Many problems to do with the Special Theory of Relativity, interestingly, can be solved using intermediate algebra - which not very many people know who aren't physicists. In this post we work toward such simpler algebraic solutions to key aspects of the special theory by first examining a few of the basic principles.
The theory itself 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 physicist Albert A. Michelson in 1881, and subsequently repeated in 1887, with the help of Edward W. Morley. The experiment came to be known as the Michelson-Morely Experiment.
The basic idea was to time the transits of light in two distinct directions: perpendicular to the Earth's orbital motion, and parallel to the orbital motion. This was to be accomplished by using an arrangement of mirrors and light beams such as depicted in the experimental arrangement shown in the (blue) highlighted link above.
Note then that 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 means confirmation of a remarkable entity called "the Ether". In effect, if light represents waves propagating through the Ether, the velocity of light 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.
Imagine the apparatus below to be moving with velocity v toward the right, then only in the event of a null result 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/ (c2 - v2)
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 (c2 - v2) e.g. see Fig. 2 at top. The same is true
for the return journey to the total time t2 for the path X-M1-X is:
Now, the recombination 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 D t was found equivalent to a path difference:
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 could be computed from:
D t = t1 - t2
= 2L/c{ 1/ (1 - v2/c2) - 1/ (1 - v2/c2)½}
This equation can be approximately expressed as:
D t » L/c (v2/c2)
Since v << c, this corresponds to a fringe shift of:
This null result flabbergasted physicists of the time. They were simply unable to conceive that light required no medium within which to propagate. 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.
In
a last ditch effort 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 - v2/c2]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
space 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 more 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 Fig. 1, 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’s easy to obtain velocities
by differentiating with respect to t:
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 demonstrated 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 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) r2 = x2 + y2 + z2 = c2 t2
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'):
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) and the observer S' at position (x', y', z', t').
Subtracting equation (2) from equation (1) and transposing terms:
x2 + y2 + z2 - c2 t2 = x’2 + y’2 + z’2 - c2 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 - x2(a2 -1)/ va
Similarly, we find for t:
(5) t = -at' + x'(a2 - 1)/ va
If we now substitute x' = a(x - vt) and equation (4) into the right hand side
of equation (2), we obtain:
(6) x2 + y2 + z2 - c2 t2 = a2(x - vt)2 + y2
+ z2
= c2[at - x/v (a2 - 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) x2 - c2 t2 = [a2 - (a2 - 1)c2/
a2v2]x2
+ 2[(a2 - 1)c2/v2 - a2] xvt - (c2
- v2)a2t2
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) (a2 - 1)c2/ v2 - a2 = 0
Solving for a:
(9) a2 = 1/ (1 - v2/c2) and a = [1/ (1 - v2/c2)]½
Finally:
(10) (a2 - 1)/a = v2/c2/ [1/ (1 - v2/c2)]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 - v2/c2)]½
and
x' = x - vt/ (1 - v2/c2)]½
(12) y = y' and y' = y
(13) z = z' and z' = z
(14) t = t' + x’v/c2/[(1 - v2/c2)]½
and
t' = t - xv/c2/[(1 - v2/c2)]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".
Then: x' = x/a - vt/a and t' = t/a - vx/ac2
and: x' + vt/a = x/a and t' + vx/ ac2 = t/a
so:
x = a(x' + vt/a) and t' = a(t' + vx/ ac2 )
Finally: x = a(x' + vt) and t = a(t' + vx/c2)
Suggested 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) 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.)
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