This article was authored by several Physics Forums members with PhDs in physics or mathematics.
Definition/Summary The four laws of black hole thermodynamics are as follows… The Zeroth Law Surface gravity [itex](\kappa)[/itex] is constant over a black holes event horizon. The First Law ‘This law deals with the mass (energy) change, dM when a black hole switches from one stationary state to another.’ The following (in natural units) applies- [tex]dM=\frac{\kappa}{8\pi}\,dA\,+\,\Omega\,dJ\,+\,\Phi\,dQ[/tex]…
Definition/Summary The impedance of a load (a combination of components) in an AC current is a complex number [itex]Z\ =\ R+jX[/itex] where [itex]R[/itex] is the resistance of the load and [itex]X[/itex] is its reactance. It can also be written in polar form: [itex]Z\ =\ |Z|e^{j\phi}[/itex], or as the phasor [itex]|Z|\angle\phi[/itex]. Impedance is the AC equivalent…
Definition/Summary Abstract from Poisson and Israel’s 1990 paper, ‘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 unbounded…
Definition/Summary Potential energy is simply another name for (minus) the work done by a conservative force. Since the work-energy theorem states that change in energy minus work done is constant, that means that for a conservative force, energy plus potential energy is constant. For example, a object of mass [itex]m[/itex] moving a height [itex]h[/itex] and…
Definition/Summary The moment of Inertia is a property of rigid bodies. It relates rotational force (torque) to rotational acceleration in the same way that mass relates ordinary (linear) force to ordinary acceleration. Moment of Inertia has dimensions of distance squared times mass ([itex]ML^2)[/itex]. The moment of Inertia is always relative to a given axis. The…
Definition/Summary A first-order polynomial equation in one variable, its general form is [itex]Mx+B=0[/itex] where x is the variable. The quantities M, and B are constants and [itex]M\neq 0[/itex]. Equations [tex]Mx+B=0[/tex] Extended explanation Since [itex]M\neq 0[/itex] the solution is given by [tex]x=-B/M\;.[/tex] The variable x does not have to be a number. For example, x and…
Definition/Summary Significant figures (commonly called “sig figs”) are the number of figures (digits) included when rounding-off a number. For example, 789.001 to 6 5 4 3 2 and 1 sig figs is 789.001 789.00 789.0 789 790 and 800 respectively (note the presence and absence of “0”s in 5 4 and 3 sig figs). By…
Definition/Summary Intuitively speaking, a fibre bundle is space E which ‘locally looks like’ a product space B×F, but globally may have a different topological structure. Extended explanation Definition: A fibre bundle is the data group [itex](E, B,\pi, F)[/itex], where [itex]E, B[/itex], and [itex]F[/itex] are topological spaces called the total space, the base space, and the…
Definition of real numbers Real numbers are a comprehensive set of numbers that encompasses all possible values on the number line. They include both rational and irrational numbers and are typically denoted by the symbol ℝ. Real numbers can be positive, negative, or zero, and they can be represented as finite decimals, recurring decimals, or…
What is a Parabola? A parabola is a U-shaped curve in mathematics that is defined by a specific set of points. It is a fundamental geometric shape that appears in various mathematical and real-world contexts. Parabolas have a distinct symmetry and are defined by a simple mathematical equation. Key characteristics of a parabola include: Symmetry:…
Definition/Summary A Lie algebra (“Lee”) is a set of generators of a Lie group. It is a basis of the tangent space around a Lie group’s identity element, the space of differences between elements close to the identity element and the identity element itself. Lie algebras include a binary, bi-linear, anti-symmetric operation: commutation. The commutator…
Definition/Summary The tangent to a curve in a plane at a particular point has the same Gradient as the curve has 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 as the surface has at that point. So if [itex]A\,=\,(a_1,a_2,\cdots…
Definition/Summary Virtual particles are a mathematical device used in perturbation expansions of the S-operator (transition matrix) of interaction in quantum field theory. No virtual particle physically appears in the interaction: all possible virtual particles, and their antiparticles, occur equally and together in the mathematics and must be removed by integration over the values of their…
According to standard cosmological models, which are based on general relativity and are found to agree well with observations, time and space did not exist before the Big Bang — or even at the time of the Big Bang, which is a point where the theory breaks down because various quantities (such as temperature and…
This Insight is part of our “Young Authors” series, where talented young students showcase their knowledge. This is how both you and I learned it back in elementary school. $$5^3 = 5*5*5$$ Using this you can raise anything to any power- as long as that power is a whole number. What about fractions and negative…
How do we know that the redshifts of galaxies and quasars are cosmological and not “intrinsic?” Evidence for the cosmological interpretation The purpose of this FAQ entry is to explain why nonstandard interpretations of redshifts are not viable, not to explain from scratch how standard cosmological models were arrived at. The following is only 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/engineering students often refer to a body as to be possessing heat, but as we shall see, to do so is completely nonsensical. We start by examining the first law and defining each of its terms. The first law…
Atoms and Coulomb Forces If one describes atoms using only the Coulomb forces, the electron and the nucleus will attract each other and no stable atoms could exist. Obviously, this is not the case. Niels Bohr was the first (1913) to propose a better model, which consisted of electrons moving around the nucleus in circular…
In the early universe, the matter was gathered together at very high density, so why wasn’t it a black hole? The first thing to understand is that the Big Bang was not an explosion that happened at one place in a preexisting, empty space. The Big Bang happened everywhere at once, so there is no…
If you believe wholeheartedly in Mach’s principle, then there is no way to test empirically for rotation of the universe as a whole, since there is nothing else for it to be rotating relative to. However, general relativity is not very Machian, and it offers a variety of ways in which an observer inside a…
The short answer is: no, there’s no violation. The longer answer can be this: Given the probabilistic aspect of quantum theory, what do we mean now, by “conservation of energy”? In quantum theory, it can be expressed in two different ways. The first way is this: A state with a precisely known energy will always…
Why do people say 1 and 0.999… are equal? Aren’t they two different numbers? No, they really are the same number, though this is often very counterintuitive to many beginning students. Here are some non-rigorous proofs that 1=0.999…: Proof #1 For any two unequal numbers, there is always another number in between them. (That is…
One common proposal for achieving faster than light 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 lightyears away. The fundamental idea is that when one end is moved the other…
This FAQ is about proofs. Proofs are central to mathematics, and writing proofs is for many people a skill that is hard to master. There are actually two separate skills that one must master: finding the proof and communicating the proof. We will try to focus on the former here, but we will try to…
It seems nowadays that we are getting swamped with science. But not the science of the valid kind. Science where people follow the scientific method and are based on sound reasoning, good experimental evidence and makes check-able predictions. No, sadly, most of the so-called science are pseudoscience, evil heinous impostors that exploit the public’s curiosity…