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I was reading the wikipedia page of the derivation of the Lorentz Transformations, paragraph "Galilean reference frames". Wiki

However, at the fourth line of the paragraph, I don't understand this sentence:

"Since space is assumed to be homogeneous, the transformation must be linear."

Why does the transformation have to be linear? Couldn't it be non-linear?

Qmechanic
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Dave Venturini
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1 Answers1

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a nonlinear transformation necessarily means that there is some point in space which is "special". If I translate $\bf{x}$ to $\bf{x}+\bf{x}_0$ and want to transform it, a nonlinear transformation will involve a product $\bf{x} \bf{x}_0$, which means that this translation depends on the specific point from which we measured everything.

Think of transformation of the type $x\to x+a$, which doesn't care about where is the origin (all points are pushed by a uniform degree), in contrast to a transformation of the type $x\to x^2/a$, which depends on what point we chose for $x=0$.