Reference Frames
A reference frame is a set of coordinate axes that help describe the position or movement of an object. There are two basic types of reference frames: inertial and accelerating. Inertial frames are those where the coordinate axes appear to follow Newton's first law in every direction from the point of view of an observer: things at rest tend to remain at rest unless acted upon by an external force. An accelerating frame, on the other hand, is one where the coordinate axis are accelerating from the point of view of an observer, such as a rotating merry go round or an accelerating rocket ship.
Two types of transformation are used when transforming or mapping coordinates from one inertial frame to another. If the constant velocity of the frame is zero or small compared to the speed of light, a Galilean transformation is appropriate. However, if the velocity is a significant portion of the speed of light, a Lorentz transformation is appropriate, since the laws of special relativity apply. The Lorentz transformation equations are actually true at any speed, but Galilean equations are mathematically valid at low speeds.
For example, take two carts, one stationary and one moving with a constant velocity. To a stationary observer in each frame, the ball on the stationary cart will fall straight down, but the ball on the moving cart will fall in a parabola. Data is taken on both balls from a non-moving view point in each frame. A Galilean transformation is performed on the stationary cart and checked against the moving cart data.
Now, suppose there are two electrons, each in its own frame of reference. One is moving with velocity v=0.6c and the other with velocity v=-0.6c. A Galilean transformation from either frame of reference to the other makes one electron moving at v=1.2c while the other is stationary. We know from special relativity that this is not possible; nothing can move faster than c, the speed of light in a vacuum. This is where we would have to use a Lorentz transformation on the data to map from one coordinate system to another.
As an example of a Lorentz transformation, let's consider a reference frame, B, moving relative to another frame, A, with a constant velocity, v=0.8c along the common direction of their x-axes. All clocks read zero as the origins coincide. An observer in B sees an explosion, E, on his or her x-axis at position x=1.5km and at time t=0.001s. To find when and where the person in A will observe this event, we must do the transformation.
Our givens are: t =0.001s, x =1.5 x 103 m, and v =0.8c =0.8(3 x 108 )m/s.
To find x,
To find t,
Notice the time and length contraction of the B frame as predicted by special relativity.
As these graphs show, events are relative; they depend on the observer's reference frame. It is very important when working in physics to define correctly the appropriate reference frame.
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