# What determine whether the dynamical equations are tensor equations or vector equations?

Newton's 2nd law which is central to Newtonian dynamics, is a vector equation

$\sum\textbf{F}_{external}=m\textbf{a}$

Same with Maxwell's equations in the covariant form.

On the other hand, general relativity is governed by the tensor equation

$\displaystyle R_{\mu\nu}-\frac{1}{2}R~g_{\mu\nu} =\frac{8\pi G}{c^4} T_{\mu\nu}$

My questions are:

1. Is there a deep reason why some dynamical equations are tensor (of the 2nd rank) equations and others are vector equations?
2. Is Schrodinger equation a scalar equation?
3. Are there dynamical equations in physics which are tensor equations of rank higher than 2?
4. Is there an upper bound on the highest rank a physics tensor equation can have? from some physical argument may be?
• A tensor is mathematically a vector, since the set of all such-and-such tensors forms a vector space. A vector is also a tensor, since it is a linear map from the dual space onto the field over which the vector space is defined. So there doesn't seem to be that much of a difference. They are all vector equations. They are also all tensor equations. I'm not posting this as an answer since it would be a pretty smart-alek. What I mean to point out is that you seem to be more interested in asking why the vector spaces are different, or alternatively why the ranks of the tensors are different – Mark Eichenlaub Oct 29 '11 at 0:58
• @MarkEichenlaub Yes that is the heart of my question, why some dynamical equations are of different ranks, and if there is an upper bound on the rank number – Revo Oct 29 '11 at 1:15
• @Mark Eichenlaub When discussing tensors vector is a rank 1 tensor. It is better not mix this with "vector spaces" which better name is "linear space": those have less structure. – Misha Oct 29 '11 at 4:45
• @Misha which axiom fails? en.wikipedia.org/wiki/Vector_space#Definition – Mark Eichenlaub Oct 29 '11 at 5:05
• @Mark Eichenlaub I said "better not", not "forbidden to". Of course tensors are elements of linear space. By saying this you throw away their transformation properties, which make them tensors. Better read more about tensors. – Misha Oct 29 '11 at 5:31

## 1 Answer

1. The number of variables in a field state. This question is like "why electron mass is less than proton mass". Just because there is a choice.
2. When discussing symmetry, it is better to discuss not Schroedinger equation but Klein-Gordon or Dirac equations which are scalar and spinor (1/2 rank tensor) equations. Schroedinger equation is a nonrelativistic limit of a second with spin neglected. It might give you a better picture than just a statement that it is scalar. Because technically it is not invariant.
3. In solid state spin 5/2 is a normal thing. Half filled d-shells of some metals behave as they are spin 5/2 particles "under normal conditions" (until you are interested in a energy range where the system starts to feel its composite nature). Probably, there are situations where you have higher momenta which are better to treat effectively as a whole, without going into details of their formation.
4. So far, no elementary particles with spin >2 are known. Consequently, equations for those particles have rank less than 2. There is no upper bound.