Observers

Coming from general relativity.
Definition. An observer is the worldline of a massive particle together with the choice of an orthonormal basis

at each $T_{\gamma (\lambda )}M$, such that $e_0(\lambda )=v_{\gamma ,\gamma (\lambda )}$.
In other words, is the lift to the orthonormal frame bundle of a smooth curve in $M$ (with conditions...).
$◼$
Related: vielbein.
Postulate 3. A clocks carried by a specific observer $(\gamma ,e)$ will measure a "time" (this is indeed the definition of time)

between the two events:

• start the clock $\gamma ({\lambda }_0)$, and
• stop the clock $\gamma ({\lambda }_1)$.
  It is called the proper time or the eigentime.

Postulate 4. Let $(\gamma ,e)$ be an observer and $\delta$ be a massive particle worldline, that is parameterized such that $g(v_{\delta ,\delta (\lambda )},v_{\delta ,\delta (\lambda )})=1$ everywhere along $\delta$. Suppose the observer and the particle meet at some $p\in \mathcal{M}$, i.e. $\gamma ({\tau }_1)=p=\delta ({\tau }_2)$. This observer measures the 3-velocity (or spatial velocity) of this particle as

where ${ϵ}^{\alpha }$ is the ${\alpha }^{\text{th}}$ component of the so-called dual basis of $e$.

Relation to Lorentz transformations

Lorentz transformations emerge as follows. Consider two observers $(\gamma ,e)$ and $(\tilde{\gamma },\tilde{e})$ with $\gamma (0)=\widetilde{\gamma (0)}$. We have two basis $e$ and $\tilde{e}$ for the same tangent space $T_{\gamma (0)}M$, so we can consider a matrix $\mathrm{\Lambda }\in GL(4)$ for the change of basis:

But since both basis are orthonormal, $\mathrm{\Lambda }\in O(1,3)$. Lorentz transformations relate the frames of any two observers at the same point of spacetime.