Index juggling refers to the practice of denoting a contraction with the metric tensor by changing the placement of the participating index. In fact, we have already encountered this practice on two occasions. Indeed, in Chapter 9, we established that the covariant and contravariant bases $\mathbf{Z}_{i}$ and $\mathbf{Z}^{i}$ are related by a contraction with the metric tensor, i.e. (Notice that the first equation was, in fact, the definition of the contravariant basis $\mathbf{Z}^{i}$.) Similarly, the contravariant and covariant components $U^{i}$ and $U_{i}$ of a vector $\mathbf{U}$ are similarly related by contraction with the metric tensor, i.e. In general, denoting the result of contracting a system $A_{j}$ with the contravariant matric tensor $Z^{ij}$, i.e. is referred to as raising the index. Meanwhile, denoting the result of contracting a system $A^{i}$ with the covariant metric tensor $Z_{ij}$, i.e. is referred to as lowering the index.

Note that the last two equations are equivalent by the inversion of the metric tensor described in Section 9.5.2. This, if $A_{i}$ is obtained from $A^{i}$ by lowering the index, then $A^{i}$ is obtained from $A_{i}$ by raising the index. In other words, the operations of lowering and raising indices are the inverses of each other. Had this not been the case, using the same letter to denote objects related by index juggling would lead to ambiguities as, for instance, the symbol $A^{i}$ would denote both an original system as well as the result of lowering and subsequent raising of the index.

For a higher-order system, any of the indices can be individually raised or lowered. For example, consider a third-order system $T_{ijk}$ with three subscripts. To raise the first index, contract $T_{ijk}$ with the contravariant metric tensor $Z^{ri}$ to produce the system $T_{jk}^{r}$ T_{jk}^{r}=Z^{ri}T_{ijk}.tag{$-$} end{equation} Note that in this situation, the use of the dot placeholder described in Section 7.2 is essential. Otherwise, it is not possible to say whether the symbol, say, $T_{bc}^{r}$ represents the raising of the first index on the system $T_{ibc}$, the second index on $T_{bic}$, or the third index on $T_{bci}$. The use of a placeholder removes this ambiguity by making the order of the indices clear. With the help of the placeholder, the above identity reads and it is clear simply by looking at the symbol $T_{\cdot jk}^{r}$ that it represents the result of raising the first index of a system $T_{ijk}$.

Contracting $T_{ijk}$ with $Z^{rj}$ -- notice the superscript $j$ in $Z^{rj}$ -- raises the second index, i.e. The indicial signature of the symbol $T_{i\cdot k}^{\cdot r}$ clearly indicates that the superscript is the second index. Similarly, the equation represents the raising of the third index of $T_{ijk}$.

Once one index of a multi-dimensional system has been juggled, another can be juggled, as well. For example, raising the second index of $T_{\cdot jk}^{r}$ yields When two indices are juggled one after another, the order in which the two steps take place is immaterial. This is a direct consequence of the commutative property of elementary multiplication since $T_{\cdot\cdot k} ^{rs}$ is also given by a double contraction, i.e. Thus, two juggling operations may be viewed as simultaneous.

Finally, any number of indices can be juggled at once. For example, raising all three indices of $T_{ijk}$ yields a system with three superscripts, i.e.

Recall that, according to the Einstein convention, dummy indices represent a contraction.Thus, dummy indices, being already involved in a summation, are not available for contraction with the metric tensor. However, index juggling allows such indices to exchange flavors. For a most basic example, let us show that The idea is to "extract" the metric tensor from $S_{i}$ by noting that $S_{i}=S^{j}Z_{ji}$: and subsequently let it get "reabsorbed" into $T^{i}$: Finally, renaming $i$ into $j$, we arrive at the desired result In summary, the flavors of the dummy indices in any contraction can be switched at will.

A word of caution is in order for situations where one of the systems appears under a differential operator of some kind. For example, suppose that systems $S_{i}$ and $T^{i}$ are defined along a trajectory $Z^{i}\left( t\right)$, and thus these systems may be considered functions $S_{i}\left( t\right)$ and $T^{i}\left( t\right)$ of $t$. Then it would be wrong to let the dummy indices exchange flavors in the contraction To see why this is so, let us attempt to repeat the logic used in the previous example. The product $S^{j}Z_{ji}$ in occurs under the derivative operator. Therefore, the next step is an application of the product rule: Since the metric tensor varies from one point to another, the derivative $dZ_{ij}/dt$ is not zero and we thus have a new term compared to our previous analysis. Multiplying out the right side, we find that and, therefore, or, having renamed $i$ into $j$, Thus, we find that due to the variability of the covariant basis. We can describe the effect we just observed by saying that dummy indices cannot be juggled across derivatives. This is something that we should be cognizant of when we begin to study the covariant derivative in Chapter 15.

In a tensor identity, a free index can be consistently raised or lowered on both sides of the equation. For example, the identity implies a similar one in which the free index $i$ appears as a superscript:

To see why this is true, contract both sides of the original identity with $Z^{ik}$: which yields Now, rename $k$ into $i$ to obtain the final form We will refer to the net effect of this manipulation as raising or lowering an index on both sides of the equation.

The same word of caution regarding index juggling in the presence of differential operators must be repeated here: free indices cannot be juggled across derivatives. For example, the superscript $i$ cannot be lowered in the identity Indeed, contracting both sides with $Z_{ij}$ yields On the left, $Z_{ij}$ combines $S^{i}$ to produce $S_{j}$. On the right, however, $Z_{ij}$ is separated from $T^{i}$ by the derivative. It can be brought inside the derivative by an application of the product rule Therefore, we find that and, once again, there is an extra term due to the variability of the metric tensor, leading to the conclusion that does not imply

Index juggling can be applied to any index in any system. Let us now apply it to the Kronecker delta $\delta_{j}^{i}$.

As we discussed in Section 7.2, the order of the indices of the Kronecker delta does not matter. Nevertheless, in order to avoid any potential ambiguities, we will count the superscript as the first index and the subscript as the second, and therefore write $\delta_{j}^{i}$ as $\delta_{\cdot j}^{i}$.

Let us now lower the superscript $i$ by contracting $\delta_{\cdot j}^{i}$ with $Z_{ki}$. Consistent with our convention of using the same letter for objects related by index juggling, denote the result by $\delta_{kj}$: We must emphasize the fact that the elements of $\delta_{kj}$ no longer equal $1$ when $j=k$ and $0$ otherwise. Those are the values of the proper Kronecker delta $\delta_{\cdot j}^{i}$. The values of $\delta_{kj}$, on the other hand, depend on the choice of coordinates and vary from one point to another.

As a matter of fact, we can determine the values of the elements $\delta_{kj}$ by looking at the combination $Z_{ki}\delta_{\cdot j}^{i}$ from a different perspective. Instead of thinking of $Z_{ki}\delta_{\cdot j}^{i}$ as $Z_{ij}$ lowering the index on $\delta_{\cdot j}^{i}$, let us think of it as $\delta_{\cdot j}^{i}$ renaming the index $i$ into $j$ on $Z_{ki}$ and thus yielding $Z_{kj}$: Matching up the equivalent outcomes of the two interpretations, we discover that the elements of $\delta_{kj}$ are precisely the same as those of $Z_{kj}$: Thus, the Kronecker delta $\delta_{\cdot j}^{i}$ and the metric tensor $Z_{ij}$ are related by index juggling and are therefore thought of as two manifestations of the exact same object.

This conclusion can be reinforced by raising one of the indices of $Z_{ij}$. Let us raise the first index of $Z_{ij}$ by contracting it with $Z^{ki}$ and denote the result by $Z_{\cdot j}^{k}$ By the very definition of the contravariant metric tensor as the matrix inverse of the covariant metric tensor, we know that Therefore and we once again observe the equivalence of the Kronecker delta and the metric tensor.

Summing up, the systems $\delta_{\cdot j}^{i}$ and $Z_{ij}$ are related by index juggling and are therefore different manifestations of the same object. Thus, the letters $\delta$ and $Z$ in the symbols $\delta_{\cdot j}^{i}$ and $Z_{ij}$ can be used interchangeably. For example, the symbols $\delta_{ij}$ and $\delta^{ij}$ are legitimate representations of the covariant and the contravariant metric tensors. Meanwhile, $Z_{\cdot j}^{i}$ is a legitimate representation of the Kronecker delta. Nevertheless, in practice, we will always use the symbol $\delta_{\cdot j}^{i}$ for the Kronecker delta and $Z_{ij}$ and $Z^{ij}$ for the metric tensors. Additionally, we will go back to using the symbol $\delta_{j}^{i}$ without a placeholder since, as we discussed in Section 7.2, this cannot lead to ambiguity.

Index juggling is a powerful feature of the tensor notation, as it greatly increases the succinctness of tensor expressions. Just compare the expressions $U_{i}V^{i}$ and $Z_{ij}U^{i}V^{j}$ that represent the dot product of the vectors $\mathbf{U}$ and $\mathbf{V}$. It is hardly debatable that the former expression is more appealing than the latter: it has fewer terms, fewer indices, fewer contractions, a sense of algebraic transparency, and a general sense of lightness and dynamism.

Thanks to index juggling, these attractive features are found in many tensor expressions. For example, it is rare for any formula to contain an explicit reference to the metric tensor since any instance of the metric tensor engaged in a contraction with another system gets absorbed into that system. For a similar reason, the identity matrix $I$ rarely appears in Linear Algebra expressions. After all, if it is featured in a matrix product, such as $IA$, it simply gets absorbed into the matrix that it multiplies, i.e. $IA=A$.

The only way a metric tensor can explicitly appear in a tensor identity is when both of its indices are free. Two notable examples of such expressions appear in this book. The first is the statement that two systems $A_{j}^{i}$ and $B_{k}^{j}$ are matrix inverses of each other, i.e. The second is the formula found in the next volume where we will give a tensor description of embedded surfaces. This formula states that the sum of the two projection operators, onto the surface and away from the surface, is the identity operator. In other words, a vector is a sum of its orthogonal projections onto and away from the surface.

Exercise 11.1Show that if we lower the index on $A^{i}$ to produce $B_{i}$ and then raise the index on $B_{i}$ to produce $C^{i}$, then This exercise confirms that the operations of lowering and raising indices are the inverses of each other. Furthermore, it shows that using the same letter for systems related by index juggling cannot lead to ambiguities.

Exercise 11.2Show that the result of raising both indices on the covariant metric tensor $Z_{ij}$ is the contravariant metric tensor $Z^{rs}.$

Exercise 11.3Consider a coordinate system in which the metric tensor $Z_{ij}$ at a given point corresponds to the matrix Suppose that $T_{\cdot jkl}^{i}$ is a fourth-order system whose sole nonzero element at the same point is Find the values of the elements $T_{1212}$ and $T_{2212}$ at that point.