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Can LIGO, VIRGO, GEO600, TAMA, AIGO or LISA Detectors Really Detect?

Pawel Tatrocki (Department of Physics, Pedagogical Academy, Cracow)

Published in physic.philica.com

Abstract
In this paper we checked how the diffraction of light on the detector’s mirrors influences the sensivity of the light interferometer detectors. Moreover, simple estimations in this paper have demonstrated that the gravitational wave detection by means of LIGO, VIRGO, GEO600, AIGO, TAMA or LISA detectors can be very complicated, or even impossible, because of atom oscillations of the mirror surfaces brought about by the laser beam photon reflection on them. Atom oscillation amplitude calculated theoretically is about a hundred times higher than the detector displacement rate caused by the passage of gravitational wave. This displacement is expected to be subjected to measurements.

Article body

Introduction

Attempts to detect gravitational waves as specific solutions to Einstein equations have taken longer than 40 years now and started with the pioneer work of J. Weber [1]. So far, however, only two types of detectors are practically fit for receiving gravitational waves: they are Michelson's laser interferometer based on ideas of F. A. E. Pirani [2], M. E. Gersenshtein [3], R. Weiss [4], G. E. Moss, L. R. Miller, and L. R. Foreward [5] and finally designed by R. E Vogt, R. W. Drewer, F. J. Raab, K. S. Thorne and R. Weiss [6-7] and electromagnetic interferent antenna developed by V. B. Braginsky, M. B. Mensky, L. P. Grishchouk, A. G. Doroshkievich, Ya. B. Zeldovitch, I. D. Novikov, M. V. Sazhin [8-9], and especially by A. Ku?ak [10]. This paper covers discussion on the operation of only one detector type - Michelson's laser interferometer. Discussion of two effects will be carried out here which may interfere with the measurement of the apparatus distortion caused by passage of the gravitational wave. They are in succession as follows:1. Seismic interference, 2. Diffraction of the laser beam on the detector mirrors and 3. Laser heating up thelight reflecting mirror surface. The first effect was discussed in [7], [12], but seems to be not deep enough, whereas the other one included only mirror distortion caused by interaction with laser beam [13-27] or they would assume in a classical way that the laser beam reflects from the whole mirror, and not from different atoms thereof. Our discussion will consequently cover effects which have not been taken into account so far. The first of discussion points refers only to LIGO, VIRGO, GEO600, AIGO and TAMA detectors while the other on involves also LISA detector. Let us then take on these problems.

Noise from the urban area

1. Did the project experimentators take into account existence of seismic noise coming from urban area located for example 100 kms away from interferometer?May these incidental ground vibrations "jam" the measurements? It concerns not only vibrations which would be transmitted only to mirrors, but also those of the whole object. For example, as a result of such vibrations, the whole vacuum tube could contract or elongate, thus possibly simulate the passage of gravitational wave. This condition is similar to the case of Apollo 13 spaceship on the Moon. At that time, seismic wave had formed, which was registered tens of kilometers away from the fall place. What does the seismic map of this terrain look like? To give an example,tramping of horses in a prairie is perceptible from anywhere from ten totwenty kilometers. What is more, the traffic of thousands of cars can addconsiderably to the detector noise within tha range between 100 Hz and 1 kHz,so within an observation range. The lack of perception of such a noise in theinterference spectrum lines may be the result of covering this effect by theeffect described in section IV (The situation from 1998 when the paper was at first submitted to the Physical Review and without any reason rejected has slightly changed. We know now that the power saws noise (from the nearby forest) and earth core plasma movement noise disturbed the measurement almost completely. The noise coming from the earth plasma core movement is 1011 as much as higher than expected at the frequency of 10-4 Hz. Thus, advanced LIGO project is infeasible).

Diffraction effects in detectors

Has the phenomenon of laser beam extension under the influence of diffraction upon optical elements been taken into consideration? Publications [16] and [19] seem to show it has not, although figure in the publication [14] suggests that an effect has been taken into account. Figure of the laser beam in [14] shows that the beam expands while traveling the distance between the mirrors, but it is being focussed after it has been reflected back. It seems unlikely that the same focussing effect has been produced by a single mirror of 600 mm in diameter (the size of the mirrors after [21]). We are convinced of it by means of elementary calculations. From [22] we obtain that the radius of mirror curvature is 3.5 km and the mirror diameter is 40 cm. Thus, the spread s of the maximum distance between the surface of a mirror and the cutting plane will be (Fig. 1.):

$s = R_{1} - l=R_{1}-\sqrt{R_{1}^{2}-R^{2}}=R_{1} \left( 1-\sqrt{1-{R^{2} \over R_{1}^{2}}} \right) \simeq \frac{R^{2}}{2R_{1}} \simeq 5.8 \times 10^{-6}m.$

To derive the formula for s we have used the expansion of a root to the first term in Taylor's series because of the small value of the R / R1 ratio. The same value s means that we would have to create the radius of curvature for invidual mirrors with the accuracy of 10-7% R1. Taking into consideration comparative data for lens curvature [28], namely the fact that the bigger the radius of curvature, the smaller the accuracy of making the lens with a 10 m curvature radius is 0.1% R1, it seems very unlikely to make a mirror of 6 km radius of curvature (VIRGO project [14]) with the accuracy of 10-7% R1. Probably such restrictions forced Japanese industry to reject scientists' offer to make a mirror of such length [29]. Let us pay attention to one more physical aspect, which will make the use of the mirror as a laser beam focusing element difficult. With this end in view let us consider Fig. 2.

That is to say, let us calculate the difference of optical paths O'A and OB. If the difference is smaller than the distance λ' calculated by means of Heisenberg uncertainty principle, a single photon will not recognize whether it is reflected from point A or from point B. Thus, the system between the points loses its focussing properties. The physical effect of reflecting a photon from a mirror will be the same in points A and B. Therefore, thearea of the mirror, in which the optical path difference O'A and OB is smaller than λ', will behave like a mirror. A considerable part of the laser beam falling onto this area will be reflected in accordance with the law of reflection and refraction, and it will go beyond the region accessible to a CCD detector the diameter of which is 5 cm. Such effect will cause the decrease in the beam reaching the detector, which will result in decreasing interference pattern intensity. Let us evaluate the size of the area between points A and B. From the figure we obtain OB=OA=R - radius of the mirror and

$O'A=\sqrt{OC^{2}-OO'^{2}}, \; \; OC-O'A=\lambda', \; \; \lambda'={\lambda \over 4\pi},$

where λ stands for the laser light length. Thus,

$\lambda'=OC-\sqrt{OC^{2}-OO'^{2}},$

that is

$OO'=\sqrt{2OC\lambda'-\lambda'^{2}} \simeq \sqrt{2OC\lambda'} \simeq 2.6\, cm,$

for λ=1.064 μm and OB=4 km. Thus, in the area of 5 cm of the CCD detector there will be a sudden decrease of interference pattern intensity, which will worsen considerably the accuracy of calculating the position of mirrors. Obviously, CCD elements can be increased; however, we will deal with the decrease of interference pattern intensity and with the effect of their vanishing as the range of interference pattern increases. Provided that making such mirrors is impossible, nothing remains but to use plane mirrors and a laser beam of the smallest angular divergence possible. The angle of a laser beam divergence is represented by the [30] formula:

$\alpha \simeq 1.22 \frac{\lambda}{D},$

where λ stands for the length of laser light wave, D means the diameter of the output aperture of the optical system. In the case of normal mirrors the reflection angle θ is decreased by the angle 2φ caused by diffraction (Fig. 3.).

Taking this fact into consideration as well as setting the data as follows: λ=1.064 μm, D=600 mm, we obtain the result that the angle of a laser beam divergence β will be equal to angle

$\beta=2(\vartheta+ \phi), \,\, \phi=1.22{\lambda \over D},$

where $\vartheta$ is the half divergence angle of the incident wave and φ is the angle due to the diffraction of the laser beam on the mirror. Thus:

$\Delta D \simeq 2(\vartheta+ \phi) l,$

where D is the diameter of the beam on the mirror (Fig. 3). Therefore, for example in the case of LIGO, the ray, after having travelled the arm of the length equal l=4 km, with the divergence angle of

$2\vartheta=2\times1.22{\lambda \over D} = 2.44 \times (1.064\times10^{-6}m)/ 0.05 m \simeq 5.2\times 10^{-5}rad$

will increase its diameter by

$\Delta D=\vartheta l \simeq 0.2 m.$

Now let's try to estimate the power loss due to decreasing of the laser beam because of its diffraction on interferometer mirrors. To simplify, let us assume that the diameter of the laser ray is about 50 mm [21] and that the radiation distribution is homogeneous along the whole diameter and it is not of the gauss type. Moreover, assuming that interferometer constructors disregarded the problem of a laser beam divergence thinking it is of no importance, we make an assumption that the ray of only 5 cm in diameter has been taken into consideration to analyse interference lines. Thus, the output power of a laser beam will be equal to the total power of input beams with regard to the phenomenon of a uniform dissipation of luminous energy over the whole increased ray area. We make an assumption that the ray is reflected in the mirrors 10 000 times, much more than in [32] and only the rays which have travelled the distance of a detector arm even number of times (thus, there are 5 000 of them) contribute to the output beam. We disregard the loss in the laser beam power in our calculations since they can be neglected in case of mirrors with the coefficient of 10-6 [23], [33] and the reflectionnumber of 10 000. Thus, the total power of the output beam is as follows:

$P=\sum_{n=1}^{5000} \frac{P_{n}d^{2}}{(d+\Delta D_{n})^{2}},$

where Pn means the power of beam for the nth passage of the beam towards the detector (to and fro), ΔD means the change of the beam diameter for the nth passage, and d stands for the diameter of this ray. The denominator, in general, is derived from the fact that the laser ray increases by ΔR each time after having travelled the doubled distance of the interferometer arm. Moreover, the accessible power of the laser beam decreases each time after traveling along mirrors because a part of the beam is out of the 5 cm spread of the initial beam. Therefore, due to the diffraction, this part of the beam is lost. Thus, we write Pn and not P0, where P0 is the power of the initial beam. The series is a very fast convergent, and spreading it even onto a million of reflections will change very little in comparison with 5000 reflections. If we insert the data from [19] into the power P formula, we will get the following power value:

$P\simeq 10^{-6} P_{theoret.},$

where Ptheoret. is the value calculated in case of not considering the laser ray divergence (in the appendix we presented a very simple program in C language to make easy the control of our calculations). A decrease in the output power of a laser beam by six orders results in the reduction of the interferometer sensitivity by three orders [32]. However, if we use phase - conjugate mirrors [34] instead of normal mirrors, then calculations made for them will differ only insignificantly from the calculations made for normal mirrors. So we adopt these calculations. The light reflected from the mirror undergoes diffraction on this mirror, in accordance with Huyghen's law. Therefore, reflection from phase-conjugate mirrors will not take place precisely along incidence lines but along lines somewhat broadened by the angle 2θ (Fig. 4.).

It is the angle by which a beam is broadened as a result of diffraction. Therefore, as a result of diffraction, the beam will increase in diameter as follows:

$\Delta D \simeq 2\theta l, \,\,\, \theta=1.22{\lambda \over D}.$

ΔD stands for the increase in the beam diameter at the point where we collect its power into the CCD detector. Assuming that the light reflects in the interferometer of the order of 104 times and excluding any absorption losses, we obtain that the accsssible power Paccess in the opening of 5 cm in diameter is:

$P_{access} \simeq 10^{-4} P_{theor}.$

Thermal noise in laser interferometers

How will the experimentators cope with atom mirror vibration generated by the laser beam reflected from the mirror? Let us try to make an estimation of such effect. Consider an atom as a harmonic oscillator oscillating under impact of photons of laser beam. The atom is kept in the crystal lattice by cohesion forces from the nearest neighbours. To estimate the recoil energy of an atom due to reflecting of photon on it we have to have a numerical value of an elastic constant of bonds in cristals. Let us try to find an estimation of the k'. Let us suppose that cohesion energy of an atom in the mirror is such that at the temperature of T = 500 K atoms oscillate with the amplitude of A= 0.1 lattice constant, which is assumed at 1 Å. We suppose that the lattice is a simple cubic, i.e. unit cell is of cubic shape with 8 atoms on its nodes. Thus, the energy of the atom in the cristal is equal to 8 energy bond values. In addition let's suppose that an atom has only three freedom degrees. Then:

$8\times{1 \over 2} k'A^{2}={3 \over 2} k T,$

where k stands for Boltzmann's constant. Then, elastic constant k' is equal to:

$k'={3 k T \over 8A^{2}}.$

Putting the numerical values for the temperature, Boltzmann's constant and dislocation into this formula we obtain

$k'={3 \times 500K \times 1.38 \times 10^{-23} {J \over K}\over 8(10^{-11}m)^{2}} \simeq 25.9 {N \over m}(1).$

And this number is accepted for our further calculations. Atom energy which any atom obtains as a result of collision with the photon from laser beam is estimated from Compton effect [38] (at the same time, electron mass is substituted by atom mass M). This effect produces energy E' of photons scattered at the angle θ equal to (Fig. 5.) :

$E'= {E m_{e} c^{2} \over m_{e} c^{2} + E(1-\cos\theta)}(2).$

Atom deflection is estimated by comparision of harmonic oscillator energy with atom recoil energy. However, single atom mirror mass must be estimated there, yet. Mass of a single proton is equal to approximately 1.67 × 10-27 kg, and for elements having atom number in order of 100 (numbers are exaggerated for simplification) with about 100 protons and 150 neutrons, assuming in addition that the neutron mass is equal to proton mass and little bond energy is neglected, thus producing :

$M=250 \times 1.67 \times 10^{-27} kg \simeq 4 \times 10^{-25}kg(3).$

As photon with frequency of 1014 Hz can give the velocity of 103 m/s to the atom with mass ≈ 10-26 kg, so we assume, that M=M'. Hence, atom kinetic energy EM, based upon energy conservation law, is equal to:

$E_{M}= E-E^{'}={E M c^{2} + 2 E^{2} (1-\cos \theta)-EMc^{2} \over Mc^{2}+2E(1-\cos \theta)}={2E^{2}(1-\cos \theta) \over Mc^{2}+2E(1-\cos \theta)}(4).$

Because the light reflects from the mirror practically under the angle of π, thus, we assume that the cosθ = -1. Let us make one approximation more, namely such that for visible radiation, with frequency of 1014 Hz, or the energy of the single photons E is equal to E=6.63 × 10-26 J, and its energy is much less than atom rest energy Es amounting to

Es=Mc2=4.18 × 10-25 kg × (3 × 108)2 m2/s ≈ 3.76 × 10-8 J, it can be omitted in denominator of (4). Therefore, the formula for Em is now as follows:

$E_{M}={4E^{2} \over Mc^{2}}(5).$

Consequently, comparing atom recoil energy accumulated in 8 neighbours bonds with incident photon energy the formula is obtained for atom deflection x :

$E_{M}=8 {1 \over 2} k' x^{2}(6),$

thus

$x=\sqrt{E_{M} \over 4k'}(7).$

Putting (6) into (7) we obtain:

$x=\sqrt{E^{2} \over Mk'c^{2}} = {E \over c \sqrt{Mk'}}={h\nu \over c \sqrt{Mk'}} = {h \over \lambda \sqrt{Mk'}}(8),$

where λ stands for wavelength of the laser beam. Substituting numerical data obtained earlier by (7) we obtain:

$x = {{6.63 \times 10^{-34}Js} \over {1.064\times10^{-6}m \times\sqrt{4.25\times10^{-25}kg\times25.9 {\over {N}{m}}}}}\simeq 1.9\times10^{-16}m(9).$

which is the value too big in comparison with the value less than 10-18 m, which we want to control in LIGO, VIRGO, GEO600, AIGO, TAMA or LISA detectors. So, why is that shifts in the range of 10-17 m were observed during physical experiments [7], [13], [20], [24] although it apparently follows from our theory that they should not be. The reasons may be as follows. Firstly, instruments may have been maladjusted. Secondly, we have to look at the way measurements are done at LIGO. The thought is like this. We assume that the mirror surface is ideally smooth as compared to the lightwave length; moreover, laser beam reflection point is perfectly located (with accuracy of fractions of atom nucleus) in space [15], [21-24]. Seeing that, the changes in interference pattern locations will be caused exclusively by the changed mirror locations to each other. This motion can be suppressed quite well [13] (see exeption in Point 1). Furthermore, this relative motion of the mirrors in relation to each other can be resolved into Fourier components and contribution of each of them to the noise specter can be analyzed. If, therefore, we say that we observe a shift of some 10-17 m in any frequency interval, then we mean Fourier component of the specter from this interval which produces contribution of just some 10-17 m. If, therefore gravitational wave would result in the shift of some   10-16 m in this frequency interval, then we might be able to receive it, and we would say that we observed gravitational wave transition, and the noise coming from thesurroundings would be determined in accordance with interference pattern oscillations which would be one order less. Simplification in the above scheme is in that we assume that laser beam reflection point is very accurately positioned in space, whereas, according to the Heisenberg relation, this point is determined with maximum accuracy equal to

$2 \Delta p \Delta x = \frac{1}{4 \pi} \: h$

or

$\Delta x = \frac{\lambda}{8 \pi},$

wherein λ is laser beam lightwave length. So it can be said that the laser beam reflection from the mirror occurs at the depth Δx of some 0.04 of the wave length. Since we use light wave of some 1 μm in the experiment, so Δx will be of some 4.0×10-8 ≈ 100 crystal layers provided that the mirror is crystal. If we were an amorphous compound, it would be 100 avarage distances between linearly arranged atoms in the mirror. Seeing that photons penetrate to the distanceof Δx, so photons with step-wise changing phases from 0 to value corresponding to Δx will be interfering with each other in the CCD camera. This effect will cause noise in interference pattern oscillations. This noise will be again amplified by atom movements of the thermal motion nature as caused by photons bouncing off atoms in a cylinder of the surface equal to beam section and the length of Δx. If the beam power is about 1 kW - 10 kW [14], then during 1 s this area will be hit by about  1022 - 1023 photons (because of recycling) resulting in every atom from this area being hit on average 103 - 104 times per second. It may also produce noise frequency of some 102 to 104 Hz which would have non-vanishing Fourier components at 10-17 m exactly in the 100 Hz to 1000 Hz range. The possibility of such effect occurence is betokened by the Fermi-Pasta-Ulam paradox wherein energy flow and increased oscillation amplitude were observed other than homogenous energy dispersion increase between all modes as expected. This effect would result in noise being very similar to that as was obtained for the 40 m prototype notwithstanding increased arm length of an interferometer. And then gravity wave transition which causes changed mirror positions by some 10-18 m could be completely dampened by the noise of such type. Correlated atom displacement by distance of some 10-18 m would be overlapped by atom motions triggered by laser beam influence and changed photon phase in the area Δx deep. Notably, complete photon absorption by the atom which occurs on average once per 106 reflections (for best mirrors [33]) is not included in the analysis. Thus, atom oscillation amplitude increases to 10-11 m. At that moment, the question makes sense what is the probability for the area of S Δx (S - beam section) in which atoms to oscillate at the amplitude of 10-11 m to shift over a year 10 times the distance of some 10-18 m in a correlated way, i e to simulate gravitational wave transition. Taking advantage of the fluctuation theory in the manual of [39], we can roughly estimate fluctuations of this area sizes. Let us examine following situation. Laser beam heats up an area volume of S Δx, wherein S is section area of the laser beam and Δx is beam penetration depth into the mirror. Assuming that this volume is in thermodynamic equlibrium with beam radiation and creates approximately isolated system. Such assumption is justified by the fact that the molecule number does not change, although energy interchanges with radiation. We assume that the system temperature T is so that

$h\nu =\frac{3}{2} kT(10),$

or that energy of atom thermal motion is equal to energy of falling photon. Futhermore, we assume that during 1 s, the area of S is hit by 1018 photons. To simplify calculus, we will assume that laser beam is of square and not circular section. Such unrealistic approximation substantially simplifies reasoning and results in only slight result deviations from real situation. One can imagine that a square being the section perpendicular to laser beam direction of the area which is hit by the light is inscribed in the circle cross-section of laser beam. We have in addition 100 layers into which radiation penetrates with equal probability, but the beam reflects between mirrors not less than 103 times, thus resulting in about 1019 photons reflected from front mirror surface as well as from each layer inside. So, internal pressure of the S Δx volume area is approximately constant and the inside temperature is also constant. Using formula [39] for volume fluctuations (ΔV) and thermodynamic equlibrium condition (10) we obtain

$(\Delta V)^{2}= - kT \left({{\partial V} \over {\partial p}} \right)_{T}\approx - {2\over 3} h\nu \left ({ {\Delta V}\over{\Delta p}}\right)_{T}(11).$

Assuming that volume changes with changed pressure Δp at each dimension by l in a way given by Hooke's law (we assume that there occurs compression, hence negative sign at formula (11))

$\Delta V= l^{3}$

and

$l= - {F X \over S E}= - {{\Delta \tilde{p}}\over {\Delta t}}{X\over {S E}},$

hence

$\Delta V = - \left({{\Delta \tilde{p}}\over{\Delta t}} \right)^{3} {{ X^{3} \over {S^{3} E^{3}}}(12),$

wherein

$\Delta \tilde{p}$

would mean photon momentum change over time, X length of the area under consideration, E - Young's modulus, S relation F/S of light pressure to area of X length. Since we assumed that the temperature in the system is constant, so formula (11) may be applied. By inserting (11) to (12)

$(\Delta V)^{2} \approx -{2\over 3} h\nu (-1) \left({{\Delta \tilde{p}}\over{\Delta t}} \right)^{3}{{X^{3}}\over{S^{2} E^{3}}} {1\over {\over{\Delta \tilde{p}}{S \Delta t}}}={2\over 3} h\nu \left( {{\Delta \tilde{p}}\over{\Delta t}} \right)^{2}{X\over {S^{2}E^{3}}}(13)$

is obtained. But

Δp = -2Nh$\nu \over c$,

wherein N is number of photons falling on S. By inserting this relation to (13),

$(\Delta V)^{2} \approx {2 \over 3} h\nu \left( {{-2 N h {{\nu}\over {c}}}\over{\Delta t}} \right)^{2} {X\over {S E}},$

or

$\Delta V \approx \left( {8\over 3} \right)^{1 \over 2} {{N h {\nu \over c}}\over {S \Delta t}} (h \nu )^{1 \over 2} \left( X E \right)^{3 \over 2}$

$\Delta V \approx \Delta X_{fluc}^{3}$,

which means that volume fluctuations are equal along x, y, z, axes, wherein      (ΔX)3fluc means area fluctuation along beam direction,

$(\Delta X)_{fluc} \approx \left( {8\over 3} \right)^{1\over 6} \left( {X\over E} \right)^{1\over 2} \left( {{N h}\over {\lambda S \Delta t}} \right)^{1\over 3} (h\nu)^{1\over 6}$

is obtained. In addition, by inserting

$\tilde N = \frac{N}{\Delta t}$,

we will obtain

$(\Delta X)_{fluc} \approx \left( {8 \over 3} h \nu{N^{2} h^{2} \over \lambda ^{2} S^{2} } {{X^{3}}\over {E^{3}}} \right)^{1\over 6}(14).$

By inserting into (14) numeric values [16], [21], [23] of X=5 × 10-8 m, E=1012 Pa, N= 1022 photons/s, h=6.63 × 10-34 Js, λ=10-6 m, S=2 × 10-3 m2, ν=3 × 1014 Hz, we will obtain

$(\Delta X)_{fluc} \: \approx 3 \times 10^{-14} m,$

Consequently, this approximation would indicate that detection by means of LIGO, VIRGO, AIGO, GEO600, TAMA or LISA interferometers is very difficult, unless we were able to filter off the noise which seems little probable. We think so because if it was gas in the Michelson's cavities and caused noise it had to pump out this gas to remove the noise, while here the source of disturbances is not possible to be removed. Our calculations are not in contradict with those made in [22] if we consider the mirror having the diameter of 5 cm and thickness of 10-7 m.

Conclusions

Following hypothetical situation pattern of the experiment stems from the above study and results of [7], [21], [30], [32]:

1)The power of the laser used in LIGO and other experiments must be decreased at least 104 times. This is because of diffraction of the laser beam on the interferometer mirrors. This effect increases the total power accumulating in the cavitites. We have to avoid diffraction by the self-focusing of the beam. Due to the diffraction the intensity of the interference pattern and the sensitivity of the interferometer increases by two ranges.

2) Mirror as a whole stands unmoved, which is indicated by instruments, while its small fraction is excited to oscillate by laser beam and moves approximately independently from the rest of the mirror, should there be additional signal noise caused by experimental results in accordance with the above reasoning. The spectral density of the noise is flat which agrees with the results obtaining by using method of the double whitening. Moreover, our results agree one with other because the laser beam reflects from approximately a hundred layers, thus, the result taken from Compton's efffect should be multiplied by 100, which gives the figure of 10-14 m got from the thermodynamic estimations. These results are in good agreement with that of A.M. Sinev [41] who received his result from quite different reasoning.

3) Measurements taken in the paper [7], [13], [20], [24], [40] may have nothing in common with the movement of a pendulum resulting from external disturbances. Such an effect can be obtained by resolving to harmonic components the noise forming in the system as a result of multiple reflection of a laser beam from perticualr atomic layers of mirrors. The distance between them is about 10-10 m, therefore they are 8 orders of magnitude bigger than the expected shifts of mirrors caused by the transition of the gravitational wave. Such a situation is more likely to happen due to the results obtained in the paper [7]. It has been observed there that short laser impulses of a much bigger intensity of energy than those applied in the normal work of an interferometer, which were used for the apparatus calibration, generate maximum locating in the range of 100-1000 Hz, which is compared to the number of reflections of such a beam between the mirrors. Thus, the continuous work of a laser of smaller intensity of the beam may generate a spectrum observed in the experiment [7]. In such a situation, transition of a very weak gravitational wave will not cause any measurable change in the decomposition of a spectrum of the vibration of interference pattern. Thus, we have shown in this paper that there exists a statistical limit below which ideal and real phenomena of light reflection from the mirror begin to differ from each other.

4) These results, however, can be reconciled with the expectations relating to the work of a detector on the condition that the movement of atoms in a mirror is averaging in such a way that only a component of the vibration resulting from the transition of a gravitation wave through the system is visible. Similarly, the change of a photon phase caused by the change of a path between the mirrors must be averaged, as a result of various reflections from various atomic layers in the mirrors. Whether such effects occurs or not should be ascertained based upon reliable computer simulations (similarly as in case of Fermi-Pasta-Ulam paradox), which so far are missing. Hence, the disscussion if interferometers LIGO, VIRGO, AIGO, GEO600, TAMA or LISA can really detect is incomplete.

Note to Editor: unluckily there was no possibility to use Insert/Edit button (it failed on my computer) thus, the paper is without any picture. Afetr removal of this fault the old paper could be withdrawn and submitted again.

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Peer-review ratings (from 2 reviews, where a score of 100 represents the ‘average’ level):
Originality = 132.24, importance = 116.73, overall quality = 135.41
This Article was published on 16th August, 2006 at 08:39:09 and has been viewed 7709 times.

1 Peer review [reviewer #2144] added 16th August, 2006 at 12:17:54

I refer to the assumption that neutrons and protons have the same rest mass. Since these experiments are measuring very small quantities, would it be okay to neglect the difference in masses of the neutron and the proton, and their mass when in motion (relativisitc velocity)?

(I am not a physicist, I am actually a bio-chemist, so pardon me if I made a mistake)

Very Comprehensive article otherwise.

Originality: 6, Importance: 6, Overall quality: 6

2 Author comment added 16th August, 2006 at 12:43:26

It depends on how your exact measuremnts should be. If they are rough estimations it is OK, but otherwise I should be more precise. Whether or to avoid rest masses of these particles is dependent how large is the momentum of theirs against thier rest mass. If it is comparable you cannot avoid the mass, otherwise yes, you can.

3 Author comment added 16th August, 2006 at 12:47:03

Here the kinetic energy of the protons and neutrons are so small that the difference between relativistic and classical mass can be omitted each of the proton and neutron in the mirror.

4 Additional peer comment [reviewer #2144] added 17th August, 2006 at 12:57:39

From what I know, (3/2)kT is the formula for kinetic energy of an ideal gas. However, there is no “gas” in the experiment, so could this formula still hold?

5 Author comment added 17th August, 2006 at 20:04:40

Yes, we can. If we consider simple Debye’s model of the cristal lattice then we get, for teperature much higher than Debye’s temeprature (which are about 100 K, here we have 13500 K) that approxiamtion of ideal gas is good. One part is the energy bound of the lattice and the second one the kinetic energy of the atoms. If we take the average values we can just divide the total energy by two and get an average value of each energy. It is very rough estimation, but works well.

6 Peer review [reviewer #9922] added 19th August, 2006 at 15:38:21

It is not clear exactly how the passage of a gravitational wave through the interferometer is expected to produce any effect While this may be explained in one or more of the references, it should be decsribed in this paper.

Originality: 4, Importance: 4, Overall quality: 6

7 Author comment added 21st August, 2006 at 07:53:57

Yes, you are right. If the article has to be self-consistent there should be such an explanation. But otherwise many papers do not do it, either. So, the explanation how the gravitational wave causes the effects that are measurable is as follows: when the gravitational wave passes through the detector it distorts the whole space-time beteewen the mirrors, but only space distortion is neasured. During the passage changes the distance between the mirrors so the ground is shrinking, either. This shortenend distance is comapring with the distance perpendicualr to the wave that does not contract. The difference in optical waves gives the optical pattern on the CCD camera that is measured (but this theory that is very idealised, which is shown in this paper that it may be not true).

8 Author comment added 21st August, 2006 at 07:56:48

The contraction takes place between the mirror placed parallel to the gravitational wave.

9 Author comment added 21st August, 2006 at 09:28:03

Mistake:… difference in optical ways… . There are two arms of the detector: one is parallel to the incoming wave, the second one is perpendicular to it.

10 Author comment added 6th October, 2006 at 16:33:13

According to the yesterday’s meeting of the Polish Physical Society my picture of detecting of the gravitational wave is wrong.
It should be repalced by the following scenario. When the gravitational wave passes through the plane of the detectors arms perpendicularly to it then both of them contract or lengthen appropriate to the space deformation (there are the two components of the gravitational wave). This deformations of the arms give the specific time distribution of the fringes on the CCD camera which is compared with the schemes gained from the numerical simulations of the phenomena which are to be detected. When we know the phenomena well there is a chance of detecting something (but what?), if not we can think of detegraw (Article No 7). And there is still possibility of confusing the schemes with the signals coming from the detected noise. So, one phenomenon has happened but another ones has been detected. But the chance of such occurrence is small (there are still another six detectors). The picture complicates when the gravitational wave does not incident perpendicularly to the arm’s plane, so each case must be recalculated additionally. According to the Russia physicists’ former works, which weren’t metioned by prof. Biozn, there is no possibility of building LIGO ADVANCED PROJECT. So detecting the gravitational wave in 2013 is infeasible. Thus, English Bookmaker Company may be sure of its money for 99,999% (they bet 500:1 for this scenario against the possibilty of the detection).

11 Author comment added 19th October, 2015 at 16:00:51

Well, after 9 years the LIGO project has still found nothing. Yesterday lasted one month of working Advanced LIGO detectors and they has found nothing. According to our calculations the shouldn’t find anything. The time is passing and probably after 10 years of further search in case of non-detection of gravitational waves the LIGO collaborators will have to admit that something is wrong. But what is wrong: theory or the detectors? I don’t think that theory is wrong in weak fields - the further observations should support that view. But time will tell who is right. It’s nice to be right, but on the other hand I would like to be wrong, because it would be nice to detect the gravitational wave. But we can’t avoid the truth and if our calculations are correct the LIGO is a waste of time, money and human effort. Maybe scientists should go through that experience?

12 Author comment added 13th February, 2016 at 11:29:26

Well, I was wrong criticizing the LIGO project or I may say that I was partially right. Some of the disturbances occured and made the previous LIGO detectors impossible to detect. But finally it turnd out that the LIGO detectors were constructed in a proper way and were able to detect the gravitational wave coming from the colliding black stars. But on the other hand it is nice to be wrong. Maybe the submitted reasoning should be reconsidered again to find its flaws, but now it is not important for the observation. I join to the people who congratulate the LIGO teams.

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