The notion of parallelism between a vector and a covector comes up naturally in the following context. Say you have a scalar field T that measures the temperature of the CMB. Then [itex]\nabla_a T[/itex] is a cosmologically preferred covector field. I would think of it as being "parallel" to the velocity vector field [itex]v^a[/itex] of the Hubble flow, but since vectors and covectors live in different vector space, there is no natural notion of parallelism between them.(adsbygoogle = window.adsbygoogle || []).push({});

One thing I could do would be to raise or lower an index, so I could say that [itex]v^a[/itex] was parallel to [itex]\nabla^a T[/itex]. Or I could say that [itex]v^a[/itex] maximizes [itex]v^a \nabla_a T[/itex] subject to the constraint [itex]v^av_a=1[/itex]. (I'm using a +--- metric.)

Are these equivalent?

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# Parallelism between a vector and a covector

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