hello
lets be clear im a relativity beleiver
but recently i had a problem with generalizing the calculations of the special relativity

when ever they explain time dialation and length contraction the take this example of the two observers one in a train and the other on the platform
and they take the example of emmiting a light signal vertically up and bouncing of a mirror and bieng detected at a detector right beside where it was emitted

but whats if we recalculate it in a case where the light signal is emitted horizontally (in the same direction of the trains motion)
i mean it was emitted from a light source at the end of the train compartment and is detected at the other end of the compartment

now for the observer inside the train the light signal has traveled a distance D at the speed C
so the time he measures to=D/C
and that is a proper time

for the outside observer
the light has traveled the length of the compartment plus the distance that the front end of the compartment has traveled during the time light has took to reach it
that means that according to what the platform observer sees the compartment length as D/GAMMA
and he sees that the front end of the compartment has traveled V*t
then he sees that the light signal has traveled D/GAMA+V*t

then the time he measures will be t=(D/GAMMA+V*t)/C
substituting for D=C*to
t=(C*to/GAMMA+V*t)/C
t=to/GAMMA+V*t/C
by separation of variables
t(1-V/C)=to/GAMMA

now according to what we know
t=GAMMA*to
so whats wrong with my calcualtions

ps: please i ask relativity non believers to stay clear ok!!!

now according to what we know
t=GAMMA*to
so whats wrong with my calcualtions

t = γ t₀ is valid only if t₀ is the difference in time of two events that occur at the same place. The situation between the two times t and t₀ is not symmetric.
In order to convince yourself take two events that occur in reference frame S₀ at the same place. The difference in time will be t₀.
In a reference frame S₁, which has a velocity v₁ with respect to S₀, the difference in time between these two events will be t₁ = γ₁t₀. If you have another reference frame S₂ that has a velocity v₂ with respect to S₀, the difference in time between the events will be given by t₂ = γ₂t₀. However you will not have t₁ = γ₁₂t₂ or t₂ = γ₁₂t₁.
In order to find the relation between the time differences of two events that do not occur at the same place, you must use the Lorentz transformation. I remind you that the Lorentz transformation of the time component, also depend on the position of the event.
In your example, the two events (one is the emmission of the light and the other is the arriving of the light at the front of the compartment), do not occur at the same place in neither reference frame that you have chosen.

nightwing,

Hopefully 1100f answered your question. This (http://casa.colorado.edu/~ajsh/sr/timebig.html) excellent link about time dilation might help too.

i did work on it and came up with the correct naswer andill post it here for revesion soon