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Minkowski's principal tool is the Minkowski diagram, and he uses it to define concepts and demonstrate properties of Lorentz transformations (e.g., proper time and length contraction) and to provide geometrical interpretation to the generalization of Newtonian mechanics to relativistic mechanics.
A Minkowski diagram is a two-dimensional graphical depiction of a portion of Minkowski space, usually where space has been curtailed to a single dimension. The units of measurement in these diagrams are taken such that the light cone at an event consists of the lines of slope plus or minus one through that event. [ 3 ]
Commonly a Minkowski diagram is used to illustrate this property of Lorentz transformations. Elsewhere, an integral part of light cones is the region of spacetime outside the light cone at a given event (a point in spacetime). Events that are elsewhere from each other are mutually unobservable, and cannot be causally connected.
Minkowski diagram of the twin paradox. There is a difference between the trajectories of the twins: the trajectory of the ship is equally divided between two different inertial frames, while the Earth-based twin stays in the same inertial frame.
Penrose diagram of an infinite Minkowski universe, horizontal axis u, vertical axis v. In theoretical physics, a Penrose diagram (named after mathematical physicist Roger Penrose) is a two-dimensional diagram capturing the causal relations between different points in spacetime through a conformal treatment of infinity.
A world line is a special type of curve in spacetime. Below an equivalent definition will be explained: A world line is either a time-like or a null curve in spacetime. Each point of a world line is an event that can be labeled with the time and the spatial position of the object at that time.
Rindler chart, for = in equation (), plotted on a Minkowski diagram.The dashed lines are the Rindler horizons. The worldline of a body in hyperbolic motion having constant proper acceleration in the -direction as a function of proper time and rapidity can be given by [16]
In Minkowski space there is only one geodesic that connects any given pair of events, and for a time-like geodesic, this is the curve with the longest proper time between the two events. In curved spacetime, it is possible for a pair of widely separated events to have more than one time-like geodesic between them.