# Clarification about local Lorentz transformation

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?

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.

• thank you for your response but It seems that your definition of global Lorentz transformation and active transformation are the same in Minkowisky space time. As I am understanding Active transformation are diffeomorphism from the manifold to itself and if it preserve the flat metric it is a global Lorentz transformation Apr 4 at 20:42
• @amiltonmoreira From a manifold perspective, a passive transformation isn't a diffeomorphism from the manifold to itself - it's just a change of chart (of course, a chart transition map is a diffeomorphism from one set of coordinates to another). The points on the manifold are left alone. If a passive transformation is global, that implies it's a change from one global chart to another, but as Andrew points out such charts need not exist in general. Apr 4 at 20:59
• @ J. Murray I am aware that change of charts are not diffeomorphism of a manifold to itself although the change of charge induces a diffeomorphism on the manifold. But what I am not understanding is the difference between global transformation and active transformation Apr 4 at 21:05
• @amiltonmoreira Global means defined on the entire manifold (as opposed to local, meaning defined in some neighborhood). Active means a diffeomorphism at the level of the manifold (as opposed to passive, meaning a change of chart). You can have any combination of these (active global, active local, passive global, passive local). Apr 4 at 21:13
• @ J. Murray I understand now. So for example suppose we have a electric field, if we rotate the electric field in the whole you universe that would be an active global Lorentz transformation. If we rotate part of it that would be an active local transformation.... Apr 4 at 21:29

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.

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

• @ J. Murray what about change in our measurer instrument would it be a passive transformation? Apr 4 at 22:05
• @amiltonmoreira If you're not physically moving the system around, that would correspond to a passive transformation. Apr 4 at 22:09
• finally what about a gauge transformation would it be a local passive transformation? Apr 4 at 22:15
• @amiltonmoreira First, gauge transformations are said to be local in a sense which differs from the definition I used here; a local gauge transformation is a transformation on some field which is different for different points in space. Secondly, gauge transformations can be either active or passive as well. For example, thinking of e.g. GR as a $\mathrm{SO}(1,3)$ gauge theory, one could think of a position-dependent rotation of the tangent bundle to a manifold as an active gauge transformation, or a position-depenent change of basis as a passive gauge transformation. Apr 4 at 23:46
• I really like this systematic characterisation of active and passive transformations! Also, there is a typo in your last scenario, where "global, active" has been written instead of "global, passive". May 9 at 9:11