Research
Here's an overview of the topics I've worked on, or am currently working on - click a heading to read more.
My PhD
Stay tuned for updates on my PhD research.
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Quantum Black Holes
Learn more about my Part III essay on quantum properties of near-extremal black holes.
Black holes are fascinating objects, not least because they seem to be so featureless. If we trust Einstein’s theory of relativity, a black hole can fully be characterised by just a few numbers, for example the angular momentum, mass, or electric charge of the black hole. This story changes once we bring quantum mechanics into the picture. For example, quantum physics predicts that black holes radiate at a so-called Hawking temperature. In my Part III essay, I gave an overview of the physics of low-temperature charged black holes and discussed the exciting and subtle quantum effects that are important in this regime.
The central tool in this case is the gravitational path integral. It allows us to calculate various physical quantities of interest perturbatively via the expansion
Z = ∫ Dgμν DAμ e−S
≈ ( det Ŝquadratic )−1/2 e−Sclassical
Two famous formulae about black holes follow directly from the tree-level path integral. Requiring the Euclidean solution to be smooth fixes the black hole temperature, while evaluating the action on the classical solution gives its entropy — following the method by Gibbons and Hawking, one finds
T = κ 2π SBH = A 4
One can push this strategy beyond tree level. At the next order in perturbation theory, one must calculate the eigenvalues of the differential operator appearing in the action. The surprising result: an infinite number of zero modes appears at exactly zero temperature. Moving slightly away from extremality lifts these modes, giving each a small eigenvalue proportional to the temperature, and it is precisely these lifted modes that make the one-loop contribution to the entropy logarithmic,
ΔSBH ∼ log T
which dominates the familiar area law at low temperatures. The reason for this behaviour is the special AdS2 × S2 near-horizon geometry of the black hole.
What does this mean for the existence and physicality of the zero temperature limit? If you are curious about the wondrous world of quantum black holes, you can download my essay here.
The essay also contains references and further resources. If you have any questions or just want to have a chat, please reach out.
Quantum Cosmology
Find out about my Bachelor's thesis on Quantum Cosmology.
Quantum Cosmology is an exciting branch of theoretical physics that applies the principles of quantum mechanics to the universe as a whole. It seeks to explain the beginning and early evolution of the universe in a way that incorporates both quantum theory and general relativity, which makes it an important testbed for candidate theories of quantum gravity.
The quantum state of the universe can be described by a wave function ψ[h, φ] that depends on the metric and matter configuration we observe at a given time. This wave function is known to satisfy the so-called Wheeler–DeWitt equation
Ĥ ψ = 0,
an analogue of the Schrödinger equation. However, this differential equation has an infinite number of solutions, so that approach requires an additional input of initial conditions to find the wave function.
A different proposal to find the wave function from a path integral was put forward by Hartle and Hawking. They suggested determining ψ by evaluating a Euclidean path integral over all compact geometries without a boundary in the past:
ψ[h, φ] ∝ ∫(h,φ) Dg Dφ e−S[g,φ]
This so-called “no boundary proposal” thus describes the creation of a self-contained universe from “nothing”. This is illustrated with the shuttlecock geometry:
Authors such as Linde and Vilenkin have put forward their own proposals for calculating the wave function from a suitable path integral. Even though they often obtain similar results to Hartle and Hawking, their wave functions disagree in one important prediction with the No-Boundary proposal: they predict a vastly different value for the inflaton field at the beginning of inflation.
More recently, Feldbrugge, Lehners and Turok (building on work of Halliwell and Louko) popularised a method to evaluate the Lorentzian path integral directly using Picard–Lefschetz theory. Depending on the choice of contour in the complex plane, they obtained wave functions of both “Hartle–Hawking-type” and “Linde–Vilenkin-type” from their model.
In my thesis, I reviewed the different approaches to calculating the wave function of the universe and explored the connection between the different results. One important part of the discussion was to relate the arising wave functions to the solutions of the Wheeler–DeWitt equation, and to analyse their behaviour for small values of the scale factor. A summary of the thesis can be found in the slides of a presentation I gave on the topic.
The slides also contain references and resources for further reading. If you have any questions or would like to discuss the topic, don’t hesitate to reach out.