Faraoni - Special Relativity

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Valerio Faraoni Undergraduate Lecture Notes in Physics Special Relativity 2013 10.1007/978-3-319-01107-3_1

Springer International Publishing Switzerland 2013

1. Fundamentals of Special Relativity

Valerio Faraoni 1

(1)

Physics Department, Bishops University, College Street 2600, Sherbrooke, QC, J1M 1Z7, Canada

Valerio Faraoni

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Abstract

The theory of Special Relativity formulated by Albert Einstein in 1905 [1, 2] is, without doubt, one of the great intellectual achievements of the 20th century. Our everyday experience is about objects moving at speeds much smaller than the speed of light in vacuo, . Newtonian mechanics was developed to describe phenomena at typical speeds and fails when speeds are not negligible in comparison with . This situation is not infrequent; for example, it is relatively easy to accelerate electrons to speeds in accelerators. However, as their velocities become closer and closer to , it becomes harder and harder to accelerate these electrons further.

Concepts that have proven useful in ordering things easily achieve such authority over us that we forget their earthly origins and accept them as unalterable givens.

Albert Einstein.

1.1 Introduction

The theory of Special Relativity formulated by Albert Einstein in 1905 [] is, without doubt, one of the great intellectual achievements of the twentieth century. Our everyday experience is about objects moving at speeds much smaller than the speed of light in vacuo, . Newtonian mechanics was developed to describe phenomena at typical speeds and fails when speeds are not negligible in comparison with . This situation is not infrequent; for example, it is relatively easy to accelerate electrons to speeds in accelerators. However, as their velocities become closer and closer to , it becomes harder and harder to accelerate these electrons further. As we will see, it is believed that the speed of light is a barrier that particles and physical signals cannot break.

The theory of Special Relativity was a one mans work which incorporated previous ideas and contributions (especially those of Poincar), but in a way that nobody was able to fully grasp before Einstein.

In our exposition of Special Relativity we will omit gravity from the picture and we will assume that inertial frames extend to spatial infinity. Special Relativity is a theory for a world in which gravity is switched off; the relativistic theory of gravity, General Relativity, contains Special Relativity as the limit for vanishing gravitational fields, and was created by Einstein in 1915. We will not consider General Relativity and gravity in this book, except for some considerations at the end of it.

1.2 The Principle of Relativity

Physical phenomena are normally described with respect to a given reference frame, which consists of a set of spatial axes and a set of synchronized clocks with respect to which one measures positions and time intervals. Newtons second law is valid in all inertial frames, which are defined by Newtons first law. According to this law, an inertial frame is one in which a free body (i.e., a body not subject to forces) is not accelerating. Any system moving with constant velocity with respect to an inertial frame is also an inertial frame.

The Principle of Newtonian (or Galilean ) Relativity states that

the laws of mechanics are invariant under change of inertial frame.

In other words, the outcome of a physical experiment performed in two different inertial frames must be the same; no experiment can detect a difference between the two frames. This statement applies not only to kinematics, but to any kind of (non-gravitational) physics. Therefore, there is no absolute motion through space, and there is no preferred inertial frame .

In pre-relativistic physics, the transformation from one inertial frame to another ( change of frame ) is given by the Galilei transformation . For simplicity, consider the motion of two inertial frames and such that the , and Cartesian axes of are parallel to the , and axes of , respectively, and such that the system moves along the -axis with constant velocity ( standard configuration ), as in Fig.. We adopt the convention that the velocity is positive if the system is moving away from .

Fig. 1.1

Two inertial frames and in standard configuration

Suppose that you synchronize clocks in the two inertial frames in such a way that the time indicated by a clock in

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