Regarding notation for Lorentz transformation

[grzz]

Difficulty regarding notation for Lorentz transformation

Please can somebody explain to me the relation between Δ$^{σ}$$_{μ}$ and Δ$_{σ}$$^{μ}$ as symbols representing a Lorentz transformation?

Thanks.

 

[Mentz114]

Those two are mutual inverses.

Have a look at post#6 in this topic https://www.physicsforums.com/showthread.php?t=705454

 

[DrGreg]

Note also the usual symbol is $\Lambda$, not $\Delta$.

For help see How to type mathematical equations here?

 

[Bill_K]

For help see How to type mathematical equations here?

Or in many cases, you can use UTF and avoid Latex entirely: Λ^μ_ν = η^μσ Λ_σ^τ η_τν

 

[grzz]

It was silly of me to use Δ instead of $\Lambda$.

Thanks everybody for the help given.

 

[vanhees71]

Or in many cases, you can use UTF and avoid Latex entirely: Λ^μ_ν = η^μσ Λ_σ^τ η_τν

No! Please use LaTeX. It's not only quicker to type but also better to read. You write for the reader and not for yourself. A bit effort to make your text as readable as possible increases to probability to be read tremendously!

 

[Bill_K]

No! Please use LaTeX. It's not only quicker to type but also better to read. You write for the reader and not for yourself. A bit effort to make your text as readable as possible increases to probability to be read tremendously!

That's why I use UTF.

 

[grzz]

So one can write

x$^{μ^{'}}$x$_{μ^{'}}$= λ$^{μ^{'}}$$_{α}$x$^{α}$ λ$_{μ^{'}}$$^{β}$ x$_{β}$ = $\delta$$^{β}_{α}$ x$^{α}$x$_{β}$ = x$^{α}$x$_{α}$

showing the invariance of x$^{α}$x$_{α}$ under the Lorentz transformation.

Hence the notation of the indices itself suggests the position of the various indices.

 

[vanhees71]

That's also a very misleading notation. Don't put the primes at the indices but on the symbol, i.e., the above equation you should write
$$x'^{\mu} x'_{\mu}={\Lambda^{\mu}}_{\alpha} {\Lambda_{\mu}}^{\beta} x^{\alpha} x_{\beta}=\delta_{\alpha}^{\beta} x^{\beta} x_{\alpha}=x^{\alpha} x_{\alpha}.$$
It can be pretty misleading to put the primes on the indices rather than the symbol, because if you rename simply dummy indices in a Einstein-convention sum it's a trivial identity:
$$x^{\alpha} x_{\alpha}=x^{\mu'} x_{\mu'}.$$
Here the dummy index on the rhs of the equation simply is called $\mu'$ and that's it.

 

[Bill_K]

No! Please use LaTeX. It's not only quicker to type but also better to read. You write for the reader and not for yourself. A bit effort to make your text as readable as possible increases to probability to be read tremendously!

What part of Λ^μ_ν = η^μσ Λ_σ^τ η_τν do you find difficult to read?

 

[dextercioby]

Not all Greek letters look that Greek: tau, nu.

 

[grzz]

...It can be pretty misleading to put the primes on the indices rather than the symbol, because if you rename simply dummy indices in a Einstein-convention sum it's a trivial identity:
$$x^{\alpha} x_{\alpha}=x^{\mu'} x_{\mu'}.$$
Here the dummy index on the rhs of the equation simply is called $\mu'$ and that's it.

Yes if one renames simply dummy indices in a Einstein-convention sum then it's a trivial identity and such use of indices IS misleading.

But denoting a Lorentz transformation by $\Lambda$$^{μ'}$$_{β}$ will show that the Lorentz transformation is between the frame of reference labelled by the primed indices and the frame of reference with the unprimed indices.

Hence if one is talking about three Lorentz transformations involving frames of reference labelled by x^μ, x^μ' and x^μ'' one can distinguish which transformation one is using.

Thanks everyone for your help. I am a starter in all this!