Clarification about local Lorentz transformation

Question

Note there are others questions about local Lorentz transformations and global Lorentz transformations but they all concerned about mathematics. Here what I am trying to understand is the link between mathematics and experiment.

The difference between Global and local Lorentz transformation is not well explained in literature. But for me local Lorentz transformations are rotation of your measurer instrument at a point (passive transformation) while global Lorentz are transformation of the objects in spacetime (active transformation).

In special relativity we confuse these two things because in special relativity we can make measure at distant point by parallel transport our bases so if we rotate our basis (measurer instrument) it is equivalent to (active) rotation in the opposite direction.

In general relativity since we can not parallel transport our basis to point uniquely so measure at distance does not make sense so we only compare measure at a point. As for global Lorentz transformation there is not because generally the Lorentz transformation are not symmetry of the metric.

Is my Idea correct?

Answer 1

Active transformation: the vectors and other geometric quantities change.

Passive transformation: the vectors (with the exception of basis vectors) and other geometric quantities do not change, but the basis does (e.g. a coordinate basis), so the components of a vector change even though the vector itself does not.

Local Lorentz transformation: the coordinates in the vicinity of some event in spacetime are changing, and we make no comment on the coordinates far from that event. This concept is always a well-defined idea in any Lorentzian manifold.

Global Lorentz transformation: the coordinates throughout spacetime change by the same Lorentz transformation applied everywhere. This is not always a well-defined idea in a curved space.

In your question you appear to have a muddle between global and active. They are different ideas.

Answer 2

Here are some concrete examples which may shed some additional light on the issue. Let's consider the manifold $\mathcal M = \mathbb R \times \mathrm S^1$, i.e. the cylinder. Points $p\in \mathcal M$ can be labeled by a triple $(z,a,b)$, where $z\in \mathbb R$ and $a^2+b^2=1$. It's critical to note that $(z,a,b)$ are not the coordinates of $p$ in some chart, since $\mathcal M$ is a 2-dimensional manifold.

---

A local, active transformation is a diffeomorphism $\phi : U \rightarrow U$, where $U\subset \mathcal M$ is some neighborhood. In our case, let's choose $$U := \big\{ (z,a,b) \in \mathbb R \times \mathrm S^1 \ \bigg| \ a > 0\}$$ $$\phi : U \ni (z,a,b) \mapsto (z+1,a,b)$$ In words, $\phi$ just pushes the points of the manifold along the cylindrical axis by one unit.

---

A local, passive transformation is a change of chart. Let $x$ be a chart map defined on the subset $U$ as before, defined by

$$x:(z,a,b) \mapsto \big(z, \tan^{-1}(b/a)\big)\in \mathbb R^2$$ Now let $y$ be a different chart map, defined by $$y : (z,a,b) \mapsto (z^3, \tan^{-1}(b/a)\big)\in \mathbb R^2$$

The chart transition map $(y \circ x^{-1}):(z,\theta) \mapsto (z^3 ,\theta)$ maps from the $x$ coordinates to the $y$ coordinates. However, this is not a map from $U\rightarrow U$; the points $p\in U$ aren't actually going anywhere, we're just choosing different labels for them.

---

A global, active transformation is a diffeomorphism $\Phi: \mathcal M\rightarrow \mathcal M$. As an example, we could let $$\Phi :\mathcal M \ni (z,a,b) \mapsto (z,-a,-b)$$

---

A global, passive transformation is a change of chart where each chart covers the entire manifold. One such chart is the following: $$ x : \mathcal M \rightarrow \mathbb R^2-\{(0,0)\}$$ $$ (z,a,b) \mapsto (e^za, e^z b)$$ Another example would be $$ y: \mathcal M \rightarrow \mathbb R^2-\{(0,0)\}$$ $$(z,a,b) \mapsto (-e^z b, e^z a)$$ The chart transition map is $(y \circ x): (\alpha,\beta) \mapsto (-\beta,\alpha)$, which corresponds to a $90^\circ$ rotation. Once again, note that this is not actually moving points in $\mathcal M$ around; it's just changing the labels.

---

Finally, a Lorentz transformation is one which preserves the Minkowski metric. An active Lorentz transformation is a (global or local) diffeomorphism which is also an isometry of the Minkowski metric; a passive Lorentz transformation is a (global or local) change of chart which preserves the form of the Minkowski metric, i.e.

$$\frac{\partial x^i}{\partial y^a} \frac{\partial x^j}{\partial y^b} \eta_{ij} = \eta_{ab}$$