How Would Physics Change Without Covariant and Contravariant Tensors?

[extrads]

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

 

[PeterDonis]

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.

 

[Mentz114]

If the OP is asking whether we could express G_uv=8πT_uv 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').

 

[WannabeNewton]

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

 

[sardar]

Covariant and contravariant tensors are important concepts in tensor analysis, which is a mathematical tool used in fields such as physics and engineering. These tensors represent linear transformations between vector spaces and are used to describe physical quantities that can change with respect to different coordinate systems.

If the notions of covariant and contravariant tensors were not introduced, it would be much more difficult to accurately describe and analyze physical phenomena. Without these concepts, the Einstein field equations, which describe the relationship between the curvature of spacetime and the distribution of matter and energy, would not be able to accurately account for the effects of different coordinate systems. In fact, the equations would likely be much more complicated and difficult to solve.

The Einstein field equations, Guv=8πTuv, would also be drastically different without the use of covariant and contravariant tensors. These tensors allow for the equations to be written in a covariant form, meaning that they are independent of the choice of coordinate system. This is important in the study of general relativity, as it allows for the equations to accurately describe the effects of gravity on spacetime in any given coordinate system.

In summary, the introduction of covariant and contravariant tensors has greatly enhanced our understanding and analysis of physical phenomena, particularly in the field of general relativity. Without these concepts, it would be much more difficult to accurately describe and solve complex equations, ultimately hindering our ability to understand the fundamental laws of the universe.