Student Projects/Newtonian Relativity

You are walking by the riverbank; you notice a bird flying nearby a tree. Now, erase all the contents in the scenery but the bird. What would be your answer if someone asks, “where is the bird?”  Quite hard to tell. We can say that we don’t have a proper quantity or a reference with which we can locate the bird.

If you have some knowledge on the graphs, then this can be a little easier. Consider yourself to be at the origin, then, the bird is at some point at a particular time with coordinates (x, y, z) at time t. There is another person beside you who is also looking at the bird. The bird is at the same place but considering him to be the origin, the coordinates change (x’, y’, z’) at t’. These coordinates with time are the frames of references (i.e., the two of you are the references with respect to which the bird is located).

For now, we consider an inertial frame of reference i.e., you (who is not accelerating) and anything that is not accelerating with respect to you.

Starting at t = t’ = 0, let you and the other person be at the same point. So, you both see the bird at (x, y, z, t). Now, the other person starts to walk with a velocity v with respect to you, then the way he sees the bird changes. But how is this change related to you. How different is the position of the bird as seen by him than as seen by you?

Let (x’, y’, z’, t’) be the coordinates of the bird as seen by the other person. Then if you want to know the position of the bird being in his shoes, but not really being there, then you use this:

X’ = x – vt , as the person is moving in the x – direction only

Y’ = y

Z’ = z

And since the two of you look at the bird at the same time (assumption) , t’ = t

And the velocity can be given by

U_x^’ = u_x – v

U_y^’ = u_y

U_z^’ = u_z

Where the primed velocities are as observed by the other person.

These equations which helps you know what the other person sees in physics point of view, is the Galilean Transformation.

Here, the length, mass, time, change in the velocity, acceleration of the bird, are all independent of the relative motion of the observer, although different inertial observers record different velocities for the same bird.

This follows that ‘the laws of mechanics and equations of motions are the same in all inertial frames of references.’

Hence, all inertial frames are equivalent as far as mechanics is concerned. No mechanical experiments performed can be used to detect the absolute velocity of the frame through empty space. We can speak of only the relative velocity of the frame, not its absolute velocity.

Electromagnetism and Newtonian relativity

Will the Newtonian relativity work for electromagnetism as it does for the mechanics?

If it did, then there would be no preferred frame. But if we take an example of electromagnetic wave moving through a medium “ether “  ( consider ether to be an inertial frame of reference ) with velocity c , then in a frame of reference S’ moving with velocity v with respect to ether , the observer will measure the velocity of light range from v+c to v-c. Hence the speed of light is not invariant.

If the transformations really do apply here then there is one inertial system in which the measured speed of light is exactly c, i.e., there is a unique inertial system in which the so-called ether is at rest. Then we would have a physical way of identifying the absolute frame. But this is not possible in Galilean transformation. So, there is something going amiss in here.

The fact that Galilean transformation does not apply to the electromagnetism may be due to the following reason:

1.      The relativity principle exists for mechanics but not for electrodynamics, i.e., there is a preferred frame that can be identified.

2.      Relativity principle exists for both mechanics and electrodynamics, but the electrodynamics given by Maxwell is wrong.

3.      Relativity principle exists for both mechanics and electrodynamics, but the laws of mechanics given by newton is not correct everywhere, therefore there might be some other transformation that applies to both.

Is the first hypothesis true?

Many scientists wanted this to be true. They believed that there is a medium and without that medium light cannot travel. There are experiments that showed this assumption is wrong. Many still wanted to prove it right and said that ether exists.

MICHELSON MORLEY EXPERIMENT

This was one of the experiments that was conducted to prove the presence of ether (an absolute frame) but failed and gave us a new light.

The Michelson interferometer is fixed on the earth. If we imagine the ether to be fixed with respect to the sun, then earth moves through the ether with velocity 30km/sec (v), in different directions, in different seasons. The beam of laboratory source is split by the partially silvered mirror M, into two coherent beams, beam 1 being transmitted and beam 2 being reflected. Beams 1 and 2 return after reflecting from M1 and M2 respectively. The returning beam 1 and 2 are partially reflected and transmitted by mirror M respectively to telescope at T where they interfere (form dark and light bands – fringe system).

There exists a phase difference between beam 1 and 2 due to the different path length travelled, l₁ and l₂ and the different speeds of travel with respect to the instrument because of the “ether wind” ,v. The time for beam 1 to travel from M to M₁ and back is

$t_{1}=\left({\frac {l_{1}}{c-v}}\right)+\left({\frac {l_{1}}{c+v}}\right)=\left({\frac {2l_{1}}{c(1-\left({\frac {v^{2}}{c^{2}}}\right)}}\right)$

And for beam 2 to travel for, M to M₂ and back, is

$t_{2}=\left({\frac {2l_{2}}{c{\sqrt {1-\left({\frac {v^{2}}{c^{2}}}\right)}}}}\right)$

The difference in transit time is:

$\Delta t=t_{2}-t_{1}=\left({\frac {2}{c}}\right)(\left({\frac {l_{2}}{\sqrt {1-\left({\frac {v^{2}}{c^{2}}}\right)}}}\right)-\left({\frac {l_{1}}{(1-\left({\frac {v^{2}}{c^{2}}}\right)}}\right))$

Now the instrument is rotated through 90⁰ and we get,

$\Delta t'=t_{2}'-t_{1}'=\left({\frac {2}{c}}\right)(\left({\frac {l_{2}}{(1-\left({\frac {v^{2}}{c^{2}}}\right)}}\right)-\left({\frac {l_{1}}{\sqrt {1-\left({\frac {v^{2}}{c^{2}}}\right)}}}\right))$

The rotation changes the difference by,

$\Delta t'-\Delta t\simeq \left({\frac {l_{1}+l_{2}}{c}}\right)\left({\frac {v^{2}}{c^{2}}}\right)$

Therefore, rotation should cause a shift in the fringe pattern, since it changes the phase relationship between beams 1 and 2.

But, on theoretical calculations and so in observations, the change in fringes were none i.e., N is 0.

But the scientists believed that ether existed, and the experiment conducted was wrong, and tried to modify the experiment.

The interferometer was mounted on a massive stone slab for stability and floated the apparatus in mercury, to be rotated smoothly about the central pin, and the same experiment was conducted, again, there was no fringe shift at all.

We took v to be the earth’s velocity with respect to a fixed ether with respect to the sun. However, the solar system is itself in motion and the experiment determines the earth’s speed with respect to ether (if it exists) and it turns out to be 0. But the velocity cannot always be zero, as the velocity of apparatus changes from day to night and with different seasons, but such changes can be easily noted experimentally, but we didn’t find the shift.

One way to interpret this is to conclude that the measured speed of light Is the same for all directions in every inertial frame, which could lead to N = 0. Such a conclusion, being incompatible with Galilean transformations, seemed to be too drastic philosophically at that time. If measure of speed of light did not depend on the motion of the observer, all inertial systems would be equivalent for a propagation of light and there could be no experimental evidence to indicate the existence of a unique inertial system.

So, to save the ether and explain the Michelson Morley results, scientists suggested alternative hypothesis.

Which of the hypothesis mentioned above, do you think is right?

For further information refer “Introduction to special relativity” by “Robert Resnick”.