# Can I rotate my coordinate system in linearized gravity?

In Linearized gravity one can perform coordinate transformations

$$x^\mu \rightarrow x'^\mu=x^\mu+\xi^\mu(x)~~~~~~\text{with the condition } \Biggl|\frac{\partial \xi^\mu(x)}{\partial x^\nu}\Biggr|\ll1$$

It seems to me that the condition prevents me from rotating my coordinate system. If I wanted to rotate around the $$x^3$$-axis the transformation for $$x^1$$ would be

$$x'^1=x^1\cos\alpha-x^2\cos\alpha=x^1+(x^1\cos\alpha-x^2\cos\alpha-x^1)=:x^1+\xi^1(x)$$

So

$$\frac{\partial\xi^1}{\partial x^1}=\cos\alpha-1$$

which is not necessarily small. This seems very weird to me, because I feel like it should not make a difference how I rotate my coordinate system without severe consequences for my theory...

• Well, your condition is satisfied for sufficiently small angles. I'm not especially knowledgeable about linearized gravity, but perhaps it is rotational invariant only to first order. This seems intuitively plausible. Commented Sep 8, 2020 at 17:12