"Lorentzian distance," is the geometry of [[spacetime]] in the theory of relativity, particularly under [[Lorentz transformations]]. ### Lorentzian Geometry In the context of relativity, "Lorentzian" pertains to the [[Lorentz transformations]] and [[Lorentzian metrics]]. The Lorentz transformation is a linear transformation that describes how the measurements of space and time by two observers are related to each other in special relativity. It reflects how time and space are not absolute but are perceived differently by observers in different states of motion. ### Lorentzian Metric The Lorentzian metric (or Lorentz metric) is used in general relativity and is a way of measuring distances that accounts for the speed of light as a constant and the maximum speed. This metric is fundamental in describing spacetime in general relativity. The metric itself is defined by having one time dimension with a negative coefficient in the metric tensor, which differentiates it from Euclidean metrics used in classical physics. ### Distance in Lorentzian Geometry In Lorentzian geometry, the notion of distance differs from traditional Euclidean geometry. For two points in spacetime, the interval $s^2$ calculated using the Lorentzian metric can be: - **Time-like**: $s^2>0$, meaning the separation between events can be bridged by an object moving slower than the speed of light. - **Light-like**: $s^2=0$, meaning the separation between events is exactly such that light can travel from one to the other. - **Space-like**: $s^2<0$, meaning no object, not even light, can travel fast enough to bridge the events, thus they are not causally related. In this context, the "distance" can refer to how events or points relate to each other through time and space under these [[causality]]-sensitive conditions, but it doesn't conform to the everyday notion of spatial distance as measured in Euclidean terms. ### Conclusion [[Lorentzian distance]] is pivotal in understanding how spacetime is structured differently from the more intuitive three-dimensional space governed by [[Euclidean geometry]]. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Lorentzian Distance") or contains(subject, "Lorentzian metrics") sort modified desc, authors, title ```