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In Riemannian geometry, it has been proven [12] that the curvature tensor (((( is the only tensor that can be constructed from the metric tensor and its first and second derivatives, and is linear in the second derivatives." Einstein identified the Ricci curvature tensor Rab (( R(a(b) as the required 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. |
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