## Gravitational radiation and large value of gravitational Planck constantGravitational 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.
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
P= dE/dt=(32/π)×G 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 10
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
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
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 M
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 CP 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.
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.
- Orbital plane defines the natural quantization axis of angular momentun. Giant graviton and all dark gravitons corresponds to n
_{a}-fold coverings of CP_{2}by M^{4}points, which means that one has a quantum state for which fermionic part remains invariant under the transformations φ→ φ+2π/n_{a}. This means in particular that the ordinary gravitons associated with the giant graviton have same spin so that the giant graviton can be regarded as Bose-Einstein condensate in spin degrees of freedom. Only the orbital part of state depends on angle variables and corresponds to a partial wave with a small value of L. - The total angular momentum of the giant graviton must correspond to the change of angular momentum in the quantum transition between initial and final orbit. Orbital angular momentum in the direction of quantization axis should be a small multiple of dark Planck constant associated with the system formed by giant graviton and source. These states correspond to Bose-Einstein condensates of ordinary gravitons in eigen state of orbital angular with ordinary Planck constant. Unless S-wave is in question the intensity pattern of the gravitational radiation depends on the direction in a characteristic non-classical manner. The coherence of dark graviton regarded as Bose-Einstein condensate of ordinary gravitons is what distinguishes the situation in TGD framework from that in GRT.
- If all elementary p"/public_html/articles/ with gravitons included are maximally quantum critical systems, giant graviton should contain r(G,S) =n
_{a}/n_{b}ordinary gravitons. This number is not an integer for n_{b}>1. A possible interpretation is that in this case gravitons possess fractional spin corresponding to the fact that rotation by 2π gives a point in the n_{b}-fold covering of M^{4}point by CP_{2}points. In any case, this gives an estimate for the number of ordinary gravitons and the radiated energy per solid angle. This estimate follows also from the energy conservation for the transition. The requirement that average power equals to the prediction of GRT allows to estimate the geometric duration associated with the transition. The condition hbar ω = E_{f}-E_{i}is consistent with the identification of hbar for the pair of systems formed by giant-graviton and emitting system.
B.3 Dark graviton as topological light raySecond 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 n
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?
The oscillations of the distance between the two masses defines a simplified picture about the receival of gravitational radiation. Now ME would correspond to n 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.
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/hbar
r= GmE/[v
Assuming m>>E
r=Gm
Note that in the interaction of identical masses ordinary hbar is possible for m≤ (v
One can interpret the formula by saying that de-coherence splits from the incoming dark graviton dark piece having energy E
E
from the total emitted energy E
The splitting process should stop when the condition r≤ 1 is satisfied and the topological light ray carrying gravitons becomes 1-sheeted covering of M
E/m≤ (2v
The value of r=hbar/hbar
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=2
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.
Consider first the model for the leakage of giant graviton to the sectors of H with smaller Planck constant.
- Giant graviton leaks to sectors of H with a smaller value of Planck constant via quantum critical points common to the original and final sector of H. If ordinary gravitons are quantum critical they can be regarded as leakage points.
- It is natural to assume that the resulting dark graviton corresponds to a radial topological light ray (ME). The discrete group Z
_{na}acts naturally as rotations around the direction of propagation for ME. The Planck constant associated with ME-G system should by the general criterion be given by the general formula already described. - Energy should be conserved in the leakage process. The secondary dark graviton receives the fraction Δ ω/4π= S/4π r
^{2}of the energy of giant graviton, where S(ME) is the transversal area of ME, and r the radial distance from the source, of the energy of the giant graviton. Energy conservation givesS(ME)/4π r ^{2}hbar(G,S)ω= hbar(ME,G)ω .or S(ME)/4π r ^{2}= hbar(ME,G)/hbar(G,S)≈ E(ME)/M(S) .The larger the distance is, the larger the area of ME. This means a restriction to the measurement efficiency at large distances for realistic detector sizes since the number of gravitons must be proportional to the ratio S(D)/S(ME) of the areas of detector and ME.
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=n For G-S system one has
r(G,S)= GME/v 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/v 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)
If the actual area of detector is smaller than d 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)≥ n
d/r≥(M(D)/M(S))
Suppose for simplicity that n
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 r
r = (E(G)M(S)/E(ME)M(D))
for the distance at which ME is created. The distances of binaries studied in LIGO are of order D=10
D.4 Some quantitative estimates for gravitational quantum transitions in planetary systems
The expressions for the energies of dark gravitons can be deduced from those of hydrogen atom using the replacements Ze
E
E
v
Bohr radius scales as
r
For v
The frequency ω(n,n-k) of the dark graviton emitted in n→n-k transition and orbital rotation frequency ω
ω(n,n-k) = v
ω 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)= mv
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 Gm
_{1}Q_{2}g^{2}/v_{0}, were g is the coupling constant of appropriate gauge interaction. v_{0} need not be same as in the gravitational case. The value of Q_{1}Q_{2}g^{2} for which perturbation theory fails defines a plausible estimate for v_{0}. The naive guess would be v_{0}≈ 1. In the case of gravitation this interpretation would mean that perturbative approach fails for GM_{1}M_{2}=v_{0}. For r>1 Planck constant is quantized with rational values with ruler-and-compass rationals as favored values. For gauge interactions r would have rather small values. The above criterion applies to the field body connecting two gauge charged systems. One can generalize this picture to self interactions assignable to the "personal" field body of the system. In this case the condition would read as Q^{2}g^{2}/v_{0}>>1.
One can imagine several applications.
- A possible application would be to electromagnetic interactions in heavy ion collisions.
- Gamma ray bursts might be one example of dark photons with very large value of Planck constant. The MEs carrying gravitons could carry also gamma rays and this would amplify the value of Planck constant form them too.
- Atomic nuclei are good candidates for systems for which electromagnetic field body is dark. The value of hbar would be r=Z
^{2}e^{2}/v_{0}, with v_{0}≈ 1. Electromagnetic field body could become dark already for Z>3 or even for Z=3. This suggest a connection with nuclear string model (see this) in which A< 4 nuclei (with Z<3) form the basic building bricks of the heavier nuclei identified as nuclear strings formed from these structures which themselves are strings of nucleons. - Color confinement for light quarks might involve dark gluonic field bodies.
- Dark photons with large value of hbar could transmit large energies through long distances and their phase conjugate variants could make possible a new kind of transfer mechanism (see this) essential in TGD based quantum model of metabolism and having also possible technological applications. Various kinds of sharp pulses suggest themselves as a manner to produce dark bosons in laboratory. Interestingly, after having given us alternating electricity, Tesla spent the rest of his professional life by experimenting with effects generated by electric pulses. Tesla claimed that he had discovered a new kind of invisible radiation, scalar wave pulses, which could make possible wireless communications and energy transfer in the scale of globe (see this for a possible but not the only TGD based explanation).
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 . |