Discussion Overview
The discussion centers on the concepts of tensor rank and type, particularly in the context of general relativity (GR). Participants explore the definitions and implications of these terms, how they relate to tensor fields, and the potential for variation in rank across different points in a field.
Discussion Character
- Technical explanation
- Debate/contested
Main Points Raised
- Some participants propose that tensors are classified by type (n, m) indicating covariant and contravariant indices, while rank relates to the number of "simple" terms needed to express a tensor.
- Others argue that when referring to the metric tensor as a rank 2 tensor, it is actually a tensor field of type (0,2), suggesting a distinction in terminology.
- One participant states that rank can be defined as the sum of covariant and contravariant indices (n+m), asserting that tensors in a field maintain the same type throughout.
- A later reply questions the consistency of the term "rank" as used in different contexts, referencing a specific paper that may define rank differently.
- Another participant reflects on the analogy with matrix rank, suggesting that the metric tensor must always be rank 4 if defined in terms of having a non-zero determinant, while also speculating that the rank of a tensor field could vary at different points.
Areas of Agreement / Disagreement
Participants express differing views on the definitions and implications of tensor rank and type, indicating that there is no consensus on these concepts. The discussion remains unresolved regarding the relationship between rank and type, and whether rank can change across a tensor field.
Contextual Notes
There are limitations in the definitions of rank and type as they are used in various contexts, and the discussion highlights the potential for confusion arising from different terminologies in the literature.