Koobee Wublee wrote:
On Feb 20, 6:29 am, Tom Roberts wrote:
[to Koobee] You just do not understand tensors, and the distinction between
tensors and their components.

You are still practicing matheMagic. After creating a matrix
where all of its elements are observer dependent, the matrix cannot
become a tensor just by waving your matheMagic wand over it a few
times.

OK, I'll try once more.

Let's drop down a level and discuss vectors, not rank-2 tensors, and use
3-space, not spacetime. This permits easy visualization of the issues.

Consider a vector of unit length pointing east.

In coordinates with x headed east, y headed north, and z headed up, it
has components (1,0,0). Those three real numbers are NOT the vector, but
they are related to it, and with proper additional structure they can be
used to compute the vector. That additional structure is the BASIS
VECTORS of the coordinate system on which the vector was projected to
obtain those components. In this case the basis vector for x is a unit
vector pointed east, and similarly for y and z.

In coordinates with x' headed north, y' headed west, and z' headed up,
that SAME VECTOR has components (0,-1,0). Those three real numbers are
NOT the vector, but they are related to it, and with proper additional
structure can be used to generate the vector. That additional structure
is the BASIS VECTORS of the coordinate system on which the vector was
projected to obtain those components. In this case the basis vector for
y' is a unit vector headed west, and similarly for x' and z'.

It should be clear that by computing the vector sum of the components of
the vector times the corresponding basis vectors, one obtains the vector
itself:
v = v_i e^i (here the {v_i} are the components of vector v, and
the {e^i} are the corresponding basis vectors.
The {v_i} are each a real number; the {e^i} are
each a vector, and the sum over i is a vector sum.)
The VECTOR is independent of the coordinate system, but the COMPONENTS
are not.


Applied to a rank 2 tensor, the "matrix" you keep referring to is a
matrix of the tensor's COMPONENTS. To obtain the tensor itself, they
must be composed with the basis vectors of the coordinate system onto
which the tensor was projected to obtain the components:
T = T_ij e^i (X) e^j (here (X) indicates a tensor product.
The {T_ij} are a matrix of real numbers
corresponding to the COMPONENTS of the
tensor T; they are not T itself. The
sum over i and j is a rank-2 tensor sum.)
The result is the same: the TENSOR is independent of the coordinate
system, but the COMPONENTS (the "matrix" you mention) are not.


There is no "matheMagic" here, just very basic tensor math. Stuff you
repeatedly refuse to learn, and/or are congenitally unable to
understand. shrug


Tom Roberts