fourth,
do not exist in a continuum of 4 dimensions.
§ 7. Multiplication of Tensors.
_Outer multiplication of Tensors_:—We get from the components of a
tensor of rank _z_, and another of a rank _z′_, the components of a
tensor of rank (_z_ + _z′_) for which we multiply all the components of
the first with all the components of the second in pairs. For example,
we obtain the tensor Τ from the tensors A and B of different kinds:—
Τ_{μνσ} = A_{μν}B_{σ},
Τ^{αβγδ} = A^{αβ}B^{γδ},
Τ_{αβ}^{γδ} = A_{αβ}B^{γδ}.
The proof of the tensor character of Τ, follows immediately from the
expressions (8), (10) or (12), or the transformation equations (9),
(11), (13); equations (8), (10) and (12) are themselves examples of the
outer multiplication of tensors of the first rank.
_Reduction in rank of a mixed Tensor._
From every mixed tensor we can get a tensor which is two ranks lower,
when we put an index of co-variant character equal to an index of the
contravariant character and sum according to these indices (Reduction).
We get for example, out of the mixed tensor of the fourth rank
A_{αβ}^{γδ}, the mixed tensor of the second rank