Black Holes

The equivalence principle is a cornerstone of general relativity. It posits that the effects of a uniform gravitational field are indistinguishable from the effects of uniform acceleration. This means that an observer in a windowless, free-falling elevator experiences weightlessness, just like an observer in deep space, far from any gravitational source, forming the conceptual basis for gravity as a geometric phenomenon.

A topological space is an ordered pair [latex](X, \tau)[/latex], where [latex]X[/latex] is a set and [latex]\tau[/latex] is a collection of subsets of [latex]X[/latex], called open sets, satisfying three axioms: 1) The empty set [latex]\emptyset[/latex] and [latex]X[/latex] itself are in [latex]\tau[/latex]. 2) The union of any number of sets in [latex]\tau[/latex] is also in [latex]\tau[/latex]. 3) The intersection of any finite number of sets in [latex]\tau[/latex] is also in [latex]\tau[/latex].

General relativity provided the first accurate explanation for the anomalous precession of Mercury's perihelion. Newtonian gravity could not fully account for the slow, gradual shift in the orientation of Mercury's elliptical orbit. Einstein's theory correctly predicted the missing 43 arcseconds per century, attributing it to the curvature of spacetime around the Sun, a major early triumph for the theory.

A black hole is a region of spacetime where gravity is so strong that nothing—no particles or even light—can escape. The boundary of this region is called the event horizon. Predicted by general relativity as a solution to the field equations, a black hole is the result of the complete gravitational collapse of a massive star.

General relativity predicts that the path of light is bent by gravity. When light from a distant source passes a massive object like a galaxy or a star, its path is deflected. This phenomenon, known as gravitational lensing, can magnify, distort, or create multiple images of the background source, acting like a cosmic telescope for observing the distant universe.

All quantum entities, such as photons and electrons, exhibit both particle and wave properties. Depending on the experimental setup, they can behave like a localized particle or a distributed wave. The de Broglie hypothesis states that any particle with momentum [latex]p[/latex] has an associated wavelength [latex]\lambda = h/p[/latex], where [latex]h[/latex] is Planck's constant.

This is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is a linear partial differential equation for the wavefunction, [latex]\Psi(x, t)[/latex]. The time-dependent version is [latex]i\hbar\frac{\partial}{\partial t}\Psi = \hat{H}\Psi[/latex], where [latex]\hat{H}[/latex] is the Hamiltonian operator, representing the total energy of the system.

The boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant. Introduced by Ludwig Prandtl, this concept simplifies fluid dynamics problems by dividing the flow into two regions: the thin boundary layer where viscosity dominates and the outer region where inviscid flow theory can be applied.

The von Mises yield criterion predicts that yielding of a ductile material begins when the second deviatoric stress invariant, [latex]J_2[/latex], reaches a critical value. It is often expressed in terms of the von Mises stress, [latex]\sigma_v[/latex], a scalar value that must be less than the material's yield strength, [latex]\sigma_y[/latex]. Yielding occurs when [latex]\sigma_v = \sigma_y[/latex].

A key prediction of general relativity is that time passes at different rates in different gravitational potentials. A clock in a stronger gravitational field (closer to a massive object) will tick slower than a clock in a weaker field. This effect, known as gravitational time dilation, has been experimentally verified and is a crucial factor in modern technology like GPS.

The Einstein Field Equations (EFE) are a set of ten coupled, non-linear partial differential equations that form the core of general relativity. They describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy. The equation is concisely written as [latex]G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}[/latex], relating spacetime geometry to its energy-momentum content.

General relativity predicts that a massive, rotating object should 'drag' the fabric of spacetime around with it, a phenomenon known as frame-dragging or the Lense-Thirring effect. This means a gyroscope orbiting the rotating body will precess, not because of any applied torque, but because spacetime itself is being twisted by the body's rotation.

A Bingham plastic is a viscoplastic material that behaves as a rigid body at low stresses but flows as a viscous fluid at high stress. It is characterized by a yield stress, [latex]\tau_0[/latex], which is the minimum shear stress required to initiate flow. Below this threshold, it does not deform. Common examples include toothpaste, mayonnaise, and clay slurries.

Shear-thinning, or pseudoplasticity, is a non-Newtonian behavior where a fluid's viscosity decreases with an increasing rate of shear stress. Many common substances, like ketchup or paint, exhibit this property. At rest, they are thick, but they flow more easily when shaken, stirred, or spread. This is often modeled by the Ostwald–de Waele power-law relationship.

Thixotropy and rheopexy describe time-dependent changes in viscosity. A thixotropic fluid's viscosity decreases over time under constant shear stress (e.g., yogurt becomes runnier when stirred). A rheopectic (or anti-thixotropic) fluid's viscosity increases over time under constant shear (e.g., some lubricants). These effects are reversible; the fluid returns to its original state after a period of rest.