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The maximum speed of propagation of interaction, usually abbreviated in some manner, or signal velocity is the theoretical maximum speed by which a physical interaction travels. It is an important parameter in any mechanical theory, affecting the underlying physics in a profound and fundamental way. Historically, the signal velocity had been taken to be infinite in most if not all physical theories. However, such a position was observed to be in contradiction to the electromagnetic field equations as they had just been formulated in the late nineteenth century; the subsequent incorporation of the change in the signal velocity to a finite scalar, as demanded by the electromagnetic field equations, led to an epochal change in physics in the early twentieth century.

Definition
Suppose there exist two physical objects displaced by a non-zero finite distance, the first of whom may interact with the second in some way. If the physical state of the first object is then changed in some way that will affect the second, it is natural to ask what is the earliest point in time that the second object will effect those changes. The signal velocity is the displacement of the objects (between the position of the first object at the instant that its physical state was changed and the position of the second object at the instant it first experienced the interaction) divided by this time.

Importance
Suppose there exists a frame of reference in which an object that is not acted upon by an external influence moves uniformly and linearly. Such a frame is called an inertial frame of reference. Experiment confirms that all frames in relative motion to an inertial frame with a constant velocity are also inertial. Furthermore, the principle of relativity holds that all the laws of physics take the same form in every inertial frame. This means that any two inertial observers must be in complete agreement of the fundamental physics of a process.

Having now formulated this important principle of relativity, it is through it that the signal velocity acquires an immediate importance in physics. Using the principle of relativity, since the signal velocity is a physical parameter (i.e. a measurable quantity) that is fundamental to the description of the interaction being considered, it's value must be the same in all inertial reference frames. For if it were not, one observer's equations for the interaction would differ from another's.

Via the principle of relativity, again, the signal velocity also puts an important restriction on the kinematics of objects in inertial frames: namely, that all motion faster than the signal velocity is proscribed. This restriction is obvious since otherwise, by means of such a motion, one could either realize an interaction speed exceeding its maximum value — a logical contradiction — or else violate the principle of relativity by making physical outcomes dependent on the observer.

Non-inertial observers
In arbitrary reference frames, experiment still affirms the equivalence of the form of the laws of physics. Thus, the principle of relativity is applicable to all frames of reference, and is therefore redefined to expand its scope.

It should be noted very clearly that the proscription above does not say that the speed of an object as measured by an arbitrary observer can never be greater than the numerical value of the signal velocity in an inertial frame. For example, an observer on Earth (who is a non-inertial observer due to their angular motion about the Earth's axis) could decide to measure the speed of Rigel, the brightest star in the constellation Orion. Over the period of several hours the observer will have measured Rigel to have moved across the sky along an approximate circle. Since Rigel is approximately 10 quadrillion kilometres away (and lying approximately on the celestial equator), the observer would measure it to have traversed a circle of an approximate circumference of 60 quadrillion kilometres in 24 hours. This gives the star a much larger apparent speed than what they would measure to be the universal speed of propagation of interactions in an inertial frame, the speed of light in a vacuum. The observer is free to come to this conclusion without violating the proscription above since, whilst non-inertial observers may be able to record the motion of some objects to be faster than the signal velocity from their perspective, the object itself cannot travel travel faster than an interaction that is traveling in the same local neighbourhood of the object. In other words, an object may appear to a non-inertial observer to travel faster than the numerical value of the signal velocity as measured in an inertial frame, but pitted against a signal traveling in the same place as the object the signal will always travel faster. With the example of the observer watching Rigel, the observer would measure any light ray emitted from the star to travel a larger distance than the star in a small instant of time (the duration must be small since otherwise the light ray and the star are no longer local to one another). Thus, the star never moves faster than the light ray.

Non-relativistic Physics



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