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I found this expression in my SR notes:

$$ (\Lambda^{-1})^{\lambda}_{\ \ \ \sigma} = g^{\lambda\mu}~\Lambda^{\rho}_{\ \ \ \mu} ~g_{\rho\sigma} = \Lambda_\sigma^{\ \ \ \lambda}$$

I know where it comes from, so I don't need a proof, but:

  1. First off, I thought that when doing matrix multiplication the dummy indices had to be next to each other, as in $\mathbf{A} \cdot \mathbf{B} = A_{ij}B_{jk} $ where $j$ is a column for $A$ and a row for $B$, while here $\mu$ is column for both $g$ and $\Lambda$....

  2. Why is this expression true? Maybe I am missing something, but what is the action of the $g$'s on $\Lambda$?

3) Is $\Lambda_\sigma^{\ \ \ \lambda}$ the transpose?

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1 Answer 1

up vote 6 down vote accepted

For doing matrix multiplication, yes, we usually put the dummy indicies together, but this is (assuming one is not working over $\mathbb{H}$) convention really.

Don't forget that this is shorthand Einstein convention.

When you write

$$ \mathbf{A} \cdot \mathbf{B} = A_{ij} B_{jk} = \sum_{j=1}^n A_{ij} B_{jk} $$

you are writing a shorthand version in Einstein convention. What this is for a single matrix element is

$$ (\mathbf{A} \cdot \mathbf{B})_{i j} = A_{ij} B_{jk} = \sum_{j=1}^n A_{ij} B_{jk} = \sum_{j=1}^n B_{jk} A_{ij} $$

assuming that your matrices are $n\times n$ matrices, since real-valued (or complex-valued) numbers commute.

However, if we always remember to obey the Einstein convention, for tensor calculus, it doesn't really matter$^{\dagger}$, since our metric in Special Relativity is symmetric.

So I could write

$$ g^{\mu \nu} = \Lambda^{\mu}_{\space \space \alpha} \Lambda^{\nu}_{\space \space \beta} g^{\alpha \beta} = \Lambda^{\mu}_{\space \space \alpha} g^{\alpha \beta} \Lambda^{\nu}_{\space \space \beta} = g^{\alpha \beta} \Lambda^{\mu}_{\space \space \alpha} \Lambda^{\nu}_{\space \space \beta} $$

say, since we know that

$$ g^{\mu \nu} = g^{\nu \mu} $$

as long as we're careful in summing over the indices as indicated, these are all the same.

Now, as for you're second question, I don't exactly know what you're asking, since you say that you understand where the expression comes from. The action of the metrix tensor is to raise and lower indices.

So for the first of your matric tensors in the second expression, $ g^{\lambda \mu} $, say, we see that it is contracted with the $\mu$ index of your Lorentz transformation tensor and it raises this index to a $\lambda$.

Since $ g^{\lambda \mu}$ is symmetric as I say, maybe you would see this more clearly if written as

$$ g^{\lambda \mu} \Lambda^{\rho}_{\space \space \mu} = g^{ \mu \lambda} \Lambda^{\rho}_{\space \space \mu} = \Lambda^{\rho \lambda} $$

Sometimes it's easier to see if the contracted indices are 'on the same side', the left in this case , and sometimes if the free index, $\lambda$ here, is on the 'same side as it will end up', again, the left in this case.

As for your final question, well, I've never thought about that before, but for $\Lambda$ a Lorentz transformation we have

$$ \Lambda \in \mbox{SO}(3,1) $$

in which case I would say yes, the inverse is just the transpose for orthogonal matrices

$$ \Lambda \Lambda^{T} = I $$

so

$$ \Lambda^{T} = \Lambda^{-1} $$

I hope that helps!

$\dagger$ I should put in a disclamer, when you're doing Supersymmetry with Weyl fermions your 'metric' is antisymmetric, so in that case it does matter, beause

$$ \epsilon^{\mu \nu} = - \epsilon^{\nu \mu} $$

and there is a distinction about summing 'North-East', $\nearrow$ and 'South-East' $\searrow$.

I realise that this doesn't apply to you here, since you're doing Special Rel., but I just didn't want you to think that this is always the case.

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1) What is $\mathbb{H}$ and why would the summation convention be different? 2) Where does the fact that $g^{\mu\nu}$ is symmetric come into play in $g^{\mu \nu} = \Lambda^{\mu}_{\space \space \alpha} \Lambda^{\nu}_{\space \space \beta} g^{\alpha \beta} = \Lambda^{\mu}_{\space \space \alpha} g^{\alpha \beta} \Lambda^{\nu}_{\space \space \beta} = g^{\alpha \beta} \Lambda^{\mu}_{\space \space \alpha} \Lambda^{\nu}_{\space \space \beta}$? Could you please write this equation or another relevant one in terms of matrices? I guess that there will be transpose matrices, but I don't know why/where –  SuperCiocia Apr 13 at 18:27
    
@harold $\mathbb{H}$ is the field of Quaternions. They anti-commute. You can ask a new question on them, or search wikipedia. –  Flint72 Apr 13 at 21:06
    
@harold The fact that $g^{\mu \nu}$ is symmetric means that North-East and South-East summation are the same, so in the second expression the summation is North-East in both $\alpha$ and $\beta$, while in the fourth expression it is South-West in both $\alpha$ and $\beta$. These things are in general not the same. –  Flint72 Apr 13 at 21:25
    
one last question: $ \Lambda^{\mu}_{\space \space \alpha} \Lambda^{\nu}_{\space \space \beta} g^{\alpha \beta} = \Lambda g \Lambda^T$ ? –  SuperCiocia Apr 15 at 19:19
    
@Flint72, Nice answer +1. But you have a small mistake in your second equation which should read: $ (\mathbf{A} \cdot \mathbf{B})_{i k} $ –  PhotonicBoom Dec 12 at 13:38

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