The Concept of Extremal Paths

For those studying physics, you may have encountered principles of least action or the concept of minimal time paths in optics. Common sayings like "the path of least resistance" resonate widely in both English and German-speaking cultures. The notion of an extremal path—whether it be the shortest or longest—feels instinctive, as many animals, such as gazelles evading predators, demonstrate this understanding. However, in the realm of geometry, defining these paths can present significant challenges.

In geometric terms, a path of extremal length—whether maximum, minimum, or an infimum/supremum—is termed a geodesic. This term originates from the Greek words for Earth and to divide, reflecting the historical context of surveying.

The renowned mathematician Carl Friedrich Gauß was tasked in 1828 with surveying Hannover, then a British territory. Although his measurements were not perfectly accurate, he advanced surveying techniques that laid the groundwork for non-Euclidean geometry. His cautious approach to publishing his findings allowed others, such as János Bolyai and Nikolai Lobachevsky, to gain recognition for their contributions to the field, though Gauß’s insights were foundational.

Central to both Riemannian and Pseudo-Riemannian geometry is the concept of a geodesic—the shortest distance between two points on a manifold. Pseudo-Riemannian manifolds are characterized by a pseudo metric, which can lead to nontrivial signatures.

Exploring Singular Hamiltonians in Geodesics

Attempting a Hamiltonian approach to identify the shortest path can lead to unexpected results, such as the trivial equation 0=0. Before we explore this anomaly in detail, let's understand the underlying issues intuitively. The definition of the shortest path as an action integral utilized in the Lagrangian Calculus of Variations is imprecise, complicating the naïve Hamiltonian approach.

Imagine navigating from point A to B in your car; you can choose various speed-time profiles, such as stopping for coffee or enjoying a scenic view. Regardless of these detours, when you arrive, you have still traveled the shortest route!

To derive our Lagrangian, we must extremize the integrand through the Euler-Lagrange Equation, incorporating the metric tensor and the velocity vector on the tangent space.

As we analyze the action integral, we can transform the path parameter, maintaining the same endpoints while altering the speed profile. The critical realization is that the speed profile remains a "dangling" degree of freedom, allowing for various detours along the shortest route.

A Cheeky Solution to the Hamiltonian Challenge

While the square root in our calculations poses a challenge, we might consider disregarding it altogether. By doing so, we can leverage the Cauchy-Schwarz inequality to simplify our problem. This leads to the conclusion that we can characterize all paths through transformations, reinforcing that in General Relativity, the physical path is one where an observer's proper time progresses uniformly.

Furthermore, the Hamiltonian formulation emerges naturally from this approach, paralleling Newtonian mechanics for a free particle. This analogy illustrates how geodesics in a Lorentzian manifold represent maximum proper time paths.

In summary, while the traditional definition of a geodesic in Riemannian or pseudo-Riemannian manifolds identifies the shortest path, it also provides an affine path parameterization, ensuring that observers experience their proper time uniformly.

The journey through geodesics intricately connects geometry with the fundamental principles of physics.

Geodesics in Action: Visualizing the Concepts

The first video, "Sometimes The Shortest Distance Between Two Points is NOT a Straight Line: GEODESICS" by Parth G, explores the nuances of geodesics and their importance in understanding the geometry of spacetime.

The second video, "Spacetime Diagrams: An Easy Way to Visualize Special Relativity" by Parth G, offers an accessible way to visualize the complexities of special relativity and its connection to geodesics.