Explanation of Einstein's Theory of Relativity - some formulas for the interested reader o.a. light


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I want to give some formulas for the interested reader.

Without Lorentz one could calculate (predict) that a person/object with a constant speed v in rest frame A, the time t' would experience in his own rest frame as t' = t . ( 1 - v/c), in which t is the time in rest frame A. This formula is not right (see Lorentz), but predicts already a time dilation. See time diagram below. Probably one looses a little piece of time during the own motion (in B lesser passing light measured from source in A, and must give c again with taken clock).

light

With the Lorentz formulas times and locations from rest frame B for an object, can be converted to rest frame A for an observator, in which the object has a constant speed v.

The Lorentz formulas for location and time are: x' = γ . (x - v. t) en t' = γ . (t - v.x/c²) for which γ = 1 / √ (1 - v²/c²). Locations x and x' can be made visual by applying a coordinate system in the rest frame from the observator. For making it simple now, I don't consider the y and z coordinates.

Suppose a person/object moves with a constant speed v in a rest frame A from the observator. With the Lorentz formulas one find the time t' in that locations for which the person/object is in rest, t' = 1 / γ . t
This means the observator sees all times from the person/object in rest frame A as t' = 1 / γ . t and all locations as x' = 0 (in locations for which the person is not in rest, the observer sees the length, here width, in the moving direction of the system shorter, called length contraction, the distance between 2 coordinates at the same time, height stay equal). Length contraction is a consequence from 2 different times in 1 direction, normal time + slower time. Perpendicular on the direction of motion of the system is no length contraction, only slower time (time dilation).

Suppose now that the moving object is a light source which emitted light. The formulas for locations and time of the light wave (how light looks like) are x = γ . x' . (1 - v/c) and t = γ . t'. (1 - v/c) for the observer standing still. This describes the relativistic Doppler effect of light (if a light source moves, the frequency of light changes, and so its color, think at the red and blue blinking stars which we see). So the observer experiences the Doppler effect. The path of light is totally smaller observed (compare with length contraction for materials), besides for the same time dilation the frequency can be different. Time dilation is in each system in motion everywhere the same, but the progress of time and location can be different observed for each object in motion. So too if you look to the progress of time for the Doppler Effect. Different written is this here t = 1/γ . t' . c/(c + v). So the time is not going slower more, but totally smaller observed (also x = 1/γ . x' . c/(c + v)).

Now to calculate this really for 1 wave of light, how you would see this standing at the right location, as follows (graphical see first link under 2.4):

at v = .5c become the points 0 and 2π, 0 and 1.1547π, so totally smaller observed (-2π and 0 become -1.1547π and 0).

It may be clear now that each location experiences its own time, so for a location must also be specified the time t as a coordinate near the x,y,z, coordinates. So with Lorentz a location (t,x,y,z) will be converted to (t',x',y',z').

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