“She had studied the universe all her life, but had overlooked its clearest message: For small creatures such as we the vastness is bearable only through love.” ― Carl Sagan, "Contact"

# Epsilon Step at a Time

## Research Interests

"I have yet to see any problem, however complicated, which, when you looked at it in the right way, did not become still more complicated." -- Poul Anderson

Theories of Gravitation: Does General Relativity Need Modification?

Einstein's general relativity celebrated its centenary in 2015. Described by S. Chandrasekhar as "probably" the most beautiful theory, general relativity has passed all observational and experimental tests thus far with flying colors. To quote from the book The Perfect Theory by Pedro Ferreira published in 2014, "One thing about general relativity that has always puzzled me is how, despite being around for almost a century, it continues to yield new results."

Indeed, general relativity has a firm mathematical foundation based on Lorentzian geometry. The same cannot be said for many modified theories of gravity, which aim to explain the accelerated expansion of the Universe without including a cosmological constant (some also try to explain "dark matter" this way -- which is far more challenging). It is not difficult to propose a new theory of gravity, what is difficult is to propose a healthy theory. More often than not, attempts to modify general relativity resulted in pathological features such as energy not bounded from below (thus resulting in severe instability), wrong sign of kinetic energy (so-called "ghost"), ill-posed Cauchy problem (namely, given an initial condition, we cannot uniquely solve for the future evolution of the system -- which means that we cannot do physics) or even the unwelcomed existence of arbitrarily small closed timelike curves (thus violating causality).

I am interested in further understanding general relativity and appreciating its subtleties; I am also interested in the mathematical structures of modified gravity, and had spent some efforts in uncovering the problems that seem to plagued some teleparallel theories like f(T) gravity, as well as massive gravity. I am especially interested in theories with torsion. General relativity is, by construct, torsion-free, but a given connection has in general, in addition to curvature, also torsion and non-metricity. Geometries with these quantities are rich and interesting, and might offer some insights into gravitational physics. If indeed gravity is only the effect of spacetime curvature, then it would also be interesting to understand why Nature chooses not to make use of torsion and non-metricity. How rigid is general relativity? Ultimately, what is gravity? This brings us to the next topic...

Black Hole Thermodynamics, Singularities, Cosmic Censorship, and Gravitational Waves

A black hole is a region of spacetime with curvature behaving in such a way that nothing, not even light, can escape from within. The boundary of no return is called an "event horizon". (Note that gravity as measured by tidal deformation is not necessarily strong at the event horizon!). There are a lot we don't understand about black holes, both at the astrophysical level and the theoretical level. The former includes questions like: when did supermassive black holes first form in the Universe, and did they play any role in the reionization of the Universe? No doubt with the recent discoveries of gravitational waves by advanced LIGO, a new era of astrophysics has begun. I am, however, more interested in the theoretical aspects.

Black holes are thermodynamical objects -- they have temperature and entropy. The nature of black hole entropy, nevertheless, has remained mysterious -- what is the underlying degrees of freedom of a black hole? Is it some kind of "spacetime atom" or other microstructure? Why does adding electrical charge or increasing angular momentum make the entropy go down? In additional, can quantum gravity "cure" the singularity inside black holes? If so, was the same mechanism at work during the Big Bang? If so, how do we explain the arrow of time -- the fact that the very early Universe has a vastly lower entropy? How can we quantify gravitational entropy? All these questions are inherently related to the nature of curvature singularities and quantum gravity. In fact, we see evidence that cosmic censorship (Penrose's proposal that no naked singularity can form from a generic initial condition) remains relevant at the semi-classical level (including low energy limit of string theory), so it is not quite obvious that singularities can be resolved in the full quantum gravity theories. In fact -- if singularities are really resolved, then why is there such a bound at the classical level that we observe to hold in astrophysical black holes (see figure below)?

As I mentioned in an interview by Scientific American in August 2021, the importance of cosmic censorship (at least recently) is not so much in proving or disproving it, rather it is what we can learn along the way, what insights we can gain, what tools we can develop. The journey will be important, not just the destination.

In the coming years and decades, we will have more data on gravitational waves, which would hopefully provide more hints on new physics beyond general relativity, or to further constrain quantum gravitational models that give rise to corrections at the horizon scales.

Black Holes: Information Paradox, Quantum Information, and Holography

A notoriously difficult problem in theoretical physics is the so-called "information paradox": what happens to the information about the stuff that falls into a black hole? Since Hawking radiation makes a black hole smaller and smaller, and (possibly) eventually disappears, the fear is that information is lost, which seems to contradict a central tenet of quantum information ("unitarity"). There are many proposed resolutions to this paradox in the literature, but none seems convincing. The problem was made worse when it was claimed in 2012 that if information leaked out from a black hole by being entangled in the Hawking radiation, then the event horizon of a black hole at late time becomes a very high energy curtain of "firewall", completely contradicting our prior knowledge about black holes.

A cartoon illustration of gravitational waves as ripples of spacetime, produced by two black holes that are about to collide and merge. Take a moment to appreciate that this is produced in a vacuum spacetime -- all there is in spacetime devoid of matter, yet energy propagates.

(Credit: Swinburne Astronomy Productions)

An old black hole might be surrounded by a blazing firewall (Credit: Equinox Graphics/SPL)

I am interested in the properties of Hawking radiation for different black holes, and the information paradox. I have investigated a particular proposal by Harlow and Hayden, concerning the enormously long time required to decode Hawking radiation, and how this might evade firewalls. In addition, I have also written a comprehensive review on the remnant scenario -- the possibility that black holes eventually stop evaporating. As part of the effort to understand black hole evaporation, I have also been involved in the research of "moving mirrors" in (1+1)-dimensional flat spacetime (just quantum field theory), which serves as a toy model for an evaporating black hole.

Another important theoretical aspect of black hole physics is in the context of holography, which is also known as the AdS/CFT correspondence (though the term "holography" is arguably more general and thus more appropriate -- most applications are not strictly about CFT anyway). According to holography, the physics of a gravitating system in anti-de Sitter (AdS) spacetime is completely equivalent to another physical system -- a quantum field theory without gravity -- that lives on the boundary of the AdS spacetime. This opens a door to understand gravity using ordinary quantum field theory (even systems one can study in the lab, such as superconductor, cold atoms, and quark-gluon plasma) and vice versa.

Holography (Credit: Tom Brown)

I am interested in a few aspects of holography. Notably, being a nontrivial correspondence between two completely different physical systems, there must be some underlying consistency conditions to holography. Uncovering these are important, and might help us to understand how general holography is -- does it work for any mathematically/physically consistent theory of gravity in asymptotically AdS spacetimes, or does it require one to be able to embed the theory into string theory?

I presented as the co-chair of the parallel session on black hole information loss paradox during the 2nd LeCosPA International Symposium “Everything about Gravity”, which was held in National Taiwan University, Taipei, December 2015.

Cosmology: Physics at the Largest Scale

I have always been fascinated by the cosmos ever since I was a child. I still vividly remember my excitement when my father gave me a pair of binoculars as birthday present when I was nine years old. I especially enjoyed looking at the Pleiades cluster from the window of my bedroom then.

The Universe we live in is a remarkable place. For one thing, it is not only expanding but accelerating, and no one is quite sure why. Perhaps it is just a cosmological constant. Maybe it is the result of our over-simplification of cosmological models, and one really has to take inhomogeneities into account. Maybe it is something else entirely, a mysterious form of "dark energy". Maybe, after all, general relativity has to be modified, and the new theory will be able to explain the accelerated expansion in a "natural" manner. It is frustrating but at the same time exciting, that about 95% of the matter-energy content (so-called "dark sector") of the Universe remains unknown: all those remarkable things like planets and stars are nothing but 5% (of which a lot we still do not understand)! What rich physics await us in the dark sector? To quote Carl Sagan, “Somewhere, something incredible is waiting to be known.”

Also, as Einstein once remarked, "The most incomprehensible thing about the world is that it is comprehensible." Indeed, it is very impressive that the human species managed to figure out the big picture of the history of the Universe, in only a couple of centuries since the beginning of modern science. Of course, there are a lot more to understand, but what we do know are already quite impressive! We may be a young species in the grand scheme of things, but make no mistake, we are ambitious, we are curious, and we will be out there among the stars figuring the Universe out. One day!

Cosmic Epochs

(Credit: NASA, ESA, and A. Field (STScI))

From high precision observations, we know that the Universe is incredibly flat, and that its temperature distribution (of the Cosmic Microwave Background) is extremely uniform. These imply that the Universe started off in a very special initial condition. Not so surprising -- of course it makes sense for the initial condition to be relatively special, since entropy increases with time (the 2nd law of thermodynamics). Understanding why entropy is low in the beginning is a difficult but -- at least to me -- an important problem. Many would agree that cosmic inflation -- an epoch during which the Universe increased its size exponentially -- played a crucial role in this (though maybe only a partial role), but the underlying mechanism for inflation remains unknown. In fact, if we trace the history of the Universe far back enough, we would eventually reach a time when quantum effects can no longer be omitted, and we have to face the Big Bang singularity seriously. In the realm of theoretical cosmology, where my current interest lies, much work remain to be done.