( 1 - ( v² / c² ) ).Links to sections:
Suppose some advanced race created Orbitsville, an idea taken up by science fiction writers such as Bob Shaw. A physicist suggested that all but a tiny proportion of a sun's energy is wasted in nurturing intelligent life on the odd planet orbiting it. A vastly more efficient way would be to engineer an egg-shaped surround at a distance of the earth's elliptic orbit, about eight minutes' light-speed distance from the sun. That way, life could exist on the entire inside surface of the egg ( or rather an ellipsoid and not too different from a sphere ).
( Orbitsville would have to be manoevrable from collisions with stellar matter. It would be a mother ship to other space-ships. )
Imagine Orbitsville and a space-ship passing by it ( or being passed by it, depending on your point of view ) with a uniform velocity, at a significant fraction of light speed. We can use this scenario to illustrate Einstein's demonstration that two observers in relative motion cannot agree on the time of an event: in the special theory of relativity, simultaneity is lost.
The sun is in the centre of Orbitsville ( we suppose, rather than at one of the two focuses of Orbitsville's elliptic hull ). The climate on its narrower part ( the minor axis ) will be warmer than on the length-wise direction ( the major axis ). From those living on the inside surface, observers are stationed at each end of the major axis. Suppose the engineers of Orbitsville have created a night-simulating mechanism, of eclipsing satellites or whatever, such that dawn breaks at exactly the same time for the observers at the far ends of Orbitsville. For instance, internal 'astronomers' on the minor axis of Orbitsville might see a simultaneous swathe of light upon either end of the major axis.
Say, both ends of Orbitsville are about eighty-nine and one-third million
miles from the sun. The speed of light is about 670 million miles per hour.
Therefore, the internal astronomers will measure the sunlight taking ( ( 89
1/3 ) / 670 ) = 2 / 15 of an hour or eight minutes, to reach either end of
Orbitsville.
( Light takes about eight minutes to reach the earth from the sun, and the
distance between them, typically quoted, is about ninety-three million miles.
)
Orbitsville might have a transparent band along the major axis of its
shell. The space-ship, out-side, is moving in line with this at a fair
fraction of light speed ( say, one-half the speed of light ). Relative to
itself, it sees Orbitsville heading off at that 1/2 of light speed. But the
space-ship doesnt agree with the internal astronomers that the shafts of
sun-light, traveling from the middle of Orbitsville, arrive the same time at
either end.
What is, from the space-ship, the forward-moving end of Orbitsville appears to
be distancing itself from the sun's rays. So, the space-ship measures a time
longer than eight minutes for the sun-light to reach the end moving forward,
relative to the space-ship.
Whereas, the aft end appears to be moving to meet the sun-light, so the
space-ship measures it taking less than eight minutes. Orbitsville's clocks
show it took the same eight minutes for light to reach either end of
Orbitsville. But the space-ship's clock will show an inequality, dependent on
how close its relative motion to light speed.
Using the figures given, this inequality can be worked out by the arithmetic of the Lorentz transformations. These transformations also show that the space-ship will see Orbitsville's sun-light has a constant speed, whether a light ray is moving from, or towards, the space-ship. This is an axiom of the special theory of relativity. We show this first.
Suppose the space ship passes the exact middle of Orbitsville, just as a night shield, covering its sun, opens before it. Light is made visible to either end of Orbitsville, at the same time. But, from the space-ship's point of view, that does not mean both ends receive the light at the same time.
The common sense ( of the Galilean transformations ) would have us believe that if the relative motion of Orbitsville and space-ship is 1/2 light speed, then we have to add or subtract that speed, with respect to the light speed, depending on whether a light ray is moving with or against the direction of Orbitsville, with respect to the space-ship.
However, the Lorentz transformations, for the addition and subtraction of velocities, ensure that the space-ship sees light speed stay the same in both cases.
Take the case of the light ray moving with Orbitsville's away-ward moving direction from the space-ship ( parallel to it ). This requires the Lorentz transformation for the addition of velocities. This is a bit more complicated than the Galilean transformation, that simply gives speed of light plus Orbitsville's relative motion of one-half the speed of light:
Say u' is the speed of away-moving light, seen from the space-ship. And u
is that light ray, which is no different, to those stationed in Orbitsville,
from the ray moving to the other end of its shell. Those standing in
Orbitsville have no relative motion to ( add to or subtract from ) the light
rays going to both ends. Which end is forward or backward has no meaning to
them. In this case ( as well as the other case, we shall deal with next ), the
velocity, u, of the light ray, within Orbitsville's stationary frame-work, can
be set at a 'c' for constant: u = c.
Let v equal the relative motion of one-half light speed or c/2.
Then, the Lorentz transformation for addition of velocities is:
u' = ( u + v ) / ( 1 + ( uv / c² )).
To simplify the working, light speed is often treated as unitary, or c = 1. This makes v = 1/2, and u = c, in this example. Therefore:
u' = ( 1 + 1/2 ) / ( 1 + ( 1 x 1/2 ) / 1 ) = 1.
Or, u' = c.
Therefore, the space-ship measures u', the speed of the light ray, moving with the away-moving Orbitsville, as just c, the constant speed of light, irrespective of an addition of relative motion.
The same applies for the light ray traveling to the opposite end of
Orbitsville. This ray is traveling against the relative motion of Orbitsville
from the spaceship. This means that the aft end of Orbitsville is coming, at
one half light speed to meet the unit light speed of the backward-moving ray,
from the space-ship's point of view.
The common sense or Galilean transformation is that the combined velocity would be one minus one-half equals a velocity of half unitary light speed for the backward ray,
relative to the space-ship.
But the Lorentz transformation, this time, for the subtraction of
velocities again ensures the space-ship's relative motion does not allow it to
see light move at other than its constant speed.
In our example, the terms have the same meaning as before:
u' = ( u - v ) / ( 1 - ( uv / c² ) )
u' = ( c - v ) / ( 1 - ( cv / c² ) )
u' = ( 1 - 1/2 ) / ( 1 - (( 1 x 1/2 ) / 1) )
u' = 1 = c.
Therefore, the Orbitsville sun-light moving with the space-ship's direction is still measured at light's constant speed. This is the same velocity the space-ship measured for the Orbitsville sun-light's away-moving beam.
The Lorentz transformations of time show why observers in relative motion,
that is a fair fraction of light speed, cannot agree on the time they observed
an event.
All standing in Orbitsville are agreed that both light rays, from centre to
either end, took ( time, t, of ) 8 minutes, or 2/15 of an hour, to cross (
distance, X, of ) 89 1/3 million miles, at light's constant speed, c, which is
about 670 million miles per hour.
But the space-ship's clock will show different times, t', that it took the two light rays to reach the ends of Orbitsville. The away-moving light-ray will be timed as taking longer than eight minutes, as the Lorentz addition of times shows:
t' = ( t + ( Xv / c² ) ) / ( 1 - ( v² / c² ) ).
Distance, X, equals velocity, u, times time, t. Also u = c. If c = 1, and v = 1/2, the square of v equals 1/4. So:
t' = 8 ( 1 + 1/2 ) / ( 1 - 1/4 ).
t' = 8 3.
That is, the space-ship's clock will record the away-moving Orbitsville
light ray as taking about 13.9 minutes ( to one decimal place ) to catch up
with the 'forward-moving' end of Orbitsville, instead of the 8 minutes that
Orbitsville clocks register.
Now we find out the time the space-ship's clock tells us it takes the light
ray, that is moving the same way as itself, to meet the aft end of
Orbitsville. From the space-ship's view, the aft is moving, at half the speed
of light, toward this ray, and so reducing the time they take to meet.
Using the Lorentz subtraction of times:
t' = ( t - ( Xv / c² ) ) / ( 1 - ( v² / c² ) ).
Following a similar procedure to the previous example for Lorentz addition of times:
t' = 8 ( 1 - 1/2 ) / ( 1 - 1/4 ).
t' = 8 / 3.
This time, t', of about 4.6 minutes ( to one decimal place ) is the
space-ship's clocking of the light ray, in the same direction as the
space-ship, whose relative motion makes the apparent aft of Orbitsville appear
to come to meet the ray.
This is not quite half the 8 minutes time, t, that Orbitsville's clock shows
for the same journey. With this other Orbitsville light ray, directed in the
opposite direction, simultaneous time has been lost, again. Because of their
high relative motion, the space-ship observers cannot agree with the
Orbitsville observers when the light ray reaches from centre to end of
Orbitsville.
Besides the disagreement over times between the space-ship and Orbitsville observers, they also disagree over distances. Distance equals velocity times time. Knowledge of any two of these variables will tell us the third. The previous two sections have already found the respective velocities and times measured by the two lots of observers. So, the respective distances, measured by their respective 'rods', follows from that information:
In Orbitsville, the distance, X, from centre to either end is eighty nine and one-third million miles. The space-ship sees one light ray appear to have to chase the end of Orbitsville moving away from the space-ship, thus elongating the distance, X', that the space-ship's rod measures.
Or, X' = u'.t' = c.t'. This equals the light speed, of 670 million miles
per hour, times ( 2/15 ) 3 hours. ( Eight minutes is 2/15 of an hour.
) Therefore, X' works out at about 154.7 million miles ( to one decimal place
).
For the sake of completeness, we can repeat this result, by the Lorentz addition of distances:
X' = ( X + vt ) / ( 1 - ( v² / c² ) ).
X' = ( ( 89 1/3 ) + ( 335 x 2/15 ) ) / ( ( 3 ) / 2 ).
X' = ( 134 x 2 ) / 3.
This again equals about 154.7 million miles for the space-ship's 'expanded' rod reading.
Now we come to the case where the space-ship's rod or spatial measure appears
to 'shrink' to less than the Orbitsville rod's registered eighty-nine and
one-third million miles. Again the simplest solution of, X', now as the
shrunken distance, is u'.t', where u' still equals c, and where t' equals 8 / 3 ( which is about 4.6 ) minutes, or 2 / ( 15 x 3 ) of an hour.
This puts X', as at about 51.6 million miles.
Had we not had the benefit of both velocity and time information, we might have had to find this result by starting with the Lorentz subtraction of distances:
X' = ( X - vt ) / ( 1 - ( v² / c² ) ). Or,
X' = t ( u - v ) / ( 1 - ( v² / c² ) ). Where u = c = 670 million
miles per hour.
X' = ( 2/15 ) ( 670 / 2 ) 2 / 3.
X' = 268 / ( 3 x 3 ).
X' ~ 51.6.
This repeats the result we got directly because we happened to have the information for X' = u'.t'.
In the above examples, we illustrated the special relativity principle of the constant speed of light, despite an observer adding or subtracting the other observer's relative motion to it. In general, the observers' respectively measured velocities, u' and u, need not be of light beams, whose constant speed they are both bound to equal, but can measure things at slower velocities, so that, like distance and time, velocities are not generally agreed by observers in relative motion, significantly approaching the velocity of light.
The value of the Interval, for our example, also illustrates its nature as a particular case. The previous web page, part two on special relativity, related the Interval to the Lorentz transformations. The transformations, as exemplified above, show that observers in high speed relative motion usually measure different distances and times for a given event.
The Interval recovers a common measurement between observers of an event. But it is a combined space-time measurement, upon which they depend for agreement as to the 'where-when' something happened.
The Interval measures a four-dimensional space-time common to observers. Our simplified example only dealt in one dimension of space, X, and the one time dimension, t. From the above sections, the Orbitsville co-ordinates are X and t, and the space-ship co-ordinates are X' and t'.
As described on the previous web page, the square of the Interval equals:
X² - c².t² = X'² - c².t'².
Our example is of the Orbitsville observers and space-ship observers of light beams. Special relativity assumes both sets of observers must measure the same velocity for light. That is both u = c and u' = c.
In this particular case, since X = ut and X' = u't', the Interval for both observers is zero. The general point is that the Interval is always the same value for both observers.
My explanations of special relativity were in terms of 'kinematics' or motion, not 'dynamics', which further involves the concept of mass or energy.
Fortunately, the conversion of the Lorentz transformations, from their kinematic form to their dynamic form, is fairly straight-forward. The Lorentz transformations correspond observers' differing distance and time measurements of an event. Call the observers' differing distances, x and x', and differing distances , t and t', respectively. Consequently, the differing velocities of a body, they observe, will be, say, u = x/t and u' = x'/t'.
If the mass of the
body is also to be considered, then an observer, who is at rest relative to
the motion of the body, will measure its 'rest mass'.
In classical mechanics, there only is a rest mass: the mass is deemed
constant. But in relativistic mechanics, the mass of a body is known to
noticably increase as its motion noticably approaches the speed of light. (
The relevant Lorentz transfomation shows that a body would have to be of
infinite mass to reach the speed of light. And this is why light speed is
nature's speed limit. )
Essentially, the Lorentz transformations for distance and for time are both multiplied by the same quantity, the mass divided by the time, to convert them into Lorentz transformations for momentum and energy, respectively.
Momentum, p, is mass times velocity. In the system of mechanics, distance has three dimensions, the x, y and z directions, to refer to all spatial motions. Therefore, velocity, as distance over time, also has direction with reference to the three co-ordinates. For simplicity, we have considered observers measuring events, which differ only in the x-direction, according to their respective measurements. That is to say where distances y = y' and z = z' between respective observers. But their respective measurements do not make x equal to x'. So, velocities in the x-direction, u and u' also need not be equal.
Consequently, the Lorentz transformation for momentum is in terms of the observers' differing measures of both mass and velocity, or: p = mu and p' = m'u'.
The differing times, observers measure, are proportionate to the differing masses they measure: t'/t = m'/m. Note, also, that E = mc² and E' = m'c², where E and E' are the respective energies measured by the observers, according to the famous equation, not here demonstrated.
Then, multiply the Lorentz transformation for distance, x, by m/t, and x' by m'/t', for the Lorentz transformation between observers' differing measures of momentum ( in the x-direction ):
mx/t = mv = p = m'(
x' + vt' )/ t'( 1 - v²/c² )
= ( p' + vE'/c²
)/( 1 - v²/c²
).
The same procedure, of multiplying by mass over time, is followed for converting the Lorentz transformation of time into the Lorentz transformation of energy:
tE/c²t = ( t'E' +
vx'E'/c² )/ c²t'( 1 - v²/c² )
This simplifies to:
E = ( E' + vu'm' )/
( 1 - v²/c² )
= ( E' + vp' )/
( 1 - v²/c² ).
The Lorentz transformation of velocity follows from dividing the Lorentz transformation for distance by that for time:
u = ( u' + v )/ ( 1 + vu'/c² ).
In dynamics, compare the result of dividing momentum by energy. The denominators of both transformations are the Lorentz contraction factor, which cancels:
p/E = ( p' + vE'/c² )/( E' + vp' ).
Or: u.m/mc² = m'( u' + v )/m'c²( 1 + u'v/c² ).
This reduces to the Lorentz transformation of velocity.
Furthermore, the Minkowski Interval, as well as the Lorentz transformations, convert from kinematics to dynamics, in a straight-forward manner. The Minkowski Interval is a four-dimensional space-time geometry. It is like Euclid's geometry in three dimensions. But time is treated like a fourth dimension of space. The only formal mathematical difference between the three space dimensions and the one time dimension is that the time is multiplied by the square root of minus one.
This factor gives the time, t, the opposite sign to the three space dimensions x, y and z, when the Interval extends Euclid's geometry, of Pythagoras theorem in three dimensions, into a 4-D space-time version of the theorem.
The Lorentz transformations are the means by which an observer changes their measurements into those of differently situated observers of the same event. The point of the Interval is that it combines their respective measurements into a 'space-time' measurement, of a given event, that is one and the same for all observers, in uniform relative motion.
As for the Lorentz transformations, the Interval is simply given here for only one dimension of space and the one dimension of time. As discussed on previous pages, this is:
t²( c² - u² ) = t'²( c² - u'² ).
Or: t²c² - x² = t'²c² - x'².
The space-time Interval changes to its momentum-energy version, as with the Lorentz transformations, thru replacing time by mass:
( mc )²t²( c² - u² )/t²c² = ( m'c )²t'² ( c² - u'² )/t'²c².
Or: ( E/c )² - p² = ( E'/c )² - p'².
Richard Lung.
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