The Relativistic Inertia Tensor in
Projective Spacetime Geometric Algebra
Eric Lengyel • November 13, 2024
This is a short post about some of the relativistic physics I’ve been studying in projective geometric algebra in my free time. A discussion of the background material would be far too long to include here, as would a complete physical interpretation of the results. My intention is only to promote awareness of some stuff that I think is very interesting. I will eventually provide a thorough treatment of the subject in the second edition of Projective Geometric Algebra Illuminated.
The setting for this post is the five-dimensional geometric algebra \(\mathbb R(3,1,1)\) that’s projective over four-dimensional spacetime. There are five vector basis elements \(\mathbf e_0\), \(\mathbf e_1\), \(\mathbf e_2\), \(\mathbf e_3\), and \(\mathbf e_4\), where the vectors \(\mathbf e_1\), \(\mathbf e_2\), and \(\mathbf e_3\) correspond to 3D space with \(\mathbf e_i \cdot \mathbf e_i = +1\) for \(i = 1,2,3\), the vector \(\mathbf e_0\) corresponds to time with \(\mathbf e_0 \cdot \mathbf e_0 = -1\), and the vector \(\mathbf e_4\) is the projective dimension with \(\mathbf e_4 \cdot \mathbf e_4 = 0\). Using these basis vectors, the spacetime position r of a particle is expressed as
\(\mathbf r = ct\,\mathbf e_0 + x\,\mathbf e_1 + y\,\mathbf e_2 + z\,\mathbf e_3 + \mathbf e_4\),
where c is the speed of light. The linear momentum p of a particle with rest mass m is then given by
\(\mathbf p = m \dfrac{d\mathbf r}{d\tau} = \gamma mc\,\mathbf e_0 + \gamma m\dot x\,\mathbf e_1 + \gamma m\dot y\,\mathbf e_2 + \gamma m\dot z\,\mathbf e_3\),
where \(\tau\) is proper time, a dot above means derivative with respect to coordinate time t, and \(\gamma = dt/d\tau\), as usual in relativity. This is a purely directional spacetime vector that we can combine with the particle’s position r using the wedge product to construct a line P through spacetime given by
\(\begin{split}\mathbf P = \mathbf r \wedge \mathbf p = \gamma m {\large[}&c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} \\[1ex] +\, &(y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} \\[1ex] +\, &c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30}{\large]}.\end{split}\)
The first four terms of P contain the linear momentum, the next three contain the angular momentum, and the last three contain the dynamic mass moment. This ten-component bivector represents everything about the momentum of a particle with respect to a particular origin in spacetime. What we would like to do is relate this momentum bivector P to some kind of ten-component velocity in a manner analogous to the classical formula \(\mathbf L = \mathcal I \boldsymbol \omega\), where \(\mathbf L\) is the angular momentum, \(\mathcal I\) is the \(3 \times 3\) moment of inertia tensor, and \(\boldsymbol \omega\) is the angular velocity.
In spacetime, physically possible continuous motions correspond to the subgroup of the Poincaré group containing all combinations of spacetime translations and proper orthochronous Lorentz transformations. These can all be implemented by motion operators (motors) in the geometric algebra such that a position \(\mathbf r_0\) (or any other quantity) is transformed according to \(\mathbf r(\tau) = \mathbf Q(\tau) \mathbin{\unicode{x27C7}} \mathbf r_0 \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}(\tau)\) for some \(\mathbf Q(\tau)\) consisting of up to 16 components having even antigrade. Without going into the details, the motor \(\mathbf Q(\tau)\) can be expressed as a product of exponentials of trivectors (with respect to the geometric antiproduct), and that allows us to write \(d\mathbf r/d\tau\) as
\(\dfrac{d\mathbf r}{d\tau} = \mathbf V(\tau) \mathbin{\unicode{x27C7}} \mathbf r(\tau) - \mathbf r(\tau) \mathbin{\unicode{x27C7}} \mathbf V(\tau)\),
where \(\mathbf V(\tau)\) is a function of the exponents that can be, in general, dependent on the proper time \(\tau\). The trivector \(\mathbf V(\tau)\) has the form
\(\mathbf V(\tau) = v_t\,\mathbf e_{321} + v_x\,\mathbf e_{230} + v_y\,\mathbf e_{310} + v_z\,\mathbf e_{120} + \omega_x\,\mathbf e_{410} + \omega_y\,\mathbf e_{420} + \omega_z\,\mathbf e_{430} + \mu_x\,\mathbf e_{423} + \mu_y\,\mathbf e_{431} + \mu_z\,\mathbf e_{412}\),
where the terms have been written in the order in which complementary basis elements appear in the bivector momentum P above. If we calculate a momentum bivector \(\mathbf r \wedge m\,d\mathbf r/d\tau\) with the formula based on \(\mathbf V(\tau)\) and match terms to the formula for P given earlier, then we can extract a \(10 \times 10\) matrix that linearly transforms from \(\mathbf V(\tau)\) to P. Doing so gives us the following inertia tensor \(\mathcal I\) in which we sum over particles of rest masses m at spacetime positions r.
\(\mathcal I = {\Large\sum}{\gamma m \left[\begin{array}{cccc|ccc|ccc} -1 & 0 & 0 & 0 & 0 & 0 & 0 & -x & -y & -z \\ 0 & 1 & 0 & 0 & 0 & z & -y & -ct & 0 & 0 \\ 0 & 0 & 1 & 0 & -z & 0 & x & 0 & -ct & 0 \\ 0 & 0 & 0 & 1 & y & -x & 0 & 0 & 0 & -ct \\ \hline 0 & 0 & -z & y & y^2 + z^2 & -xy & -zx & 0 & ctz & -cty \\ 0 & z & 0 & -x & -xy & z^2 + x^2 & -yz & -ctz & 0 & ctx \\ 0 & -y & x & 0 & -zx & -yz & x^2 + y^2 & cty & -ctx & 0 \\ \hline -x & -ct & 0 & 0 & 0 & -ctz & cty & c^2t^2 - x^2 & -xy & -zx \\ -y & 0 & -ct & 0 & ctz & 0 & -ctx & -xy & c^2t^2 - y^2 & -yz \\ -z & 0 & 0 & -ct & -cty & ctx & 0 & -zx & -yz & c^2t^2 - z^2 \end{array}\right]}\)
This is a symmetric matrix, meaning that the 45 entries below the diagonal are equal to the 45 entries above the diagonal. Of those 45 entries, 21 are always zero, leaving 34 meaningful numbers when we include the entries on the diagonal. Ignoring sign, there are several repeated values, and it turns out that only 17 of the 100 entries in the full matrix are unique.
With this inertia tensor \(\mathcal I\), we can write \(\mathbf P = \mathcal I\,\mathbf V(\tau)\). The upper-left \(4 \times 4\) portion of \(\mathcal I\) corresponds to the linear momentum, and the Minkowski metric is clearly visible. The classical \(3 \times 3\) inertia tensor can be found in the center of \(\mathcal I\), and it corresponds to the angular momentum. The parts of \(\mathcal I\) not in the upper-left \(7 \times 7\) portion involve the dynamic mass moment.
I know a lot of details have been skipped over in this post, but the goal for now has been only to write about the existence of \(\mathcal I\). As my own understanding of the physical interpretation of all this develops further, I will be certain to write much more.