# Action of metric tensor on Levi-Civita symbol

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}

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.

• What is meant by saying that the expression $g_{mn}\varepsilon^{npqr}$ is totally degenerate? Jun 22 at 10:22

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)

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

• Do you happen to know why it is defined so? Also, as written, the LHS is a tensor that takes many vectors (or 1-forms), but the RHS seems to be a scalar quantity after contraction. Jun 21 at 2:49
• @emir sezik, I know the difference between the Levi-Civita tensor and Levi-Civita tensor density. $\epsilon^{npqr}$ in my original post is the Levi-Civita tensor and not the tensor density. Jun 21 at 4:57