How to define the components of the Poincare group?

Question

I know that the Poincare group/inhomogeneous Lorentz group can be defined as:
$$
x^mu = (t,-x) 
t rightarrow t^prime = gamma x + delta t + b^0 
x rightarrow x^prime = alpha x + beta t + b^1 
x^{mu prime} = R(b) L x
$$
and that it has to be invariant under Minkowski Metric
$$
ds^2 = c^2dt^2 - dx^2 = {ds^2}^prime = c^2{dt^2}^prime - {dx^2}^prime
$$
Usually I could use $$ x^prime = 0  x= vt$$ and use this in the metric and get the components of the Lorentz transformation, but I am not sure that using translation works too.
Can please someone try to explain to me how to get the Lorentz transformation assuming translation?

J. Murray · Answer

Lorentz transformations relate two reference frames whose origins coincide.  If there is a spacetime translation involved, then those two frames are not related via Lorentz transformation.

Starting from an inhomogeneous Lorentz transformation $x'^mu = Lambda^mu_{  nu} x^nu + b^mu$, consider an infinitesimal displacement $dx^mu$.  In the transformed coordinates, the inhomogeneous term falls away and you are left with $dx'^mu = Lambda^mu_{  nu} dx^nu$, since
$$x'^mu-y'^mu = (Lambda^mu_{  nu} x^nu + b^mu) - (Lambda^mu_{  nu}y^nu + b^mu) = Lambda^mu_{  nu}(x^mu-y^mu)$$
From there, you can proceed as usual to derive the form of $Lambda$ from physical considerations.

How to define the components of the Poincare group?

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