Derive Eqs. (25.98a) for the trace-reversed metric perturbation outside a stationary (time-independent), linearized source of gravity. More specifically, do the following.

(a) First derive  h̅₀₀. In your derivation identify a dipolar term of the form 4D_jx^j/r³, and show that by placing the origin of coordinates at the center of mass, Eq. (25.96), one causes the dipole moment D_j to vanish.

(b) Next derive h̅_0j. The two terms in Eq. (25.97) should give rise to two terms. The first of these is 4P_j/r, where Pj is the source’s linear momentum. Show, using the gauge condition h̅^0μ ,μ = 0 [Eq. (25.89)], that if the momentum is nonzero, then the mass dipole term of part (a)must have a nonzero time derivative, which violates our assumption of stationarity. Therefore, the linear momentum must vanish for this source. Show that the second term gives rise to the h̅_0j of Eq. (25.98a).

(c) Finally derive h̅_ij.

Equations.

Transcribed Image Text:

huv," = 0. (25.89)