Why do we say that the position 4-vector, [tex]x^{\mu}[/tex], is naturally contravariant and that the del operator, [tex]\partial_{\mu}[/tex], is naturally covariant?(adsbygoogle = window.adsbygoogle || []).push({});

The only thing I could come up with is that the contravariant del components [tex]\partial^{\mu} = (-c^{-1}\partial_t,\nabla)[/tex] have a negative sign in front of the c. Is this 'unnatural'?

The relationship between time, [tex]t[/tex], and proper time, [tex]\tau[/tex], is simply [tex]t = \gamma\tau[/tex]. Which makes [tex]t[/tex] a function of velocity. Now the velocity 4-vector:

[tex]\frac{dx^{\mu}}{d\tau}[/tex]

is definately a 4-vector right? Im sure it is. But is

[tex]\frac{dx^{\mu}}{dt}[/tex]

a 4-vector? Could the fact that [itex]t[/itex] is not a scalar have an influence on this question?

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# Some questions on tensors

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