Action of metric tensor on Levi-Civita symbol

Question

We know that a metric tensor raises or lowers the indices of a tensor, for e.g. a Levi-Civita tensor. If we are in $4D$ spacetime, then \begin{align} g_{mn}\epsilon^{npqr}=\epsilon_{m}{}^{pqr}\tag{1} \end{align} where $g_{mn}$ is the metric and $\epsilon^{npqr}$ is the Levi-Civita tensor.

The Levi-Civita symbol, which we can denote by $\varepsilon^{npqr}$, is not a tensor. It obeys the relation \begin{align} \varepsilon^{npqr}=\varepsilon_{npqr} \tag{2}. \end{align}

What happens if the metric tensor is multiplied with the Levi-Civita symbol $\varepsilon^{npqr}$? \begin{align} g_{mn}\varepsilon^{npqr}=\, \tag{3}? \end{align}

Answer 1

If i understand correctly the Levi-Civita symbol is the function $\varepsilon : \{0, \dots, 3\}^4 \to \{-1,0,1\}$ that that sends a permutation of $\{0,\dots,3\}$ to its sign and non-permutations to 0.

Since it is a function and not a tensor (and not even defined on the manifold) its "upper" or "lower" indices have no actual meaning and are only a tool used for convenience to use the summation convention. We define the Levi-Civita symbol with any combination of upper and lower indices to be the same function that sends the indices to $\varepsilon( \mathrm{indices})$.

Now $g_{mn}$ is not the metric tensor, rather it is a component function of the metric tensor that is to say it is a smooth function $g_{mn} : M \to \mathbb{R}$, where $M$ is our manifold. So that locally $g = g_{mn} dx^m \otimes dx^n$ where $x^0, \dots , x^3$ is some coordinate system and $g$ is the metric tensor.

Now the expression $g_{mn} \varepsilon^{npqr}$ is totally degenerate because it depends on the choosen basis. It is simply a function (that for fixed $m, p,q,r$) sends a point $s$ on $M$ to the number $g_{mn}(s) \varepsilon^{npqr}$.

The index lowering/increasing isomorphism acts on (non degenerate) tensor fields and not on functions and especially not on functions that are not defined on the manifold.

Answer 2

Your equation $2$ is not correct. It is only true for flat spaces. For the purposes of this question, I shall work with Lorentzian manifolds. The object you call Levi-civita tensor is actually called a volume form and is not really a tensor but a tensor density. It tells you the volume density in your space-time. Furthermore, like the Levi-civita tensor in flat space it is completely ani-symmetric. It is usually defined as

$$ \epsilon_{12...n} = \sqrt{-g}$$

where $g = det(g_{\mu \nu})$. A straightforward calculation shows (remembering the Leibniz definition of determinant)

$$ \epsilon^{12...n} = \frac{-1}{\sqrt{-g}}$$