An earthquake isn't just a point on a map. It's a complex physical process: a rupture that starts somewhere and spreads, sometimes for hundreds of kilometers. To understand this, we need to model the physics of a fault as it breaks. Seismologists use two main approaches: quasi-dynamic and elastodynamic modeling.

Quasi-dynamic models treat the process slowly. They focus on the balance of forces on the fault, assuming things change so gradually that inertia—the resistance to changes in motion—can be ignored. This is useful for studying the long-term stress buildup over years or centuries, but it can't capture the violent, split-second reality of an earthquake.

For that, we need elastodynamic models. These models embrace the chaos. They solve the full wave equation, accounting for how the slipping fault radiates seismic waves and how those waves, in turn, affect the fault's ongoing rupture. This is where the 'dynamic' part of rupture dynamics comes from. Inertial effects are critical; the ground's own resistance to acceleration and deceleration shapes how the earthquake unfolds.

At the heart of any rupture model is friction. The simple idea of a constant coefficient of friction, where resistance is just proportional to the normal stress holding the two sides together, isn't enough to explain why faults stick and slip. Real fault friction is far more complex.

Rate-and-state friction laws capture two key observations. First, there's a direct effect: when a fault starts slipping faster, its frictional resistance momentarily increases. Second, there's an evolution effect: as the surfaces slide against each other, the 'state' of the fault contact changes, which in turn alters the friction. This state can be thought of as the 'maturity' or 'healing' of the contact points between the two rock surfaces. These competing effects determine whether a fault slip will accelerate into a full-blown earthquake or die out.

When the parameter $a-b$ is negative, the system is unstable. A small increase in slip velocity ultimately leads to a decrease in friction, causing the slip to accelerate further. This is called velocity-weakening behavior, and it's the fundamental ingredient for earthquake instability.

Once a rupture starts, how fast does it spread? The speed is limited by how quickly information about the stress change can travel through the rock. This 'speed limit' is the shear-wave velocity ($V_s$), the speed at which seismic S-waves propagate. For a long time, it was thought that ruptures couldn't travel faster than this speed.

However, we now know that some earthquakes can break this rule. Under certain conditions, a rupture can jump to a 'super-shear' speed, traveling faster than the S-waves it generates. This is analogous to a sonic boom. When this happens, the rupture front is preceded by a powerful shock wave, which can cause much more intense shaking. Super-shear ruptures are more likely on long, straight, and relatively simple fault segments.

Earthquakes are fundamentally about energy. Over centuries, elastic strain energy builds up in the rock around a fault. During an earthquake, this stored energy is suddenly released. But where does it go?

A key concept is the stress drop ($\\\Delta \sigma$), which is the difference between the stress on the fault before the earthquake and the stress after. This change in stress, integrated over the fault area, determines the total energy released. This energy is partitioned into two main categories: radiated energy, which travels outwards as seismic waves, and fracture energy, which is the work done to break the rock and overcome friction at the rupture front.

This leads us to the physical basis for earthquake magnitude: the seismic moment. It's the most reliable measure of an earthquake's total size because it's directly tied to the physical properties of the fault slip.

Unlike older magnitude scales, the seismic moment isn't just an empirical number; it's a physical quantity measured in Newton-meters. It tells us the total work done by the fault. By studying seismic waves recorded around the world, we can work backward to figure out the seismic moment and infer what happened on the fault itself. This process, known as dynamic source inversion, allows us to create detailed maps of how slip, stress drop, and rupture velocity varied across the fault plane during an earthquake, giving us an unprecedented view into the engine of the Earth.