failure of Galilean transformation btech 1 year
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The failure of the Galilean Transformation
Maxwell's equations, which summarise electricity and magnetism, cause the Galilean Transformation to fail on two counts ....
They predict the speed of light is independent of the inertial reference frames instead of ($c'=c+v$) as required by Galilean Relativity.
They are not invariant under the Galilean Transformation. (This is stated without proof in this course.)
More sophisticated experiments (specifically, experiments on the behaviour of light and experiments that dealt with fast moving particles) indicated that Galilean Relativity was approximately correct only for velocities much smaller than the speed of light.
What a conundrum !
Shall we throw out all the theories of Electro-magnetsim ? This is hard to do. If these theories seem to have no flaw in their derivation, are firmly based in
experiments, their predictions are verified by further experiments, and we all use our cell phones with impunity, then its hard to fault Maxwell's equations !
The problem must lie somewhere else ....
But where ?
Enter Special Relativity, which was first developed by Einstein (1905). This theory treats inertial reference frames in a way
that is compatible with all measurements so far. Later on, came General Relativity which is able to deal with non-inertial reference frames and also to provide a geometrical way of dealing with gravity.
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Answer:
Suppose there are two reference frames (systems) designated by S and S' such that the co-ordinate axes are parallel (as in figure 1). In S, we have the co-ordinates $\{x,y,z,t\}$ and in S' we have the co-ordinates $\{x',y',z',t'\}$. S' is moving with respect to S with velocity $v$ (as measured in S) in the $x$ direction. The clocks in both systems were synchronised at time $t=0$ and they run at the same rate.
Figure 1: Reference frame S' moves with velocity $v$ (in the x direction) relative to reference frame S.
\includegraphics[width=0.7\textwidth]{ref_frames.eps}
We have the intuitive relationships
$\displaystyle x'$ $\textstyle =$ $\displaystyle x-vt$
$\displaystyle y'$ $\textstyle =$ $\displaystyle y$
$\displaystyle z'$ $\textstyle =$ $\displaystyle z$
$\displaystyle t'$ $\textstyle =$ $\displaystyle t$
This set of equations is known as the Galilean Transformation. They enable us to relate a measurement in one inertial reference frame to another. For example, suppose we measure the velocity of a vehicle moving in the in $x$-direction in system S, and we want to know what would be the velocity of the vehicle in S'.
\begin{displaymath}
v_x' = \frac{dx'}{dt'} = \frac{d(x-vt)}{dt} = v_x-v
\end{displaymath} (1)
This is the result our intuition is familiar with.
We have stated the we would like the laws of physics to be the same in all inertial reference frames, as this is indeed our experience of nature. Physically, we should be able to perform the same experiments in different reference frames, and find always the same physical laws. Mathematically, these laws are expressed by equations. So, we should be able to ``transform'' our equations from one inertial reference frame to the other inertial reference frame, and always find the same answer.
Suppose we wanted to check that Newton's Second Law is the same in two different reference frames. (We know from experiment that this is the case.) We put one observer in the un-primed frame, and the other in the primed frame, moving with velocity $v$ relative to the un-primed frame. Consider the vehicle of the previous case undergoing a constant acceleration in the $x$-direction,
$\displaystyle f' = m'a'$ $\textstyle =$ $\displaystyle m'\frac{d^2x'}{dt'^2}$
Explanation: