Do Gravitational Waves Confirm Hawking's Area Law?
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Do gravitational waves confirm Hawking’s area law? Galina Weinstein The Department of Philosophy, University of Haifa, Haifa, the Interdisciplinary Center (IDC), Herzliya, Israel. July 23, 2021 Abstract Recently an experiment has been performed for the purpose of ”test- ing the area Law with GW150914 data”. As the experimenters put it, the experiment presents ”observational confirmation” of Hawking’s area law based on the GW150914 data. It is the purpose of this paper to philosoph- ically examine the test of the area law and to show that the area law is not confirmable yet is falsifiable. Accordingly, the GW150914 data do not confirm Hawking’s area law. What has been tested with positive results was the hypothesis A3 > A1 + A2, where A3 = GW150914 remnant and A1 + A2 = GW150914 merger. But this single instance does not provide observational confirmation of Hawking’s area law. 1 Introduction In 2016, Stephen Hawking commented on the first observation of gravitational waves, the GW150914 gravitational wave event. The gravitational wave sig- nals were observed by the LIGO’s two detectors in Hanford and Livingston on 14/09/2015. Hawking said: ”This discovery is the first detection of a black hole binary system and the first observation of black holes merging. The observed properties of this system is consistent with predictions about black holes that I made in 1970 here in Cambridge. The area of the final black hole is greater than the sum of the areas of the initial black holes as predicted by my black holes area arXiv:2107.10680v1 [physics.hist-ph] 21 Jul 2021 theorem. The properties of these black holes are also consistent with the black hole no-hair theorem, which says that they should be uniquely characterized by their mass and spin” [11]. After Hawking’s death, experiments were performed for the purpose of testing Hawking’s theorems. The goal of this paper is not to reconstruct a historical picture of the detec- tion of gravitational waves. This has been done in great detail both historically and technically by other writers. See for instance [16] and [21]. The primary 1 aim of this paper is to analyse from a philosophical point of view the tests of the no-hair theorem and the area law. I begin the paper by describing in Section 2.1 the tests performed by the LIGO Scientific and Virgo Collaborations. It was demonstrated by the collab- orations that the Kerr black hole hypothesis is consistent with the GW150914 data. I then discuss in Section 2.2, the test of the no-hair theorem with the GW150914 data. Next, Section 2.3 presents the test of the area theorem using the GW150914 data. In Section 2.3, I include a discussion of the contradic- tion between the area law and Hawking radiation. The paper then proceeds in Section 3 to a discussion of confirmation theory and degree of confirmation. Maximiliano Isi and colleagues who have performed the test write in their paper: ”We present observational confirmation of Hawking’s black-hole area theorem based on data from GW150914, finding agreement with the prediction [...] when we model the ringdown” phase [15], p. 1. It is tempting to agree with this state- ment. I will explain the precise sense in which hypotheses may be confirmable and falsifiable and then show that what has been achieved was not confirmation of Hawking’s area law. 2 Detection of gravitational waves 2.1 Testing the Kerr black hole hypothesis Two stellar-mass black holes that merge, form a single distorted compact object that gradually settles to a final stationary form. Gravitational waves are emitted throughout the entire process, at each moment carrying information about the evolving compact object. There are three stages in the coalescence of the two black holes [23], pp. 18-19: 1) Inspiral: a long phase in which the two black holes are still quite far one from each other but are slowly orbiting one another in a quasi-circular orbit. Since we are dealing with weak gravitational fields, this phase allows an analytic description using the post-Newtonian expansion (approximation). During this phase, the amplitude and frequency of the gravitational waves are increasing. 2) Merger: the distance between the two black holes gets smaller, the or- bit gradually shrinks and the two bodies finally merge. Numerical relativity (numerical simulations) is required. That is because the gravitational field is so strong and also time-dependent. During this phase, the amplitude of the gravitational waves gradually increases until it reaches a maximum. 3) Ringdown: the newly created object releases its final gravitational wave signals away into endless space. It wobbles and oscillates during which it rings like a bell with characteristic frequencies and damping times determined entirely by the mass and spin of the black hole. This stage is treated analytically using perturbation theory. A linearly perturbed Kerr black hole emits gravitational waves in the form of exponential damped sine waves, with specific frequencies and decay rates determined exclusively by the black hole’s mass and spin. 2 The ringdown phase consists of a superposition of damped sinusoids, quasi- normal modes. Each quasi-normal mode has a characteristic complex angular frequency: the real part is the angular frequency and the imaginary part is the inverse of the damping time. These modes are distinguished by their longitu- dinal and azimuthal indices, l and m respectively, as well as by their overtone number n. Ringdown overtones are the quasi-normal modes with the fastest decay rates. Each mode has a particular frequency and decay rate which are functions of the Kerr black hole parameter spin and total mass of the black hole that is being perturbed. Numerical simulations have demonstrated that the fundamental mode l = m = 2 and n = 0 seems to dominate the ringdown signal [3], p. 124018-18, p. 124018-21. In 2016, the LIGO Scientific and Virgo Collaborations announced the first joint detection of gravitational waves (GW150914) with the LIGO detectors. In 2018, they reported of several quantitative tests made on the gravitational-wave data from the detection of GW150914. The purpose of the tests was to show that the Kerr black hole hypothesis (two black holes form a remnant Kerr black hole) is consistent with the data. The result of the tests indicated that the entire GW150914 inspiral-merger-ringdown (IMR) waveform does not deviate from the predictions of a binary black-hole merger in classical general relativity [18], pp. 221101-3-221101-5. It was found that the detected gravitational-wave signal increases in fre- quency and amplitude in about eight cycles from 35 to 150 Hz, where it reaches a peak amplitude. After a time around 0.42s, the amplitude drops rapidly, and the frequency appears to stabilize. After the peak gravitational wave amplitude is reached, the signal makes one to two additional cycles, continuing to rise in frequency until reaching about 250 Hz, while dropping sharply in amplitude. According to the LIGO Scientific and Virgo Collaborations, the most plausi- ble explanation for this empirical evolution is gravitational-waves emission from two orbiting masses in the inspiral and merger phases. The drop in amplitude is in accord with a Kerr black hole. The LIGO Scientific and Virgo Collaborations concluded that the signal in the data was fully consistent with the final object being a Kerr black hole with a mass MBH3 = 65M⊙ and a dimensionless spin parameter a3 = 0.7. For a Kerr black hole, the ringdown is expected to have a damping time roughly equal to the period of oscillation (inversely equal to the frequency). For a black hole ≈ 65M with spin a3 = 0.7, a ringdown frequency of 260 Hz M ⊙ and a damping M time of 4ms 65M were calculated. It was then written: ”Such a final mass ⊙ [MBH3 = 65M⊙] is consistent with the merger of two black holes of ∼ 35M⊙ each, after accounting for the energy emitted as gravitational waves” [19], pp. 2-3, pp. 12-13; [23], pp. 19-20. More specifically, the following parameters were extracted for the masses and spins of the two progenitor black holes: +5.2 +0.5 +3.7 +0.5 MBH1 = 36.2 3.8M⊙,a1 =0.3 0.3,MBH2 = 29.1 4.4M⊙,a2 =0.5 0.4, (1) − − − − 3 and the measured mass and spin for the remnant that is produced by the coa- lescence of the binary is [17], p. 241102-1, p. 241102-6-241102-7: +3.7 +0.05 MBH3 = 62.3 3.1M⊙,a3 =0.68 0.06, (2) − − The above masses are estimated in the source frame and not in the detector frame. Two additional parameters are the energy radiated by the inspiral and merger of the two black holes: E =3 ± 0.5M ⊙ c2, (3) and the redshift of the source: +0.03 z =0.09 0.04. (4) − The LIGO Scientific and Virgo Collaborations explain that the initial be- havior of the gravitational-wave signal ”cannot be due to a perturbed system returning back to stable equilibrium, since oscillations around equilibrium are generically characterized by roughly constant frequencies and decaying ampli- tudes”. On the other hand, the GW150914 data ”demonstrate very different behavior. During the period when the gravitational wave frequency and ampli- tude are increasing, orbital motion of the two bodies is the only plausible ex- planation: there, the only ’damping forces’ are provided by gravitational wave emission, which brings the orbiting bodies closer (an ’inspiral’), increasing the orbital frequency and amplifying the gravitational wave energy output from the system” [19], p. 3. It is concluded: ”our inspiral–merger–ringdown test shows no evidence of discrepancies with the predictions of GR” [18], p.