Discussion Overview
The discussion revolves around the non-zero components of the stress-energy tensor, specifically why the off-diagonal components \( T^{ij} \) can be non-zero for \( i \neq j \). Participants explore the implications of the definition of the tensor and the conditions under which its components are evaluated.
Discussion Character
- Technical explanation
- Debate/contested
Main Points Raised
- One participant questions the non-zero nature of \( T^{ij} \) for \( i \neq j \), suggesting that the i-th component of momentum should be tangent to the surface of constant \( x_j \) and thus its flux should be zero.
- Another participant references the principle axis theorem, explaining that while there are basis vectors for which \( T^{ij} \) is diagonal (and thus zero for \( i \neq j \)), this does not apply generally as the basis vectors may not align with the principal axes, allowing for non-zero off-diagonal entries.
- A different participant expresses confusion regarding the use of basis vectors not aligned with the coordinate surfaces, indicating a lack of clarity in the literature on this point.
- One participant clarifies that \( T^{ij} \) represents the i-th component of the momentum flux across the surface of constant \( x_j \), emphasizing that the momentum flux can point in any direction, which supports the possibility of non-zero values.
- A later reply indicates a moment of realization and understanding from one participant after the clarification provided.
Areas of Agreement / Disagreement
Participants express differing views on the conditions under which the off-diagonal components of the stress-energy tensor can be considered non-zero, indicating that the discussion remains unresolved with multiple competing perspectives.
Contextual Notes
There are limitations in the assumptions regarding the alignment of basis vectors and the interpretation of momentum flux across surfaces, which are not fully explored in the discussion.