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a)So I'm reading over my notes and they say that under the Lorentz transformation L, [itex]\phi \rightarrow \phi'[/itex] where [itex]\phi'(x)=\phi(x')[/itex] where [itex]x'^\mu = (L^{-1})^\mu{}_\nu x^\nu[/itex]

I don't really understand why this is true.

Why is it not just [itex]\phi'(x)= L \phi(x)[/itex]

Clearly this fails because the LHS is a scalar and the RHS should have indices on the L and so it won't be a balanced tensor equation but to my mind this is the right "form" that the equation should have.

Can anyone explain this to me?

b)However, earlier in my notes when Lorentz transformations were introduced it says

A Lorentz transformation is a linear transformation on space and time

[itex]x^\mu \rightarrow x'^\mu = L^\mu{}_\nu[/itex]

which preserves the relativistic line element [itex]ds^2[/itex]

and so

[itex]L^\mu{}_\sigma L^\nu{}_\tau g^{\sigma \tau} = g^{\mu \nu}[/itex] (*)

Why does it define the transformation as [itex]x^\mu \rightarrow x'^\mu = L^\mu{}_\nu[/itex] here and as [itex]x'^\mu = (L^{-1})^\mu{}_\nu x^\nu[/itex] in the other part of my notes (as I mentioned in part a) of my post) - are these equivalent? If so, how?

And how do we derive the equation (*)

Thanks a lot.

I don't really understand why this is true.

Why is it not just [itex]\phi'(x)= L \phi(x)[/itex]

Clearly this fails because the LHS is a scalar and the RHS should have indices on the L and so it won't be a balanced tensor equation but to my mind this is the right "form" that the equation should have.

Can anyone explain this to me?

b)However, earlier in my notes when Lorentz transformations were introduced it says

A Lorentz transformation is a linear transformation on space and time

[itex]x^\mu \rightarrow x'^\mu = L^\mu{}_\nu[/itex]

which preserves the relativistic line element [itex]ds^2[/itex]

and so

[itex]L^\mu{}_\sigma L^\nu{}_\tau g^{\sigma \tau} = g^{\mu \nu}[/itex] (*)

Why does it define the transformation as [itex]x^\mu \rightarrow x'^\mu = L^\mu{}_\nu[/itex] here and as [itex]x'^\mu = (L^{-1})^\mu{}_\nu x^\nu[/itex] in the other part of my notes (as I mentioned in part a) of my post) - are these equivalent? If so, how?

And how do we derive the equation (*)

Thanks a lot.

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