Proper Lorentz transformations from group theory?

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PeroK
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The requirement that the set of Lorentz transformations forms a group comes from the basic properties that we expect of spacetime and inertial frames of reference.

The transformations must be linear, otherwise the transformation would depend on a particular reference point for ##(t, x, y, z) = (0,0,0,0)##.

There's a derivation from homogeneity and isotropy here:

http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf
 
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vanhees71
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The group structure comes from the demand that transformations between different inertial reference frames should be invertible. The transformation from a frame to itself, i.e., doing in fact nothing, for sure is a symmetry and doing two transformations is also just a transformation from one inertial reference frame to another one.

The transformations should be linear because it should map any uniform motion of any point particle in one frame to such a motion in any other.

To get Minkowski space and the Poincare transformations you have to assume in addition also that any inertial observer describes his space as Euclidean.
 

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