Abstract
We show that the diffeomorphisms, which preserve the null nature for a generic null metric very near to the null surface, provide noncommutative Heisenberg algebra. This is the generalization of the earlier work Majhi (2017) [21], done for the Rindler horizon. The present analysis revels that the algebra is very general as it is obtained for a generic null surface and is applicable for any spacetime horizon. Finally using these results, the entropy of the null surface is derived in the form of the Cardy formula. Our analysis is completely off-shell as no equation of motion is used. We believe present discussion can illuminate the paradigm of “gravity as an emergent phenomenon” and could be a candidate to probe the origin of gravitational entropy.
Highlights
The works of Bakenstein and Hawking [1,2,3] led to the conclusion that the black holes are the thermodynamic objects which has entropy, proportional to the surface area of the horizon
We have briefly described about the Gaussian null coordinates (GNC) in which we have made our whole analysis
As we have shown that one can formulate the non-commutative Heisenberg algebra for the metric in GNC, the result will hold for any null surface– including the horizon of a Kerr black hole or any other Rindler horizon as well
Summary
The works of Bakenstein and Hawking [1,2,3] led to the conclusion that the black holes are the thermodynamic objects which has entropy, proportional to the surface area of the horizon. We shall show that the algebra between the conserved charges, defined on some particular basis, is non-commutative These charges are generated due to the null horizon preserving symmetry. Present analysis does not use any information of gravitational field equations of motion and, the metric (1) in general may not be a solution of a particular gravity theory, i.e. we are considering the whole set of null surfaces. We call this as an off-shell analysis
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