Difference between revisions of "Minkowski space"
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| − | '''Minkowski space''' is 4-dimensional [[Euclidean space]] (i.e., the set of points <math>(x,y,z,t)</math>) together with the [[metric]]: | + | '''Minkowski space''' is 4-dimensional quasi-[[Euclidean space]] (i.e., the set of points <math>(x,y,z,t)</math>) together with the [[metric]]: |
<math> | <math> | ||
c^2dt^2-dx^2-dy^2-dz^2 | c^2dt^2-dx^2-dy^2-dz^2 | ||
</math> | </math> | ||
| + | |||
| + | It is called quasi-Euclidean because the metric coefficients for space and time are different, rather than always being positive for a true Euclidean space. Whether the time coefficient or the space coefficients should be the negative ones has been debated for a century. It doesn't matter in practice. The convention above (time is positive and space negative) was the one preferred by Einstein. | ||
A [[vector]] ''v'' in Minkowski space is said to be [[time]]-like if <math>||v||^2 > 0</math>, [[light]]-like if <math>||v||^2 = 0</math> and [[space]]-like if <math>||v||^2 < 0</math>. The [[isometry|isometries]] of Minkowski space are the [[Lorentz transformation]]s. | A [[vector]] ''v'' in Minkowski space is said to be [[time]]-like if <math>||v||^2 > 0</math>, [[light]]-like if <math>||v||^2 = 0</math> and [[space]]-like if <math>||v||^2 < 0</math>. The [[isometry|isometries]] of Minkowski space are the [[Lorentz transformation]]s. | ||
Latest revision as of 21:04, November 22, 2018
Minkowski space is 4-dimensional quasi-Euclidean space (i.e., the set of points <math>(x,y,z,t)</math>) together with the metric:
<math> c^2dt^2-dx^2-dy^2-dz^2 </math>
It is called quasi-Euclidean because the metric coefficients for space and time are different, rather than always being positive for a true Euclidean space. Whether the time coefficient or the space coefficients should be the negative ones has been debated for a century. It doesn't matter in practice. The convention above (time is positive and space negative) was the one preferred by Einstein.
A vector v in Minkowski space is said to be time-like if <math>||v||^2 > 0</math>, light-like if <math>||v||^2 = 0</math> and space-like if <math>||v||^2 < 0</math>. The isometries of Minkowski space are the Lorentz transformations.