Abstract
A recent attempt to arrive at a quantum version of Raychaudhuri’s equation is looked at critically. It is shown that the method, and even the idea, has some inherent problems. The issues are pointed out here. We have also shown that it is possible to salvage the method in some limited domain of applicability. Although no generality can be claimed, a quantum version of the equation should be useful in the context of ascertaining the existence of a singularity in the quantum regime. The equation presented in the present work holds for arbitrary n+1 dimensions. An important feature of the Hamiltonian in the operator form is that it admits a self-adjoint extension quite generally. Thus, the conservation of probability is ensured.
Highlights
General relativity is the most successful theory of gravity so far, the existence of a singularity in a classical spacetime is inevitable in this theory as demonstrated by the Penrose-Hawking singularity theorems [1,2]
Physical laws break down and the spacetime geometry is pathological at a singularity
A general expectation is that quantum effects which come into the picture in strong gravity regime may alleviate this problem
Summary
General relativity is the most successful theory of gravity so far, the existence of a singularity in a classical spacetime is inevitable in this theory as demonstrated by the Penrose-Hawking singularity theorems [1,2]. The Raychaudhuri equation has been useful in showing the avoidance of singularities in loop quantum cosmology [26,27,28,29] This is possible due to the existence of repulsive terms which arise due to quantum effects. We arrive at an equation which is valid for an arbitrary dimension This is done with the help of a more rigorous approach by using the well-known Helmholtz conditions [37,38,39,40,41,42] regarding the Lagrangian formulation of a problem. The perspective is to find a quantum version of geodesic flow equation primarily following basic canonical approach very similar to the formulation of Wheeler–DeWitt quantization scheme. Some possible arena for applications are discussed in the final section
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