# What are all objects that are invariant under Lorentz transformations?

I am interested in the following question: What are all objects that are invariant under Lorentz transformations? And, once a list is provided, how to justify that these are indeed ALL such objects?

Here is a paper in which it is shown that "up to multiplication by a scalar, the Minkowski metric tensor is the only second-order tensor that is Lorentz-invariant". However, my question does not concern only second-order tensors, but all objects.

Here is a thread that gets very close to what I am interested in. It is claimed that the only Lorentz-invariant objects are the metric, the inverse metric, the Kronecker delta, and the Levi-Civita tensor. So we have the list. But what is the justification that this list is complete? One answer in this thread refers to a book by Spivak, where one can find a theorem concerning scalar invariants (M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. V, theorem 35 on page 327 in third edition). Now, I have two questions:

(1a) According to the answer in the mentioned thread, to conclude from this theorem anything about tensors one needs to observe that "an invariant tensor becomes a scalar invariant when contracted with enough vectors, and any such scalar invariant arises from an invariant tensor". But I do not see why the second part of this claim holds, i.e., why any scalar invariant needs to arise from an invariant tensor. What about e.g. mass? Does it arise from any tensor? And how to justify the claim in general?

(1b) Is there any way of justifying that the four mentioned tensors are the only Lorentz invariants without referring to the mentioned theorem in Spivak's book?

(2) Is it true that only tensors are candidates for invariant objects, i.e., it makes no sense to ask whether some non-tensorial object is Lorentz-invariant? If yes, why is it so? If no, can one extend the reasoning that works for tensors to encompass non-tensorial objects as well?

Concerning terminology, here it is claimed that only scalars can be called Lorentz invariants in the proper sense, but tensors other than scalars can have components that do not change under Lorentz transformations. Of course, when asking about all Lorentz invariants, I mean Lorentz invariance in a broader sense, encompassing also cases of non-scalars whose components don't change under Lorentz transformations.

• Is "object" a mathematical term? boots are objects. Nov 24, 2020 at 12:00
• Possibly interesting: mathshistory.st-andrews.ac.uk/Extras/Geometric_object Nov 24, 2020 at 13:51
• "Objects" in the sense of mathematical objects. I use this term to stress the fact that what I'm interested in are not only tensors. There are non-tensorial objects that also have determinate transformation rules, e.g. Christoffel symbols in GR. Nov 24, 2020 at 19:32