Efficient trace-free decomposition of symmetric tensors of arbitrary rank

In our investigation of gravitational lensing phenomena [11, 12, 13], we utilized the representation of the scalar gravitational potential via an infinite series of symmetric trace-free (STF) tensor mass multipole moments [4]. The STF mass multipole moments of an object of arbitrary shape can be obtained by integrating the appropriate STF coordinate combinations over the volume of the object, as was demonstrated in [13]. Such a representation is useful to study not just gravitational lensing but also electromagnetism, gravitational radiation, geodesy and celestial mechanics [14, 8, 5, 15, 6, 7].

Although the STF representation of a scalar potential is equivalent to the representation using spherical harmonics, it simplifies the integration of wave propagation equations [11, 12, 13] and offers the advantage of coordinate independence [2]. This, in turn, makes it possible to use a Cartesian coordinate system for actual calculations, which offers great convenience over spherical coordinates in many scenarios. On the other hand, the STF formalism can be computationally intensive, which may explain why it remains somewhat less well known and less frequently used than spherical harmonics.

As we shall see below, the trace-free representation of tensors is unique. Obtaining the decomposition, however, requires solving a potentially very large linear system of equations [4]. The complexity of the system of equations makes it difficult to use the results in analytical derivations. While it is computationally straightforward to solve a linear system of equations numerically, such solutions are opaque, offer little insight, and are useless when the goal is to obtain closed form analytical or semianalytical solutions of a physical problem.

On the other hand, the problem can be greatly simplified when the tensor in question is symmetric in all indices [9]. This is the case in particular when it comes to tensorial STF mass multipole moments. This investigation led us to a new, direct, coordinate-independent method of trace-free decomposition of fully symmetric tensors of arbitrary rank. After implementing this method using computer algebra software, we realized that a more efficient, closed-form representation is also achievable. Guided by earlier efforts to obtain such a representation specifically in the three-dimensional case, we were able to derive a closed-form result in arbitrary dimensions.

In the present paper, we offer a brief introduction to the trace-free representation in Sec. 2. In Section 3 we explore the fully symmetric case. In Section 5 we present a closed-form representation for the STF coordinate combinations. We offer a brief discussion in Sec. 6. We demonstrate the utility of this result in the Appendix by way of a simple use case, computing the coordinate combinations that represent STF mass multipole moments for rank 5 through 8.

To introduce our approach, let us first consider the simplest case, the trace of a rank-2 tensor:

where $\eta_{ab}$ is the covariant metric. (In case of Euclidean space with Cartesian coordinates, $\eta_{ab}=\delta_{ab}$, but our approach is not confined to this special case; we assume a generic metric manifold with a nondegenerate metric.) The trace-free representation amounts to decomposing the generic tensor $Q^{ab}$ into a linear combination of a trace-free tensor $Q_{0}^{ab}$ and the trace $Q$ multiplied by the metric:

where $n$ is the number of dimensions. It is well known and easy to see that this representation is unique.

For rank-3 tensors the decomposition becomes a bit more interesting. These tensors have three distinct traces, $\eta_{ab}Q^{abc}$, $\eta_{bc}Q^{abc}$ and $\eta_{ca}Q^{abc}$. The decomposition then follows:

To determine the values of the quantities ${\cal A}^{a}$ through ${\cal C}^{a}$, we can take the trace of (3) and obtain three linear tensor equations in three sets of unknowns:

These can be readily solved to yield

thus

This establishes the generic pattern. As we can see, the representation is unique. For rank-$k$ tensors, the representation involves $\tfrac{1}{2}k(k-1)$ unknown tensors of rank $(k-2)$:

which, after taking the traces, yields $\tfrac{1}{2}k(k-1)$ equations in the form

The structure of this system of equations is more complex than it appears at first sight, as it sometimes involves the unknown tensors, sometimes their traces. In the general case, this is a linear system of equations in $\tfrac{1}{2}k(k-1)n^{2}$ unknowns. In principle, this is trivially solvable, but the result is a complex expression, not readily tractable.

When $Q^{ab..k}=Q^{(ab..k)}$ is symmetric in all indices, the situation is greatly simplified, since ${\cal A}={\cal B}=...={\cal X}$ in (11). In particular, for $k=3$ we get

Taking the trace with respect to the $ab$ indices, we get

Solving for ${\cal A}^{c}$ and substituting back into (14), we obtain

Similarly, for $k=4$, we obtain the equation

Taking the trace with respect to the $ab$ indices, we get

Taking the trace again, with respect to $cd$, we obtain

which we can substitute back into (18), to get

Finally, we can substitute this result back into (17):

where we used round brackets in indices to indicate symmetrization. This nice, clean result suggests that a generalization to arbitrary rank $k$ might be possible. To investigate this, let us first consider the $k=5$ case following the same approach:

However, generalizing to $k=6$ introduces a new twist, the need to take the trace not twice but three times:

Even as, with higher rank, we need to calculate more and more traces, this establishes a general pattern that allows us to compute the trace-free representation of fully symmetric tensors:

This approach can be readily continued to arbitrary rank, as required.

The generic procedure to obtain the trace-free representation of a symmetric tensor can be summarized as follows:

1. 1.

   Start with the equation

   $\displaystyle Q^{ab..k}=Q_{0}^{ab..k}+\eta^{(ab}{\cal A}^{c..k)}.$

   (37)

2. 2.

   Form subsequent traces with respect to $\eta_{ab}$, $\eta_{cd}$, etc., until only a vector or scalar expression remains.

3. 3.

   Solve the final equation for ${\cal A}$ (if $k$ is even) or ${\cal A}^{a}$ (if $k$ is odd) in terms of $Q$ or $Q^{a}$.

4. 4.

   Substitute the result into the preceding (rank-2 or rank-3) equation and solve for ${\cal A}^{ab}$ (for even $k$) or ${\cal A}^{abc}$ (for odd $k$).

5. 5.

   Repeat iteratively until solutions for all the indexed ${\cal A}$ quantities are obtained in this manner.

6. 6.

   Finally, substitute these solutions into (37), which is solved trivially for the trace-free $Q_{0}^{ab..k}$.

It may be tempting to think that computing the trace-free representation of a rank-$k$ tensor can utilize results from the rank-$(k-1)$ or rank-$(k-2)$ computation. This, however, is not the case. At each rank, the computation relies on unique terms, despite the apparent, formal similarities between expressions.

We implemented this algorithm using the Maxima computer algebra system, as presented in A. Although we have also obtained a closed form representation (see Sec. 5 below) the iterative algorithm presented here may be more convenient in some situations. This implementation can compute the symbolic form of the decomposition for tensor ranks up to rank-8 on desktop computers with typical hardware and memory resources. (Higher rank tensors exhaust the memory allocation of the LISP language implementation underlying the Maxima language. The symbolic result in any case becomes too complex to be of practical use.)

This Maxima implementation of our trace-free decomposition algorithm can also be used either directly for numerical computation within the Maxima system, or it can serve as a model for numerical implementations. The Maxima session shown in B demonstrates using this algorithm to explicitly compute matrix elements, in this case the rank-3 STF tensor characterizing the octupole gravitational potential of a point mass $m$ located at coordinates $(0,0,z)$ in a Cartesian coordinate system.

The trace-free nature of the result is readily seen. By assigning values to the parameters $m,z$ the final expression can be numerically evaluated. Specific cases, in principle up to arbitrary rank, in actual practice up to rank-8, can be just as readily evaluated. It is not necessary to display large, untractable interim expressions; a few additional lines of Maxima code can be used, as needed, to compute results numerically and then print only unique nonzero elements of the resulting tensor.

Recurring patterns in this solution appear to suggest that a closed-form can be obtained. Indeed, such a representation has been derived for the case of $n=3$ dimensions in [10] for use in the study of gravitational radiation; see also [8, 1, 3].

Following the approach in [10], in the general $n$-dimensional case, we obtain by induction the expression

The product term in the denominator can, in turn, be expressed using gamma functions and simplified:

leading to the final trace-free representation of the fully symmetric, $n$-dimensional, rank-$k$ tensor $Q^{a_{1}a_{2}..a_{k}}$ in the form

This closed-form representation can also be readily implemented using computer algebra, as shown in C. To demonstrate the practical applicability of our results, in D we present the coordinate combinations needed to compute the STF mass moments of rank $k=5..8$ that are not readily available in the literature.

Symmetric trace-free tensors provide a versatile alternative to spherical harmonics in the investigation of many problems in physics, including the topic that inspired our investigation, gravitational lensing [11, 12, 13]. The primary advantage of the STF tensor representation is its coordinate independence. This allows us to work using the coordinate system of our choice, which includes the convenient Cartesian coordinate system. Nonetheless, the STF representation is less well known than that with the spherical harmonics. This is likely due to the fact that their use, especially the use of higher-rank STF tensors, can be computationally daunting, and that symbolic results suitable for analytical calculations are not readily available.

The STF representation often begins by obtaining a tensor, symmetric in all indices, but not yet trace-free. The corresponding STF tensor is obtained using the unique trace-free representation of this tensor. In principle, finding this representation requires solving a potentially large system of linear equations. The result that we developed relies on the unique properties of fully symmetric tensors to compute this decomposition efficiently in a coordinate-independent form. This may be done not only to arbitrary order, but also in the geometries with arbitrary dimensions.

Though the principles are well known, our result presents several advantages. First, it is valid in an arbitrary metric manifold with a nondegenerate metric, in arbitrary dimensions. Second, it is a closed-form expression: instead of relying on generic methods like matrix inversion or linear equation solvers, it relies on the unique properties of symmetric tensors. As a consequence, the results are readily presented in symbolic form, suitable for use in analytical expressions and for use in computer algebra.

As such, this method is usable in many applied contexts, including electromagnetism, geodesy and celestial mechanics, as well as the subject of our investigation, gravitational lensing. As the solution works in arbitrary dimensions, it may also find use in the development and application of new theories such as higher-dimensional theories of gravitation. Our results can also help solve wave propagation equations in many areas of wave physics.

In D we present the STF coordinate combinations that are required to compute the STF mass multipole moments for rank $r=5..8$, which can be used to study gravitational lensing by complex mass distributions. The work on using these results is ongoing; results, when available, will be published elsewhere.

This work in part was performed at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. VTT acknowledges the generous support of Plamen Vasilev and other Patreon patrons.