cosmology - Is energy conserved in general relativity? Does $nabla_aT^{ab}_{rm matter}=0$ represent the conservation of energy and momentum?

1. 
   

   For example, the radiation dominated cosmology, the energy density of radiation is proportional to $a^{-4}$ and the volume is proportional to $a^3$, where $a$ is the scale factor. So the total energy of radiation is propotional to $a^{-1}$. So where is the loss of energy of radiation? Is it because the gravitational field has the energy?

   
   


2. 
   

   Does $\nabla_aT^{ab}_{\rm matter}=0$ represent the conservation of energy and momentum of matter field in GR?