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One traditional representation of the stress tensor among relativists is a rank-2 fully contravariant tensor, associating a contravariant force per unit area $t^i$ to a unit normal $n_j$ defined on a surface created by the coordinate system, leading the unit normal to transform covariantly:

$$t^i = \sigma^{ij} n_j$$

This is consistent with the definition used by most physicists in the field of electromagnetics and relativity, and is used in many books such as A. J. M. Spencer's Continuum Mechanics.

However, continuum mechanicists often define the stress tensor as associating a contravariant force per unit area with a unit normal defined on a coordinate-independent surface, causing this normal vector to be contravariant and leading to the expression:

$$t^i = \sigma^i_j n^j$$

The stress tensor is defined this way in many textbooks, such as Frankel's Geometry of Physics.

These definitions of the stress tensor should represent the same thing, but are numerically different in general coordinate systems by a metric tensor factor.

Is there a straightforward way to describe the physical effect of that metric tensor factor on the Cauchy stress tensor, and how does one spot when someone is using the contravariant values or the mixed values?

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