# William Unruh

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## Graduate Student Supervision

##### Doctoral Student Supervision (Jan 2008 - May 2021)

This doctoral thesis explores semiclassical effects on black hole physics. Semiclassical theory refers to the application of quantum field theory in curved, classical background geometries, which respond to the expectation value of the regularised stress-energy tensor of the quantum matter.Among the original findings, I develop a few useful techniques to help regularise the stress-energy tensor in two dimensions. I apply them to a model of stellar collapse to analyse the importance of quantum mechanical effects in the collapse itself. I find an explicit example showing that the behaviour of the late-time Hawking radiation does not depend on the details of the collapse and argue that any quantum mechanical effect is negligible for the collapse of an astrophysical object (whose mass is comparable to the solar mass).In the realm of black hole thermodynamics, I prove the first law for stationary black holes and propose a definition for the entropy in piecewise stationary black holes which I show to obey the generalised second law of thermodynamics. After also discussing the zeroth law, it becomes clear that this set of laws is rooted in semiclassical physics and give the hypotheses which are necessary for it to hold. My derivation of the laws of black hole thermodynamics also contributes towards the answer to the long-standing question of interpreting the Bekenstein-Hawking entropy. My work suggests that it is understood from the information perspective as accounting for the information hidden behind the horizon.

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We investigate the gravitational property of the quantum vacuum by treating its large energy density predicted by quantum field theory seriously and assuming that it does gravitate to obey the equivalence principle of general relativity. We find that the quantum vacuum would gravitate differently from what people previously thought. The consequence of this difference is an accelerating universe with a small Hubble expansion rate $H\propto \Lambda e^{-\beta\sqrt{G}\Lambda}\to 0$ instead of the previous prediction $H=\sqrt{8\pi G\rho^{vac}/3}\propto\sqrt{G}\Lambda^2\to\infty$ which was unbounded, as the high energy cutoff $\Lambda$ is taken to infinity. In this sense, at least the ``old'' cosmological constant problem would be resolved. Moreover, it gives the observed slow rate of the accelerating expansion as $\Lambda$ is taken to be some large value of the order of Planck energy or higher. This result suggests that there is no necessity to introduce the cosmological constant, which is required to be fine tuned to an accuracy of $10^{-120}$, or other forms of dark energy, which are required to have peculiar negative pressure, to explain the observed accelerating expansion of the Universe.

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In this thesis we present the first numerical study of gravitational collapse in braneworld within the framework of the single brane model by Randall-Sundrum (RSII). We directly show that the evolutions of sufficiently strong initial data configurations result in black holes (BHs) with finite extension into the bulk. The extension changes from sphere to pancake (or cigar, as seen from a different perspective) as the size of BH increases. We find preliminary evidences that BHs of the same size generated from distinct initial data profiles are geometrically indistinguishable. As such, a no-hair theorem of BH (uniqueness of BH solution) is suggested to hold in the RSII spacetimes studied in this thesis—these spacetimes are axisymmetric without angular momentum and non-gravitational charges. In particular, the BHs we obtained as the results of the dynamical system, are consistent with the ones previously obtained from a static vacuum system by Figueras and Wiseman. We also obtained some results in closed form without numerical computation such as the equality of ADM mass of the brane with the total mass of the braneworld.The calculation within the braneworld requires advances in the formalism of numerical relativity (NR). The regularity problem in previous numerical calculations in axisymmetric (and spherically symmetric) spacetimes, is actually associated with neither coordinate systems nor the machine pre- cision. The numerical calculation is regular in any coordinates, provided the fundamental variables (used in numerical calculations) are regular, and their asymptotic behaviours at the vicinity of the axis (or origin) are compatible with the finite difference scheme. The generalized harmonic (GH) formalism and the BSSN formalism for general relativity are developed to make them suitable for calculations in non-Cartesian coordinates under non-flat background. A conformal function of the metric is included into the GH formalism to simulate the braneworld.

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To investigate the possibility that an intrinsic form of gravitational decoherence can be theoretically demonstrated within canonical quantum gravity, we develop a model of a self-gravitating interferometer, and analyze the WKB regime of its reduced phase space quantization. We search for evidence in the resulting interference pattern that general relativity necessarily places limits on coherence, due to the inherent ambiguity associated with forming superpositions of geometries. We construct the "beam" of the interferometer out of WKB states for an infinitesimally thin shell of matter, and work in spherical symmetry to eliminate the occurrence of gravitational waves. For internal consistency, we encode information about the beam optics within the dynamics of the shell itself, by arranging an ideal fluid on the surface of the shell with an equation of state that enforces beam-splitting and reflections.The interferometric analysis is performed for single-mode inputs, and coherence is shown to be fully present regardless of gravitational self-interaction. Next we explore the role coordinate choices play in our description of the interferometer, by considering a family of generalized coordinate systems and their corresponding quantizations. Included in this family are the Painleve-Gullstrand coordinates, which are related to a network of infalling observers that are asymptotically at rest, and the Eddington-Finkelstein coordinates, which are related to a network of infalling observers that travel at the speed of light. We then introduce another model, obtained by adding to the shell a harmonic oscillator as an internal degree of freedom. The internal oscillator evolves with respect to the local proper time of the shell, and therefore serves as a clock that ticks differently depending on the shell's position and momentum. If we focus only on the external dynamics, we must trace out the clock degree of freedom, and this results in a form of intrinsic decoherence that shares some features with a recently-proposed "universal" decoherence mechanism attributed to gravitational time dilation. We discuss some variations of this proposal, and point out a way to bootstrap the gravitational contribution to the time dilation decoherence with self-gravitation. We interpret this as a fundamentally gravitational intrinsic decoherence effect.

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In this thesis, we present numerical studies of models for the accretion of fluids and magnetofluids onto rotating black holes. Specifically, we study three main scenarios, two of which treat accretion of an unmagnetized perfect fluid characterized by an internal energy sufficiently large that the rest-mass energy of the fluid can be ignored. We call this the ultrarelativistic limit, and use it to investigate accretion flows which are either axisymmetric or restricted to a thin disk. For the third scenario, we adopt the equations of ideal magnetohydrodyamics and consider axisymmetric solutions. In all cases, the black hole is assumed to be moving with fixed velocity through a fluid which has constant pressure and density at large distances. Because all of the simulated flows are highly nonlinear and supersonic, we use modern computational techniques capable of accurately dealing with extreme solution features such as shocks.In the axisymmetric ultrarelativistic case, we show that the accretion is described by steady-state solutions characterized by well-defined accretion rates which we compute, and are in reasonable agreement with previously reported results by Font and collaborators [1,2,3]. However, in contrast to this earlier work with moderate energy densities, where the computed solutions always had tail shocks, we find parameter settings for which the time-independent solutions contain bow shocks. For the ultrarelativistic thin-disk models, we find steady-state configurations with specific accretion rates and observe that the flows simultaneously develop both a tail shock and a bow shock. For the case of axisymmetric accretion using a magnetohydrodynamic perfect fluid, we align the magnetic field with the axis of symmetry. Preliminary results suggest that the resulting flows remain time-dependent at late times, although we cannot conclusively rule out the existence of steady-state solutions. Moreover, the flow morphology is different in the magnetic case: additional features are apparent that include an evacuated region near the symmetry axis and close to the black hole.

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##### Master's Student Supervision (2010 - 2020)

In 2007, based on stochastic gravity, Bei-Lok Hu and Albert Roura claimed that black hole fluctuations grow and eventually become important when a black hole has evaporated for a sufficiently long period of time but well before the Planckian regime is reached. In this thesis, we investigate their derivation of the black hole fluctuations and propose two analogue examples to better understand the mechanism of large fluctuations. Our analysis clearly shows the classical nature of the large black hole fluctuations. On the other hand, to test the validity of stochastic gravity, we calculate the centerpiece of stochastic gravity, the noise kernel, first for a perfect reflecting mirror and then for a more realistic mirror which becomes transparent at high frequencies. We find that one of the noise kernel components which corresponds to the fluctuations of energy flux is negative and thus seems to give "imaginary" fluctuations. We also perform calculations of both the fluctuations of forces acting on mirrors and of the fluctuations of energy flux radiated by them and divergent results are obtained. We try to give interpretations of these negative and divergent values. They need further investigation in the future.

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