tensor. If Rab includes no other first order sum, the exact equation would be

Rab = X(2)ab - 4((c-2((T(m)ab + (T(m)gab, (13)

where T(m) (= gcdT(m)cd) is the trace, X(2)ab is a second order unknown tensor chosen by Einstein to be zero. However, a non-zero X(2)ab may be needed to ensure eq. (12) as an approximation of eq. (13) [7].
Now, let us examine Rab further whether the above physical requirement can be valid. Let us decompose

Rab = R(1)ab + R(2)ab , (14a)
where
R(1)ab = (c( c (ab - (( c((b(ac ( (a (bc( ( (a(b ( , (14b)

and R(2)ab consists of higher order terms. If eq. (12) provides the first order approximation, the sum of other linear terms must be of second order. To this end, let us consider eq. (12a), and obtain K = 8((c-2 and

(c( c(( a(ab) = (K((( aT(m)ab + ((b(m)( . (15a)

From (cT(m)cb = 0, it is clear that K ( cT(m)cb is of second order but K(b(m) is not. However, one may obtain a second order term by a suitable linear combination of ( c(cb and (b (. From (15a), one has

(c( c(( a(ab ( C (b() = (K ((( aT(m)ab + (( + 4C( + C()(b(m)( . (15b)

Thus, simply choosing the harmonic coordinates (i.e., (( a(ab ( (b(/2 " 0), can lead to inconsistency. It follows eq. (14b) and eq. (12b) that, for the other terms to be of second order, one must have

( ( 4C( + C( = 0, 2C + 1 = 0, and ( + ( = 1. (15c)

The solution of eq. (15c) is C = -1/2, ( = 2, and ( = -1. Thus, for the first order approximation,

(c( c (ab = (K (T(m)ab + (m) (ab( , (16)

which is equivalent to eq. (10c), has been determined to be the field equation of massive matter.
This derivation is independent of the exact form of an Einstein equation. An implicit gauge condition is that the flat metric (ab is the asymptotic limit. Eq. (16) is compatible with the equivalence principle as demonstrated by Einstein [2] in his calculation of the bending of light. Thus, the derivation is self-consistent.
One might argue that Einstein equation (3) could be ?erived" from a linear equation more general than eq. (12a), if one regards the gravitational field as a spin-2 field coupled to the energy tensor [19,33]. However, such a ?ure" theoretical approach is not really consistent with Newton? theory and related observations because the notion of gauge is used. Moreover, in such a ?roof", the existence of bounded dynamic2) solutions for eq. (3) must be invalidly assumed.
Note that Einstein obtained the same values for ( and ( by considering eq. (13) after assuming X(2)ab = 0 [34]. The present approach makes it possible to obtain from eq. (13) an equation with an additional second order term, i.e.,

Gab ( Rab ( gabR = - K(T(m)ab ( Y(1)ab(, (17)
where
KY(1)ab = X(2)ab - g ab(X(2)cd gcd(

is of second order. The conservation law (cT(m)cb = 0 and (cGcb ( 0 implies also (a Y(1)ab = 0. If Y(1)ab is identified as the gravitational energy tensor t(g)ab , eq. (10) is reaffirmed.


The anti-gravity coupling of t(g)ab that explains the dynamical failure of eq. (1), is due to Einstein? radiation formula [7]. However, Pauli [10] was the first to point out explicitly the possibility of such an antigravity coupling. Moreover, the existence of such a coupling is, in a way, implicitly suggested by the singularity theorems, which show that if all the couplings are of the same sign [21], the existence of unrealistic spacetime singularities would b inevitable. The need of an anti-gravity coupling was first discovered in calculating the gravity of an electromagnetic wave [6].
Moreover, it was Einstein and Rosen [35] who first discover that the 1915 equation may not have a propagating wave solution. In 1953 Hogarth [36] conjectured that this equation does not have a dynamic solution. A definitive indication of this is the non-existence of the plane-wave solution [8]. Note that the ?lane waves" proposed by Bondi, Pirani, and Robinson [37], are actually unbounded although they believe that a plane-wave is an idealization of a weak wave from a distant isolated source. These unbounded solutions satisfy the condition of planeness but are not related to any weak wave. Also, although Misner, Thorne & Wheeler [13] conclude correctly that the plane-waves are bounded, their equation for plane-waves actually has no bounded wave solution [8]. This illustrates that over confidence may lead to careless, and result in inconsistency.
In short, the theoretical framework of general relativity permit an additional term Y(1)ab ( 0 whose existence is required by the dynamic cases. The 1915 equation is only an over simplified special choice of Einstein. Note, however, such a choice is consistent with the equivalence principle is known only for the static case.
5. Validity of a Space-Time Metric and the Equivalence Principle
Einstein proposed that the equivalence principle is satisfied in a physical space-time1). In fact, the equivalence principle is satisfied, if and only if the space-time manifold is physically realizable, since a satisfaction of the equivalence principle requires that the geodesic represent a physical free fall. Thus, although defining a coordinate system for the purpose of calculation is only a mathematical step, choosing a space-time coordinate system requires physical considerations.
It will be shown that not all mathematical coordinate systems are equivalent in physics as claimed by Bergmann [14] and Liu [15]. For clarity, this will be illustrated with a few simple Lorentz metrics without gravity.
Example 1. To see the need of considering beyond the metric signature, consider the artificial metric,

ds2 = (2dt2 - dx2 - dy2 - dz2, (18a)

the time unit of t is second, the space unit is cm, and ( (( 2c, c = 3x1010 cm/sec). If the equivalence principle were valid, ds2 = 0 would imply the light speed to be (. Immediately, there is a contradiction, and thus the equivalence principle cannot be valid.
Nevertheless, one might argue that metric (18a) can be transformed to



ds2 = c2dt'2 - dx'2 - dy'2 - dz'2, (18b)

by the following diffeomorphism,

x' = x, y' = y, z' = z, and t' = t(/c. (19a)

Eq. (19a) implies