(adsbygoogle = window.adsbygoogle || []).push({}); What sort of "vector" is this?

A contravariant transforms as [tex]A^\rho \rightarrow {\Lambda^\rho}_\sigma A^\sigma[/tex].

A covariant vector [tex]A_\mu[/tex] can be built from [tex]A^\rho[/tex] by [tex]A_\mu=\eta_{\mu\rho}A^\rho[/tex]. Then it transforms according to

[tex]A_\mu\rightarrow\eta_{\mu\rho}{\Lambda^\rho}_\sigma A^\sigma=\Lambda_{\mu\sigma}A^\sigma={\Lambda_\mu}^\sigma A_\sigma[/tex].

(Indeed, [tex]{\Lambda_\mu}^\sigma {\Lambda^\rho}_\sigma=\delta^\rho_\mu[/tex] can be considered the definition of a Lorentz transformation, that is, one which leaves [tex]A^\mu A_\mu[/tex] invariant.)

But are there (covariant?) vectors whose components transform according to

[tex]B_\sigma \rightarrow {\Lambda^\rho}_\sigma B_\rho[/tex]

?

What sort of a "vector" is B? It is not dual to a contravariant vector in the ordinary sense. And (why I put this in relativity forum and not a math forum) what does this type of vector signify within relativity theory? Are there any physical quantities which transform like this?

Edit:

Ok. I think I get it. B is a covariant vector and the transformation being questioned represents transforming it by the inverse of the given Lorentz transform.

But please correct if I am wrong. Thanks!

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# What sort of vector is this?

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