# Differential geometric perspectives on Lorentz invariance

Consider a general spacetime $$(M,g)$$. In a differential geometric language, Lorentz transforms (other than being a part of the isometry group of Minkowski space) can be recovered as the transformations at a point $$p \in M$$ of the (diff. geometric) frames of two observers: sections over future-directed timelike curves of the orthonormal frame bundle.

Now consider two observers whose worldlines intersect at $$p \in M$$. The frames associated with them, being orthonormal, are clearly both bases for $$T_p M$$. Thus they must be related by a $$\mathrm{GL}(4)$$ matrix $$\Lambda$$. With this information, it is easy to see we have $$\eta_{ab} = \Lambda^{m}_{a} \Lambda^{n}_{b} \eta_{mn}$$ where $$a$$,$$b$$ are indices with respect to one basis and $$m$$,$$n$$ are with respect to the other. Hence, we see $$\Lambda \in \mathrm{O}(1,3)$$ and so have recovered the Lorentz transform.

Yet, in special relativity, where the pseudo-Riemannian manifold is $$(\mathbb{R^{1,3}}, \eta)$$, why is it that Lorentz transformations can be viewed as changes of coordinates as well? And hence why is then the Lorentz invariance of say an action in field theory is not just equivalent that it is constructed of geometric objects: i.e. its covariance?

• I didn't get the question. Are you referring to the fact that $T_pM\cong_{iso}\mathbb{M}$? I.e. local ON tetrad transformation at $p\in M$ can be extended to global co-ordinate transformation if $M\cong\mathbb{M}$?
– KP99
Jan 28 at 22:26