Vectors in Space-Time
The development of relativistic physics requires the use of a geometry adequate to account for the structure of spacetime. Whereas special relativity describes the non-quantum physical world in a nongravitational context in which spacetime is flat, the gravitational effects generated by general relativity must also account for the additional complexity of the curvature of spacetime.
Four-dimensional vectors thus take into account the special requirements of space-time geometry, i.e., the fact that space-time has four dimensions, as opposed to the three dimensions of Euclidean space. They accordingly have several applications in the context of relativistic physics. These applications include the use of translation vectors, which represent displacements in the four-dimensional continuum space-time model, the more abstract consequences of special relativity such as time dilation, length contraction or the addition of relativistic velocities, and the more elaborate descriptions of space and time by observers measuring time and space differently because they are moving relative to each other. In this case, the relevant quantities are assembled into 4-vector representations which transform linearly when their values are compared for the observers in different states, say, of uniform motion. A 4-vector is then a quantity with four components which changes like spacetime coordinates under a coordinate transformation. 4-vector representations are obtained using the same approach as the one used, for example to generate a 3-component vector xi from a set of three spatial coordinates:
Similarly, a 4-vector space-time coordinate for an eventcan then be written as:
By convention, Greek letters are used to specify space-time components and Roman letters are used for spatial components.
An example of the use of four-dimensional vectors in the spacetime context is provided by the calculation of relative speeds. If an observer A measures two objects B and C to be travelling at velocities u = (u, u, u) and v = (v, vy , v) respectively, the relative speed between B and C would be given in classical relativity by: w2 = (u-v)˙(u-v) = (u- v)2 + (u - v)2 + (u - v)2 .
In special relativity the relative speed is instead given by the formula:
w2 =[(u-v)˙(u-v) - (uXv)2 /c2 ]/ (1 - (u ˙v)/c2 )2 .
where u-v = (u - v, u - v , u - v) is the vector difference of u and v, u ˙v = u v + u v + u v is the inner product of u and v and uXv is the vector product for which (uXv)2 = (u˙u)(v˙v) - (u˙v)2 . And when u = u = v = 0, the formula reduces to: w = │u - v│/ (1 - u v/c2 ).
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