The Two fundamental Postulates are,
1. The laws of Physics are same in all inertial reference frames.
2. The speed of light is always a constant independent of the motion of the observer.
As a student When I heard these postulates for the first time, I really never thought that there would be so much in it to understand. But the implications are quite significant to explain the Nature. Let us see, how it changes the universe...
The most important term is that "the speed of light". We know the velocity follow Galilean addition rules withing one frame to another frame of reference.
And we know how to make this Galilean transformation between two inertial reference frames.
Now, with surprise the postulate is given as the speed of light is constant in all reference frames, which means we cannot apply our previous Galilean transformations. It should be changed!!
But we are already using these transformation and getting best results. That means we can be sure that these laws can never proved to be completely wrong.
All it can be done is some alteration without affecting the known phenomena. For the mathematics, let us take a case..
As usual we take two inertial reference frames S and S' with relative speed "v". We want to keep the structure of Galilean equations at speed lower than c.
So, we assume that there is an extra factor comes into play with Galilean equations, which has the special property that it gets the value "1" when v<<c. Let us assume the factor is $ \gamma $
Rewriting the known Galilean transformation with the new factor "gamma".. $$ x' = \gamma (x - vt)..............eq.(1) $$ For convenience we always used to take the motion to be only in x-direction. And it will never going to change anything in y or z- direction. It is always y'=y and z'=z.
Inverse transformation similarly given as, $$ x = \gamma (x'+vt)..............eq.(2) $$ So far, we never used any information from the postulates.
Now from the second postulate, for a light wave speed is always the constant - c. Let us imagine that a light wave is allowed to propagate in the positive x direction, when the both reference frames starts from the common origin.
[ Why I am taking the case of Light wave?
We assume the Universe is same for everything else whether it is a light or a particle. As the postulate defined at light speed, if we could find the value of the gamma factor at light speed, it will eventually obey for all velocities. Intrinsically, we believe that the Universe won't have difference structure at different velocities.
Simply, it is finding the function by applying the boundary conditions.]
For the light wave, the equations become, x' = ct' and x = ct
[We now thrown away the concept of space and time being flat and universal to everyone]
Applying in eq.(1) and (2).. $$ ct' = \gamma (ct-vt)...............eq.(3)$$
and $$ ct = \gamma (ct'+vt')................eq.(4)$$ eq.(3) becomes $$ t' = \gamma \frac{(c-v)}{c}t $$ and eq.(4) becomes $$ t = \gamma \frac{(c+v)}{c} t' $$
Combining the (4) with (3) we can cancel the time variables..
$$ t = \gamma \frac {(c+v)}{c} ( \gamma \frac{(c-v)}{c}t ) $$
$$ t = \gamma^2 \frac{(c^2 - v^2)}{c^2} t $$
Cancelling the 't' on both sides..
$$ \gamma^2 = \frac{1}{1 - (v^2/c^2)}$$ Thus we finally get the value of "gamma" as $$ \gamma = \frac {1}{\sqrt{1-(v^2/c^2)}}$$
Remember, we took a special case at light speed and got the value for gamma - $\gamma$ factor. Now, we need to substitute this in the normal case i.e. eq.(1) and (2) which is given a new name - Lorentz Transformation equations. $$ x' = \frac{x - vt}{\sqrt{1 - (v^2/c^2)}} $$ and the inverse transformation is $$ x = \frac{x' + vt'}{\sqrt{1 - (v^2/c^2)}} $$
To derive the general transformation equation for time, substitute eq.(2) in eq.(1) $$ x' = \gamma ((\gamma (x' + vt')) - vt) \\~\\ x' = \gamma^2 (x' + vt') - \gamma vt $$ Taking t in one side and the remaining in the other gives..$$ \gamma vt = (\gamma^2 - 1) x' + \gamma^2 vt' $$ Finally we get t in terms of x' and t'.. $$ t = \frac{\gamma^2 x' (v^2/c^2)} {\gamma v} + \frac{\gamma^2 vt'}{\gamma v} \\~\\ t = \gamma (t' + (x'v/c^2) = \frac{t' + (x'v/c^2)}{\sqrt{1 - (v^2/c^2)}} $$
Similarly, the inverse tranformation can be obtained as, $$ t' = \frac {t - (xv/c^2)}{\sqrt{1 - (v^2/c^2)}} $$
I think.. we finished getting the most basic Lorentz transformation equations from one inertial frame of reference to other which moves at a relative speed "v".
From these transformation equations we may able to understand all those amazing, significant physical nature of Space and Time.
The most important term is that "the speed of light". We know the velocity follow Galilean addition rules withing one frame to another frame of reference.
And we know how to make this Galilean transformation between two inertial reference frames.
Now, with surprise the postulate is given as the speed of light is constant in all reference frames, which means we cannot apply our previous Galilean transformations. It should be changed!!
But we are already using these transformation and getting best results. That means we can be sure that these laws can never proved to be completely wrong.
All it can be done is some alteration without affecting the known phenomena. For the mathematics, let us take a case..
As usual we take two inertial reference frames S and S' with relative speed "v". We want to keep the structure of Galilean equations at speed lower than c.
So, we assume that there is an extra factor comes into play with Galilean equations, which has the special property that it gets the value "1" when v<<c. Let us assume the factor is $ \gamma $
Rewriting the known Galilean transformation with the new factor "gamma".. $$ x' = \gamma (x - vt)..............eq.(1) $$ For convenience we always used to take the motion to be only in x-direction. And it will never going to change anything in y or z- direction. It is always y'=y and z'=z.
Inverse transformation similarly given as, $$ x = \gamma (x'+vt)..............eq.(2) $$ So far, we never used any information from the postulates.
Now from the second postulate, for a light wave speed is always the constant - c. Let us imagine that a light wave is allowed to propagate in the positive x direction, when the both reference frames starts from the common origin.
[ Why I am taking the case of Light wave?
We assume the Universe is same for everything else whether it is a light or a particle. As the postulate defined at light speed, if we could find the value of the gamma factor at light speed, it will eventually obey for all velocities. Intrinsically, we believe that the Universe won't have difference structure at different velocities.
Simply, it is finding the function by applying the boundary conditions.]
For the light wave, the equations become, x' = ct' and x = ct
[We now thrown away the concept of space and time being flat and universal to everyone]
Applying in eq.(1) and (2).. $$ ct' = \gamma (ct-vt)...............eq.(3)$$
and $$ ct = \gamma (ct'+vt')................eq.(4)$$ eq.(3) becomes $$ t' = \gamma \frac{(c-v)}{c}t $$ and eq.(4) becomes $$ t = \gamma \frac{(c+v)}{c} t' $$
Combining the (4) with (3) we can cancel the time variables..
$$ t = \gamma \frac {(c+v)}{c} ( \gamma \frac{(c-v)}{c}t ) $$
$$ t = \gamma^2 \frac{(c^2 - v^2)}{c^2} t $$
Cancelling the 't' on both sides..
$$ \gamma^2 = \frac{1}{1 - (v^2/c^2)}$$ Thus we finally get the value of "gamma" as $$ \gamma = \frac {1}{\sqrt{1-(v^2/c^2)}}$$
Remember, we took a special case at light speed and got the value for gamma - $\gamma$ factor. Now, we need to substitute this in the normal case i.e. eq.(1) and (2) which is given a new name - Lorentz Transformation equations. $$ x' = \frac{x - vt}{\sqrt{1 - (v^2/c^2)}} $$ and the inverse transformation is $$ x = \frac{x' + vt'}{\sqrt{1 - (v^2/c^2)}} $$
To derive the general transformation equation for time, substitute eq.(2) in eq.(1) $$ x' = \gamma ((\gamma (x' + vt')) - vt) \\~\\ x' = \gamma^2 (x' + vt') - \gamma vt $$ Taking t in one side and the remaining in the other gives..$$ \gamma vt = (\gamma^2 - 1) x' + \gamma^2 vt' $$ Finally we get t in terms of x' and t'.. $$ t = \frac{\gamma^2 x' (v^2/c^2)} {\gamma v} + \frac{\gamma^2 vt'}{\gamma v} \\~\\ t = \gamma (t' + (x'v/c^2) = \frac{t' + (x'v/c^2)}{\sqrt{1 - (v^2/c^2)}} $$
Similarly, the inverse tranformation can be obtained as, $$ t' = \frac {t - (xv/c^2)}{\sqrt{1 - (v^2/c^2)}} $$
I think.. we finished getting the most basic Lorentz transformation equations from one inertial frame of reference to other which moves at a relative speed "v".
From these transformation equations we may able to understand all those amazing, significant physical nature of Space and Time.
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