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

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• Kostik
In summary: Finally, the covariant derivative of a tensor is just the gradient of its adjoint: $$D^\mu\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(no Einstein summation)}$$In summary, Dirac says 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
Kostik
TL;DR Summary
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?

Last edited:
Kostik said:
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
$$T = A^\nu \otimes B_\nu$$

vanhees71
I'm not familiar with your notation, I wonder if Dirac's decomposition can be explained using only his definition of tensors.

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Kostik said:
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
Ibix said:
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.

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).

## 1. What is a rank-2 tensor in the context of General Theory of Relativity?

A rank-2 tensor in General Theory of Relativity is a mathematical object that describes the curvature of spacetime. It has two indices and represents the relationship between two vectors in a given coordinate system. In the context of General Theory of Relativity, these tensors are used to describe the gravitational field and its effects on matter and energy.

## 2. How do you decompose a rank-2 tensor in General Theory of Relativity?

To decompose a rank-2 tensor in General Theory of Relativity, you can use the covariant derivative operator to split the tensor into its symmetric and antisymmetric parts. This process is known as the Ricci decomposition and is used to simplify the equations of General Theory of Relativity and make them easier to solve.

## 3. What is the significance of decomposing rank-2 tensors in General Theory of Relativity?

The decomposition of rank-2 tensors in General Theory of Relativity allows for a more intuitive understanding of the equations and concepts involved. It also simplifies the equations and makes them easier to solve, which is crucial in the study of this complex theory.

## 4. Can you give an example of decomposing a rank-2 tensor in General Theory of Relativity?

One example of decomposing a rank-2 tensor in General Theory of Relativity is the decomposition of the stress-energy tensor, which describes the distribution of matter and energy in spacetime. The symmetric part of this tensor represents the energy density and pressure, while the antisymmetric part represents the flow of energy and momentum.

## 5. How does the decomposition of rank-2 tensors relate to Dirac's equation?

Dirac's equation is a fundamental equation in quantum mechanics that describes the behavior of particles with spin. The decomposition of rank-2 tensors in General Theory of Relativity is important in the study of Dirac's equation as it allows for a better understanding of the relationship between gravity and quantum mechanics.

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