- #1

Athenian

- 143

- 33

- Homework Statement
- Given the components of two vector fields, ##u^\alpha, v^\beta##, show that ##u^\alpha v^\alpha = u^0 v^0 + u^1 v^1 + u^2 v^2 + u^3 v^3## is not a scalar invariant under a Lorentz-transformation.

- Relevant Equations
- Refer below ##\rightarrow##

**"My" Attempted Solution**

To begin, please note that a lot - if not all - of the "solution" is largely based off of @eranreches's solution from the following website: https://physics.stackexchange.com/questions/369352/scalar-invariance-under-lorentz-transformation.

With that said, below is my attempt to apply what @eranreches's did in his solution for this problem while trying to understand how the solution works. I would sincerely appreciate it if anyone in the community could help either solidify my understanding or perhaps correct any mistakes I may have made below. Thank you very much for all your time and assistance!

Since - as @eranreches's stated - the quantity ##u^\alpha v^\beta## has a tensor of rank ##(2,0)##, it can then be expressed by a ##4\times 4## matrix as seen below:

##\begin{pmatrix}u^0v^0&u^0v^1&u^0v^2&u^0v^3\\u^1v^0&u^1v^1&u^1v^2&u^1v^3\\u^2v^0&u^2v^1&u^2v^2&u^2v^3\\u^3v^0&u^3v^1&u^3v^2&u^3v^3\end{pmatrix}##

where the diagonal elements refer specifically to ##u^\alpha v^\alpha## (i.e. ##u^0 v^0, u^1 v^1, u^2 v^2, u^3 v^3##).

In the case of this question, the quantity ##u^\alpha v^\beta## would transform like below (also based off of @eranreches's answer):

$$u^{\alpha^\prime} v^{\beta^\prime}=\frac{\partial x^{\alpha^\prime}}{\partial x^\alpha}\frac{\partial x^{\beta^\prime}}{\partial x^\beta}u^\alpha v^\beta$$

but, specifically, for the diagonal elements this time around, we would get ##\Rightarrow##

$$u^{\alpha^\prime} v^{\alpha^\prime}=\frac{\partial x^{\alpha^\prime}}{\partial x^\alpha}\frac{\partial x^{\alpha^\prime}}{\partial x^\beta}u^\alpha v^\beta$$

After writing the above equations, the author came to the conclusion that "In general, [the above equation in the question] is not invariant under the Lorentz transformations".

However, what I have a hard time understanding is how does the above equation prove that "##u^\alpha v^\alpha = u^0 v^0 + u^1 v^1 + u^2 v^2 + u^3 v^3## is not a scalar invariant under a Lorentz-transformation"?

Is it because the equation for the diagonal elements right above didn't have ##u^\beta v^\beta## as equaling to ##u^\alpha v^\alpha## - as in, kind of similar to what is written below?

$$u^{\alpha^\prime}v_{\alpha^\prime}=\frac{\partial x^{\alpha^\prime}}{\partial x^\beta}u^\beta\frac{\partial x^{\gamma}}{\partial x^{\alpha^\prime}}v_\gamma=\frac{\partial x^{\gamma}}{\partial x^\beta}u^\beta v_\gamma=\delta^{\gamma}_{\beta}u^\beta v_\gamma=u^\beta v_\beta$$

To wrap up, if anyone could correct any of my potential mistakes or help clear up any confusion I have, I would sincerely appreciate it. Thanks in advance!