the principle of covariance (as will be shown necessarily be restricted to space-time coordinate systems which are compatible with the equivalence principle.) and 3) the field equation whose source can be modified. Note that eq. (10c) is invariant with respect to the Lorentz transformations. Moreover, eq. (10c) is compatible with the notion of weak gravity. Thus, eq. (10c) as an approximation for a specified coordinate system, is compatible with the requirement of covariance and compatibility with weak gravity. It remains to show that eq. (10c) is derivable from the equivalence principle.
The equivalence principle and the principle of general relativity imply that the geodesic equation (2) is the equation of motion for a neutral particle [2,3]. In comparison with Newton�r theory, Einstein [2] obtains the gravitational potential,

( " c2g00/2. (11)

Since ( satisfies the Poisson equation (( = 4(((, according to the correspondence principle, one has the field equation, (g00/2 = 4((c-2T00, where T00 "(, the mass density and ( is the coupling constant.
Then, according to special relativity and the Lorentz invariance, one has

(c( cgab = (c( c (ab = -4((c-2((T(m)ab + ((m)(ab(, (12a)
where
( + ( = 1, (m) = (cd T(m)cd , (12b)

T(m)ab is the tensor for massive matter, (ab is the Minkowski metric, and ( and ( are constants. Eq. (12) is a field equation for the first order approximation (as assumed) for weak gravity of moving particles. An implicit gauge condition is that the flat metric (ab is the asymptotic limit at infinity. To have the exact equation, since the left hand side of eq. (12a) does not satisfy the covariance principle, one must search for a tensor whose difference from (c( c (ab/2 is of second order in (c-2.
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