BRS: Random Comments on Some Recent PF Threads

[Chris Hillman]

OK, hope you remind them of the importance of LTB and CPW (and maybe Ernst vacuums and related families) if they don't mention these.

 

[Chris Hillman]

BRS: the bump function hat from MTW

Forgot to say: Schutz's textbook includes a discussion of the Heintzmann ssspf, another of the better known ssspf solutions. The review by Lake and the papers by Visser at al. on ssspf solutions mention some other interesting examples, including a rather recent solution of Martin.

Re

www.physicsforums.com/showthread.php?t=460495

the notions of first and second order contact may help:

• the tangent line to curve C at point P makes first order contact with C, and if P is an inflection point or if the path curvature vanishes there for any reason, the tangent makes second order contact
• the tangent plane to euclidean surface S at point P makes first order contact with S, with quadratic deviations related to Gaussian curvature of S at P

This generalizes to higher dimensions, and the intrinsic curvature turns out not to depend upon the embedding, although this is not obvious from this approach.

In any case, the first order contact of tangent line with a curve C doesn't prevent the extrinsic curvature of C from varying, nor does the fact that the tangent space at any point P of any manifold M makes first order contact with M prevent the intrinsic or extrinsic curvature of M from varying.

A good reference is Berger's book on Riemannian geometry. See also the BRS on euclidean surfaces of revolution.

The surface of revolution described by the OP is very similar to an example in MTW. It has positive Gaussian curvature on 0 < r < 3^{-1/4}, negative Gaussian curvature on 3^{-1/4} < r < 1, and vanishes outside the unit disk; see the figures below. On the latitude circle r=3^{-1/4} \approx 0.7599 the tangent planes make second order contact with the surface of revolution, so the Gaussian curvature vanishes along this circle.

(To make uploadable images, I used the plot2d command of Maxima to plot the curves and then used ksnapshot (with capture mode "window under cursor") to export the Maxima figures as .png files. Ideally, SA/Ms would often provide figures this way.)

Figures:

• the height z=\exp(-1/(1-r^2)) of the surface of revolution z=f(r) considered by the OP (zero outside the unit circle),
• the Gaussian curvature on 0 < r < 1 (zero outside the unit circle); notice the curvature is negative on 0.76 &lt; r &lt; 1.

 

[Chris Hillman]

BRS: Why do first covariant derivatives of the metric tensor vanish?

Again re "Metric tensor of a non-homogeneous universe"

www.physicsforums.com/showthread.php?t=460495

If I understand you correctly the covariant derivative of a tensor tells us whether or not the tensor is changing relative to the metric tensor. In other words it tells us the changes in the tensor "net of any changes attributable to the change in the metric tensor". So no matter how many different types of spaces we "sew" together, with all sorts of associated radical changes to the metric between different parts of the manifold, the covariant derivative of the metric will always be zero because it is "changes in the metric tensor net of any changes attributable to changes in the metric tensor", which will be zero by definition.

Think of a McLaurin series expanding the metric wrt any event on any Lorentzian four-manifold. To first order it should always look just like the metric tensor of E^{1,3}; curvature (which distinguishes between Lorentzian manifolds which are not locally isometric to each other) represents second order deviations from the metric tensor of flat spacetime.

The very same issue arises the same way for Riemannian manifolds; this has nothing to do with physics per se, but with the assumptions Riemann made in formulating the notion of a Riemannian (or Lorentzian) manifold! See the excellent discussion of Riemannian two-manifolds in Berger, A Panorama of Riemannian Geometry, a readable romp through a wonderful subject. (Note that many of the topics discussed by Berger in this book are special to Riemannian viz. Lorentzian geometry, but there is also considerable overlap.)

 

[Chris Hillman]

BRS: Kleinian Geometry, anyone? ... No? ... Sigh ...

Re "Mapping Class Group and Path-Component of Id"

www.physicsforums.com/showthread.php?t=460570

quasar misunderstood the question, I think.

The identity component of a (nontrivial) topological group G is the connected component of the identity, which is always much larger than the trivial subgroup (consisting of the identity element e in G)! It is always a closed normal subgroup; see Cohn, Lie Groups, Theorem 2.4.1. (Since the mapping class group is a group and the identity map is its identity, this greatly generalizes the result desired by the OP.) In the case when G is a Lie group, this implies that it is a Lie subgroup of G. Example: SO(3) is the identity component of O(3). The identity component has the same dimension as G when G is a finite dimensional Lie group.

Note: in general topology, the path component of a point p in X need not be quite the same thing as the topological component of p. (See a good textbook on topology for the standard definitions of these notions.) But IIRC, the distinction doesn't much matter in this context.

Re

www.physicsforums.com/showthread.php?t=460573

Let M be an n-manifold and N an n-dim. subspace of M, of possibly lower dimension than M , then M is knotted, or N is a knot in M if there are non-isotopic embeddings of N in M.

This definition would regard both an "unknotted" circle and the trefoil knot as knotted one-dimensional submanifolds of S^3, for example, which seems somewhat perverse. Surely one should say that "N is knottable in M" if at least two nonisotopic embeddings exist. Or some such terminology. Because the property of a submanifold being knotted in its parent is extrinsic, not intrinsic.

conversely, if f,g, are two non-isotopic embeddings of N in M, does it follow that H_n(M) is not trivial?

Consider N = S^1.

Re "Affine plane, block design"

www.physicsforums.com/showthread.php?t=460788

The website of Peter Cameron (author of numerous books on combinatorics, group theory, &c) includes at least two complete sets of lecture notes dealing with the geometries associated with classical groups, including their finite dimensional relatives

www.maths.qmul.ac.uk/~pjc/
www.maths.qmul.ac.uk/~pjc/pps/

The unfinished (indeed, hardly even begun) BRS on the Rubik group is also devoted to finite geometries from the point of view of Klein's Erlangen Program.

According to Klein, each geometry on a "naked set" X is associated with a "generalized isometry" group G, consisting of those bijective transformations X \rightarrow X which preserve the geometrical structure. Subgroups of G correspond to "more rigid" alternative geometrical structures which can be placed on X. And conjugacy classes of subgroups enumerate the "geometrical concepts" of a geometry and display the interrelationships between these elements.

In particular, it helps to start with G=PGL(d+1,q), where q = p^n for some prime p, which is the "isometry group" for the d-dimensional projective geometry over field GF(q). This has a subgroup H=AGL(d,q) which is the "isometry group" for d-dimensional affine geometry over GF(q). (More precisely, G has two conjugacy classes of subgroups isomorphic to H, which turns out to be geometrically significant). H has further interesting subgroups defining even more rigid geometries, e.g. analogs of euclidean metrical geometry.

We can consider various actions by these groups and study the resulting lattice of stabilizers and its Galois dual, the lattice of fixsets. In the case of geometries which arise by "decorating" (adding structure to) projective geometry, it helps to start with the action on the points of projective space. Studying the stabilizer lattice we can identify certain fixsets which correspond to the k-flats. We can consider these as subsets and study the action on the k-flats, which provide additional orbits in the action on points, lines, 2-flats,... (d-1)-flats (or hyperflats).

When we restrict from G to H, some of these orbits break up into smaller orbits, and the stabilizer lattice becomes more complicated under the restricted action. That is, as a rule, more rigid geometries have more "geometrical concepts" than less rigid geometries. For example, in projective geometry all points are equivalent, and all hyperflats are equivalent, but in affine geometry, there are two kinds of points, ordinary points and ideal points, and also two kinds of hyperflats, ordinary hyperflats and a unique ideal hyperflat (the "hyperflat at infinity").

Studying the stabilizer lattice of the action of the basic geometrical elements (various kinds of k-flats for k=0...d-1) by H=AGL(d,q), we can identify a conjugacy class of stabilizers which corresponds to the parallel lines. The stabilizers in this class correspond to the sets of parallel lines, and also shows how each such set relates to various other geometrical concepts of affine geometry.

In general, because group theory provides so many powerful tools, as Galois himself recognized, it is very helpful to transform counting questions (and other questions) into the realm of the lattice of subgroups of some group, where such questions are usually easier to answer. This is one advantage of Klein's point of view, but there are many others--- I have long been intrigued by the way that simply writing down an action by a group leads naturally to the correct "geometrical concepts" and their interrelationships, and indeed to an information theory generalizing the highly successful information theory of Shannon and in some sense unifying it with classical Galois theory.

Examples of useful facts from the theory of groups include:

• Given H a finite index subgroup of G, [G] is the size of the right coset space H\G, so if H is the stabilizer of some "geometric configuration" implicit in some action by G on some set X, there are [G] geometric motions of the given configuration.
• The number of conjugates of H in G is [G:N_G(H)], where N_G(H) is the normalizer in G of H, so if H is the stabilizer of some "geometric configuration" implicit in some action by G on some set X, there are [G:N_G(H)] configurations which are geometrically equivalent to the given configuration.
• H is normal in N_G(H), and the right coset space (also a factor group) H\N_G(H) is the internal symmetry group of the configuration. So in the chain H < N_G(H) < G, the right coset space N_G(H)\G corresponds to "external motions" while the right coset space (a group in its own right) H\N_G(H) corresponds to "internal motions".

andreass asked about AGL(2,2), but finite geometries over GF(2) are "geometrically atypical" in many ways (basically, because "lines have too few points"), so I suggest studying the stabilizer lattice of PGL(3,3) and its subgroup AGL(2,3) instead. Especially in conjunction with PGL(3,R) and AGL(2,R), the corresponding real projective and affine geometries.

In the figures below, you can see that the conjugacy class of the stabilizer of 3 parallel (ordinary) lines has 4 elements, so there are 4 sets of 3 parallel lines. Furthermore, each line is included in a unique such triple, and each triple contains three lines. Well, duh!, in this case, but we also see relations which might not be so obvious when we study more complicated geometries using Klein's approach. In the figure, notice that the class consisting of 36 C2 is the pointwise stabilizer of a line (which arises in the action on points) whereas the conjugacy class of 12 S3^2 is the setwise stabilizer of a line. And the class of the stabilizers of the 4 triples of parallel lines is intermediate between the pointwise and setwise stabilizers of a line--- which makes sense! Also, |AGL(2,3)| = 432, so there are 432/18 = 24 motions of each triple of parallel lines, consisting of translations by various amounts ("distance" is not a concept of affine geometry, so I am avoiding that word) in various directions, plus certain "reflections". Again, comparing with AGL(2,R) is helpful!

For actions by finite groups G on finite sets X, the logarithm of the indices [G] (i.e. the logs of the sizes of the right coset spaces H\G) behave just like Shannon entropies. For actions by finite dimensional Lie groups G on finite dimensional manifolds X, the dimensions of the right coset spaces H\G behave just like Shannon entropies. To each stabilizer H < G which arises in some action by G on some set X, there corresponds the right coset space H\G, or complexion, which measures the variety of motions of the "geometric element" or "geometric configuration" which is stabilized by H. So these entropies measure our uncertainty about which of the possible motions will be chosen in some "random process". The coset space formed from the intersection of two stabilizers measures our uncertainty about the joint motion of two configurations, and restricting from the action by G to the action by one of the stabilizers H on this joint coset space gives a conditional complexion, where the corresponding conditional entropy measures our uncertainty concerning a motion of the second configuration after we are told the chosen motion of the first configuration. Generally, in finite geometries, such motions are not entirely independent (due, if you like, to the failure of the Hilbert hotel phenomenom familiar from bijections on infinite sets), so this is actually significant information.

Furthermore, if \varphi: X \rightarrow Y is some G-hom (morphism in the category of G-sets, for a given group G; compare the category of R-modules, for a given ring R, for example), the stabilizer of preimages of a point \varphi(x) \in Y under the given action on Y is a subgroup of the stabilizer of x in X under the given action on X. Even better, it is a normal subgroup, so if we combine two transitive G-sets, one the G-homomorphic image of the other, into a single G-set, which means that we regard \varphi as an G-endomorphism mapping one orbit onto another orbit, then conditional complexion measuring our uncertainty about the motions preimages of any point y in the second orbit given the motions of y generalizes the notion of Galois group from classical Galois theory. Similarly for other G-endomorphisms. This is also related to the notion of cellular automata and certain shift spaces studied in symbolic dynamics.

Returning to the figure, we can see other notable stabilizer subgroups. For example, the class of 12 C3 consists of groups of shears which fix one line pointwise and fix the two parallel lines setwise. The class of 36 D6 corresponds to the 36 flags (ordinary point on an ordinary line), while the class of 54 D2 corresponds to the 54 intersections of a pair of ordinary lines.

The class of 9 GL(2,3) give the stabilizers of the 9 ordinary points, and choice of one of them corresponds to choice of origin, which then implies a restriction from affine transformations to linear transformations. In a linear geometry on the affine plane, one ordinary point is distinguished as "the origin", a concept which simply makes no sense in affine geometry.

Taking a wider view, there happen to be two conjugacy classes of subgroups isomorphic to AGL(2,3) in PGL(3,3), each consisting of 13 subgroups. These correspond to the stabilizers in the action on projective points and projective lines, so as you would expect from projective duality, the stabilizer-fixset lattice of these two actions look the same. Choice of a particular stabilizer in the action on lines corresponds to choosing one of the 13 lines of the projective plane over GF(3) as the ideal line, leaving 12 ordinary lines. This also chooses one of the equivalent "affine structures" which can be placed on the projective plane. The orbit of 13 points under PGL(3,3) breaks up under our chosen subgroup isomorphic to AGL(2,3) into 4 ideal points (the points lying on the ideal line) and 9 ordinary points.

Furthermore, there is a conjugacy class in PGL(3,3) of 117 subgroups isomorphic to GL(3). In projective geometry, these are the stabilizers of a configuration consisting of a line and a point off that line. Choosing one of them amounts to

• choosing an ideal line (placing an affine structure on the projective plane)
• designating a particular ordinary point as "the origin" (placing a linear structure on the just defined affine plane).

Again, this sketchily illustrates why we say that linear geometry is more rigid than affine geometry, which is in turn more rigid than projective geometry.

There are in all 46 conjugacy classes of subgroups of AGL(2,3), so in the action on points and lines, only a small fraction of the totality of subgroups appear as stabilizers. However, we can consider many derived actions, e.g. actions of subsets of various sizes, and in this way, by going back and forth between the geometry and the abstract structure of its isometry group, we can eventually identify each subgroup with some possibly subtle geometrical property or configuration, e.g. a labeled or colored configuration of some kind.

GAP4 can be very useful in exploring small finite geometries. If you want to explore,

• PrimitiveGroup(9,7) gives the action by AGL(2,3) on the 9 ordinary points of the affine plane over GF(3)
• TransitiveGroup(12,157) gives the action by AGL(2,3) on the 12 ordinary lines of the affine plane over GF(3)

and you can write a routine to combine these two orbits into an intransitive permutation group. (Make sure you use the correct generators in constructing the combined action--- IsomorphismGroups is the tool you need--- and of course you will need to reindex one of the orbits.)

The OP asked about block designs, a notion introduced by R. A. Fisher, who needed to design experiments which would efficiently explore all possible relationships between certain variables in agricultural experiments. However, the notion turns up in many places in combinatorics, and turns out to have many unexpected applications in science (and the original applications remains quite important, e.g. in medial research). There are indeed many relationships between finite geometries and block designs, and this has been a major topic of research in combinatorics for almost a century! Cameron is an expert on this subject so his website should be very helpful.

There are also many relationships between the classical groups PGL(d+1,F), AGL(d,F), etc., and the theories of linear representations, invariants, Lie algebras, reflection groups, regular polytopes, Schubert calculus, multiply-transitive groups, and finite simple groups. Many of the best of the expository series of John Baez, This Week in Mathematical Physics, were devoted to exploring one or another aspect of these relationships. One which is particularly relevant here is the q-calculus, in which one constructs q-analogs (for q = p^n as above) for binomial coefficients. The q-analog of Pascal's triangle then counts the number of k-flats in d-dimensional projective space over GF(q), for given q. This is then closely related to things like Bruhat partial order on Young diagrams, invariant tori, parabolic subgroups, Schubert calculus...

More generally, enumerative geometry can be approached via the theory of structors (certain functors, also called "combinatorial species"), and it turns out that the theory of structors is very closely related to the theory of finite permutation groups (which is in turn very closely related to the theory of actions by finite groups on finite sets). And it greatly generalizes the wonderful counting formula of Polya. There is a natural generalization to actions on infinite sets, the oligomorphic actions (see the website of Peter Cameron for more about these). And this is in turn closely related to model theory and various topics in mathematical logic. And to close the circle, notions from topology such as compactness turn up naturally here and play an important role. The unity of mathematics is a wondrous if sometimes bewildering thing!

It is intriguing that I seem to see more questions relating in some way to Kleinian geometry in the past six months at PF, and unfortunate that there seems to be little interest among the SA/Ms in exploring this wonderful topic, which happens to be a more serious interest of mine than general relativity--- I only yak about gtr so much because I happen to know that theory and people ask about it constantly at PF, and their naive questions so often require considerable sophistication to answer well, so inevitably I seem to keep getting sucked back into trying to help students learn more about gtr.

Figures: for the action of AGL(2,3) on the points and lines of the affine plane over GF(3) (orbits: 9 ordinary points and 12 ordinary lines; not shown: orbit of 4 ideal points and trivial orbit of 1 ideal line)

• Stabilizer subgroups, up to conjugacy (inclusion runs upwards)
• Fixsets, up to affine motions (inclusion runs downwards)

Notice the Galois duality of these two lattices, which generalizes the well known duality between subgroups and fixed fields from classical Galois theory.

 

[Chris Hillman]

BRS: I'd rather be yakking about Kleinian geometry!

Re

www.physicsforums.com/showthread.php?t=462413

Such a project would be much too hard for an undergraduate. But looking towards graduate work, he should make sure to learn about symplectic integrators for Newtonian multibody simulations, and if his mathematical sophistication were up to integro-differential equations (which apparently is not the case), he can study the theory of the Vlasov equation in a Newtonian context.
The idea here is to model stellar clusters/galaxies as "dust, Newtonian style", using the probablistic methods of statistical mechanics. Try
C. C. Lee, "Dynamics of Self-Gravitating Systems: Structure of Galaxies", in
Studies in Applied Mathematics,
edited by A. H. Taub,
MAA Studies in Mathematics, 1971.

(A. H. Taub is the Taub in Taub-NUT vacuum.)

Re

www.physicsforums.com/showthread.php?t=457968

and
See

www.physicsforums.com/showthread.php?t=456626

I think Goldbeetle and Dark_knight90 need to be warned (I predict Goldbeetle will ignore the warning but others might not) that context is crucial in reading scientific papers, which is the most important reason why trying to learn gtr by following the history guarantees failure. One must master gtr as currently understood from modern textbooks, and read some good modern papers, before possibly reading a book like Pais's biography which does attempt to trace the historical development of gtr. But to really understand the history, one must study the leading textbooks a generation prior to Einstein 1916, and the most important scientific papers in the years around 1916, and one must understand the issues which concerned physicists of the time both in modern terms and in terms of how contemporaries understood or misunderstood them. Indeed, to really understand the history, one must also know something about the politics of the day (the early development of gtr occured, after all, in the context of a global war, then post-war chaos, then a global depression...). And one must come to know the personalities involved as closely as is possible after so many years. (Pais was personally acquainted with Einstein and knew another leading figure well, Niels Bohr.)

But I can't stress this enough: one must master the modern theory of gtr before (possibly) attempting any historical analysis. The productions of would-be historians of science who have written dreadful nonsense owing to failing to obey this rule speaks for itself. Indeed, rather incredibly, there are even historians who have overlooked the political context of the early development of gtr.

I really don't know where so many people seem to have obtained the false notion that one can avoid mathematics by studying the historical development of gtr. It's really sad that PF doesn't do more to try to prevent students or interested laypersons from going down this path, because nothing good can come of that.

I often vehemently disagree with Cleonis, but in his Post #10 in the Goldbeetle thread, he does make an important and valid point: the modern understanding of the Principle of Equivalence is that this is a simple consequence of the assumption that "spacetime" should be modeled by a Lorentzian manifold, and it simply amounts to saying that the tangent space to each event E is a real vector space equipped with E^{1,3} binary form, which is identified with the metric tensor at E.

I generally disagree with the historical interpretations of Cleonis, but in his Post #6 he again surprised me somewhat by drawing attention to a valid point about the interaction between Nordstrom and Einstein; from the modern point of view, N was quite correct to challenge AE to refine his overly vague understanding of the POE. In the early days, it was far from clear that gtr was above all a theory of gravitation, or even that the Riemann curature tensor defines "the gravitational field". AE wanted gtr to be more than a theory of gravitation; in a sense, the modern understanding has eventually come to a superficially similar point of view: roughly speaking, there are two phenomena common to all (classical) theories of fundamental physics, energy and gravitation; the metatheory of energy conversion/transport which applies to all theories of specific interactions is thermodynamics; the metatheory of gravity is gtr. In order to use thermodynamics, one adopts a model of "matter" (e.g. an equation of state for a gas); in order to use gtr, one writes down a Lagrangian, obtains a model of matter, specific non-graviational fields, "exotic matter", etc., finds the appropriate energy-momentum tensor, and attempts to find spacetime models in which the Einstein tensor matches the energy-momentum tensor and all the fields satisfy their field equations (use covariant derivatives!) on the resulting curved spacetime. This is the modern way of understanding the "universal" character Einstein sought, and it makes it plausible that there should be a deep relation between thermodynamics and gravitation.

Re

www.physicsforums.com/showthread.php?t=462327

My gosh... good illustration, sad to say, of what I meant when I said that the kooks are using Maple/Maxima too, so that there is an arms race. Clearly this poster is hopelessly confused by mathematical notation, has no knowledge of classical physics, much less relativistic physics, and probably has a language barrier too. And has no clue about CODE tags.

IMO, the only reasonable reply is a polite reformulation of this:

Your post is incoherent, but clearly shows that you lack the math/physics background to use the computer tools you are trying to use. You'll never be able to say anything useful/intelligible, so give up your interest in physics forthwith, or expect to be labeled a "cranky ignoramus" if you persist.

Re "Coordinate radius [r]"

www.physicsforums.com/showthread.php?t=462307

Another earnest newbie, who also suffers from a deficit of mathematical sophistication (and a language barrier), but maybe fixable if he/she can be persauded to study hard modern textbooks for several years.

It is my understanding that the coordinate radius [r] is defined in terms of the ‘reduced circumference’, i.e.
coordinate radius r = LaTeX Code: circumference/2 \\pi
As such, a number of texts seem to describe calculating [r]

I think PF should tell such posters

If you cannot be bothered to cite your sources you are too lazy to either learn or converse about mathematical physics. If you insist on keeping secret what textbooks you are reading, you will raise the suspicion that you are trying to pull a fast one on your professors. In any case, what you claimed your textbook says makes no sense.

Re "Tensor Rank of Stress Tensors"

www.physicsforums.com/showthread.php?t=462258

this poster is confusing simple two-forms (a subclass of antisymmetric second rank tensors) with symmetric second rank tensors. Happy to say that in this case I see hope that this student can overcome his difficulties.

The stress tensor is commonly given in terms of a rank two tensor - the tensor appears to be composed with the components of the force density vector over a given differential area, and *the normal vector of that differential area*.

I think I know what he is trying to say, and he is confused, yet shows some insight here. One possible response would be to suggest that if he learns to use frames and to compute frame components of tensors, his immediate confusion will go away and he'll understand stress tensors much better. In a coordinate basis, the two-dimensional area element does occur as a denominator in many general expressions for coordinate basis components for curvature quantitites, but this is irrelevant to physical understanding at the appropriate level of stress tensors.

the stress tensor should be a rank three tensor, with one rank for the force vector, and the other two for the differential area two-vector. When I set it up this way, I can transform the stress arbitrarily, and get results that make physical sense

This is wrong. Instead of the force acting on a particle, consider the acceleration as a function defined on the world line of that particle. This is simply the path curvature, which is a vectorial quantity defined along the world line, which is given by the covariant derivative of the tangent vector along itself. Technically, it is much easier to consider your world line just one integral curve of a timelike congruence and to work with the congruence.

Re "How to make two frames purely Galilean"

www.physicsforums.com/showthread.php?t=462154

the OP is evidently a fringer hoping to "prove" [sic] that Galilean relativity can replace str. Mathematically and geometrically, Galilean relativity can be understood as a sensical mathematics/geometry associated with a degenerate bilinear form E^{0,3} and a symmetric group E(0,3), and this group even arises naturally as the point-symmetry group associated with certain simple ODEs, as does the distinct symmetry group E(1,3), the Poincare group, which is associated an indefinite but nondegenerate bilinear form E^{1,3}. But Galilean and Minkowski geometries are completely different as Kleinian geometries, so their "curved manifold" elaborations (as Cartanian geometries) are also distinct. Thus, there is no hope of "showing" [sic] that they are mathematically or geometrically equivalent, and thus no hope of "showing" [sic] that they are "physically equivalent" [sic].

User:chinglu1998 is explicity cranky. User:grav-universe is IMO simply being coy and I have no doubt the underlying crankiness will soon become evident.

Re

www.physicsforums.com/showthread.php?t=461941

ShiroSato needs to give some context, e.g. by quoting from a cited book or paper. All one can say without context is that "degrees of freedom" is a somewhat archaic term for the dimension of some kind of parameter space. E.g. the Lorentz group has "six degrees of freedom" which can be understood as consisting of three independent rotations and three independent boosts. The Poinare group has "ten degrees of freedom", adding time translation and three spatial translations.

Re

www.physicsforums.com/showthread.php?t=461717

I think TrickyDicky is trying to ask: what does the contribution of the CMBR to the energy-momentum tensor look like? Answer: a superposition (using weak-field theory is appropriate here) of null dust terms associated with "waves" coming from all directions, so adding up to a very tiny radiation fluid term, I think. Checking this carefully would be a good exercise.

I note that above Kleinian geometry made another appearance, and reiterate that this is really more my thing anyway...