length Contraction

An important implication of time dilation is that length in one inertial frame of reference (A) measured by an observer in another inertial frame of reference (B) moving relative to it is less than the length which would be measured by an observer actually in frame A. This can be seen most simply by considering a radioactive particle travelling in a fixed frame of reference, for example a laboratory. An observer in the laboratory will see it travel a certain distance before it decays. This distance is simply its speed multiplied by its lifetime us measured in the laboratory which, remember, is longer than its lifetime if it were at rest. However, an observer travelling with the particle sees the same span of laboratory pass at the speed of the particle but, since the particle is at rest relative to him/her, its lifetime is shorter than that measured in the laboratory. In turn, the distance this observer sees covered in the laboratory is also shorter. Thus, length in the laboratory as measured by the moving observer is less than length measured by the fixed observer length contraction has taken place because of the motion. Distance, like time, is not absolute-it depends on the frame of reference from which it is measured, but, again, as with time this is a relativistic effect and plays no part in the nature of mechanics and motion unless the entitities concerned are travelling with speeds comparable to the speed of light. Clearly, however, if these effects are taken into account, then the associated mathematical theory will differ from that describing non-relativistic phenomena. Nevertheless, it should be expected that the mathematics describing motion should revert back to classical form when dealing with low-speed phenomena. To illustrate this consider the way in which speeds were added. Here a problem arose when adding the speed of the car to the speed of light since it gave a value greater than the speed of light in the frame of reference fixed to the ground which is forbidden by relativity theory. It turns out that the relativistic formula for the addition of two speeds u and v is rather different from the classical formula. It is

total speed = (U + v )/(1 + uv/c²).

If u and v are much smaller than c (the speed of light) then the second term in the denominator can be neglected and we revert back to the simple expression u + v for the total speed. At the other extreme, if v, for example, is equal to c, then it requires no great algebraic skill* to see that the total speed is simply c as

*Total speed = (U + c)/(l + UC/C*) = (U + c)/(l + d c ) = c(u + c)/(u + c) = c.

required. Whatever speed is added to c, the resultant speed is still c.

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