Black hole thermodynamics consists of four laws that mirror the ordinary laws of thermodynamics: a zeroth law of uniform surface gravity, a first law relating mass to horizon area, a second law of non-decreasing horizon area, and a third law forbidding zero surface gravity in finite steps. Together they connect black hole geometry to entropy…
Impedance is the AC (alternating current) equivalent of resistance. It is a complex number, Z = R + jX, combining resistance R and reactance X, and it is used in the AC form of Ohm’s Law, V = IZ, where voltage and current are represented as complex phasors. Impedance is measured in ohms (Ω) and,…
Mass inflation is the phenomenon in which the internal mass parameter of a rotating or charged black hole grows without bound near the Cauchy horizon, driven by the infinite blueshift of infalling radiation. This causes spacetime curvature to reach Planckian scales on a spacelike surface just inside the black hole, where classical general relativity breaks…
Potential energy is the negative of the work done by a conservative force on an object, representing mechanical energy stored due to position within a force field. It is a scalar quantity with the same dimensions as energy, ML²/T², measured in joules (J). Common forms include gravitational, elastic (spring), electric, and magnetic potential energy, each…
What Is Moment of Inertia? Moment of inertia is a rigid body property that relates torque to rotational acceleration, playing the same role that mass plays in Newton’s second law for linear motion. It has units of mass times distance squared (M·L²) and is always defined with respect to a specific axis, meaning the same…
Definition / Summary This article summarizes the first-order (linear) polynomial equation in one variable, its solution, and natural extensions to matrices and operators. Definition A first-order polynomial (linear) equation in one variable has the general form [tex]Mx + B = 0[/tex], where [tex]x[/tex] is the variable, [tex]M[/tex] and [tex]B[/tex] are constants, and [tex]M \neq 0[/tex]….
Definition / Summary Significant figures (commonly called “sig figs”) are the digits in a number that are considered when rounding a value to reflect its precision. They show how accurately a quantity is known. Examples: Rounding to significant figures For example, the number 789.001 rounded to 6, 5, 4, 3, 2 and 1 significant figures…
A fibre bundle is a mathematical structure consisting of a total space that looks locally like a simple product of two spaces but may have a different global shape. The formal definition uses four pieces of data written (E, B, π, F): a total space E, a base space B, a projection map π, and…
Real numbers are the complete set of values on the continuous number line, denoted ℝ, and they include both rational numbers (fractions and repeating or terminating decimals) and irrational numbers (non-repeating, non-terminating decimals such as √2, π, and e). Real numbers can be positive, negative, or zero, and they satisfy a completeness property that makes…
A parabola is a U-shaped curve where every point is equidistant from a fixed point called the focus and a fixed line called the directrix. Its standard form is y = ax² + bx + c, and it opens upward when a is greater than zero or downward when a is less than zero. Parabolas…
A Lie algebra is the vector space of infinitesimal generators attached to a Lie group at its identity element, equipped with a bilinear antisymmetric commutator operation. Lie algebras are typically easier to analyze than the Lie groups that generate them, and much of Lie group representation theory is built directly from the algebra rather than…
A tangent line or tangent hyperplane to a curve or surface at a given point shares the same gradient, or first-order linear approximation, as that curve or surface at that exact point. This concept extends from two-dimensional curves to n-dimensional surfaces, and can be derived either from an implicit equation using partial derivatives or from…
What Is a Virtual Particle? A virtual particle is a mathematical device, not a physical entity, that appears in perturbation expansions of the S-operator (the transition matrix used in quantum field theory to calculate interaction probabilities). Virtual particles never physically appear in an interaction. All possible virtual particles and their antiparticles occur together in the…
The Big Bang did not happen at a single point in space, so it has no center. Standard cosmological models based on general relativity treat the Big Bang as the origin of space and time itself, not an explosion occurring at one location within a preexisting space. Observations of the cosmic microwave background show the…
What Is the General Definition of Exponentiation? Exponentiation is defined for all real and imaginary exponents using the formula xy = ey ln(x), where ex is defined by the infinite series 1 + x/1! + x2/2! + x3/3! + … This definition replaces “repeated multiplication,” which only works for whole-number exponents, and extends naturally to…
Galaxy and quasar redshifts are classified as cosmological, not “intrinsic,” because they match predictions from general relativity’s expanding-universe solutions, because independent tests (quasar-host redshift matches, the Lyman-alpha forest, the Gunn-Peterson trough, and gravitational lensing geometry) all confirm the same picture, and because every specific observational claim for an alternative “intrinsic redshift” mechanism has been shown…
The observable universe has a radius of about 46 billion light-years, far larger than its age of roughly 14 billion years. This happens because space itself has been expanding since the Big Bang, so light traveling toward us covers a proper distance greater than ct would suggest under special relativity alone. General relativity, not special…
Heat is not something a body possesses. Heat is a transfer mechanism: the non-mechanical exchange of energy between a system and its surroundings that occurs because of a temperature difference. A body has internal energy; it does not “have heat.” This distinction matters because the first law of thermodynamics, ΔU = Q + W, treats…
Atoms are stable because electrons do not behave as classical orbiting particles. Quantum mechanics describes electrons using wavefunctions and stationary energy states (orbitals), which are time-independent and do not involve continuous acceleration, so the classical prediction that electrons must radiate energy and spiral into the nucleus does not apply. Key Takeaways Niels Bohr proposed his…
The Big Bang was not a black hole because it happened everywhere in space at once rather than at a single point, and because the near-perfect homogeneity of the early universe meant tidal forces were effectively zero everywhere. Black holes require an asymptotically flat spacetime and nonzero tidal forces around a localized singularity, conditions the…
All current astronomical observations are consistent with zero global rotation of the universe. Solar-system data set a model-independent upper limit of roughly 10−7 radians per year on cosmic rotation (Clemence 1957), while cosmic microwave background analyses push that limit far tighter, to as low as 10−15 rad/yr under some model assumptions (Barrow, Juszkiewicz & Sonoda…
Direct answer: No, the Heisenberg time-energy uncertainty relation does not violate conservation of energy in quantum mechanics. The relation describes limits on how precisely energy can be measured within a given time interval, or how much energy a measurement interaction can transfer to a system. It does not permit energy to appear from nowhere or…
Yes, 1 and 0.999… are exactly the same real number, not merely two numbers that happen to be extremely close. This surprises many people because 0.999… looks like a process of approaching 1 rather than a fixed value, but treated as a complete infinite decimal, it equals 1 exactly. Several independent, intuitive arguments confirm this,…
A perfectly rigid rod cannot send faster-than-light (FTL) signals because no such rod exists. Every real material transmits mechanical disturbances at a finite speed called the speed of sound in that material, and this speed is always far below the speed of light (c = 299,792,458 m/s), even in principle. Key Takeaways No material, however…
A mathematical proof is a logical argument, built only from definitions, axioms, and previously established results, that shows a statement is true in every case with no exceptions. Checking examples, no matter how many, is never enough. Proofs use deductive reasoning, and the main techniques are direct proof, proof by contradiction, and proof by induction….
Genuine science follows the scientific method: testable predictions, sound reasoning, and reproducible evidence. Pseudoscience borrows scientific-sounding language while ignoring these safeguards. The clearest warning signs include personal attacks replacing argument, theories built to be unfalsifiable, appeals to authority or faith instead of evidence, and misuse of technical jargon. Key Takeaways A scientific theory must be…