Consequences of special relativity

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

  • The time lapse between two events is not invariant from one 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 traveling 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 traveling 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.

[edit] 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, say "horizontally". The photon must always travel at the speed of light (c). Even when the clock is moving (at velocity vclock), the light photon must move at exactly the speed of light. Now if the direction the clock is moving is "vertical" (perpendicular to the original path of the photon), the photon's velocity can be represented by a vector whose direction is the hypotenuse of the right triangle formed by the horizontal (vphoton) and vertical (vclock) components. Since the length of this hypotenuse (the resultant velocity) must not exceed c, the "horizontal" speed of the photon (the length of vphoton) will be less than c. This can be determined using the Pythagorean theorem (a2 + b2 = c2):

v_\mathrm{photon}^2+v_\mathrm{clock}^2=c^2

which can be solved as:

v_\mathrm{photon}=\sqrt{c^2-v_\mathrm{clock}^2} .

To check, substituting 0 for vclock makes:

vphoton = c   (which makes sense).

If the distance between the clock mirrors were wclock and the clock were not moving, then based on the definition of speed, the photon would travel once between the mirrors in the amount of time, tstationary, given by:

tstationary = distance / speed = wclock / c .

tstationary is the time interval measured by the stationary clock -- its basic unit (or "tick" interval).

However, if the clock were moving, then the "horizontal" speed of the photon towards the opposite mirror (vphoton) calculated above would be \sqrt{c^2-v_\mathrm{clock}^2}. Hence, the time interval measured by the moving clock will be:

t_\mathrm{moving}=w_\mathrm{clock}/\sqrt{c^2-v_\mathrm{clock}^2}

(Note that the above works because wclock is the ("horizontal") width of the clock in a dimension in which it is not moving, and the clock's width is not relativistically affected by the movement in an orthogonal dimension.)

Since we want to know the effect of moving the clock on its basic time interval, we rearrange the equation, tstationary = wclock / c to:

w_\mathrm{clock}=t_\mathrm{stationary}\cdot c

and substitute that into the equation above to get:

t_\mathrm{moving}=t_\mathrm{stationary}\cdot c/\sqrt{c^2-v_\mathrm{clock}^2}.

c can be factored out of the denominator by factoring the c2 out of the (c^2-v_\mathrm{clock}^2) to get \sqrt{c^2(1-v_\mathrm{clock}^2/c^2)} which leads to c\cdot \sqrt{1-v_\mathrm{clock}^2/c^2}. The c's in the numerator and the denominator then cancel out to make

t_\mathrm{moving}=t_\mathrm{stationary}/\sqrt{1-v_\mathrm{clock}^2/c^2} = \gamma \cdot t_\mathrm{stationary}

where \gamma = 1 / \sqrt{1-v_\mathrm{clock}^2/c^2} is known as the Lorentz factor.

This means that the faster the clock moves, the longer its "tick interval" relative to a stationary clock. In effect, time measured by the moving clock has slowed down!

Extrapolating this, 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 logical or experimental way of saying which occupants are "right", so they can both be said to be correct.