Our professor defined a rank $(k,l)$ tensor as something that transforms like a tensor as follows:
$$T^{\mu_1' \mu_2'…\mu_k'}{}_{\nu_1'\nu_2'…\nu_l'} ~=~ \Lambda^{\mu_1'}{}_{\mu_1}…\Lambda^{\mu_k'}{}_{\mu_k}~\Lambda^{\nu_1}{}_{\nu_1'}…\Lambda^{\nu_l}{}_{\nu_l'}~T^{\mu_1\mu_2…\mu_k}{}_{\nu_1\nu_2…\nu_l}$$
Where $\Lambda$ are the Lorentz transformation matrices (translations, rotations, or boosts). I'm not sure if this is only for SR or if also for GR since we've only been talking about SR thus far, though GR is something we'll be covering soon.
He wrote on the whiteboard: if $S_{\mu\nu\rho}=S_{\nu\mu\rho}$ then $S$ is symmetric in $\mu$ and $\nu$.
But let's just talk about a rank (2,0) symmetric contravariant tensor for a second, denoted $S^{\mu\nu}$ and equals $S^{\nu\mu}$. How would we prove this is a tensor? Our book uses $R$ in place of $\Lambda$ in their formulations above, where $R$ might just be rotations. I'm sure General Tensors would have any jacobian and inverse jacobians are matrices rather than just the Lorentz transformations. This is question in Prof. Zee's, "Einstein's Gravity in a Nutshell", Chapter I.4 Exercises 2.
Also if you want to give a student like myself who is new to tensor some advice on learning tensors, and some tensor properties, and how to work with them, be my guest 🙂
Also, are all transformations homogeneous linear transformations? – These can be read about at: http://www.math.ucla.edu/~baker/149.1.02w/handouts/e_htls.pdf
http://www.lecture-notes.co.uk/susskind/special-relativity/lecture-6/rank-two-tensors/
This lecture gives a nice matrix form of what a symmetric (2,0) tensor looks like. I think this may help, thinking of these tensors as matrices themselves, visually. Basically they're symmetric matrices of the form $A^T=A$.
Also we can think of $\Lambda$ if it's a rotation matrix having the property $\Lambda^{-1}=\Lambda$.
Does this picture below do anything for the problem?
Best Answer
A tensor is not particularly a concept related to relativity (see e.g. stress tensor), but is a more general concept that describes the linear relationships between objects, independent of the choice of coordinate system.
This coordinate independence results in the transformation law you give where, $\Lambda$, is just the transformation between the coordinates that you are doing. For special relativity this is the Lorentz transform, but in classical physics in may be a simple rotation. The point is that the tensor holds regardless of the change in coordinates.
Therefore to show something is a tensor you just have to show that it obeys the transformation equation and that your transformed answer is still a valid result and can be transformed back to the original by doing the inverse transform.