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"/public_html/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"/public_html/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"/public_html/articles/ (dark matter) and their field bodies (dark energy).
For more details see the chapter Quantum Astrophysics .