Spacetime and Euclidean geometry

  title={Spacetime and Euclidean geometry},
  author={D. Brill and T. Jacobson},
  journal={General Relativity and Gravitation},
Using only the principle of relativity and Euclidean geometry we show in this pedagogical article that the square of proper time or length in a two-dimensional spacetime diagram is proportional to the Euclidean area of the corresponding causal domain. We use this relation to derive the Minkowski line element by two geometric proofs of the spacetime Pythagoras theorem. 

Figures from this paper

From Einstein's 1905 postulates to the geometry of flat space‐time
Minkowski diagrams in 1+1 dimensional flat space-time are given a strictly geometric derivation, directly from two gedanken experiments incorporating the principle of the constancy of the velocity ofExpand
Relativity on Rotated Graph Paper
It is shown that many quantitative results can be read off a spacetime diagram by counting boxes, using a minimal amount of algebra, in the spirit of the Bondi k-calculus. Expand
Alice and Bob and Hendrik
This paper offers an alternative approach to discussing both the principle of relativity and the derivation of the Lorentz transformations. This approach uses the idea that there may not be aExpand
Physics and the Pythagorean Theorem
Pythagoras' theorem lies at the heart of physics as well as mathematics, yet its historical origins are obscure. We highlight a purely pictorial, gestalt-like proof that may have originated duringExpand
Plane Geometry in Spacetime
Minkowski’s spacetime diagrams are extracted directly from Einstein’s 1905 postulates, using only some very elementary plane geometry. I have spent a significant part of my career looking at familiarExpand
The cissoid of Diocles in the Lorentz-Minkowski plane
This article presents the cissoid of Diocles and the cissoid of two circles with respect to origin in the Lorentz-Minkowski plane.


Black Holes and Wormholes in 2+1 Dimensions
Vacuum Einstein theory in three spacetime dimensions is locally trivial, but admits many solutions that are globally different, particularly if there is a negative cosmological constant. TheExpand
Mathematical and quantum aspects of relativity and cosmology : proceedings of the second Samos Meeting on Cosmology, Geometry and Relativity held at Pythagoreon, Samos, Greece, 31 August - 4 September 1998
Global Wave Maps on Curved Space Times.- Einstein's Equations and Equivalent Hyperbolic Dynamical Systems.- Generalized Bowen-York Initial Data.- The Reduced Hamiltonian of General Relativity and theExpand
Space–time intervals as light rectangles
Two inertial observers in relative motion must each see the other’s clock running at the same rate. The representation of this symmetry of the Doppler effect in a two-dimensional space–time diagramExpand
A Theory of Time and Space
THE appearance of Dr. Robb's treatise, if such a word can be applied to a volume which opens out a new field of philosophical inquiry on the basis of modern physical science, is a very welcome event,Expand
The Absolute Relations of Time and Space
The arrangement corrects color amplitude errors in a video playback system which utilizes a plurality of transducers for recovering recorded information in sequential segmented manner from a recordExpand
Black holes and wormholes in 2+1 dimensions, " in Mathematical and Quantum Aspects of Relativity and Cosmology
  • Lecture Notes in Physics
  • 2000