Theoretical Considerations for a Geosynchronous, Earth-Based Gravity Wave Interferometer

Transcription

1 AMERICAN JOURNAL OF UNDERGRADUATE REEARCH VOL. 4, NO.1 (5) Theoretical Considerations for a Geosynchronous, Earth-ased Gravity Wave Interferometer William P. Griffin Department of Physics University of Northern Iowa Cedar Falls, Iowa UA Received: March 1, 5 Accepted: May 6, 5 ATRACT We investigated theoretical considerations in the design of an Earth-ased laser interferometer for detecting gravitational waves. Our design envisages a ground-ased tracking station in communication with two geosynchronous satellites. We assumed linearized gravitational waves in a chwarzschild spacetime geometry outside the Earth. Our initial calculations show that such a design is sufficiently sensitive to successfully detect gravitational waves near Earth. I. INTRODUCTION When Einstein developed special relativity, the separate parameters of space and time ecame inextricaly intertwined. The resulting concept of spacetime was further developed when Einstein understood the key to general relativity: gravity is the manifestation of curvature in spacetime. One result of this conceptualization is the existence of gravity waves, propagating ripples of spacetime caused y matter in non-spherical, non-uniform motion. Indirect experimental evidence for their existence has een oserved in careful astronomical measurements of a inary pulsar. No direct detection of a gravity wave has een confirmed, ut some projects with this goal are underway. The weak coupling to matter of gravity waves makes direct detection challenging. One potential method is to use interferometers (see Figure 1). A laser, in red, is shot from L to a eam splitter,, where it travels to the mirrors that are acting as test masses, M, at the end of each arm. The reflected eams are then recomined L M D Figure 1. A schematic of an interferometer. and ecause of a property of light, when light waves are comined the sum of the light waves is the result. The intensity of the resulting light wave can e measured y the detector, D. The conditions for constructive and destructive interference are, respectively, L L L n λ (1) M 7

2 AMERICAN JOURNAL OF UNDERGRADUATE REEARCH VOL. 4, NO.1 (5) L L L ( n + 1 )λ () for n, 1, and light wavelength λ. ince the speed of light is constant, if the arms change length the path of the light will lengthen or shorten depending on the change. This change will cause the sum of the light waves and the amount of interference to change. In order to understand how gravity waves will affect the interferometer, it is useful to first know more aout them. Metrics of geometries close to flat spacetime, g α β, can e written ( x) h ( x) g η + α β α β (3) α β where h α β (x) is a metric perturation to the flatspace metric, η α β. Gravity waves are only perturations to flat spacetime so their forms can e well approximated using the linearized approximation to the vacuum Einstein equation [1]. The resulting linearized gravitational wave perturation is h α β a ι ω ( z t ) ( x) e f f + ( t z) f ( t z) ( t z) f ( t z) a + If arms of equal flatspace length, L *, are lined up along the x - and y -axes in the z plane, then the cross polarization ecomes inapplicale and the plus polarization simplifies to f () t a sin ( ω t +δ ). If the initial time is + chosen such that δ, g α β, then can e used to determine L: ( ω t ) L L a sin (5) * This equation shows that longer arms will cause greater changes in the interference pattern allowing the interferometer to e more sensitive. everal projects with the intent of detecting gravity waves are in various stages of completion. An interferometer similar to the one descried aove called LIGO, short for Laser Interferometer Gravitational (Wave) Oservatory, is currently in its initial operational stages []. The disadvantage of this interferometer is that the arms are relatively short, although this has een improved upon y adding partially reflecting mirrors that effectively lengthen the arms of the interferometer. Another gravitational wave oservatory ased on interferometric principles that solves this prolem called LIA, short for Laser Interferometer pace Antenna, has een proposed for launch into space in 11 []. In space, the lengths of the arms are only limited y the power of the laser. The disadvantage of this interferometer is that once it is launched, it will e very hard to reach if prolems occur or it needs technological updates. The interferometer proposed in this paper was intended to solve oth of these issues. This interferometer would e ased on the interferometer in Figure 1, ut the test masses at the ends of the arms would e satellites with mirrors in geosynchronous orits. This allows the arms to e aout 4, kilometers long while keeping most of the equipment on or relatively near Earth where it can e easily repaired or updated. II. EARTH PACETIME CURVATURE The reason the spacetime curvature produced y Earth won t prevent the operation of the interferometer is that the paths of the lasers are equivalent. To understand this, the design of the interferometer must e more clearly defined. Let Earth e centered on the origin of a Cartesian coordinate system with the equator on the z (see Figure ). The point on the Earth s equator where eam splitter lies, along with the Earth s center, defines the x-axis. Let the satellites,, e placed at points in the Earth s equatorial plane that are equidistant from the eam splitter,, such that angle is a right angle. 8

3 AMERICAN JOURNAL OF UNDERGRADUATE REEARCH VOL. 4, NO.1 (5) R Figure. Two satellites () orit in the equatorial plane at equal distances from an Earth-ased station (), to form an Earth-ased gravity wave interferometer. R is a satellite s orital radius. To a very good approximation, the Earth can e treated as spherically symmetric. A special metric exists for spacetime near a spherically symmetric mass M, called the chwarzschild metric: t r θ ϕ g α β M t 1 r r θ ϕ M 1 r 1 r r sin θ (6) When the origin of such a coordinate system is centered on Earth, θ corresponds to latitude in radians with θ pointing at one pole and θ π at the other, while ϕ corresponds to longitude in radians with ϕ pointing at the positive x -axis. The radius measured from the origin is r. Figure is a cutaway of the θ π/ plane. The line element that corresponds to this metric is ds 1 1 dt r M r M r ( dθ + sin θ dϕ ) dr (7) 9

4 AMERICAN JOURNAL OF UNDERGRADUATE REEARCH VOL. 4, NO.1 (5) This line element can e used to calculate distances in the chwarzschild geometry of spacetime. If t θ ϕ, this calculation simplifies to a function of the radial distance: L 1 / M dr r () r 1 (8) ince the flatspace distance from the origin to each satellite is the same, equation (8) implies the distances in chwarzschild space must also e the same. The light reflected from the satellites also ends at the same point, the eam splitter. ecause of the symmetry of this design, ϕ along each arm must also e exactly the same, ut it should also e clear that distance in the chwarzschild metric is only dependant on the change in ϕ, not the specific values of ϕ. This implies the arms must e of exactly the same length and eams traveling down each arm must e traveling exactly equivalent paths. This means that the spacetime curvature produced y Earth will not effect the operation of this interferometer. III. LENGTH OF ARM IN FLATPACE To determine the sensitivity of the interferometer, it is necessary to find the length of the arms in flatspace. The radius of a geosynchronous orit, R, must first e found y setting the centripetal force [3] acting on a satellite of mass m traveling at a tangential velocity v equal to the Newtonian gravitational force [3] on it due to the mass of Earth, M: m v G M m (9) R ( R ) Relativistic mechanics are not necessary for this calculation ecause space is flat and speeds involved are not significant compared to the speed of light. The satellites are geosynchronous, so they orit Earth once per sidereal day, a day with respect to the distant stars instead of the sun. A sidereal day is aout four minutes shorter than a solar day [4]. This information can e used to calculate the angular velocity of the satellite, ω π/438 radians/s. The tangential velocity is simply the angular velocity multiplied y the radius of orit [3]. ustituting for v into equation (9) allows the orital radius to e solved for, R 1 π ( G M ) m (1) This result is otained y letting M kg [1] and G N m /kg [3]. Now the length of the arms can e calculated geometrically y choosing the origin of a Cartesian coordinate system to e at the eam splitter with the arms along the x- and y-axes in the z plane (see Figure 3). Let vector c (c, c, ) point from the eam splitter to the center of Earth. The lengths of the arms were defined to e equal, so let constant k e this length so that a (k,, ) and (, k, ) represent the arms. The distances R ( k c ) ( ) + c a c (11) R ( c ) + ( k c ) c (1) are of course equal. etting these equations equal to each other and simplifying gives the result c c. These components can e calculated y setting the equatorial radius of Earth, R, equal to c and solving: ( c ) + ( c ) ( c ) R. (13) olving equation (13) we get c ± R/. Comparing this to Figure 3 and letting R m [1], it is clear that 7 c c m. ustituting this result into either equation (11) or equation (1) and solving for k results in two solutions. The negative solution can e discarded leaving R R k + R 1 (14) ( R ) m. 3

5 AMERICAN JOURNAL OF UNDERGRADUATE REEARCH VOL. 4, NO.1 (5) R c a R Figure 3. Geometry of an Earth-ased gravity wave interferometer, with the coordinate axes rotated and shifted (from Figure ) such that the origin is now at the eam splitter,. IV. LENGTH OF ARM IN CHWARZCHILD GEOMETR To see how the mass of Earth affects the lengths of the arms, these lengths will e calculated in the chwarzschild geometry. This calculation will e eased y allowing the center of a Cartesian coordinate system to coincide with the center of Earth and with one of the satellites on the x-axis, as shown in Figure 4. Then let vector - c, of magnitude R, point from the origin to the eam splitter and vector of magnitude R points from the origin to the satellite on the x-axis. Another useful tool in this calculation is a line of the form y mx + that descries the straight path through eam splitter and the satellite on the x-axis. For reference, the parameter for the flat space lengths of the arms, k, has een included in Figure 4. It has already een noted that the laser paths are identical in the chwarzschild coordinates, so finding the new value for length k in this geometry of the arm near the x-axis will e sufficient for oth. ince dt dθ, the chwarzschild line element in equation (7) simplifies to 1 G M 1 dr r ϕ ds + c r ( sin θ d ) (15) where the right-hand side is now written in G c 1 units, as will e used from here on. If r is the radial position of a point along arm length k, then we note that R r R implies G M (16) c r a quantity much, much less than 1. This means the right-hand side of equation (15) can e linearly approximated in GM/(c r) G M c r ( sin θ d ) ds 1 + dr + r ϕ (17) Taking the square root of equation (17) and rearranging gives the result, GM dr ds r c r dϕ. (18) Although this is not immediately solvale, if the line of the eam can e parameterized in terms of r and ϕ then it will e possile to integrate the equation. efore the line can e written in spatial spherical polar coordinates, it must e found in spatial Cartesian coordinates. Point (R,, ) on the line is the position of a satellite. The coordinates of another point 31

6 AMERICAN JOURNAL OF UNDERGRADUATE REEARCH VOL. 4, NO.1 (5) R y mx + k k Figure 4. The layout of the gravity wave interferometer, with the coordinate origin at the Earth s center; the distance etween the eam splitter and each satellite, k, is calculated for the chwarzschild geometry in section IV. correspond to the components of (,, ). ince the magnitude of this vector is known, ( ) ( ) R + (19) can e written in terms of, ( ) R + ( ) () It is also clear that ( R ) ( ) k +. (1) ustituting equation () into this equation and solving for results in ( R ) + R k m. R () ustituting this result into equation (), solving for, and choosing the positive solution gives the result 6 ( ) m. R + (3) ince the coordinates of the eam splitter and the satellite are now known, the slope of the line that they are on, m, can e calculated: m R.16 (4) We also know that y(r ) m R +, so the y-intercept,, can e solved for: 6 mr m. (5) If we recall the coordinate transformations from spherical polar to Cartesian coordinates, r cosϕ (6) r sinϕ (7) The line can e written in spatial spherical polar coordinates y sustituting the coordinate transformations into the equation of the line. olving this equation for r and taking the derivative with respect to ϕ gives the results: r dr dϕ sinϕ m cosϕ (8) ( cosϕ + msinϕ ) ( sinϕ m cosϕ ). (9) 3

7 AMERICAN JOURNAL OF UNDERGRADUATE REEARCH VOL. 4, NO.1 (5) These results can e sustituted into equation (18). Algeraically manipulating this result and integrating it over the values ϕ ϕ, where ϕ is the ϕ coordinate of the eam splitter, gives a formula for the length of the arm in the chwarzschild geometry, L : L ϕ cosϕ + msinϕ G M sinϕ mcosϕ m cosϕ c ( sinϕ mcosϕ ) dϕ (3) The coordinate φ can e found y dividing equation (7) y equation (6), solving for ϕ, and transforming the spatial Cartesian coordinates of the eam splitter to otain tan 1 (31) ϕ.668 radians. Equation (3) is not easily solvale y analytic techniques. It is easier to numerically solve when it is separated into a term that descries the flatspace length and one that descries the additional length caused y spacetime curvature due to the Earth s mass. The second term can e identified y the presence of the gravitational constant. This only appears in one term in equation (3), so if the equation could e separated into a part with G and a part without it, the term without G would represent the flatspace length of the arm, k, and the second would e the additional length due to spacetime curvature. This can e done if the equation is rearranged into the form where L ϕ 1 + q cosϕ + m sinϕ 1 + sinϕ mcosϕ m cosϕ dϕ (3) q G M cosϕ + msinϕ mcosϕ cosϕ + m sinϕ 1+ mcosϕ ( sinϕ mcosϕ) c (33) The limits φ φ.668 radians imply q «1, so the inomial approximation, (1 + q) 1/ 1 + ½ q, would e accurate [5]. ustituting this approximation into equation (3), separating the result into two terms, and letting the term without G e equal to k ecause it is the flatspace term gives the result L L k ϕ cosϕ + m sinϕ G M mcosϕ cosϕ m sinϕ c mcosϕ sinϕ cosϕ + m sinϕ 1 + dϕ. mcosϕ (34) This equation is manageale y a computer, so it was programmed into the software, MATLA, and solved numerically using the quadl command [6] with the result L.44 m. << k. The length of the arm in the chwarzschild geometry was then calculated to e L Z L Z + k k m. 33

8 AMERICAN JOURNAL OF UNDERGRADUATE REEARCH VOL. 4, NO.1 (5) V. ENITIVIT OF THE INTERFEROMETER To find out if the interferometer can detect gravity waves, its sensitivity must e calculated. It has een shown the spacetime curvature caused y Earth is insignificant in the lengths of the arms. Another consideration is what affect the rotation of Earth would have on this length. It can e shown rotation s effects on curvature are smaller than those found using the chwarzschild metric, so these effects can e ignored. This means the spacetime curvature near Earth is small enough that spacetime can e approximated to e flat. Thus the gravitational wave equations already calculated in section four are accurate in this situation. To calculate the sensitivity of the interferometer, return to the arrangement of the interferometer with respect to the coordinate system in Figure 1. Assume that this is the z plane and that a gravity wave is propagating in the z-direction. Choose the initial time, t, such that δ and let the flatspace length of the arms e L * 1 k. Let a 1 since that is the expected amplitude of gravity waves that will e detectale on Earth [1]. In this situation, equation (4) can e used to calculate the maximum difference in the lengths of the arms, L MA, when a gravity wave passes with the maximum occurring when sin(ωt) 1. It is more useful to use the wavelength of the laser, λ, to calculate the fraction of a wavelength the lengths of the arms will change, f. For a typical Helium-Neon laser, λ m, the fraction is L λ L a * λ MA 8 f. (35) ecause of the partially reflecting mirrors, the effective lengths of LIGO s arms are L * m [7]. ustituting this value into equation (35) gives the result f LIGO It is reported [1] that the initial LIGO detector will e ale to detect f LIGO, reported 1-9, so these values are consistent. If the proposed interferometer were operating with equipment similar to LIGO s such that it was capale of detecting f 1-9, then it would e ale to detect gravity waves of a ~ 1-1. Tale 1 is a partial reproduction of the tale found on page 171 of artusiak [7] of expected rates of gravitational wave detections that LIGO will make. CONCLUION The proposed interferometer can e expected to successfully make a gravitational wave detection given enough time according to the theoretical considerations made. ince it is more sensitive than LIGO, it would also make more detections than those expected as listed in Tale 1. It is also a safer expenditure than LIA ecause it is repairale and updatale. Thus the interferometer is at least worth further investigation. The next step in making this interferometer a reality would e to consider the experimental issues. Many of these have already een solved in the process of creating LIGO and LIA, ut it will also provide new challenges. LIGO s arms are contained in tues that are not practical for this interferometer. On the other hand, LIA s arms will e in the vacuum of space. Interactions with Earth s atmosphere will introduce difficulties that neither of these other two interferometers encountered. Event Region of pace Detection Rate upernova Within Milky Way 1 to 3 per century lack Hole/lack Hole Merger 3 million light-years 1 per 1, years to 1 per year Neutron tar/ Neutron tar Merger 6 million light-years 1 per 1, years to 1 per century Neutron tar/ Neutron tar Merger 13 million light-years 1 per 1, years to 1 per century Tale 1. LIGO s Expected Detection Rates of Gravitational Wave Detections [7]. 34

9 AMERICAN JOURNAL OF UNDERGRADUATE REEARCH VOL. 4, NO.1 (5) olving the experimental difficulties such as this one could make this interferometer a practical solution for a gravitational wave oservatory of the future. ACKNOWLEDGEMENT I wish to thank Dr. C. C. Chancey, those involved in the Gravitational Wave Astronomy ummer chool, and the University of Northern Iowa Physics Department for their guidance, advice, and support. REFERENCE 1. Hartle, James. (3). Gravity: An Introduction to Einstein s General Relativity. an Francisco, CA: Addison Wesley.. Hughes, cott A. (4, June). Listening to the Universe with Gravitational-Wave Astronomy. Paper presented at the meeting of the 1 st Gravitational Wave Astronomy ummer chool, outh Padre Island, T. University of Northern Iowa 3. Tipler, Paul A. (1999). Physics for cientist and Engineers (4 th ed.). New ork, N: W. H. Freeman and Company. 4. Pasachoff, Jay M. and Filippenko, Alex. (1). The Cosmos: Astronomy in the New Millenium. Pacific Grove, CA: rooks/cole Thomson Learning. 5. Taylor, John R., Zafiratos, Chris R., and Duson, Michael A. (4). Modern Physics for cientists and Engineers ( nd ed.). Upper addle River, NJ: Prentice Hall. 6. Hanselman, Duane and Littlefield, ruce. (1). Mastering MATLA 6: A Comprehensive Tutorial and Reference. Upper addle River, NJ: Prentice Hall. 7. artusiak, Marcia. (3). Einstein s Unfinished ymphony: Listening to the ounds of pace-time. New ork, N: erkley ooks. University of Northern Iowa Physics Department

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