Graduate Decomposing Rank-2 Tensors in Dirac's "General Theory of Relativity

[Kostik]

TL;DR

Dirac says that a general rank-2 tensor $T^{\mu\nu}$ can be decomposed as $A^\mu B^\nu + A'^\mu B'^\nu + A''^\mu B''^\nu + \cdots\,$. Is this obvious?

Dirac's book "General Theory of Relativity" says on p. 2 that a general rank-2 tensor can be written as a sum of outer products: $$ T^{\mu\nu} = A^\mu B^\nu + A'^\mu B'^\nu + A''^\mu B''^\nu + \cdots $$ Importantly, he repeats this on p. 18, in developing the covariant derivative, where he mentions that a tensor $T_{\mu\nu}$ is "expressible as a sum of terms like $A_\mu B_\nu$".

Is this obvious? Can someone show or explain this?

 

[Orodruin]

TL;DR Summary: Dirac says that a general rank-2 tensor can be decomposed: $T^\mu\nu = A^\mu B^\nu + A'^\mu B'^\nu + A''^\mu B''^\nu + ...$. Is this obvious?

Dirac's book "General Theory of Relativity" says on p. 2 that a general rank-2 tensor can be written as a sum of outer products:

$$T^\mu\nu = A^\mu B^\nu + A'^\mu B'^\nu + A''^\mu B''^\nu + ...$$

Importantly, he repeats this on p. 18, in developing the covariant derivative, where he mentions that a tensor $T_\mu\nu is "expressible as a sum of terms like A_\muB_\nu$".

Is this obvious? Can someone show or explain this?

By definition, a tensor of rank two can be written as
$$
T = T^{\mu\nu} e_\mu \otimes e_\nu
$$
We can introduce the vectors $A^\nu = T^{\mu\nu} e_\mu$ and $B_\nu = e_\nu$ (note that here $\nu$ is being used as a counter rather than a component index) and therefore
$$
T = A^\nu \otimes B_\nu
$$

 

[Kostik]

I'm not familiar with your notation, I wonder if Dirac's decomposition can be explained using only his definition of tensors.

 

[Ibix]

I added the missing braces, but the LaTex still doesn't seem to be working in the original post.

It's a known issue when you make the first post to use LaTeX (OP or reply) on a page. The parser doesn't get loaded until you refresh the page. Your LaTeX looks fine to me, and will look fine to you once you've hit refresh.

 

[Kostik]

It's a known issue when you make the first post to use LaTeX (OP or reply) on a page. The parser doesn't get loaded until you refresh the page. Your LaTeX looks fine to me, and will look fine to you once you've hit refresh.

Aha, yes, I see it now.

 

[Kostik]

Oh, I think it's actually fairly straightforward. Write (showing the summation explicitly): $$T^{\mu\nu}=\sum_{\lambda,\kappa}T^{\lambda\kappa}{\delta_\lambda}^\mu{\delta_\kappa}^\nu \,\,\,\,\text{(no Einstein summation)}$$ Then $A^\mu = T^{\lambda\kappa}{\delta_\lambda}^\mu$ (not summed over $\lambda$) and $B^\nu={\delta_\kappa}^\nu$. (Regard $\lambda$ and $\kappa$ as fixed.) Since everything in sight is a tensor, the $A^\mu$ and $B^\nu$ are obviously vectors (no need to worry about constructing a non-vector).