The Theory of Special Relativity

The Special Theory of Relativity

David Brückner

Lancing College, West Sussex, United Kingdom

March 2012

Until the end of the nineteenth century, the simple Galilean principle of relativity was used to relate physical observations in one frame of refer- ence to another moving relative to it. When the phenomena of electro- magnetism and light where unified in Maxwell’s equations, this principle was first called into question as it stood in conflict with the idea of ab- solute time and motion. The most famous experiment that attempted to determine the absolute motion of the earth, the Michelson-Morley ex- periment, will be discussed here. Subsequently, the ideas and postulates contained in Einstein’s first paper on relativity will be introduced and hence the kinematic transformations based on the principles will be de- rived and their implications on the relativity of space and time as well as on Newtonian mechanics will be stated.

1 Galilean Invariance

The principle of relativity was first formulated by Galilei in 1661 in his book ”Dialogues on two world systems” where he makes observations on the invari- ance of physical events on a moving boat as opposed to a stationary one [?], in other words, physical phenomena are unaffected by the choice of reference frame from which they are observed [?]. The principle assumes that there is a universal time shared by all reference frames. Considering F = ma leads to the restriction to unaccelerated, so called inertial frames. Only then will the motion of a particle be correctly described in all reference frames [?, ?].

To test whether an equation obeys the principle of relativity, a transforma- tion that translates the values of physical observables from one frame to another is required. Consider two frames K and K′ with K′ moving at velocity v rela- tive to K in the x-direction as shown in Figure 1. The Galilean transformation relates the spacetime-point (t, x) in K to the same point (t′, x′) in K′, provided the systems coincided when t = t′ = 0 in the following fashion:

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Figure 1: Two inertial frames K and K’ which moves at speed v relative to K.

These have long been thought to be consistent with Newton’s laws as experiments demonstrate that the mass and force are constant in all inertial frames, unless one is dealing with speeds close to that of light [?]. The transformation should hence yield a = a’ and indeed

(2)

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(3)

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As this simple principle works so nicely in Newtonian mechanics, it was for long seen as being self-evident. However, when at the end of the nine- teenth century the long investigations into the phenomena of light, electricity and magnetism culminated in Maxwell’s equations of the electromagnetic field, first published in 1865 [?, ?], which describe these phenomena in one uniform system, these transformations were called into question as Maxwell’s equations did not seem to obey the principle of relativity [?, ?]. That is, if the transfor- mations are substituted into Maxwell’s equations, they do not remain the same which would imply for example, that electrical and optical phenomena could be used to determine the absolute speed of a spaceship without looking at the surroundings which violates the principle of relativity.

One of the implication of Maxwell’s equations is that light propagates at a constant speed c ≈ 3 × 10⁸ ms−¹, independent of the motion of the source [?]. This brings up an interesting problem: Consider a spaceship moving with speed v and light from the rear travelling at speed c. According to (2), the speed measured in the spaceship should hence be c − v and not c. It should therefore be possible to measure the absolute velocity of the earth with respect to a universal reference frame, the hypothetical aether that was supposed to pervade all space.

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Figure 2: Schematic representation of the Michelson-Morley experiment.

2 The Michelson-Morley Experiment

The most famous of the many experiments that have been performed on this matter is the experiment conducted by Michelson and Morley in 1887 [?, ?] using an interferometer as shown schematically in Figure 2. Their idea was to measure the difference in the time of travel of a light beam in two perpendic- ular directions [?]. The apparatus consists of a sodium light source A shining monochromatic light onto a semi-silvered glass plate B wich splits the incoming beam into two beams continuing in mutually perpendicular perpendicular di- rections to the mirrors C and E, where they reflect back to B. On returning to B, the are joined into two superimposed beam, D and F. If the time of travel for the beams are equal, the waves will be in phase but if they differ slightly, interference will occur [?, ?].

It can be shown, that if the apparatus is at rest in the aether, no interference will occur but if it is moving at a velocity v to the right, the times should differ. First, let us calculate the time taken to travel from B to E and back. Let the time to travel forth be t1 and back t2. As the apparatus moves a distance vti during the time of travel, it can be said that

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In order to account for the possible difference in l1 and l2, the apparatus was turned through 90◦ to record the shift in interference [?]. Taking this into account, it was expected that there would be interference as th = tv making it possible to measure the absolute velocity of the earth. However, no interference was found - the result of the experiment was zero [?, ?]. The experiment was repeated many times and other experiments were performed but they all gave the same result [?]. The velocity of the earth through the aether could not be detected.

3 Einstein’s Postulates

The question why Nature would apparently yield up no information about our motion with respect to a hypothetical fundamental frame of reference troubled the minds of some of the best physicists of the nineteenth century. Most of them took the view that the aether existed but that special mechanisms were at work that would undo every phenomenon that would permit a measurement of the absolute velocity v [?]. The first fruitful idea for such a mechanism came from Lorentz and Fitzgerald (independently) in 1892. They suggested that material bodies contract along their d√ction of motion through the aether and if this contraction is by a factor of 1 − v² /c², the zero fringe shift follows directly [?, ?].

However, this result seemed to be too artificial, designed solely for explaining away the difficulties. The French mathematician Poincaré was the first one who suggested that there is such a law of nature, that it is impossible to discover an absolute velocity by experiment [?].

It was Einstein who, instead of imposing preconceived ideas on the facts, brought a grand clarity of outlook in his 1905 paper ”Zur Elektrodynamik be- wegter Körper”. He pointed out that the analysis of motion had always been Hence, the total time of horizontal travel th is

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based on the assumption that there was a universal, absolute time. The following quote from the paper [?] describes the starting point of his argument better than any paraphrase could:

We need to consider that all our judgements in which time plays a part are always judgements of simultaneous events. If, for example, I say ”That train arrives here at 7 o’clock,” I mean something like this: ”The pointing of the hand of my watch to 7 and the arrival of the train are simultaneous events.”

This is of course almost trivial but Einstein goes on arguing that the case be- comes problematic if it concerns the relationship between events that occur at different locations in space. Consider two observers with a clock at points A and B that can both make time-related observations in their surroundings. Ac- cording to Einstein’s analysis, it is not possible to make comparisons of events in A and B without further assessment as, even though there is an A-time and a B-time, no common time has been defined. This can only be done by defining that the time light takes to travel from A to B is equal to the time it takes to travel the way back. Consider a light beam that leaves A when the A-time reads tA, reaches B at B-time tB , is reflected back to A and reaches A at tA. Then, by Einstein’s definition, the clocks run simultaneous if

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A corollary of this definition is that the velocity of the light signal, given by

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will have the same value in all circumstances, hence V = c [?]. This leads to the definition of the following two postulates [?]:

Postulate 1: All inertial frames are equivalent with respect to all the laws of physics.

Postulate 2: The speed of light in empty space always has the same value c.

It is striking that a whole new dynamics can be built on these two short statements that will fully resolve the inconsistencies of theory and experiment with an exalted simplicity which could only have been discerned by a genius mind like Einstein.

4 The Lorentz Transformations

In order to test these postulates on Newton’s and Maxwell’s equations, a new kinematic transformation from one frame to another moving relative to it is re- quired. These transformations were first formulated by Lorentz in 1904, a year

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Figure 3: (a) Space-time diagram showing an experiment to define simultaneity at A and C which are at rest in this reference frame K. (b) Equivalent experiment with A, B and C at rest in K’ which is moving relative to K as seen in K. before Einstein’s paper, when he worked out that Maxwell’s equations would remain unchanged under them [?, ?].

Here, they shall be derived from first principles (based on a method from [?]) like Einstein did independently of Lorentz. To derive the Galilean trans- formations (1), one only had to look at the geometry in the space dimensions as a universal time was presumed. Now space-time geometry has to be consid- ered as, like it could be seen in Einstein’s thought experiments, our judgement of time and simultaneity are a function of the particular reference frame used [?].

Consider again the inertial frames K and K′ with K′ moving at speed v along the x-axis relative to K. Three observers A, B and C are at rest equally spaced along the x-axis of K. Their world-lines are vertical as their spacial positions are constant. Now consider a light signal sent out at B at t = 0 whose world line will be x = xB ± ct. The arrival of the signal at the other observers is given by the intersection of the world lines of A and C and the world line of the signal, A1 and C1 (see Figure 3a). Simultaneity in the positions A and C is hence given by the line A1C1.

Now consider A, B and C at rest in K′. Their world lines with respect to K′ are now at an angle as they change position relative to K at a rate v. The arrival of the signal at A′¹ andC¹ isnotsimultaneousinKasA¹ C¹ isnot parallel of the x-axis in K (see Figure³ b). As a consequence of the¹. Postulate however, A1 and C1 are simultaneous events in K′. This now allows us to add the coordinate axes of K′: The axis of x′ is the line that is parallel to A1C1 since this is the line defining simultaneity and any line t′ = λ where λ is a constant is parallel to the x′-axis. The t′-axis is simply the world line of the origin of K′.

The point event P can therefore be expressed alternatively by x and t or

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Figure 4: The point event P as seen in K and K’.

by x′ and t′ (see Figure 4). The relativity principle implies that the kinematic transformations must be bilinear, meaning that x′ and t′ are both linear func- tions of x and t and vice versa, because otherwise, a motion at uniform velocity in K would not be seen as at constant velocity in K′. This symmetry implies that:

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The velocity of K′ relative to K in terms of λ and µ can be found by looking at the motion of the origins by substituting x = 0 into (6.1) and x′ = 0 into (6.2). This gives v = µ/λ.

The link between the two frames is the light beam as it is the only motion invariant under motion. A signal sent of at O will be described as x = ct and x′ =

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can be obtained. By elimination of t and t′ and using µ = λv one finds c² = λ² (c² − v² )

and hence

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The required transformation can now be found by using λ = γ(v) and again µ = λv in equation (6):

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which reduce to the Galilean transformation (1) at low speeds, when v/c → 0. By simple rearranging, the transformation in time can now also be found:

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It follows directly from the 1. Postulate that the quantities that measure dis- tance transverse to the direction of motion are equivalent in all frames because otherwise, there would be ways of detecting absolute motion and displacement [?, ?], so:

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The equations (8), (9) and (10) together form the Lorentz transformations which express any K′ coordinates in terms of K and vice versa.

These transformations have consequences that go against the assumptions of classical physics. From the point of view of an observer at rest, a clock in motion will have go slower than his and a rod in motion will have a shorter lenght than the same rod at rest [?]. These corollaries are called time dilation and length contraction.

The time dilation can be derived by considering two events (x0, t1) and (x0, t2) in K. The time coordinates in K′, moving at v relative to K, are t′¹ = γ(t1 − vx0/c² ) and t′² = γ(t2 − vx0/c² ). The lapse of time in K′ in terms of the observed difference in K is therefore

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This is often written with[Abbildung in dieser Leseprobe nicht enthalten] as

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The length contraction is easily found by considering a body whose two ends are marked by x1 and x2 in K and consequentially has length l0 = x2 − x1. In K′ this distance at the same instance is judged as[Abbildung in dieser Leseprobe nicht enthalten] where,by transformation, [Abbildung in dieser Leseprobe nicht enthalten]Thus,

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and hence,

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which is exactly the contraction proposed by Lorentz and Fitzgerald.

We can conclude, that the time passes more slowly for a moving observer than a stationary one and that the measured length of a material body is less in moving frame than in the rest frame [?, ?]. This has been verified by many experiments and observations for instance by the elongation of the half-life of muons travelling at a speed close to c [?].

[...]

¹ G. Galilei, Dialogues on two world systems. Translated by T. Salusbury, 1661, available at http://archimedes.mpiwg-berlin.mpg.de/cgi-bin/ toc/toc.cgi?step=thumb&dir=galil_syste_065_en_1661, accessed 22. February 2012.

² R. P. Feynman, Six Not-So-Easy Pieces. Penguin Books, 1st ed., 1999.

³ E. Taylor, J. Wheeler, Spacetime Physics: Introduction to Special Relativity. 2nd ed., 2001.

⁴ A. P. French. Special Relativity. M.I.T. Introductory physics series. Chapman and Hall, 1st ed., 1968.

⁵ H. Lipson, The Great Experiments in Physics. Oliver and Boyd, 1st ed., 1968.

⁶ B. Ridley, Time, space and things. Penguin Books, 2nd ed., 1984.

⁷ A. Einstein, Relativity - The Special and General Theory. Translated by R. W. Lawson, Routledge, 1st ed., 1960.

⁸ W. Rindler, Introduction to Special Relativity. Oxford Science Publications, 1991.