Explaining Tensors in Special Relativity

[Wminus]

Hey! I'm reading Special Relativity right now and I am stuck trying to understand tensors. Can you kind people please explain to me the difference between the following 3 tensors?

$$A^{\alpha \beta}$$ $$A_{\alpha \beta}$$ $$A^{\alpha}_{\beta}$$

 

[Orodruin]

The difference lies in how these tensor components transform under coordinate transformations. In relativity you always have a metric to relate different types of tensor components to the underlying meaning of co- and contravariant properties often gets lost. A good first step to understanding the difference is to focus on understanding the difference between covariant and contravariant vectors.

 

[Nugatory]

There's a good introduction at http://preposterousuniverse.com/grnotes/grtinypdf.pdf

 

[Wminus]

The difference lies in how these tensor components transform under coordinate transformations. In relativity you always have a metric to relate different types of tensor components to the underlying meaning of co- and contravariant properties often gets lost. A good first step to understanding the difference is to focus on understanding the difference between covariant and contravariant vectors.

I understand how to do the component transforms ("up/down transforms"), and I understand the concept of dual spaces/one forms (same as bra's and ket's). What I do not understand is the physical meaning of all the tensors where $(n,p)$ where the $n$ and $p$ are not equal to 1. I mean, $L^{\alpha}_{\beta}$ is the good old boost matrix that acts on 4-vectors, but what the heck is a $L^{\alpha \beta} = L^{\alpha}_{\mu} g^{\beta \mu}$ ?

And also how come $g^{\beta \mu}$ is a physical matrix (it's the minkowski metric matrix which you use to find the scalar product of two 4-vectors), while $L^{\alpha \beta}$ isn't ?! The two tensors are of the same rank, after all!

Please keep in mind that I only know and think in terms of engineering linear algebra (i.e. vector/hilbert spaces, linear independence and matrices). I've never studied differential geometry! Do I need to pick up a book in that to get a real understanding of Einstein's two theories of relativity?

 

[haushofer]

L, the Lorentz transformation, is defined as a coordinate transformation. As such it has two indices. This coordinate transformation then acts on an index of a tensor.

Let's look upon it from a fundamental perspective. As a theoretical physicist you look at the fundamental building blocks of nature. These building blocks are particles and the forces between them, also mediated by particles. So it's particles everywhere. These particles are described by fields. These fields can be described by how their components change when you change your perspective as an observer. Think as an analogy about geometrical objects like points, lines, cubes, etc in 3-space. You could say that the more complicated the field transforms, the more indices it gets. Maybe you know about the concept of spin; well, there is a relation between the number and nature of indices a field has, and the amount of spin you can assign to it. Fields without indices are called scalars and have spin 0, fields with one index are called vectors and have spin 1, and on top of that you have half-integer spin fields, which have different kind of indices (called spinor indices). In the end it is just a matter of labeling.

Maybe it also helps to look at the physical interpretation of the energy-momentum tensor, a tensor with two indices; it describes the energy- and momentum flux of spacetime surfaces.

A Lorentz transformation with two upper indices is just a mathematical expression of the matrix product of a Lorentz transformation with a metric. That's (as I understand) all there is to it.

 

[PeterDonis]

while LαβLαβL^{\alpha \beta} isn't ?! The two tensors are of the same rank, after all!

The Lorentz transformation matrix is not a tensor. It happens to be written using similar index notation, but that doesn't mean it's the same kind of thing.

I've never studied differential geometry! Do I need to pick up a book in that to get a real understanding of Einstein's two theories of relativity?

Sean Carroll's online lecture notes give a good introduction to differential geometry as it is used in relativity.

http://arxiv.org/abs/gr-qc/9712019