- In econonometrcs, it arises when one or more of the explanatory variables is jointly determined with the dependent variable, typically through an equilibrium mechanism.
- In criminal law, for a criminal violation to be established, it usually must be shown that there was simultaneity of actus reus and mens rea.
- In mathematics, a system of equations or a set of simultaneous equations share variables; a solution is a set of variable values for which all these equations are satisfied together.
- In music, see Simultaneity (music) in Wikipedia.
- In physics, see Relativity of simultaneity (below).
- In marketing, simultaneity is one of the characteristics of a service which differentiates it from a product. It refers to the idea that the production and consumption of a service occur simultaneously, making it impossible to produce and store a service prior to consumption.
Relativity of simultaneityEvent B is simultaneous with A in the green reference frame, but it occurred before in the blue frame, and will occur later in the red frame.
A, B, and C occur in different order depending on the motion of the observer. The white line represents a plane of simultaneity being moved from the past to the future.
In physics, the relativity of simultaneity is the concept that simultaneity–whether two events occur at the same time–is not absolute, but depends on the observer’s reference frame.
According to the special theory of relativity, it is impossible to say in an absolute sense whether two distinct events, occur at the same time if those events are separated in space, such as a car crash in London and another in New York. The question of whether the events are simultaneous is relative: in some reference frames the two accidents may happen at the same time, in other frames (in a different state of motion relative to the events) the crash in London may occur first, and in still other frames the New York crash may occur first. If the two events are causally connected ("event A causes event B"), then the relativity of simultaneity preserves the causal order (i.e. "event A causes event B" in all frames of reference).
If we imagine one reference frame assigns precisely the same time to two events that are at different points in space, a reference frame that is moving relative to the first will generally assign different times to the two events. This is illustrated in the ladder paradox, a thought experiment which uses the example of a ladder moving at high speed through a garage.
A mathematical form of the relativity of simultaneity ("local time") was introduced by Hendrik Lorentz in 1892, and physically interpreted (to first order in v/c) as the result of a synchronization using light signals by Henri Poincare in 1900. However, both Lorentz and Poincaré based their conceptions on the aether as a preferred but undetectable frame of reference, and continued to distinguish between "true time" (in the aether) and "apparent" times for moving observers. It was Albert Einstein in 1905 who abandoned the (classical) aether and emphasized the significance of relativity of simultaneity to our understanding of space and time. He deduced the failure of absolute simultaneity from two stated assumptions:
- the principle of relativity -- the equivalence of inertial frames, such that the laws of physics apply equally in all inertial coordinate systems;
- the constancy of the speed of light detected in empty space, independent of the relative motion of its source.
In 1892 and 1895, Hendrik Lorentz used a mathematical tool called "local time" t' = t – v x/c2 for explaining the negative aether drift experiments. However, Lorentz gave no physical explanation of this effect. This was done by Henri Poincare who already in 1898 emphasized the conventional nature of simultaneity and who argued that it is convenient to postulate the constancy of the speed of light in all directions. However, this paper does not contain any discussion of Lorentz's theory or the possible difference in defining simultaneity for observers in different states of motion. This was done in 1900, when he derived local time by assuming that within the aether the speed of light is invariant. Due to the "Principle of relative motion" also moving observers within the aether assume that they are at rest and that the speed of light is constant in all directions (only to first order in v/c). So if they synchronize their clocks by using light signals, they will only consider the transit time for the signals, but not their motion in respect to the aether. So the moving clocks are not synchronous and do not indicate the "true" time. Poincaré calculated that this synchronization error corresponds to Lorentz's local time. Also in 1904 Poincaré emphasized the connection between the principle of relativity, "local time", and light speed invariance, however, the reasoning in that paper was presented in a qualitative and conjectural manner.
Albert Einstein 1905 used a similar method to derive the time transformation for all orders in v/c, i.e. the complete Lorentz transformation (also Poincaré got the full transformation in 1905 but in those papers he did not mention his synchronization procedure). This derivation was completely based on light speed invariance and the relativity principle, so Einstein noted that for the electrodynamics of moving bodies the aether is superfluous. Thus the separation into "true" and "local" times of Lorentz and Poincaré vanishes–all times are equally valid and therefore the relativity of length and time is a natural consequence. In 1908 Herman Minkowski used a quadratic form to present a primitive model of spacetime near a given event taken as the origin of a four-dimensional coordinate system. This model displays relativity of simultaneity through the concept of a simultaneous hyperplane that depends on the velocity of the traveler through the event.