I have quite a few questions resulting from my recent foray into the
mathematical and conceptual foundations of STR. I realise there is a
Google group devoted to this subject but, to be frank, there are only
one or two minds on that forum whose answers I have come to respect,
and it is not always easy to get their attention, so I'm seeking
'kindred minds' here. I hope it's the right place.

In STR a tensor is generally introduced in one of two ways: either as
an indexed family of components that transform in a certain way under
Lorentz transformations, or as a multilinear form from several copies
of the underlying vector space and its dual into the reals.

I am confused about the presumed equivalence of these two definitions.
In particular, the 'multilinear algebra' approach (my personal
preference) makes no mention whatsoever of a Lorentz transformation.
Wouldn't an equivalent 'coordinatised' definition therefore need to
assert the behaviour of tensor components under *any* linear
transformation whatsoever?

My confusion deepens when I think about the logical precedence of the
Lorentz transformation in either scheme, for isn't such a
transformation defined in terms of its behaviour with respect to the
underlying metric tensor? And doesn't this mean that we are then
defining the concept of a tensor in terms of an object which is itself
defined in terms of a tensor?

In STR a tensor is generally introduced in one of two ways: either as
an indexed family of components that transform in a certain way under
Lorentz transformations, or as a multilinear form from several copies
of the underlying vector space and its dual into the reals.

On May 15, 10:51 pm, "Vonny N." <von... at (no spam) gmail.com> wrote:

In STR a tensor is generally introduced in one of two ways: either as
an indexed family of components that transform in a certain way under
Lorentz transformations, or as a multilinear form from several copies
of the underlying vector space and its dual into the reals.

Yes, the latter is the proper way to define a tensor, which depends
only on the differential structure of the manifold and from which the
transformation properties arise naturally, without need for
mentioning the Lorentz transformations. Lorentz transformations arise
with more structure, as you said, when one inputs a metric on the
underlying manifold. Then one has the concept of an isometry under the
action of (a certain representation of) the symmetry group, in this
case, the Lorentz. Finding the duals with respect to this metric
determines the representation of the group of isometries on the dual
vectors. The confusion sometimes arise due to the triviality of the
Minkowski spacetime, this is one case where I believe a bird's eye
view makes things clearer. Generalize and you shall see.

I am confused about the presumed equivalence of these two definitions.
In particular, the 'multilinear algebra' approach (my personal
preference) makes no mention whatsoever of a Lorentz transformation.
Wouldn't an equivalent 'coordinatised' definition therefore need to
assert the behaviour of tensor components under *any* linear
transformation whatsoever?

The concept of tensor is indeed independent of a metric, as you see from

My confusion deepens when I think about the logical precedence of the
Lorentz transformation in either scheme, for isn't such a
transformation defined in terms of its behaviour with respect to the
underlying metric tensor? And doesn't this mean that we are then
defining the concept of a tensor in terms of an object which is itself
defined in terms of a tensor?

I really want to master this subject deeply, but as you can see, I
have become confused on merely opening the front door.

Vonny N.