2 1 8 D O C . 2 1 9 A N E W U N I F I E D F I E L D T H E O R Y § 1. The Underlying Field Law. For variations of the field potentials h μa (and h α respectively), which vanish at the boundaries of a region, let the variation of a Hamilton integral vanish: … (1) , … (1a) [5] where the quantities h ( = |h | ), gμν, and Λα μν are defined in Eqs. (9) and (10), loc cit. Let the h field describe both the electromagnetic and the gravitational fields. A “pure gravitational field” would then be present when in addition to the validity of eq. (1), also the quantities … (2) vanish. This implies a covariant and rotationally invariant condition.1 § 2. The Field Law to First Approximation. If the manifold is the Minkowski world of special theory of relativity, then we can choose the coordinate system in such a way that h 11 = h 22 = h 33 = 1, and (= ), and all the other vanish. This value system for the is somewhat inconvenient for calculations. Therefore, for the calculations in this sec- tion, we prefer to choose the x 4 coordinate to be purely imaginary then, in fact, the Minkowski world (the lack of any field at all for a suitable choice of coordinates) can be described by … (3) The case of infinitesimally weak fields can be conveniently represented by …, (4) where the are small quantities of first order. By neglecting the quantities of se- cond and third order, one can then replace (1a), taking account of (10) and (7a), loc. cit. by ___________________________ 1. Here, there is still a certain indeterminacy in the interpretation, since one could also characterize the pure gravitational field by the vanishing of the .[6] [p. 225] d H 0 = H hg = a = h 44 i = –1 h a h a h a a = h a a k + = k x x –