How is the approximate gravitational wave stress energy momentum tensor not 0?

How is the approximate gravitational wave stress energy momentum tensor not 0?

In section 35.7 of MTW, an effective stress energy momentum tensor for gravitational waves is defined. TGW_mu nu frac132 pi langle barh_alpha beta, mu barhalpha beta_spacespace,nu - frac12barh_,mubarh_nu - barhalpha beta_space space ,beta barh_alpha mu, nu - barhalpha beta_space space, beta barh_alpha nu,murangle The brackets indicate an averaging over a region of space much larger than a wavelength of the wave. The text then says that on a background space time with Einstein tensor GB_mu nu, GB_mu nu 8 pi TGW_mu nu Trm other fields_mu nu But for a gravitational wave propagting through empty space, with RB_mu nu 0, wouldnt this then imply that TGW is 0 What am I missing