D O C . 1 G R A V I T A T I O N A L W A V E S 1 1

If this equation is multiplied by and added to equation (2a), then the second

term on the right-hand side of (2a) cancels out. The left-hand side can be written

more comprehensively if one introduces, instead of the , the functions

. (3)

The equation then takes the form:

. (4)

However, this equation can be simplified considerably if one demands the

to satisfy not only equation (4) but also the relations

. (5)

On first sight it seems strange that the 10 equations (4) for 10 functions

should allow for 4 additional and arbitrary conditions without becoming overdeter-

mined. But the justification of this procedure is seen from the following. Equations

(2) are covariant with respect to arbitrary substitutions, i.e., they are satisfied for an

arbitrarily chosen coordinate system. Upon the introduction of a new coordinate

system, the of the new system depend upon 4 arbitrary functions which define

the transformation of the coordinates. These 4 equations can now be chosen such

that the of the new system satisfy four arbitrarily prescribed relations. We

imagine them to be such that they transform into equations (5) for the approxima-

tions we are interested in. These latter equations, therefore, represent our prescrip-

tion according to which the coordinate system has to be chosen. One obtains in

place of (4), due to (5), the simple equations

. (6)

δμν

γμν

γμν

′

γμν

1

2

--δμν∑γαα -

α

– =

∂2γμν

′

∂xα 2

α

∑-------------

∂2γμα

′

-----------------

α

∑∂xν∂xα

–

∂2γνα-

′

-----------------

α

∑∂xμ∂xα δμν∑∂xα∂xβ

∂2γαβ

′

-----------------

αβ

+ – 2κTμν =

γ′μν

∂γμα

′

∂xα

α

∑-----------

0 =

γμν′

gμν

[p. 156]

gμν

[6]

∂2γμν ′

∂xα 2

α

∑-------------

2κTμν =