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


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I want to give some formulas for the interested reader (12/10/20) and what graphs for time dilation and length contraction from 2014 below, moved from another webpage. Also an own visual simple to follow mathematical derivation of √(1 - v²/c²).

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). (23/10/20) Now I know in 2020 that if the time is going slower, you are seeing the passing light or materials from a faster going time slow downed, that's nature, an observer from that faster time can argue this, you in the slower time are that not aware. I had in 2014 the feelings that I could explain one and the other visually very well with my drawing below and so could derivate the formula of time dilation, I didn't succeed in 2014 (see below), but now after a few days thinking I succeeded, so the drawing is correct. So I am going to explain now what the observer sees and what the person sees who is in motion. But the standpoint is that the speed of light must be the same observed by everybody and that is c per (slower/faster) second, so the travelled path of light can be used as the converted local time. The observer sees that the person in motion, sees the quantity of passing light (from the faster time of the observer) decreased with (1 - v/c), because of his/her motion is a piece of passing light missing. So recalculated sees the person in motion a slower time t' = t . (1 - v/c). If you turn the direction of light, the person in motion would see a faster time t' = t . (1 + v/c). But the direction of light may not give a difference, so an observer must see on balance the same quantity of light passing for the person in motion, and the person in motion must see the same quantity independent of the direction of light. Therefore you must as observer first observe the balance of the quantity of passing light at the person in motion by letting go the light in 2 directions mirrored, in one direction that would be (1 - v/c) of the passing light, and in the other direction would be left the balance (1 + v/c) . (1 - v/c). If that person in motion would be observed in the opposite direction, would that balance being (1 - v/c) . (1 + v/c), so exactly the same, this balance can be written as (1 - v²/c²). The person in motion must see in both directions the same quantity of light and must be for the observer the same balance as already calculated. Mathematically seen this can be only the known time dilation factor √ (1 - v²/c²), the balance for the observer is also the same because that is √ (1 - v²/c²) . √ (1 - v²/c²) = (1 - v²/c²) (mathematically seen it looks a kind of constant inner product).
(04/01/20) This again commented. The light from the faster time of the observer is being mirrored, according the observer is lesser light on the way being observed by the person in motion and more from that light on the way back. Graphically (and also mathematically) is to prove (I don’t do this now) that the person in motion these two different factors sees as two equal factors on the way and way back from that light, this factor is being called the time dilation factor. That person in motion sees that light with the same velocity on the way and way back, but slower than the faster light from the observer. The above balance must not be seen as the total dilation (that are the separate and equal time dilation factors) but a balance what is being constant apparently on the argued way, for which I use the factor on the way back for the remaining light left on the way, but considered as a piece faster light from the observer.
(10/04/21) Also Einstein / Lorentz calculated an average time dilation, also there was the travelled path of light shorter on the way there than the way back. So what I argue here above is exactly the same.
(10/01/21) The reader of part (2) must skip the above mentioned, but for the meaning of this piece of theory, see extra explanation (technical proof in this case) at the bottom in part (1), at 4).


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 in location 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) x' = γ . x . (1 - v/c) and t = γ . t'. (1 - v/c) 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) t' = γ . t . (c - v)/c So the time is not going slower more, but totally smaller observed (also x = 1/γ . x' . c/(c + v) x' = γ . x . (c - v)/c).

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 2.4under):

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). (12/10/20) Normally show the Lorentz formulas for time and location, how a motion looks in a system in motion for as well an observer inside that system (who don't experience time dilation) as outside that system (so stationary), length contraction ignored for now. The principle for Lorentz is the constant speed of light under all circumstances. If one have a light wave in a system in motion, an observer inside that system sees that light wave as normally as when that system was not in motion, that observer observes no time dilation. But an observer outside that system in motion, so stationary, sees that light wave different, with another frequency (Doppler effect), so in this case of light both observers see not the same.

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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Correction text second graph: Light wave from system S2 observed in S1 (Doppler effect, S2 or light source going to the right)
Correction text third graph: Light wave from system S2 observed in S1 (Doppler effect, S2 or light source coming from the left), and v = .5c

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