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+ + - = 0
,11 ,22 ,33 ,44
Here the system consists of only one differential equation for one field variable. We assume to
be expanded in a Taylor series in the neighborhood of a point P (which presupposes the analytic
character of ). The totality of its coefficients describes then the function completely. The
number of nth order coefficients (that is, the nth order derivatives of at the point P) is equal to
4
4.5 ... n + 3 ëøöø
ëø öø
abbreviated , and all these coefficients could be freely chosen if the
ìø÷ø
ìø ÷ø
1.2 ... n n
íø øø
íøøø
differential equation did not imply certain relations between them. Since the equation is of second
order, these relations are found by ( n - 2) fold differentiation of the equation. We thus obtain for
4
ëø öø
the nth order coefficients conditions. The number of nth order coefficients remaining
ìø ÷ø
n - 2
íø øø
free is therefore
4 4
ëø öø ëø öø
z = - (1)
ìø ìø
n÷ø n - 2÷ø
íø øø íø øø
""2
*
In the following the comma will always denote partial differentiation; thus, for example, = , = etc.
,i
"xi ,11 "x1"x1
78
This number is positive for any n. Hence, if the free coefficients for all orders sma ller than n
have been fixed, the conditions for the coefficients of order n can always be satisfied without
changing the coefficients already chosen.
Analogous reasoning can be applied to systems consisting of several equations. If the
number of free nth order coefficients does not become smaller than zero, we call the system of
equations absolutely compatible. We shall restrict ourselves to such systems of equations. All
systems known to me which are used in physics are of this kind.
Let us now rewrite equation (1). We have
4 4 n
ëø öø ëø öø -1 n 4
ëø öøëø z1 z2
== 1- + + ...öø
ìø ìø
n - 2÷ø ìø n÷ø n + 2 n + 3 n÷øìø n n2 ÷ø
íøøø
íø øø íø øø íø øø
where z1 = +6.
z2
If we restrict ourselves to large values of n, we may neglect the terms etc. in the
n2
parenthesis, and we obtain for (1) asymptotically
44
ëø öø ëø öø
z1 6
z = (1a)
ìø ìø
n÷ø n n÷ø n
íø øø íø øø
We call z1 the "coefficient of freedom," which in our case has the value 6. The larger this
coefficient, the weaker is the corresponding system of equations.
Second example: Maxwell's equations for empty space
is
= 0; + + = 0
,s ik,l kl,i li,k
i k
results from the antisymmetric tensor by raising the covariant indices with the help of
ik
ëø -1
öø
ìø÷ø
-1
ik
ìø÷ø
=
ìø÷ø
-1
ìø÷ø
ìø
1÷ø
íøøø
These are 4 + 4 field equations for six field variables. Among these eight equations , there
exist two identities. If the left -hand sides of the field equations are denoted by Gi and Hi kl
respectively, the identities have the form
G,ii a" 0; Hikl,m - Hklm,i + Hlmi, k - Hmik, l = 0
In this case we reason as follo ws.
79
The Taylor expansion of the six field components furnishes
4
ëø öø
6ìø
n÷ø
íø øø
coefficients of the nth order. The conditions that these nth order coefficients must satisfy are
obtained by (n - 1)fold differentiation of the eight field equations of the first order. The number of
these conditions is therefore
4
ëø öø
8ìø
n -1÷ø
íø øø
These conditions, however, are not independent of each other, since there exist among the eight
equations two identities of second order. They yield upon ( n - 2)fold differentiation
4
ëø öø
2ìø
n - 2÷ø
íø øø
algebraic identities among the conditions obtained from the field equations. The number of free
coefficients of nth order is therefore
4 îø 4 4 ùø
ëø öø ëø öø ëø öø
z = 6ìø -- 2ìø
n÷ø ïø8ìø n -1÷ø n - 2÷øúø
íø øø íø øø íø øø
ðøûø
z is positive for all n. The system of equations is thus "absolutely compatible." If we extract the
4
ëø öø
factor on the right-hand side and expand as above for large n, we obtain asymptotically
ìø ÷ø
n
íø øø
4 îøùø
n -1 n
ëø öø n
z = - 8 + 2
úø
ìø
n÷ø ïø6 n + 3 n + 2 n + 3
íø øø
ðøûø
4
ëø öø îø 36 ùø
öø öø
- 8ëø1- + 2ëø1-
ìø ÷ø ìø ÷ø
n÷ø ïø6 ìø nn
íø øø íø øøúø
íø øø ðøûø
4
ëø öø 12
îø0 + ùø
ìø
úø
n÷ø ïø n
ðø ûø
íø øø
Here, then, z1 = 12. This shows that -and to what extent- this system of equations determines the
field less strongly than in the case of the scalar wave equation ( zl = 6). The circumstance that in
both cases the constant term in the parenthesis vanishes expresses the fact that the system in
question does not leave free any function of four variables.
Third example: The gravitational equations for empty space.
We write them in the form
s
Rik = 0; gik ,l - gsk“il - gis“s = 0
lk
80
The Ri k contain only the “ and are of first order with respect to them. We treat here the g and
“ as independent field variables. The second equation shows that it is convenient to treat the “ as
quantities of the first order of differentiation, which means that in the Taylor expansion
“=“+“ xs +“ xsxt +...
s st
0 1 2
we consider “ to be of the first order, “ of the second order, and so on. Accordingly, the Ri k
s
0 1
must be considered as of second order. Between these equations, there exist the four Bianchi
identities which, as a consequence of the convention adopted, are to be considered as of third order.
In a generally covariant system of equations a new circumstance appears which is essential for a
correct enumeration of the free coeffi cients: fields that result from one another by mere coordinate
transformations should be considered only as different representations of one and the same field.
Correspondingly, only part of the
4
ëø öø
10ìø
n÷ø
íø øø
nth order coefficients of the gi k serves to characterize essenti ally different fields. Therefore, the
number of expansion coefficients that actually determine the field is reduced by a certain amount
which we must now compute.
In the transformation law for the gi k ,
"xa "xb
gik* = gab
* *
"xi "xk
gab and gik* represent in fact the same field. If this equation is differentiated times with respect
to the x*, one notices that all ( n+1) st derivatives of the four functions x with respect to the x*
4
ëø öø [ Pobierz caÅ‚ość w formacie PDF ]

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