Abstract
The effects on Raychaudhuri’s equation of an intrinsically-discrete or particle nature of spacetime are investigated. This is done through the consideration of null congruences emerging from, or converging to, a generic point of spacetime, i.e., in geometric circumstances somehow prototypical of singularity issues. We do this from an effective point of view, that is through a (continuous) description of spacetime modified to embody the existence of an intrinsic discreteness on the small scale, this adding to previous results for non-null congruences. Various expressions for the effective rate of change of expansion are derived. They in particular provide finite values for the limiting effective expansion and its rate of variation when approaching the focal point. Further, this results in a non-vanishing of the limiting cross-sectional area itself of the congruence.
Highlights
An effective metric, or qmetric, bitensor qab has been introduced [1,2,3], capable of implementing the existence of an intrinsic discreteness or particle nature of spacetime at the microscopic scale, while keeping the benefits of a continuous description for calculus [4]. qab acts like a metric in that it provides a squared distance between two generic spacelike or timelike separated events P and p, which approaches the squared distance as of an ordinary gab metric when P and p are far away
When these geodesics are meant as histories of ultrarelativistic or massless particles, we are led to singularity formation issues
Exploiting the fact that an affine parameter λ, assigned with a null geodesics γ, is a distance as measured along γ by suitable canonical observers parallelly-transported along it, the qmetric is introduced as something that leads to replacing λ(p, P) (having λ(P, P) = 0) with an effective parameterization [λ]q = λ (λ), which depends on the characterizing scale L
Summary
An effective metric, or qmetric, bitensor qab has been introduced [1,2,3], capable of implementing the existence of an intrinsic discreteness or particle nature of spacetime at the microscopic scale, while keeping the benefits of a continuous description for calculus [4]. qab acts like a metric in that it provides a (modified) squared distance between two generic spacelike or timelike separated events P and p (considered as the base and field point, respectively), which approaches the squared distance as of an ordinary gab metric when P and p are far away. In [5], an extension of this qmetric approach to include the case of null separated events has been considered, and an expression of qab for them has been provided.
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