On September 14, 2015 09:50:45 UTC, the two laser interferometers of the LIGO program simultaneously observed a first gravitational wave signal called GW150914. With the commissioning of the VIRGO interferometer in 2017, two other detections, GW170814 and GW170817, were observed and their positions given accurately by LIGO and VIRGO. In this article, I argue that the photons circulating in the cavities of the three interferometers of LIGO and VIRGO were sensitive to the field of attraction of the planets of our Solar System and more particularly to that of the Sun, and would not be due to a coalescence of black hole or neutron stars. The shape of the signals obtained by my interaction model (called GEAR) between the photons in the interferometer cavity and the gravitational field of the Sun is very similar to that of a compact binary coalescence, identical to those obtained by general relativity. Solving the equations of GEAR also gives the exact positions and pseudo-date of the coalescences of all the LIGO and VIRGO detections detected so far, and probably those that will come at the end of 2018 and beyond.
The theory of general relativity shows us that a pair of black holes in orbit around each other loses energy in the form of gravitational waves (GWs). The two black holes approach each other, a phenomenon that can last for billions of years, before accelerating suddenly in what is called compact binary coalescence. In a fraction of a second, the two black holes then collide at a speed close to that of light and merge into a single black hole. As this one is lighter than the sum of the two initial black holes, a part of their mass (the equivalent of three Suns in the case of the detection called GW150914) is converted into GWs according to Einstein’s formula E = m c 2 . It is these GWs that LIGO and VIRGO may have observed. General relativity, in its gravitomagnetic form [
My argument is based in part on earlier work of Einstein [
γ = G ⋅ M r 2 ⋅ 1 1 − ν / c (1)
Here, ν is the projection of the velocity ν along r. Note that in this study, if the gravitational constant, G, and M (here, the mass of the Sun) are relativistic and depend on speed ν, then r does not depend on this speed and remains constant. An interesting characteristic of Equation (1) is that acceleration is no longer independent of the sign of the velocity v of the test particles making up the mass DM of the photons in the interferometer cavity.
I argue that for all the dates of detections of GWs given by the institutions LIGO and VIRGO, the Sun was positioned according to the graph in
With GTSMHGW defined by the International Astronomical Union as the Greenwich sidereal time for the specific GW on a given day, and with ϕ s = − 32.50 ∘ , right ascension (RA) and declination (dec.) of the Sun on the celestial sphere, the projection of the x, y, z unit vector of the Sun’s position (as a function of time) at the time of detecting the GW is given by S l a t ≡ dec . and Slong, where Slat is the projection of the latitude in radians and Slong, given by Equation (2), is the longitude in degrees, equal to
S l o n g = [ − ( G T S M H G W ( t ) ⋅ s + ϕ s 15 ⋅ 3600 s ) 3600 s + R A ] ⋅ 15 + 180 ⋅ i (2)
Here, i takes the value of 1 between June 21 and December 22 and 0 for the rest of the year. This is because the calculation of Slong is shifted by π with respect to the vernal reference point, i.e., the point of the spring equinox.
( X s u n G W Y s u n G W Z s u n G W ) = [ sin ( π 2 − S l a t ) ⋅ cos ( S l o n g ⋅ π 180 ) sin ( π 2 − S l a t ) ⋅ sin ( S l o n g ⋅ π 180 ) cos ( π 2 − S l a t ) ] (3)
The direction of the X arm taken as reference for LLO and the Y arm for LHO is given by [
( X L H O Y Y L H O Y Z L H O Y ) ≈ ( X L L O X Y L L O X Z L L O X ) = ( − 0.95457412153 − 0.1415807734 − 0.26218911324 )
Finally, we can calculate the cos θ of the angle θ between the axis r of the Sun and the axis of the cavities of the X arm of LLO and the Y arm of LHO at zero time of the coalescence of the GW, with ω e a r t h the rotational speed of the Earth, t the time since zero time of the coalescence detection of the GW,
ϕ = 22.586 ∘ × π 180 and Δ ϕ ≈ 2 ∘ × π 180 , the rotation matrix:
( cos ( ω e a r t h ⋅ t + ϕ + Δ ϕ ) − sin ( ω e a r t h ⋅ t + ϕ + Δ ϕ ) 0 sin ( ω e a r t h ⋅ t + ϕ + Δ ϕ ) cos ( ω e a r t h ⋅ t + ϕ + Δ ϕ ) 0 0 0 1 )
the angles f and fs are constants that intervene only for an exact calculation of the Sun’s position, and Δ ϕ is a correction angle f dependent on the position of the Sun on the ecliptic and not exceeding the order of magnitude of 2˚.
The rotation matrix applied to the direction vector of the X arm of LLO allows for a relatively short time to consider whether the relationship cos ( XLLOGW ( t ) ) (Equation (4)) is sufficiently accurate. Indeed, I consider that the position x, y, z of the Sun during this time t remains relatively constant, or the relationship:
cos ( XLLOGW ( t ) ) = ( X s u n G W Y s u n G W Z s u n G W ) ⋅ ( cos ( ω e a r t h ⋅ t + ϕ + Δ ϕ ) − sin ( ω e a r t h ⋅ t + ϕ + Δ ϕ ) 0 sin ( ω e a r t h ⋅ t + ϕ + Δ ϕ ) cos ( ω e a r t h ⋅ t + ϕ + Δ ϕ ) 0 0 0 1 ) ⋅ ( X L L O X Y L L O X Z L L O X ) (4)
Note that Equation (4) can have the sign ± cos ( XLLOGW ( t ) ) because of the Sun’s position (see
cos θ ( t ) ≡ cos ( XLLOGW (t))
e ( E p x , s i g , h , t ) = G ⋅ M s u n r 2 ⋅ E p x c 2 N r t ∫ 0 h cos θ ( t ) 1 − s i g ⋅ cos θ ( t ) ⋅ [ 1 1 − ( s i g ⋅ cos θ ( t ) ) 2 ] 0.5 ⋅ d h (5)
In this relationship of energy absorbed during a trip e ( E p x , s i g , h , t ) , r represents the distance between the Sun and the cavity of the LIGO or VIRGO interferometer, M s u n is the mass of the Sun, E p x is the energy of the circulating photon in the cavity (laser photon), ± s i g is the sign of speed, ± h is the distance between the two mirrors of the cavity of the interferometer, N r t is the number of roundtrips, and t is the time around the zero point of coalescence. The balance for a roundtrip is:
Δ E G W = e ( E p x , s i g , h , t ) + e ( E p x , − s i g , − h , t )
The function Δ E G W is similar to the energy of the GW detected by LIGO and VIRGO, from 2014 to 2017. With the exact time of the detection of coalescence, we can draw the graph in
The graph in
To complete our demonstration of the supposed localizations (see
cos ( π 2 − S l a t ) ⋅ cos ( π 2 − B H l a t ) + sin ( π 2 − S l a t ) ⋅ sin ( π 2 − B H l a t ) ⋅ cos ( ( S l o n g − B H l o n g ) ⋅ π 180 ) = 0 (6)
cos ( π 2 − S l a t ) ⋅ cos ( π 2 − N S l a t ) + sin ( π 2 − S l a t ) ⋅ sin ( π 2 − N S l a t ) ⋅ cos ( ( S l o n g − N S l o n g ) ⋅ π 180 ) = 0 (7)
cos ( π 2 − B H l a t ) ⋅ cos ( π 2 − N S l a t ) + sin ( π 2 − B H l a t ) ⋅ sin ( π 2 − N S l a t ) ⋅ cos ( B H l o n g − N S l o n g ) = 0 (8)
arccos [ cos ( π 2 − B H l a t ) 2 + sin ( π 2 − B H l a t ) 2 ⋅ cos [ ( L L O l o n g − B H l o n g ) ⋅ π 180 ] ] ⋅ 180 π ⋅ 1 cos ( B H l a t ) + L L O l o n g = B H l o n g (9)
In Equations (6)-(9), BHlat and NSlat represent the latitude of the black holes and neutron stars, respectively, and BHlong and NSlong are their longitude, respectively, on the celestial sphere. LLOlat and LLOlong are the coordinates of the LLO X arm on the celestial sphere, and by way of example for the detection event GW170817 (with i = 1 ):
a s t z ( t ) = ( G T S M H G W ( t ) 24 ⋅ 3600 ⋅ 360 + ϕ s ) ⋅ π 180 − π
I obtained for LLOlong
L L O l o n g = ( a x t z ( t ) + ϕ + Δ ϕ ) ⋅ 180 π − wholepartof ( ( a x t z ( t ) + ϕ + Δ ϕ ) ⋅ 180 π 360 ) ⋅ 360 − ( 90 + 46 60 + 27 3600 ) (10)
The term ( 90 + 46 60 + 27 3600 ) in Equation (10) represents the longitude of LLO [
For GW170817 and for t = 0 (coalescence), I find with L L O l a t = 0.533 rad or 30 .539 ∘ : L L O l o n g = 47.297 ∘ .
After solving Equation (9), we obtain the value of BHlat which allows us to find the value of BH long from Equation (6). This pair of coordinates is the point of reference, the “master” location of the black holes and neutron stars. After the rotation of the vector position of the master location around the axis of the Sun
( r ) (where the date coordinates are those of the GW in question) of π 2 , then π, and finally by 3 2 π , we obtain the only four possible positions (solutions to Equations (6)-(9)) of the stars or pseudo-coalescences. Finally, we find for the detection events GW170814 and GW170817,
The calculation, using the GEAR model, of the signal effectively received by LIGO and VIRGO is identical to a coalescence of black holes or neutron stars.
This signal is obtained with GEAR (
To obtain the coalescence signal of
In
± cos ( XLLOGW ( t ) ) according to the position of the Sun. We thus have the possibility of knowing the order of arrival of the signal at the interferometers of LIGO. With a negative sign for Equation (4), the signal arrives first at LLO and later at LHO; in the case of a positive sign, the signal arrives at LHO before arriving at LLO. See
GW | Date (dd/mm/yyyy) | Hour UTC | Arrival time from LHO to LLO (ms) | Sign of cos ( XLLOGW ( t ) ) |
---|---|---|---|---|
2015 | ||||
GW150914 | 14/09/2015 | 9:50:45 | −7 | − |
LVT151012 | 12/10/2015 | 9:54:43 | − | |
GRB150522 | 22/05/2015 | 10:23:50 | + | |
GW151002 | 02/10/2015 | 09:50:45 | − | |
GW151226 | 26/12/2015 | 03:38:53 | +1.1 | + |
2017 | ||||
GW170104 | 04/01/2017 | 10:11:59 | +3 | + |
GW170814 | 14/08/2017 | 10:30:43 | −8 (LLO) −14 (VIRGO) | − |
GW170608 | 08/06/2017 | 02:01:16.49 | +7 | + |
GW170817 | 17/08/2017 | 12:41:04 | −0.5 - ? (VIRGO) | − |
In this article, with four different reasons developed here, I have shown that it is likely that the signals detected by the LIGO and VIRGO interferometers originate from the interaction between the Sun’s gravitational field and the photons present in the interferometers’ cavities. In fact, LIGO and VIRGO probably did not detect signals from black hole or neutron star coalescence as defined by Einstein’s general relativity. And Einstein may have been right in 1936 when he wrote to his friend Max Born [
“Together with a young collaborator, I arrived at the interesting result that gravitational waves do not exist, though they had been assumed a certainty to the first approximation. This shows that the non-linear general relativistic field equations can tell us more or, rather, limit us more than we have believed up to now.”
Elbeze, A.C. (2018) On Gravitational Waves: Did We Simply Detect the Gravitational Effect of the Sun on the Photons Moving in the Cavity of Interferometers LIGO and VIRGO? Journal of Modern Physics, 9, 1281-1290. https://doi.org/10.4236/jmp.2018.96076