Towards resolving the black hole information paradox!

Based on the paper The black hole information puzzle and the quantum de Finetti theorem

Black holes are some of the most fascinating objects in the universe. They are extreme deformations of space and time, formed from the collapse of massive stars, with a gravitational pull so strong that nothing, not even light, can escape it. Apart from the astrophysical aspects of black holes (which are bizarre enough to warrant their own study), they provide the ideal theoretical laboratory for exploring various aspects of quantum gravity, the theory that seeks to unify the principles of general relativity with those of quantum mechanics.

One definitive way of making progress in this endeavor is to resolve the infamous black hole information paradox [1], and through a series of recent exciting developments, it appears that we might be closer to achieving this than we have ever been before [5, 6]! Paradoxes in physics paradoxically tend to be quite useful, in that they clarify what we don’t know about what we know. Stephen Hawking’s semi-classical calculations of black hole radiation treat the matter in and around black holes as quantum fields but describe them within the framework of classical general relativity. The corresponding results turn out to be in disagreement with the results obtained from a purely quantum theoretical viewpoint. The information paradox encapsulates this particular discrepancy. According to the calculations of Hawking and Bekenstein in the 1970s [3], black hole evaporation via Hawking radiation is completely thermal. This simply means that the von Neumann entropy S(R) of radiation (a measure of its thermality or our ignorance of the system) keeps growing with the number of radiation quanta, reaching a maximum when the black hole has evaporated completely. This corresponds to a complete loss of information, whereby even a pure quantum state entering a black hole, would be transformed into a mixed state of Hawking radiation and all previous information about it would be destroyed. This conclusion is in stark contrast to what one would expect when regarding the black hole from the outside, as a quantum system that must obey the laws of quantum mechanics. The fundamental tenets of quantum mechanics are determinism and reversibility, the combination of which asserts that all information must be conserved. Thus if a black hole is formed by collapsing matter in a pure state, the state of the total system including the radiation R must remain pure. This can only happen if the entropy S(R) that first increases during the radiation process, ultimately decreases to zero when the black hole has disappeared completely, corresponding to a final pure state [4]. This quantum mechanical result is depicted in the famous Page curve (Fig.1)

Certain significant discoveries in the recent past showing that the Page curve is indeed the correct curve and can in fact be reproduced by semi-classical approximations of gravitational path integrals [7, 8] may finally hold the key towards the resolution of this paradox. These calculations rely on the replica trick [9, 10] and take into account contribution from space-time geometries comprising of wormholes that connect various replica black holes. This simple geometric amendment to the gravitational path integral is the only factor different from Hawking’s calculations and yet leads to diametrically different results! The replica trick is a neat method that enables the computation of the entropy of the radiation field by first considering n identical copies of a black hole, calculating their Renyi entropies and using the fact that this equals the desired von Neumann entropy in the limit of n \rightarrow 1. These in turn are calculated using the semi-classical gravitational path integral under the assumption that the dominant contributions come from the geometries that are classical solutions to the gravity action, obeying Zn symmetry. This leads to two distinct solutions:

  • The Hawking saddle consisting of disconnected geometries (corresponding to identical copies of a black hole).
  • The replica wormholes geometry consisting of connections between the different replicas.

Upon taking the n \rightarrow 1 limit, one finds that all that was missing from Hawking’s calculations and the quantum compatible Page curve was the inclusion of the replica wormhole solution in the former.
In this paper, the authors attempt to find the reason for this discrepancy, where this extra information is stored and the physical relevance of the replica wormholes using insights from quantum information theory. Invoking the quantum de Finetti theorem [11, 12], they find that there exists a particular reference information, W and the entropy one assigns to the black hole radiation field depends on whether or not one has access to this reference. While Hawking’s calculations correspond to measuring the unconditional von Neumann entropy S(R), ignoring W, the novel calculations using the replica trick calculate the conditional von Neumann entropy S(R|W), which takes W into account. The former yields the entropy of the ensemble average of all possible radiation states, while the latter yields the ensemble average of the entropy of the same states. They also show that the replica wormholes are a geometric representation of the correlation that appears between the n black holes, mediated by W.
The precise interpretation of W and what it might appear to be to an observer falling into a black hole remains an open question. Exploring what it could represent in holographic theories, string theories and loop quantum gravity could open up a dizzying array of insights into black hole physics and the nature of space-time itself. It appears that the different pieces of the quantum gravity puzzle are slowly but surely coming together to what will hopefully soon give us the entire picture.


  1. The Entropy of Hawking Radiation
  2. Black holes as mirrors: Quantum information in random subsystems
  3. Particle creation by black holes
  4. Average entropy of a subsystem
  5. The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole
  6. Entanglement Wedge Reconstruction and the Information Paradox
  7. Replica wormholes and the black hole interior
  8. Replica Wormholes and the Entropy of Hawking Radiation
  9. Entanglement entropy and quantum field theory
  10. Generalized gravitational entropy
  11. Locally normal symmetric states and an analogue of de Finetti’s theorem
  12. Unknown quantum states: The quantum de Finetti representation

Can we measure black hole kicks using gravitational waves?

Article: Black hole kicks as new gravitational wave observables
Authors: Davide Gerosa, Christopher J. Moore
Reference: arXiv:1606.04226Phys. Rev. Lett. 117, 011101 (2016)

On September 14 2015, something really huge happened in physics: the first direct detection of gravitational waves happened. But measuring a single gravitational wave was never the goal—.though freaking cool in and of itself of course!  So what is the purpose of gravitational wave astronomy?

The idea is that gravitational waves can be used as another tool to learn more about our Universe and its components. Until the discovery of gravitational waves, observations in astrophysics and astronomy were limited to observations with telescopes and thus to electromagnetic radiation. Now a new era has started: the era of gravitational wave astronomy. And when the space-based eLISA observatory comes online, it will begin an era of gravitational wave cosmology. So what is it that we can learn from our universe from gravitational waves?

First of all, the first detection aka GW150914 was already super interesting:

  1. It was the first observation of a binary black hole system (with unexpected masses!).
  2. It put some strong constraints on the allowed deviations from Einstein’s theory of general relativity.

What is next? We hope to detect a neutron star orbiting a black hole or another neutron star.  This will allow us to learn more about the equation of state of neutron stars and thus their composition. But the authors in this paper suggest another exciting prospect: observing so-called black hole kicks using gravitational wave astronomy.

So, what is a black hole kick? When two black holes rotate around each other, they emit gravitational waves. In this process, they lose energy and therefore they get closer and closer together before finally merging to form a single black hole. However, generically the radiation is not the same in all directions and thus there is also a net emission of linear momentum. By conservation of momentum, when the black holes merge, the final remnant experiences a recoil in the opposite direction. Previous numerical studies have shown that non-spinning black holes ‘only’ have kicks of ∼ 170 km per second, but you can also have “superkicks” as high as ∼5000 km per second! These speeds can exceed the escape velocity of even the most massive galaxies and may thus eject black holes from their hosts. These dramatic events have some electromagnetic signatures, but also leave an imprint in the gravitational waveform that we detect.

Fig. 1: This graph shows two black holes rotating around each other (without any black hole kick) and finally merging during the final part of the inspiral phase followed by the very short merger and ringdown phase. The wave below is the gravitational waveform. [Figure from 1602.03837]
The idea is rather simple: as the system experiences a kick, its gravitational wave is Doppler shifted. This Doppler shift effects the frequency f in the way you would expect:

Doppler shift from black hole kick.

with v the kick velocity and n the unit vector in the direction from the observer to the black hole system (and c the speed of light). The black hole dynamics is entirely captured by the dimensionless number G f M/c3 with M the mass of the binary (and G Newton’s constant). So you can also model this shift in frequency by using the unkicked frequency fno kick and observing the Doppler shift into the mass. This is very convenient because this means that you can use all the current knowledge and results for the gravitational waveforms and just change the mass. Now the tricky part is that the velocity changes over time and this needs to be modelled more carefully.

A crude model would be to say that during the inspiral of the black holes (which is the long phase during which the two black holes rotate around each other – see figure 1), the emitted linear momentum is too small and the mass is unaffected by emission of linear momentum. During the final stages the black holes merge and the final remnant emits a gravitational wave with decreasing amplitude, which is called the ringdown phase. During this latter phase the velocity kick is important and one can relate the mass during inspiral Mi with the mass during the ringdown phase Mr simply by

Mass during ringdown related to mass during inspiral.

The results of doing this for a black hole kick moving away (or towards) us are shown in fig. 2: the wave gets redshifted (or blueshifted).

Fig. 2: If a black hole binary radiates isotropically, it does not experience any kick and the gravitational wave has the black waveform. However, if it experiences a kick along the line of sight, the waveform can get redshifted (when the system moves away from us) as shown on the left of blueshifted (when system moves toward us) as shown on the right. The top and lower panel correspond to the two independent polarizations of the gravitational wave.[Figure taken from this paper]
Fig. 2: If a black hole binary radiates isotropically, it does not experience any kick and the gravitational wave has the black waveform. However, if it experiences a kick along the line of sight, the waveform can get redshifted (when the system moves away from us) as shown on the left of blueshifted (when system moves toward us) as shown on the right. The top and lower panel correspond to the two independent polarizations of the gravitational wave. [Figure from 1606.04226]
This model is refined in various ways and the results show that it is unlikely that kicks will be measured by LIGO, as LIGO is optimized for detecting black hole with relatively low masses and black hole systems with low masses have velocity kicks that are too low to be detected. However, the prospects for eLISA are better for two reasons: (1) eLISA is designed to measure supermassive black hole binaries with masses in the range of 105 to 1010 solar masses, which can have much larger kicks (and thus are more easily detectable) and (2) the signal-to-noise ratio for eLISA is much higher giving better data. This study estimates about 6 detectable kicks per year. Thus, black hole (super)kicks might be detected in the next decade using gravitational wave astronomy. The future is bright 🙂

Further Reading