Hi everybody! The last chapter of my course named "Advanced Mechanics and Special Relativity" treats of Lorentz transformations, but the script of my teacher does not explain much about the notation used and it's getting quite confusing for me without understanding it fully.(adsbygoogle = window.adsbygoogle || []).push({});

So far we've considered an inertial frame ##\sum '## moving "away" from inertial frame ##\sum## at velocity ##v## in the ##x##-direction. We derived the Lorentz transformation for that change of inertial frame:

##\Lambda = \begin{pmatrix} \cosh \eta & - \sinh \eta & 0 & 0 \\ - \sinh \eta & \cosh \eta & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}##

so that ##(ct', x', y', z') = \Lambda (ct, x, y, z)## with ##\tanh \eta = \frac{v}{c}##. Our Minkowski metric is defined as

##\eta = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}##.

Then without any explanation he suddenly introduces that ##\Lambda^T \eta \Lambda = \eta## can equivalently be written as ##\Lambda_{\rho}^{\mu} \eta_{\mu \nu} \Lambda_{\kappa}^{\nu} = \eta_{\rho \kappa}##! From this point he uses only this notation without describing what it means...

So I've done some research to try to make sense of it, but it seems rather complicated. Here are some assumptions I have made with the help of Google and some definitions given in my script:

1. ##x^{\mu}## is known as the contravariant 4-vector and can be defined as ##x^{\mu} = (x^0, x^1, x^2, x^3) = (ct, x, y, z)## (at least in the case of our example above). So ##\mu## seems to refer to the components of the vector. Is that right? I would also assume without any certainty at all that the index up means the coordinates are not given using the Minkowski metric, because:

2. ##x_{\mu} = \eta_{\mu \nu} x^{\nu}## is known as the covariant 4-vector and when doing the matrix multiplication I get ##x_{\mu} = (-ct, x, y, z)##. That seems to be the spacetime coordinates using the Minkowski metric, since it is the multiplication of ##\eta## with ##x^{\nu}##. Is that assumption correct?

3. I've noticed that notations such as ##\eta_{\mu \nu} x^{\nu}## refer to the Einstein summation convention, and in fact mean ##\sum_{\nu = 0}^{3} \eta_{\mu \nu} x^{\nu}##. But I actually don't see the difference with a regular matrix multiplication. I read that a few times already though, does that mean it is different to matrix multiplication but I will realise that only later down the road? But still, what confuses me the most are the indexes. ##{\mu}## seems to refer to the columns of the matrix ##\eta##, and ##{\nu}## to the rows. Why would we even write them then if we are not dealing with components? Or do we write them just to show that the index is down, so that we are not in the Minkowski metric (if my first assumption about those indexes was right in the first place...)?

4. I get the same problem about the notation for the matrix ##\Lambda_{\nu}^{\mu}##. They seem to refer to the components of the matrix (##\mu## would be the rows and ##\nu## the columns?), but I don't understand again why we would write them and what does it mean then when say I have two matrices ##\Lambda_{\nu}^{\mu}## and ##\Lambda_{\kappa}^{\rho}##...

Those are my first questions about the notation in SR... The script of my teacher directly derives relations and goes to infinitesimal Lorentz transformations after that (using this notation of course). Apart from what I wrote above, internet gave me very complicated answers about the matter. The special relativity represents only 3 chapters out of over 60 in that course, so I'm not gonna start (yet) reading a book about SR just to understand that notation when I have an exam coming up in 3 weeks...

Thank you very much in advance for your help, and sorry my message was so long... I think some examples illustrating the uses of this notation would be very helpful for me to understand it.

Julien.

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# I Some clarifications needed about SR notation

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