Lorentz transformations and rest frame

Question

So a Lorentz transformation leaves the internal product invariant in the Minkowski spacetime. Let's say I do a Lorentz transformation, either a boost or a spatial rotation and I choose the reference frame at rest with this new observer. If I want to describe the metric in these new coordinates ($x'^{mu}$) the metric is not the simple Minkowski metric anymore, right? I have that
$$eta'= J^{T} eta J$$
where $J$ is the Jacobian of the transformation (in this case is just the boost/rotation matrix $Lambda^{mu}{}_{nu}$).
How can I see that it is indeed the same spacetime?
Is there any difference between the above metric and the metric I would obtain with the same procedure but using a more general transformation that leads me to a non-inertial reference frame?
(Maybe in this case the metric is not diagonal anymore?)

Richard Myers · Accepted Answer

Let $Lambda$ be any Lorentz transformation and define $x^{prime}=Lambda x$ for some coordinates $x$. Then by definition of the Lorentz transformation, the Minkowski inner product is left invariant by this transformation:
$$
x^{prime T}eta x^prime=x^TLambda^TetaLambda x=x^Teta x.
$$
Since this must hold for any $x$, it follows that
$$
Lambda^TetaLambda=eta.
$$
In fact, this is really the definition of the Lorentz group.
In any case, observe that the Jacobian of the Lorentz transformation $x^prime=Lambda x$ is
$$
J=Lambda.
$$
using this, your question of invariance of the metric follows immediately from the definition of the Lorentz transformations.

Lorentz transformations and rest frame

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