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Consequences of Special Relativity

Special relativity has several consequences that struck many people as bizarre, among which are:

The time lapse between two events is not invariant from observer to another, but is dependent on the relative speeds of the observers' reference frames. (See Lorentz transformation equations)
Two events that occur simultaneously in different places in one frame of reference may occur at different times in another frame of reference (lack of absolute simultaneity).
The dimensions (e.g. length) of an object as measured by one observer may differ from the results of measurements of the same object made by another observer. (See Lorentz transformation equations)
The twin paradox concerns a twin who flies off in a spaceship travelling near the speed of light. When he returns he discovers that his twin has aged much more rapidly than he has (or he aged more slowly).
The ladder paradox involves a long ladder travelling near the speed of light and being contained within a smaller garage.
Velocities do not combine by simple addition, but instead by a relativistic velocity addition formula.
Fast moving objects will appear to be distorted by Terrell rotation.
The inability for matter or information to travel faster-than-light.

The Effect on Time

The fact that light travels at a constant speed has a distinct effect on time.

Imagine a clock that measured time by bouncing a photon (a particle of light) between two mirrors that are its walls. The photon must always travel at the speed of light (c). Even if the ship were moving (at velocity $v s h i p$ ), the light would move at exactly the speed of light. This can be represented by a vector, whose magnitude is c and whose sideways magnitude were $v s h i p$ . The vectors c, $v s h i p$ , and the speed of the photon upward or downward form a right triangle and therefore, the speed at which the photon moves upward or downward ( $v p h o t o n$ ) could be figured using the Pythagorean Theorem ( $a 2 + b 2 = c 2$ ):

$v_{photon}^2+v_{ship}^2=c^2$

Rearranged to:

$v_{photon}=\sqrt{c^2-v_{ship}^2}$

To check, substituting 0 for $v s h i p$ makes:

v p h o t o n = c

(which makes sense)

If the length of the ship were $h s h i p$ and the ship were not moving, then based on the definition of speed:

t s t a t i o n a r y = h s h i p / c

where $t s t a t i o n a r y$ is the time interval measured by the stationary clock.

If the clock were moving, then the speed of the photon towards the opposite mirror ( $v p h o t o n$ ) is $\sqrt{c^2-v_{ship}^2}$ . Then, the time interval measured by the moving clock will be:

$t_{moving}=h_{ship}/\sqrt{c^2-v_{ship}^2}$

Since we don't want to deal with the length of the ship, but rather, the time interval, we rearrange the equation, $t s t a t i o n a r y = h s h i p / c$ to:

$h_{ship}=t_{stationary}\cdot c$

We can substitute that into the equation above to get:

$t_{moving}=t_{stationary}\cdot c/\sqrt{c^2-v_{ship}^2}$

$c$ can be factored out of the $\sqrt{c^2-v_{ship}^2}$ by factoring the $c 2$ out of the $\sqrt{c^2-v_{ship}^2}$ to get $\sqrt{c^2(1-v_{ship}^2/c^2)}$ factor the $c 2$ out to get $c\cdot \sqrt{1-v_{ship}^2/c^2}$ The $c$ s in the numerator and the denominator cancel out to make

$t_{moving}=t_{stationary}/\sqrt{1-v_{ship}^2/c^2}$

The reason that this is so definitely correct is the fact that $h s h i p$ is the length of the ship in a dimension in which it is not moving, so the ship in that dimension cannot be affected. That means that because the ship is not moving in the direction of $h s h i p$ , $h s h i p$ is not changed by relativity.

Because all motion is relative, if ship A is moving relative to ship B, occupants of ship A see the time of occupants of ship B running slow and occupants of ship B see the time of occupants of ship A running slow. There is no experimental way of finding out which occupants are right, so they can both be said to be correct.

Categories: Special relativity

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03-10-2013 05:06:04
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