Re:The Geometric Tool That Solved Einstein's Relativity Problem
by thechao on Monday August 12, 2024 @03:48PM
(#41228526)
> (individual numbers (0 dimensions), vectors (1 dimension), matrixes (2 dimensions), and so on)
You're using the word "dimension" in two (distinct) ways. Instead, use the word "rank":
> (individual numbers (0 rank), vectors (1 rank), matrixes (2 rank), and so on)
Now, we can talk about a 4-dimensional rank-1 tensor, e.g., a 4-element vector.
Now, think about a 4x4 matrix: if we multiply the matrix by a 4-vector, we get a 4-vector out: in some ways, the multiplication has "eaten" one of the ranks of the matrix; but, the dimension of the resulting object is the same. If we had a 3x2 matrix, and we multiplied it by a 3-vector, then both the rank has changed (from 2 to 1) and the dimension has changed (from 3 to 2).
A tensor has any number of rank.
More importantly, the ranks of a tensor come in two "flavors": a vector and a one-form. The concepts are pretty darn general, but one way to get a feel for how they're related is that the transpose of a vector can be its dual. This gets into things like pre- and post- multiplication; or, whether we 'covary' or 'contravary' with respect to the tensor.
Frankly, tensor products are a beast to deal with, mechanically, so the literature mostly deals with them as opaque objects. The modern tensor software libraries and high performance computing has seen a sea-change in the use of GR.
You're using the word "dimension" in two (distinct) ways. Instead, use the word "rank":
> (individual numbers (0 rank), vectors (1 rank), matrixes (2 rank), and so on)
Now, we can talk about a 4-dimensional rank-1 tensor, e.g., a 4-element vector.
Now, think about a 4x4 matrix: if we multiply the matrix by a 4-vector, we get a 4-vector out: in some ways, the multiplication has "eaten" one of the ranks of the matrix; but, the dimension of the resulting object is the same. If we had a 3x2 matrix, and we multiplied it by a 3-vector, then both the rank has changed (from 2 to 1) and the dimension has changed (from 3 to 2).
A tensor has any number of rank.
More importantly, the ranks of a tensor come in two "flavors": a vector and a one-form. The concepts are pretty darn general, but one way to get a feel for how they're related is that the transpose of a vector can be its dual. This gets into things like pre- and post- multiplication; or, whether we 'covary' or 'contravary' with respect to the tensor.
Frankly, tensor products are a beast to deal with, mechanically, so the literature mostly deals with them as opaque objects. The modern tensor software libraries and high performance computing has seen a sea-change in the use of GR.