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THE LIGHT BEAM RIDER
 
 

Now, for a ray of light going away in the form of sinusoidal waves, we can choose l = (x-ct), by observing the particular x which is l far away from ct. So, l˘ = x˘ - ct˘ = l[Ö{(c+v)/(c-v)}]. Hence, n˘ = n[Ö{(c-v)/(c+v)}]. This is called relativistic Doppler’s effect. The notion of relative velocity also changes in relativity. From Lorentz transformation,

                                x˘/t˘ = (x-vt)/(t-vx/c2)

Put x/t = u, velocity of a particle measured in frame L. We get u˘, the velocity of the same particle measured in frame L˘, as:

u˘ = (u-v)/(1-uv/c2)

These relations are sufficient to take a very different form. In his papers, Einstein gave another important result, the energy-mass relativity.

Mass transforms as:

                                m ® m0/{Ö(1-v2/c2)}

We can easily derive the relation for energy from it:

                                E = m0c2/{Ö(1-v2/c2)}

by using dE = Fdx = (dP/dt)dx = vdP and p = m0v/{Ö(1-v2/c2)}. 

The quantity m0c2 is looked at as the potential energy of the particle. So, kinetic energy = E - m0c2 = m0c2[1/{Ö(1-v2/c2)} – 1].

For approximation, kinetic energy = m0c2[1 + (1/2)(v2/c2) – 1] = (1/2)m0v2. 

These quantities - momentum and energy also follow transformation rules:

                                p ® (p-vE)/ /{Ö(1-v2/c2)}

                                E˘ ® (E-vp)/{Ö(1-v2/c2)}

And, they follow the relation:

                                p2 = E2/c2 = constant,

                                                where, constant = -m02c2 (as we see when velocity = 0).

This implies that E2 = p2c2 + m02c4.

Now, look at the relation x˘2 - c2t˘2 = x2 - c2t2. A quantity x2 - c2t2, which is a function of x and t, i.e., coordinates) is invariant in relativity. Doesn’t this look something like some form of Pythagoras Theorem? This was the insight of Hermann Minkowski, the former teacher of Einstein. He said that x2 - c2t2 represented length in some form of space-time system. So,             

ds2 = dx2 – c2(dt)2

We can identify 1, –c2 to be the metric of this space. This space is called Minkowski space and is the gateway to general relativity.

There is a remarkable symmetry in such a space when it comes to Lorentz transformation. Lorentz transformation is simply rotation of Minkowski space because distance s2 = x2 - c2t2 has to be constant, which allows simply rotation of coordinates.
 

 

We can obtain Lorentz transformation equation by going back from here using concepts of rotation (coordinate transformation). There is a concept called light cone. Suppose light is emitted from the origin. It will follow a path for which x2 - c2t2 = 0. This is called null geodesic.
 

 

The concept of null geodesic is always useful in studying the path of light. Minkowski space is a space with a metric given by (1, 1, 1, -1), where we have assumed the speed of light to be one; we can always do that since we can adjust the scale of measurement the way we wish. This space is also called pseudo-Euclidean space since the Euclidean metric (1, 1, 1, 1) is changed. Still, the Minkowski space is flat since for whole space, the basis vector remains unchanged.

Special relativity provided a clear view that there was going to be a radical change in the laws of Physics. After Minkowski provided the geometrical interpretation of space-time, it was the time for our great hero to attack a greater feat, something which Planck initially dubbed ‘impossible’.

 

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