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Space is Relative |
From a time-varying three-dimensional (TV3D) perspective, space consists of a three-dimensional world containing matter and energy that interact with one another as time progresses. At any particular instant of time, various particles of matter occupy particular points in space. The location of these points may be specified mathematically through the use of a three-dimensional coordinate system, the most common of which is the rectangular Cartesian coordinate system denoted by (x, y, z).
Although other coordinate systems can be used by various observers, coordinate transformation equations always exist to transform those other coordinates into the x, y, z-system of one particular observer in one particular frame of reference. Therefore, the x, y, z-coordinate system of any observer located in space is sufficient to completely specify the distribution of matter in the space. In other words, the coordinates x, y, z completely span the space and completely describe its contents at any instant in time.
Mathematically, you could even say that the set of coordinates x, y, z define space. So, if you want to know what space is like, look carefully at x, y, and z. If you do, then you will notice that as time progresses and the distribution of matter changes, the x, y, z-coordinates change. Mathematically, we say that x, y, and z are functions of time: x = x(t), y = y(t), and z = z(t). Physically we say that space is a function of time or that space changes with time. Of course, by this we really mean that the distribution of matter in space changes with time.
But the Star Trek paradox demonstrates clearly that the distribution of matter in space and the way it changes in time are different from one observer’s TV3D perspective to another’s. The Lorentz transformation equations quantify this difference.
Since y = y’ and z = z’, we conclude that space perpendicular to the direction of motion is exactly the same for both observers. Therefore, there is no contraction or expansion of a moving object perpendicular to its direction of motion. Width contraction and height contraction do not occur.
But length contraction does occur.
If we define the proper length of an object moving with velocity v to be
the length Lo of the object in its direction of motion as
measured by an observer moving along with the object, and the relative length
of the same object to be the length L measured by an observer at rest,
then the two lengths are related by the
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Length Contraction Equation: |
L = Lo/g |
In other words, the length of a moving rod contracts in the direction of motion by a factor of gamma.
It follows immediately that volume contraction also occurs. In terms of the
proper volume Vo and the relative volume V of an
object moving with velocity v, one obtains the
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Volume Contraction Equation: |
V = Vo/g |
In other words, the volume of a moving object contracts by a factor of gamma.
Both length contraction and volume contraction are real phenomena. They are not simply an illusion resulting from the high velocity of the object and the finite speed of light. Objects do not just appear to contract, they really do contract.
Their appearance is considerably more complicated than a simple contraction in the direction of motion. This is because the visual appearance of an object is determined by the light arriving at the observer at a particular instant in time. Because of the high velocity of the object and the finite velocity of light, the light arriving at an observer at a given instant must have left different parts of the object at different times in the past. This means that the visual appearance of an object depends not only on its actual shape but also on where different parts of the body were located at different times in the past. (See Serway p.19 and Tipler pp.38-39 for more information on this subject.)
It also means that the view out the window of a spaceship moving through the
galaxy close to the speed of light will be nothing like that depicted in any
science fiction movie to date. Some day someone is going to write a computer
program to simulate the correct view. Among the surprising effects will be (1)
the forward motions of the surrounding stars as the spaceship accelerates
toward the speed of light, (2) the disappearance of certain stars as they shift
their spectra into the ultraviolet ahead of the spaceship and into the infrared
behind the spaceship, and (3) the beautiful rainbow ring of moving stars seen
through the front window of the spaceship.
P1. Length Contraction
(a) What is the speed of a meter stick whose length is measured to be 25 cm.
(b) What is the length of a meter stick moving at 0.99 c?
P2. Cubical Volume Contraction
A cubical Borg spaceship is 1000 m on a side. What are its (a) length, (b)
width, (c) height, and (d) volume if it is traveling at 0.999 c?
P3. Spherical Volume Contraction
The sun has a radius of 6.96x108 m. (a) What is its proper volume?
(b) What is its volume according to a cosmic ray traveling at 0.95 c?
(c) What is its shape according to the cosmic ray?
P4. Pilot’s View
The pilot of a spaceship looks out his cockpit window before and after
accelerating to 0.9 c. Before accelerating, he sees a star out his side
window making an angle of 90o with his intended flight path. (a)
After accelerating, what is the angle between that star and his flight path?
(b) According to the pilot, which direction has the star moved as he
accelerated forward? Did the star move forward or backward?
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