by Parampreet Singh.

Parampreet Singh with a young student who often asks him the most difficult and so far unanswerable questions on the resolution of singularities. Dr Parampreet Singh is Associate Professor at Department of Physics and Astronomy at Louisiana State University.

Einstein’s theory of classical general relativity breaks down when spacetime curvature
becomes extremely large near the singularities. To answer the fundamental questions
about the origin of our Universe or what happens at the central singularity of the black holes thus lies beyond the validity of Einstein’s theory. Our research deals with discovering the framework which guarantees resolution of singularities.

It has been long expected that quantum gravitational effects tame the classical singularities leading to insights on the above questions. A final theory of quantum gravity is not yet there but the underlying techniques can be used to understand whether quantum gravitational effects resolve cosmological and black hole singularities. Our goal is to answer such questions about singularity resolution using techniques of loop quantum gravity.

Sahil Saini is a graduate student at Louisiana State University working with Dr. Singh on proving various aspects of singularity resolution in loop quantum cosmology.

About a decade ago, one of the very first rigorous quantizations of a cosmological spacetime in loop quantum cosmology was performed. Along with my collaborators, we were able to show that non-perturbative quantum gravitational effects as understood in loop quantum gravity replace the classical big bang with a big bounce [1]. The primary reason deals with the properties of quantum geometry as found in loop quantum gravity. Kinematical operators such as area of a surface or volume of a region have discrete eigenvalues. Classical differential geometry is replaced by a discrete quantum geometry. Classical continuum spacetime emerges when spacetime curvature becomes small compared to Planck scale. However, near the Planck scale the physics of quantum spacetime is very different from the classical spacetime. Quantum geometric discreteness effectively makes gravity repulsive, causing a collapsing spacetime approaching an inevitable singularity to bounce.

These early results on singularity resolution opened an avenue for extensive studies on singularity resolution in loop quantum cosmology. A variety of spacetimes and phenomenological scenarios were investigated with increasing complexity. The basic recipe was simple. Consider your favorite matter model, apply techniques of loop quantum cosmology and using numerical simulations show that classical singularity is resolved. No fine tuning needed. However, it was found that for a certain exotic equation of state a singularity exists in loop quantum cosmology [2]. For such cases, quantum gravity bounded the energy density and the Hubble rate, yet there existed divergences of spacetime curvature. The myth that quantum gravity effects in loop quantum cosmology always bind the spacetime curvature was broken.

The resolution lies in noting the strength of the singularity. In their seminal works in the classical theory, Tipler, Krolak, and others provided a classification of singularities in terms of tidal forces [3]. Singularities such as the big bang and central singularity in black holes are strong singularities. No detector, irrespective of its characteristics, can escape such a singularity. But there are also weak singularities, such as sudden singularities in certain cosmological spacetimes. These are examples of the weak singularities. If one is able to construct a sufficiently strong detector, it just passes unharmed through the weak singularity. These events allow geodesics to be extended. In a way, weak singularities are not really the singularities to worry about even classically.

Soon after the result on this lone singularity in loop quantum cosmology, I showed that for the spatially flat isotropic and homogeneous spacetimes singularities can exist in loop quantum cosmology but they are only weak singularities [4]. The singularity found earlier in loop quantum cosmology, belonged to this class. An interesting feature of quantum gravity was being deciphered. All strong singularities were eliminated but the weak ones were spared. Quantum gravitational effects completely ignored them.

To understand genericness of singularity resolution in this homogeneous setting, these results needed to be generalized in two directions: inclusion of spatial curvature and of anisotropies. The generalization to isotropic models with spatial curvature was straightforward with similar results [5]. In contrast, the generalization to anisotropic spacetimes is tedious and till recently was performed for Bianchi-I spacetime where almost identical results appear [6]. All strong singularities are eliminated, and all weak ones are generally ignored for a wide class of matter content. However, interestingly it was recently  found that  a very mild non-existent weak singularity also gets resolved in a flat vacuum Bianchi-I spacetime [7].

The complexities increase when spatial curvature and anisotropies are present together, such as in the Kantowski-Sachs spacetimes which in absence of matter capture the Schwarzschild interior, or the Bianchi-IX model. The reason is tied to certain technical aspects of quantization due to richness of interplay of intrinsic curvature and anisotropic shear. The associated difficulties for long did not allow existing proofs for singularity resolution for the isotropic [4] and the Bianchi-I spacetime [6] to be extended further. In a work with my graduate student, Sahil Saini, which has recently appeared in CQG we have provided this crucial breakthrough for the Kantowski-Sachs spacetime with arbitrary matter content. We find that no strong singularities can exist in a  finite proper time evolution. Only weak ones can exist — which are harmless. The proof we present not only can be constructed for the isotropic and the Bianchi-I spacetime, providing an alternative to the previous ones [4,6], but is a promising approach for proving a generic resolution of singularities for all the homogeneous spacetimes in loop quantum cosmology.

This is just a beginning towards proving generic resolution of singularities. So far the results have been found in the homogeneous setting. The big challenge is to include inhomogeneities. The hope is that similar to the case in Einstein’s theory where results on singularities were first proved for homogeneous isotropic and anisotropic spacetimes, before the celebrated singularity theorems, perhaps the above results on the resolution of singularities provide the building blocks of a non-singularity theorem in quantum gravity.

[1] Ashtekar A., Pawlowski T. and Singh, P., “Quantum Nature of the Big Bang,”   Phys. Rev. Lett. 96, 141301 (2006)

[2] Cailleteau T., Cardoso A., Vandersloot K. and Wands D., “Singularities in loop quantum cosmology,” Phys. Rev. Lett. 101, 251302 (2008)

[3] Tipler F. J., “Singularities in conformally flat spacetimes,” Phys. Lett. 64A, 8 (1977);  Krolak A., “Towards the proof of the cosmic censorship hypothesis,” Class. Quant. Grav. 3, 267 (1986)

[4] Singh P., “Are loop quantum cosmos never singular?” Class. Quant. Grav. 26, 125005 (2009)

[5] Singh P., and Vidotto F., “Exotic singularities and spatially curved Loop Quantum Cosmology,” Phys. Rev. D 83, 064027 (2011)

[6] Singh P., “Curvature Curvature invariants, geodesics and the strength of singularities in Bianchi-I loop quantum cosmology,” Phys. Rev. D 85, 104011 (2012)

[7] Singh P., “Is classical flat Kasner spacetime flat in quantum gravity?”,  Int. J. Mod. Phys. D, 25, 1642001 (2016)

Read the full article in Classical and Quantum Gravity:
Geodesic completeness and the lack of strong singularities in effective loop quantum Kantowski–Sachs spacetime
Saini & Singh 2016 Class. Quantum Grav. 33 245019

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