Relativity is not a game
Interviewee: Martin Roelfs, Flanders Make, Doctor of Science: Physics, University of Groningen
During your talk, you presented Projective Geometric Algebra (PGA) as a framework for understanding special relativity. What does this approach clarify compared to the traditional formulation?
A central advantage lies in how naturally it captures transformations in spacetime, particularly Lorentz boosts – the operations that relate observers moving at different velocities. In a conventional course, these transformations are typically introduced through hyperbolic trigonometry and matrix calculations. While formally correct, this quickly becomes computation-heavy, and the underlying geometry can fade into the background.
PGA shifts the focus entirely. Instead of working through layers of algebra, one reasons directly in terms of geometric objects in spacetime – for example, the ‘line’ an object follows. Transformations are then expressed through simple algebraic products that directly encode this geometry. The result is immediate: rather than deriving formulas step by step, the transformation emerges from the geometry itself. The emphasis is no longer on calculation, but on understanding what is happening geometrically.
Does this mainly improve intuition, or does it also simplify the mathematics itself?
It does both. Conceptually, it provides a clearer picture, but it also streamlines the mathematics in a very concrete way. The key idea in Geometric Algebra is that the structure is built into the elements themselves, rather than imposed through external constraints. In more traditional approaches, one often must ensure manually that equations satisfy certain conditions or produce the correct transformations. In PGA, those properties are inherent: the algebra guarantees that operations behave correctly.
This reduces the likelihood of errors and allows expressions to remain at a higher, more geometric level. An instructive comparison is with programming: instead of explicitly computing sines and cosines, one writes down the geometric relationships, and the algebra handles the rest. In fact, these trigonometric functions need not appear at all – the transformations arise directly from the algebraic structure. So, the simplification is not just conceptual; it is computational, as well.
Should special relativity then be taught using Geometric Algebra instead of the traditional approach?
There is a balance to strike. There is still value in working through the traditional calculations at least once, as it builds a concrete understanding of how everything operates at a lower level. However, it would be beneficial to introduce the geometric viewpoint much earlier and avoid losing sight of it.
In practice, a hybrid approach makes sense: students can still perform some calculations by hand to develop algebraic intuition, but they do not need to remain at that level indefinitely. The geometric framework can take over much sooner, providing both clarity and efficiency without abandoning rigour.
Is the main obstacle to such an approach the need to fall back on traditional methods?
The deeper issue is that Geometric Algebra itself is not yet widely part of standard curricula. Students are typically trained in Linear Algebra and trigonometry, but not in Projective or Geometric Algebra. As a result, introducing PGA within a special relativity course requires teaching an entirely new mathematical language alongside the physics.
If this foundation were introduced earlier in education, the situation would change significantly. Special relativity could then be taught directly within this geometric framework from the outset, without the need for translation between formalisms. Developments such as the introduction of PGA courses at Howest point in a promising direction, suggesting that this shift may gradually become more feasible.
For someone already familiar with special relativity in the traditional sense, what new insights does PGA offer?
One of the most striking differences is the role of the observer. In standard treatments, calculations are often tied to a specific reference frame, giving the observer a somewhat privileged status. With PGA, this dependence largely disappears. One can compute relationships between events directly in spacetime without switching perspectives. The framework is inherently covariant: the laws and operations take the same form regardless of the chosen frame.
Additionally, PGA provides a richer set of geometric operations. Beyond the equivalent of matrix multiplication, one can, for instance, combine two events to form a line or intersect lines to obtain a point – all within the same algebraic system. These operations transform correctly under changes of frame, preserving their physical meaning.
This expanded toolkit allows physical laws to be expressed more compactly and geometrically than in traditional formalisms. Rather than working with a limited set of algebraic operations, one has access to a broader, intrinsically geometric language. The result is not just a reformulation, but a shift in perspective: special relativity becomes less about manipulating equations and more about understanding the structure of spacetime itself
Human interest interview conducted by Maxime Arnolis
Technical interview conducted by Anthony Pirolo