## Tuesday, August 9, 2011

### An Introduction to Special Relativity (1)

As promised, we'll now commence a six part blog on a basic introduction to Einstein's theory of special relativity. This will also include some basic problems at the end of each segment and subsequently the solutions. Not much more than basic algebra will be needed to follow most of the development.

In the popular mind, at least, the word "relativity" usually conjures up visions of space travelers returning youthful to Earth after journeys of many decades (measured on Earth). Of course, there is much more to relativity than this. As a matter of fact many aspects of relativity as in the case of motions, aren't new at all. Basically, it merely entails the assertion that the laws of physics appear to be the same in given reference frames or coordinate systems. This is made more accurate in Einstein's special relativity by referring to inertial reference frames.

Though Henri Poincare came close to discovering the principles of relativity, it was Albert Einstein first and foremost who ruthlessly and relentlessly pursued the basic principles to their utterly logical conclusions, including that lengths shrink as velocities tend toward the speed of light, c, and time slows or "dilates".

Like Ernst Mach before him, Einstein adopted the view that in considering two objects in relative motion, it is futile and meaningless to attempt to decide which object is "really" in motion and which is at rest. If you are in a Jumbo jet traveling at a speed of 900 km/hr relative to the Earth, it makes no difference whether someone says you are moving at that speed, or the Earth is moving at that speed. In either case, the operation of the laws of physics in your jet and on the Earth will be the same. Balls will still drop, and you will still fall off your chair if not careful. There exist no absolute frame of reference to contradict you.

The Special Theory of Relativity, interestingly, may be said to have had its origins in the null result of an experiment the primary aim of which was to detect relative motion. This experiment was first carried out by the American phuysicist Albert A. Michelson in 1881, and subsequently repeated in 1887, with the help of Edward W. Morley. The experiment has thus come to be known as "the Michelson-Morley experiment".

The basic idea was to time the transits of light in two distinct directions: perpendicular to the Earth's orbital motion, and parellel to the orbital motion. This was to be accomplished by using an arrangement of mirrors and light beams such as depicted in Fig. 1. A difference in light velocities (transit distance/ time) would reveal itself by a delicate interference pattern formed by two separate beams after rejoining each other.

The implication of a difference in velocities would mean the confirmation of a remarkable entity called "the Ether". In effect, if light represents waves propagating through the Ether, the velocity of light as recorded by instruments on Earth's surface must be distorted by the motion of the Earth through space. An analogous principle is that a swift river must retard a swimmer's combined upstream and downstream speeds more than his cross-current speed. Similarly, a large difference in light velocities (along the two different paths) should show Earth is moving rapidly through the Ether, while a small difference would show it's moving slowly.

Astonishingly, on each occasion the experiment was conducted the result was virtually negative. There was no evidence of any Ether flowing past the Earth in any direction. Any minor deviations fell within the purview of the experimental errors.

Now, imagine the apparatus in Fig. 1 to be moving with velocity v toward the right, then only in the event of a null result should there should there also be a relative velocity of the Ether of magnitude v to the left. To get a positive result the apparatus needs to move a speed v relative to the Ether. The round trip time for a light beam following path X-M2-X is:

t1 = 2Lc/ (c^2 - v^2)

Meanwhile, the light beam traveling from X to M1 must have a component of velocity v along X-M2, relative to the hypothesized Ether or it will not strike the mirror at M1. Since the velocity of the light relative to Ether is c, subtracting the preceding component leaves a velocity of (c^2 - v^2) E.g. see Fig. 2. The same is true for the return journey to the total time t2 for the path X-M1-X is:

t2= 2L/ (c^2 - v^2)^½

Now, the recomibination of two light beams (originally split in two) will produce interference fringes as depicted in Fig. 3. Any difference in the times taken to traverse the paths will show up as a shift in the position of the bright and dark fringes since it will indicate different path lengths.

Hence, a difference in time delta t is equivalent to a path difference:

c (delta t)/ lambda (d)

where d = the width of one fringe and lambda is the wavelength of the light (taken to be 6 x 10^-7 m).

From this, the time difference between the two beams can be computed from:

delta t = t1 - t2

= 2L/c{ 1/ (1 - v^2/c^2) - 1/ (1 - v^2/c^2)^½}

This equation can be approxiamtely expressed as:

delta t ~ L/c (v^2/c^2)

Since v << c, this corresponds to a fringe shift of:

delta d - c (delta t)/ lambda = Lc^2/ lambda (c^2)

The measurement of the fringe shift is accomplished by rotating the whole apparatus through 90 degrees. The effect of this is to interchange the arms, X-M1 and X-M2, thereby reversing the sign of the fringe shift so that an overall shift of 2 (delta d) should be observed.

For the Michelson -Morley experiment of 1887, L = 11 m, v = 0.0001c.

Using the path difference equation for delta d, we arrive at:

delta d = 0.183

or, approximately 0.2 fringe.

The overall shift expected was:

2 x (delta d) = 2 x (0.2) = 0.4 fringe

But the largest shift actually detected was only 0.01 fringe within the experimental error.

This null result flabbergasted physicists of the time. They were simply unable to conceive that light required no medium within which to propagate. Hence, it's not difficult to see why so many clung to the Ether mcguffin for so long, even after it was disproven. Thus, fanciful and elaborate schemes were thought up to explain the null result, much like today's intelligent design proponents have confected fanciful explanations to try and disavow Darwinian evolution.

For his own part, Michelson naturally assumed that the local ether had to be adhering to the Earth, travelling with it through space. All other scientists were incredulous that an orbital velocity of 30 km/s in relation to the Sun could not generate the tiniest ether "breeze". To many, the situation was not unlike a ship maintaining constant speed and direction in the sea, irrespective of current changes.

Problems

1. In the Michelson-Morley experiment, the length L of each arm of the interferometer was 11 meters. Sodium light of wavelength 5.9 x 10^-7 m (590 nm) was used. The experiment would have revealed any fringe shift > 0.005 fringe.

What upper limit does this place on the Earth's velocity through the supposed Ether?

2. Using Fig.1, say the time of travel to the right is: t(r) = L/(c - v) and the time of travel to the left is t(L) = L/(c + v).

a) Find the "total time of travel" by adding both left and right contributions.

b) find the time consumed for "a half-trip".

c) Find the time consumed for a round trip.

d) Add the two "half trips" and what do you obtain?

e) Why does this not agree with the value obtained for (a)?

Next: How Fitzgerald attempts to rescue the situation.