The Galilean group is a double isometry. It leaves invariant the following two quadratic forms:
$$dt^2, \quad \left(\frac{∂}{∂x}\right)^2 + \left(\frac{∂}{∂y}\right)^2 + \left(\frac{∂}{∂z}\right)^2,$$
as well as
$$dx \frac{∂}{∂x} + dy \frac{∂}{∂y} + dz \frac{∂}{∂z} + dt \frac{∂}{∂t}.\tag{1}\label{1}$$
The first two of these correspond, respectively, to a covariant metric and an independent contravariant metric, while the third is an invariant generic to differentiable manifolds. The two metrics are no longer inverses of one another, though their contraction is still a multiple of the identity - namely: the zero multiple. They have zero contraction.
When the Galilean group is centrally extended to the Bargmann group, the resulting group has the following as its invariants:
$$dx^2 + dy^2 + dz^2 + 2 dt du, \quad dt,$$
and
$$\left(\frac{∂}{∂x}\right)^2 + \left(\frac{∂}{∂y}\right)^2 + \left(\frac{∂}{∂z}\right)^2 + 2\frac{∂}{∂t}\frac{∂}{∂u}, \quad \frac{∂}{∂u},$$
with ($\ref{1}$) extended to and replaced by
$$dx \frac{∂}{∂x} + dy \frac{∂}{∂y} + dz \frac{∂}{∂z} + dt \frac{∂}{∂t} + du \frac{∂}{∂u}.\tag{2}\label{2}$$
(Edit: For strict equivalence, the linear invariants $dt$ and $∂/∂u$ should be made quadratic $(dt)^2$ and $(∂/∂u)^2$, otherwise there will be disagreement on which characterizations allow for time reversal $t → -t$, or reversal in the other coordinates. The same with the other linear invariants, below.)
By comparison, the Poincaré group has the following as its invariants
$$dt^2 - \frac{1}{c^2}\left(dx^2 + dy^2 + dz^2\right), \quad \left(\frac{∂}{∂x}\right)^2 + \left(\frac{∂}{∂y}\right)^2 + \left(\frac{∂}{∂z}\right)^2 - \frac{1}{c^2}\left(\frac{∂}{∂t}\right)^2,$$
as well as ($\ref{1}$). The contraction of the two metrics is the $-1/c^2$ multiple of the identity, so they are conventionally treated as inverses of one another.
Corresponding to the Bargmann group is the (trivial) central extension of the Poincaré group, which has - as its invariants:
$$
dx^2 + dy^2 + dz^2 + 2 dt du + \frac{1}{c^2}du^2, \quad dt + \frac{1}{c^2}du,\\
\left(\frac{∂}{∂x}\right)^2 + \left(\frac{∂}{∂y}\right)^2 + \left(\frac{∂}{∂z}\right)^2 + 2\frac{∂}{∂t}\frac{∂}{∂u} - \frac{1}{c^2}\left(\frac{∂}{∂t}\right)^2, \quad \frac{∂}{∂u},$$
and ($\ref{2}$).
Of note: the Poincaré group is not the non-relativistic limit of the Bargmann group. It is the extended Poincaré group, which is. At the same time, the Galilean group is not what yields the familiar representations for bodies with mass. Instead, it is the Bargmann group, which does.
The Lie brackets for all of the above-mentioned groups can be succinctly written in vector form as:
$$\begin{align}
[Λ,Λ']
&= \left(𝞈×𝞈' - α𝞄×𝞄'\right)·𝐉 + \left(𝞈×𝞄' + 𝞄×𝞈'\right)·𝐊 + \left(𝞈×𝝴' + 𝝴×𝞈'\right)·𝐏\\
&+ \left(𝞄·𝝴'-𝝴·𝞄'\right)M + \left(τ𝞄' - 𝞄τ'\right)·𝐏
\end{align}$$
where
$$
Λ = 𝞈·𝐉 + 𝞄·𝐊 + 𝝴·𝐏 - τH + ψμ,\\
Λ' = 𝞈'·𝐉 + 𝞄'·𝐊 + 𝝴'·𝐏 - τ'H + ψ'μ,\\
M = μ + αH.
$$
Using $‹⋯›$ to denote the group generated by $⋯$, the following cases apply:
- The Galilean group is $‹𝐉,𝐊,𝐏,H›$, with $μ = 0$ and $α = 0$.
- The Bargmann group is $‹𝐉,𝐊,𝐏,H,μ›$, with $α = 0$.
- The Poincaré group is $‹𝐉,𝐊,𝐏,E›$, with $E = Mc^2$ and $α = 1/c^2$.
- The extended Poincaré group is $‹𝐉,𝐊,𝐏,H,μ›$, with $α = 1/c^2$.
The coordinate representations of the groups are given by the following infinitesimal transforms:
$$Δ𝐫 = 𝞈×𝐫 - 𝞄t + 𝝴,\quad Δt = -α𝞄·𝐫 + τ,\quad Δu = 𝞄·𝐫 + ψ,$$
where
$𝐫 = (x,y,z)$, when written in terms of Cartesian coordinates. Using
$Δ(d⋯) = d(Δ⋯)$, the corresponding transforms on the coordinate differentials can be derived from this, and are given by:
$$Δ(d𝐫) = 𝞈×d𝐫 - 𝞄dt,\quad Δ(dt) = -α𝞄·d𝐫,\quad Δ(du) = 𝞄·d𝐫.$$
From this, using the invariant (
$\ref{2}$), the transforms on the differential operators are given by:
$$Δ(∇) = 𝞈×∇ + 𝞄\left(α\frac{∂}{∂t} - \frac{∂}{∂u}\right),\quad Δ\left(\frac{∂}{∂t}\right) = 𝞄·∇,\quad Δ\left(\frac{∂}{∂u}\right) = 0.$$
The infinitesimal rotations and boosts are, respectively, $𝞈$ and $𝞄$; the infinitesimal spatial and time translations are, respectively, $𝝴$ and $τ$. The infinitesimal translation $ψ$ for the extra coordinate $u$ is what appears in the extension of the Galilei group to the Bargmann group.
For the Galilean group, $u$ and $ψ$ drop out of the picture, and $∂/∂u = 0$.
For the extended Poincaré group, the "absolute time" of the Galilei and Bargmann groups lives on as $s = t + αu$, whose transforms are:
$$Δs = σ = τ + αψ, \quad Δ(ds) = 0.$$
Under conversion from $(t,u)$ to $(s,t)$, the differential operators convert to:
$$\left(\frac{∂}{∂t}\right)_s = \left(\frac{∂}{∂t}\right)_u - \frac{1}{α}\left(\frac{∂}{∂u}\right)_t,\quad \left(\frac{∂}{∂s}\right)_t = \frac{1}{α}\left(\frac{∂}{∂u}\right)_t.$$
Their transforms for the coordinate differentials convert to:
$$Δ(∇) = 𝞈×∇ + α𝞄\left(\frac{∂}{∂t}\right)_s,\quad Δ\left(\frac{∂}{∂t}\right)_s = 𝞄·∇,\quad Δ\left(\frac{∂}{∂s}\right)_t = 0.$$
The invariants expressed in terms of $(s,t)$ convert to:
$$
dx^2 + dy^2 + dz^2 - \frac{1}{α}dt^2 + \frac{1}{α}ds^2, \quad ds,\\
\left(\frac{∂}{∂x}\right)^2 + \left(\frac{∂}{∂y}\right)^2 + \left(\frac{∂}{∂z}\right)^2 - α \left(\frac{∂}{∂t}\right)_s^2 + α\ \left(\frac{∂}{∂s}\right)_t^2, \quad \left(\frac{∂}{∂s}\right)_t,
$$
and
$$dx \frac{∂}{∂x} + dy \frac{∂}{∂y} + dz \frac{∂}{∂z} + ds \left(\frac{∂}{∂s}\right)_t + dt \left(\frac{∂}{∂t}\right)_s.\tag{3}\label{3}$$
The reduction from extended Poincaré to Poincaré corresponds to eliminating $s$ by reducing it to the proper time, i.e. by the constraint:
$$0 = dx^2 + dy^2 + dz^2 + 2 dt du + α du^2 = dx^2 + dy^2 + dz^2 - \frac{dt^2}{α} + \frac{ds^2}{α},$$
i.e.
$$ds^2 = dt^2 - α\left(dx^2 + dy^2 + dz^2\right),$$
and eliminating $∂/∂s$ from the picture.
The conversion between $(s,t)$ and $(t,u)$ is the geometric underpinning to the Foldy-Wouthuysen transform. The extended geometry with all five coordinates $(x,y,z,t,u)$, when $α = 0$, is the Bargmann geometry. When $α ≠ 0$, it has no standard name; but the curved versions of it appear in some papers on 5D cosmology on ArXiv, with the true nature of the geometries that are being dealt with in those papers, unbeknownst to the authors. If I find links, I'll add them in a later edit or comment.
