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Principle of relativity
Historically, the first principle of relativity that was formulated was a principle of relativity of uniform motion suggested by the observation that there doesn't seem to be a phenomenon in dynamics that will allow an observer to establish a zero point of velocity, nor a preferred direction.
Every choice of a zero point of velocity, a choice necessary in order to perform a calculation, constitutes a choice of reference frame. All reference frames that move with respect to each other with constant velocity and in a straight line are called inertial reference frames. The circularity of this definition is a necessity, since there is no preferred inertial reference frame.
In Galilean relativity, reference frames are related to each other in an intuitive way: to transform the velocity of an object from one frame to another, the vector representing the velocity of the object is added to the vector representing the velocity difference between the two reference frames. Such a transformation is called a Galilean transformation. The geometry of space is assumed to be Euclidian, and the measurement of time is assumed to be the same for all observers.
Another way of formulating the observation that there is no phenomenon in dynamics that will allow an observer to establish a zero point of uniform velocity, is to state that the laws of motion are equally valid in all inertial reference frames. For example the following property of motion: the common center of mass of two objects will move in uniform motion and it will also remain in uniform motion when the two objects collide or bounce against each other. This is valid in all inertial reference frames.
Einstein saw, as did his contemporaries, that if one assumes that both the Maxwell equations are valid, and that Galilean transformation is the appropriate transformation, then it should be possible to measure velocity absolutely. Einstein saw that if one assumes that the Lorentz transformations are the appropriate transformations for transforming between inertial reference frames, then that constitutes a principle of relativity that is compatible with the Maxwell equations.
Special relativity restored a principle of relativity in physics. The Maxwell equations had led to ether-theories, in which the nature of electrostatic and magnetic forces depend on the velocity of an object. Special relativity brought back the interpretation that in all inertial reference frames the same physics is going on. Thus there is no phenomenon that would allow an observer to pinpoint a zero point of velocity.
The assumption that the Lorentz transformations are the appropriate transformations has vast implications. The intuitive assumption that time is universal has to be relinquished.
When accelerated motion is involved, there are phenomena that will allow an observer to establish a zero point, there are phenomena that determine a preferred reference frame. For example: rotation. If a space station in orbit around Earth is not rotating then everything inside is weightless. If objects that are released tend to move away from an axis (of rotation) of the space-station then it is clear that it is rotating.
General relativity unifies the description of gravity and the description of inertia. General relativity is a theory of gravity that describes the properties of the mediator of gravitational interaction, in general relativity the mediator of gravitational interaction is deformation of space-time geometry. Gravity alters the rate of progression of time.
For an object to move inertially is to remain in the same Lorentz frame. To be accelerated by a force is to be accelerated with respect to the local Lorentz frame. If viewed sufficiently locally, the situations of being accelerated by a force and of gravity being opposed by a force are fundamentally the same physics. (To be precise: it is completely the same only in the limit of infinitisimally small volumes of space-time.) At larger size-scales this equivalence does not hold. The deformation of space-time due to gravity decreases in proportion to the inverse square of the distance to the center of gravity. When viewed from a sufficiently large distance the local deformation of space-time due to gravity is recognizable and thus can be accounted for as gravity in a description that is based on as much relevant data as possible.
Einstein's original aim was to show that a theory can be formulated in which the same type of relativity as in special relativity also holds for non-inertial motion. In the end, following where demands of consistency led him, Einstein formulated a theory that moves the descriptions of gravity and inertia to a deeper level, and unifies them. The name 'general relativity' reflects Einstein's original aim, not what it eventually turned out to be.
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