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Is there a connection between preferred extremals and AdS_{4}/CFT correspondence?

The preferred extremals satisfy Einstein Maxwell equations with a cosmological constant and have negative curvature for negative value of Λ. 4-D space-times with hyperbolic metric provide canonical representation for a large class of four-manifolds and an interesting question is whether these spaces are obtained as preferred extremals and/or vacuum extremals.

4-D hyperbolic space with Minkowski signature is locally isometric with AdS_{4}. This suggests a connection with AdS_{4}/CFT correspondence of M-theory. The boundary of AdS would be now replaced with 3-D light-like orbit of partonic 2-surface at which the signature of the induced metric changes. The metric 2-dimensionality of the light-like surface makes possible generalization of 2-D conformal invariance with the light-like coordinate taking the role of complex coordinate at light-like boundary. AdS would presumably represent a special case of a more general family of space-time surfaces with constant Ricci scalar satisfying Einstein-Maxwell equations and generalizing the AdS_{4}/CFT correspondence.

For the ordinary AdS_{5} correspondence empty M^{4} is identified as boundary. In the recent case the boundary of AdS_{4} is replaced with a 3-D light-like orbit of partonic 2-surface at which the signature of the induced metric changes. String world sheets have boundaries along light-like 3-surfaces and space-like 3-surfaces at the light-like boundaries of CD. The metric 2-dimensionality of the light-like surface makes possible generalization of 2-D conformal invariance with the light-like coordinate taking the role of hyper- complex coordinate at light-like 3-surface. AdS_{5}× S^{5} of M-theory context is replaced by a 4-surface of constant Ricci scalar in 8-D imbedding space M^{4}× CP_{2} satisfying Einstein-Maxwell equations. A generalization of AdS_{4}/CFT correspondence would be in question. Note however that the accelerated expansion of the Universe requires positive value of Λ and favors De Sitter Space dS_{4} instead of AdS_{4}.

These observations give motivations for finding whether AdS_{4} or dS_{4} or both allow an imbedding as vacuum extremal to M^{4}× S^{2}⊂ M^{4}× CP_{2}, where S^{2} is a homologically trivial geodesic sphere of CP_{2}. It is easy to guess the general form of the imbedding by writing the line elements of, M^{4}, S^{2}, and AdS_{4}.

- The line element of M
^{4}in spherical Minkowski coordinates (m,r_{M},θ,φ) reads as

ds

^{2}= dm^{2}-dr_{M}^{2}-r_{M}^{2}dΩ^{2}.

- Also the line element of S
^{2}is familiar:

ds

^{2}=- R^{2}(dΘ^{2}+sin^{2}(θ)dΦ^{2}) .

- By visiting in Wikipedia one learns that in spherical coordinate the line element of AdS
_{4}is given by

ds

^{2}= A(r)dt^{2}-(1/A(r))dr^{2}-r^{2}dΩ^{2},

A(r)= 1+y

^{2}, y = r/r_{0}.

- From these formulas it is easy to see that the ansatz is of the same general form as for the imbedding of Schwartschild-Nordstöm metric:

m= Λ t+ h(y) , r

_{M}= r ,

Θ = s(y) , Φ= ω× (t+f(y)) .

The non-trivial conditions on the components of the induced metric are given by

g

_{tt}= Λ^{2}-x^{2}sin^{2}(Θ) = A(r) ,

g

_{tr}= 1/r_{0}[Λ dh/dy -x^{2}sin^{2}(θ) df/dr]=0 ,

g

_{rr}= 1/r_{0}^{2}[(dh/dy)^{2}-1- x^{2}sin^{2}(θ)(df/dy)^{2}- R^{2}(dΘ/dy)^{2}]= -1/A(r) ,

x=Rω .

- For Θ(r) the equation for g
_{tt}gives the expression

sin

^{2}(Θ)= P/x^{2},

P= Λ

^{2}-A =Λ^{2}-1-y^{2}.

The condition 0≤ sin

^{2}(Θ)≤ 1 gives the conditions

(Λ

^{2}-x^{2}-1)^{1/2}≤ y≤ (Λ^{2}-1)^{1/2}.

Clearly only a spherical shell is possible.

- From the vanishing of g
_{tr}one obtains

dh/dy = ( P/Λ)× df/dy ,

- The condition for g
_{rr}gives

(df/dy)

^{2}=[r_{0}^{2}/AP]× [A^{-1}-R^{2}(dΘ/dy)^{2}] .

Clearly, the right-hand side is positive if P≥ 0 holds true and RdΘ/dy is small.

From this condition one can solved by expressing dΘ/dy using chain rule as

(dΘ/dy)

^{2}=x^{2}y^{2}/[P (P-x^{2})] .

One obtains

(df/dy)

^{2}= [Λ r_{0}^{2}y^{2}/AP]× [(1+y^{2})^{-1}-x^{2}(R/r_{0})^{2}[P(P-x^{2})]^{-1})] .

The right hand side of this equation is non-negative for certain range of parameters and variable y.

Note that for r_{0}>> R the second term on the right hand side can be neglected. In this case it is easy to integrate f(y).

The conclusion is that AdS_{4} allows a local imbedding as a vacuum extremal. Whether also an imbedding as a non-vacuum preferred extremal to homologically non-trivial geodesic sphere is possible, is an interesting question. The only modification in the case of De Sitter space dS_{4} is the replacement of the function A= 1+y^{2} appearing in the metric of AdS_{4} with A=1-y^{2}. Also now the imbedded portion of the metric is a spherical shell. This brings in mind TGD inspired model for the final state of the star which is also a spherical shell. p-Adic length scale hypothesis motivates the conjecture that stars indeed have onion-like layered structure consisting of shells, whose radii are consistent with p-adic length scale hypothesis. This brings in mind also Titius-Bode law.

For details and background see the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Physics as Infinite-dimensional Geometry", or the article with the title "Do geometric invariants of preferred extremals define topological invariants of space-time surface and code for quantum physics?".

## 22 Comments:

http://backreaction.blogspot.com/2012/12/adscft-predicts-quark-gluon-plasma-is.html

AdS/CFT predicts the quark gluon plasma is unstable

The gauge-gravity duality is a spin-off from string theory and has attracted considerable attention for its potential to describe the quark gluon plasma produced in heavy ion collisions. The last news we heard about this was that the AdS/CFT prediction for the energy loss of quarks or gluons passing through the plasma does not agree with the data. The AdS/CFT community has so far been disappointingly silent on this issue, which has now been known for more than a year.

http://backreaction.blogspot.com/2011/10/adscft-confronts-data.html

I start to write summary of my understanding in TGD in a simple form, step by step. It helps me to understand TGD better. It doesn’t mean I want to teach TGD because I know very little about the giant topic! I request from Matti, to correct and complete my wrong sentences. Although i try to write as simple as i can, but It is needed for the readers to know some basic understandings of physics and mathematics too. If they don’t know, they can send to my mail Dehghanihamed@gmail.com their questions about the bases and I try to answer them if I can. I welcome for the suggestions.

I encourage you to make why before every sentence and try to answer it :-). For example, it is important to know why Noether's theorem leads to one-one correspondence between conservation laws with symmetries. You can think for alternatives for every idea and ask to yourself: Is it possible to think in this manner?!

TGD notes1

The first thing is energy problem in general relativity and how TGD can solve it. I try to write minimum requested subjects of physics or mathematics that is needed for understanding it at this level.

1-Special relativity is a theory that unifies space and time in the space-time. The space-time has 4 dimensions and is called 4D-Minkowski space-time or M4. It is a flat space-time.

2- By Noether's theorem, conservation laws are in one-one correspondence with symmetries. It relates classical conservation laws (conservation of energy, momentum, and angular momentum) to

Symmetries of M4.

3-General relativity is a theory for describing gravitation .properties of the gravity is deduced from curvature of space-time. Therefore space-time is not flat as in special relativity and ordinary intuition. . In very short review, in general relativity, space-time tells matter how to move; matter tells space-time how to curve.

In General relativity, space-time is flat only locally, this means intuitionally if you zoom on the surface of a sphere, you can see it flat roughly. Therefore, one can say in general relativity, space-time is M4 locally and special relativity is only locally correct.

4-Because of the curvature of space-time in general relativity, the symmetries of the empty M4 are lost as are lost also the corresponding conservation laws, in particular the conservation of energy. Physicists put under the rug this problem!

To be continued...

5- For solving the problem it is needed to adhere on the symmetries of empty M4 and obviously it is not obtain it unless we adhere on the M4 itself. But in the other hand we need the curvature of space-time for describing gravitation in according to general relativity. How is it possible to have both curved space-time and flat space-time!!!

For this reason, TGD started from something that is very interesting and important. M4 is different from ordinary space-time that we live in!!! In really space-time is a subspace of M4*S. S is some compact space. The M4*S is called the imbedding space in TGD. Space-time can be curve in the imbedding space, but M4 is flat. The curvature is because of the way it is imbedded in the M4*S that is in contrary to general relativity that there isn’t any extra dimension for making curvature.

One can say when an object is moving in space-time, it is moving in M4 too. only when the space-time is flat and parallel to M4, it can be seen the same as M4.

6- I must be note that extra-dimensions in KaluzaKlein or string theories are different with the idea of TGD. Because in KaluzaKlein approach space-time itself has more than 4 dimensions! Therefore it lost Poincare symmetries again, because of it’s curvature.

7-if one can understand distinctions between the two concepts of space-time and M4, the solving of the conservation problem in general relativity will be simple:

Instead of moving a point along space-time surface, isometrics move the entire space-time surface in imbedding space. Although there aren’t arbitrary symmetries for moving a particle along space-time surface as like in general relativity, but there are symmetries for the entire space-time in the embedding space. Therefore one can have exact Poincare symmetries even when the space-time has curvature!

8- M4 can be seen as a privileged framework!!! But we can adhere to special relativity too! I hope you understand and enjoy the taste of this beauty :). There are some main differences with Newtonian privileged framework, Absolute space in the Newtonian sense means absolute time coordinate and M^4xS does not have it.

This was only an introduction to TGD as a Poincare invariant theory of gravitation. I must to note that solving the energy problem in TGD is very richer than I explained here. More explanations are needed by introducing kahler action. I hope that I explain them in my after TGD notes.

To Ulla:

I think I wrote about the discrepany of AdS/CFT model for quark gluon plasma as a commentary to a very nice posting by Sabine Hossenfelder.

In higher energy QCD one would expect quark gluon plasma. This prediction is definitely inconsistent with the observations about correlated charged particle pairs moving in same or opposite directions and strongly bringing in mind a decay of string like object. The problem is that string like objects make sense only in low energy QCD, not at very high energies in proton proton collisions!

TGD explanation in its original form was based on the identification of string like objects as color magnetic flux tubes containing quarks in liquid like phase - color class. Only recently I realized that these flux tubes could be nothing but the hadrons of M_89 hadron physics so that RHIC would have observed the basic prediction of TGD for - was it seven years ago! If I am not wrong (usually the physicist who speculates is wrong;-)), particle physics cannot avoid a scandal: people are desperately searching standard SUSY and other signals for new physics and new physics is in front of their eyes!

Dear Hamed,

a very nice and concise summary about basic ideas of TGD as far as general relativity and Poincare invariance are considered. Maybe it is possible to learn TGD after all;-). As one sees, a careful consideration of the notion of symmetry plays a key role.

There is a second bunch of ideas related to the relation of TGD and particle physics: again careful consideration of what we really know about symmetries. How standard model quantum numbers find explanation and how classical gauge fields find a geometrization are the basic questions.

At the technical side the basic challenge is the construction of the quantum TGD, a work still in progress. I remember how my friends and relatives asked for decades ago with a tone of worry in their voice whether it is perhaps done soon --- namely, this construction of TOE... I gave no promises, and good so! My comment to Stephen tells about the basic ideas relate to this challenge.

http://www.sciencedaily.com/releases/2012/12/121212205617.htm

Still to Ulla about AdS/CFT. I want clarify the distinctions between the meaning of this word in string theory context and its possible meaning in TGD framework.

AdS_5xS^5 correspondence is used in M-theory inspired approach to model quark-gluon plasma and one obtains color glass picture: liquid like state of partons instead of plasma and explaining the observed correlations. This approach has some problems about which Bee wrote earlier and also I did so.

M-theory inspired AdS_5xS^5 correspondence is very different from strong form of holography of TGD. In TGD framework constant Ricci scalar property would suggest that strong form of holography can be seen as a **generalization** of AdS_4 correspondence. This posting demonstrated that AdS_4 defines a piece of vacuum extremal so that this conjecture might make sense.

There are important differences which should not be forgotten.

a) In TGD 10-D AdS_5xS^5 is replaced with the real 4-D space-time realized as 4-surface in M^4xCP_2. There is no need to replace the original physical system with black-holes in 10-D. Strong form of holography is the basic principle which can be seen as a generalization of AdS_4/CFT correspondence.

b) String world sheets in 4-D space-time carrying fermion fields (right handed neutrino is exception) are real physical objects would replace the strings in fictive 10-D target space.

c) In AdS_5xS^5 one has 4-D Minkowski space-time as a conformal boundary of AdS_5. Now M^4 is replaced by light-like 3-surface at which the signature of the induced metric changes and which represents line of generalized Feynman diagram. By strong form of holography (of general coordinate invariance) one can say that it is replaced with space-like 3-surfaces at the end of causal diamonds.

d) In AdS_5 conformal symmetries of M^4 defined conformal symmetries of CFT. In TGD conformal symmetries of light-like 3-surfaces forming infinite-D group are the conformal symmetries.

This is huge extension of conformal symmetries of 4-D QFT in M^4 and very closely related to stringy conformal symmetries.

e) Last but not least: the origin of standard AdS/CFT correspondence in its standard form is mystery. In TGD framework the mystery disappears: strong form of general coordinate invariance implies its counterpart as strong form of holography.

Dear Matti,

some questions:

A comparison between TGD and GR:

In GR, space time is distinguished from matters or physical fields. Gravitational field is not a physical field, but a geometric field that is the origin of space-time. In the framework, Can one say “space-time tells matter how to move; matter tells space-time how to curve”.

In TGD, space time is distinguished from kahler field, because the origin of space time is from induced metric on the sub-manifold of the imbedding space that is the same as gravitation, but the origin of Kahler field is from induced kahler form on the sub-manifold. Therefore Can one say “space-time tells the dynamics of kahler field and kahler field tells space-time how to curve”?

This kahler field is only superposition of the effects of physical fields on the submanifold not themselves(?). In similarly gravitational field or induced metric on the submanifold is superposition of effects of gravitational fields of matters? But I don’t see any room for placing these superpositions in the kahler field or gravitational field, because these fields are just induced from imbedding space.

I use the word submanifold instead of space-time because kahler field is not something over space-time as like physical fields in GR are over space-time. In other words space-time is not most fundamental than kahler field. Induced metric makes space time over the submanifold and separately induced kahler form makes kahler field over the submanifold. is there any wrong here?

Because kahler action doesn’t contain gravitational field, one should write another Lagrangian density for gravity in addition to it? just as in Einstein formalism there was a term for gravity “scalar curvature”’ and other terms related to lagrangian of physical fields.

Can one call Einstein tensor, just the energy momentum tensor of gravitation field? it Is deduced from the Lagrangian of gravity “curvature Scalar” as the same as energy momentum tensor of Kahler field is deduced from Lagrangain of Kahler field? Therefore intuitionally Einstein equation tells that sum of energy momentum tensors of gravity and other physical fields should be zero! What is wrong in my arguments? Or one can say this is just a little view of zero energy ontology!?

If you see, Einstein tensor depends on Riemannian tensor as very similar to energy momentum of kahler field depends on the kahler field((induced kahler form).

Dear Hamed,

thank you for excellent questions. I add comments to your text. This makes my response long and I must split it in pieces.

[Hamed] In GR, space time is distinguished from matters or physical fields. Gravitational field is not a physical field, but a geometric field that is the origin of space-time. In the framework, Can one say “space-time tells matter how to move; matter tells space-time how to curve”.

[MP]One can distinguish between matter and geometric fields. The dream of Einstein was however the geometrization of physics so that both gravitational and other fields would reduce to geometry. In the attempts to quantize GRT the vision of Einstein was given up and one treated gravitational field as one particular massless field. Also in superstring models one gave up the geometrization of classical fields and assign them to massless modes of string.

[Hamed] In TGD, space time is distinguished from kahler field, because the origin of space time is from induced metric on the sub-manifold of the imbedding space that is the same as gravitation, but the origin of Kahler field is from induced Kahler form on the sub-manifold. Therefore Can one say “space-time tells the dynamics of kahler field and kahler field tells space-time how to curve”?

[MP] Your are right in the sense that Kahler action defines the dynamics via its preferred extremals.

You are however wrong in that also in TGD Einstein's original statement is true in the sense that preferred extremals are solutions of Einstein's equations with a cosmological term. The only difference from Einstein's theory *in gravitational sector* is that metric is induced metric: this means extremely strong restriction to allowed 4-geometries. For instance, overcritical and critical cosmologies have a finite duration.

There are several induced structures. Both the metric of imbedding space and Kahler form of CP_2 are induced to space-time surface. One can induce also the components of spinor connection of CP_2 just by projecting them to space-time surface and obtain classical electroweak gauge potentials. The projections of CP_2 Killing vector fields define candidates for classical gluons. Gamma matrices and spinor structure can be induced. This means that classical dynamics reduces to that for the shape of space-time surface classically and to that for induced spinor fields at elementary particle level.

To be continued...

Dear Hamed,

I continue...

[Hamed] This kahler field is only superposition of the effects of physical fields on the submanifold not themselves(?). In similarly gravitational field or induced metric on the submanifold is superposition of effects of gravitational fields of matters? But I don’t see any room for placing these superpositions in the kahler field or gravitational field, because these fields are just induced from imbedding space.

[MP] There is some confusion here. Induced Kahler form is purely classical and geometric notion: it is the analog of classical Maxwell field but not equal to classical em field but to its U(1) part. Preferred extremal property means that there are very few Bohr orbit like patterns of classical gauge fields - psychologist might speak of "archetypes". This would lead to contradiction with what we know without the many-sheetedness of the space-time. Test particle (small 3-surface) topologically condenses (form topological sum contacts) to all sheet having non-trivial projection to the region of M^4 in which it resides, and particle experiences the sum of the classical fields associated with them. This visual argument is "must". Superposition for fields is replaced with that for their effects - I have said this many times but this indeed an overall important difference from field theories. This is what allows enormous reduction of dynamical field like variables: just 4 imbedding space coordinate when 4 of them are eliminated using general coordinate invariance.

[Hamed]

I use the word submanifold instead of space-time because kahler field is not something over space-time as like physical fields in GR are over space-time. In other words space-time is not most fundamental than kahler field. Induced metric makes space time over the submanifold and separately induced kahler form makes kahler field over the submanifold. is there any wrong here?

[MP] There is;-). Induced Kahler form is like any classical field defined in space-time. It is however not a primary field variable: imbedding space coordinates take this role. What is fundamental is the identification of space-time as a 4-surface. This reduces the dynamics to that of 4-D drum membrane. Every classical field is induced by that dynamics as a byproduct.

[Hamed] Because kahler action doesn’t contain gravitational field, one should write another Lagrangian density for gravity in addition to it? just as in Einstein formalism there was a term for gravity “scalar curvature”’ and other terms related to lagrangian of physical fields.

[MP] Kahler action contains induced metric and thus classical gravitational field! Kahler action density is

J^{munu}J_{munu} sqrt(g_4)

J^{munu}= g^{mualpha}J^{nubeta} J_{alphabeta} is obtained by index raising from J_munu which indeed does not contain gravitational field. The metric determinant sqrt(g_4) also contains gravitational field. Only in the approximation that induced metric is flat M^4 metric the dependence on classical gravitational field disappears.

There is no need to add curvature scalar. Preferred extremals satisfy Einstein Maxwell equations with cosmological term and also minimal surface equations. This is solely due to the generalized conformal structure which I call Hamilton Jacobi structure in Minkowskian regions. This does not of course mean that Kahler action would reduce to Einstein- Maxwelll action with cosmological term!! The only condition needed to get E-M equations is the requirement that energy momentum tensor for Kahler field has a vanishing covariant divergence: this condtion is needed for preferred extremal property. Same condition can be used to derive Einstein's equations without mentioning action at all.

To be continued...

Dear Hamed,

and still....

[Hamed] Can one call Einstein tensor, just the energy momentum tensor of gravitation field? It is deduced from the Lagrangian of gravity “curvature Scalar” as the same as energy momentum tensor of Kahler field is deduced from Lagrangian of Kahler field? Therefore intuitionally Einstein equation tells that sum of energy momentum tensors of gravity and other physical fields should be zero! What is wrong in my arguments? Or one can say this is just a little view of zero energy ontology!?

If you see, Einstein tensor depends on Riemannian tensor as very similar to energy momentum of kahler field depends on the kahler field((induced kahler form).

[MP] What goes wrong is that in TGD one has just Kahler action, and field equations imply Einstein-Maxwell equations with Lambda for preferred extremals (so I strongly believe!). They are *not* obtained from a variation of Einstein Maxwell actions.

Indeed, in Einstein's approach one would have Maxwell action plus curvature scalar plus volume term. This would give by Noether's theorem for the action of diffeomorphisms (gauge transformations by general coordinate invariance) T- kG-lambda g =0 so that all conserved quantities associated with diffeomorphisms vanish as they should- in particular, they vanish for local Poincare transformations whose identification is far from unique.

This is the Noetherian catastrophe of GRT and leads to endless attempts to identify conserved quantities in some ad hoc manner. This is avoided in TGD framework.

We can accuse Hilbert for leading us to wrong path by discovering that Einstein's equations are obtained by adding to the action the curvature scalar;-)!

There are very strong objections against Einstein Maxwell action. For instance, curvature scalar does not have definite sign and this leads to problems even if one makes Wick rotation and integrates over metrics with Euclidian signature in path integral.

There are however speculations that supersymmetric variant without matter term might give rise to a divergence-free perturbation theory. The idea that gravitation could be in some sense a square of gauge theory is behind this thinking. In TGD one goes further: by bosonic emergence bosons as quantal objects are "squares" of fermions residing at string world sheets. Gravitons are actually fourth powers of fermions;-).

Dear Matti,

Thanks a lot.

From your answer:

“Test particle (small 3-surface) topologically condenses (form topological sum contacts) to all sheet having non-trivial projection to the region of M^4 in which it resides, and particle experiences the sum of the classical fields associated with them. This visual argument is "must". Superposition for fields is replaced with that for their effects.”

I don’t understand yet how this effects is contributed to the Kahler action? kahler action is just from lagrangian density of Kahler field that is a function of induced metric and induced kahler form. How these effects appear in the Lagrangian density?

For example if we have some distribution of matters on the space-time, In GR one can calculate energy momentum tensor from the distribution of matters and Einstein equation gives us metric of space-time.

How do you the same thing in TGD?

[Dear Hamed,

this is good question since I see this point as enormously important.

a) In practice all field theories are linearized in the lowest order approximation. If one wants to calculate interactions one forms the sum of various fields generated by the particles and calculates the effects of the resulting field on various sources: F= q(E +vxB)= q*\sum_i(E_i+v\tme B_i) for electrodynamics!.This procedure works nicely for discrete set of sources but for continuous sources one meets problems with self-interactions.

b) In TGD framework there is no linear superposition of classical fields at any level. Not for induced metric, various gauge potentials nor induced Kahler form since they are nonlinear in the primary field variables and their gradients. Primary fields are mbedding space coordinates from which 4 can be eliminated by taking them space-time coordinates. Linear superposition is also impossible for primary field variables.

c) How does one get the physical counterpart of linear superposition? The crucial observation is that we observe only the superposition for effects of fields, not for the fields themselves. This fact has been forgotten by theoreticians long time ago although basic text books (such as my old Alonso-Finn emphasizes it). The linear superposition of fields is quite too strong a hypothesis.

d) What one obtains in the recent case then? The linear superposition for fields is replaced with the set theoretic union of space-time sheets carrying the fields. This is the king idea: + goes to U!

One has space-time sheets: think of 2-D space-time sheet in E^3 with extremely small distance of order CP_2 radius between them. By their vicinity they tend to touch each other by forming wormhole contacts and therefore interact. These wormhole contacts do not carry magnetic charge and are therefore unstable so that they do not represent elementary particles.

These wormhole contacts mean interaction between particle like space-time sheet and bigger space-time sheets carrying effects of external sources. The effect on particle is in the first approximation just the sum of effects - say gauge forces and gravitational acceleration.

There is of course feedback as in the standard description. Particle modifies also the space-time sheets a little bit since the generation of wormhole contact changes the preferred extremals in question a little bit.

e) Consider now your example. In the case of the gravitation the effective deviation of metric experienced by particle would be estimated as follows. Use common (say linear) M^4 coordinates for the space-time sheets. One sums up the deviations of the corresponding induced metrics from flat M^4 metric at various space-time sheet. This defines the effective metric giving rise to the gravitational acceleration experienced by the test particle idealized to point particle. Particle experiences sum of gravitational accelerations just as in standard description. Similar prescription applies to classical gauge forces.

f) The effects do not appear in the form of Lagrangian density: there is no need to add any interaction terms in Kahler action. The value of Kahler action of course changes by terms which can be interpreted as interaction terms: wormhole contacts are created and space-sheets are slightly deformed in the interaction by the condition of being preferred exremal.

Dear Hamed,

you asked earlier: Can one say “space-time tells the dynamics of kahler field and kahler field tells space-time how to curve”?

I did not get your point completely. Kahler field indeed replaces matter understood classical fields so that in this sense you are correct. What is new that both Kahler field as a "matter field" and metric are dictated by the surface property in terms of imbedding space coordinates and gradients. This means a reduction to a deeper level completely analogous to that in string models.

Do not however forget that also induced spinor fields are present. The consistency of their couplings to the purely geometric degrees of freedom requires that one must use in Dirac action modified gamma matrices which satisfy the condition D_alpha Gamma^alpha=0, which implies field equations for Kahler action. This also implies super-symmetry: there exists an infinite number of conserved fermionic charges.

For induced spinor fields for which conservation of electric charge implies restriction to 2-D string world sheets (right handed neutrino is an obvious exception) so that a very close connection with string theory in 4-D space-time emerges.

Never say never, Ulla. AdS depends integrally on negative stress, as in percolation, but, yes, hard to rationalize in the vacuum - Hoyle and Narlikar were on a "continuous creation" beat.

Now here's a reframe of the problem, working from a surface (!) to a volume element, and the effect in the vacuum of its expansion. Via the only *identity element* I could find in arxiv Physics.

http://arxiv.org/abs/hep-th/0112169v3

I've scraped some time to work on Generalized Uncertainty, and find it roots in Bohr correspondence on just Kepler's 3 laws and Newton's 4th - fluid dynamics - where the identity element appears, in the relativistic appearance.

As for Justin Bieber, I see he was knocked of the all-time top spot on YouTube by a ... peace video!

Cheers!

Negative stress? What is that?

Stress is force per unit area, i.e. pressure (as drives percolation) but in physics more often referring ot field forces. Weyl did fields as stresses, and Tij in Gr is a stress tensor. Negative stress implies a deficit relative to background, so in cosmology that means creationism or multiverse, which are hard to distinguish at this point.

There are some candidate multiverse bubble-traces, which might give background torsion. All very speculative like cosmology is. But I'm not surprised that fractal structure does not reach the largest scales.

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Dear Tahir Sumar and Muhammed Ibraheem,

my sincere request is that you would stop infecting this blog by your silly commercials.

Matti Pitkanen

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