Dual to Tensor: Meaning & -1/6 Explained

[yukcream]

In my lecture note about tensor it states:
As element of integration over the hypersurface it is more convenient to use the 4-vector dS^i , dual to the tensor dS^ikl:

dS^i = -1/6 e^iklmdS_klm ,dS_klm = e_iklmdS^i

What is the meaning of dual?
How the -1/6 comes from?

yukyuk

 

[robphy]

You have contracted over three indices involving totally antisymmetric tensors. 3!=6. The (-1) comes from the metric signature. See https://www.physicsforums.com/showthread.php?t=90239 for a related discussion.

 

[Entropia]

The concept of duality in mathematics refers to the relationship between two objects that are related in a symmetric way. In this context, the dual of a tensor is a related object that is defined by the same set of components but with a different transformation rule. In simpler terms, the dual of a tensor is a different way of representing the same mathematical object.

In the context of the lecture notes, the dual of the tensor dS^ikl is the 4-vector dS^i, which is defined as -1/6 times the tensor dS^ikl multiplied by the Levi-Civita tensor e^iklm. This transformation is necessary in order to properly integrate over a hypersurface, as it simplifies the calculation and makes it more convenient.

The -1/6 factor comes from the specific definition of the Levi-Civita tensor, which is a mathematical object used to describe the orientation of a coordinate system in four dimensions. In this case, it represents the orientation of the hypersurface over which the integration is being performed. The value of -1/6 is a result of the specific definition of the Levi-Civita tensor and is necessary to properly relate the tensor and its dual.

Overall, the concept of duality is important in mathematics and physics as it allows for different perspectives and representations of the same underlying mathematical objects. In the case of tensors and their duals, it provides a more convenient and efficient way to perform calculations and solve problems.