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The Lorentz Group is defined as the group of all transformations that leaves the 4-dim. scalar product invariant. An implication of this definition is that the absolute value of the first matrix element ${\Lambda^0}_0$ for every transformation must be equal or bigger than 1.

I assume this has to do something with time dilatation, since ${\Lambda^0}_0$ connects the old time coordinate $t$ with the new one $t'$, right? (Although it's not 100% clear to me, because there is no other constraint for ${\Lambda^1}_0, {\Lambda^2}_0$ and ${\Lambda^3}_0$ to be equal to zero - which would make my assumption trivial...)

My first question is, if this assumption is true. And if yes, is there a better explaination?

My second question is: If this is true, then there is for sure some other constraint about the other diagonal matrix elements, that follows directly from the definition of the Lorentz group, so that length contraction is ensured, right?

Qmechanic
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Xhorxho
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1 Answers1

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${\Lambda^0}_0$ and ${\Lambda^1}_1$ are both $\gamma$ ($\geq 1$) for a Lorentz boost along the x-direction. This expresses the fact that moving clocks are dilated and contracted, irrespective of whether they move away from or towards the observer.

Edit: I am only considering proper Lorentz transformations here.

Sidd
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