# Additions to the stress-energy tensor that leave equations of motion the same

In short:

For a stress-energy tensor $$T^{\mu\nu}$$, what are possible additions that will leave the tensor equations of motion $$\nabla_\nu T^{\mu\nu} = 0$$ unchanged?

Context:

Any modification, $$T^{\mu\nu} \rightarrow T^{\mu\nu} + A^{\mu\nu}$$ where $$\nabla_\nu A^{\mu\nu} = 0$$ as an identity, would work.

In various publications and text books, it appears that people say one can add something that looks like

$$A^{\mu\nu} = \nabla_\lambda B^{\mu\nu\lambda}$$, where $$B^{\mu\nu\lambda}=-B^{\mu\lambda\nu}.$$

See for instance, here for a discussion in the context of curved space-time (beginning of Section 2).

However, I calculate (using the antisymmetry of $$B$$ and Eq. 3.68 from Carroll's Lecture Notes on GR)

$$\nabla_\nu A^{\mu\nu} = \nabla_\nu \nabla_\lambda B^{\mu\nu\lambda} = \frac{1}{2}R^\mu_{\lambda\nu\alpha}B^{\lambda\nu\alpha}$$ ($$R$$ is the curvature tensor)

so it wouldn't effect the equations of motion in flat space-time, but in curved space-time, it does affect them.

1. Am I missing something here? Some identity that makes the expression above zero even in curved space-time? The derivations only a few lines, and I don't believe I made a mistake.

1. If this is not zero in curved space-time, is there a general form that describes the family of tensors one can add to the stress-energy tensor without changing the equations of motion?

PS: This is not a question about how to derive the stress-energy tensor in GR (varying the Lagrangian with respect to the metric), or its uniqueness. In GR, it seems to me the stress-energy tensor is uniquely defined by varying Lagrangian with respect to the metric. Any non-zero addition would add to the curvature of space-time, which implicitly changes the equations of motion by changing the covariant derivative. This is just a question about portions of the stress-energy tensor that contribute to the curvature, but not the equations of motion explicitly; I could rephrase as: "Given a stress-energy tensor, what portions of the tensor do not explicitly contribute to the tensor equations of motion for matter?"

Applying the covariant derivative $$\nabla_\nu$$ to $$A^{\mu \nu}$$ you have
$$\nabla_\nu \nabla_\lambda B^{\mu \nu \lambda}.$$
You then have a contraction between two tensors in the $$\nu, \lambda$$ indices. The first tensor $$\nabla_\nu \nabla_\lambda$$ is obviously symmetric in these two indices since covariant derivatives commute. Whereas $$B$$ is anti-symmetric in these two indices. The contraction of a symmetric and anti-symmetric tensor vanishes, hence $$\nabla_\nu A^{\mu\nu}=0$$.