Spacetime and Euclidean geometry

@article{Brill2004SpacetimeAE,
  title={Spacetime and Euclidean geometry},
  author={D. Brill and T. Jacobson},
  journal={General Relativity and Gravitation},
  year={2004},
  volume={38},
  pages={643-651}
}
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. 

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References

SHOWING 1-10 OF 11 REFERENCES
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
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