Friday, August 12, 2011

An Introduction to Special Relativity (2)

In a last ditch efort to satisfactorily explain the riddle of the null result (from the Michelson-Morley experiment), a fantastic idea was put forward by George F. Fitzgerald in 1890. Using the analogy of a rubber ball which is deformed upon striking a wall, Fitzgerald conceived that the ether would distort matter.

This distortion would take the form of a contraction of length in the direction of the motion through the ether. Such a contraction would explain the null result of the Michelson -Morley experiment. That is, the arm L, of the apparatus, moving against the ether would be shortened by "ether pressure" just enough to compensate for the slowing down of light by the ether wind.

A similar but more mathematical theory was worked out by the physicist Hendrik A. Lorentz, who expressed the length contraction in the direction of motion by:

L' = L[1 - v^2/c^2]^½

This famous equation became known as the "Lorentz-Fitzgerald contraction".

Understandably, the hypothesis of Lorentz and Fitzgerald gradually gained general acceptance, except for a Swiss Patent clerk, who refused to be deluded by such a contrived idea (based as it was on absolute motion). The clerk's name was Albert Einstein, born in Ulm, Bavaria, in 1879.

Einstein had graduated as a physics major, but he managed to produce such a poor impression as a teacher that he was dismissed from three teaching jobs in succession. Having been reduced to a 'hand-to-mouth' existence, he was lucky to find a job processing patent applications in Bern, Switzerland. Fortunately, he also has a leisurely schedule that permitted him to while away a good many hours contemplating space, time and energy. (A good thing his time was nearly a century ago, otherwise - today- he'd be accused of "gold bricking"!)

After much thought, Einstein was forced to conclude that motion is never observable as motion with respect to spact and that there is no basis for the introduction of "absolute motion". In Einstein's mind, the only kind of motion was relative rest perceived from different viewpoints (e.g. "reference frames"). Einstein called this insight the Principle of Relativity. According to Einstein's Principle of Relativity: "All the laws of physics are the same in all inertial reference frames."

It's instructive at this point to commence a quantitative approach to see exactly how Einstein reasoned. This necessitates we first get accustomed to the idea of a coordinate system. Basically, the purpose of any coordinate system is to enable us to identify a particular point in space. The standard procedure (for 3-dimensional space) is to take three mutually perpendicular axes with coordinates in x, y and z, representing distanced to where the axes meet at the origin, or 0. In special relativity such coordinate systems are typically designated: S, S', S" etc. or alternatively, S1, S2, S3.

In Fig. 1 we have two coordinate systems, S and S', with S having coordinate x, y and S' having coordinates x', y'. (We confine the systems to 2 dimensions here for initial simplicity.) Thus, S' has its origin at 0' and S at 0. In the figure, S' is shown moving with constant velocity v in the x - x' direction (which obviously coincide for S and S').

Keep in mind though we arbitrarily assigned S' as moving this depends on which coordinate system is taken to be at rest. An observer attached to system S may very well consider himself moving and S' stationary. This would resemble the well-known example of a passenger in a stationary train observing a train parallel to his through his window and deducing he is moving, though it is actually the train parallel to his. The key point is that the systems S and S' have a constant relative velocity. Such coordinate systems occupy a special place in relativity and are called "inertial reference frames" or "inertial coordinate systems". Their primary feature is that they lack and acceleration of one to the other.

Now, think about this carefully: if the origins O and O' coincide at time t = 0, and we are observers in S', then we will see 0' move along OX with velocity v. Then the two sets of coordinates, representing the same point (in S and S') are related by:

x' = x - vt and y' = y

If we chose we could append a 3rd axes (z) i.e. coming out of the page, and have also:

z' = z

The preceding equations describe the "Galilean transformation". From these it is very easy to obtain velocities, and we need only differentiate with respect to t:

E.g.

dx'/dt = dx/dt = v

dy'/dt = dy/dt

dz'/dt = dz/dt

(Note here that never once did we write t' = t since the notion of an absolute, universal time was the very cornerstone of Newtonian theory!)

We now start with the preceding transformations and see how Einstein reasoned. Reference may be made to Fig. 2 which displays a three-dimensional perspective of the relative motion and hence is a bit more complicated. Our aim in using it is to find an alternative to the Galilean transformations - which obviously can't be correct for all situations.

For example, according to the Galilean transformations, a pulse o flight sent out from S' would move with the velocity c, the speed of light, as measured from S', but with the velocity (c + v) measured from S. But this directly contradicts the demopnstrated fact (Michelson-Morley) that the speed of light is always constant when computed from one reference frame relative to another.

Einstein thus began afresh by using not only the Principle of Relativity (just stated, i.e. the laws of physics are the same in all inertial reference frames)) but also:

The speed of light is always found to have the same value no matter what the motion for the source or the observer.

From these two postulates, Einstein deduced a number of surprising results which would have been totally unacceptable to a more conservative mind.

Start then with the two systems depicted in Fig. 2 which are coincident in space at at the instant a flash bulb (say) is set off when the origins coincide. The observer S sees S' moving in the x-direction with the velocity v and observer S' sees S moving with the same velocity in the opposite (-x) direction.

Both observers see the flash of light travel away from the origin with the same velocity c. The distance the light travels in the S system during time t, is ct. This must be true in any direction. If the light spreads out equally in all directions, then by the end of time t it has extended to fill a sphere whose radius is r, and we can write:

(1) r^2 = x^2 + y^2 + z^2 = c^2 t^2

The exact same equation holds in the S' system since the flash was set off when O coincided with O. Thus, while the 2 systems are separating from each other, the observer in S' also sees the light fill a spherical shell in his own system, so writes (for r'):

(2) r'^2 = x'^2 + y'^2 + z'^2 = c^2 t'^2

Note the velocity of light is the only thing that's the same in both systems. The two preceding equations thus describe a point lying on the spherical shell. Even though they describe the same point in space, the observer in S sees the point at position (x, y,z, t) abd the observer S' at position (x', y', z', t').

Subtracting equation (2) from equation (1) and transposing terms:

x^2 + y^2 + z^2 - c^2 t^2 = x'^2 + y'^2 + z'^2 - c^2 t'^2

We now look for a transformation similar to the Galilean transformation, but which will allow c to be the same in both S and S'. Since the y and z coordinates of the position are not affected by the motion in the x-direction we can say y' = y and z' = z. For the x-coordinate, we try a transformation of the form: x = a(x' + vt') and x = a(x - vt), where a is an invariant determined by the two fundamental postulates (i.e. the same quantity is used in going from x to x' as from x' to x).

Further, we expect a to depend on the velocity v in such a way that it becomes equal to 1 when v becomes very small compared with the speed of light. When this happens, the x and x' transformations become the same as the ordinary Galilean transformations.

So we begin by using x = a(x' + vt') to solve for t' and obtain:

(3) t' = 1/v (x/a - x')

For x' above, we now insert the value for x' (e.g. x' = a(x- vt)):

(4) t' = 1/v(x/a - ax - avt) = at - x^2(a^2 -1)/ va

Similarly, we find for t:

(5) t = -at' + x'(a^2 - 1)/ va

If we now substitute x' = a(x - vt) and equation (4) into the right hand side of equation (2), we obtain:

(6) x^2 + y^2 + z^2 -c^2t^2 = a^2(x - vt)^2 + y^2 + z^2

= c^2[at - x/v (a^2 - 1)^2/a]

Now, re-arrange terms and cancel the z and y terms, which are the same on both sides of the equation, to get:

(7) x^2 - c^t^2 = [a^2 - (a^2 - 1)c^2/ a^2v^2]x^2

+ 2[(a^2 - 1)c^2/v^2 - a^2]xvt - (c^2 - v^2)a^2t^2

If the preceding is to hold true for any value of x and t, each term on the left side must equal each term on the right. Since there are no terms with the combination xt on the left, the xt term on the right must be zero. This means:

(8) (a^2 - 1)c^2/v^2 - a^2 = 0

Solving for a:

(9) a^2 = 1/ (1 - v^2/c^2) and a = 1/ (1 - v^2/c^2)^½

Finally:

(10) (a^2 - 1)/a = v^2/c^2/ (1 - v^2/c^2)^½

Substituting the preceding into our x, x' transformation equations and equation (3), we arrive at the following transformations to replace the Galilean:

(11)

x = x' + vt'/(1 - v^2/c^2)^½

and

x' = x - vt/ (1 - v^2/c^2)^½

(12) y = y' and y' = y

(13) z = z' and z' = z

(14) t = t' + x'v/c^2/ (1 - v^2/c^2)^½

and

t' = t - xv/c^2/(1 - v^2/c^2)^½

Equations (11)- (14) are known as the Lorentz transformation or the Lorentz-Einstein transformation.

Note the important feature is that the time must be given as well as the position, because the respective clocks in S and S' will cease to read identical times after they have parted from one another. This is the significance of (14) in the above set. Also, the fact that time is given the same importance as space (i.e. as another dimension) shows there's nothing special or mystical about "the fourth dimension".

Problems:

1- Given that x' = 1/a (x - vt) and t' = 1/a (t - vx/c^2), derive similar equations for x and t in terms of x' and t'.
(Recall: 1/a = (1 - v^2/c^2)^½)

2- An event in space-time occurs at x' = 60 m, t = 8 x 10^-8 s, in a frame S' (y' = 0, z' = 0). The frame S' has a velocity of 0.6c along the x-direction with respect to a frame S. The origins O and O' coincide at time t = t' = 0. Find the space-time coordinates of the event in S.

3- Suppose an astronaut is traveling at 0.9c in a space ship with respect to the Earth. How long a time interval will his clock indicate when the Earth has revolved once around the Sun? (Take the duration of one standard revolution of Earth around the Sun to be 365 ¼ days.)