How Would Physics Change Without Covariant and Contravariant Tensors?

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Discussion Overview

The discussion revolves around the implications of not having covariant and contravariant tensors in physics, particularly in the context of the Einstein field equations (EFE). Participants explore theoretical consequences, alternative formulations, and the fundamental nature of tensors in physics.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants question the feasibility of expressing the Einstein equations without tensors, suggesting that the tensorial nature is essential.
  • Others argue that while covariant and contravariant tensors are distinct, one could still maintain the concept of tensoriality without explicitly using them.
  • A participant proposes that the notion of contravariant and covariant tensors is inherent and can be represented in different notations, including index-free notation.
  • One participant introduces the idea that the extension of tensors could lead to the concept of spinors, indicating a broader framework for understanding these mathematical objects.

Areas of Agreement / Disagreement

Participants express differing views on whether the Einstein equations can be formulated without tensors, with some asserting that tensoriality is indispensable while others suggest alternative representations are possible. The discussion remains unresolved regarding the implications of omitting these concepts.

Contextual Notes

There are limitations in the assumptions made about the necessity of tensors and the definitions of tensoriality. The discussion does not resolve how these concepts interact with the broader framework of physics.

extrads
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If the notions of covariant and contravariant tensors were not introduced,what would happen?E.g. what form will the Einstein E.q. Guv=8πTuv be changed into ?
 
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extrads said:
If the notions of covariant and contravariant tensors were not introduced,what would happen?

I'm not sure what you mean by this. Covariant and contravariant tensors represent distinct kinds of physical things; if you know the metric, you can compute correspondences between them, but they are still distinct concepts. So if you're going to use tensors at all, you need both kinds.
 
If the OP is asking whether we could express Guv=8πTuv without tensors, I would have to say that it can be done, but the central property of coordinate independence would still be there ( ie 'tensoriality').
 
The notion of contravariant and covariant is always there. It is made explicit in index notation but you can just as well write it in index-free notation as ##G = 8\pi T## but you cannot get rid of the tensorial nature of the classical EFEs. The extension of the concept of a tensor is a spinor: http://en.wikipedia.org/wiki/Spinor
 

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