This is clearly symmetric

[physlad]

I was reading about the momentum-energy tensor (or stress-energy tensor), at one point the author says,
"
$$\theta^{\mu\nu} = (\partial^\mu\phi)(\partial^\nu\phi) - g^{\mu\nu}L$$

This is clearly symmetric in $$\mu$$ and $$\nu$$."

$$\theta^{\mu\nu}$$: is the stress-energy tensor
$$\phi$$ is a scalar field
$$g^{\mu\nu}$$ is the metric (+---)
L is the lagrangian density

My question (I'm not an expert in tensors) is how do you see that it's "clearly" symmetric? another silly question: when do we need to symmetrize and antisymmetrize tensors?

Please, tell me guys if this isn't the right place for my question.

 

[cristo]

It's symmetric because:

- g is symmetric since it's the metric, thus the second term is symmetric.
- The first term is the product of two derivatives, so is symmetric (i.e. $\partial^0\phi \partial^1\phi \equiv \partial^1\phi\partial^0\phi$, and similarly for other values of mu and nu.)

 

[astros]

... when do we need to symmetrize and antisymmetrize tensors?...

Sometimes one has a product of a symmetric tensor with another tensor which is not symmetric nor antisymmetric, then one can show that the antisymmetric part of the second is killed by the first, the same thing occurs for the antisymmetric case, this is why we need to antisymmetrise and symmetrise tensors: to see which part remains and which part is killed...

 

[matteo137]

Suppose to have a generic tensor T, which isn't symmetric or antisymmetric.
You can ALWAYS write it as sum of its symmetric part with its antisymmetric part.

e.g. for a rank-2 tensor

$$T^{\mu\nu}=\frac{1}{2}(T^{\mu\nu}+T^{\nu\mu}) + \frac{1}{2}(T^{\mu\nu}-T^{\nu\mu})$$

On the right-hand side, the first term $\left(\bullet+\bullet\right)$ is a symmetric tensor, while the second $\left(\bullet-\bullet\right)$ is antisymmetric!


Hence, in general: $T=T_S+T_A$.


Now...if you have a product between tensors, and you know that one of them is explicitly symmetric $S$ or antisymmetric $A$...you can write:

$ST=ST_S+ST_A$

$AT=AT_S+AT_A$

Of course the product between a symmetric and an antisymmetric tensor is zero, i.e. $SA=AS=0$!

And thus

$ST=ST_S+ST_A=ST_S$

$AT=AT_S+AT_A=AT_A$.

 

[blue_raver22]

I can confirm that the statement "This is clearly symmetric" is accurate. To understand why, let's first define what symmetry means in this context. A tensor is symmetric if it is unchanged when its indices are interchanged. In other words, if we swap the positions of \mu and \nu, the resulting tensor should be the same as the original.

Now, looking at the equation provided, we can see that the first term on the right-hand side is a product of two partial derivatives, each with one index of \mu and one index of \nu. This means that interchanging the indices will not change the result, as both derivatives will still act on the same scalar field \phi. The second term, on the other hand, has a metric tensor that is symmetric by definition, so swapping the indices will not change the result either. Therefore, the entire expression is symmetric in \mu and \nu.

To answer your second question, we need to symmetrize or antisymmetrize tensors when working with equations that require them to have certain symmetries. For example, in the context of general relativity, the stress-energy tensor must be symmetric in order to satisfy the conservation of energy-momentum equation. Similarly, in electromagnetism, the electromagnetic field tensor must be antisymmetric in order to satisfy Maxwell's equations. These symmetries are necessary for the equations to accurately describe physical phenomena. I hope this helps to clarify your understanding of tensors.