What is a fourth rank tensor
summed up according to this index (“rejuvenation”). One wins
so z. B. from the mixed tensor of fourth rank A the
mixed tensor of the second order
and from this, again by rejuvenation, the tensor
zeroth rank A = A = A.
Proof that the result of rejuvenation is effective
Lich has a tensor character, results either from the
Tensor representation according to the generalization of (12) in
Connection with (6) or from the generalization of (13).
Inner and mixed multiplication of tensors. These
consist in the combination of the outer multiplication with
Examples. - From the second-order covariant tensor
A. and the contravariant first-order tensor B form
we get the mixed tensor by external multiplication
By tapering according to the indices , the co-
variant quad vector
We also refer to this as the inner product of the tensors
A. and B. Analogously, A is formed from the tensors and
B. by external multiplication and tapering twice
the inner product AB.. Through external product formation
and one-time taper is obtained from A and Bthe
mixed tensor of second rank D = AB.. One can
aptly refer to this operation as a mixed one; because
it is an external one with respect to the indices and , an inner one
regarding the indices and.
We now prove a theorem that proves the
Tensor character is often usable. After the just presented
is AB. a scalar if Aand B Tensors
are. But we also claim the following. If AB.For
any choice of tensor B is an invariant, then A has Tensor
Proof. - It is according to requirement for any
But after reversing (9),
This, inserted into the above equation, yields:
This can be done with any choice of B' only then met
be when the bracket disappears, from which with reverse
Looking at (11) the assertion follows.
This theorem applies to arbitrary tensors
Rank and character; the proof must always be carried out analogously.
The proposition can also be proved in the form: are
B. and C arbitrary vectors, and with every choice the-
same the inner product
a scalar, so is A. a covariant tensor. This latter
Theorem still applies even if only the more specific statement
it is true that with any choice of the four-vector B the
is a scalar if one also knows that A the sym-
operating condition A. = A enough. Because on the earlier
given paths one proves the tensor character of
, from which then because of the symmetry property
the tensor character of A itself follows. This sentence too
can be easily generalized to the case of covariant and
contravariant tensors of any rank.
Finally follows from what has been proven that
dear tensors generalizable proposition: if the sizes
A.B. with any choice of the four-vector B a tensor
form first rank, then A a second rank tensor.
Is C an arbitrary four-vector, so is because of the
Tensor character AB. the inner product AC.B. at
any choice of the two four-vectors C and B a
Scalar, from which the claim follows.
§ 8. Some things about the fundamental tensor of g.
The covariant fundamental tensor. In the invariant
Expression of the square of the line element
plays d x the role of a freely selectable contravariant
Vector. Since also = g, it follows after the considerations
of the last paragraph from this that g a covariant tensor
second rank is. We call it the “fundamental tensor”.
In the following we derive some properties of this tensor
ab, which are inherent in every tensor of the second rank; but
the special role of the fundamental tensor in our theory,
which their in the peculiarity of the gravitational effects
has a physical reason, it means that the
winding relations only for the fundamental tensor for
matter to us.
The contravariant fundamental tensor. Is one formed in that
Determinant scheme of g to each g the sub-determi-
nante and divides this by the determinant g = the
G, one obtains certain quantities g(= g) of which we
want to prove that they form a contravariant tensor.
According to a well-known set of determinants is
where the sign Means 1 or 0, as the case may be =
or is. Instead of the above expression for d s2 can
or according to (16) also
write. But now form according to the multiplication rules
of the previous paragraph the sizes
a covariant four-vector, namely (because of the will-
free choice of d x) any freely selectable
Four-vector. By introducing it into our expression
Since this with any choice of the vector d a scalar
is and g according to its definition in the indices and sym-
is metric, it follows from the results of the previous para-
graph that g is a contravariant tensor. From (16)
still follows that too is a tensor we mixed the
Can call fundamental tensor.
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