Every observation we make and every measurement we perform occur within some frame of reference. Normally we do not even think about this, because the frame of reference is usually some point that we consider to be fixed on the surface of the Earth. Surveyors and navigators of ocean liners or airplanes must be certain that they understand the location of the frame of reference they are using or they risk disastrous consequences.

The oft-cited example of an object floating in a stream illustrates the importance of carefully specifying the frame of reference. If we state that a boat has a certain velocity (specifying both magnitude and direction) with respect to the stream we also need to know the velocity of the stream with respect to the shore before we can state what the boat's velocity is with respect to the shore. If the velocity of the boat is parallel to the velocity of the stream we note whether it is moving in the same or the opposite direction in which the stream is moving and add the two velocities algebraically with due regard to sign. If the velocity of the boat is perpendicular to that of the stream we must use the tail-to-head vector addition method considered in Appendix C on page 474 in the text. These ideas conform to our experiences and form the basis of the system of mechanics developed by Galileo and Newton that we considered in earlier chapters.

As long as the reference systems that we are using do not accelerate with respect to each other we can use the principle of relativity as expressed by Galileo and Newton. The expression of relativity presented by Galileo states that the laws of physics are the same in any inertial frame of reference. An inertial frame of reference is a reference frame that is not experiencing an acceleration.

When we studied waves in Chapter 15 we found that all waves except electromagnetic waves required a medium in which to travel. Something is required to do the "waving," as it were. Until nearly the end of the nineteenth century physicists thought that some medium was also required for electromagnetic waves, particularly for light. One such medium that was postulated was a luminiferous ether that was thought to permeate all of space. The famous experiment of Michelson and Morley showed that there was no such ether and that electromagnetic waves, light in particular, did not need a medium in which to travel. As often happens in science, this discovery raised additional questions regarding the relative velocity of light; that is, if a light source is moving with respect to some frame of reference what is the speed of the light emitted by the moving source with respect to that frame of reference? Do we add the speed of the moving source to the speed light has in a stationary system to obtain the speed of light in the moving system? Einstein offered a answer to this question when he presented his two postulates of special relativity. The first postulate was the same as the statement of relativity offered by Galileo and Newton, i.e. that the laws of physics are the same in any inertial frame of reference. The second postulate was much more radical in that it stated that the velocity of light in a vacuum is the same in any inertial frame of reference, regardless of the relative motion of the source and observer. This postulate produced great changes in how we view time and space, as it led to the ideas of time dilation (moving clocks run slower) and length contraction (lengths measured by moving observers are shorter). Einstein's theory of special relativity provides mathematical formulas for calculating the changes in time and distance measurements. The equations show that these effects become significant only when the speeds become close to the speed of light. Obviously Galileo and Newton did not have any laboratory devices capable of achieving such speeds, so they did not anticipate special relativity. Indeed it takes special equipment or experiments with subatomic particles to see any noticeable differences between the predictions of Newtonian mechanics and special relativity. Another way of phrasing this is to state that special relativity reduces to Newtonian mechanics when the velocities involved are small compared to the velocity of light.

A major consequence of special relativity is the discovery of the equivalence of mass and energy. This is most commonly expressed in terms of the rest-mass energy E_o = m c² .

If the velocities involved are small compared to the velocity of light we may use Newtonian mechanics, and we may use simple vector addition to combine velocities in inertial reference frames. If, however, one of the velocities is comparable to the speed of light we must use special relativity for analysis.

The first postulate of special relativity states that the laws of physics have the same form in any inertial frame of reference. This statement conforms with our expectations from Newtonian mechanics. The second postulate's assertion that the velocity of light is the same in any inertial frame of reference, regardless of the motion of the source provides more conceptual difficulty. By this postulate Einstein asserted that a beam of light emitted from a stationary source has the same velocity for all observers in all inertial reference frames regardless of the velocity of the source.

The equations which Einstein derived to use in special relativity all involve the factor g = (1 - v² /c² )^-1/2 where v is the velocity involved and c is the speed of light. This expression is always greater than or equal to one. It is equal to one only for the case of v = 0. The appearance of this factor in the expressions for combining velocities, for calculating momentum, and determining kinetic energy leads to the predictions in special relativity that differ from the predictions of Newtonian mechanics. One major consequence is the realization that mass and energy are equivalent.

Experiments have confirmed these predictions by Einstein's theory. Some of the experiments include those with nuclear fission and nuclear fusion that were discussed in Chapter 19. For velocities that are small compared to the velocity of light (remember that c = 3 x 10⁸ m / s and the 55 mile per speed limit represents 24.6 m / s) the factor g is nearly equal to one and special relativity reduces to Newtonian mechanics.

Einstein's theory of special relativity deals with inertial frames of reference that are moving with constant velocity relative to each other. His theory of general relativity considers the case where the frame of reference is accelerated the general theory asserts that an acceleration of a frame of reference cannot be distinguished from the presence of a gravitational field.