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Physics in Many-Sheeted Space-Time
Note: Newest contributions are at the top!
There was a very interesting link at Thinking Allowed Original in this morning - a lot of thanks for Ulla. The link was to two old anomalies discovered in the solar system: Pioneer anomaly and Flyby anomalies with which I worked for years ago. I remember only the general idea that dark matter concentrations at orbits of planets or at spheres with radii equal that of orbit could cause the anomaly. So I try to reconstruct all from scratch and during reconstruction become aware of something new and elegant that I could not discover for years ago.
The article says that Pioneer anomaly is understood. I am not at all convinced about the solution of Pioneer anomaly. Several "no new physics" solutions have been tailored during years but later it has been found that they do not work.
Suppose that dark matter is at the surface of sphere so that by a well-known text book theorem it does not create gravitational force inside it. This is an overall important fact, which I did not use earlier. The model explains both anomalies and also allow to calculate the total amount of dark matter at the sphere.
A simple TGD model for the sphere containing dark matter could be in terms of boundary defined by gigantic wormhole contact (at its space-time sheet representing "line of generalized Feynman diagram" one has deformation of CP2 type vacuum extremal with Euclidian signature of induced metric) with radius given by the radius of Bohr orbit with gravitational Planck constant equal to hgr =GMm/v0. This radius does not depend on the mass of the particle involved and is given by rn= GM/v03 where 2GM is Schwartschild radius equal to 3 km for Sun. One has v0/c≈ 2-11.
An interesting possibility is that also Earth-Moon system contains a spherical shell of dark matter at distance given by the radius of Moon's orbit (about 60 Earth's radii). If so the analogs of the two effects could be observed also in Earth Moon system and the testing of the effects would become much easier. This would also mean understanding of the formation of Moon. Also interior of Earth (and also Sun) could contain spherical shells containing dark matter as the TGD inspired model for the spherically symmetric orbit constructed for more than two decades ago suggests. One can raise interesting questions. Could also the matter in mass scale systems be accompanied by dark matter shells at radii equal to Bohr radii in the first approximation and could these effects be tested? Note that a universal surface density for dark matter predicts that the change of acceleration universally be given by Hubble constant H.
For details see the chapter TGD and Astrophysics or the article Pioneer and Flyby anomalies for almost ten years later.
One of the basic predictions of TGD is the variation of effective light velocity determined in terms of the time taken to travel from point A to B. This time depends on the space-time sheet along which the photon, neutrino or some other elementary particle propagates (note that neutrinos are not quite massless). This effect is one of the key signatures of many-sheeted space-time manifesting as anomalies of general relativity.
The space-time of general relativity corresponds to effective space-time obtained by replacing the sheets of space-time with single region of Minkowski space with metric replaced with the sum of empty space Minkowski metric with the deviations of the induced metrics of the space-time sheets (effects of classical fields on space-time sheets on test particle sum up since it touches all the space-time sheets: linear superposition of fields is replaced with that for their effects).
SN1987A supernova provides first evidence for the presence of several space-time sheets. Neutrinos came as two bursts and before photons. The variation of effective light-velocity was of order Δ c/c ≈ 2× 10-9. Opera experiment claimed much larger variation of order: Δ c/c ≈ 10-5: unfortunately there was an error in the analysis of the experiments. Now Lubos has a posting about galactic blackhole Sagittarius A as neutrino factory. Chandra X-ray observatory and also Nustar and Swift Gamma-Ray Burst Mission detected some X-ray flares from Sagittarius A. 2-3 hours earlier IceCube detected high energy neutrinos by IceCube on the South Pole. As a good conservative Lubos of course denies the effect as he denies also climate warming. He did not however claim that experimenters are communists;-).
Could neutrinos arrive from galactic center? If they move with the same (actually somewhat lower) velocity than photons, this cannot be the case. The neutrinos did the same trick as SN1987A neutrinos and arrived 2-3 hours before the X-rays! What if one takes TGD seriously and estimates Δ c/c for this event? The result is Δ c/c ∼ (1.25-1.40 )×10-8 for 3 hours lapse using the estimate r= 25,900+/- 1,400 light years (see this). Δ c/c is by a factor 4 larger than for SN1987A at distance about 168,000 light years (see this). This distance is roughly 8 times longer. This would suggests that the smaller the space-time sheets the nearer the velocity of neutrinos is to its maximal value. For photons the reduction from the maximal signal velocity is larger.
For details see the chapter TGD and potential anomalies of GRT.
Lubos has written a highly rhetoric, polemic, and adrenaline rich comments posting about the mediabuzz related to supernova SN 1987 A. The target of Lubos is the explanation proposed by James Franson from the University of Maryland for the findings discussed in Physics Archive Blog. I do not have any strong attitude to Franson's explanation but the buzz is about very real thing: unfortunately Lubos tends to forget the facts in his extreme orthodoxy.
What happened was following. Two separate neutrino bursts arrived from SN1987 A. At 7.35 AM Kamionakande detected 11 antineutrons, IMB 8 antineutrinos, and Baksan 5 antineutrinos. Approximately 3 hours later Mont Blanc liquid scintillator detected 5 antineutrinos. Optical signal came 4.7 hours later.
The are several very real problems as one can get convinced by going to Wikipedia:
In TGD framework the explanation would be in terms of many-sheeted space-time. In GRT limit of TGD the sheets of the many-sheeted space-time time are lumped to single sheet: Minkowski space with effective metric defined by the sum of Minkowski metric and deviations of the metrics of the various sheets from Minkowski metric. The same recipe gives effective gauge potentials in terms of induced gauge potentials.
Different arrival times for neutrinos and photons would be however a direct signature of the many-sheeted space-time since the propagation velocity along space-time sheets depends on the induced metric. The larger the deviation from the flat metric is, the slower the propagation velocity and thus longer the arrival time is. Two neutrino bursts would have explanation as arrivals along two different space-time sheets. Different velocity for photons and neutrinos could be explained if they arrive along different space-time sheets. I proposed for more than two decades ago this mechanism as an explanation for the finding of cosmologists that there are two different Hubble constants: they would correspond to different space-time sheets.
The distance of SN1987A is 168,000 light- years. This means that the difference between velocities is Δ c/c ≈ Δ T/T≈ 3 hours/168 × 103≈ 2× 10-9. The long distance is what makes the effect visible.
I proposed earlier sub-manifold gravity as an explanation for the claimed super-luminality of the neutrinos coming to Gran Sasso from CERN and mentioned in this context also SN1987A but did not compare the deviations from the light velocity. In this case the effect would have been Δ c/c≈ 2.5× 10-5 and thus four orders of magnitude larger than four supernova neutrinos. It however turned out that the effect was not real.
For details see the chapter TGD and potential anomalies of GRT.
The earlier attempts to understand the relationship between TGD and GRT have been in terms of solutions of Einstein's equations imbeddable to M4× CP2 instead of introducing GRT space-time as a fictive notion naturally emerging from TGD as a simplified concept replacing many-sheeted space-time. This resolves also the worries related to Equivalence Principle. TGD can be seen as a "microscopic" theory behind TGD and the understanding of the microscopic elements becomes the main focus of theoretical and hopefully also experimental work some day.
The understanding of Kähler Dirac action has been second long term project. How can one guarantee that em charge is well-defined for the spinor modes when classical W fields are present? How to avoid large parity breaking effects due to classical Z0 fields? How to avoid the problems due to the fact that color rotations induced vielbein rotation of weak fields? The common answer to these questions is restriction of the modes of induced spinor field to 2-D string world sheets (and possibly also partonic 2-surfaces) such that the induced weak fields vanish. This makes string picture a part of TGD.
For details see the chapter TGD and GRT or the article Further progress concerning the relationship between TGD and GRT and Kähler-Dirac action.
Quantum criticality is the TGD counterpart of the inflation and the flatness of 3-space follows from the condition that no local dimensional quantities are present in 3-geometry. Also the imbeddability fo M4 is an important piece of story and restricts the set the parameters of imbeddable cosmologies dramatically.
One can try to understand the situation microscopically in terms of the cosmic strings which gradually develop higher than 2-D M4 projection during cosmic evolution and become magnetic flux tubes carrying magnetic monopole fluxes explaining the presence of magnetic fields in cosmology.
At microscopic level magnetic flux tubes are the key structural elements. The phase transitions increasing Planck constant for the matter associated with flux tubes and thus also the lengths of magnetic flux tubes should be important as also the phase transitions increasing p-adic prime and reducing Planck constant originally emerged in the modelling of TGD inspired quantum biology are highly suggestive. First transitions would mean adiabatic expansion with no heat generation and latter transitions would liberate magnetic field energy since flux conservation forces field strength to be reduced and leads to liberation of magnetic energy producing ordinary matter and dark matter. Dark energy in turn is identifiable as magnetic energy.
The key question concerns the mechanism causing the isotropy and homogeny of the cosmology. There are two possible identifications.
The formulas used to make back of the envelope (see this) calculations in inflation theory discussed in a guest posting in Lubos's blog given some idea about TGD counterpart for the generation of gravitons. Inflationary period is replaced with essentially unique critical cosmology containing only its duration as a free parameter. The fluctuations in the duration of this parameter explain scalar temperature fluctuations assoiated with CMB.
How the local polarization of CMB is generated?
There is a nice discussion about the mechanism leading to the generation of CMB polarization (see this). The polarization is generated after the decoupling of CMB photons from thermal equilibrium and is due to the scattering of photons on free electrons during decoupling. This scattering is known as Thomson scattering. The page in question contains schematic illustrations for how the polarization is generated. The scattering from electrons polarizes the photons in direction orthogonal to the scattering plane. In thermal equilibrium the net polarization of scattered radiation vanishes. If however the scattered photons from two perpendicular directions have different intensities a net polarization develops.
Polarized photons could be produced only during a short period during recombination scattering from free electrons was still possible and photons could diffuse between regions with different temperature. Polarized photons were generated when electrons from hot and cold regions where scattering on same electrons. CMB polarization indeed varies over sky but not in long length scales since photons could not diffuse for long lengths.
So called quadrupole anisotropy of CMB temperature contains information about the polarization. There are three contributions: scalar, vector, and tensor.
How the polarization anisotropies could be generated in TGD Universe?
One can try to understand microscopically how the polarization anisotropies are generated in TGD framework using poor man's arguments.
Back on the envelope calculations in TGD framework
One can modify the back on the envelope calculations of John Preskill (see this) in Lubos's blog to see what could happen in TGD framework. Now one however starts from the critical cosmology fixed apart from its duration and looks what it gives rather than starting from Higgs potential for inflaton field. The obvious counterpart for inflaton scalar field would be magnetic field intensity having same dimension but one should avoid too concrete correspondences.
The key question is whether the critical period generates the rapid expansion smoothing out inhomogenities or whether it generates them. The original guess that it smooths them out turns out be wrong in closer examination.
The possibility of very rapid expansion near a=aF<a1 leading to radiation dominated cosmology should have some deep meaning. The following tries to catch this meaning.
TGD inspired critical cosmology (see this) relies on the identification of 3-space as a= constant section, where a is Lorentz invariant cosmological time defined by the light-cone proper time a=(m0)2-rM2)1/2, and from the assumption that (quantum) criticality corresponds to a vanishing 3-curvature meaning that 3-space is Euclidian.
The condition that the induced metric of the a= constant section is Euclidian, fixes the critical cosmology apart from its duration a0 from the existence of its vacuum extremal imbedding to M4× S2, where S2 homologically trivial geodesic sphere:
ds2 = gaada2 -a2 (dr2 +r2dΩ2) ,
gaa= (dt/da)2=1- ε2 /(1-u2) , u=a/a0 , ε=R/a0 .
sin(Θ)= +/- u , Φ= f(r) ,
1/(1+r2) -ε2(df/dr))2=1 .
From the expression for dt/da one learns that for the small values of a it is essentially constant equal to dt/da=(1 ε2)1/2. When a/a0 approaches to (1-ε2)1/2, dt/da approaches to zero so that the rate of expansion becomes infinite. Therefore critical cosmology is analogous to inflationary cosmology with exponential expansion rate. Note that the solution is defined only inside future or past light-cone of M4 in accordance with zero energy ontology.
After this a transition to Euclidian signature of metric happens (also a transition to radiation dominated cosmology is possible): this is something completely new as compared to the general relativistic model. The expansion begins to slow down now since dt/da approaches infinity at a/a0=1. In TGD framework the regions with Euclidian signature of the induced metric are good candidates for blackhole like objects. This kind of space-time sheets could however accompany all physical systems in all scales as analogs for the lines of generalized Feynman diagrams. For sin(Θ)=1 at a/a0=1 the imbedding ceases to exist. One could consider gluing together of two copies of this cosmology together with sin(Θ)= sin(π-Θ)= a/a0 to get a closed space-time surface. The first guess is that the energy momentum tensor for the particles defined by wormhole contacts connecting the two space-time sheets satisfies Einstein's equations with cosmological constant.
Quantum criticality would be associated with the phase transitions leading to the increase of the length and thickness of magnetic flux tubes carrying Kähler magnetic monopole fluxes and explaining the presence of magnetic fields in all length scales. Kähler magnetic energy density would be reduced in this process, which is analogous to the reduction of vacuum expectation value of the inflation field transforming inflaton vacuum energy to ordinary and dark matter.
At the microscopic level one can consider two phase transitions. These phase transitions are related to the hierarchy of Planck constants and to the hierarchy of p-adic length scales corresponding to p-adic primes near powers of 2.
Although this picture is only an artist's vision and although one can imagine many alternatives, I have the feeling that the picture might contain the basic seeds of truth.
A natural question is whether TGD could allow inflationary cosmology. In the lowest order this would require imbedding of the De Sitter space. De Sitter space allows two basic coordinate slicings.
In TGD framework also the imbedding of space-time as surfaces matters besides the metric which is purely internal property. The most general ansatz for the imbedding of De Sitter metric into M4× CP2 is as a vacuum extremal for for Kähler action with the understanding that small deformation carries energy momentum tensor equal to Einstein tensor so that Einstein's equations would old true in statistical sense.
The cautious conclusion is that sub-manifold cosmology neither excludes nor favors inflationary cosmology and that critical cosmology is more natural in TGD framework. In TGD Universe De Sitter metric looks like an ideal model for the interior of a stationary star characterized by its radius just like blackhole is characterized by its radius. It seems that TGD survives the new findings at qualitative and even partially quantitative level.
BICEP2 team has announced a detection of gravitational waves via the effects of gravitational waves on the spectrum on polarization of cosmic microwave background (CMB). What happens that gravitational waves (or possibly some other mechanism) transforms so called E modes which correspond the curl free part of polarization field expressible as gradient to B modes responsible for the divergenceless part of polarization field expressible as curl of vector field.
Interaction of photons with gravitons would induce this polarization changing transformation: this is discussed in earlier post by Lubos. The signal is unexpectedly strong constraints on possible models, in particular to the inflationary models which are currently in fashion. There is excellent popular summary of the physics behing scalar, vector, and tensor perturbations of CMB here. The map produced by BICEP describes the vorticity of the polarization field at the sky and one can clearly see it.
There has been a lot of pre-hype about the finding as proof for inflation, which it is not. Even Scientific American falls in the sin of inflationary hyping which is a pity. Inflationary theory is only the dominating theory which might be able to explain the finding.
In the following the findings are discussed in the framework of TGD based cosmology in which the flatness of 3-space is interpreted in terms of quantum criticality rather than inflation. The key role is played by gradually thickening cosmic strings carrying magnetic monopole flux, dark energy as magnetic energy and dark matter as large heff phases at cosmic strings. Very thin cosmic strings dominate the cosmology before the emergence of space-time as we know it and quantum criticality is associated with the phase transition between these two phases. Later cosmic strings serve as seeds of various cosmological structures by decaying partially to ordinary matter somewhat like inflaton fields in inflationary cosmology. Cosmic strings also explain the presence of magnetic fields in cosmos difficult to understand in standard approch. The crucial point is that - in contrast to ordinary magnetic fields - monopole fluxes do not require for their creation any currents coherent in long scales.
Liam McAllister's summary about the findings of BICEP2 team
Liam McAllister from Cornell University has written an excellent posting about the discovery and its implications in Lubos's blog. McAllister discusses the finding from several points of view. Can one trust that the finding is real? How should one interpret the result? What are its implications? A brief summary is in order before going to details.
Comparison of inflationary models and TGD
Further conclusions depend on the cosmological model adopted and McAllister considers the situation in the framework of inflationary models and lists the basic aspects of inflationary model.
Fluctuations of gravitational field
McAllister gives a nice overall summary about the physics involved if given by inflationary models.
The natural expectation is that any theory explaining the findings in terms of gravitons produces similar prediction but with the energy density of scalar field replaced with something else. In TGD the energy density assignable to cosmic strings so that the square root of the energy density of cosmic string multiplied by some numerical factor should be the relevant parameter now.
Inflation should begin at GUT mass scale
The first implication of the findings is that if inflation explains the findings, it should have begun in GUT scale 1016 GeV, which is very high. The findings cut off a gigantic portion of the parameter space of inflationary models and leaves only inflation potentials that are approximately translationally invariant.
In TGD framework one expects that the energy scale corresponds to that in which quantum critical period begins after string dominated primordial period. This scale should be given by CP2 mass scale apart from some numerical factor. CP2 mass corresponds to m(CP2)=hbar/R(CP2), where R(CP2) is CP2 radius. p-Adic mass calculations predict the value of electron mass and assign to electron the largest Mersenne prime M127 having the property that the p-adic length scales kenosqrtpR(CP2) is not completely super-astronomical. This fixes R(CP2) and m(CP2). The outcome is m(CP2)∼ 4× 1015 GeV.
A numerical constant can be present in the estimate for the energy scale at which quantum critical period begins. In particular, the factor 1/αK1/4 should be present since Kähler action is proportional to 1/αK, which by simple argument is in excellent approximation equal to the inverse of the fine structure constant equal to 137. This would rise the estimate for the energy scale to about 1016 GeV if the same formula for it is used also in TGD (which might of course be wrong!). With a considerable dose of optimism one could say that TGD allows to understand why the measured value of r is what it is.
Difficulties of the inflationary approach
What is nice that McAllister discusses also so the difficulties of inflationary approach.
Also the ordinary Higgs mechanism is plagued by the loss of naturalness and predictivity by the fact that the Higgs particle has too low mass and SUSY has not been found in low enough mass scales to stabilize Higgs mass. In TGD framework the string tension of string like objects assignable to elementary particles would give the dominating contribution to gauge boson masses and p-adic thermodynamics in its original form the dominating contribution to fermion masses (see this and this). The couplings of fermions to Higgs are gradient couplings and the coupling is same for all fermions in accordance with naturality and universality.
The overall conclusion is that TGD survives the new findings at qualitative and even partially quantitative level.