Gravitational Waves Summary of the presentation for the Proseminar Theoretical Physics

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1 Gravitational Waves Summary of the presentation for the Proseminar Theoretical Physics Nehir Schmid Contents 1 Introduction 1 2 Theoretical Background Linearized Theory Transverse Traceless Gauge Quadrupole Radiation Strain Magnitudes 3 4 Method of Measurement 4 5 The Example of GW Compact Binary Systems Estimates on the involved mass Compactness of the System Conclusion

2 1. INTRODUCTION University of Zürich, HS Introduction In 2015 the first direct measurement of gravitational wave was taken by LIGO, the Laser Interferometer Gravitational Wave Observatory situated in Hanford (Washington) and Livingston (Louisiana). Two years later, in 2017, Rainer Weiss, Barry C. Barish and Kip S. Thorne were awarded the Nobel Prize in Physics for their decisive contributions to the LIGO Observatory and the detection of gravitational waves. This short summary contains some theoretical concepts from which physicists deduced that there should be gravitational waves and it would be worth searching for them. It will briefly explain the method of detection and finally discuss the sort of information, one can gain from such a measurement. 2 Theoretical Background 1 In general relativity (GR) the presence of mass, or in general energy and momentum of matter, is coupled to a curvature of spacetime, and the force of gravity corresponds to an inertial motion within that curved spacetime. Gravitational waves are a local disturbance in the geometry of space-time. This means spatial lengths and time durations change as a wave passes through. The waves propagate at the speed of light through vacuum. Their source is accelerated mass. How can we show that they exist? Einstein s field equations (1) describe the relation between the mass and energy distribution in space and the curvature of space-time. R µν 1 2 g µνr = 8πG c 4 T µν (1) The energy-momentum tensor T µν on the right hand side contains all the information about the distribution and the flux of mass and energy. The left hand side depends on g µν the metric of curved space-time. (The Ricci tensor R µν also depends on depends on g µν.) Einstein predicted gravitational waves in 1916, as their local disturbances in the curvature of space-time, which travel through vacuum at the speed of light, are a solution of (1). The disturbances are represented in the metric g µν, which contains the information about the curvature. Finding a solution of (1) is one of the main problems in GR. The field equations are 16 coupled differential equations, which are difficult to solve. Exact solutions are in fact only possible for highly symmetric problems, for which the degrees of freedom are reduced e.g. for a non-spinning black hole. Normally approximations are used to simplify the problem. The technique used in the derivation of gravitational waves is linearized theory. 2.1 Linearized Theory The idea is to express the metric g µν of curved space-time as the metric η µν of flat space-time plus some perturbation h µν : g µν = η µν + h µν. (2) All the information about the disturbances caused by a gravitational wave is now contained in h µν. Then we assume that the perturbation is small. This approximation is okay for an observer on earth, because there are no such heavy objects nearby that would curve space-time strong enough that its effects could compete with the curvature caused by a typical cosmological source of gravitational waves that we could hope to detect. Since these typical waves only cause weak curvature on earth. the approximation is legitimate. The next step is to expand Einstein s field equations in h µν and only keep 1 st order terms. This leads to a simplified equation that is linear in h µν where h µν + η µν ρ σ hρσ ρ µρ ρ µ hνρ = 16πG c 4 T µν (3) h µν = h µν 1 2 η µνh. (4) 1 For further information: Michele Maggiore, Gravitational Waves [1] 1

3 2. THEORETICAL BACKGROUND University of Zürich, HS2018 By using some of the gauge freedoms by moving to Hilbert gauge Einstein s field equations become a simple wave equation ν hµν = 0. (5) h µν = 16πG c 4 T µν. (6) The comparison to the wave equation for electromagnetic (EM) waves (7) shows that both equations are of exactly the same form. The EM waves are described in the value A α and it s source is the electric current density j α, while the gravitational waves are in the perturbation h µν and the source term is proportional to the energy-momentum tensor T µν. A α = 4π c jα (7) From the form of the wave equation one can directly see that there is a solution for waves propagating at the speed of light in vacuum, where T µν = 0. Continuing with the analogy to EM waves, we can solve the wave equation using Green s functions. 2 The Green s function that not only solves the equation but also satisfies the causality condition is the retarded Green s function. G( x x 1 ) = 4π x x δ( x0 ret x 0 ) (8) This leads to a solution (10) of Einstein s field equation in linearized theory. With the retarded Green s function, (10) depends on the integral over the energy-momentum tensor at retarded time, which means it depends at the mass and energy distribution and flux at a time t ret as much before the time of measurement t, as information needs to travel from a source at x to the point x. h µν (x) = 16πG c 4 h µν (t, x) = 4G c 4 d 4 x G(x x )T µν (x ) (9) ( d 3 x 1 x x T µν t x ) x, x c (10) 2.2 Transverse Traceless Gauge Normally, the solution of the wave equation contains not only the information about the radiation itself, but also about the relative position of an observer to the source. To get rid of this additional information, we move to transverse-traceless (TT) gauge, where we have gotten rid of all the gauge freedoms. In TT-gauge, the perturbation h µν of the metric of curved space-time is traceless and perpendicular to the direction of travel of the wave. For a cartesian coordinate system in which the wave propagates in z-direction, and the x- and y-axes span the plane perpendicular to the wave vector, the perturbation is given by h T T ij (t, z) = h + h 0 h h ab [ ( cos ω t z )]. (11) c The h 0ν and h µ0 are not shown, as those entries become 0 in TT-gauge in linearized theory. As the perturbation tensor is perpendicular to the xy-plane, all the changes in distances caused by the wave lie within this plane. For a chosen reference frame there are two possible polarisations: plusand cross-polarisation. In Figure 1 you can see the effects of a passing wave in each of those polarisations. Observe that as much as the lengths stretch in one direction, as much they squeeze in the direction perpendicular to it. This could be expected for a traceless perturbation tensor. 2 The concept of Green s functions is explained here: (2015) 2

4 3. STRAIN MAGNITUDES University of Zürich, HS2018 Figure 1: The effects of a passing gravitational wave in plus- or cross polarization on a ring of test masses. Source: The First Detection of Gravitational Waves [2] 2.3 Quadrupole Radiation To determine the form of the waves an observer could see for a specific source, it is necessary to solve the integral in eq. (10) which is not trivial. One possibility is to expand the energy-momentum tensor in mass multipoles (eq. (12)) and drop higher order terms. With each higher order comes an additional factor of v/c. v is the typical velocity in the source. Therefore this approximation is valid for nonrelativistic sources like typical cosmological sources are. The lowest order term is the quadrupole term. The constant monopole does not contribute to the radiation and the dipole term in the expansion of the energy-momentum tensor is proportional to the derivative of the total momentum with respect to time, which of course is zero. ( T kl t r c + x ˆn ) (, x T kl t r c c, x ) + x i n i 0 T kl + 1 c 2c 2 x i x j n i n j 0T 2 kl +... (12) In this approximation the perturbation of the metric of curved space-time is given by [h T ij T (t, x)] quad = 1 2G r c 4 Λ ij,kl(ˆn) M kl (t r/c), (13) where M kl is the second mass moment, the normal moment of inertia, of the source. This can be calculated in any reference frame in Hilbert gauge, and is then projected to TT-gauge by the projector Λ ij,kl wich depends on the direction of propagation ˆn of the wave. With this formula we can calculate the gravitational waves created by any source of which we know the mass distribution and how that changes over time. Once having found h T ij T (t, x), we can determine further properties of the waves. For example how much power is radiated P quad = G 5c 5 M ij M ij 1 3 ( M... kk) 2 = G de GW Q 5c 5 ij Q ij dt (14) 3 Strain Magnitudes How big a perturbation h do we expect to deal with? h gets smaller with the distance to the source. Typical cosmological sources of gravitational waves cause extremely small perturbations on earth. They are of magnitudes smaller than h L L 1 d L h is also proportional to the relative change of distances, the strain. When measuring the effects of a gravitational wave on earth, we need to measure absolute distances. The absolute change in distance L 3

5 4. METHOD OF MEASUREMENT University of Zürich, HS2018 needs to be large enough for us to be able to detect it. In order to measure large absolute changes for small relative changes in distance, the total distance L over which one measures should be as large as possible. In LIGO it is chosen to be L = 4 km. Over this length, the L measured in GW150914, the famous first direct measurement of gravitational waves, were smaller than 1/1000 of the size of a proton. Or in other words: The waves in GW in LIGO were equivalent to changing the distance to the nearest star by one hair s width. 4 Method of Measurement Figure 2: Laser interferometer to measure the changes in distance caused by gravitational waves. Source: /PhysRevLett [3] The tiny strains from gravitational waves are measured using laser interferometers. The basic concept is: There is a laser producing monochromatic light, which is then split in two at the beam splitter. At the end of each of the 4 km arms is a mirror, which corresponds to the test masses that we want to measure the distance to. The beams are reflected backwards and merge again at the beam splitter. In LIGO the length of the paths travelled by the two beams is such that, as long as no wave passes, the beams interference is negative and no light reaches the photodetector. When a gravitational wave passes, it changes the proper length of the two arms. The path lengths travelled by the two beams change and the light does not interfere negatively any longer, but in a detectable interference pattern. From this pattern, a specific sequence of different intensities, the exact changes in length can be determined. LIGO consists of two such interferometers in the U.S. that are situated far enough apart to allow to exclude a lot of measured fluctuations as noise created by local sources. Only measurements made by both detectors can be counted as signal from gravitational waves. There is also collaboration with an other, similar detector, VIRGO in Italy. Because none of the detectors alone is particularly sensitive to the direction the waves travel in, one uses triangulation to determine, where on the sky the source of the radiation lies. As shown in Fig. 3, a third measurement improves the results. Whether we can detect a gravitational wave or not does not only depend on the amplitude but also the frequency of the wave. Different types of detectors are sensitive in different frequency regions. All the detectors we have today that are able to detect gravitational waves are terrestrial interferometers. 4

6 4. METHOD OF MEASUREMENT University of Zu rich, HS2018 Figure 3: The sky around earth with the localization of the sources of the waves from different detections. For GW data from VIRGO was included. Source: /PhysRevLett [4] They are sensitive in the rather high frequency end of the spectrum (see Fig. 4). Different types of sources typically radiate waves in typical frequency regions. This means we can mainly hope to detect signals from rotating neutron stars and supernovas as well as from compact binaries within or galaxy and beyond. Luckily, the latter are the types of sources we can model the best. This helps in finding out whether a signal in the interferometer really came from a gravitational wave or not, and by comparing the measurement to the model, we can determine some parameters of the source. Figure 4: The spectrum of gravitational waves with typical sources radiating in specific frequency ranges. Image source: NASA 5

7 5. THE EXAMPLE OF GW University of Zürich, HS The Example of GW Let us discuss the sort if information that can be gained from a measurement of gravitational waves using the example of LIGO s first measurement from 2015: GW In this summary we restrict ourselves to some rather rough, classical approximations, which can already tell quite a lot about the source of the radiation without needing too much knowledge about GR. We will for example be able to tell what kind of objects were involved in the system and estimate their mass Compact Binary Systems The process of a compact binary system can be divided into 3 phases: The inspiral, the merger and the ringdown. In the inspiral phase the two objects are in orbit around each other. It can be approximated with a simple Keplerian orbit. Because this is an accelerated motion of rather heavy objects, they emit gravitational radiation, which causes the system to loose energy over time. This energy is mainly taken from the orbital movement. So the radius of the orbit decreases as the system looses potential energy. When this happenes, the kinetic energy increases, because for smaller radii objects need to move faster for stable Keplerian orbits. de orb = GMµ dt 2r 2 ṙ (15) For faster rotations, the frequency and the amplitude of the gravitational radiation increase. The theoretical form of the gravitational wave of such a source is shown in Figure 5. Figure 5: Theoretical waveform of gravitational radiation emitted by a compact binary system. Source: /PhysRevLett [3] There is a distance, beyond which, when the bodies get closer to each other, the Keplerian orbits and the resulting waveforms cannot describe the system well enough anymore. Beyond this point, the two objects start plunging towards each other. The distance at which this change happenes is approximately the ISCO, the Innermost Stable Circular Orbit in the Schwartzschild geometry. After plunging together, the objects coalesce and the merger phase is completed. The newly formed object is still distorted. It has not yet reached its final equilibrium state where no mire gravitational radiation is emitted. There is still 3 More on this can be found in: The basic physics of the binary black hole merger GW [5] 6

8 5. THE EXAMPLE OF GW University of Zürich, HS2018 movement of mass. The distortions linearise over time and in the end there is a superposition of quasinotmal-modes which leads to the ringdown ending in several cycles of almost harmonic, damped oscillations. The stains in GW represent the expected waveform very well as can be seen when comparing 5 and 6. It is clearly a signal from a compact binary system. 5.2 Estimates on the involved mass Figure 6: Top: Strains measured in the two LIGO detectors for GW Bottom: Amplitude of the signal given as a function of time and frequency. Source: /PhysRevLett [3] Approximating the two involved objects as non-relativistic point masses on a Keplerian orbit and using eq. (14) We can calculate the power radiated by a binary system as gravitational waves. The formula only depends on time derivatives of the second mass moment, which means the radiates power is given by the involved masses, their position in space and how that changes over time. Because of conservation of energy, the power radiated has to be equal to be the negative loss of orbital energy. de GW dt = 32G 5c 5 µ2 r 4 ω 6 = GMµ 2r 2 ṙ = de orb dt Both sides of the equation depend on the involved masses, their position in space, namely the radius of the orbit, and how that changes over time. Kepler s laws link the orbital radius to the orbital period and thus the frequency of the orbital motion as well as the graviational radiation. Eq. (16) is therefore a relation of the two involved masses, the orbital frequency and its time derivatives. Solving for the masses gives an expression for the chirp mass M. Its general definition is shown in eq. (18). M = c3 G (16) ( ( ) ) 3 1/5 5 π 8 (f GW ) 11 ( f 96 GW ) 3 30 M (17) The frequency and how it changes can be read directly from the measured data shown in 6. The result is a chirp mass of approximately 30 M for the binary system in GW

9 5. THE EXAMPLE OF GW University of Zürich, HS2018 System R Mercury Orbit Binary Orbit Cyg X HM Cancri (RX J0806) Neutron Stars (just touching) 2 to 5 GW Black Holes (just touching) 1 Table 1: Source: /andp [5] The chirp mass is a value that contains information about the sum of the involved masses as well as their ratio. No direct statement about the magnitude of the two individual masses can be made from it. M = (m 1m 2 ) 3/5 (m 1 + m 2 ) 1/5 = (µ3 M 2 ) 5 (18) 5.3 Compactness of the System The chirp mass alone cannot tell us exactly which kind of objects were involved in the binary system, even though M 30 M already tells us that the object were rather heavy, most probably in the mass regions of black holes. Knowing that the most dense objets are black holes followed by neutron stars and then other types of stars, it is insightful to look at the compactness of the system to learn more about the types of objects involved. The compactness is given as the compactness ratio R which is defined as the separation of the objects, the radius, divided bi the sum of the smallest possible radii of each object, which would be the Schwarzschild radius of a non rotating black hole of according mass. With this definition, the compactness ratio of two black holes that are just touching each other is 1. R = R R SS1 + R SS2 (19) As a first approximation we can determine R for two objects of the same mass. In this case we can get the masses from eq. (18) and ure it to determine the R SS. This gives a compactness ratio of 1.7 for GW Comparing this to the values in Table 1, one sees that it is more compact than a system of two neutron stars that are just touching. This would mean that under the estimation of two objects of the same mass, at least one of them would be a black hole. Of course the two masses do not have to be the same. What happens when they differ and the mass ratio q differs from 1 is illustrated in Figure??. The black contour line with R = 1.0 indicates where the system becomes more compact than two black holes that are joust touching each other. Of course no binary system can be more compact than that. This means q is limited to a value a bit smaller than 10. We can neglect the effect of the change in eccentricity not only because it is rather small, but also because gravitational radiation causes the orbit of a binary system to become more and more circular over time. This happenes because the energy radiated depends on the speed and acceleration of the object on an elliptical orbit, which changes with the position. It causes the orbit to decrease at different rates on different parts of the ellipse, which leads to a decrease in eccentricity. For a highly elliptical orbit one would expect alternating high and low amplitudes in the radiation, which was not observed for GW150914, leading to the assumption that the orbit was already practically circular when the signal was measured. The limitation in the mass ratio q combined with the known chirp mass M leads to a lower limit of the mass m 2 of the lighter object. m 2 11M (20) m 2 neutron star limit (21) It is still much heavier than any known neutron star and 3-4 times heavier than the neutron star limit. This indicates that both objects involved in the binary system of GW were black holes. 8

10 References University of Zürich, HS2018 Figure 7: The compactness Ratio of a binary system of Chirp Mass M 30 M in dependence of the ratio of the two involved masses and the eccentricity of the orbit. Source: /andp [5] 5.4 Conclusion By using very simple mathematical concepts and approximations, we could tell from the signal of GW that it was created by a compact binary system of two black holes and estimate their masses. These results are in good agreement with more complex analyses. With the proper methods one can of course derive much more information about the signal s source, like its distance from earth. In general the measurement of gravitational waves has lead to many new findings. It has allowed scientists to test Einstein s theory of general relativity and has given raise to new astronomical questions and insights. References [1] Michele Maggiore. Gravitational Waves, Volume 1: Theory and Experiments, volume 1 of Gravitational Waves. Oxford University Press, ISBN URL id=aqvpqgaacaaj. [2] Andrzej Królak and Mandar Patil. The first detection of gravitational waves. Universe, 3, doi: /universe URL [3] B. P. Abbott et al. Observation of gravitational waves from a binary black hole merger. Phys. Rev. Lett., 116:061102, Feb doi: /PhysRevLett URL doi/ /physrevlett [4] B. P. Abbott et al. Gw170814: A three-detector observation of gravitational waves from a binary black hole coalescence. Phys. Rev. Lett., 119:141101, Oct doi: /PhysRevLett URL [5] P. B. Abbott et al. The basic physics of the binary black hole merger gw Annalen der Physik, 529, doi: /andp URL /andp