Spacetime with zero point length is two-dimensional at the Planck scale
It is generally believed that any quantum theory of gravity should have a generic feature—a quantum of length. We provide a physical ansatz to obtain an effective non-local metric tensor starting from the standard metric tensor such that the spacetime acquires a zero-point-length (Formula presented.) of the order of the Planck length (Formula presented.). This prescription leads to several remarkable consequences. In particular, the Euclidean volume (Formula presented.) in a D-dimensional spacetime of a region of size (Formula presented.) scales as (Formula presented.) when (Formula presented.) , while it reduces to the standard result (Formula presented.) at large scales ((Formula presented.)). The appropriately defined effective dimension, (Formula presented.) , decreases continuously from (Formula presented.) (at (Formula presented.)) to (Formula presented.) (at (Formula presented.)). This suggests that the physical spacetime becomes essentially 2-dimensional near Planck scale.
Raychaudhuri equation with zero point length
The Raychaudhuri equation for a geodesic congruence in the presence of a zero-point length has been investigated. This is directly related to the small-scale structure of spacetime and possibly captures some quantum gravity effects. The existence of such a minimum distance between spacetime events modifies the associated metric structure and hence the expansion as well as its rate of change deviates from standard expectations. This holds true for any kind of geodesic congruences, including time-like and null geodesics. Interestingly, this construction works with generic spacetime geometry without any need of invoking any particular symmetry. In particular, inclusion of a zero-point length results into a non-vanishing cross-sectional area for the geodesic congruences even in the coincidence limit, thus avoiding formation of caustics. This will have implications for both time-like and null geodesic congruences, which may lead to avoidance of singularity formation in the quantum spacetime.
Topical collection: in memory of professor T Padmanabhan
Professor Thanu Padmanabhan contributed significantly to a broad spectrum of topics in astrophysics, cosmology, and classical and quantum aspects of gravitation. In his memory, the Editorial Board of General Relativity and Gravitation invited us to edit a Topical Collection “In Memory of Professor T Padmanabhan”. This collection gives a snapshot of ongoing research in the subjects he worked on.