# Variations of tensors are tensors?

Recently I posted a question about variation of metric. I thought I understood it and talked with my friend about it. After that he said he's not convinced because he can't prove variation of metric are tensor as well as orginal one. I considered a lot but I can't get an answer, so plz let me ask a question.

He says: Where $$D^{\alpha}_{\beta} = ∂x^{\alpha}/∂x'_{\beta}$$, $$\delta g_{\mu \nu} = \delta (g_{\mu \nu})$$ and under a Lorenz transformation $$\delta g’_{\mu \nu} = \delta(D^{\rho}_{\mu} D^{\sigma}_{\nu}g_{\rho \sigma}) \neq D^{\rho}_{\mu} D^{\sigma}_{\nu} \delta g_{\rho \sigma}$$ in general so $$\delta g_{\mu \nu}$$ is not a tensor anymore. And he suggests some examples:

For flat Minkowski metric, $$\delta g = g(x+\delta x) - g(x) = 0$$ but for radial coordinates $$\delta g \neq 0$$ so it's not a tensor (indices are omitted here).

However, I think:

1. We can't define sum of fields that have different properties under the transformation so the varation should be tensor as well. e.g. scalar + vector = ?

2. We should treat $$\delta g_{\mu \nu}$$ as $$(\delta g)_{\mu \nu}$$ (i.e. ($$\mu, \nu$$)-component of tensor $$\delta g$$), in other words it's tensor if this condition $$(\delta g)'_{\mu \nu} = D^{\rho}_{\mu} D^{\sigma}_{\nu} (\delta g)_{\rho \sigma}$$ is satisfied. However, I can't give him explanation about this point anymore.

So my question is: $$\delta g$$ is a tensor again? If so/not, please tell me what is wrong in our discussion.

Update:

• I think the problem is whether $$\delta g$$ is equal to $$g(x+\delta x) - g(x)$$ or just a arbitrary tensor .

• I think function is also functional when $$x$$ of $$g(x)$$ is a particular point. In other words, $$g_{\delta}[g] \equiv g_{\mu\nu}(x_0)= \int_{M} \delta(x-x_0) g_{\mu\nu}(x)$$ so we can take arbitrary 2-rank tensor as its variation, but if we think $$x$$ as a function of proper time $$\tau$$ on world-line, then we can think $$g$$ is a functional of $$x$$: $$g_{\mu \nu}[x]$$ so $$\delta x$$ resurges.

• Corrected some mistakes in my question.