\section{Comparison with existing results}
 
  Before we begin the actual comparison it is useful to point out that because $h^{\mu\nu}$ is a rank 3 degenerate metric, its spatial sector $h^{ij}$ is nondegenerate. Now defining $h_{ij}$ and $h^{ij}$ from (\ref{degeratematric1}) and (\ref{degeratematric}), respectively, as,
 \begin{eqnarray}
 h_{ij}&=& \Lambda_{i}{}^a \Lambda_{j}{}^a\nonumber\\
 h^{ij}&=& \Sigma_{a}{}^i \Sigma_{a}{}^j
 \label{metric}
 \end{eqnarray}
 it is easy to verify that one is the inverse of the other,
 \begin{eqnarray}
 h^{ij}h_{jk} = \delta^i_k
 \end{eqnarray}
 
 In flat space $h^{ij}$ and $h_{ij}$ reduce to the Kroneckar deltas. Let us now look at the form of the action valid for time independent transformations, given in the celebrated paper,
 \begin{equation}
S = 
\int
\Big(\dfrac{m}{2\Theta}\Big)\dfrac{ h_{ij}\dfrac{dx^{i}}{d\lambda}\dfrac{dx^{j}}{d\lambda}}{\bigg(\dfrac{ dx^{0}}{d\lambda}\bigg)} ~d\lambda
\label{actionm4}
\end{equation}

Unfortunately this expression is not space-time covariant. It is only space covariant. Not unsurprisingly, it does not have the Newton-Cartan element as a coupling. Incidentally, the paper gives separate actions for time independent and time dependent transformations, so that a unified picture is again lost. However before we come to that aspect, let us remind that in our case no distinction is made between time independent or time dependent transformations and there is a single form for the action.

To see the invariance of the action (\ref{actionm1}) under general transformations, we explicitly present the variation of the individual terms. The denominator is invariant due to (\ref{vanish}). Likewise, the variation of the numerator can be calculated from (\ref{gt}), \ref{degeratematric1}) and (\ref{master}). It is,
\begin{equation}
\delta \Bigg( h_{\nu\rho}\dfrac{dx^{\nu}}{d\lambda}\dfrac{dx^{\rho}}{d\lambda}\Bigg)= -2\Theta\frac{dt}{d\lambda}\frac{d}{d\lambda}(x_\mu u^\mu)
\label{actionm5}
\end{equation}

Combining everything the final answer turns out to be,
\begin{equation}
\delta S = -m\int d\lambda \frac{d}{d\lambda}(x_\mu u^\mu)
\label{final}
\end{equation}
so that the action is quasi-invariant- a result that was deduced earlier in (\ref{final1}) from general considerations also. For time independent transformations the boosts do not appear and hence the invariance is exact.

Let us now make contact with (\ref{actionm4}).		
 If the set of Galilean transformations do not include the boosts and if further during localisation the parameters are made only 
  function of position, then the vielbeins $\Lambda_0^a$ transform as
 \begin{eqnarray}
 \delta \Lambda^{a}_{0} = w^{a}_{b} \Lambda^{b}_{0}
\end{eqnarray}
obtained from (\ref{lambdatrans}) with the subsidiary conditions $u^a = 0$ and $\xi^k$ independent of time.
We see that in these circumstances,
\begin{equation}
 \Lambda^{a}_{0} = 0
 \end{equation}
is a valid choice, though not the most general one.  The corresponding action that is obtained from (\ref{actionm1}) is just (\ref{actionm4}). Obviously complete space-time covariance is lost since the temporal part of the vielbein has been dropped. 


 
 \section{Conclusion}
 
 Galilean gauge theory was mooted by us in \cite{BMM1, BMM2} to develop a complete algorithm for cou[ling a non relativistic matter theory
  with gravity which was (and still is) infested with various riddles. After that in many examples the efficacy of the GGT has been established. Applications of GGT to galilean invariant models revealed Newton Cartan structures, the geometry of which was discovered long ago by Cartan to formulate Newtonian gravity as a geometric theory. A nice account of which is given in \cite{MTW} which clearly shows that curvature appears only when gravitation is viewed as a space time process, So spatial diffeomorhism will appear as spacetime phenomena, where the diffeomorphism is restricted to be spatial. Another
  point is the special role of time. In relativity the metric is instrumental in the foliation of space time. On the contrary, in NC geometry timelike vectors determine the foliation. The assumptions of GGT automatically ensure that dynamics is described in such foliation.
  
  GGT is inspired by Utiyama's approach of gauging the symmetry of a dynamical model. A differen6t approach is to gauge the symmetry algebra. This may be important in other contexts but for the present problem it offers no clues. WE have at several places pointed out this limitation. The model chosen in this paper reveals the limitations of the gauging of algebra approach, as we have shown here. We also presented the results from GGT, clearing the confusions that still persists in the literature.
 


































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