Universality of Black Hole Entropy

A range of supermassive black holes lights up this new image from NASA’s NuSTAR. All of the dots are active black holes tucked inside the hearts of galaxies, with colors representing different energies of X-ray light.

It was not until the past few decades that physicists have made remarkable experimental advancements in the study of black holes, such as with the Event Horizon Telescope and the Laser Interferometer Gravitational-Wave Observatory.

On the theoretical side, there are still lingering questions regarding the thermodynamics of these objects.  It is well known that black holes have a simple formula for their entropy. It was first postulated by Jacob Bekenstein and Stephen Hawking  that the entropy is proportional to the area of its event horizon. The universality of this formula is quite impressive and has stood the test of time.

However, there is more to the story of black hole thermodynamics. Even though the entropy is proportional to its area, there are sub-leading terms that also contribute. Theoretical physicists like to focus on the logarithmic corrections to this formula and investigate whether it is just as universal as the leading term.

Examining a certain class of black holes in four dimensions, Hristov and Reys have shown such a universal result may exist. They focused on a set of spacetimes, that asymptote for large radial distance, to a negatively curved spacetime, called Anti-de Sitter.  These Anti-de Sitter spacetimes have been at the forefront of high energy theory due to the AdS/CFT correspondence.

Moreover, they found that the logarithmic term is proportional to its Euler Characteristic, a topologically invariant quantity, and a single dynamical coefficient, that depends on the spacetime background. Their work is a stepping stone in understanding the structure of the entropy for these asymptotically AdS black holes.

Strings 2021 – an overview

Strings 2021 Flyer

It was that time of year again when the entire string theory community comes together to discuss current research programs, the status of string theory and more recently, the social issues common in the field. This annual conference has been held in various countries but for the first time in its 35 year-long history has been hosted in Latin America at the ICTP South American Institute for Fundamental Research (ICTP-SAIFR).

One positive aspect of its virtual platform has been the increase in the number of participants attending the conference. Similar to Strings 2020 held in South Africa, more than two thousand participants were registered for the conference. In addition to research talks on technical subtopics, participants were involved in daily informal discussions on topics such as the black hole information paradox, ensemble averaging, and cosmology and string theory. More junior participants were involved in the poster sessions and gong shows, held in the first week of the conference.

One particular discussion session I would like to point out was panel discussion on the 4 generations of women in string theory, featuring women from different age groups and how they have dealt with issues of gender and implicit bias in their current or previous roles in academia.

To say the very least, the conference was a major success and has shown the effectiveness of virtual platforms for upcoming years, possibly including Strings 2022 to be held in Vienna.

For the string theory enthusiasts reading this, recordings of the conference can be found here.

String Dualities and Corrections to Gravity

Based on arXiv:2012.15677 [hep-th]

Figure 1: a torus is an example of a geometry that has T-duality

Physicists have been searching for ways to describe the interplay between gravity and quantum mechanics – quantum gravity – for the last century. The problem of finding a consistent theory of quantum gravity still looms physicists to this day. Fortunately, string theory is the most promising candidate for such a task. 

One of the strengths of string theory is that at low energies, the equations arising from string theory are shown to be precisely Einstein’s theory of general relativity. Let’s break down what this means. First, we must make sure we know the definition of a coupling constant. Theories of physics are typically described by some parameter that signifies the strength of the interaction. This parameter is called the coupling constant of that theory. According to quantum field theory, the value of the coupling constant depends on the energy. We often plot the logarithm of the energy and the coupling constant to understand how the theory behaves at a certain energy scale. The slope of this plot is called the beta function and when this function is zero, that point is called a fixed point. These fixed points are interesting since they imply that the quantum theory does not have any notion of scale.

Back to string theory, its coupling constant is called α′ (said as alpha-prime). At weak coupling, when α′ is small, we can similarly find the beta function for string theory. At the quantum level, string theory must have a vanishing beta function. At the corresponding fixed point, we find that the Einstein’s equations of motion emerge. This is quite remarkable!

We can go even further. Due to the smallness of α′, we can expand the beta function perturbatively. All the subleading terms in α′, which are infinite in number, are considered to be corrections to general relativity. Therefore, we can understand how general relativity is modified via string theory. It becomes technically challenging to compute these corrections and little is known about what the full expansion looks like.

Fortunately for physicists, string theories are interesting in other ways that could help figure us out these corrections to gravity. Particularly, the string energy spectrum that has radii R and radii α′/R look exactly the same. This relation is called T-duality. An example of a geometry that has this duality is the torus, see Figure 1. Because we know that certain dualities for strings must hold, we can use this to guess what the higher order correction must look like. Codina, Hohm and Marques took advantage of this idea to find corrections to the third power of α′. Using a simple scenario where the graviton is the only field in the theory, they were able to predict what the corrections must be.

This method can be applied at higher orders in α′ as well as a theory with more fields than the graviton, although technical challenges still arise. Due to the structure of how T-duality was used, the authors can also use their results to study cosmological models. Finally, the theory result confirms that string theory should be T-duality at all orders of α′.


Inside the Hologram

This is from arXiv:2009.04476 [hep-th] Inside the Hologram: Reconstructing the bulk observer’s experience


Figure 1 (adapted from arXiv:2009.04476 [hep-th]):  The setup showing the reference system coupled to a black hole

The holographic principle in high energy physics has been proved to be quite rigorous. It has allowed physicists to understand phenomena that were too computationally difficult beforehand. The principle states that a theory describing gravity is related to a quantum theory. The gravity theory is typically called “the bulk” and the quantum theory is called “the boundary theory,” because it lives on the boundary of the gravity theory. Because the quantum theory is on the boundary, it is one dimension less than the gravity theory. Like a dictionary, physical concepts from gravity can be translated into physical concepts in the quantum theory. This idea of holography was introduced a few decades ago and a plethora of work has stemmed from it.

Much of the work regarding holography has been studied as seen from an asymptotic frame. This is a frame of reference that is “far away” from what we are studying, i.e. somewhere at the boundary where the quantum theory lives.

However, there still remains an open question. Instead of an asymptotic frame of reference, what about an internal reference frame? This is an observer inside the gravity theory, i.e. close to what we are studying. It seems that we do not have a quantum theory framework for describing physics for these reference frames. The authors of this paper explore this idea as they answer the question: how can we describe the quantum physics for an internal reference frame?

For classical physics, the usual observer is a probe particle. Since the authors want to understand the quantum aspects of the observer, they choose to have the observer be made up of a black hole that is entangled with a reference system. One way to see that black holes have quantum behavior is by studying Stephen Hawking’s work. Particularly, he showed that black hole can emit thermal radiation. This prediction is now known as Hawking radiation.

The authors proceed to measure the proper time between and energy distribution of the observer. Moreover, the researchers propose a new perspective on “time,” relating the notion of time in General Relativity to the notion of time in the quantum mechanics point of view, which may be valid outside the scope of holography.

The results have proven to be quite novel as it fills some of the gaps we have about our knowledge of holography. It is also a step towards understanding what the notion of time means in holographic gravitational systems.