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On the real Klein-Gordon field solution space with conserved symplectic form , a compatible real-linear map obeys , preserves , and makes positive definite. Its eigenspace in the complexified space has positive Klein-Gordon inner product with the convention . It defines the one-particle Hilbert space and a free-field Fock vacuum.
On a scalar, the covariant wave operator is . For the metric signature it has the flat-space form . This divergence expression is convenient for separation of variables and for integrating the Klein-Gordon equation by parts.
For real Klein-Gordon field solutions with suitable support or boundary conditions, is a conserved symplectic form. The field equation makes its current divergence-free. Its complexification gives , the Klein-Gordon inner product. A compatible complex structure on the Klein-Gordon solution space is additional data needed for a particle representation.
For a twice differentiable position away from the origin, write with radial unit vector . Its unit-length constraint gives and , separating radial and directional contributions to speed and kinetic energy. The identity is kinematic and does not require a particular force law.
A degenerate Killing horizon has zero surface gravity. In the radial quotient of an extremal black hole, a double horizon root produces a pole in the tortoise coordinate rather than the logarithm of a simple root. Such a horizon has no bifurcation surface of the usual nondegenerate type.
For a Rademacher sum in a normed vector space, any two fixed positive moments of its norm are comparable with constants depending only on those moments. The sharp Rademacher second-moment inequality gives the sharp first-to-second moment comparison for arbitrary coefficients.
The irreducible mass of a four-dimensional uncharged Kerr black hole is . Its mass satisfies . Hawking's area theorem prevents a classical process from decreasing under the theorem's energy and global hypotheses.
Ingoing Kerr coordinates cancel the explicit inverse- singularities of the Boyer-Lindquist coordinates metric. They give a smooth, nondegenerate Kerr metric at the future outer horizon when . Its stationary and axial Killing vector fields are and . The advanced chart need not cover every horizon branch of the maximal extension.
The outer Killing horizon of a Kerr black hole is generated by , with the stationary Killing vector field normalized at infinity and the axial generator having -periodic orbits. In ingoing Kerr coordinates its generators satisfy . This is the horizon's angular velocity relative to the nonrotating asymptotic stationary frame.
For the massless Klein-Gordon equation on a Kerr black hole, the covariant wave operator separates into radial and angular ordinary differential equations. With and angular separation constant , the radial equation is . The angular equation is . No extra radial factor is included in the displayed ansatz.
An orbit about a star. Dust on such an orbit can experience radiation pressure and Poynting–Robertson drag, in addition to gravitational perturbations.
An orbit about a planet. Stellar tides, radiation pressure and other perturbations can limit its stability; a sufficiently small semi-major axis compared with the Hill sphere permits a local approximately planet-centred Kepler orbit.
For an attractive inverse-square force, , where is true anomaly, is orbital eccentricity, and is the semilatus rectum. The specific angular momentum relation converts this geometric equation into an angular evolution law.
In the centre-of-mass frame of two Newtonian point masses, total orbital energy is reduced mass times specific orbital energy: . The factor multiplies both kinetic and potential contributions, while is the relative gravitational parameter.
For unit mass in , conservation of angular momentum splits the velocity norm into . Conservation of energy then gives . It separates radial dynamics from the reconstruction of the orbital angle.
For an instantaneous in-plane change of velocity of magnitude on a circular orbit, with angle measured from the forward tangential direction, the new specific orbital energy gives . The specific angular momentum gives . The signed semi-major axis becomes negative for an unbound hyperbolic Kepler orbit.
For positive chemical concentration , taking makes the chemotaxis drift velocity . Cells then respond to a relative chemical gradient. This singular response can remain finite where the concentration tends to zero if its logarithmic derivative has a finite limit.
The neutral flavour states and can mix through second-order weak interactions. Box diagrams with W bosons and internal up-type quarks induce a strangeness-changing-by-two four-quark operator. The GIM mechanism removes its flavour-independent part.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact