- #1
Phrak
- 4,267
- 6
Background.
We define vectors in general relativity as the differential operators
[tex]\frac{\cdot}{d\lambda}=\frac{dx^\mu}{d\lambda}\frac{\cdot}{\partial x^\mu}[/tex]
which act on infinitessimals--dual vectors,
[tex]df=\frac{\partial f}{\partial x^\mu} dx^\mu \ ,[/tex]
as linear maps to reals.
However, both vectors and dual vectors are elements of their own vector spaces which
recognize no distinction between vectors and dual vectors.
So.
Call the manifold which does recognized vectors as vectors and dual vectors as dual vectors, M. It would seem natural to look for a manifold M* where the vectors on M are dual vectors on M* and the dual vectors on M are vectors on M*.
Is this sort of dual manifold definable?
Observations.
It may be rather difficult to come up M* all at once, if at all. However finding the relationships for a single point, Mp* from a point Mp may be the best place to start.
The units of the manifold M* would be inverted. That is, (t,x,y,z) on M would become (\omega, kx, ky, kz) on M*.
We define vectors in general relativity as the differential operators
[tex]\frac{\cdot}{d\lambda}=\frac{dx^\mu}{d\lambda}\frac{\cdot}{\partial x^\mu}[/tex]
which act on infinitessimals--dual vectors,
[tex]df=\frac{\partial f}{\partial x^\mu} dx^\mu \ ,[/tex]
as linear maps to reals.
However, both vectors and dual vectors are elements of their own vector spaces which
recognize no distinction between vectors and dual vectors.
So.
Call the manifold which does recognized vectors as vectors and dual vectors as dual vectors, M. It would seem natural to look for a manifold M* where the vectors on M are dual vectors on M* and the dual vectors on M are vectors on M*.
Is this sort of dual manifold definable?
Observations.
It may be rather difficult to come up M* all at once, if at all. However finding the relationships for a single point, Mp* from a point Mp may be the best place to start.
The units of the manifold M* would be inverted. That is, (t,x,y,z) on M would become (\omega, kx, ky, kz) on M*.
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