The objection generalizes also to induced gauge fields expressible solely in terms of CP2 coordinates and their gradients. This argument is not so strong as one might think first since in standard model only classical electromagnetic field plays an important role.
- Any electromagnetic gauge potential has in principle a local imbedding in some region. Preferred extremal property poses strong additional constraints and the linear superposition of massless modes possible in Maxwell's electrodynamics is not possible.
- There are also global constraints leading to topological quantization playing a central role in the interpretation of TGD and leads to the notions of field body and magnetic body having non-trivial application even in non-perturbative hadron physics. For a very large class of preferred extremals space-time sheets decompose into regions having interpretation as geometric counterparts for massless quanta characterized by local polarization and momentum directions. Therefore it seems that TGD space-time is very quantal. Is it possible to obtain from TGD what we have used to call classical physics at all?
The imbeddability constraint has actually highly desirable implications in cosmology. The enormously tight constraints from imbeddability imply that imbeddable Robertson-Walker cosmologies with infinite duration are sub-critical so that the most pressing problem of General Relativity disappears. Critical and over-critical cosmologies are unique apart from a parameter characterizing their duration and critical cosmology replaces both inflationary cosmology and cosmology characterized by accelerating expansion. In inflationary theories the situation is just the opposite of this: one ends up with fine tuning of inflaton potential in order to obtain recent day cosmology.
Despite these and many other nice implications of the induced field concept and of sub-manifold gravity the basic question remains. Is the imbeddability condition too strong physically? What about linear superposition of fields which is exact for Maxwell's electrodynamics in vacuum and a good approximation central also in gauge theories. Can one obtain linear superposition in some sense?
- Linear superposition for small deformations of gauge fields makes sense also in TGD but for space-time sheets the field variables would be the deformations of CP2 coordinates which are scalar fields. One could use preferred complex coordinates determined about SU(3) rotation to do perturbation theory but the idea about perturbations of metric and gauge fields would be lost. This does not look promising. Could linear superposition for fields be replaced with something more general but physically equivalent?
- This is indeed possible. The basic observation is utterly simple: what we know is that the effects of gauge fields superpose. The assumption that fields superpose is un-necessary! This is a highly non-trivial lesson in what operationalism means for theoreticians tending to take these kind of considerations as mere "philosphy".
- The hypothesis is that the superposition of effects of gauge fields occurs when the M4 projections of space-time sheets carrying gauge and gravitational fields intersect so that the sheets are extremely near to each other and can touch each other ( CP2 size is the relevant scale).
A more detailed formulation goes as follows.
- One can introduce common M4 coordinates for the space-time sheets. A test particle (or real particle) is identifiable as a wormhole contact and is therefore pointlike in excellent approximation. In the intersection region for M4 projections of space-time sheets the particle forms topological sum contacts with all the space-time sheets for which M4 projections intersect.
- The test particle experiences the sum of various gauge potentials of space-time sheets involved. For Maxwellian gauge fields linear superposition is obtained. For non-Abelian gauge fields gauge fields contain interaction terms between gauge potentials associated with different space-time sheets. Also the quantum generalization is obvious. The sum of the fields induces quantum transitions for states of individual space time sheets in some sense stationary in their internal gauge potentials.
- The linear superposition applies also in the case of gravitation. The induced metric for each space-time sheet can be expressed as a sum of Minkowski metric and CP2 part having interpretation as gravitational field. The natural hypothesis that in the above kind of situation the effective metric is sum of Minkowski metric with the sum of the CP2 contributions from various sheets. The effective metric for the system is well-defined and one can calculate a curvature tensor for it among other things and it contains naturally the interaction terms between different space-time sheets. At the Newtonian limit one obtains linear superposition of gravitational potentials. One can also postulate that test particles moving along geodesics in the effective metric. These geodesics are not geodesics in the metrics of the space-time sheets.
- This picture makes it possible to interpret classical physics as the physics based on effective gauge and gravitational fields and applying in the regions where there are many space-time sheets which M4 intersections are non-empty. The loss of quantum coherence would be due to the effective superposition of very many modes having random phases.
The effective superposition of the CP2 parts of the induced metrics gives rise to an effective metric which is not in general imbeddable to M4× CP2. Therefore many-sheeted space-time makes possible a rather wide repertoire of 4-metrics realized as effective metrics as one might have expected and the basic objection can be circumvented In asymptotic regions where one can expect single sheetedness, only a rather narrow repertoire of "archetypal" field patterns of gauge fields and gravitational fields defined by topological field quanta is possible.
The skeptic can argue that this still need not make possible the imbedding of a rotating black hole metric as induced metric in any physically natural manner. This might be the case but need of course not be a catastrophe. We do not really know whether rotating blackhole metric is realized in Nature. I have indeed proposed that TGD predicts new physics new physics in rotating systems. Unfortunately, gravity probe B could not check whether this new physics is there since it was located at equator where the new effects vanish.
For background and more details see either the article Could the measurements trying to detect absolute motion of Earth allow to test sub-manifold gravity? or the chapter TGD and GRT of "Physics in Many-Sheeted Space-time".
17 comments:
This was good for me, thanks. I struggle with Poincare and Einstein and their spacetimes. Look, jumping matter and jumping light.
http://www4.rcf.bnl.gov/brahms/WWW/pubs/NPA757_1.pdf
http://www.relativitycalculator.com/articles/Bose-Einstein-Condensate/Quantum_Quirk.html
Evolution and dust in cosmology http://dark.nbi.ku.dk/research/publications/theses/phd/christa_galls/Christa_Galls.pdf/
Stephen Cowley: http://www-thphys.physics.ox.ac.uk/people/AlexanderSchekochihin/wpi/workshop4_pdfs/cowley.pdf
Dear matti,
some questions:
When I see an object on my hand, in really there are a lot of constant curvature 3-sufaces? ;-), then why do I see a classical space time sheet as the object?
One can respect to every space-time a kahler action? Then One can assign infinite space-times to every constant curvature 3-surfaces and classical space-time is a space-time with absolute minimum kahler action for some preferred 3-surface? Then what is the role of other 3-surfaces? (non-preferred 3-surfaces)
At “Space-time as orbit of particle like 3-surfaces.”
Does the particle like 3-surfaces and space-time are the same as x3 and x4 in x4(x3)?
The space-time is classical space-time?
Is your purpose from the word “orbit” is the same as the word in lie group theory? Then In action of what group the 3-surface define the space time?
Are x4 in x4(x3) and m4 in m4*cp2, same?
A mistake (I know m4 and x4 are very different, you must laugh with my latest question ;-).)
It can be better that I replace latest question with something like what is the role of m4 in the approach of symmetric spaces.
Dear Hamed,
thank you for questions. I decided to split them to pieces to make answering simpler.
Q: When I see an object on my hand, in really there are a lot of constant curvature 3-surfaces? ;-), then why do I see a classical space time sheet as
the object?
A: I will comment the "constant curvature" below. When I look around me I see everywhere 2-dimensional surfaces which I interpret as outer surfaces of 3-dimensional objects that I have used to call matter.
One of the king ideas of TGD is that matter -or rather its shape- is topologized: material objects correspond to space-time sheets. Space-time topology is pure science fiction even in everyday length scales. Not only in Planck length scale (quantum foam hypothesis).
i) One interpretation for the outer surface would be as a genuine boundary.
ii) Second- more plausible but practically equivalent interpretation- would be as a 2-D surface at which the signature of the induced metric changes from Minkowskian to Euclidian.
I cannot say for certain which interpretation is correct but at this moment I prefer option ii) so that I will continue by something about it.
Even my own physical body would have the space-time sheet. Next you need the notion of topological sum: topological sum means that two surfaces touch each other and fuse together. One can think that one drills hole to both surfaces and connects them with a tube. Draw the 2-D case to get the idea. At this space-time sheet would topologically condense (topological sum) smaller space-time sheets representing body parts, at these the space-time sheets representing organs. This hierarchy of space-time sheets would continue: organelles, cells, molecules, atoms.... down to the level of quarks and leptons.
Q: One can respect to every space-time a kahler action? Then One can assign infinite space-times to every constant curvature 3-surfaces and classical space-time is a space-time with absolute minimum kahler action for some preferred 3-surface?
A: Space-time sheets are not constant curvature surfaces! (Vy the way, soap bubbles would be 2-D constant curvature surfaces whereas soap films spanned by frames are minimal surfaces).
In string theory one has minimal surfaces: string orbit would be analogous to soap film. The action minimized is the surface area for string world sheet.
In TGD the direct generalization would be volume of space-time surface but this action does not seem to be physical. What I minimize is Kahler action which is Maxwell action but for a Maxwell field which is purely geometric. This means that the dynamics of gauge fields reduces to the analog of the dynamics of vibrating membrane - now however 3-dimensional. In aether hypothesis one identifies fields as vibrations of mysterious substance called aether. In TGD gauge fields result as vibrations of 3-D membrane defining 3-space.
To be continued....
Dear Hamed,
here is the continuation of my answer.
Q: Then what is the role of other 3-surfaces? (non-preferred 3-surfaces)
When I talk about preferred extremal, I mean 4-D surfaces. So that I take the liberty of replacing "3-" by "4-" in your question;-).
*General Coordinate Invariance of the metric of WCW ( infinite-D (!) space of all possible 3-surfaces) requires that the definition of the WCW metric assigns to each 3-D surface a unique 4-D surface- space-time surface at which 4-D general coordinate transformations act.
*What is this 4-D surface? Quantum classical correspondence gives the idea: it must be a solution to field equations analogous to Maxwell equations determining the dynamics for the analog of 3-D soap membrane. Kahler action is the action principle. Classical dynamics reduces to the geometry of WCW in full spirit with the geometrization program of entire physics.
*Space-time surface must be a preferred exremal of Kaehler action with which I mean following. It is certainly a solution of field equations but this is not enough since there is infinite number of solutions "going through" a given 3-surface (analogous to the boundary frame of soap film) .
*Preferred extremal must satisfy additional conditions so that it is very much like Bohr orbit in Bohr's model of atom for which these additional conditions lead to energy and angular momentum quantization. In fact, General Coordinate Invariance implies the counterparts of Bohr rules through this preferred extremal condition so that totally unexpected connection emerges.
*What these conditions for preferred extremals is a non-trivial question: I have made several guesses and I think that I understand the problem now. I will not go to this here because it is quite too technical.
To be continued....
And still more....
Q: ?Space-time as orbit of particle like 3-surfaces.? Does the particle like 3-surfaces and space-time are the same as x3 and x4 in x4(x3)? The space-time is classical space-time?
A: To visualize think of a 1-D curve which moves in 3-D space. Its orbit defines a 2-D surface. In the same manner the orbit of 3-D surface in 8-D space-time defines 4-D surface identified as space-time surface.
Many-sheeted space-time unifies the notions of particle and space-time. Particles are in this picture quanta of space-time having finite size. When one looks with microscope the interior of particle one sees just space-time. When one looks particle space-time sheet in birds' eye of view one sees it as a particle: a surface with outer boundary. Particle or space-time: this depends on the scale in which you look at the situation.
Q: Is your purpose from the word ?orbit? is the same as the word in lie group theory? Then In action of what group the 3-surface define the space time?
A: Orbit in Lie group theory is analogous to 1-D orbit of point-like particle (0-dimensional) object. Now the object is 3-dimensional and orbit has dimension 3+1=4.
Q: Are x4 in x4(x3) and m4 in m4*cp2, same?
A: Again we can consider a simplified example. Consider 1-D curve instead of X^4 in 3-D space RxR^2 instead of M^4xCP_2. R could be z-axis and R^2 the 2-D plane. This 1-D curve is not R except in very special case that the curve is straight line.
In the same manner one can have imbedding of M^4 in M^4xCP_2 by putting CP_2 coordinates constant and allowing M^4 coordinates to vary. One obtains in this case X^4=M^4. By performing deformations of this surface, one can obtain more interesting space-time surfaces for which X^4=M^4 is not true.
As I started to work with TGD I read the old classic of Eisenhart about sub-manifold geometry. Certainly there are many other books about sub-manifold geometry and it is probably easy to find them either from math library or web.
Matti:
An earlier comment that I'm not sure you noticed. If you would so oblige...
Do you believe space to be a crucible?
The origin of the inward gravitational motion and the motion of the outward expansion of space and time are one and the same. Together they interact forming both the large structure of the universe, and the microcosmic structure of matter. The origin of gravity, the expansion of the universe and the cohesion of solids are all produced by the same theoretical motion that constitutes radiation and matter itself. The existence of this extremely simple yet powerful mechanism is the necessary consequence of the nature of the proportions of space and time in the equation of motion.
The reasons, therefore, why gravity cannot be detected except in its effects, and why it cannot be screened-off, or modified in any way, is simple and straightforward: it is because the same motion that constitutes mass and inertia is also producing the action of gravity; the three-dimensional inward scalar motion of matter opposes the three-dimensional outward scalar progression of space and causes each mass aggregate to independently move inward towards all space locations and thus towards all other mass aggregates in sufficient proximity. But the three-dimensional displacement is also distributed three-dimensionally in extension space and thus attenuated by the inverse square law so that at a certain distance, which Larson calls the "gravitational limit," the three-dimensional outward Progression of space is greater than the inward motion of mass, and thus commences to disperse the locations of space, maintaining the separation of heavenly bodies.
Regards.
So thanks Matti, I studied your comments very carefully, my intuition of the world increased very much! What I saw and what I see now ;-).
I’m going to study as you said sub-manifold geometry but first on the topic of symmetric spaces And at the same time I think about decomposition of configuration space into union of coset spaces in TGD. Have you better suggestion in my process?
Dear Anonymous,
I noticed but forgot. Apologies.
I do not quite understand what you mean by crucible: I looked from web the meaning and understood that space-time would be some kind of vessel containing matter.
I do not certainly propose this. Space-time *is* the matter: both in the sense of metric geometry- the analogs of Einstein equations- and also in the sense of coding for the shape of material objects, which in TGD framework are space-time sheets. This means unification of the notions of particle and space-time: whether one sees a piece of space-time or particle as surface depends on the scale of resolution.
The model for gravitation must be able to produce precise quantitative predictions: General Theory of Relativity defines quite an impressive standard here. I do not know whether you have counterparts for Einstein's equations so that I cannot comment your proposal further.
Dear Hamed,
the notion of configuration space- I prefer to say world of classical worlds- is rather abstract. It took 12 years and many futile attempts for me to end up with it;-).
Maybe it is best to gain first good understanding of about TGD based space-time concept. The understanding of the notion of surface is certainly essential. One should gain kind of intuitive phenomenological understanding. Requires just some hard work.
Induction of various geometric structures is the core of sub-manifold geometry.
a) The basic notion is induced metric (measuring distances using imbedding space metric). Induced metric is projection of imbedding space metric to space-time surface. One encounters also second fundamental form defining a generalization of acceleration vector for a curve , its trace vanishing for minimal surfaces so that in some sense minimal surfaces are generalizations of geodesics.
b) One can generalize the induction of metric to parallel translation using induced connection. This gives rise to a geometrization of gauge potentials and selects CP_2. Induce gauge potentials are just projections of CP_2 spinor connection to space-time surface.
c) A further important notion is induction of spinor structure. One uses spinors of M^4xCP_2 and obtains geometrization of electroweak quantum nimbers.
Dear Hamed,
I think that M^4 as a non-compact space cannot be regarded in as a symmetric space of form G/H in any interesting manner. In any case, M^4 as a flat space allows in well-defined sense maximal symmetries which is the characteristic of symmetric spaces which are also constant curvature spaces as also M^4 in trivial manner.
In fact, CP_2 is constant curvature space and CP_2 type vacuum extremals which have light-like random curve as M^4 projection are locally isometric with CP_2 and locally constant curvature spaces. The 4-D lines of generalize Feynman graphs correspond to deformations of these spaces.
M^4 is unique because its light-cone has 3-D like surface as boundary. This space is metrically 2-D and allows infinite-D group of conformal symmetries extending the ordinary 2-D conformal symmetries. This makes the imbedding space with 4 large dimensions a unique choices.
Space-time surfaces have also dimension 4 and the reason is same light-like 3-surfaces defining the wormhole throats at which the signature of the induced metric changes allow infinite-D conformal invariance.
Matti:
Firstly, re General Relativity and Mercury's orbital eccentricity. Here are a few statements on the subject:-
"...aside from the perihelia motion of the planets, Newcomb, in the late 19th century, testified to several other disturbances in the solar family. Two of these were the variation in the eccentricity of Mercury and the motion of the line of nodes of Venus. In this regard, General Relativity becomes useless since it is unable to even explain in mathematical terminology both the secular distortions in Mercury's eccentricity and the nodes of Venus. These distortions amount to the exceeded values of -0.88" and +10.2" arc per century, respectively."
"Relativity does not predict the excess perturbations in the orbits of the planets other than those of the motions of their perihelia. Still, of all the secular perturbations observed, General Relativity [closely] agrees only with the motion of the orbit of Mercury. It fails to account for more or less than this perturbation since there exists no flexibility in the equations of Einstein, no uncertainty in the calculations, and no room for compensation.
"Take the case of Venus for example. Relativity estimates an excess orbital rotation rate of + 8.6" arc per century with a forward motion. Yet the computed excess rotational rate was found to be -7.3" arc per century with a retrograde motion! "[Therefore it appears] that the forward rotation of Mercury's orbit is a mere coincidence since Relativity theory predicts the correct anomaly for this sole case....however, relativists stress this coincidence as one of the four conclusive proofs for the curved geometry of space and General Relativity.
"The motion of the other perihelia has not been the only fallacy found in Einstein's theories. The alleged deflection of light in the vicinity of a massive body.....(etc.,etc.)". (Poor)
Dear Anonymous,
I am unable to say much about these statements because I am not a specialist.
In any case, already for Mercury the effect is small and the effects are expected to be much smaller for other planets. I would guess that they go like GM/r and planetary perturbations could well mask them. Mercury would be in very special position as far testing of GRT effects is considered.
An interesting question is of course where these statements are made and whether every specialist agrees with them.
Matti
This feels familiar :)
Can it be used as ref?
http://arxiv.org/PS_cache/arxiv/pdf/0908/0908.0738v2.pdf
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