# Definition of energy in curved spacetime

1. 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?

2. 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?

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.

• So energy and momentum are just tensor terms, more precisily they are linked to a combination of derivatives of terms of the fundamental tensor. The energy and momentum terms are locally conserved quantities, the conserve also globally right?
– Anto
Commented Aug 8, 2021 at 8:44
• Yes energy and momentum (along with flux) make up all the components of $T^{\mu\nu}$ - to see details, just refer wikipedia. Yes it is conserved globally only because the curvature is 'conserved' due to Bianchi identities. Commented Aug 8, 2021 at 8:52
• Isn't curvature something point related? I don't get how "Yes it is conserved globally only because the curvature is 'conserved' due to Bianchi identities."
– Anto
Commented Aug 8, 2021 at 9:01
• Curvature is only realized when you parallel transport a vector on the spacetime. For example look at pictures and text from Sean Carroll's notes Of course, the vectors and differential forms are defined only locally. I should have quoted the word 'globally'. All we require is a covariant conservation. And that is what I meant to say. Commented Aug 8, 2021 at 9:21