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."

Continue reading on didacticful

No comments:

Post a Comment