Minimum-length Ricci scalar for null separated events

Abstract

We consider spacetime endowed with a zero-point length, i.e. with an effective metric structure which allows for a (quantum-mechanically arising) finite distance $L_0$ between events in the limit of their coincidence. Restricting attention to null separated events, we find an expression for the Ricci (bi)scalar in this zero-point-length metric; this is done for when geometric circumstances are such that the collection of all null geodesics emerging from a point $P$ has all the information needed to fix the value of scalar curvature at $P$. Taking then the coincidence and further $L_0 \to 0$ limits, we find that this expression does not reduce to the Ricci scalar $R$ of the ordinary metric but to $(D-1) R_{ab} l^a l^b$ in $D$-dimensional spacetime ($D \ge 4$), where $R_{ab}$ and $l^a$ are the ordinary Ricci tensor and tangent vector to the null geodesics. This adds nicely to the existing results for time and space separations. This finding seems to give further support to the view that the quantity $R_{ab} l^a l^b$, ubiquitous in horizon thermodynamics, embodies something which remains as a relic/remnant/memory of a quantum underlying structure for spacetime in the limit of (actual detectability of) this quantumness fading away, and which as such should enter the scene when aiming to derive/motivate the field equations. Further, it turns out to be the same quantity used in an existing derivation of field equations from a thermodynamic variational principle, thus adding further evidence of an origin as quantum-spacetime relic for the latter.