Definition of energy in curved spacetime 
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*everywhere everyone (on internet) talking about energy in general relativity considers mostly energy conservation, but i have a question wich comes before: is there really a way to express energy in general relativity? How can energy conserve if there is not really a way of defining it? We have the energy-stress tensor but the conservation laws are about a pseudotensor wich terms obviusly vanish or change with the change of coordinates. How can energy be an invariant?


*the energetic term of energy-stress tensor is it really energy? How can we define it energy if we cannt define energy as says in point 1? Isn't energy something wich arises in the special/newtonian limit of general relativity?
 A: We represent energy density and momentum and flux (which source the gravitational field) using a tensorial quantity $T^{\mu\nu}$. Conservation of energy is a ill defined notion because we cannot yet calculate all the the things which contribute to $T^{\mu\nu}$ - for example the gravitons, which we believe has to do with LHS (the gravitational/curvature part), of Einstein equation can interact with itself (higher order perturbations) which will contribute to $T^{\mu\nu}$.
Hence, we cannot directly define covariant conservation of Energy momentum tensor $\nabla_\mu T^{\mu\nu} = 0$. To our rescue, we have Bianchi identities which are mathematical way of implying conservation of curvature(LHS) tensor. From this identity we can deduce covariant conservation of energy momentum tensor. However, we can always define local conservation $\partial_\mu T^{\mu\nu} = 0$ independent of such identities.
Vectors and differential forms are defined locally hence $T^{\mu\nu}$ is a local quantity.  In general, we do not have a globally conserved energy-momentum. On flat spacetime, we could add the local current vectors or energy densities to define global conservation of quantities. On the other hand, tensors (on curved spaces) which are defined in different tangent spaces cannot be added. Hence we define local covariant conservation.
