Foundations of STR #1: On the alternative formulations of tensors

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?

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.

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Relevant Pages

  • Re: Proposal: How to deal with Spaceman
    ... |> I disagree that a Lorentz transformation fits the definition of a tensor ... | the trace of a proper Lorentz transform as ... | other literature as well, though admittedly never in a book specifically ...
    (sci.physics.relativity)
  • Re: Proposal: How to deal with Spaceman
    ... given k dual vectors and l ordinary ... > fix all but one of the vectors or dual vectors it is a linear map on the ... coordinate transformation and a tensor. ... a Lorentz transformation gives new coordinates (with ...
    (sci.physics.relativity)