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Lorentz invariance of the relativistic dot product

Perhaps you are familiar with the idea of the dot product and its invariance. That is, for the dot product, $a \cdot b$, it is true that $a \cdot b = a' \cdot b'$, where $a'$ and $b'$ are the vectors $a$ and $b$ after some translation or rotation operation has been applied. In other words, the dot product between two vectors doesn't change if you rotate or translate $a$ and $b$ together -- the dot product is invariant under such transformations. The reason for this invariance is because these two transformations do not alter the length of $a$ and $b$ nor the angle between the vectors $\theta$, so the value of the dot product is unaffected by such transformations. To see this, write: $ a \cdot b = ab \cos\theta = a' b' \cos\theta' = a' \cdot b' $

It turns out that there is an analog to the dot product with regards to four-vectors. That is, there is a quantity that we can calculate that is unaffected by a certain kind of transformation. We call this quantity the "relativistic dot product" and this transformation the  "Lorentz transformation." This form of invariance is often referred to as "Lorentz invariance."

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