According to https://arxiv.org/abs/2207.03560, for rigid-body motion in arbitrary dimensions, the relationship between the angular momentum two-form $L$ and angular velocity two-form $\omega$ is given by $$L_{ij} = \frac{1}{2} I_{ijkl} \omega_{kl},$$ where $I$ is the rank-four tensor of inertia defined by $$I_{ijkl} := 4 \int dm \left( x_{[j} \delta_{i][k} x_{l]} \right).$$

Interestingly, the tensor of inertia is an algebraic curvature tensor, meaning that it has the same index symmetries (and therefore the same algebraic degrees of freedom) as the Riemann curvature tensor. This paper (PDF) tries to explain the reason behind this surprising parallel, but it uses some obscure math involving the geometric algebra, and I personally don't find it very illuminating.

On any algebraic curvature tensor, we can perform the Ricci decomposition, which decomposes the tensor as a sum of three terms, which contain the same information as (a) the doubly-contracted scalar invariant $I_{ijij}$, (b) the traceless part of the singly-contracted rank-two tensor $I_{kikj}$, and (c) everything else not contained within that rank-two tensor, respectively. The last term, the Weyl tensor, turns out to be conformally invariant, but can only be nonzero in $d>3$ dimensions.

The Weyl tensor summand in the Ricci decomposition of the inertia tensor should reflect the existence of some qualitatively new physical phenomena about rigid-body rotation that only appears in $d>3$, just as the Weyl tensor in general relativity reflects the existence of gravitational waves, black holes with extended influence, etc. in general relativity with $D>3$. What's the physical interpretation of this piece of the tensor of inertia in $d>3$ dimensions?


1 Answer 1


Interesting question. I've also wondered about rigid-body mechanics in n-dimensions via this fourth-order tensor.

However, I don't have an answer to your specific question.
But I can offer a reference which may be of use. (Interestingly, none of the two papers (Jensen https://arxiv.org/abs/2207.03560 and Berrondo https://doi.org/10.1119/1.4734014 ) you link to mention this reference.)

From my answer to Inertia tensor as two times contravariant pseudotensor

Synge and Schild's Tensor Calculus (1949, 1969, Dover:1978, ISBN 9780486636122 ). Do a "Google book search" in it for "moment of inertia" and get to page 161 to read about the fourth-order moment of inertia tensor in N-dimensions.

These may also be helpful


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