PoS(HEASA2018)029 http://pos.sissa.it/ ∗ [email protected] [email protected] Speaker. Gravitational waves (GWs) are one of the key(GR). predictions Scientists of have been Einstein’s theory looking of for evidence general of relativity The GWs since first their prediction direct by detection Einstein in was 1916. merging achieved of by two LIGO black in holes 2015, inindependent 100 a polarization years binary states. after into In one. their this prediction, According paper,along from to the we GR two study polarization gravitational the states waves power from have radiated a two compact and binary the system strain with of its GW general orbital properties. ∗ Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). High Energy Astrophysics in Southern Africa1-3 - August, HEASA2018 2018 Parys, Free State, South Africa S. Razzaque Centre for Astro-Particle Physics (CAPP) andJohannesburg, Department Auckland of Park Physics, 2006, University South of Africa E-mail: L. Nyadzani Centre for Astro-Particle Physics (CAPP) andJohannesburg, Department Auckland of Park Physics, 2006, University South of Africa E-mail: Polarization of Gravitational Waves from Binary Systems PoS(HEASA2018)029 , y + (2.2) (2.1) µν η = L. Nyadzani µν and vector- g x ) becomes 2.1 ) under the harmonic gauge µν in ( h µ ∂ µν g µν µν h GT µν π 8 GT − π = 16 1 R − direction [12]. However in GR, only two states of µν z = g 2 1 µν and their derivatives ( ¯ h −  µν . The spacetime metric µν h R µν η , is “the trace-reversed tensor” of h ) by assuming the spacetime far away from the source to be a Minkowski µν ) in powers of η 2.1 1 2 2.1 − (flat). Let the GW passing through the observer be represented by the perturbation µν h µν = η µν ¯ h Gravitational waves are one of the key predictions of Einstein’s theory of It was not until 2015 when the first direct detection of GWs (GW150914) was achieved by This paper starts with an introduction to the theory of GW generation and propagation in on a background spacetime . Expanding ( µν µν h condition, we obtain the following wave equation [3]. where spacetime h (hereafter GR) [1, 2]. In GR,ples accelerated radiate massive away bodies from generate ripples theirradiation in source carry spacetime. at information These about the rip- their speed origin,first of indirect GWs light evidence also of [3]. carry the information existence The about ofknown GW their same as was origin. way PSR a The the binary 1913+16 system electromagnetic [4]. discoveredThis by decay They Hulse was and consistent observed Taylor, with that the thetion expected as orbital orbital predicted decay period by of of the a standard this system quadrupole emitting system formula gravitational was radia- in decaying. GR [5]. LIGO. This event wasGW150914 event, due there have to been 5 inspiraling more confirmedGW151226, detections orbit GW170104, made by GW170608, and LIGO GW170814 and merger and VIRGO namely GW170817scientists of [6–11]. have two been With able black these detections, to holesfield probe [6]. regime some fundamental of physics gravity, Since in asstudy the the predicted GWs highly is by dynamical the the and polarization GR strong- statesdict using of up these to GW waves. six signals. polarization In states, general, One namely alternative plus, of theories cross, of the breathing, gravity key longitudinal, pre- vector- tools used to GW Polarizations 1. Introduction assuming that the GW is propagating in the We solve equation ( spacetime. Then review theGW polarization power states generated of and strain GW oforbit as this with described general wave by configuration. in spacetime GR for and two finally compact study objects the spiraling in an 2. GW Theory In the theory of general relativity, the Einstein’s field equations are given by, polarizations are allowed, the plus andpolarizations the from cross GR will modes. be In discussed. this paper, only these two independent PoS(HEASA2018)029 Ω d (3.1) (3.2) (2.4) (2.5) . Let φ ). w × e L. Nyadzani and ˆ θ w , ˆ ˆ k are orthogonal to the ) and cross ( φ + e w and ˆ θ is the orbital angular frequency, w ) with the above plane wave we ) ω x 2.3 · 2 k φ θ  0, and we have ˆ ˆ , namely plus ( w w i j − e t = µν j ⊗ ⊗ i j e ω α 3 φ φ ( µν Q ˆ ˆ m w w T 3 dt i 0 (2.3) d α cos − + =  m θ φ µν 5 2 ˆ ˆ w w c α µν ∑ G ae ¯ h π ⊗ ⊗ 8 =  = θ θ . ˆ ˆ i j w w = is the GW amplitude, 1 µν Q a = = h Ω × + = dE e e dtd µν ¯ h is the position. Solving equation ( x is [13], The effect of the plus and cross polarizations on a ring of particles in space. i j e ) can be evaluated to find the power radiated by a system of masses undergoing 2 2.2 is the quadrupole moment tensor given by , is the polarization unit tensor, Figure 1: 1 i j Q µν = e α is the wave vector, and be the direction of propagation of the GW such that unit vectors ˆ ˆ and are represented graphically in Figure propagation of this GW. In this coordinate system, the two polarization modes are defined as, k Equation ( where The above equation has a plane wave solution, 3. GW Polarization Consider a GW with orthonormal coordinates constructed by three unit vectors find that there are only two independent components of with polarization and arbitrary motion. Using quadrupole approximation, the power radiated into the solid angle GW Polarizations In vacuum, where there is no matter present we have Here k PoS(HEASA2018)029 i θ ) ) 3 0 , θ 3.4 1 (3.5) (3.4) (3.6) (3.3) 2 cos , 0  4 cos e = (  ) and ( 4 ˆ L. Nyadzani y e 47 , 128 , eccentricity ) 3.3 ). 0 ) + 47 , 128 π 2 0 i 2.4 , e , + φ 0 1 ( 2 95 32 e θ = ( . + cos4 95 32 2 x i 1 θ φ  + 2 3 1 1  θ cos θ cos4 4 ) + 2 e ) + θ sin ) cos ˆ 5 z θ θ ˆ z 25 4 2 128 − + cos φ is given by equation ( + cos cos φ 1 5 orbiting around a common center of gravity we can integrate equation ( θ Ω + in a closed orbit around a common center + 2 sin + 2 sin d φ 1 1 2 m θ i θ ( ( θ . m 4 θ  cos , ω e 4 and the observer is located as described above 3 cos and  a φ cos sin e ˆ 4 ( a y ˆ 1 y and 3 e 25  cos 512 m + 4 1 51 cos + 256  e ˆ 49 64 y φ 4 m − φ e + + φ + 51 2 2 256 cos from the source with coordinates ˆ e 64 49 φ cos e θ θ + R + 99 64 cos2 sin 97 16 2 2 ) ˆ e x e cos sin + θ + ˆ ˆ x − x 4 1 2 99 64 2 97 16 = = = + h h + cos ˆ k φ θ ) ) 1 2 ˆ ˆ 2 2 2 w w − 2 2 and semi-major axis ), and semi-major axis / / m m 1 h h 7 7 2 e ( ) ) ) ) + + m 2 2 2  2 2 2 / e e 4 1 1 / 7 m m e 7 ) m m − − ) and the orbital angular frequency 2 ( ( and + + 1 2 e 1 1 16 2 2 2 2 a e 1 1 1 ( ( m m 5 5 − m m m + 2 1 2 1 − a a ( ( . Then the coordinate systems of the wave and observer are related by, 1 2 are the conventional polar coordinates, as shown in Figure . The total power radiated into the cone 1 2 2 2 2 ) m m to get power radiated as a function of the inclination angle ( e ( 2 1 5 5 5 φ m m 5 ) 4 4 , a c c 2 1 2 1 a 13 32 0 π 3 G G π π , m m 2  Binary system consisting of two masses 0 , and 5 5 4 4 0 = = c c + , masses ( ( θ G G π π = ( ) × + φ Ω Ω 1 z , = = d d dP dP 0 × + ( P Assuming that the emission of GW is symmetric over over e P Figure 2: and in Figure GW Polarizations If we consider an observer at a distance where 3.1 Power We now consider a binary systemof of mass masses with eccentricity with semimajor axis Powers radiated into the two polarization components are given by [14], and ˆ PoS(HEASA2018)029 (3.7) (3.8) shows ) on a µν h L. Nyadzani top right panel ; ) π , 0 ( . ) a ( is the quadrupole moment. The ) (blue solid line) shows polarization i j shows power as a function of the mass × Q + × e e i j ¨ Q × × r 1 i j i j 4 from the source, one obtains that, the disturbance h h bottom left panel = r ; = = ) i j e h + × ( h h (red dashed line) and cross ( (+) ) at a distance µν η shows power as a function of the inclination angle has only two independent components, the plus and cross polarization, and shows power as a function of the semi-major axis i j e we plot these powers as different functions of parameters. 3 Top left panel Power radiated into plus is the distance form the source to the observer and r In Figure bottom right panel in space is given by the following formula [3], 3.2 GW Strain If one solves the Einstein’sflat field spacetime equation background by ( assuming a small spacetime perturbation ( power radiated as a function ofand the eccentricity Figure 3: components. GW Polarizations where polarization tensor the strains for these components are defined as PoS(HEASA2018)029 ) θ 3.8 (3.9) (3.14) (3.13) (3.15) (3.10) (3.11) (3.12) cos ) which )  1 )    ) into ( L. Nyadzani ψ + 0 0 3.10 2 ) ψ 3.7 1 − 2 φ + 2 ) and ( ( ψ 2sin ψ 2 − 3.9 sin 2sin ) ψ ) 2sin ψ φ 2 ) + − , and the semimajor axis is cos ) we substitute ( ) and differentiate twice with e ψ φ ω − ψ  2 i j 2cos + ) − 2.5 ¨ ψ φ Q − φ 2 cos + φ ψ ( 2 2 ) e 3 ψ + cos ) ( φ 2 − 2 ) 0 in equation ( + ( cos 2 ψ ψ 1 cos m ) ψ sin 3 3 ψ = e / − 2 θ + e 3 cos 2 + 1 sin ( 2 e 2 ) 5 a ) + 1 ψ cos ψ 2 φ 2 2 θ e 2 m sin cos ( e m 2 cos ( 3 e ) + θ sin + / )( ( G + + 2 sin ) 2 ψ using equation ( φ cos 1 2sin e 1 3 θ ω ψ 2 cos s 5 ( 2 ( m + − ( 3 3 π 3 3 − − / / / / 3 1 2 cos cos ψ 2 2 1 1 / ( φ 1 ) ) ), we obtain the following, 3 2 2 2 2cos ψ ω ω = − 2 2 ( G 2 2 m m + m m θ 1 4 4 1 1 π 3.7 cos sin m m ω e 2 = 2 sin 1 1 ) e m m + + + 2 2 ) we obtain the following, e a 2 rac rac + 1 1 m m ψ )( = ) in terms of orbital frequency 3 3 cos ψ G G m m / / θ ψ 2 2 4 ( ( 5 5 φ T 2 2 ) + 4 4 3.12 2 G G − − φ 3.12 2cos rc rc 4 2 2 cos cos 2sin 0 0 0 sin = = ( + ( − − e φ − e + × 2 )( − ) and ( = = h h sin ) show the relationship between GW strain and the four parameters ψ 1 + 2  ) and ( × + 3 cos e ) + ψ h h 1 3.11 2  2 3.10 − θ cos ) m 3.11 2 − φ e 1 1 2 cos 2 2 m e ) in ( e m − cos ( 2 ( 1 ( 2 r sin for a system described in Figure G    e ( 2 4 ) and ( m ( 2 2 i j 3.13 r − + ac = G Q 3.9 4 2 i j = ¨ ac is the retarded position of the system. Having obtained Q + h − ψ = × h Substituting ( where give, Then to obtain the following, namely, mass, distance traveled, eccentricity,position. inclination These are angle, general semimajor equations for axis arbitrary and orbital the parameters. retarded From here on we will study circular systems by letting related to the orbital period by Kepler’s law of orbits. Kepler’s law of orbits GW Polarizations Calculating respect to time, as required by equation ( Equations ( It is better to write ( PoS(HEASA2018)029 ) × A . If ) such ) and 3.19 , then π (3.16) (3.17) (3.18) (3.19) × the plus dt g = 2 the / 3.16 π / θ ψ L. Nyadzani ( π ) and ( and d is + = g 3.18 θ , ω × A , + A with the inclination ) is zero or multiples of θ × θ 2 ). ) are the amplitude of the GW along A 5 / × 3 cos A θ  3.17 , and f + + t and when the inclination + 1 ∂ A cos ( A 3 × 3 3 / / / A ) and ( 2 2 from the earth (size of our galaxy), ( ) 11 ) f φ ω ω φ 3 3 3 6 2 5 2 / / / 3.16 kLy 8 5 5 4 4 2. The GW strain hold information about the origin − − for both π / t rc rc t the two amplitudes are not equal. For example for a is twice M M π ω 24 96 3 3 ω π 2 / / + − 2 6= ( 5 5 ( A G G 10 θ = 4 2 π cos sin × is the , ( 4 ). Thus, the effect of GW on spacetime decreases as the wave . M Hz and 100 5 = = = = 0 ) are the GW sinusoidal waveform. Equations ( / 5 35 1 . × + + × × − , where, ) = 3.17 g g g A A 2 × , 10 g θ m + × = g ) are valid for objects moving with very small speed compared to the + A shows that, power radiated into the plus polarization is more than that 1 f ) and ( , 2 = m (

3.15 ) can be used to constrain the distance to the source. For a system with × remains. to be constant with time. We define new quantities / M h 5 3.16 + / 5 ω . 3 A 3.17 ) 1 2 and ) and ( = m + 1 2 g m m + 3.14 ) and ( , A

= ( , assuming = M t 3.16 2 + M ) give the amplitude to be 3 ω h from equations ( = = r 1 / 3.17 4. Summary In this paper we haveinto calculated the plus how and the cross power polarizationsthe radiated is angle dependent by of on a inclination the with binary masses, respectin system eccentricity, to determining semi-major in the properties axis observer. the of and Studying form the GW systemsto of polarization that the GW states generate observer. is them, important such Figure as, the inclination angle relative vanishes and only the inclination angle is notsystem a described above multiple with of Once the polarization modes are separated suchtions that ( the inclination angle can be constrained, equa- ( radiated into the cross polarization for m of the wave, such as theamplitude inclination of angle, the mass, wave orbital is frequency determinedsome and from of the the the distance detectors system traveled. such properties If as using the LIGO, equations it ( can be used to determine the two polarizations, and ( show that the GW frequency is twice1 that of the orbital frequency and thepropagate GW away amplitude from falls the off source. as When the angle of inclination . The semimajorconstant axis over decreases a slowly, long and period the orbital of frequency time. can The be orbital taken to angular be frequency is given by, GW Polarizations Equations ( and the cross amplitudes arethe orbital equal. frequency can If be the determinedrelation observer from [15], their is data a and detector the chirp such mass as can the be calculated LIGO using and VIRGO, ψ that Here PoS(HEASA2018)029 195 (1964) , . (2017) 136 L. Nyadzani Preuß. Physical Physical , , , (2017) 118 119 (2016) 061102. (2018) 001 GW151226: Observation of (1997) . GW170608: 116 GW170817: observation (Dec., 2017) L35 GW170104: observation of Physical Review (2016) 041015. , 851 The Astrophysical Journal 6 , HEASA2017 ApJL , Physical Review Letters PoS , Bulletin of the Astronomical Society of , , Physical Review Letters , Physical Review Letters , Physical Review X , arXiv preprint gr-qc/9712019 , 7 (2017) 042001. 96 (1918) . Sitzungsberichte der Königlich Preußischen Akademie der Probing dynamical gravity with the polarization of continuous , Discovery of a pulsar in a binary system Gravitational radiation from point masses in a keplerian orbit Physical Review D , (2016) 241103. (1916) . ]. 116 Lecture notes on general relativity Gravitational radiation and the motion of two point masses Näherungsweise integration der feldgleichungen der gravitation. sitz.-ber. k Über gravitationswellen The Hulse-Taylor PSR 1913+ 16. (1963) 435. 688 Current status of observations (1995) 77–83. 131 23 1711.05578 (1975) L51–L53. India gravitational waves from a binary black hole merger Observation of a 19 Solar-mass Binary Black Hole Coalescence 221101. Observation of gravitational waves from aReview 22-solar-mass Letters binary black hole coalescence mergers in the first Advanced LIGO observing run of gravitational waves from a binary161101. inspiral a 50-solar-mass binary black hole coalescence at 0.2 gravitational waves Review B1224. Wissenschaften (Berlin), Seite 154-167. Akad. Wiss [ [9] B. Abbott, R. Abbott, T. Abbott, F. Acernese, K. Ackley, C. Adams et al., [8] B. P. Abbott, R. Abbott, T. Abbott, F. Acernese, K. Ackley, C. Adams et al., [5] C. Sivaram, [6] B. P. Abbott, R. Abbott, T. Abbott, M. Abernathy, F. Acernese, K. Ackley et al., [3] S. M. Carroll, [4] R. A. Hulse and J. H. Taylor, [7] B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams et al., [1] A. Einstein, [2] A. Einstein, [10] B. P. Abbott, R. Abbott, T. Abbott, M. Abernathy, F. Acernese, K. Ackley et al., [11] B. Abbott, R. Abbott, T. Abbott, M. Abernathy, F. Acernese, K. Ackley et al., [12] M. Isi, M. Pitkin and A. J. Weinstein, [14] P. C. Peters, [15] N. Bishop, [13] P. Peters and J. Mathews, GW Polarizations Acknowledgement This work was supported by grants from the National Research Foundation (NRF)References South Africa.