I'm slightly confused by the difference between covariant and contravariant 4-vectors and how they transform under Lorentz boosts. I'm aware that [itex]x_{\mu} = (-x^0 ,x^1, x^2, x^3) = (x_0 ,x_1, x_2, x_3)[/itex], but when I do a Lorentz transform of the covariant vector, it seems to transform exactly like a contravariant vector would:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]x_{\mu} \Lambda^{\mu}_{\nu} = \pmatrix{\gamma & -\gamma\beta & 0 & 0\\-\gamma\beta & \gamma & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1} \pmatrix{x_0\\x_1\\x_2\\x_3} = \pmatrix{\gamma(x_0 -\beta x_1)\\\gamma(x_1 -\beta x_0)\\x_2\\x_3}[/tex]

But I've heard people say that they transform differently; so am I doing this wrong?

I was also hoping someone would explain how differential operators behave/transform as well (does [itex]\frac{\partial}{\partial x^\mu} = \partial_\mu[/itex] transform like a covariant vector? What would [itex]\frac{\partial}{\partial x_\mu}[/itex] mean?)

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# Transforming co- and contravariant 4-vectors

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