Covariance in Equations: How to Identify and Understand It

[tiger_striped_cat]

How can you tell if an equation is covariant just by looking at it. Please try and keep explaniation to text more than equations.

 

[dextercioby]

How can you tell if an equation is covariant just by looking at it. Please try and keep explaniation to text more than equations.

There are basically two criteria:
1.The equation must contain only quantities with the same type of greek/spacetime indices summed over.Wrt to indices,the equation must be 'balanced',that is the tensor rank of the RHS must be equal to the tensor rank of the LHS.
2.The sides of the equation must have the same 'tensor quality' (i made it up).You cannot have an equality between a tensor (e.g.in the LHS) and a nontensor (in the LHS).

Daniel.

PS.The equation
$$A^{i}=F^{\mu i}B_{\mu}$$
is not covariant.

 

[tiger_striped_cat]

Thank you for your great explaniation. But could you explain the example:

$$A^{i}=F^{\mu i}B_{\mu}$$


I think I'm having problems due to my lack of understanding with tensors and covariance, no fault of your explaniation.

 

[dextercioby]

It's basically the "F" 'animal'.The way it's given,it's not a tensor because:
a) one index takes 4 values and the other only 3.
b) both indices should behave the same at a general coordinate transformation,but the trouble is that one index transforms with the normal matrix (4*4),while the other with another one,which has only (3*3) components.

Daniel.