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Physics in Many-Sheeted Space-Time
Note: Newest contributions are at the top!
The introduction of cosmological constant seems to be the only manner to explain accelerated expansion and related effects in the framework of General Relativity. As summarized in the previous posting, TGD allows different explanation of these effects. I will not however go to this here but represent comments about the notion of vacuum energy and the possibility to describe accelerated expansion in terms of cosmological constant in TGD framework.
The term vacuum energy density is bad use of language since De Sitter space is a solution of field equations with cosmological constant at the limit of vanishing energy momentum tensor carries vacuum curvature rather than vacuum energy. Thus theories with non-vanishing cosmological constant represent a family of gravitational theories for which vacuum solution is not flat so that Einstein's basic identification matter = curvature is given up. No wonder, Einstein regarded the introduction of cosmological constant as the biggest blunder of his life.
De Sitter space is representable as a hyperboloid a2-u2= -R2, where one has a2=t2-r2 and r2=x2+y2+z2. The symmetries of de Sitter space are maximal but Poincare group is replaced with Lorentz group of 5-D Minkowski space and translations are not symmetries. The value of cosmological constant is Λ= 3/R2. The presence of non-vanishing dimensional constant is from the point of view of conformal invariance a feature raising strong suspicions about the correctness of the underlying physics.
1. Imbedding of De Sitter space as a vacuum extremal
De Sitter Space is possible as a vacuum extremal in TGD framework. There exists infinite number of imbeddings as a vacuum extremal into M4×CP2. Take any infinitely long curve X in CP2 not intersecting itself (one might argue that infinitely long curve is somewhat pathological) and introduce a coordinate u for it such that its induced metric is ds2=du2. De Sitter space allows the standard imbedding to M4×X as a vacuum extremal. The imbedding can be written as u= ±[a2+R2]1/2 so that one has r2< t2+R2. The curve in question must fill at least 2-D submanifold of CP2 densely. An example is torus densely filled by the curve φ = αψ where α is irrational number. Note that even a slightest local deformation of this object induces an infinite number of self-intersections. Space-time sheet fills densely 5-D set in this case. One can ask whether this kind of objects might be analogs of D>4 branes in TGD framework. As a matter fact, CP2 projections of 1-D vacuum extremals could give rise to both the analogs of branes and strings connecting them if space-time surface contains both regular and "brany" pieces.
It might be that the 2-D Lagrangian manifolds representing CP2 projection of the most general vacuum extremal, can fill densely D> 3-dimensional sub-manifold of CP2. One can imagine construction of very complex Lagrange manifolds by gluing together pieces of 2-D Lagrangian sub-manifolds by arbitrary 1-D curves. One could also rotate 2-Lagrangian manifold along a 2-torus - just like one rotates point along 2-torus in the above example - to get a dense filling of 4-D volume of CP2.
The M4 projection of the imbedding corresponds to the region a2>-R2 containing future and past lightcones. If u varies only in range [0,u0] only hyperboloids with a2 in the range [-R2,-R2+u02] are present in the foliation. In zero energy ontology the space-like boundaries of this piece of De Sitter space, which must have u02>R2, would be carriers of positive and negative energy states. The boundary corresponding to u0=0 is space-like and analogous to the orbit of partonic 2-surface. For u02<R2 there are no space-like boundaries and the interpretation as zero energy state is not possible. Note that the restriction u02>R2 plus the choice of the branch of the imbedding corresponding to future or past directed lightcone is natural in TGD framework.
2. Could negative cosmological constant make sense in TGD framework?
The questionable feature of slightly deformed De Sitter metric as a model for the accelerated expansion is that the value of R would be same order of magnitude as the recent age of the Universe. Why should just this value of cosmic time be so special? In TGD framework one could of course consider space-time sheets having De Sitter cosmology characterized by a varying value of R. Also the replacement of one spatial coordinate with CP2 coordinate implies very strong breaking of translational invariance. Hence the explanation relying on quantization of gravitational Planck constant looks more attractive to me.
It is however always useful to make an exercise in challenging the cherished beliefs.
For details see the chapter Quantum Astrophysics.
Bohr orbit model for astrophysical objects suggests that also galactic halo should have a modular structure analogous to that of planetary system or the rings of Saturn rather than that predicted by continuous mass distribution. Quite recently it was reported that the halo of Milky Way - earlier thought to consist of single component - seems to consist of two components (see the article of Carolle et al in Nature. See also this and this).
Even more intriguingly, the stars in these halos rotate in opposite directions. The average velocities of rotation are about 25 km/s and 50 km/s for inner and outer halos respectively. The inner halo corresponds to a range 10-15 kpc of orbital radii and outer halo to 15-20 kpc. Already the constancy of rotational velocity is strange and its increase even stranger. The orbits in inner halo are more eccentric with axial ratio rmin/rmax≈ .6. For outer halo the ratio varies in the range .9-1.0. The abundances of elements heavier than Lithium are about 3 times higher in the inner halo which suggests that it has been formed earlier.
Bohr orbit model would explain halos as being due to the concentration of visible matter around ring like structures of dark matter in macroscopic quantum state with gigantic gravitational Planck constant. This would explain also the opposite directions of rotation.
One can consider two alternative models predicting constant rotation velocity for circular orbits. The first model allows circular orbits with arbitrary plane of rotation, second model and the hybrid of these models only for the orbits in galactic plane.
The big difference in the average rotation velocities < vφ> for inner and outer halos cannot be understood solely in terms of the high eccentricity of the orbits in the inner halo tending to reduce < vφ>. Using the conservation laws of angular momentum (L= mvminρmax) and of energy in Newtonian approximation one has < vφ>= ρmaxvmin< 1/ρ>. This gives the bounds
vmin< < vφ>< vmax= vmin [ρmax/ρmin]≈ 1.7 vmin .
For both models v=v0= k1/2, k=TG, (T is the effective string tension) for circular orbits. Internal consistency would require vmin<< vφ>≈.5v0<vmax≈ 1.7 vmin. On the other hand, vmax<v0 and thus vmin>.6v0 must hold true since the sign of the radial acceleration for ρmin is positive. Obviously 0.5v0>v>sub>min>.6v0 means a contradiction.
The big increase of the average rotation velocity suggests that inner and outer halos correspond to closed cosmic string like objects around which the visible matter has condensed. The inner string like object would create an additional gravitational field experienced by the stars of the outer halo. The increase of the effective string tension by factor x corresponds to the increase of < vφ> by a factor x1/2. The increase by a factor 2 plus higher eccentricity could explain the ratio of average velocities.
For details see the new chapter Quantum Astrophysics.
There are several pieces of evidence for accelerated expansion, which need not mean cosmological constant, although this is the interpretation adopted here. It is interesting to see whether this evidence is indeed consistent with TGD based interpretation.
A. The four pieces of evidence for accelerated expansion
A.1. Supernovas of type Ia
Supernovas of type Ia define standard candles since their luminosity varies in an oscillatory manner and the period is proportional to the luminosity. The period gives luminosity and from this the distance can be deduced by using Hubble's law: d= cz/H0, H0 Hubble's constant. The observation was that the farther the supernova was the more dimmer it was as it should have been. In other words, Hubble's constant increased with distance and the cosmic expansion was accelerating rather than decelerating as predicted by the standard matter dominated and radiation dominated cosmologies.
A.2 Mass density is critical and 3-space is flat
It is known that the contribution of ordinary and dark matter explaining the constant velocity of distance stars rotating around galaxy is about 25 per cent from the critical density. Could it be that total mass density is critical?
From the anisotropy of cosmic microwave background one can deduce that this is the case. What criticality means geometrically is that 3-space defined as surface with constant value of cosmic time is flat. This reflects in the spectrum of microwave radiation. The spots representing small anisotropies in the microwave background temperature is 1 degree and this correspond to flat 3-space. If one had dark matter instead of dark energy the size of spot would be .5 degrees!
Thus in a cosmology based on general relativity cosmological constant remains the only viable option. The situation is different in TGD based quantum cosmology based on sub-manifold gravity and hierarchy of gravitational Planck constants.
A.3 The energy density of vacuum is constant in the size scale of big voids
It was observed that the density of dark energy would be constant in the scale of 108 light years. This length scale corresponds to the size of big voids containing galaxies at their boundaries.
A.4 Integrated Sachs-Wolf effect
Also so called integrated Integrated Sachs-Wolf effect supports accelerated expansion. Very slow variations of mass density are considered. These correspond to gravitational potentials. Cosmic expansion tends to flatten them but mass accretion to form structures compensates this effect so that gravitational potentials are unaffected and there is no effect of CMB. Situation changes if dark matter is replaced with dark energy the accelerated expansion flattening the gravitational potentials wins the tendency of mass accretion to make them deeper. Hence if photon passes by an over-dense region, it receives a little energy. Similarly, photon loses energy when passing by an under-dense region. This effect has been observed.
B. Comparison with TGD
The minimum TGD based explanation for accelerated expansion involves only the fact that the imbeddings of critical cosmologies correspond to accelerated expansion. A more detailed model allows to understand why the critical cosmology appears during some periods.
B.1. Accelerated expansion in classical TGD
The first observation is that critical cosmologies (flat 3-space) imbeddable to 8-D imbedding space H correspond to negative pressure cosmologies and thus to accelerating expansion. The negativity of the counterpart of pressure in Einstein tensor is due to the fact that space-time sheet is forced to be a 4-D surface in 8-D imbedding space. This condition is analogous to a force forcing a particle at the surface of 2-sphere and gives rise to what could be called constraint force. Gravitation in TGD is sub-manifold gravitation whereas in GRT it is manifold gravitation. This would be minimum interpretation involving no assumptions about what mechanism gives rise to the critical periods.
B.2 Accelerated expansion and hierarchy of Planck constants
One can go one step further and introduce the hierarchy of Planck constants. The basic difference between TGD and GRT based cosmologies is that TGD cosmology is quantum cosmology. Smooth cosmic expansion is replaced by an expansion occurring in discrete jerks corresponding to the increase of gravitational Planck constant. At space-time level this means the replacement of 8-D imbedding space H with a book like structure containing almost-copies of H with various values of Planck constant as pages glued together along critical manifold through which space-time sheet can leak between sectors with different values of hbar. This process is the geometric correlate for the the phase transition changing the value of Planck constant.
During these phase transition periods critical cosmology applies and predicts automatically accelerated expansion. Neither genuine negative pressure due to "quintessence" nor cosmological constant is needed. Note that quantum criticality replaces inflationary cosmology and predicts a unique cosmology apart from single parameter. Criticality also explains the fluctuations in microwave temperature as long range fluctuations characterizing criticality.
B.3 Accelerated expansion and flatness of 3-cosmology
Observations 1) and 2) about super-novae and critical cosmology (flat 3-space) are consistent with this cosmology. In TGD dark energy must be replaced with dark matter because the mass density is critical during the phase transition. This does not lead to wrong sized spots since it is the increase of Planck constant which induces the accelerated expansion understandable also as a constraint force due to imbedding to H.
B.4 The size of large voids is the characteristic scale
The TGD based model in its simplest form model assigns the critical periods of expansion to large voids of size 108 ly. Also larger and smaller regions can express similar periods and dark space-time sheets are expected to obey same universal "cosmology" apart from a parameter characterizing the duration of the phase transition. Observation 3) that just this length scale defines the scale below which dark energy density is constant is consistent with TGD based model.
The basic prediction is jerkwise cosmic expansion with jerks analogous to quantum transitions between states of atom increasing the size of atom. The discovery of large voids with size of order 108 ly but age much longer than the age of galactic large voids conforms with this prediction (see this). One the other hand, it is known that the size of galactic clusters has not remained constant in very long time scale so that jerkwise expansion indeed seems to occur.
B.5 Do cosmic strings with negative gravitational mass cause the phase transition inducing accelerated expansion
Quantum classical correspondence is the basic principle of quantum TGD and suggest that the effective antigravity manifested by accelerated expansion might have some kind of concrete space-time correlate. A possible correlate is super heavy cosmic string like objects at the center of large voids which have negative gravitational mass under very general assumptions. The repulsive gravitational force created by these objects would drive galaxies to the boundaries of large voids. At some state the pressure of galaxies would become too strong and induce a quantum phase transition forcing the increase of gravitational Planck constant and expansion of the void taking place much faster than the outward drift of the galaxies. This process would repeat itself. In the average sense the cosmic expansion would not be accelerating.
For details see the chapter Quantum Astrophysics.
TGD predicts that cosmic expansion at the level of individual astrophysical systems does not take place continuously as in classical gravitation but through discrete quantum phase transitions increasing gravitational Planck constant and thus various quantum length and time scales. The reason would be that stationary quantum states for dark matter in astrophysical length scales cannot expand. One would have the analog of atomic physics in cosmic scales. Increases of hbar by a power of two are favored in these transitions but also other scalings are possible.
This has quite far reaching implications.
The obvious question - that I did not ask - is whether this kind of phase transition might have occurred for Earth and led from a completely granite covered Earth -Pangeia without seas- to the recent Earth. Neither it did not occur to me to check whether there is any support for a rapid expansion of Earth during some period of its history.
Situation changed when my son Paavo visited me last Saturday and told me about a Youtube video by Neal Adams, an American comic book and commercial artist who has also produced animations for geologists. We looked the amazing video a couple of times and I looked it again yesterday. The video is very impressive (no wonder!) but in the lack of references skeptic probably cannot avoid the feeling that Neal Adams might use his highly developed animation skills to cheat you. I found also a polemic article of Adams but again the references were lacking. Perhaps the reason of polemic tone was that the concrete animation models make the expanding Earth hypothesis very convincing but geologists dare not consider seriously arguments by a layman without a formal academic background.
1. The claims of Adams
The basic claims of Adams were following.
2. The critic of Adams of the subduction mechanism
The prevailing tectonic plate theory has been compared to the Copernican revolution in geology. The theory explains the young age of the seafloor in terms of the decomposition of the litosphere to tectonic plates and the convective flow of magma to which oceanic tectonic plates participate. The magma emerges from the crests of the mid ocean ridges representing a boundary of two plates and leads to the expansion of sea floor. The variations of the polarity of Earth's magnetic field coded in sea floor provide a strong support for the hypothesis that magma emerges from the crests.
The flow back to would take place at so called oceanic trenches near continents which represent the deepest parts of ocean. This process is known as subduction. In subduction oceanic tectonic plate bends and penetrates below the continental tectonic plate, the material in the oceanic plate gets denser and sinks into the magma. In this manner the oceanic tectonic plate suffers a metamorphosis returning back to the magma: everything which comes from Earth's interior returns back. Subduction mechanism explains elegantly formation of mountains (orogeny), earth quake zones, and associated zones of volcanic activity.
Adams is very polemic about the notion of subduction, in particular about the assumption that it generates steady convective cycle. The basic objections of Adams against subduction are following.
After I had decided to check the claims of Adams, the first thing that I learned is that Expanding Earth theory, whose existence Adams actually mentions, is by no means new. There are actually many of them.
The general reason why these theories were rejected by the main stream community was the absence of a convincing physical mechanism of expansion or of growth in which the density of Earth remains constant.
TGD based model differs from the tectonic plate model but allows subduction which cannot imply considerable back flow of magma. Let us sum up the basic assumptions and implications.
5. Did intra-terrestrial life burst to the surface of Earth during Cambrian expansion?
Intra-terrestrial hypothesis is one of the craziest TGD inspired ideas about the evolution of life and it is quite possible that in its strongest form the hypothesis is unrealistic. One can however try to find what one obtains from the combination of the IT hypothesis with the idea of pre-Cambrian granite Earth. Could the harsh pre-Cambrian conditions have allowed only intra-terrestrial multicellular life? Could the Cambrian explosion correspond to the moment of birth for this life in the very concrete sense that the magma flow brought it into the day-light?
To sum up, TGD would provide only the long sought mechanism of expansion and a possible connection with the biological evolution. It would be indeed fascinating if Planck constant changing quantum phase transitions in planetary scale would have profoundly affected the biosphere.
For more details see the chapter Quantum Astrophysics.
The definition of kilogram is not the topics number one in coffee table discussions and definitely not so because it could lead to heated debates. The fact however is that even the behavior of standard kilogram can open up fascinating questions about the structure of space-time.
The 118-year old International Prototype Kilogram is an alloy with 90 per cent Platinum and 10 per cent Iridium by weight (gravitational mass). It is held in an environmentally monitored vault in the basement of the BIPM�s House of Breteuil in S�vres on the outskirts of Paris. It has forty copies located around the world which are compared with Sevres copy with a period of 40 years.
The problem is that the Sevres kilogram seems to behave in a manner totally in-appropriate taking into account its high age if the behaviour of its equal age copies around the world is taken as the norm (see Wikipedia article and the more popular article here). The unavoidable conclusion from the comparisons is that the weight of Sevres kilogram has been reduced by about 50 μg during 118 years which makes about
for Sevres copy or relative increase of same amout for its copies.
Specialists have not been able to identify any convincing explanation for the strange phenomenon. For instance, there is condensation of matter from the air in the vault which increases the weight and there is periodic cleaning procedure which however should not cause the effect.
1. Could the non-conservation of gravitational energy explain the mystery?
The natural question is whether there could be some new physics mechanism involved. If the copies were much younger than the Sevres copy, one could consider the possibility that gravitational mass of all copies is gradually reduced. This is not the case. One can still however look what this could mean.
In TGD Equvalence Principle is not a basic law of nature and in the generic case gravitational energy is non-conserved whereas inertial energy is conserved (I will not go to the delicacies of zero energy ontology here). This occurs even in the case of stationary metrics such as Reissner-Nordström exterior metric and the metrics associated with stationary spherically symmetric star models imbedded as vacuum extremals (for details see this).
The basic reason is that Schwartschild time t relates by a shift to Minkowski time m0:
such that the shift depends on the distance r to the origin. The Minkowski shape of the 3-volume containing the gravitational energy changes with M4 time but this does not explain the effect. The key observation is that the vacuum extremal of Kähler action is not an extremal of the curvature scalar (these correspond to asymptotic situations). What looks first really paradoxical is that one obtains a constant value of energy inside a fixed constant volume but a non-vanishing flow of energy to the volume. The explanation is that the system simply destroys the gravitational energy flowing inside it! The increase of gravitational binding energy compensating for the feed of gravitational energy gives a more familiar looking articulation for the non-conservation.
Amusingly, the predicted rate for the destruction of the inflowing gravitational energy is of same order of magnitude as in the case of kilogram. Note also that the relative rate is of order 1/a, a the value of cosmic time of about 1010 years. The spherically symmetric star model also predicts a rate of same order.
This approach of course does not allow to understand the behavior of the kilogram since it predicts no change of gravitational mass inside volume and does not even apply in the recent situation since all kilograms are in same age. The co-incidence however suggests that the non-conservation of gravitational energy might be part of the mystery. The point is that if the inflow satisfies Equivalence Principle then the inertial mass of the system would slowly increase whereas gravitational mass would remain constant: this would hold true only in steady state.
2. Is the change of inertial mass in question?
It would seem that the reduction in weight should correspond to a reduction of the inertial mass in Sevres or its increase of its copies. What would distinguish between Sevres kilogram and its cousins? The only thing one can imagine is that the cousins are brought to Sevres periodically. The transfer process could increase the kilogram or stop its decrease.
Could it be that the inertial mass of every kilogram increases gradually until a steady state is achieved? When the system is transferred to another place the saturation situation is changed to a situation in which genuine transfer of inertial and gravitational mass begins again and leads to a more massive steady state. The very process of transferring the comparison masses to Sevres would cause their increase.
In TGD Universe the increase of the inertial (and gravitational) mass is due to the flow of matter from larger space-time sheets to the system. The additional mass would not enter in via the surface of the kilogram but like a Trojan horse from the interior and it would be thus impossible to control using present day technology. The flow would continue until a flow equilibrium would be reached with as much mass leaving the kilogram as entering it.
3. A connection with gravitation after all?
Why the in-flow of the inertial energy should be of same order of magnitude as that for the gravitational energy predicted by simple star models? Why Equivalence Principle should hold for the in-flow alhough it would not hold for the body itself? A possible explanation is in terms of the increasing gravitational binding energy which in a steady situation leaves gravitational energy constant although inertial energy could still increase.
This would however require rather large value of gravitational binding energy since one has
Δ Egr=ΔMI/M .
The Newtonian estimate for E
TGD predicts that gravitational constant is proportional to p-adic length scale squared
G propto Lp2.
Ordinary gravitation can be assigned to the Mersenne prime M127 associated with electron and thus to p-adic length scale of L(127)≈ 2.5×10-14 meters. The open question has been whether the gravities corresponding to other p-adic length scales are realized or not.
This question together with the discrepancy encourages to ask whether the value of the p-adic prime could be larger inside massive bodies (analogous to black holes in many respects in TGD framework) and make gravitation strong? In the recent case the p-adic length scale should correspond to a length scale of order 108L(127). L(181)≈ 3.2× 10-4 m (size of a large neuron by the way) would be a good candidate for the p-adic scale in question and considerably smaller than the size scale of order .1 meter defining the size of the kilogram.
This discrepancy brings in mind the strange finding of Tajmar and collaborators suggesting that rotating super-conductors generate a gravimagnetic field with a field strength by a factor of order 1020 larger than predicted by General Relativity. I have considered a model of the finding based on dark matter (see this). An alternative model could rely on the assumption that Newton's constant can in some situations correspond to p larger than M127. In this case the p-adic length scale needed would be around L(193)≈ 2 cm.
For more details see the chapter TGD and GRT.
MAGIC collaboration has found evidence for a gamma ray anomaly. Gamma rays are different energy ranges seem to arrive with different velocities from Mkn 501 (see this). The delay in arrival times is about 4 minutes. The proposed explanation is in terms of broken Lorentz invariance. TGD allows to explain the finding in terms of many-sheeted space-time and there is no need to invoke breaking of Lorentz invariance.
1. TGD based explanation at qualitative level
One of the oldest predictions of many-sheeted space-time is that the time for photons to propagate from point A to B along given space-time sheet depends on space-time sheet because photon travels along lightlike geodesic of space-time sheet rather than lightlike geodesic of the imbedding space and thus increases so that the travel time is in general longer than using maximal signal velocity.
Many-sheetedness predicts a spectrum of Hubble constants and gamma ray anomaly might be a demonstration for the many-sheetedness. The spectroscopy of arrival times would give information about how many sheets are involved.
Before one can accept this explanation, one must have a good argument for why the space-time sheet along which gamma rays travel depends on their energy and why higher energy gamma rays would move along space-time sheet along which the distance is longer.
2. Quantitative argument
A quantitative estimate runs as follows.
It seems that one can fairly well say that standard cosmology is making a crash down while TGD is making a breakthrough after breakthrough as the interpretation becomes more and more accurate. TGD is patiently waiting;-). Interesting to see how long it still will take before sociology of science finally gives up and the unavoidable happens.
For details and background see the chapter The Relationship Between TGD and GRT.
I have considered two models for Allais effect. The first model was constructed for several years ago and was based on classical Z0 force. For a couple of weeks ago I considered a model based on gravitational screening. It however turned that this model does not work. The next step was the realization that the effect might be a genuine quantum effect made possible by the gigantic value of the gravitational Planck constant: the pendulum would act as a highly sensitive gravitational interferometer.
One can represent rather general counter arguments against the models based on Z0 conductivity and gravitational screening if one takes seriously the puzzling experimental findings concerning frequency change.
The above findings allow to make some important conclusions about the nature of Allais effect.
2. Model for interaction via gravitational MEs with large Planck constant
Restricting the consideration for simplicity only gravitational MEs, a concrete model for the situation would be as follows.
The assumption of the scaling law
is very natural in light of conformal invariance and masslessness of gravitons and allows to make the model more explicit. With the choice λ0=rS the anomaly term can be expressed in the form
Δ agr≈ (GMS/rSrM) × (22k+1/v0)×(MM/MS)k × R0(S,P)× R0(M,P)× ∑n=1∞ ((-1)n/n2k)× cos[nπK] ,
K= x× (rM/rS)× (yM/yS).
The normalization condition reads in this case as
R02=v0/[2π∑n (1/n)2k+1]=v0/πζ(2k+1) .
Note the shorthand v0(S/M,P)= v0. The anomalous gravitational acceleration is given by
Δagr=(GMS/rS2) × X Y× ∑n=1∞ [(-1)n/n2k]×cos[nπK] ,
X= 22k × (rS/rM)× (MM/MS)k ,
It is clear that a reasonable order of magnitude for the effect can be obtained if k is small enough and that this is essentially due to the gigantic value of gravitational Planck constant.
The simplest model consistent with experimental findings assumes v0(M,P)= v0(S,P) and Φ(n)=0 and gives
Δagr/gcos(Θ)=(GMS/rS2g)× X Y× ∑n=1∞ [(-1)n/n2k]×cos(nπ K) ,
X= 22k × (rS/rM)× (MM/MS)k,
Y=1/π ∑n (1/n)2k+1 =1/πζ(2k+1) ,
K=x× (rM/rS)× (yM/yS) , x=MS/MM .
Θ denotes in the formula above the angle between the direction of Sun and horizontal plane.
4. Numerical estimates
To get a numerical grasp to the situation one can use MS/MM≈ 2.71× 107, rS/rM≈ 389.1, and (MSrM/MMrS)≈ 1.74× 104. The overall order of magnitude of the effect would be
Δ g/g≈ XY× GMS/RS2gcos(Θ) ,
(GMS/RS2g) ≈6× 10-4 .
The overall magnitude of the effect is determined by the factor XY.
For k=1 and 1/2 the effect is too small. For k=1/4 the expression for Δ agr reads as
(Δagr/gcos(Θ))≈1.97× 10-4∑n=1∞ ((-1)n/n1/2)×cos(nπK),
K= (yM/yS)u , u=(MS/MM)(rM/rS)≈ 6.95671837× 104 .
The sensitivity of cosine terms to the precise value of yM/yS gives good hopes of explaining the strong variation of Δf/f and also the findings of Jeverdan. Numerical experimentation indeed shows that the sign of cosine sum alternates and its value increases as yM/yS increases in the range [1,2].
The eccentricities of the orbits of Moon resp. Earth are eM=.0549 resp. eE=.017. Denoting semimajor and semiminor axes by a and b one has Δ=(a-b)/a=1-(1-e2)1/2. ΔM=15× 10-4 resp. ΔE=1.4× 10-4 characterizes the variation of yM resp. yM due to the non-circularity of the orbits of Moon resp. Earth. The ratio RE/rM= .0166 characterizes the range of ΔyM =ΔrM,P/rM< RE/rM due to the variation of the position of the laboratory. All these numbers are large enough to imply large variation of the argument of cosine term even for n=1 and the variation due to the position at the surface of Earth is especially large.
5. Other effects
1. Chern-Simons as helicity invariant
Tao mentions helicity as an invariant of fluid flow. Chern-Simons action defined by the induced Kähler gauge potential for lightlike 3-surfaces has interpretation as helicity when Kähler gauge potential is identified as fluid velocity. This flow can be continued to the interior of space-time sheet. Also the dual of the induced Kähler form defines a flow at the light-like partonic surfaces but not in the interior of space-time sheet. The lines of this flow can be interpreted as magnetic field lines. This flow is incompressible and represents a conserved charge (Kähler magnetic flux). The question is which of these flows should define number theoretical braids. Perhaps both of them can appear in the definition of S-matrix and correspond to different kinds of partonic matter (electric/magnetic charges, quarks/leptons?,...). Second kind of matter could not flow in the interior of space-time sheet. Or could interpretation in terms of electric magnetic duality make sense?
Helicity is not gauge invariant and this is as it must be in TGD framework since CP2 symplectic transformations induce U(1) gauge transformation which deforms space-time surface an modifies induced metric as well as classical electroweak fields defined by induced spinor connection. Gauge degeneracy is transformed to spin glass degeneracy.
2. Maxwell hydrodynamics
In TGD Maxwell's equations are replaced with field equations which express conservation laws and are thus hydrodynamical in character. With this background the idea that the analogy between gauge theory and hydrodynamics might be applied also in the reverse direction is natural. Hence one might ask what kind of relativistic hydrodynamics results if assumes that the action principle is Maxwell action for the four-velocity uα with the constraint term saying that light velocity is maximal signal velocity.
There are strong similarities with TGD which suggests that the proposed model might provide a toy model for the dynamics defined by Kähler action.
For the construction of extremals of Kähler action see the chapter Basic Extremals of Kähler action.
Allais effect is a fascinating gravitational anomaly associated with solar eclipses. It was discovered originally by M. Allais, a Nobelist in the field of economy, and has been reproduced in several experiments but not as a rule. The experimental arrangement uses so called paraconical pendulum, which differs from the Foucault pendulum in that the oscillation plane of the pendulum can rotate in certain limits so that the motion occurs effectively at the surface of sphere.
The "../articles/ Should the Laws of Gravitation Be Reconsidered: Part I,II,III? of Allais here and here and the summary article The Allais effect and my experiments with the paraconical pendulum 1954-1960 of Allais give a detailed summary of the experiments performed by Allais.
A. Experimental findings of Allais
Consider first a brief summary of the findings of Allais.
B. TGD inspired model for Allais effect
The basic idea of the TGD based model is that Moon absorbs some fraction of the gravitational momentum flow of Sun and in this manner partially screens the gravitational force of Sun in a disk like region having the size of Moon's cross section. Screening is expected to be strongest in the center of the disk. The predicted upper bound for the change of the oscillation frequency is slightly larger than the observed change which is highly encouraging.
1. Constant external force as the cause of the effect
The conclusions of Allais motivate the assumption that quite generally there can be additional constant forces affecting the motion of the paraconical pendulum besides Earth's gravitation. This means the replacement g→ g+Δg of the acceleration g due to Earth's gravitation. Δg can depend on time.
The system obeys still the same simple equations of motion as in the initial situation, the only change being that the direction and magnitude of effective Earth's acceleration have changed so that the definition of vertical is modified. If Δ g is not parallel to the oscillation plane in the original situation, a torque is induced and the oscillation plane begins to rotate. This picture requires that the friction in the rotational degree of freedom is considerably stronger than in oscillatory degree of freedom: unfortunately I do not know what the situation is.
The behavior of the system in absence of friction can be deduced from the conservation laws of energy and angular momentum in the direction of g+Δ g.
2. What causes the effect in normal situations?
The gravitational accelerations caused by Sun and Moon come first in mind as causes of the effect. Equivalence Principle implies that only relative accelerations causing analogs of tidal forces can be in question. In GRT picture these accelerations correspond to a geodesic deviation between the surface of Earth and its center. The general form of the tidal acceleration would thus the difference of gravitational accelerations at these points:
Δg= -2GM[(Δ r/r3) - 3(r•Δ rr/r5)].
Here r denotes the relative position of the pendulum with respect to Sun or Moon. Δr denotes the position vector of the pendulum measured with respect to the center of Earth defining the geodesic deviation. The contribution in the direction of Δ r does not affect the direction of the Earth's acceleration and therefore does not contribute to the torque. Second contribution corresponds to an acceleration in the direction of r connecting the pendulum to Moon or Sun. The direction of this vector changes slowly.
This would suggest that in the normal situation the tidal effect of Moon causesgradually changing force mΔg creating a torque, which induces a rotation of the oscillation plane. Together with dissipation this leads to a situation in which the orbital plane contains the vector Δg so that no torque is experienced. The limiting oscillation plane should rotate with same period as Moon around Earth. Of course, if effect is due to some other force than gravitational forces of Sun and Earth, paraconic oscillator would provide a manner to make this force visible and quantify its effects.
3. What happens during solar eclipse?
During the solar eclipse something exceptional must happen in order to account for the size of effect. The finding of Allais that the limiting oscillation plane contains the line connecting Earth, Moon, and Sun implies that the anomalous acceleration Δ |g| should be parallel to this line during the solar eclipse.
The simplest hypothesis is based on TGD based view about gravitational force as a flow of gravitational momentum in the radial direction.
C. What kind of tidal effects are predicted?
If the model applies also in the case of Earth itself, new kind of tidal effects (for normal tidal effects see this) are predicted due to the screening of the gravitational effects of Sun and Moon inside Earth. At the night-side the paraconical pendulum should experience the gravitation of Sun as screened. Same would apply to the "night-side" of Earth with respect to Moon.
Consider first the differences of accelerations in the direction of the line connecting Earth to Sun/Moon: these effects are not essential for tidal effects proper. The estimate for the ratio for the orders of magnitudes of the these accelerations is given by
|Δgp(Sun)|/|Δgp(Moon)|= (MS/MM) (rM/rE)3≈ 2.17.
The order or magnitude follows from r(Moon)=.0026 AU and MM/MS=3.7× 10-8. The effects caused by Sun are two times stronger. These effects are of same order of magnitude and can be compensated by a variation of the pressure gradients of atmosphere and sea water.
The tangential accelerations are essential for tidal effects. The above estimate for the ratio of the contributions of Sun and Moon holds true also now and the tidal effects caused by Sun are stronger by a factor of two.
Consider now the new tidal effects caused by the screening.
The intuitive expectation is that the screening is maximum when the gravitational momentum flux travels longest path in the Earth's interior. The maximal difference of radial accelerations associated with opposite sides of Earth along the line of sight to Moon/Sun provides a convenient manner to distinguish between Newtonian and TGD based models:
|Δ gp,N|=4GM ×(RE/r)3 ,
|Δ gp,TGD|= 4GM ×(1/r2).
The ratio of the effects predicted by TGD and Newtonian models would be
|Δ gp,TGD|/|Δ gp,N|= r/RE ,
rM/RE =60.2 , rS/RE= 2.34× 104.
The amplitude for the oscillatory variation of the pressure gradient caused by Sun would be
Δ|gradpS|=v2E/rE≈ 6.1× 10-4g
and the pressure gradient would be reduced during night-time. The corresponding amplitude in the case of Moon is given by
Δ |gradpS|/Δ|gradpM|= (MS/MM)× (rM/rS)3≈ 2.17.
Δ |gradpM| is in a good approximation smaller by a factor of 1/2 and given by
Thus the contributions are of same order of magnitude.
One can imagine two simple qualitative killer predictions.
D. An interesting co-incidence
The measured value of Δ f/f=5× 10-4 is exactly equal to v0=2-11, which appears in the formula hbargr= GMm/v0 for the favored values of the gravitational Planck constant. The predictions are Δ f/f≤ Δ p/p≈ 6× 10-4. Powers of 1/v0 appear also as favored scalings of Planck constant in the TGD inspired quantum model of bio-systems based on dark matter (see this). This co-incidence would suggest the quantization formula
gE/gS= (MS/ME) × (RE/rE)2= v0
for the ratio of the gravitational accelerations caused by Earth and Sun on an object at the surface of Earth.
E. Summary of the predicted new effects
Let us sum up the basic predictions of the model.
To sum up, the predicted anomalous tidal effects and the explanation of the limiting oscillation plane in terms of stronger dissipation in rotational degree of freedom could kill the model.
For details see the chapter The Relationship Between TGD and GRT.
I have updated "TGD Inspired Cosmology". Here is the updated abstract.
A proposal for what might be called TGD inspired cosmology is made. The basic ingredient of this cosmology is the TGD counter part of the cosmic string. It is found that many-sheeted space-time concept; the new view about the relationship between inertial and gravitational four-momenta; the basic properties of the cosmic strings; zero energy ontology; the hierarchy of dark matter with levels labelled by arbitrarily large values of Planck constant: the existence of the limiting temperature (as in string model, too); the assumption about the existence of the vapor phase dominated by cosmic strings; and quantum criticality imply a rather detailed picture of the cosmic evolution, which differs from that provided by the standard cosmology in several respects but has also strong resemblances with inflationary scenario.For details see the updated chapter TGD Inspired Cosmology.
I am continuing the updating of the chapters related to the relationship of TGD and GRT. The updatings are due to the zero energy ontology, the hierarchy of dark matter labelled by Planck constants, and due to the progress in the understanding of Equivalence Principle. I just finished the elimination of the worst trash from "The Relationship between TGD and GRT" and attach the abstract below.
In this chapter the recent view about TGD as Poincare invariant theory of gravitation is discussed. Radically new views about ontology were necessary before it was possible to see what had been there all the time. Zero energy ontology states that all physical states have vanishing net quantum numbers. The hierarchy of dark matter identified as macroscopic quantum phases labelled by arbitrarily large values of Planck constant is second aspect of the new ontology.For details see the updated chapter The Relationship Between TGD and GRT.
Cosmic strings belong to the basic extremals of the Kähler action. The upper bound for string tension of the cosmic strings is T≈.5× 10-6/G and in the same range as the string tension of GUT strings and this makes them very interesting cosmologically although TGD cosmic strings have otherwise practically nothing to do with their GUT counterparts.
1. Basic ideas
The understanding of cosmic strings has developed only slowly and has required dramatic modifications of existing views.
2. Critical and over-critical cosmologies involve accelerated cosmic expansion
In TGD framework critical and over-critical cosmologies are unique apart from single parameter telling their duration and predict the recently discovered accelerated cosmic expansion. Critical cosmologies are naturally associated with quantum critical phase transitions involving the change of gravitational Planck constant. A natural candidate for such a transition is the increase of the size of a large void as galactic strings have been driven to its boundary. During the phase transitions connecting two stationary cosmologies (extremals of curvature scalar) also determined apart from single parameter, accelerated expansion is predicted to occur. These transitions are completely analogous to quantum transitions at atomic level.
The proposed microscopic model predicts that the TGD counterpart of the quantity ρ+3p for cosmic strings is negative during the phase transition which implies accelerated expansion. Dark energy is replaced in TGD framework with dark matter indeed predicted by TGD and its fraction is .74 as in standard scenario. Cosmological constant thus characterizes the density of dark matter rather than energy in TGD Universe.
The sizes of large voids stay constant during stationary periods which means that also cosmological constant is piecewise constant. p-Adic length fractality predicts that Λ scales as 1/L2(k) as a function of the p-adic scale characterizing the space-time sheet of void. The order of magnitude for the recent value of the cosmological constant comes out correctly. The gravitational energy density described by the cosmological constant is identifiable as that associated with topologically condensed cosmic strings and of magnetic flux tubes to which they are gradually transformed during cosmological evolution.
3. Cosmic strings and generation of structures
4. Cosmic strings, gamma ray bursts, and supernovaeDuring year 2003 two important findings related to cosmic strings were made.
The flow of matter along Z0 magnetic (rotation) axis generates synchrotron radiation, which escapes as a precisely targeted beam along magnetic axis and leaves the star. The identification is as the rotating light beam associated with ordinary neutron stars. During the core collapse leading to the supernova this beam becomes gamma ray burst. The mechanism is very much analogous to the squeezing of the tooth paste from the tube. The fact that all nuclei are fully ionized Z0 ions, the Z0 charge unbalance caused by the ejection of neutrinos, and the radial compression make the effect extremely strong so that there are hopes to understand the observed incredibly high polarization of 80+/- 20 per cent.
TGD suggests the identification of p"../articles/ of mass m≈2me accompanying dark matter as lepto-pions formed by color excited leptons, and topologically condensed at magnetic flux tubes having thickness of about lepto-pion Compton length. Lepto-pions would serve as signatures of dark matter whereas dark matter itself would correspond to the magnetic energy of topologically condensed cosmic strings transformed to magnetic flux tubes.
For details see the updated chapter Cosmic Strings.
In the comment section of Not-Even-Wrong 'island' gave a link to an article about the observation of a new anomaly in cosmic microwave background. The article Extragalactic Radio Sources and the WMAP Cold Spot by L. Rudnick, S. Brown, and L. R. Williams tells that a cold spot in the microwave background has been discovered. The amplitude of the temperature variation is -73 microK at maximum. The authors argue that the variation can be understood if there is a void at redshift z≤ 1, which corresponds to d≤ 1.4× 1010 ly. The void would have radius of 140 Mpc making 5.2× 108 ly.
In New Scientist, there is a story titled Cosmologists spot a 'knot' in space-time about Neil Turok�s recent talk at PASCOS entitled �Is the Cold Spot in the CMB a Texture?�. Turok has proposed that the cold spot results from a topological defect associated with a cosmic string of GUT type theories.
1. Comparison with sizes and distances of large voids
It is interesting to compare the size and distance of the argued CMB void to those for large voids. The largest known void has size of 163 Mpc making 5.3×108 ly which does not differ significantly from the size 8×6.5×108 ly of CMB void. The distance is 201 Mpc making about 6.5×108 ly and roughly by a factor 1/22 smaller than CMB void.
Is it only an accident that the size of CMB void is same as that for largest large void? If large voids follow the cosmic expansion in a continuous manner, the size of the CMB void should be roughly 1/22 time smaller. Could it be that large voids might follow cosmic expansion by rather seldomly occurring discrete jumps? TGD based quantum astrophysics indeed predicts that expansion occurs in discrete jumps.
2. TGD based quantum model for astrophysical systems
A brief summary of TGD based quantum model of astrophysical systems is in order.
Concerning the explanation of CMB void one can consider two options.
In the former chapter "General View About Physics in Many-Sheeted Space-Time" the notion of many-sheeted space-time concept and the understanding of coupling constant evolution at space-time level were discussed with emphasis on the notions of topological condensation and evaporation. The notion of many-sheeted space-time used was roughly that as it was around 1990 and 17 years is a long time.
The fusion of real and various p-adic physics to single coherent whole by generalizing the notion of number, the generalization of the notion of the imbedding space to allow a mathematical representation of dark matter hierarchy based on dynamical and quantized Planck constant, parton level formulation of TGD using light-like 3-surfaces as basic dynamical objects, and so called zero energy ontology force to generalizes considerably the view about space-time.For these reasons I decided to add a chapter in which the picture about many-sheeted space-time is completed by a summary of the new rather dramatic developments in quantum TGD occurred during last few years.
For more details and background see the new chapter General View About Physics in Many-Sheeted Space-Time: Part II.
Tommaso Dorigo has an interesting posting about blackhole production at LHC. I have never taken this idea seriously but in a well-defined sense TGD predicts blackholes associated with super-canonical gravitons with strong gravitational constant defined by the hadronic string tension. The proposal is that super-canonical blackholes have been already seen in Hera, RHIC, and the strange cosmic ray events (see the previous posting). Ordinary blackholes are naturally replaced with super-canonical blackholes in TGD framework, which would mean a profound difference between TGD and string models.
Super-canonical black-holes are dark matter in the sense that they have no electro-weak interactions and they could have Planck constant larger than the ordinary one so that the value of αs=αK=1/4 is reduced. The condition that αK has the same value for the super-canonical phase as it has for ordinary gauge boson space-time sheets gives hbar=26×hbar0. With this assumption the size of the baryonic super-canonical blacholes would be 46 fm, the size of a big nucleus, and would define the fundamental length scale of nuclear physics.
1. RHIC and super-canonical blackholes
In high energy collisions of nuclei at RHIC the formation of super-canonical blackholes via the fusion of nucleonic space-time sheets would give rise to what has been christened a color glass condensate. Baryonic super-canonical blackholes of M107 hadron physics would have mass 934.2 MeV, very near to proton mass. The mass of their M89 counterparts would be 512 times higher, about 478 GeV. The "ionization energy" for Pomeron, the structure formed by valence quarks connected by color bonds separating from the space-time sheet of super-canonical blackhole in the production process, corresponds to the total quark mass and is about 170 MeV for ordinary proton and 87 GeV for M89 proton. This kind of picture about blackhole formation expected to occur in LHC differs from the stringy picture since a fusion of the hadronic mini blackholes to a larger blackhole is in question.
An interesting question is whether the ultrahigh energy cosmic rays having energies larger than the GZK cut-off (see the previous posting) are baryons, which have lost their valence quarks in a collision with hadron and therefore have no interactions with the microwave background so that they are able to propagate through long distances.
2. Ordinary blackholes as super-canonical blackholes
In neutron stars the hadronic space-time sheets could form a gigantic super-canonical blackhole and ordinary blackholes would be naturally replaced with super-canonical blackholes in TGD framework (only a small part of blackhole interior metric is representable as an induced metric).
Sp= (M/m(CP2))2× log(p),
where m(CP2) is CP2 mass, which is roughly 10-4 times Planck mass. M corresponds to the contribution of p-adic thermodynamics to the mass. This contribution is extremely small for gauge bosons but for fermions and super-canonical p"../articles/ it gives the entire mass.
Sp= k log(2)×(M/m(CP2))2 ,
m(CP2)=hbar/R, R the "radius" of CP2, corresponds to the standard value of hbar0 for all values of hbar.
S= hbar×A/4G = hbar×πGM2.
For the p-adic variant of the law Planck mass is replaced with CP2 mass and klog(2)≈ log(p) appears as an additional factor. Area law is obtained in the case of elementary p"../articles/ if k is prime and wormhole throats have M4 radius given by p-adic length scale Lk=k1/2RCP2, which is exponentially smaller than Lp.
For macroscopic super-canonical black-holes modified area law results if the radius of the large wormhole throat equals to Schwartschild radius. Schwartschild radius is indeed natural: I have shown that a simple deformation of the Schwartschild exterior metric to a metric representing rotating star transforms Schwartschild horizon to a light-like 3-surface at which the signature of the induced metric is transformed from Minkowskian to Euclidian (see this).
v0=2-11 is the preferred value of v0. One could argue that the value of gravitational Planck constant is such that the Compton length hbargr/M of the black-hole equals to its Schwartshild radius. This would give
hbargr/hbar0= GM2/v0 , v0=1/2 .
This is a natural generalization of the Nottale's formula to gravitational self interactions. The requirement that hbargr is a ratio of ruler-and-compass integers expressible as a product of distinct Fermat primes (only four of them are known) and power of 2 would quantize the mass spectrum of black hole. Even without this constraint M2 is integer valued using p-adic mass squared unit and if p-adic length scale hypothesis holds true this unit is in an excellent approximation power of two.
Schwartschild horizon for a rotating blackhole like object as a 3-D lightlike surface defining wormhole throat
The metric determinant at Schwartschild radius is non-vanishing. This does not quite conform with the interpretation as an analog of a light-like partonic 3-surface identifiable as a wormhole throat for which the determinant of the induced 4-metric vanishes and at which the signature of the induced metric changes from Minkowskian to Euclidian.
An interesting question is what happens if one makes the vacuum extremal representing imbedding of Schwartshild metric a rotating solution by a very simple replacement Φ→ Φ+nΦ, where Φ is the angle angle coordinate of homologically trivial geodesic sphere S2 for the simplest vacuum extremals, and Φ the angle coordinate of M4 spherical coordinates. It turns out that Schwartschild horizon is transformed to a surface at which det(g4) vanishes so that the interpretation as a wormhole throat makes sense. If one assumes that black hole horizon is analogous to a wormhole contact, only rotating black hole like structures with quantized angular momentum are possible in TGD Universe.
For details see the chapter TGD and GRT.
Kea commented about transition to quantum chaos and gave a link to the article Quantum Chaos by Martin Gurtzwiller in Matthew Watkins's home page devoted to Riemann Zeta. Occasionally even this kind of a masterpiece of scientific writing manages to stimulate only an intention to read it more carefully later. When you indeed read it again few years later it can shatter you into a wild resonance. Just this occurred at this time.
The article introduces the division of classical systems into regular (R) and chaotic (P in honor of Poincare) ones. Besides this one has quantal systems (Q). There are three transition regions between these three realms.
2.1 The level of stationary states
At the level of energy spectrum this means that the energy of system which correspond to sums of virtually independent energies and thus is essentially random number becomes non-random. As a consequence, energy levels tend to avoid each other, order and simplicity emerge but at the collective level. Spectrum of zeros of Zeta has been found to simulate the spectrum for a chaotic system with strong correlations between energy levels. Zeta functions indeed play a key role in the proposed description of quantum criticality associated with the phase transition changing the value of Planck constant.
2.2 The importance of classical periodic orbits in chaotic scattering
Poincare with his immense physical and mathematical intuition foresaw that periodic classical orbits should have a key role also in the description of chaos. The study of complex systems indeed demonstrates that this is the case although the mathematics and physics behind this was not fully understood around 1992 and is probably not so even now. The basic discovery coming from numerical simulations is that the Fourier transform of a chaotic orbits exhibits has peaks the frequencies which correspond to the periods of closed orbits. From my earlier encounters with quantum chaos I remember that there is quantization of periodic orbits so that their periods are proportional to log(p), p prime in suitable units. This suggests a connection of arithmetic quantum field theory and with p-adic length scale hypothesis. Note that in planetary Bohr orbitology any closed orbit can be Bohr orbits with a suitable mass distribution but that velocity spectrum is universal.
The chaotic scattering of electron in atomic lattice is discussed as a concrete example. In the chaotic situation the notion of electron consists of periods spend around some atom continued by a motion along along some classical periodic orbit. This does not however mean loss of quantum coherence in the transitions between these periods: a purely classical model gives non-sensible results in this kind of situation. Only if one sums scattering amplitudes over all piecewise classical orbits (not all paths as one would do in path integral quantization) one obtains a working model.
2.3. In what sense complex systems can be called chaotic?
Speaking about quantum chaos instead of quantum complexity does not seem appropriate to me unless one makes clear that it refers to the limitations of human cognition rather than to physics. If one believes in quantum approach to consciousness, these limitations should reduce to finite resolution of quantum measurement not taken into account in standard quantum measurement theory.
In the framework of hyper-finite factors of type II1 finite quantum measurement resolution is described in terms of inclusions N subset M of the factors and sub-factor N defines what might be called N-rays replacing complex rays of state space. The space M/N has a fractal dimension characterized by quantum phase and increases as quantum phase q=exp(iπ/n), n=3,4,..., approaches unity which means improving measurement resolution since the size of the factor N is reduced.
Fuzzy logic based on quantum qbits applies in the situation since the components of quantum spinor do not commute. At the limit n→∞ one obtains commutativity, ordinary logic, and maximal dimension. The smaller the n the stronger the correlations and the smaller the fractal dimension. In this case the measurement resolution makes the system apparently strongly correlated when n approaches its minimal value n=3 for which fractal dimension equals to 1 and Boolean logic degenerates to single valued totalitarian logic.
Non-commutativity is the most elegant description for the reduction of dimensions and brings in reduced fractal dimensions smaller than the actual dimension. Again the reduction has interpretation as something totally different from chaos: system becomes a single coherent whole with strong but not complete correlation between different degrees of freedom. The interpretation would be that in the transition to non-chaotic quantal behavior correlation becomes complete and the dimension of system again integer valued but smaller. This would correspond to the cases n=6, n=4, and n=3 (D=3,2,1).
The Bohr orbit model for the planetary orbits based on the hierarchy of dark matter relies in an essential manner on the idea that macroscopic quantum phases of dark matter dictate to a high degree the behavior of the visible matter. Dark matter is concentrated on closed classical orbits in the simple rotationally symmetric gravitational potentials involved. Orbits become basic structures instead of points at the level of dark matter. A discrete subgroup Zn of rotational group with very large n characterizes dark matter structures quite generally. At the level of visible matter this symmetry can be broken to approximate symmetry defined by some subgroup of Zn.
Circles and radial spokes are the basic Platonic building blocks of dark matter structures. The interpretation of spokes would be as (gravi-)electric flux tubes. Radial spokes correspond to n=0 states in Bohr quantization for hydrogen atom and orbits ending into atom. Spokes have been observed in planetary rings besides decomposition to narrow rings and also in galactic scale. Also flux tubes of (gravi-)magnetic fields with Zn symmetry define rotational symmetric structures analogous to quantized dipole fields.
Gravi-magnetic flux tubes indeed correspond to circles rather than field lines of a dipole field for the simplest model of gravi-magnetic field, which means deviation from GRT predictions for gravi-magnetic torque on gyroscope outside equator: unfortunately the recent experiments are performed at equator. The flux tubes be seen only as circles orthogonal to the preferred plane and planetary Bohr rules apply automatically also now.
A word of worry is in order here. Ellipses are very natural objects in Bohr orbitology and for a given value of n would give n2-1 additional orbits. In planetary situation they would have very large eccentricities and are not realized. Comets can have closed highly eccentric orbits and correspond to large values of n. In any case, one is forced to ask whether the exactly Zn symmetric objects are too Platonic creatures to live in the harsh real world. Should one at least generalize the definition of the action of Zn as symmetry so that it could rotate the points of ellipse to each other. This might make sense. In the case of dark matter ellipses the radial spokes with Zn symmetry representing radial gravito-electric flux quanta would still connect dark matter ellipse to the central object and the rotation of the spoke structure induces a unique rotation of points at ellipse.
3.3. Dark matter structures as generalization of periodic orbits
The matter with ordinary or smaller value of Planck constant can form bound states with these dark matter structures. The dark matter circles would be the counterparts for the periodic Bohr orbits dictating the behavior of the quantum chaotic system. Visible matter (and more generally, dark matter at the lower levels of hierarchy behaving quantally in shorter length and time scales) tends to stay around these periodic orbits and in the ideal case provides a perfect classical mimicry of quantum behavior. Dark matter structures would effectively serve as selectors of the closed orbits in the gravitational dynamics of visible matter.
As one approaches classicality the binding of the visible matter to dark matter gradually weakens. Mercury's orbit is not quite closed, planetary orbits become ellipses, comets have highly eccentric orbits or even non-closed orbits. For non-closed quantum description in terms of binding to dark matter does not makes sense at all.
The classical regular limit (R) would correspond to a decoupling between dark matter and visible matter. A motion along geodesic line is obtained but without Bohr quantization in gravitational sense since Bohr quantization using ordinary value of Planck constant implies negative energies for GMm>1. The preferred extremal property of the space-time sheet could however still imply some quantization rules but these could apply in "vibrational" degrees of freedom.
3.4 Quantal chaos in gravitational scattering?
The chaotic motion of astrophysical object becomes the counterpart of quantum chaotic scattering. By Equivalence Principle the value of the mass of the object does not matter at all so that the motion of sufficiently light objects in solar system might be understandable only by assuming quantum chaos.
The orbit of a gravitationally unbound object such as comet could define the basic example. The rings of Saturn and Jupiter could represent interesting shorter length scale phenomena possible involving quantum scattering. One can imagine that the visible matter object spends some time around a given dark matter circle (binding to atom), makes a transition along radial spoke to the next circle, and so on.
The prediction is that dark matter forms rings and cart-wheel like structures of astrophysical size. These could become visible in collisions of say galaxies when stars get so large energy as to become gravitationally unbound and in this quantum chaotic regime can flow along spokes to new Bohr orbits or to gravi-magnetic flux tubes orthogonal to the galactic plane. Hoag's object represents a beautiful example ring galaxy. Remarkably, there is also direct evidence for galactic cart-wheels. There are also polar ring galaxies consisting of an ordinary galaxy plus ring approximately orthogonal to it and believed to form in galactic collisions. The ring rotating with the ordinary galaxy can be identified in terms of gravi-magnetic flux tube orthogonal to the galactic plane: in this case Zn symmetry would be completely broken at the level of visible matter.
For more details see the new chapter Quantum Astrophysics .
There are two objections against planetary Bohr orbitology.
In the previous posting I proposed a simple model explaining why inner and outer planets must have different values of v0 by taking into account cosmic string contribution to the gravitational potential which is negligible nowadays but was not so in primordial times. Among other things this implies that planetary system has a finite size, at least about 1 ly in case of Sun (nearest star is at distance of 4 light years).
I have also applied the quantization rules to exoplanets in the case that the central mass and orbital radius are known. Errors are around 10 per cent for the most favored value of v0=2-11 (see this). The "anomalous" planets with very small orbital radius correspond to n=1 Bohr orbit (n=3 is the lowest orbit in solar system). The universal velocity spectrum v= v0/n in simple systems perhaps the most remarkable prediction and certainly testable: this alone implies that the Bohr radius GM/v02 defines the universal size scale for systems involving central mass. Obviously this is something new and highly non-trivial.
The recently observed dark ring in MLy scale is a further success and also the rings and Moons of Saturn and Jupiter obey the same universal length scale (n≥ 5 and v0→ (16/15)×v0 and v0→ 2×v0).
There is a further objection. For our own Moon orbital radius is much larger than Bohr radius for v0=2-11: one would have n≈138. n≈7 results for v0 →v0/20 giving r0≈ 1.2 RE. The small value of v0 could be understood to result from a sequence of phase transitions reducing the value of v0 to guarantee that solar system participates in the average sense to the cosmic expansion and from the fact inner planets are older than outer ones in the proposed scenario.
Remark: Bohr orbits cannot participate in the expansion which manifests itself as the observed apparent shrinking of the planetary orbits when distances are expressed in terms Robertson-Walker radial coordinate r=rM. This anomaly was discovered by Masreliez and is discussed here. Ruler-and-compass hypothesis suggests preferred values of cosmic times for the occurrence of these transitions. Without this hypothesis the phase transitions could form almost continuum.
2. How General Coordinate Invariance and Lorentz invariance are achieved?
One can use Minkowski coordinates of the M4 factor of the imbedding space H=M4×CP2 as preferred space-time coordinates. The basic aspect of dark matter hierarchy is that it realizes quantum classical correspondence at space-time level by fixing preferred M4 coordinates as a rest system. This guarantees preferred time coordinate and quantization axis of angular momentum. The physical process of fixing quantization axes thus selects preferred coordinates and affects the system itself at the level of space-time, imbedding space, and configuration space (world of classical worlds). This is definitely something totally new aspect of observer-system interaction.
One can identify in this system gravitational potential Φgr as the gtt component of metric and define gravi-electric field Egr uniquely as its gradient. Also gravi-magnetic vector potential Agr and and gravimagnetic field Bgrcan be identified uniquely.
3. Quantization condition for simple systems
Consider now the quantization condition for angular momentum with Planck constant replaced by gravitational Planck constant hbargr= GMm/v0 in the simple case of pointlike central mass. The condition is
m∫ v•dl = n × hbargr.
The condition reduces to the condition on velocity circulation
∫ v•dl = n × GM/v0.
In simple systems with circular rings forced by Zn symmetry the condition reduces to a universal velocity spectrum
so that only the radii of orbits depend on mass distribution. For systems for which cosmic string dominates only n=1 is possible. This is the case in the case of stars in galactic halo if primordial cosmic string going through the center of galaxy in direction of jet dominates the gravitational potential. The velocity of distant stars is correctly predicted.
Zn symmetry seems to imply that only circular orbits need to be considered and there is no need to apply the condition for other canonical momenta (radial canonical momentum in Kepler problem). The nearly circular orbits of visible matter objects would be naturally associated with dark matter rings or more complex structures with Zn symmetry and dark matter rings could suffer partial or complete phase transition to visible matter.
4. Generalization of the quantization condition
The mechanisms behind the formation of planetary systems, galaxies and larger systems are poorly understood but planar structures seem to define a common denominator and the recent discovery of dark matter ring in a galactic cluster in Mly scale (see this) suggest that dark matter rings might define a universal step in the formation of astrophysical structures.
Also the dynamics in planet scale is poorly understood. In particular, the rings of Saturn and Jupiter are very intricate structures and far from well-understood. Assuming spherical symmetry it is far from obvious why the matter ends up to form thin rings in a preferred plane. The latest surprise is that Saturn's largest, most compact ring consist of clumps of matter separated by almost empty gaps. The clumps are continually colliding with each other, highly organized, and heavier than thought previously.
The situation suggests that some very important piece might be missing from the existing models, and the vision about dark matter as a quantum phase with a gigantic Planck constant (see this and this) is an excellent candidate for this piece. The vision that the quantum dynamics for dark matter is behind the formation of the visible structures suggests that the formation of the astrophysical structures could be understood as a consequence of Bohr rules.
1. General quantum vision about formation of structures
The basic observation is that in the case of a straight cosmic string creating a gravitational potential of form v12/ρ Bohr quantization does not pose any conditions on the radii of the circular orbits so that a continuous mass distribution is possible.
This situation is obviously exceptional. If one however accepts the TGD based vision (see this) that the very early cosmology was cosmic string dominated and that elementary p"../articles/ were generated in the decay of cosmic strings, this situation might have prevailed at very early times. These cosmic strings can transform to strings with smaller string tension and magnetic flux tubes can be seen as their remnants dark energy being identifiable as magnetic energy. If so, the differentiation of a continuous density of ordinary matter to form the observed astrophysical structures would correspond to an approach to a stationary situation governed by Bohr rules and in the first approximation one could neglect the intermediate stages.
Cosmic string need not be infinitely long: it could branch into n return flux tubes, n very large in accordance with the Zn symmetry for the dark matter but also in this case the situation in the nearby region remains the same. For large distances the whole structure would behave as a single mass point creating ordinary Newtonian gravitational potential. Also phase transitions in which the system emits magnetic flux tubes so that the contribution of the cosmic string to the gravitational force is reduced, are possible.
What is of utmost importance is that the cosmic string induces the breaking of the rotational symmetry down to a discrete Zn symmetry and in the presence of the central mass selects a unique preferred orbital plane in which gravitational acceleration is parallel to the plane. This is just what is observed in astrophysical systems and not easily explained in the Newtonian picture. In TGD framework this relates directly to the choice of quantization axis of angular momentum at the level of dark matter. This mechanism could be behind the formation of planar systems in all length scales including planets and their moons, planetary systems, galaxies, galaxy clusters in the scale of Mly, and even the concentration of matter at the walls of large voids in the scale of 100 Mly.
The Zn symmetry for the dark matter with very large n suggests the possibility of more precise predictions. If n is a ruler-and-compass integer it has as factors only first powers of Fermat primes and a very large power of 2. The breaking of Zn symmetry at the level of visible matter would naturally occur to subgroups Zm subset Zn. Since m is a factor of n, the average number of matter clumps could tend to be a factor of n, and hence a ruler-and-compass integer. Also the hexagonal symmetry discovered near North Pole of Saturn (see this)could have interpretation in terms of this symmetry breaking mechanism.
2. How inner and outer planets might have emerged?
The Bohr orbit model requires different values of the parameter v0 related by a scaling v0→v0/5 for inner and outer planets. It would be nice to understand why this is the case. The presence of cosmic string along rotational axis implied both by the model for the asymptotic state of the star and TGD based model for gamma ray bursts might allow to understand this result.
One can construct a simple modification of the hydrogen atom type model for solar system by including the contribution of cosmic string to the gravitational force. For circular orbits the condition identifying kinetic and gravitational radial accelerations plus quantization of angular momentum in units of gravitational Planck constant are used. The prediction is that only a finite number of Bohr orbits are possible. One might hope that this could explain the decomposition of the planetary system to inner and outer planets.
String tension implies anomalous acceleration of same form as the radial kinetic acceleration implying that for given radius kinetic energy per mass is shifted upwards by a constant amount. This acceleration anomaly is severely bounded above by the constant acceleration anomaly of space-crafts (Pioneer anomaly) and for the recent value of the cosmic string tension the number of allowed inner planets is much larger than 3.
The situation was however different in the primordial stage when cosmic string tension was much larger and gradually reduced in phase transitions involving the emission of closed magnetic flux tubes. Primordial Sun could have emitted the seeds of the two planetary systems related by scaling and that this might have happened in the phase transition reducing magnetic flux by the emission of closed magnetic flux tube structure.
3. Models for the interior of astrophysical object and for planetary rings
Using similar quantization conditions one can construct a very simple model of astrophysical object as a cylindrically symmetric pancake like structure. There are three basic predictions which do not depend on the details of the mass distribution.
The following catched my attention during this morning's webwalk.
NASA Hubble Space Telescope Detects Ring of Dark Matter
"Rings" puts bells ringing! In TGD Universe dark matter characterized by a gigantic value of Planck constant making dark matter a macroscopic quantum phase in astrophysical length and time scales. Rotationally symmetric structures - such as rings- with an exact rotational symmetry Zn, n very very large, of the "field body" of the system, is the basic prediction. In the model of planetary orbits the rings of dark matter around Bohr orbits force the visible matter at the Bohr orbit (see this).
TGD based model for dark matter inspires the hypothesis that it corresponds to Bohr orbit for macroscopically quantum coherent dark matter with gigantic value of Planck constant predicted by the model. The article about finding is now in archive and contains the data making possible to test the model. I am grateful for Kea for providing the link. The ring corresponds with a good accuracy to the lowest Bohr orbit for v0= 3×2-11, which is 3 times the favored value but allowed by the general hypothesis for the favored values of Planck constant.
I add the little calculation here to give an idea about what is involved. The number theoretic hypothesis for the preferred values of Planck constants states that the gravitational Planck constant
equals to a ruler-and-compass rational which is ratio q= n1/n2 of ruler-and-compass ni integers expressible as a product of form n=2k∏ Fs, where all Fermat primes Fs are different. Only four of them are known and they are given by 3, 5, 17, 257, 216+1. v0=2-11 applies to inner planets and v0=2-11/5 to outer planets and the conditions from the quantization of hbar are satisfied.
The obvious TGD inspired hypothesis is that the dark matter ring corresponds to Bohr orbit. Hence the distance would be
r= n2 r0,
where r0 is Bohr radius and n is integer. n=1 for lowest Bohr orbit. The Bohr radius is given
where M the total mass in the dense core region inside the ring. This would give distance of about 2000 times Schwartschild radius for the lowest orbit for the preferred value of v0=2-11.
This prediction can be confronted with the data since the article Discovery of a ringlike dark matter structure in the core of the galaxy cluster C1 0024+17 is in the archive now.
The conclusion would be that the ring would correspond to the lowest possible Bohr orbit for v0=3× 2-11. I would have been really happy if the favored value of v0 had appeared in the formula but the consistency with the ruler-and-compass hypothesis serves as a consolation. Skeptic can of course always argue that this is a pure accident. If so, it would be an addition to long series of accidents (planetary radii in solar system and radii of exoplanets). One can of course search rings at radii corresponding to n=2,3,... If these are found, I would say that the situation is settled.
For more details see the new chapter Quantum Astrophysics .
Gravitational waves has been discussed on both Lubos's blog and Cosmic Variance. This raised the stimulus of looking how TGD based predictions for gravitational waves differ classical predictions. The article Gravitational Waves in Wikipedia provides excellent background material which I have used in the following. This posting is an extended and corrected version of the original.
The description of gravitational radiation provides a stringent test for the idea about dark matter hierarchy with arbitrary large values of Planck constants. In accordance with quantum classical correspondence, one can take the consistency with classical formulas as a constraint allowing to deduce information about how dark gravitons interact with ordinary matter. In the following standard facts about gravitational radiation are discussed first and then TGD based view about the situation is sketched.
A. Standard view about gravitational radiation
A.1 Gravitational radiation and the sources of gravitational waves
Classically gravitational radiation corresponds to small deviations of the space-time metric from the empty Minkowski space metric (see this). Gravitational radiation is characterized by polarization, frequency, and the amplitude of the radiation. At quantum mechanical level one speaks about gravitons characterized by spin and light-like four-momentum.
The amplitude of the gravitational radiation is proportional to the quadrupole moment of the emitting system, which excludes systems possessing rotational axis of symmetry as classical radiators. Planetary systems produce gravitational radiation at the harmonics of the rotational frequency. The formula for the power of gravitational radiation from a planetary system given by
This formula can be taken as a convenient quantitative reference point.
Planetary systems are not very effective radiators. Because of their small radius and rotational asymmetry supernovas are much better candidates in this respect. Also binary stars and pairs of black holes are good candidates. In 1993, Russell Hulse and Joe Taylor were able to prove indirectly the existence of gravitational radiation. Hulse-Taylor binary consists of ordinary star and pulsar with the masses of stars around 1.4 solar masses. Their distance is only few solar radii. Note that the pulsars have small radius, typically of order 10 km. The distance between the stars can be deduced from the Doppler shift of the signals sent by the pulsar. The radiated power is about 1022 times that from Earth-Sun system basically due to the small value of R. Gravitational radiation induces the loss of total energy and a reduction of the distance between the stars and this can be measured.
A.2 How to detect gravitational radiation?
Concerning the detection of gravitational radiation the problems are posed by the extremely weak intensity and large distance reducing further this intensity. The amplitude of gravitational radiation is measured by the deviation of the metric from Minkowski metric, denote by h.
Weber bar (see this) provides one possible manner to detect gravitational radiation. It relies on a resonant amplification of gravitational waves at the resonance frequency of the bar. For a gravitational wave with an amplitude h≈10-20 the distance between the ends of a bar with length of 1 m should oscillate with the amplitude of 10-20 meters so that extremely small effects are in question. For Hulse-Taylor binary the amplitude is about h=10-26 at Earth. By increasing the size of apparatus one can increase the amplitude of stretching.
Laser interferometers provide second possible method for detecting gravitational radiation. The masses are at distance varying from hundreds of meters to kilometers(see this). LIGO (the Laser Interferometer Gravitational Wave Observatory) consists of three devices: the first one is located with Livingston, Lousiana, and the other two at Hanford, Washington. The system consist of light storage arms with length of 2-4 km and in angle of 90 degrees. The vacuum tubes in storage arms carrying laser radiation have length of 4 km. One arm is stretched and one arm shortened and the interferometer is ideal for detecting this. The gravitational waves should create stretchings not longer that 10-17 meters which is of same order of magnitude as intermediate gauge boson Compton length. LIGO can detect a stretching which is even shorter than this. The detected amplitudes can be as small as h≈ 5× 10-22.
B. Gravitons in TGD
B.1 Gravitons in TGD
Unlike the naive application of Mach's principle would suggest, gravitational radiation is possible in empty space in general relativity. In TGD framework it is not possible to speak about small oscillations of the metric of the empty Minkowski space imbedded canonically to M4× CP2 since Kähler action is non-vanishing only in fourth order in the small deformation and the deviation of the induced metric is quadratic in the deviation. Same applies to induced gauge fields. Even the induced Dirac spinors associated with the modified Dirac action fixed uniquely by super-symmetry allow only vacuum solutions in this kind of background. Mathematically this means that both the perturbative path integral approach and canonical quantization fail completely in TGD framework. This led to the vision about physics as Kähler geometry of "world of classical worlds" with quantum states of the universe identified as the modes of classical configuration space spinor fields.
The resolution of various conceptual problems is provided by the parton picture and the identification of elementary p"../articles/ as light-like 3-surfaces associated with the wormhole throats. Gauge bosons correspond to pairs of wormholes and fermions to topologically condensed CP2 type extremals having only single wormhole throat.
Gravitons are string like objects in a well defined sense. This follows from the mere spin 2 property and the fact that partonic 2-surfaces allow only free many-fermion states. This forces gauge bosons to be wormhole contacts whereas gravitons must be identified as pairs of wormhole contacts (bosons) or of fermions connected by flux tubes. The strong resemblance with string models encourages to believe that general relativity defines the low energy limit of the theory. Of course, if one accepts dark matter hierarchy and dynamical Planck constant, the notion of low energy limit itself becomes somewhat delicate.
B.2 Model for the giant graviton
Detector, giant graviton, source, and topological light ray will be denoted simply by D, G, and S, and ME in the following. Consider first the model for the giant graviton.
Second kind of dark graviton is analog for plane wave with a finite transversal cross section. TGD indeed predicts what I have called topological light rays, or massless extremals (MEs) as a very general class of solutions to field equations ((see this, this, and this).
MEs are typically cylindrical structures carrying induced gauge fields and gravitational field without dissipation and dispersion and without weakening with the distance. These properties are ideal for targeted long distance communications which inspires the hypothesis that they play a key role in living matter (see this and this) and make possible a completely new kind of communications over astrophysical distances. Large values of Planck constant allow to resolve the problem posed by the fact that for long distances the energies of these quanta would be below the thermal energy of the receiving system.
Giant gravitons are expected to decay to this kind of dark gravitons having smaller value of Planck constant via de-decoherence and that it is these gravitons which are detected. Quantitative estimates indeed support this expectation.
At the space-time level dark gravitons at the lower levels of hierarchy would naturally correspond to na-Riemann sheeted (r=GmE/v0=na/nb for m>>E) variants of topological light rays ("massless extremals", MEs), which define a very general family of solutions to field equations of TGD (see this). na-sheetedness is with respect to CP2 and means that every point of CP2 is covered by na M4 points related by a rotation by a multiple of 2π/na around the propagation direction assignable with ME. nb-sheetedness with respect to M4 is possible but does not play a significant role in the following considerations. Using the same loose language as in the case of giant graviton, one can say that r=na/nb copies of same graviton have suffered a topological condensation to this kind of ME. A more precise statement would be na gravitons with fractional unit hbar0/na for spin.
C. Detection of gravitational radiation
What is the value of dark gravitational constant which must be assigned to the measuring system and gravitational radiation from a given source? Is the detection of primary giant graviton possible by human means or is it possible to detect only dark gravitons produced in the sequential de-coherence of giant graviton? Do dark gravitons enhance the possibility to detect gravitational radiation as one might expect? What are the limitations on detection due to energy conservation in de-coherence process?
C.1 TGD counterpart for the classical description of detection process
The oscillations of the distance between the two masses defines a simplified picture about the receival of gravitational radiation. Now ME would correspond to na-"Riemann-sheeted" (with respect to CP2)graviton with each sheet oscillating with the same frequency. Classical interaction would suggest that the measuring system topologically condenses at the topological light ray so that the distance between the test masses measured along the topological light ray in the direction transverse to the direction of propagation starts to oscillate.
Obviously the classical behavior is essentially the same as as predicted by general relativity at each "Riemann sheet". If all elementary p"../articles/ are maximally quantum critical systems and therefore also gravitons, then gravitons can be absorbed at each step of the process, and the number of absorbed gravitons and energy is r-fold.
C.2. Sequential de-coherence
Suppose that the detecting system has some mass m and suppose that the gravitational interaction is mediated by the gravitational field body connecting the two systems.
The Planck constant must characterize the system formed by dark graviton and measuring system. In the case that E is comparable to m or larger, the expression for r=hbar/hbar0 must replaced with the relativistically invariant formula in which m and E are replaced with the energies in center of mass system. This gives
r= GmE/[v0(1+β)(1-β)1/2], β= z(-1+(1+2/x))1/2) , x= E/2m .
Assuming m>>E0 this gives in a good approximation
r=Gm1 E0/v0= G2 m1mM/v02.
Note that in the interaction of identical masses ordinary hbar is possible for m≤ (v0)1/2MPl. For v0=2-11 the critical mass corresponds roughly to the mass of water blob of radius 1 mm.
One can interpret the formula by saying that de-coherence splits from the incoming dark graviton dark piece having energy E1= (Gm1E0/v0)ω, which makes a fraction E1/E0= (Gm1/v0)ω from the energy of the graviton. At the n:th step of the process the system would split from the dark graviton of previous step the fraction
from the total emitted energy E0. De-coherence process would proceed in steps such that the typical masses of the measuring system decrease gradually as the process goes downwards in length and time scale hierarchy. This splitting process should lead at large distances to the situation in which the original spherical dark graviton has split to ordinary gravitons with angular distribution being same as predicted by GRT.
The splitting process should stop when the condition r≤ 1 is satisfied and the topological light ray carrying gravitons becomes 1-sheeted covering of M4. For E<<m this gives GmE≤ v0 so that m>>E implies E<<MPl. For E>>m this gives GE3/2m1/2 <2v0 or
E/m≤ (2v0/Gm2)2/3 .
C.3. Information theoretic aspects
The value of r=hbar/hbar0 depends on the mass of the detecting system and the energy of graviton which in turn depends on the de-coherence history in corresponding manner. Therefore the total energy absorbed from the pulse codes via the value of r information about the masses appearing in the de-coherence process. For a process involving only single step the value of the source mass can be deduced from this data. This could some day provide totally new means of deducing information about the masses of distant objects: something totally new from the point of view of classical and string theories of gravitational radiation. This kind of information theoretic bonus gives a further good reason to take the notion of quantized Planck constant seriously.
If one makes the stronger assumption that the values of r correspond to ruler-and-compass rationals expressible as ratios of the number theoretically preferred values of integers expressible as n=2k∏sFs, where Fs correspond to different Fermat primes (only four is known), very strong constraints on the masses of the systems participating in the de-coherence sequence result. Analogous conditions appear also in the Bohr orbit model for the planetary masses and the resulting predictions were found to be true with few per cent. One cannot therefore exclude the fascinating possibility that the de-coherence process might in a very clever manner code information about masses of systems involved with its steps.
C.4. The time interval during which the interaction with dark graviton takes place?
If the duration of the bunch is T= E/P, where P is the classically predicted radiation power in the detector and T the detection period, the average power during bunch is identical to that predicted by GRT. Also T would be proportional to r, and therefore code information about the masses appearing in the sequential de-coherence process.
An alternative, and more attractive possibility, is that T is same always and correspond to r=1. The intuitive justification is that absorption occurs simultaneously for all r "Riemann sheets". This would multiply the power by a factor r and dramatically improve the possibilities to detect gravitational radiation. The measurement philosophy based on standard theory would however reject these kind of events occurring with 1/r time smaller frequency as being due to the noise (shot noise, seismic noise, and other noise from environment). This might relate to the failure to detect gravitational radiation.
D. Quantitative model
D.1. Leakage of the giant graviton to sectors of imbedding space with smaller value of Planck constant
Consider first the model for the leakage of giant graviton to the sectors of H with smaller Planck constant.
D.2. The direct detection of giant graviton is not possible for long distances
Primary detection would correspond to a direct flow of energy from the giant graviton to detector. Assume that the source is modellable using large hbar variant of the Bohr orbit model for hydrogen atom. Denote by r=na/nb the rationals defining Planck constant as hbar= r×hbar0.
For G-S system one has
r(G,S)= GME/v0 =GMmv0× k/n3 .
where k is a numerical constant of order unity and m refers to the mass of planet. For Hulse-Taylor binary m≈ M holds true.
For D-G system one has
r(D,G)=GM(D) E/v0 = GM(D)mv0× k/n3 .
The ratio of these rationals (in general) is of order M(D)/M.
Suppose first that the detector has a disk like shape. This gives for the total number n(D) of ordinary gravitons going to the detector the estimate
n(D)=(d/r)2 × na(G,S)= (d/r)2× GMmv0× nb(G,S)× k/n3 .
If the actual area of detector is smaller than d2 by a factor x one has
n(D)→ xn(D) .
n(D) cannot be smaller than the number of ordinary gravitons estimated using the Planck constant associated with the detector: n(D)≥ na(D,G)=r(D,G)nb(D,G). This gives the condition
Suppose for simplicity that nb(D,G)/nb(G,S)=1 and M(D)=103 kg and M(S)=1030 kg and r= 200 MPc ≈ 109 ly, which is a typical distance for binaries. For x=1,k=1,n=1 this gives roughly d≥ 10-4 ly ≈ 1011 m, which is roughly the size of solar system. From energy conservation condition the entire solar system would be the natural detector in this case. Huge values of nb(G,S) and larger reduction of nb(G,S) would be required to improve the situation. Therefore direct detection of giant graviton by human made detectors is excluded.
D.3. Secondary detection
The previous argument leaves only the secondary detection into consideration. Assume that ME results in the primary de-coherence of a giant graviton. Also longer de-coherence sequences are possible and one can deduce analogous conditions for these.
Energy conservation gives
S(D)/S(ME)× r(ME,G) = r(D,ME) .
Using the expression for S(ME) this gives an expression for S(ME) for a given detector area:
S(ME)= r(ME,G)/r(D,ME) × S(D)≈ E(G)/M(D)× S(D) .
From S(ME)=E(ME)/M(S)4π r2 one obtains
r = (E(G)M(S)/E(ME)M(D))1/2×S(D)1/2
for the distance at which ME is created. The distances of binaries studied in LIGO are of order D=1024 m. Using E(G)≈ Mv02 and assuming M=1030 kg and S(D)= 1 m2 (just for definiteness), one obtains r≈ 1025(kg/E(ME)) m. If ME is generated at distance r≈ D and if one has S(ME)≈ 106 m2 (from the size scale for LIGO) one obtains from the equation for S(ME) the estimate E(ME)≈ 10-25 kg ≈ 10-8 Joule.
The expressions for the energies of dark gravitons can be deduced from those of hydrogen atom using the replacements Ze2→4π GMm, hbar →GMm/v0. I have assumed that second mass is much smaller. The energies are given by
En= 1/n2E1 , E1= (Zα)2 m/4= (Ze2/4π×hbar)2× m/4→m/4v02.
E1 defines the energy scale. Note that v0 defines a characteristic velocity if one writes this expression in terms of classical kinetic energy using virial theorem T= -V/2 for the circular orbits. This gives En= Tn= mvn2/2= mv02/4n2 giving
vn=(v0/21/2)/n . Orbital velocities are quantized as sub-harmonics of the universal velocity v0/2-1/2=2-23/2 and the scaling of v0 by 1/n scales does not lead out from the set of allowed velocities.
Bohr radius scales as r0= hbar/Zα m→ GM/v02.
For v0=211 this gives r0= 222GM ≈ 4× 106GM. In the case of Sun this is below the value of solar radius but not too much.
The frequency ω(n,n-k) of the dark graviton emitted in n→n-k transition and orbital rotation frequency ωn are given by
ω(n,n-k) = v03/GM× (1/n2-1/(n-k)2)≈ kωn.
The emitted frequencies at the large n limit are harmonics of the orbital rotation frequency so that quantum classical correspondence holds true. For low values of n the emitted frequencies differ from harmonics of orbital frequency.
The energy emitted in n→n-k transition would be
E(n,n-k)= mv02× (1/n2-1/(n-k)2) ,
and obviously enormous. Single spherical dark graviton would be emitted in the transition and should decay to gravitons with smaller values of hbar. Bunch like character of the detected radiation might serve as the signature of the process. The bunch like character of liberated dark gravitational energy means coherence and might play role in the coherent locomotion of living matter. For a pair of systems of masses m=1 kg this would mean Gm2/v0≈ 1020 meaning that exchanged dark graviton corresponds to a bunch containing about 1020 ordinary gravitons. The energies of graviton bunches would correspond to the differences of the gravitational energies between initial and final configurations which in principle would allow to deduce the Bohr orbits between which the transition took place. Hence dark gravitons could make possible the analog of spectroscopy in astrophysical length scales.
E. Generalization to gauge interactions
One can imagine several applications.
E.2 In what sense dark matter is dark?
The notion of dark matter as something which has only gravitational interactions brings in mind the concept of ether and is very probably only an approximate characterization of the situation. As I have been gradually developing the notion of dark matter as a hierarchy of phases of matter with an increasing value of Planck constant, the naivete of this characterization has indeed become obvious.
If the proposed view is correct, dark matter is dark only in the sense that the process of receiving the dark bosons (say gravitons) mediating the interactions with other levels of dark matter hierarchy, in particular ordinary matter, differs so dramatically from that predicted by the theory with a single value of Planck constant that the detected dark quanta are unavoidably identified as noise. Dark matter is there and interacts with ordinary matter and living matter in general and our own EEG in particular provide the most dramatic examples about this interaction. Hence we could consider the dropping of "dark matter" from the glossary altogether and replacing the attribute "dark" with the spectrum of Planck constants characterizing the p"../articles/ (dark matter) and their field bodies (dark energy).
For more details see the chapter Quantum Astrophysics .
Gravity Probe B experiment tests the predictions of General Relativity related to gravimagnetism. Only the abstract of the talk C. W. Francis Everitt summarizing the results is available when I am writing this. Here is a slightly reformatted abstract of the talk.
The NASA Gravity Probe B (GP-B) orbiting gyroscope test of General Relativity, launched from Vandenberg Air Force Base on 20 April, 2004, tests two consequences of Einstein's theory:
The Confrontation between General Relativity and Experiment gives an excellent summary of various test of GRT. The predictions tested by GP-B relate to gravitomagnetic effects. The field equations of general relativity in post-Newtonian approximation with a choice of a preferred frame can in good approximation (gij=-δij) be written in a form highly reminiscent of Maxwell's equestions with gtt component of metric defining the counterpart of the scalar potential giving rise to gravito-electric field and gti the counterpart of vector potential giving rise to the gravitomagnetic field.
Rotating body generates a gravitomagnetic field so that bodies moving in the gravitomagnetic field of a rotating body experience the analog of Lorentz force and gyroscope suffers a precession similar to that suffered by a magnetic dipole in magnetic field (Thirring-Lense efffect or frame-drag). Besides this there is geodetic precession due to the motion of the gyroscope in the gravito-electric field present even in the case of non-rotating source which might be perhaps understood in terms of gravito-Faraday law. Both these effects are tested by GP-B.
In the following something general about how TGD and GRT differs and also something about the predictions of TGD concerning GP-B experiment.
1. TGD and GRT
Consider first basic differences between TGD and GRT.
There are excellent reasons to expect that Maxwellian picture holds true in a good accuracy if one uses Minkowski coordinates for the space-time surface. In fact, TGD allows a static solutions with 2-D CP2 projection for which the prerequisites of the Maxwellian interpretation are satisfied (the deviations of the spatial components gij of the induced metric from -δij are negligible).
Schwartschild and Reissner-Norströom metric allow imbeddings as 4-D surfaces in H but Kerr metric assigned to rotating systems probably not. If this is indeed the case, the gravimagnetic field of a rotating object in TGD Universe cannot be identical with the exact prediction of GRT but could be so in the Post-Newtonian approximation. Scalar and vector potential correspond to four field quantities and the number of CP2 coordinates is four. Imbedding as vacuum extremals with 2-D CP2 projection guarantees automatically the consistency with the field equations but requires the orthogonality of gravito-electric and -magnetic fields. This holds true in post-Newtonian approximation in the situation considered.
This raises the possibility that apart from restrictions caused by the failure of the global imbedding at short distances one can imbed Post-Newtonian approximations into H in the approximation gij=-δij. If so, the predictions for Thirring-Lense effect would not differ measurably from those of GRT. The predictions for the geodesic precession involving only scalar potential would be identical in any case.
The imbeddability in the post-Newtonian approximation is however questionable if one assumes vacuum extremal property and small deformations of Schwartschild metric indeed predict a gravitomagnetic field differing from the dipole form.
3. Simplest candidate for the metric of a rotating star
The simplest situation for the metric of rotating start is obtained by assuming that vacuum extremal imbeddable to M4 × S2, where S2 is the geodesic sphere of CP2 with vanishing homological charge and induce Kähler form. Use coordinates Θ,Φ for S2 and spherical coordinates (t,r,θ,φ) in space-time identifiable as M4 spherical coordinates apart from scaling and r-dependent shift in the time coordinate.
4. Comparison with the dipole field
The simplest candidate for the gravitomagnetic field differs in many respects from a dipole field.
5. Consistency with the model for the asymptotic state of star
In TGD framework natural candidates for the asymptotic states of the star are solutions of field equations for which gravitational four-momentum is locally conserved. Vacuum extremals must therefore satisfy the field equations resulting from the variation of Einstein's action (possibly with cosmological constant) with respect to the induced metric. Quite remarkably, the solution representing asymptotic state of the star is necessarily rotating (see this).
The proposed picture is consistent with the model of the asymptotic state of star. Also the magnetic parts of ordinary gauge fields have essentially similar behavior. This is actually obvious since CP2 coordinates are fundamental dynamical variables and the field line topologies of induced gauge fields and induced metric are therefore very closely related.
As already discussed, the physicists M. Tajmar and C. J. Matos and their collaborators working in ESA (European Satellite Agency) have made an amazing claim of having detected strong gravimagnetism with gravimagnetic field having a magnitude which is about 20 orders of magnitude higher than predicted by General Relativity. Hence there are some reasons to think that gravimagnetic fields might have a surprise in store.
Addition: Lubos Motl's blog tells that the error bars are still twice the size of the predicted frame-dragging effect. Already this information would have killed TGD inspired (strongly so) model unless the satellite had been at equator!
For details and background see the chapter TGD and GRT.
Machian Principle has not played any role in the development of TGD. Hence it is somewhat surprising that this principle allows several interpretations in TGD framework.
1. Non-conserved gravitational four-momentum and conserved inertial momentum at 4-D space-time level
Consider first the situation at the level of classical theory identifiable in terms of classical dynamics for space-time surfaces.
The first question is how non-conserved gravitational and conserved inertial four-momentum relate to each other. Certainly Equivalence Principle in a strong form cannot hold true.
A deeper level description of the situation is achieved at parton level. For light-like partonic 3-surfaces the dynamics is defined by almost topological QFT defined by Chern-Simons action for the induced Kähler form. The extrema have 2-D CP2 projection. Light-likeness implies the replacement of "topological" with "almost topological" by bringing in the notions of metric and four-momentum.
TGD allows several interpretations of Machian Principle and leads also to a generalization of the Principle.