# Special Relativity: Kinematics

### Postulates and the Loss of Simultaneity

#### The Relativity Principle

There are many possible places to start in Special Relativity. That is, it is a matter of choice what is to be called a 'postulate' and what a 'theorem.' However, it is best to start with the most believable and simple. With this in mind, our first postulate states:

All inertial reference frames are equivalent.

In inertial reference frame (see the introduction to this topic for an explanation of reference frames) is simply in which Newton's First Law holds (that is, the law of inertia: all bodies remain at rest or in constant motion unless a force acts upon them). This means any non-accelerating reference frame is an inertial one. In an accelerating (including rotating) frame we have to invent 'imaginary forces,' such as the (coriolis or centrifugal forces) in order for the law of inertia to be valid. Our postulate then says that any reference frame at rest or in constant motion is as good as any other--there is no absolute frame. More precisely it says that an observer on the ground (or someone at rest in outer space) does not have a preferred or 'more realistic' view of events than an observer whizzing past on a train; their different ways of viewing and measuring things are equally valid. It makes no sense to say something is moving. It only makes sense to say something is moving relative to something else. For this reason this postulate is sometimes called the relativity principle.

The relativity principle also asserts that if the laws of physics hold in one inertial frame, they must also hold in all the others. This arises because there is no preferred frame: what is true in one frame must be true in all the others, otherwise one could pick a particular frame as 'special.' It is also crucial to understand that this postulate implies that if there are two frame F and F' , then F must view events in F' in exactly the same was as F' views events in F . This has profound consequences for the laws of physics.

#### The Speed of Light

The second postulate states:

The speed of light has the same value in any inertial reference frame.

The first postulate was simple enough, but this one should be very worrying indeed. If a spaceship is hurtling towards you at 1.5×108 m/s and it fires a golf-ball from its nose at a speed relative to the spaceship of 1×108 m/s, then you would measure the speed of the golf-ball as (1.5 + 1)×108 = 2.5×108 m/s. However, if instead of a golf-ball the spaceship fires a photon of light at c = 3.0×108 relative to the spaceship, our postulate tells us that we would not measure the speed of the photon as 1.5×108 +3×108 = 4.5×108 m/s, but at 3.0×108 m/s, the same speed as in the spaceship's frame.

This may appear somewhat confusing. Clearly light does not behave like a golf-ball. No matter how fast the spaceship is traveling, either towards us, or away from us, or at any angle (so long as it remains in an inertial frame), we will measure the speed of the light reaching us to be c . Moreover, any nearby observer in any inertial frame will see the light traveling at the same speed. The veracity of this remarkable claim has been confirmed by numerous experiments, the first of which was performed by Michelson and Morley (referred to in the introduction to this topic); their careful experiments found no difference in the speed of light as the earth rotated around the sun (changing its direction of motion).

#### The Loss of Simultaneity

The most fundamental effect of the postulates of Special Relativity is to demonstrate that simultaneity (the notion of events occurring at the same time) is relative (dependent on frame of reference). This concept is best understood by way of example. Consider an observer O A standing equidistant between two light sources. O A has measured the distance to each source and found it to be l A . The sources each emit an instantaneous flash; the flashes reach O A at the same time, so he concludes that the flashes were emitted at the same time.

Now consider observer O B , moving past the sources and O A at speed v from the right (see ). In O B 's rest frame (that is the reference frame in which O B is at rest) they see the sources whizzing past to the left at speed v . O B has measured the distance between the sources and found it to be 2l B , and also that O A is standing exactly half way between the sources (a distance l B from each). From O B 's point of view, the light from the right source travels at a speed c (from our second postulate) with respect to O B , but with speed c + v with respect to the source. Similarly the light from the left source travels at a speed c - v with respect to its source. Thus the time the light takes to reach O A from the right source is t r = and from the left source it is t l = . If v 0 , then these times are different, and O B concludes that the light emitted by the sources reached O A at different times and so the flashes could not possibly have been simultaneous.

Figure %: Observers in different reference frames have different concepts of simultaneity.

Thus the whole concept of 'things happening at the same time' is relative. One must specify a reference frame to say that events are simultaneous. This will have especially important consequences for the measurement of length. Different symbols are used for the distance measured by each observer between the sources because the distances are indeed different, as we will see in Section 3, but this is not important in the example above. Finally, it is important to point out that the different times taken for the light to reach O A in O B 's frame has nothing to do with the time taken for light to reach an observers eye; rather the sources really emit their flashes at different time in O B 's frame (his frame is just as good as O A 's according to our first postulate). The time taken for light to reach an observer's eye will never be taken into consideration in this topic (it would be trivial to calculate them: simply add on an amount of time distance/c, but this just complicates things unnecessarily).

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