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##x'^{u} = \Lambda ^{u}## ##_{a} x^{a} + a^{u} ## [1], where ##a^{u}## is a constant vector and ##\Lambda^{uv}## is such that it leaves the minkowski metric, ##g_{ab}## invariant.

By invariance I have:

##ds^{2}=g_{uv}dx^{u}dx^{v}=ds'^{2}=dx'^{u}dx'^{v}g_{u'v'}## [2]

from [1] ## dx'^{u} = \Lambda ^{u}## ##_a ## ## dx^{a} ##

Plugging this into [2] I have:

##ds^{2}=g_{uv}dx^{u}dx^{v}=g_{uv}\Lambda^{u}## ##_{m} dx^{m} \Lambda ^{v} ## ##_{n}dx^{n}##

and then we can simply cancel ##g_{uv}## this was my working. and I can't see where it is flawed.

I am also able to follow the lecture notes where the only difference is the choice of indicies and get a different expression as follows:

##g_{uv}dx^{u}dx^{v}=g_{ab}dx'^{a}dx'^{b}##

## =g_{ab}\Lambda^{a}## ##_{\theta}dx^{\theta} \Lambda^{b} ## ##_{\phi} dx^{\phi}##

Now I rename ##\theta=u, \phi=v## and get

##g_{uv}=g_{ab}\Lambda^{a}## ##_{u} \Lambda^{b}## ##_{v}## as in my lecture notes.

Have I broke any index notation rules in the first one? such that my metrics cancel when they should not etc? struggling to see where I've gone wrong. I know there is a rule where you shouldn't have an index appear more than twice but I thought this was only true on the same side of the equation and in the same term, i.e. a 'multiplicative' term not summing separate terms...

Many thanks in advance.