On the Nature of Discrete Space-Time: The distance formula, relativistic time dilation and length contraction in discrete space-time

David Thomas Crouse, Joseph Skufca


In this work, the relativistic phenomena of Lorentz-Fitzgerald contraction and time dilation are derived using a modified distance formula that is appropriate for discrete space.  This new distance formula is different than the Pythagorean theorem but converges to it for distances large relative to the Planck length.  First, four candidate formulas developed by different people over the last 70 years will be discussed.  Three of the formulas are shown to be identical for conditions that best describe discrete space.  It is shown that this new distance formula is valid for all size-scales - from the Planck length upwards - and solves two major problems historically associated with the discrete space-time (DST) model. One problem it solves is the widely believed anisotropic nature of most discrete space models.  Just as commonly believed is the second problem - the incompatibility of DST's concept of an immutable atom of space and the length contraction of this atom required by special relativity.  The new formula for distance in DST solves this problem. It is shown that length contraction of the atom of space does not occur for any relative velocity of two reference frames.  It is also shown that time dilation of the atom of time does not occur.  Also discussed is the possibility of any object being able to travel at the speed of light for specific temporal durations given by an equation derived in this work. Also discussed is a method to empirical verify the discreteness of space by studying any observed anomalies in the motion of astronomical bodies, such as differences in the bodies' inertial masses and gravitational masses. The importance of the new distance formula for causal set theory and other theories of quantum gravity is also discussed. 


Ashcroft, N. W., Mermin, N. D. (1976). pp 151, ‘Solid State Physics’. Saunders, Philadelphia.

Bergson, H. (1926), ‘Duré et simultanéité: à propos de la théorie d'Einstein’.

Brightwell, Graham, and Ruth Gregory,’Structure of random discrete spacetime’, Physical review letters, 66(3), 260.

Canales, J. (2015), The physicist and the philosopher: Einstein, Bergson, and the debate that changed our understanding of time, Princeton University Press.

Carroll, Sean (2006). 2006C-SPAN broadcast of Cosmology at Yearly Kos Science Panel, Part 1.

Collins, J., Perez, A., Sudarsky, D., Urrutia, L., Vucetich, H. (2004), ‘Lorentz invariance and quantum gravity: an additional fine-tuning problem?’, Physical review letters, 93(19), 191301.

Crouse, David T. (2016a), ‘The lattice world, quantum foam and the universe as a metamaterial’, Applied Physics A, 122(4), 472.

Crouse, David T. (2016b), ‘On the nature of discrete space-time: The atomic theory of space-time and its effects on Pythagoras's theorem, time versus duration, inertial anomalies of astronomical bodies, and special relativity at the Planck scale’, arXiv preprint arXiv:1608.08506.

Cullity, B.D. and S.R Stock, (2001), Elements of X-ray Diffraction, 3rd Edn., Prentice-Hall Inc., New Jersey.

Davies, Paul, (2006), The Goldilocks Enigma, First Mariner Books.

Eberhard, Ph H., and W. Schommers. ‘Quantum theory and pictures of reality'’. in Quantum Theory and Pictures of Reality. (1989), 232.

Finkelstein D. (1969). ‘Space-time code’, Physical Review, 184(5), 1261.

Forrest, P. (1995), ‘Is space-time discrete or continuous? -- An empirical question’, Synthese, 103(3), 327-354.

Ghosh, Amitabha (2000). Origin of inertia: extended Mach's Principle and cosmological consequences, Affiliated East-West Press.

Gudder, S. (2017). ‘Reconditioning in discrete quantum field theory’, International Journal of Theoretical Physics, 1-14.

Hagar, A. (2014), Discrete or continuous?: the quest for fundamental length in modern physics, Cambridge University Press.

Helliwell, T. M. (2010), Special relativity, Univ Science Books.

Henson, J. (2008), ‘The causal set approach to quantum gravity’, In D. Oriti, editor, Approaches to Quantum Gravity: Towards a New Understanding of Space and Time, Cambridge University Press.

Kaye, Richard (2015), ‘Triangle Inequality’, Retrieved from http://web.mat.bham.ac.uk/R.W.Kaye/seqser/triangleineq.html

Kragh, H., Carazza, B. (1994), ‘From time atoms to space-time quantization: the idea of discrete time, ca 1925-1936’, Studies in History and Philosophy of Science Part A, 25(3), 437-462.

Krause, E. F. (2012), Taxicab geometry: An adventure in non-Euclidean geometry, Courier Corporation.

Maimonides, Moses. (1190), The Guide for the Perplexed, Print.

March, Arthur. (1936), Zeitschrift für Physik, 104, 161-168.

Milonni, Peter W. The quantum vacuum: an introduction to quantum electrodynamics, Academic press, 2013.

Misner, C.W., K.S. Thorne, and J.A. Wheeler. (1973), Gravitation, Freeman. San Francisco, p. 1190-1194.

Myrheim, J. (1978), ‘Statistical geometry’. No. CERN-TH-2538.

Ng, Y., Dam, H. V. (1995), ‘Limitation to Quantum Measurements of Space-Time Distances’, Annals of the New York Academy of Sciences, 755(1), 579-584.

Pullin, J., Gambini, R. (2011), A First Course in Loop Quantum Gravity, Oxford University Press.

Reid, David D. (1999), ‘Introduction to causal sets: an alternate view of spacetime structure’, arXiv preprint gr-qc/9909075.

Rideout, David, and Petros Wallden. ‘Spacelike distance from discrete causal order’, Classical and Quantum Gravity, 26(15), 155013.

Riggs, S. (2009), The Origin of The Planck Length, Planck Mass and Planck Time: A New Candidate For Dark Matter, CreateSpace Independent Publishing Platform.

Rovelli, C., Speziale, S. (2003), ‘Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction’, Physical Review D, 67(6), 064019.

Schild, E.g. A., ‘Discrete Space-Time and Integral Lorentz Transformations’, Canadian Journal of Mathematics, 1 (1949), 29-47.

Sciama, D.W. (1969), ‘The Physical Foundations of General Relativity’, Doubleday US.

Sorkin, R. D. (1983), ‘On the Entropy of the Vacuum Outside a Horizon’, in Proceedings of 10th Int. Conf. on General Relativity and Gravitation, Padua Italy, 734-736.

Sorkin, Rafael D., ‘Causal sets: Discrete gravity’, In Lectures on quantum gravity, pp. 305-327. Springer US, 2005. Harvard

't Hooft, Gerard. ‘Quantum gravity: a fundamental problem and some radical ideas’, Recent Developments in Gravitation, Cargèse 1978, 323-345.

Van Bendegem, J. P. (1987), ‘Zeno's Paradoxes and the Tile Argument’, Philosophy of Science, 54(2), 295-302.

Van Bendegem, J. P. (1995), ‘In defence of discrete space and time’, Logique et Analyse, 38(150/152), 127-150.

Van Bendegem, Jean Paul,’Finitism in Geometry’, The Stanford Encyclopedia of Philosophy (Spring 2017 Edition), Edward N. Zalta (ed.), URL = .

Van Bendegem, J. P. (2000) How to tell the continuous from the discrete. In: François Beets & Eric Gillet (réd.), Logique en Perspective. Mélanges offerts à Paul Gochet. Brussels: Ousia, 2000, pp. 501-511.

Vasileska, Dragica, ‘Emperical Pseudopotential Method: Theory and Implementation Details’, URL = .

Weyl, H., (1949), Philosophy of Mathematics and Natural Sciences, Princeton University Press, Princeton.

Wheeler, J. A. (1957), ‘On the nature of quantum geometrodynamics’, Annals of Physics, 2(6), 604-614.


  • There are currently no refbacks.