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Re

Actually, there is a move back to writing multiplication left to right. And while it is not wrong that this purely a matter of convention and symmetric within a narrow context, in a wider context, this notational choice acquires very real consequences. IMO, a major stumbling block to developing certain fields in mathematics has always been that, due to historical accidents like a seemingly innocuous notational choice made in a narrow context, it is impossible to find a notation which "behaves nicely" with

Some pros:

Re

ditto Lavinia, but even more emphatically! IMO, the principle reason why people start off wrong and get more and more confused when they try to learn, say, Lorentzian geometry, is that they haven't first studied curves in E^2, E^{1,1}, E^3, E^{1,2} and surfaces in E^3, E^{1,2}. And as just one example where classical surface theory continues to provide inspiration for topics of current interest, consider the classic problem of constructing uniform negative curvature surfaces in E^3, which has inspired many late 20th century developments related to the theory of solitons and is now being tied up with string theory and other physics stuff. See Rogers and Schief,

That is of course completely incorrect, and in fact the classical setting is a good place to explore differential forms. See the classic textbook by Flanders,

Re

This is a classic question closely related to a subject initiated by Erdos and Turan: the distribution of various properties (such as cycle decomposition, order) of randomly chosen elements of S_n, n large. A few hints can be found in section 14.4. of Bollobas,

Years ago I argued (in Wikipedia policy discussion pages) that although Wikipedia's model has proven very successful at growing an alleged "encyclopedia", it is fundamentally flawed wrt reliability, which rather munges the whole point of making the alleged "encyclopedia". So how to design a wiki which can grow rapidly while maintaining high quality? I suggested that the best approach may be to building a "universal on-line encyclopedia" may be to continuously aggregate and render in uniform "wikiskin" articles drawn from thousands of

In the past few years, some projects have appeared which follow my advice about the neccesity of restricting article creation/modification to recognized subject matter experts (the other half, about the essential role of the "editor" in the true sense of that word, which wikipedia.org has largely succeeding in debasing by conflating it with "author"). One problem here is that the existing solution to authenticating identity/credentials on-line, the gpg "web of trust", is underutilized in academic circles, which is tragic because this is hampering grass roots efforts such as the projects I am describing. Some projects have even attempted to emulate traditional peer review (Citizendium), although I tend to feel that simply restricting authorship to known individuals with good credentials, and restricting editorship to recognized leaders, may be enough, at least initially, to ensure good growth coupled with good quality. Other projects, such as the subwiki.org "wikispedias", have focused on the fact that it is possible to manage ("edit" in the true sense of the word) a specialist wikipedia (e.g. by ensuring reasonably coherent notation/terminology, making wise choices for internal linkings, resolving any differences of opinion on how to present scientific controversies); compare the chaos at Wikipedia.

I hesitate to suggest linking to the subwiki.org wikispedias in the public areas, because like some other promising projects, they are currently open to anyone, and could quickly be ruined. Also, I don't want to overload their servers. But it's a very promising project and I hope that if we cautiously spread the word to serious people, eventually the universities/government will fund the project to allow for servers keeping pace with growth. (It would be tragedy of they vanished behind paywalls.)

I love the fact that graduate students can produce hilarious essays like

But imagine the mess when the cranks discover that anyone can sign up and create/modify articles in this wiki, oh my! One of the nice things about keeping out the idgits is that authors can write with style, while still keeping within the bounds of terminological/notational conventions.

Given the importance in so many areas of mathematics of actions by large finite groups, and the properties of "generic" elements of large symmetric groups in particular, why is it so hard to find good information on-line? One might almost think there must some kind of suppression. And here's a surprise, maybe--- there

Re

Lavinia should have said "all two-dimensional Riemannian manifolds" to avoid possible confusion. It saves time in the end to try to write out a bit more to avoid confusion.

Hmm...some more possible future candidates for SA: lavinia, shoehorn, Martin Rattigan? (shoehorn and I seem to disagree on the value of open source, but hopefully that is not a serious conflict!)

Re

It's important to raise from specific to general when confronting an OP based upon multiple misconceptions. In this case, the most important misconception is that a black hole of mass m actively sucks stuff straight in like a powerful vacuum cleaner, rather than attract like any other object of mass m. In gtr, rougly speaking, the gravitational field itself gravitates, which turns out to mean that any object of mass m attracts other mass-energy just a bit more strongly than in Newtonian theory. This effect is neglible in must situations, so that a black hole of mass m interacts with other objects pretty much like any other object of mass m (think elliptical and hyperbolic trajectories as in Newtonian gravitation), unless the other object approaches it very closely. But because compact objects of mass m (neutron stars and black holes) are so much smaller than ordinary stars of mass m, another object can get much closer to a compact object and thus experience a stronger gravitational field, where the "extra attraction" becomes significant. That said, the answer the OP probably wanted is that, roughly speaking, in the range 2m < r < 6m, independent of mass m, the effects of gtr become highly significant. Further out, it becomes harder and harder to detect the difference.

Examples:

Black holes cannot be pulled apart by tidal disruption, although in a sense their horizons can be distorted during a close encounter. When two black holes collide, they merge. When they have a close encounter which does not result in a collision, their trajectories may behave in non-Newtonian ways as mentioned above. Astronomers are studying black holes which appear to have been kicked out of their parent galaxies by spin-spin effects (this can happen as the result of a close encounter, but also when two holes merge in such a way that the initial burst of strong gravitational radiation is highly asymmetric--- think action and reaction).

Re

Oh my, passionflower arguing with Chalnoth, scratch flower off the list of potential future SA candidates.

In linearized gtr we can treat approximately the generatation of gravitational radiation by configurations of mass-energy and mass-energy currents (momentum and angular momentum). Then it turns out that the decisive role is played by the multipole moments of configuration as a function of time. The monopole moment gives the "Coulomb field" while the dipole moment can be removed by choosing an appropriate comoving coordinate system (this is possible because of the spin two character of gravitational radiation). The second time derivative of the

An excellent review is

Schutz and Ricci

Gravitational Waves, Sources and Detectors

The study of tensor multipole moments, especially covariant definitions of these (the weak-field theory just described uses sensible-in-context but noncovariant notions of multipole moments), is highly developed and fascinating. Unfortunately, I know of no adequate review paper on-line.

Re

oh good, Mentz114 already posted the necessary correction!

(Before you ask, I think that some years back, Mentz114 declined SA-ship, and I think shoehorn may also have done so. Even worse, I fear I might have been unhelpfully involved in at least one of those :sad:)

Re

Terrible terminology because this has nothing to do with gtr per se, and conflicts with terminology used for gtr effects.

This appears to be a murky reference to gravitational torque--- a small aspherical object can experience a torque as it moves through an ambient gravitational field, and thus change its orientation)--- which is best studied in a Newtonian context before tackling gtr. If the small object is spinning about some axis, this gets much more complicated even in Newtonian theory.

I think oldman may be trying to suggest that under some circumstances, the deformation of a spinning perfect fluid body or elastic solid due to "centrifugural forces" might briefly just cancel the tidal deformation due to an ambient gravitational field. If so, he should run some computations, in Newtonian gravitation.

Tidal coupling, not what oldman appears to think.

A possible reference is Murray and Dermott,

Code:

`www.physicsforums.com/showthread.php?t=403487`

*all*other standard notations.Some pros:

- GAP uses this convention
- much easier to read Cayley and Schreier diagrams (e.g. for combinatorial group theory) if use this
- generally, neater interface with category theory and other topics
- most operations no harder to notate

- much harder to use with "functional notation"
- some operations become harder to notate

Re

Code:

`www.physicsforums.com/showthread.php?t=408525`

*Backlund and Darboux Transformations*, Cambridge University Press.petergreat said:In addition, classical differential geometry lacks the techniques that are widely applied in theoretical physics, such as differential forms.

That is of course completely incorrect, and in fact the classical setting is a good place to explore differential forms. See the classic textbook by Flanders,

*Differential Forms with Applications to the Physical Sciences*, which contains chapters on curve theory and surface theory.Re

Code:

`www.physicsforums.com/showthread.php?t=405899`

*Random Graphs*, 2nd edition, Cambridge University Press, 2001. There is a huge literature on these topics which is unfortunately hard to find on-line, but you can try
Code:

`groupprops.subwiki.org/wiki/Probability_distribution_of_number_of_cycles_of_permutations`

**specialist encyclopedic wikis**("wikispedias"?) which focus on specialized technical subjects, such as "group theory", authored by graduate students and faculty in those subjects, and edited by leading experts.In the past few years, some projects have appeared which follow my advice about the neccesity of restricting article creation/modification to recognized subject matter experts (the other half, about the essential role of the "editor" in the true sense of that word, which wikipedia.org has largely succeeding in debasing by conflating it with "author"). One problem here is that the existing solution to authenticating identity/credentials on-line, the gpg "web of trust", is underutilized in academic circles, which is tragic because this is hampering grass roots efforts such as the projects I am describing. Some projects have even attempted to emulate traditional peer review (Citizendium), although I tend to feel that simply restricting authorship to known individuals with good credentials, and restricting editorship to recognized leaders, may be enough, at least initially, to ensure good growth coupled with good quality. Other projects, such as the subwiki.org "wikispedias", have focused on the fact that it is possible to manage ("edit" in the true sense of the word) a specialist wikipedia (e.g. by ensuring reasonably coherent notation/terminology, making wise choices for internal linkings, resolving any differences of opinion on how to present scientific controversies); compare the chaos at Wikipedia.

I hesitate to suggest linking to the subwiki.org wikispedias in the public areas, because like some other promising projects, they are currently open to anyone, and could quickly be ruined. Also, I don't want to overload their servers. But it's a very promising project and I hope that if we cautiously spread the word to serious people, eventually the universities/government will fund the project to allow for servers keeping pace with growth. (It would be tragedy of they vanished behind paywalls.)

I love the fact that graduate students can produce hilarious essays like

Code:

`groupprops.subwiki.org/wiki/The_promise_of_freedom`

Given the importance in so many areas of mathematics of actions by large finite groups, and the properties of "generic" elements of large symmetric groups in particular, why is it so hard to find good information on-line? One might almost think there must some kind of suppression. And here's a surprise, maybe--- there

*is*some kind of suppression! Can you think why? In a future BRS I may explain, maybe even tell you a few things citizens need to know, but are not "allowed" to know. Mathematical censorship can hurt you!Re

Code:

`www.physicsforums.com/showthread.php?t=406011`

Hmm...some more possible future candidates for SA: lavinia, shoehorn, Martin Rattigan? (shoehorn and I seem to disagree on the value of open source, but hopefully that is not a serious conflict!)

Re

Code:

`www.physicsforums.com/showthread.php?t=405407`

Examples:

- light bending near the limb of our Sun is a small but detectable effect, but near the limb of a neutron star (or near the "cross section" of a black hole, the "dark disk" astronomers are trying to detect for Sag A*) optical effects can be much more dramatic.
- when two ordinary stars pass close by each other, their shape becomes deformed and one or both may even be pulled apart (tidal disruption); there are many factors involved here, not all directly involving gtr, but very roughly speaking, when an ordinary star happens to pass close by a compact object, parts of the ordinary star may encounter a strong gravitational field and thus are more likely to be pulled off; astronomers are studying examples of real stars which are apparently being disrupted by specific supermassive black holes
- when two compact objects happen to have a close encounter, an interesting gtr phenomenon which can have a dramatic effect is "spin-spin" interaction, which can result in the two objects having a highly non-Newtonian interaction, as if they had been "kicked" during the close encounter.

seto6 said:black hole can destroy them self when they interact whit each other, there are other possibility too like merge,,one gets kicked out of orbit

Black holes cannot be pulled apart by tidal disruption, although in a sense their horizons can be distorted during a close encounter. When two black holes collide, they merge. When they have a close encounter which does not result in a collision, their trajectories may behave in non-Newtonian ways as mentioned above. Astronomers are studying black holes which appear to have been kicked out of their parent galaxies by spin-spin effects (this can happen as the result of a close encounter, but also when two holes merge in such a way that the initial burst of strong gravitational radiation is highly asymmetric--- think action and reaction).

Re

Code:

`www.physicsforums.com/showthread.php?t=404319`

This is not even wrong due to ambiguity of what buckethead means by "changes direction" and "accelerates".buckethead said:Let me restate that to saya body will radiate grav waves when and only if it either changes direction or linearly accelerates. So this would include an orbiting body as well as a linearly accelerating body.

In linearized gtr we can treat approximately the generatation of gravitational radiation by configurations of mass-energy and mass-energy currents (momentum and angular momentum). Then it turns out that the decisive role is played by the multipole moments of configuration as a function of time. The monopole moment gives the "Coulomb field" while the dipole moment can be removed by choosing an appropriate comoving coordinate system (this is possible because of the spin two character of gravitational radiation). The second time derivative of the

*mass quadrupole moment*gives rise to the strongest gravitational radiation. Examples (using Newtonian language to suggest the intended picture):- two pointlike objects falling directly toward each other
- two pointlike objects in circular orbits around their COM
- an object rotating in a nonaxisymmetric manner (e.g. a rod rotating about an axis making nonzero angle with its central axis)

- a spherically pulsating star
- an axisymmetric disk rotating about its axis of symmetry

*mass octupole moment*and the*current quadrupole moment*. The latter is interesting because it can arise from Rossby waves in neutron stars, and may be detectable by LISA. Gtr makes rather precise predictions about various effects which should dominate as a young neutron star cools. Roughly speaking, in far field theory, when using a chart obeying the "Lorenz gauge condition", the angular momentum wrt "the center of mass" determines the Komar spin and the major frame-dragging/gravitomagnetism effects, but the current dipole is transformed away when using a chart obeying the stronger "transverse traceless condition".An excellent review is

Schutz and Ricci

Gravitational Waves, Sources and Detectors

Code:

`arXiv.org/abs/1005.4735`

The study of tensor multipole moments, especially covariant definitions of these (the weak-field theory just described uses sensible-in-context but noncovariant notions of multipole moments), is highly developed and fascinating. Unfortunately, I know of no adequate review paper on-line.

Re

Code:

`www.physicsforums.com/showthread.php?t=408367`

(Before you ask, I think that some years back, Mentz114 declined SA-ship, and I think shoehorn may also have done so. Even worse, I fear I might have been unhelpfully involved in at least one of those :sad:)

Re

Code:

`www.physicsforums.com/showthread.php?t=408273`

RP is describing the tidal deformation of a small initially spherical cloud of test particles by the Coulomb field due to a stationary massive object.oldman said:First consider a uniform spherical cloud of non-interacting test masses falling radially toward a central mass. As it falls the sphere will become distorted by tidal accelerations that change

inter-particle separations, into an ellipsoid of revolution whose axis is radial, as described and

illustrated by Roger Penrose in The Road to Reality,Section 17.5, p.396,397.

Not that simple. A good exercise is to run the computation (see my old thread "What is the Theory of Elasticity?"oldman said:If instead of a cloud of test particles the sphere were a isotropic solid, it would be strained by tidal forces (to a degree depending on its proximity to the central mass) into an ellipsoid of revolution

oldman said:The internal stresses that develop are compressions perpendicular to the ellipsoid axis and tensions along this axis. It looks to me that the radial compressive tidal forces are very like (but opposite in direction) the centripetal forces that would make the solid rotate about its radial axis (say spin about this axis), and that the tensile forces are very like (but opposite in direction) the centripetal forces that would make the solid rotate about any axis perpendicular to the solid’s radial axis. I’ll take the liberty of labeling these tidal forces anti-spin and anti-rotation forces because that's what they look like to me.

Terrible terminology because this has nothing to do with gtr per se, and conflicts with terminology used for gtr effects.

This appears to be a murky reference to gravitational torque--- a small aspherical object can experience a torque as it moves through an ambient gravitational field, and thus change its orientation)--- which is best studied in a Newtonian context before tackling gtr. If the small object is spinning about some axis, this gets much more complicated even in Newtonian theory.

oldman said:If the solid were to rotate with an appropriate angular velocity about an axis perpendicular to its radial ellipsoid-of-revolution axis, the centripetal accelerations generated by such rotation might exactly cancel the tensile tidal accelerations.

I think oldman may be trying to suggest that under some circumstances, the deformation of a spinning perfect fluid body or elastic solid due to "centrifugural forces" might briefly just cancel the tidal deformation due to an ambient gravitational field. If so, he should run some computations, in Newtonian gravitation.

oldman said:As in the case of The Moon presenting the same side to us as it orbits the Earth?

Tidal coupling, not what oldman appears to think.

A possible reference is Murray and Dermott,

*Solar System Dynamics*.
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