Tensor #32. Tensor Density Examples텐서 #32. 텐서 밀도의 예

These are my notes on XylyXylyX’s What is Tensor lectures. This post covers lecture 32 of the playlist, Tensor density examples. (REDUX). This time we check the tensor density defined in the previous post with two examples. One is the Levi-Civita symbol and the other is the determinant of the metric.

Recap of the Levi-Civita symbol

On an $n$-dimensional vector space, the Levi-Civita symbol $\varepsilon$ is a rule that takes $n$ indices and gives a number. If the sequence $i_1 \cdots i_n$ is a permutation $\sigma$ of $1, \dots, n$ it gives the sign, and if the indices repeat it gives 0.

It is $+1$ for an even permutation, $-1$ for an odd one, and 0 when indices repeat. Whether the indices are written up or down, the values are the same numbers. It is called a symbol to keep it apart from a tensor. This distinction is the subject of this post.

The symbol is not a tensor

In one coordinate system, let us build an $(n, 0)$ tensor whose components are the Levi-Civita symbol.

It is real coefficients multiplied by basis elements and added up, so $\mathcal{E}$ itself is certainly a tensor. Looking at the components in new coordinates, an inverse matrix is multiplied in for each upper index.

Here we need a known fact about the Levi-Civita symbol. Applying any matrix $A$ to each index brings the determinant out front.

This holds because the left side is completely antisymmetric in the new indices, and its value at the indices $1, \dots, n$ is $\det A$ straight from the definition of the determinant. (This reason is added for understanding and was not covered in the lecture. The lecture brings it in as a fact seen in the earlier lecture on determinants.) Putting in $A = \Lambda^{-1}$ gives this.

The new components changed as a tensor are not the symbol values themselves but $(\dL)^{-1}$ times them. We cannot expect the components to stay equal to the symbol in new coordinates. The components of a tensor must change by a rule when the coordinates change, but the symbol is the same numbers $0, \pm 1$ in every coordinate system. So the Levi-Civita symbol is not a tensor.

Q. If the values of the symbol are the same in every coordinate system, isn’t that a nice object that does not depend on coordinates?
A. That a tensor does not depend on coordinates means that the components change by the rule so that the object itself stays the same. Components fixed to the same numbers cannot follow this rule. The symbol has fixed values, so it is not a tensor.

Multiplying in a wedge basis to make a density

We can repair $\mathcal{E}$ by the method of the previous post. After the basis, multiply in one more top-degree wedge basis.

This wedge basis changes as $dx^1 \wedge \cdots \wedge dx^n = \dL \; dx'^1 \wedge \cdots \wedge dx'^n$, so the new components contain both the earlier $(\dL)^{-1}$ and this $\dL$.

The two factors cancel, so the components are the symbol itself in every coordinate system. Putting $w = 1$ into the definition of the previous post gives the same result, so the symbol with upper indices is a tensor density of weight $+1$. The $\widetilde{\mathcal{E}}$ made a density this way is an object whose components are the Levi-Civita symbol in every coordinate system, but it is not what is usually called the Levi-Civita tensor. We build that one later.

The symbol with lower indices follows by the same computation. Covariant components satisfy $\Lambda^{i_1}{}_{i'_1} \cdots \Lambda^{i_n}{}_{i'_n} \, \varepsilon_{i_1 \cdots i_n} = \dL \; \varepsilon_{i'_1 \cdots i'_n}$, so changing them only by the tensor rule gives a factor $\dL$. To cancel this factor we must multiply by $(\dL)^{-1}$, so the symbol with lower indices is a tensor density of weight $-1$, and it is obtained by multiplying in the vector-side wedge basis $\partial_1 \wedge \cdots \wedge \partial_n$ of the top degree. (This explanation moves the computation for upper indices to lower indices and was not covered in the lecture.)

Q. Is an object made a density this way the Levi-Civita tensor?
A. No. This one is a density made by multiplying in a wedge basis so that the components are the symbol itself in every coordinate system. The Levi-Civita tensor is an ordinary tensor obtained below by multiplying by the metric determinant, and its components differ from coordinate system to coordinate system.

The metric determinant is a scalar density

The second example is the metric tensor. The metric $g = g_{\mu\nu} \, dx^\mu \otimes dx^\nu$ is a symmetric $(0, 2)$ tensor, a special tensor that is fixed once an inner product is given on the vector space. Its components change by the rule for two lower indices.

In matrix form this is $g' = \Lambda^{T} g \, \Lambda$, and taking the determinant of both sides with $\det \Lambda^T = \det \Lambda$ gives this.

Abbreviating the determinant of the components as $g = \det g_{\mu\nu}$ gives $g' = (\dL)^2 g$. This number has no indices, so it has no tensor part, and the transformation factor is $(\dL)^2$. So $g$ is a scalar density, scalar meaning no indices, and its weight is 2. The end of lecture 31 speaks of this as if it were a $(0, 2)$ density, but as this lecture corrects, it is a scalar density of weight 2 with no indices.

The square root and the reciprocal

A scalar density is a number, so we can also take its square root.

So $\sqrt{|g|}$ is a scalar density of weight 1. The metric in relativity has diagonal entries of mixed signs like $(-,+,+,+)$, so $g$ is often negative, and for that reason textbooks write $\sqrt{-g}$ or $\sqrt{|g|}$. All three forms appear. Taking the reciprocal flips the sign of the weight.

In summary, from one metric we get for free $g$ of weight 2, $\sqrt{|g|}$ of weight 1, $1/\sqrt{|g|}$ of weight $-1$, and $1/g$ of weight $-2$. For any density from now on, multiplying by the suitable one of these numbers brings the weight to 0.

Q. Why attach an absolute value or a minus sign to the determinant?
A. The metric in relativity has a negative determinant. A negative number has no real square root, so we change it to $-g$ or $|g|$ before taking the root.

Strictly, $\sqrt{(\dL)^2} = |\dL|$, so $\sg' = |\dL| \sg$, and for a coordinate change with a negative determinant the sign disagrees with the formula in the lecture. If we use only orientation-preserving changes ($\dL > 0$) it equals the formula in the lecture. Handling orientation-reversing changes belongs to the story of oriented manifolds and pseudotensors, which the lecture postpones. (This distinction is added for understanding and was not covered in the lecture.)

In the toy below, drag the two new basis vectors and check these transformation factors. Once you pick a metric, you also get $\det g$, $\sqrt{|g|}$, and the new components of the Levi-Civita symbol.

Correcting a density into a true tensor

A scalar density is used to turn a density back into an ordinary tensor. Look at the symbol with lower indices multiplied by $\sqrt{|g|}$.

This time we did not attach a wedge basis to the basis; we only multiplied one function of the coordinates into the coefficient. Finding the new components by the tensor rule, the symbol part becomes $\dL \; \varepsilon_{i'_1 \cdots i'_n}$ as seen before, and the multiplied $\sg$ satisfies $\sg' = \dL \sg$, so we get this.

In the new coordinates it has the same shape, namely the symbol multiplied by $\sqrt{|g'|}$ of the new metric. So this set of components $\sg \, \varepsilon_{i_1 \cdots i_n}$ follows the tensor rule exactly and is a true tensor, which we call the Levi-Civita tensor. The symbol with lower indices has weight $-1$ and we multiplied it by $\sqrt{|g|}$ of weight 1, so the weight became $-1 + 1 = 0$. The upper-index version is obtained the same way by multiplying by $1/\sqrt{|g|}$. Its components differ from coordinate system to coordinate system, but as the lecture says, this is the tensor actually used in general relativity.

Summary

What we checked in this post is the following.

• If we change an object whose components are the Levi-Civita symbol as a tensor, the components leave the symbol values. So the symbol is not a tensor.
• Multiplying in a top-degree wedge basis makes a density whose components are the symbol itself in every coordinate system. The symbol with upper indices is a density of weight $+1$.
• The determinant of the metric $g$ is a scalar density of weight 2, and $\sqrt{|g|}$ has weight 1, $1/\sqrt{|g|}$ has $-1$, and $1/g$ has $-2$.
• Multiplying by a scalar density of the right weight corrects a density into an ordinary tensor, and what we get this way is the Levi-Civita tensor.