Newton s Universal Law of Gravitation says that: any two particle of mass in the space are attracted toward each other with a force formulated as:

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1 How to measre the gravity speed? Ahmet Yalcin B.Sc., M.Sc., Control Engineer March 7 A. History A.1. Gravity Speed Newton s Universal Law of Gravitation says that: any two particle of mass in the space are attracted toward each other with a force formlated as: F m m r 1 = G, where: F is the magnitde of the gravitational force between the two point masses G is the niversal gravitational constant m 1 is the mass of the first point mass m is the mass of the second point mass r is the distance between the two point masses This force cases every object on Earth to fell down and the planets to move in an orbit arond the sn. Newton assmed that, the speed of this force is instantaneos which means that in the case of any position change in one gravitating mass the force formlated above is pdated immediately. On the other hand Einstein's relativity theory says that nothing, not even the inflence of gravity itself, can travel faster than light. Althogh it is not experimentally discovered yet, modern qantm field theory says that the gravitational force is cased by a massless elementary particle called as graviton. As a particle, according to relativity theory, its speed cannot exceed the light speed. However, there is no a consenss abot the speed of the gravity amongst the scientists. In fact, if we add a delay parameter to the eqations orbits become nstable. So it is scientifically very important to solve the problem experimentally. There is one and only attempt to measre the gravity speed which was in by Sergei Kopeikin, a physicist at the University of Missori-Colmbia. 1

2 He sed the National Science Fondation's Very Long Baseline Array (VLBA), a continentwide radio-telescope system, to make an extremely precise observation when the planet Jpiter passed nearly in front of a bright qasar J on September 8th,. In this experiment, a very slight bending of the radio waves coming from the qasar de to the gravitational effect of Jpiter was recorded. From this data, the actal position of the qasar fond slightly different from its apparent position. From this experiment, Mr. Kopeikin fonded that gravity's propagation speed is eqal to the speed of light within an accracy of percent. Bt, this measrement did not satisfy some of the scientists becase: 1. The method Kopeikin sed was not a repeatable one, for a second attempt one shold wait another decade.. The method Kopeikin sed had a lot of falt sorce hence the measrement gives very low percentage of accracy. 3. Some scientists are not sre abot the trth of the method. A direct measrement of the gravity speed will end all sort of disptes. If the gravity speed is fond, for instance, to be infinity or twice the speed of light, than modern qantm theory and astrophysical sciences, in some aspect, have to be revised radically. With this opinion, I do not claim that the gravity propagates faster than light; I only say that, it is possible to measre gravity speed in laboratory environment relative to light speed easily. The only tool yo need is a precise Real Time g measrement device which can be achievable with the technology otlined below. B. An alternative proposal to measre the gravity speed Before to go frther, I mst imply that it seems to me logical to se a mass object in the gravitational field and make measrements on that object regarding the gravity. It looks like to test the behavior of the electron or any other charged particle in the electrical or magnetic field. In or experiment the sorce for the gravitational field is Sn and Moon and the object to make measrement on it is Earth. We will assme that we have a device with which we can measre the acceleration de to the gravity on Earth srface in real time with sfficient accracy. Any particle of mass on Earth srface is nder the inflence of the gravitational fields created by Earth, Sn, Moon, other planets and the stars. The particle will be forced by the resltant force created by those gravitation fields. The acceleration de to the gravitation is a measre of gravitational field intensity. This can be calclated for each field sorce sing Newton s second law. The formla is: mi g i = G, where: ri g i is the gravity acceleration on Earth srface de to field sorce m i G is the niversal gravitational constant m i is the mass of the gravitation field sorce r i is the distance between the particle and m i

3 Using the above formla the gravity acceleration g e de to Earth (G= m 3 kg -1 s -, m e =5, kg, r e = 6378,137 km) is: 9,89 m s - or 98,9 cm s -. De to the flattening this calclated vale is a bit lover at the eqator and goes higher towards to the poles. Similarly, gravity acceleration at the measrement point on Earth srface de to the Sn and Moon can be calclated. Bt, althogh they are also almost fixed in magnitde, their directions differ according to the Sn and Moon s position. Before calclation of gravitation field of the Sn and Moon, we had better refresh or memories abot motion of Earth and Moon. Figre 1, smmarizes Earth s annal motion arond the Sn. f Figre 1 We know that, looking down on Earth s north position, Earth is rotating conterclockwise with abot 4 hors per trn. Earth also revolves arond the Sn in an orbit in the same conterclockwise direction which is nearly circlar. This orbit defines a plane in the space called the plane of the ecliptic. There is 3.5º between the normal vector of the ecliptic plane which points to orbital north and Earth s rotation axis or eqivalently, Earth s eqator is at an angle of 3.5º to the ecliptic plane. De to this tilt, the Sn will be on the zenith at noon time at Eqator dring eqinoxes, and will be on the zenith at noon time at Tropic of Cancer and Tropic of Capricorn dring smmer and winter solstices respectively. Figre illstrates the sitation for the observer on the eqator dring an eqinox time. Here the observer is at the centre. As Earth rotates arond its axis, the Zenith of the observer follows Celestial Eqator. Bt, as we keep the observer on its horizontal plane fixed in the above pictre, eqivalently, the Sn s apparent position, with respect to the observer, follows the ecliptic clockwise. At noon time, the Sn will be at the zenith of the observer, and T hor later the Sn s apparent position will be on the point stated in the pictre, where φ=(π/4)t. If we choose coordinate system as zenith (x-axis), east and north (y and z-axes) directions, the Sn s apparent position can be calclated easily. The nit vectors were specified as indicated in the pictre. In this case E the Ecliptic plane normal nit vector: E Sinθ j Cosθ k ; where θ=3.5º 3

4 4 Figre Let φ be nit vector of intersection of T hor celestial meridian and Celestial Eqator. Than: j i Sin - Cos Let N φ be T hor celestial meridian normal nit vector than: j i k N Cos Sin Finally let S represent Sn s apparent normal position vector, than: E E S N N or:

5 Elevation (radian) S Cos Cosθ i Sin Cosθ j Sin Sinθ k Cos θ Sin θ Sin From above expression the elevation Elevation angle α of the verss Sn for timethe observer will be: 1,5 1,5 -,5-1 -1,5 - α acos Cos Cosθ Cos θ Sin Figre 3 θ Sin Snset Time (hors) Figre 3 shows elevation angle verss time for one day. It looks like a triangle wave. It almost swaps eqal angles with eqal time. There is a slight bend arond observer s meridian. This is de to the 3.5º of angle between the Ecliptic plane and Celestial Eqator. This is not a tre position graph of the Sn. Becase we assmed that Earth is stationary arond the Sn, we ignored revolving of Earth while it rotates arond its own axis. Frthermore, it is only apparent bt not a real position graph of the Sn, becase the actal position of the Sn is abot 8.3 mintes ahead of the apparent position. This is de to propagation time of the light from the Sn to Earth. This means that, if the Sn s gravitational field propagates instantaneosly, it will be effective on Earth srface abot 8.3 mintes before the Sn s apparent position. We selected the measrement point on any point on the Eqator at an Eqinox; the other points on the eqator will observe the Sn exactly same way with a time difference. We will have similar sitation on Tropic of Cancer and Tropic of Capricorn at smmer and winter solstices. Those are ideal points to observe the Sn s gravity effect, becase the apparent position of the Sn coincides twice a day with the observer s zenith. As Earth revolves arond the Sn the ideal observation point moves between the tropics. As the observation point moves towards the poles we will have poor sitations. Any point on Earth, in Figre, by defining the Observation point s zenith nit vector the apparent point of the Sn with respect to new observation point can be calclated. There are also very good programs, free of charge in the Internet, calclating coordinates of any celestial body anywhere and any time on Earth. Figre 4 gives elevation angles of the Sn verss time at Greenwich Observatory at an eqinox and Figre 5 gives elevation angles verss time in Ankara on Jly 4 th. It is seen that as we move toward to the poles the elevation amplitde of the apparent position of the Sn decreases. Snrise 5

6 Elevation (radian) Elevation (radian) Elevation of Sn verss time (Greenwitch, at eqinox) 1, ,5-1 Snset Time (hor) Snrise Figre 4 Sn elevation verss time (Ankara, Jly 4th) 1,5 1,5 -, Snset Time (hor) Snrise Figre 5 At this step, let s now calclate the gravity effect of the Sn on a measrement point on the Eqator. Figre 6a 6

7 We know that gravity acceleration de to Earth is abot 98 cm/sec. As Earth rotates, this fixed acceleration will vary de to the Sn and Moon s gravity effect. The mean distance between Earth and Sn is km. At noon time and midnight this mean distance will be and km respectively. As the mass of the Sn is x 1 3 kg, the gravity acceleration at noon time de to the Sn is 5,931 cm/sec and at midnight it is 5,93 cm/sec. There is a very small vale between those two accelerations meaning that every point on Earth is attracted towards the Sn almost eqal force. The other celestial body affecting Earth s gravitation field is the Moon. The plane of the orbit of the Moon is tilted abot 5 degrees with respect to the plane of Earth's orbit arond the Sn. Frthermore it will not be in the same phase with the Sn, hence, while we calclate the Moon s gravity effect we will not mention abot the time of the day bt near side and far side instead. The average distance of the Moon from Earth is km and the Moon mass is 4,3483 x 1 kg. Hence, the gravity accelerations de to the Moon on near and far side on Earth are,343 and,31 cm/sec respectively. It is seen that the effect of the Sn is abot 175 times higher than the effect of the Moon. However, the effect difference on near and far side of the Moon is mch more than the effect difference of the Sn at noon and midnight time. This is why the Moon is dominant for tidal inflence. For or prpose we will ignore the gravity acceleration on Earth de to the Moon. Figre 6b In Figre 6a, total gravity acceleration de to the gravity is g=g E +g S. Here g E is fixed in amplitde and for a measrement point the direction is always toward to Earth center. As it is seen in Figre 6b, as Earth rotates arond its axis, the g S changes its direction with its almost fixed amplitde. In Figres 6a and b the gravity acceleration component de to the Sn is exaggerated. From Figre, it is seen that the effect of the Sn remains in the Ecliptic plane (almost east-west direction). On the other hand, we have to place the gravity acceleration measrement device into north direction. This means that the device will be effective only in g E direction component of the resltant gravity acceleration. Hence the effective component in g E direction is g E x Sinα. Where g E is amplitde of g E and α is elevation angle of the Sn. The amplitde of total gravity acceleration will be g=g E +g S x Sinα.. On the measrement point the real time g device will measre this total acceleration. If the device has sfficient accracy, we will be able to distingish and filter the Sn s gravity effect. 7

8 Normalized Elevation and gs verss time Snset Snrise Time (hor) NormElev gs Figre 7 Figre 7, shows the Sn s apparent position and gravity effect verss time. We sed - g E rather than g E to avoid 18º of phase angle. Here y axis is acceleration for gravity effect and elevation angle for position. As the phase of the elevation is important for s we enlarged its amplitde to have a better view. Here apparent position graph (elevation angle) represents the light speed and g E variation graph represents gravity speed. If both speeds are eqal both graphics will be in the same phase (maximm and minimm points and zero cross times are same). If there is 8.3 mintes phase difference ( g E shifted left) between two graphics then the gravity speed will be instantaneos. In general if we have Δt mintes phase difference between elevation angle and gravity acceleration de to the Sn, than the gravity speed v g in terms of light speed c will be: v g c Δt There is one more point to discss. We have to check how mch phase difference we can distingish with or gravimeter. If the gravity speed is infinity than, g E mst be shifted maximm 8.3 mintes back. If g E shifts forward than gravity will be less than light speed. Let s check how mch phase difference we can distingish. If we shift g E 8.3 mintes back we will have normal and shifted graphs as in Figre 8. It is almost visally hard to distingish both graphics. If we get the difference of the normal and shifted gravity effect graphics, than we obtain Figre 9. Figre 9 shows that 8.3 mintes shift case maximm abot, cm/sec difference in gravity acceleration. We get maximm deviation in acceleration at Snset and Snrise time. If the real time gravimeter accracy is,1 cm/ sec, which can be achievable with on the shelf prodcts, than to distingish 8.3 mintes of shift will not be a problem. The algorithm to measre gravity acceleration will be as follow: 1. Choose a measrement point and year of the day having extreme Sn elevation angles.. Precisely calclate the Sn s elevation angle verss time. (Record the Snset and Snrise time representing arrival time of the light from the Sn to Earth) 3. Record daily acceleration de to the gravity sing a real time high precision gravimeter. 8

9 Sn gravity effect Snset Time (hor) Snrise Figre 8, Gravity difference verss time, ,1 -, Snset Time (hor) Srise Figre 9 4. Observe daily periodic change in gravity. (Total acceleration de to the Sn and Earth, minimm at noon time and maximm at midnight.) 5. Get the daily average of the gravity which gives fixed gravity acceleration de to Earth exclding Sn effect. 6. Calclate the difference between total acceleration and fixed gravity acceleration de to Earth. (Sn effect only) 7. Get the time where the Sn effect is zero which is arrival of the Sn s gravity effect to Earth. 8. Compare the light arrival time (apparent Snset and Snrise time), with the gravity effect arrival time. It is simple and straightforward. The only tool yo need is a high precision real time gravimeter. 9