This article was authored by several Physics Forums members with PhDs in physics or mathematics.
Overview Black hole thermodynamics is summarized by four laws that closely parallel the laws of ordinary thermodynamics. Below I present each law, the key formulas (in geometric and SI units where relevant), and brief physical explanations with links to background material. Figure: Summary illustration of black hole thermodynamics. (Image: Physics Forums Insights) The Zeroth Law…
Definition / Summary The impedance of a load (a combination of components) in an AC circuit is a complex number [itex]Z = R + jX[/itex], where [itex]R[/itex] is resistance and [itex]X[/itex] is reactance. Polar form The same impedance can be written in polar form: [itex]Z = |Z|e^{j\phi}[/itex], or as the phasor [itex]|Z|\angle\phi[/itex]. Units and frequency…
Definition and summary Abstract from Poisson and Israel (1990), “Internal structure of black holes”: “The gravitational effects associated with the radiative tail produced by a gravitational collapse with rotation are investigated. It is shown that the infinite blueshift of the tail’s energy density occurring at the Cauchy horizon of the resulting black hole causes classically…
Definition / Summary Potential energy is the negative of the work done by a conservative force. It represents stored mechanical energy associated with position in a force field. The work–energy theorem relates work and kinetic energy. For conservative forces this leads to an energy conservation statement in which mechanical energy plus potential energy is constant….
Definition / Summary The moment of inertia is a property of a rigid body that relates torque (rotational force) to rotational acceleration in the same way mass relates linear force to linear acceleration. It has units of mass times distance squared (M·L²) and is always defined with respect to a specific axis: the same rigid…
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. For example, the number 789.001 rounded to 6, 5, 4, 3, 2 and 1 significant figures becomes: 6 sig figs: 789.001…
Definition / Summary Fibre bundle — intuitively, a fibre bundle is a space E that locally looks like a product B × F but may have a different global topological structure. This article uses the British spelling “fibre”. Extended explanation Definition A fibre bundle is the data (E, B, π, F), where E, B and…
Definition of real numbers Real numbers are the set of all values that can appear on the continuous number line. They include rational numbers (fractions and terminating or repeating decimals) and irrational numbers (non-repeating, non-terminating decimals). The real numbers are usually denoted by the symbol ℝ. They can be positive, negative, or zero, and they…
What is a Parabola? A parabola is a U-shaped curve that appears frequently in mathematics, physics and engineering. It is a conic section defined by a simple geometric property: every point on the curve is equidistant from a fixed point (the focus) and a fixed line (the directrix). Parabolas are symmetric and can be described…
Definition / Summary A Lie algebra (pronounced “Lee”) is the tangent-space algebra of a Lie group at its identity element. Concretely, it is a vector space of generators that describes infinitesimal transformations near the identity. Lie algebras carry a binary bilinear antisymmetric operation called the commutator, and they are closed under that operation: the commutator…
Definition / Summary The tangent to a curve in a plane at a particular point has the same gradient as the curve at that point. More generally, the (n−1)-dimensional tangent hyperplane to an (n−1)-dimensional surface in n-dimensional space at a particular point has the same gradient (the same first-order linear approximation) as the surface at…
Definition / Summary Virtual particles are a mathematical device used in perturbation expansions of the S-operator (transition matrix) for interactions in quantum field theory. No virtual particle physically appears in the interaction: all possible virtual particles and their antiparticles occur together in the mathematics and are removed by integration over their momenta. In the coordinate-space…
Did the Big Bang Have a Center? What the standard models say According to standard cosmological models based on general relativity — and which agree well with observations — time and space did not exist before the Big Bang, nor at the moment of the Big Bang. In these models the Big Bang is a…
Introduction This is how both you and I learned exponents back in elementary school: 53 = 5 × 5 × 5 Using repeated multiplication you can raise anything to a whole-number power. But what about fractions, negatives, irrationals, or imaginary exponents? Fractional and negative exponents are introduced in algebra: 5-3 = 1 / 53 and…
How do we know that the redshifts of galaxies and quasars are cosmological and not “intrinsic”? Evidence for the cosmological interpretation Historical context The purpose of this FAQ entry is to explain why nonstandard interpretations of redshifts are not viable; it is not a full history of how standard cosmological models developed. Below is a…
The radius of the observable universe is about 46 billion light-years, which is considerably greater than its age of about 14 billion years. The radius of the observable universe is defined by the greatest distance from which light would have had time to reach us since the Big Bang, so you might think that it…
One of the most frustrating misconceptions in thermodynamics centers on the actual definition of heat. Many science and engineering students refer to a body as possessing heat, but as we will show below, that phrasing is misleading and technically incorrect. First Law of Thermodynamics The first law is a statement of the principle of conservation…
Atoms and Coulomb forces If atoms were described only by Coulomb attraction, the electron and nucleus would attract each other and, according to classical reasoning, no stable atoms could exist. Niels Bohr (1913) proposed an improved model in which electrons occupy discrete circular orbits, each corresponding to a specific energy level. Bohr’s model introduced quantized…
Why the Big Bang Is Not a Black Hole In the early universe, the matter was gathered together at very high density — so why wasn’t it a black hole? Big Bang vs. a Local Explosion The first thing to understand is that the Big Bang was not an explosion that happened in one place…
Can we tell whether the universe is rotating? If you believe wholeheartedly in Mach’s principle, there is no way to test empirically for the rotation of the universe as a whole, since there would be nothing else for it to rotate relative to. General relativity (GR), however, is not very Machian: GR provides several local,…
The short answer is: No — there is no violation. The longer answer Given the probabilistic aspect of quantum theory, what do we mean now by “conservation of energy”? Two ways energy conservation appears in quantum theory In quantum theory, energy conservation is commonly expressed in two different but compatible ways. First: a state with…
Why do people say 1 and 0.999… are equal? Aren’t they two different numbers? No — 1 and 0.999… really are the same number, although that can feel counterintuitive at first. Below are several accessible (non‑rigorous) proofs, followed by answers to common objections. I preserved the original voice and meaning while improving structure, clarity, and…
Why a rigid rod can’t send faster-than-light signals One common proposal for achieving faster-than-light (FTL) communication is to use a long, perfectly rigid object and mechanically send signals to the other end by pushing, pulling, or tapping it. For instance: a hypothetical rigid rod linking two people several light‑years away. The fundamental idea is that…
Introduction This FAQ is about proofs. Proofs are central to mathematics, and writing proofs is a skill many people find hard to master. There are two separate skills to develop: finding the proof and communicating the proof. This article focuses on finding proofs, but it also gives guidance on communicating proofs clearly. What is a…
It often feels like we’re being swamped with “science” these days — but much of it is not real science. Genuine science follows the scientific method, uses sound reasoning and robust experimental evidence, and makes testable predictions. Pseudoscience, by contrast, borrows the language of science while ignoring its methods and safeguards. Sadly, many sensational claims…