radiation, which is generated by accelerating charged particles. Contrary to EM radiation, however, there are no dipole GWs [73].
Instead, the simplest possible GWs are created as quadropole radiation, that is, from a constellation of matter for which the second time-derivative of the quadrupolemoment is non-vanishing.
Already in [33], Einstein provided a formula for the amount of energy radiated by such a system.
In this section, we provide a short review of these fundamentals, following closely the formulation presented in [38]. There, this formula is applied to the case of a body rotating around one of its main axes of inertia with an (almost) constant angular velocityΩ. The radiated energy per unit time is called the luminosity, and in this case is given by
LGW = 32
5 · Gc5 ·Ω6·(I1−I2)2. (2.1) Here,Gis the gravitational constant,cthe speed of light in a vacuum, and I1, I2are the moments of inertia along the axis prependicular to the rotation axis.
2.1.1 Gravitational waves from binary systems
A likely candidate to produce measurable amounts of gravitational radiation is a gravitationally bound system. Inspecting eq. (2.1), we observe that to maximize the amount of gravitational radiation being emitted from a system, it should be rotating with a large angular velocity,Ω, while having a large moment of inertia along one axis. The ratio between these two quantities is governed by orbital mechanics in this case.
We can consider two equal point masses orbiting each other at a separation D. In this scenario, we simply get I1 = 12MD2 and I2 = 0. Equation (2.1) becomes
LGW = 8
5·cG5 ·Ω6·M2D4. (2.2)
Neglecting relativistic effects for the moment, we can use Kepler’s third law to compute the angular velocity based on the masses and separation, to get
Ω2= 2GM
D3 . (2.3)
Inserting this into eq. (2.2), yields LGW = 64
5 · Gc5 ·G3MD55. (2.4)
2.1 gravitational wave luminosity and indirect detection 7 The only free parameters are the masses of the two test masses and their separation. To maximize the amount of GWs, we are looking for massive objects on tight orbits. This implies that these objects should not only be massive, but also compact, such that they are able to closely orbit each other without colliding or being ripped apart from tidal effects. Possible candidates could therefore be the remnants of collapsed stars, such as white dwarfs, neutron stars or black holes.
A useful parameter to study such compact objects is the Schwarzschild radius, RS= 2MG
c2 , (2.5)
which defines the radius of the event horizon of a black hole of the given mass.
Inserting this expression into eq. (2.2) gives LGW = 2
Note that the result is given as the ratio between the Schwarzschield radius and the separation to the fifth power, amplified by cG5 ≈ 3.6×1052W. This means that compact, tightly orbiting binaries, for which the ratio RS/Dis within a few orders of magnitude of 1, could indeed emit large amounts gravitational radiation.
As an upper bound for the most luminous events we can expect, we can consider two black holes immediately before merger, such that their event horizons are almost touching. In this extreme limit we would simply have
RS≈ D, and the Note that some
assumptions we made – such as the application of Kepler’s law – are certainly invalid in this scenario. Still, the calculated value is within a few orders of magnitude of what has been observed using direct detection of GWs, see
section2.3.
power of the gravitational radiation would be 1052W.
It is not easy to put this number into context. For example, a typical star like our sun has a luminosity of ’just’ 1026W. One of the most luminous stars observed to date, RMC136a1, radiates 2.3×1033W [25], still far below the energy emitted in a black hole merger. It is estimated that there are around 1022-1024 stars in the observable universe1, such that at least for the short time at which the two black holes are merging, they would emit more energy in GW radiation than all stars combined emit as EM radiation2.
However, as we will see in section2.2, despite the large amount of gravitational radiation emitted in such events, the interaction of GWs with matter is so weak that it is still extremely hard to detect them directly.
Instead, the first detections of gravitational waves were indirect, and relied on continous observation of systems which also emit electromagnetic radia-tion.
1 This estimate is given on http://www.esa.int/Science_Exploration/Space_Science/
Herschel/How_many_stars_are_there_in_the_Universe.
2 Stars can have luminosities ranging from 10−5 to 106 that of our sun, as visible on a Hertzsprung-Russell Diagram (e.g., see https://www.eso.org/public/austria/images/
eso0728c/). We assumed for this estimate that the average star emits around 1026W.
8 gravitational wave astronomy
Figure 2.1:Observed change in the orbital period of the Hulse-Taylor binary com-pared to the prediction from general relativity.
Figure from [91], itself based on data from [98].
0 5 10 15 20 25 30 35 40
Cumulative period shift (s)
1975 1980 1985 1990 1995 2000 2005
Year 2.1.2 Indirect detection
As discussed above, compact binary systems can emit significant amounts of energy in the form of gravitational radiation. This energy is not created out of nothing, but instead converted from the kinetic energy and mass of the binary. Therefore, the orbital period of such a system will not be constant, but will slowly decrease with time as the two stars radiate away their kinetic energy and get closer together.
This phenomenon was first observed in the Hulse-Taylor binary, which con-sists of a neutron star and a pulsar [99]. Pulsars are rapidly rotating, highly magnetized neutron stars which emit large amounts of electromagnetic radia-tion with each rotaradia-tion. In the case of the Hulse-Taylor binary, the pulsar was rotating17times per second. In the case of this pulsar, they could observe periodic variations in the arrival time of those pulses, and could conclude that these are caused by a non-pulsing companion star, with an orbital period of7.75hours.
Pulsars are usually highly stable in their rotational period. This allows observation of fluctuations in the orbital period of this binary star system over long timescales. And indeed, over3decades of observations, the Hulse-Taylor binary has been slowing down at a rate almost perfectly predicted by GR [98], see fig.2.1.
The two neutron stars are believed to be of almost equal mass of 1.4M, and follow a highly eccentric orbit with an eccentricity of≈0.6 and a semi-major axis of 1 950 100 km. Due to this high eccentricity, we would not expect our
2.1 gravitational wave luminosity and indirect detection 9 simplified eq. (2.2), which is based on a circular orbit, to produce accurate results. Still, it’s interesting to compare how close we can get with such a simple model. Putting in the masses of the two binaries and assuming we can use the semi-major axis as our separation yields a luminosity of
LGW ≈6.2×1023W . (2.7)
This falls short by about one order of magnitude from the real value 7.35×1024W computed from a full relativistic treatment, see for example [101].
2.1.3 Gravitational waves from the laboratory
Figure 2.2:A naive in the text, the amount of radiation is incredi-bly small.
It would be interesting if we could study GWs by directly generating them in a laboratory here on earth. The problem in generating GWs in a laboratory is the extremely small scaling factor G/c5 ≈ 2.7×10−53W−1 appearing in eq. (2.1). This factor must be overcome byΩ6·(I1−I2)2.
To get an idea of the orders of magnitude involved, we make a simplified thought experiment: We consider two massive spheres, which we will again model as point masses with mass MS. While the binary in the previous example was held together by gravity, we instead link our two spheres by a cable of radius r and length L. For simplicity, we will work under the assumption that we have a thin cable withr L. As before, the whole setup is rotated around the spheres’ common center of mass (see fig. 2.2).
The moments of inertia along the non-rotating axes are then I1 ≈ 121(MC+ 6MS)L2 and I2 ≈ 0, with MC as the mass of the cable. Inserting this into eq. (2.1) gives
LGW = 2
45·cG5 ·Ω6·(MC+6MS)2L4. (2.8) Even if we assume that our idealized system does not lose kinetic energy by any other means, it should eventually slow down due to the energy lost by gravitational radiation.
A possible laboratory experiment to prove the existence of gravitational waves could therefore be to put such a setup into a frictionless environment, bring it to rotation around its center of mass, and observe the rate at which it slows down. Observing that the angular velocity Ω enters eq. (2.8) to the sixth power, we want to rotate our system as fast as possible to maximize the amount of radiated GWs.
The speed at which we can rotate this system will be limited by the cables ultimate tensile strength, Smax, which has to counteract the centrifugal force pulling the system apart.
The tension acting on the cable is highest at it’s center, where it is given as the sum of the centrifugal force due to the cable itself and the test mass, divided by the cross sectional area, A, of the cable. We get
S= LΩ2
8A (4MS+ALρ), (2.9)
10 gravitational wave astronomy
withρas the density of the cable.
This means the highest angular velocity we can achieve before the cable breaks is given by
Ω2= 8ASmax
L(4Ms+ALρ). (2.10)
Inserting this into eq. (2.8), and replacing the mass of the cable by MC =ρAL gives
LGW = 1024
45 ·cG5 ·A3LS3max·(6MS+ALρ)2
(4MS+ALρ)3. (2.11)
We observe thatMSappears to the third order in the denominator, but only up to the second order in the numerator. This means that mounting heavy masses at the end of the cable actuallydecreasesthe amount of gravitational radiation. This tells us it is better to remove the spheres and rotate just the cable, which allows us to achieve a higher angular velocity.
This can be modelled by replacing MS →0 in the previous equations, which leads to the same simplified formula given in [38],
LGW = 1024
45 ·cG5 · A2Sρ3max. (2.12)
The parameters we can adjust to construct our experiment are the cross-sectional area, A, and the material properties,Smax andρ. Ideally, we would need a material with high tensile strength and low density, i.e., a material of high specific strength. A=πr2can in principle be made arbitrarily large.
However, for these equations to remain valid, we have to stay within the assumption that the cable is indeed thin, i.e., thatL2r2.
The commercially available materials with the highest specific strength are car-bon fibres, with an extremely high tensile strength of up toSmax =7000 MPa combined with a comparatively low density ofρ =1.79 g cm−3 [88]. Typically, carbon fibres are very thin, with diameters of the order of a few µm. For the sake of argument, we will assume that it is possible to create a composite cable out of these carbon fibres with A =1 cm2 while preserving the same specific strength. Assuming this cable has a length ofL=1 m, eq. (2.10) tells us that we could spin it up to a
Given asν=2πΩ. frequency of more than 5 kHz before it breaks.
Inserting these values into eq. (2.12) gives a maximum luminosity of 10−33W.
By comparison, the total kinetic energy of the rotating cable would be given as
Ekin = 1
2I1Ω2 ≈2×105J . (2.13)
This means that even if we started our experiment at
Assuming an age of the universe of13.8 billion years.
the dawn of time, it would only have radiated a factor 2×10−21 of it’s total energy.
This is obviously far below any observable power level.
Theoretically, novel materials such as colossal carbon tubes could achieve similar tensile strengths of Smax = 6.9 GPa, but with densities as low as
2.2 a short review of general relativity 11