Do I sum over these indices?

Question

Question 2.11 from A First Course In General Relativity, 3rd Edition by Bernard Schutz, asks the reader to verify the following equation:

$$ \Lambda^\nu_\beta(v) \Lambda^\beta_\alpha(-v) = \delta^\nu_\alpha$$

After reintroducing the sums and iterating over $\beta$ we have:

\begin{align} & \sum\ \Lambda^\nu_\beta(v) \Lambda^\beta_\alpha(-v) \\[8pt] = {} & \Lambda^\nu_0(v) \Lambda^0_\alpha(-v) + \Lambda^\nu_1(v) \Lambda^1_\alpha(-v) + \Lambda^\nu_2(v) \Lambda^2_\alpha(-v) + \Lambda^\nu_3(v) \Lambda^3_\alpha(-v) \end{align}

I am only somewhat familiar with the Einstein Summation Convention at this point, so I am a little confused. Am I supposed to now sum each term over $\nu$ and $\alpha$? Since $\delta^\nu_\alpha$ is the Kronecker Delta clearly there is going to be a lot of cancellation, but I am not entirely sure how to move forward, and how do I tell the next step from the notation.

For reference I am teaching my self GR from this book, so I do not have reliable access to anyone fluent in GR. For this reason I want to be sure my understanding is sound before I move on.

Answer 1

Do not sum over $\alpha$ and $\nu$. To clarify, here is the Einstein summation convention in its most accurate form:

If a covariant index is followed by the same contravariant index and vice versa, we must sum over that particular index. It is needed to make sure that after the summation, the indices of the LHS must coincide with the indices of the RHS. Any violation of this means that there was some mistake in the calculations.

In short, you must not sum over indices other than $\beta$, because they are separated by an "=" sign and they don't follow the rule mentioned above.

The equations would go like this: $$\Lambda^0_\beta\Lambda^\beta_1 = \delta^0_1$$ where you sum over $\beta$, this was one of the components. In this way, you verify all the components one by one, inserting values to $\alpha$ and $\nu$ one by one.

Answer 2

Do not sum over $\nu,\alpha$. You would only sum over them if they were repeated. The next step would be to plug in the specific formulas for each matrix element and see the cancellation explicitly. It's not that bad if you just write out the explicit matrices for each boost and do that matrix multiplication.