APPENDIX
D
T H E T R A N S F O R M A T I O N OF T H E EINSTEIN TENSOR (a) The Transformation of the Second Rank Curvature Tensorf W e begin with the transformation of the covariant second rank curvature tensor and show that in. the expression for this tensor, it is possible to separate off terms containing the d'Alembert operator applied to that component of the metric tensor which has the same suffixes. By definition we have ^ v = ^ « , 3 v
(D.01)
where, by (44.08), the fourth rank curvature tensor is
(D.02) (this equation differs from (44.08) onlv bv the naming of the indices). Therefore
(D.03) For brevity we put
r = a r°
(D.04;
where in agreement with the definition (41.15) (D.05) In (D.03) the first term is already in the form of the d'Alembert operator applied to g^. The remaining terms may be so transformed that the second derivatives of components of the metric tensor only appear through the first derivatives of the r . To perform this transformation we need the relations (D.06)
(D.07) and (D.08) t See also the papers by de Donder [16] and by Lanczos [17]. 422
Appendix
423
D
which were derived in Section 41. From these relations it follows that (D.09) Differentiating (D.06) with respect to x , we get v
(D.10) We note that owing to the symmetry of (D.10) in p and v we have (D.ll) Differentiating (D.09) with respect to x^ we get (D.12) «—
r
\~
P"
In the last term on the right we may replace dg^/dx^ by (D.13) where T , is the usual Christoffel symbol of the first kind (38.30). Equation (D.12) then appears as va p
(D.14) We interchange the indices u. and v and also the summation indices a and on the left. Then we can write
(D.15) Thus the sum of the expressions (D.10), (D.14) and (D.15), half of which enters (D.03), is equal to
(D.16) (as a result of ( D . l l ) the remaining terms cancel). W e introduce the notation (D.17) v
n'
The quantities. are formed m the same way as the symmetrized covariant derivatives of a vector, although I \ is not a vector. As a result of (D.16) the expression (D.03) for R then becomes
(D.18)
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The Theory of Space Time and Gravitation
To simplify the calculations to be performed on the terms involving first derivatives we shall use not only the notation r and for the usual Christoffel symbols but also the notation &
and for the corresponding quantities with raised indices.
W e note that jy*® is
given by (D.21) W e consider the expression (D.22) whose last term coincides with the last term in (D.18). W e write this expression in the form (D.23) W e insert (D.24) and then use the equation (D.25) Then we get (D.26) Using the notation of (D.19) and relabelling, we may put (D.27) Since the coefficient is symmetric in a and (3 we can replace the factor multiplying it by its symmetric part, which is equal to ^ty^Ox^. For the same reason we can replace dg^ldx^ by (D.13). Doing this we get (D.28) But, as is easy to verify, (D.29) Therefore (D.30) Equating the expressions (D.22) and (D.30) for the
we obtain the relation
(D.31)
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425
D
which enables us to write the expression for i?
(AV
in the following final form
(D.32) From this it is easy to obtain an equation for the contravariant components of the second order curvature tensor. By differentiating relations of the form (D.07) it is simple to derive the equation
(D.33) Using this we can write R^ as
(D.34) If we then raise the indices p, and v we get the following simple expression for
(D.35) The quantity P* is obtained from
by raising the indices according +o
v
r^
=
^vo
(D.36)
r p o
and may be expressed directly in terms of the T defined by (D.08) as follows: a
(D.37) (b) Transformation of the Invariant W e now form the invariant of the curvature tensor : R = 9 ^
v
(D.38)
W e have (D.39) where T denotes the quantity (D.40) Using the relation
(D.41) we obtain from (D.37) (D.42) and by virtue of (41.16) (D.43)
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The Theory' of Space Time and Gravitation
Differentiating equation (D.41) with respect to x„ we obtain, in analogy with (D.10) (D.44) Inserting (D.44) into (D.39) we get (D.45) W e see that the second derivatives of the metric tensor enter the expression for R only through the second derivatives of log (—g) and through I \ The terms involving first derivatives can be transformed with the aid of the relation (D.46) which is easily deducible from the equation (D.47) We now get the result (D.48) lhis expression can be written in the form (D.49) where
(D.50)
Remembering the equation (D.51) which gives the d'Alembertian of any function y, we may write R = • log ^(g)
 T  L
(D.52)
Of course the quantity y  log \ (—g) is not a scalar, but formally the operator of (D.51) may be applied to it. We note that both the first and the second term in (D.52) represent a sum of derivatives of some quantities with respect to coordinates, divided by V(~9)This fact is of importance in the formulation of a variational principle for Einstein's equations, when the quantity L defined by (D.50) plays the part of the Lagrangian (Section 60). Apart from (D.50) there are several other ways of writing the Lagrangian L. We quote the following f
(D.53) and also (D.54) the last form is most commonly encountered in the literature.
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427
D
(c) Transformation of the Einstein Tensor The foregoing equations enable us to write down an expression for the divergencefree Einstein tensor flfcv =
 \g^R
(D.55)
We shall see that the second derivatives of the metric tensor enter G^ only through second derivatives of the quantity \ / ( — g) g^ and through first derivatives of T . It is, therefore, convenient to introduce a special symbol for \ / ( ~ 9) times the contravariant components of the metric tensor. W e put v
tr = y/(g).gr
(DM)
Then equation (41.16) may be written as (D.57) For the following it is convenient to transform all equations in such a way that they involve only derivatives of the g^ . In performing the transformation we shall encounter derivatives of the quantity v
y = lo
g A
/(<7)
(D.58)
which we shall write as follows : (D.59) According to (41.07) we have For the quantities obtained from the y by raising indices we introduce the notation a
just as for a tensor (although of course the
are not a vector). W e have also
y* = T f
(D.62)
where T£ has the meaning defined by (D.19). W e shall denote the second derivatives of y by y : 0
a p
(D.63) According to (D.21) the P**" are bilinear functions of the components and of their first derivatives. Inserting the expression for g^ in terms of into that relation, we obtain pn,
=
+
A
where
(D.65)
and
(D.66)
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We shall denote the corresponding quantities with lower indices by ITj^ and W e evaluate the determinant of the g**" : Det 9^ =
{\/(9)V^9^
=9 
=9
2
(D.67)
9 Thus the determinant of the Q* is equal to the determinant of the g v
Detg^ = D e t ^ = ^
: (D.68)
v
From equation (D.66) we obtain
a
= y*
0
=
g^>
it follows that
g^ll**
(D.69)
=
a p
(D.70) + tp
(D.71)
ftp"  J
(D.72)
The last expression is also equal to ^, r* +y»=
We go on to the transformation of the Einstein tensor guv
=
^
R
_
I ^ V J J
p
5
5
^
Our starting point is from the equations __^a3
=
+ ix
^
a
3
(D.35)
and also equation (D.49) for R, which we write in the form * = The second derivative of g
fAV

r
"y«  r  z
(D.73)
is
(D.74) and therefore
(D.75) On the other hand we have
*"(r«y + r + L)  r* + r*. v
a
(D.76)
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Appendix D
By comparing the last two equations we see that apart from the terms in T v and the same combinations of second derivatives occur in both. The calculation gives
r»*
(D.77) As always, the transformation of the terms involving first derivatives is the most complicated. W e have
iv, «3r; = n*. «&n^ + A * . n ^ + A*. «*n«fe + A * . A ^ a 0
(D.78)
A 0
3
Using (D.71) we get
A^n*
+ A .«en*3 = v
3
y (w>»* + Q
 j ^ r * + r») y
y
v (D.79)
and as a result of (D.80) we have
(D.81) Further (D.82) Hence
,D.83) These relations allow us to rewrite (D.77) as
(DM) Here we put (D.85) and and we express the g ® in terms of the g in the term involving the d'Alembertian. Then the expression for the Einstein tensor becomes 0
a 0
(D.87)
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The Theory of Space Time and Gravitation
Since, according to (D.68), the determinant # can be expressed directly in terms of the g^ we can take it that in (D.87) all terms except L are expressed in terms of the g and their derivatives. It remains for us to express the Lagrangian L also in terms of the same quantities. By the definition (D.50) we have aP
(D.88) Here we insert F^ —
+ AJ
0
(D.89)
where, by (D.66) (D.90) Using the relation (D.91) which is easy to deduce from (D.90), we get (D.92) Here we express dcf^/dx^ in terms of (D.93) *nd use equation (D.71), which may be written in the form (D.94) This gives, finally (D.95) Now only the g
aP
remain under the sign of differentiation.