Can You Correctly Lower Tensor Indices Using the Metric Tensor?

[beans73]

Homework Statement

I have a tensor X$^{μ\nu}$ and I want to make this into X$_{μ\nu}$. Can I do this by simply saying X$_{μ\nu}$=$\eta_{μ\nu}$$\eta_{μ\nu}$ X$^{μ\nu}$ ??

 

[DimReg]

Homework Statement

I have a tensor X$^{μ\nu}$ and I want to make this into X$_{μ\nu}$. Can I do this by simply saying X$_{μ\nu}$=$\eta_{μ\nu}$$\eta_{μ\nu}$ X$^{μ\nu}$ ??

You have the right idea, but your indices are incorrect (it's ambiguous which indices are to be contracted with which other indices). Try introducing some new indices for the contracted ones, see if you can make it unambiguous.

for example, you could write:

$V^{\mu}$ = $\eta^{\mu \nu} V_{\nu}$

Which makes it clear which index of eta is contracted, and which isn't. In this case, eta is symmetric so it doesn't really matter, but for a general tensor with two or more indices it does matter.

To explain why it's ambiguous, consider this:

Does X$_{μ\nu}$=$\eta_{μ\nu}$$\eta_{μ\nu}$ X$^{μ\nu}$ mean X$_{μ\nu}$=$\eta_{μ\nu}$$(\eta_{μ\nu}$ X$^{μ\nu})$? In that case you would get a tensor proportional to eta. This is obviously not what you were thinking, but if someone saw that expression and didn't know that you were trying to make X$_{\mu\nu}$, they might think you meant to make a tensor proportional to eta.

So just a good rule of thumb, never repeat an index unless it is supposed to be summed.