are any of Euclid's 5 postulates false in Minkowski spacetime ?

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I often hear that Minkowski spacetime is non-euclidean. Euclidean geometry is characterized by Euclid's five postulates being true. Which of those postulates are untrue in Minkowski spacetime (if any), and what physical consequences do we observe from them?






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    I often hear that Minkowski spacetime is non-euclidean. Euclidean geometry is characterized by Euclid's five postulates being true. Which of those postulates are untrue in Minkowski spacetime (if any), and what physical consequences do we observe from them?






    special-relativity metric-tensor distance







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      I often hear that Minkowski spacetime is non-euclidean. Euclidean geometry is characterized by Euclid's five postulates being true. Which of those postulates are untrue in Minkowski spacetime (if any), and what physical consequences do we observe from them?






      special-relativity metric-tensor distance







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      I often hear that Minkowski spacetime is non-euclidean. Euclidean geometry is characterized by Euclid's five postulates being true. Which of those postulates are untrue in Minkowski spacetime (if any), and what physical consequences do we observe from them?






      special-relativity metric-tensor distance




      special-relativity metric-tensor distance






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      marjimbel

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          The Pythagorean distance formula doesn't hold for arbitrary shapes, thanks to the negative sign in the metric. It's also pretty easy to say that boosts obey hyperbolic angle addition rules rather than circular ones. Since the postulate about the congruency of right angles is needed to prove the Pythagorean distance relation, and angle addition rules for timelike intervals are different than those for spacelike intervals, one would conclude that the "all right angles are congruent" postulate doesn't hold-- the "right angle" between two null directions is different than that between two spacelike directions.






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            Minkowski space with the Minkowski metric has non-Euclidean geometry (as Jerry stated in his answer, it has hyperbolic properties). This means that Euclid's parallel postulate is violated: basically, if the parallel postulate does hold for a given geometry, then the sum of the angles of a triangle is $pi$ radians since the separation between parallel lines is constant. The geometry is non-Euclidean when the sum of the angles of a triangle is greater than (spherical) or less than (hyperbolic) $pi$ radians, since the separation between parallel lines increases or decreases, respectively.






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              The Pythagorean distance formula doesn't hold for arbitrary shapes, thanks to the negative sign in the metric. It's also pretty easy to say that boosts obey hyperbolic angle addition rules rather than circular ones. Since the postulate about the congruency of right angles is needed to prove the Pythagorean distance relation, and angle addition rules for timelike intervals are different than those for spacelike intervals, one would conclude that the "all right angles are congruent" postulate doesn't hold-- the "right angle" between two null directions is different than that between two spacelike directions.






              share|cite|improve this answer
























                up vote
                4
                down vote













                The Pythagorean distance formula doesn't hold for arbitrary shapes, thanks to the negative sign in the metric. It's also pretty easy to say that boosts obey hyperbolic angle addition rules rather than circular ones. Since the postulate about the congruency of right angles is needed to prove the Pythagorean distance relation, and angle addition rules for timelike intervals are different than those for spacelike intervals, one would conclude that the "all right angles are congruent" postulate doesn't hold-- the "right angle" between two null directions is different than that between two spacelike directions.






                share|cite|improve this answer






















                  up vote
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                  up vote
                  4
                  down vote









                  The Pythagorean distance formula doesn't hold for arbitrary shapes, thanks to the negative sign in the metric. It's also pretty easy to say that boosts obey hyperbolic angle addition rules rather than circular ones. Since the postulate about the congruency of right angles is needed to prove the Pythagorean distance relation, and angle addition rules for timelike intervals are different than those for spacelike intervals, one would conclude that the "all right angles are congruent" postulate doesn't hold-- the "right angle" between two null directions is different than that between two spacelike directions.






                  share|cite|improve this answer












                  The Pythagorean distance formula doesn't hold for arbitrary shapes, thanks to the negative sign in the metric. It's also pretty easy to say that boosts obey hyperbolic angle addition rules rather than circular ones. Since the postulate about the congruency of right angles is needed to prove the Pythagorean distance relation, and angle addition rules for timelike intervals are different than those for spacelike intervals, one would conclude that the "all right angles are congruent" postulate doesn't hold-- the "right angle" between two null directions is different than that between two spacelike directions.







                  share|cite|improve this answer












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                  answered 1 hour ago









                  Jerry Schirmer

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                  30.4k255102




















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                      Minkowski space with the Minkowski metric has non-Euclidean geometry (as Jerry stated in his answer, it has hyperbolic properties). This means that Euclid's parallel postulate is violated: basically, if the parallel postulate does hold for a given geometry, then the sum of the angles of a triangle is $pi$ radians since the separation between parallel lines is constant. The geometry is non-Euclidean when the sum of the angles of a triangle is greater than (spherical) or less than (hyperbolic) $pi$ radians, since the separation between parallel lines increases or decreases, respectively.






                      share|cite|improve this answer
























                        up vote
                        0
                        down vote













                        Minkowski space with the Minkowski metric has non-Euclidean geometry (as Jerry stated in his answer, it has hyperbolic properties). This means that Euclid's parallel postulate is violated: basically, if the parallel postulate does hold for a given geometry, then the sum of the angles of a triangle is $pi$ radians since the separation between parallel lines is constant. The geometry is non-Euclidean when the sum of the angles of a triangle is greater than (spherical) or less than (hyperbolic) $pi$ radians, since the separation between parallel lines increases or decreases, respectively.






                        share|cite|improve this answer






















                          up vote
                          0
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                          up vote
                          0
                          down vote









                          Minkowski space with the Minkowski metric has non-Euclidean geometry (as Jerry stated in his answer, it has hyperbolic properties). This means that Euclid's parallel postulate is violated: basically, if the parallel postulate does hold for a given geometry, then the sum of the angles of a triangle is $pi$ radians since the separation between parallel lines is constant. The geometry is non-Euclidean when the sum of the angles of a triangle is greater than (spherical) or less than (hyperbolic) $pi$ radians, since the separation between parallel lines increases or decreases, respectively.






                          share|cite|improve this answer












                          Minkowski space with the Minkowski metric has non-Euclidean geometry (as Jerry stated in his answer, it has hyperbolic properties). This means that Euclid's parallel postulate is violated: basically, if the parallel postulate does hold for a given geometry, then the sum of the angles of a triangle is $pi$ radians since the separation between parallel lines is constant. The geometry is non-Euclidean when the sum of the angles of a triangle is greater than (spherical) or less than (hyperbolic) $pi$ radians, since the separation between parallel lines increases or decreases, respectively.







                          share|cite|improve this answer












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                          answered 52 mins ago









                          N. Steinle

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