Chapter 1: The Nature of Laws in Physics

In the realm of physics, the laws that govern our universe are crucial for understanding how everything operates. One particular theory that emerged in 1905 revolutionized scientific thought: Albert Einstein's special relativity. This theory presents astonishing predictions that often seem paradoxical, becoming a staple of popular culture. For instance, the length of an object is influenced by its speed, and even the concept of time is not absolute. The faster an object moves, the slower it ages.

To illustrate this, consider the center of our galaxy, which is 50,000 light years distant. Light, traveling at its maximum speed, takes 50,000 years to reach it. However, a spacecraft moving at 99.9999% of light speed could make the journey in approximately 70 years. This raises questions and apparent contradictions, but the logic and implications of special relativity are consistent. The crew aboard the spacecraft experiences only 70 years, while on Earth, over 50,000 years pass. Notably, special relativity stands as one of the most precisely validated theories in physics. So, why is it labeled as not a law of nature?

Let’s clarify what is meant by a "law of nature." It typically refers to a principle that dictates how entities in the universe behave. From this viewpoint, Maxwell's equations in electrodynamics qualify as a law of nature because they specify how electromagnetic fields are generated and transformed over time. Yet, special relativity transcends this definition. It is not merely a law; it serves as a meta-law, dictating how all laws of nature should function. Any proposed law that contradicts even a single requirement of special relativity cannot be considered valid or fundamental.

The two core principles of special relativity are: 1. All laws of nature must maintain the same structure across all inertial frames (i.e., coordinate systems moving at constant velocities). 2. The speed of light remains constant in all inertial frames.

The first principle seems reasonable for a meta-law, while the second, although experimentally validated multiple times, is less intuitive. Together, these principles impose strict constraints on the potential form of any law of nature. This is captured in a sophisticated 4-dimensional framework known as Lorentz covariance, which warrants further exploration.

Section 1.1: Examples of Non-Fundamental Laws

To illustrate how special relativity eliminates certain laws of nature, let’s examine two prominent examples: the Schrödinger equation and Newton’s law of gravitation.

Subsection 1.1.1: The Schrödinger Equation

The Schrödinger equation governs the behavior of particles in quantum mechanics. It is a wave equation that yields results in alignment with experimental data for systems like the hydrogen atom. However, despite its apparent success, it does not comply with special relativity's requirements. In special relativity, time and space are treated equally, but in the Schrödinger equation, they are treated differently: time appears with a first derivative, whereas space is represented with second derivatives. Hence, this equation cannot be regarded as fundamental. Quantum field theory emerged to address this limitation, indicating that while the Schrödinger equation has its place, it is confined to specific scenarios involving low-energy, slowly moving particles.

Subsection 1.1.2: Newton's Law of Gravitation

Now, consider Newton's law of gravitation, which states that the force (F) acting on a mass (m) at a distance (r) from another mass (M) is defined by the equation:

However, if mass M starts to move away from mass m, special relativity dictates that no information can travel faster than light. According to Newton's law, mass m would instantly perceive the movement of mass M, implying that information would need to travel at infinite speed—an impossibility. Thus, this law cannot be deemed fundamental. The advanced theory that replaced Newton's law is Einstein's general relativity, which offers a more comprehensive understanding.

Chapter 2: The Unique Position of Special Relativity

In summary, special relativity is not merely a law of nature that dictates how objects behave; it functions as a meta-law that establishes the standards all laws of nature must meet to be considered valid. Through examples like the Schrödinger equation and Newton's law of gravitation, we see how their failure to comply with special relativity’s criteria disqualifies them as fundamental laws. This ability to delineate the boundaries of what constitutes a true law of nature is what sets special relativity apart from other theories.

The first video, "Marco Giovanelli: Special Relativity as a Theory of Principles," explores the foundational concepts of special relativity and its implications.

The second video, "Why The Theory of Relativity Doesn't Add Up (In Einstein's Own Words)," delves into Einstein's perspective on relativity and its complexities.