**2. Special relativity: a quick review**

#### **2.1 A brief history**

Towards the end of the nineteenth century there was one major inconsistency plaguing the structure of theoretical physics. Newton's <sup>1</sup> equations described very well the mechanics of moving objects ranging from tiny objects on earth to the orbits of planets in space. Maxwell<sup>2</sup> had successfully completed the theoretical framework of electromagnetism in 1865, a monumental task that crowned the gradual understanding of electromagnetism throughout the eighteenth and nineteenth centuries through the work of scientists such as Coulomb<sup>3</sup> , Ampère<sup>4</sup> and Faraday<sup>5</sup> . Despite the

<sup>1</sup> Sir Isaac Newton: English mathematician, physicist, engineer, philosopher, astronomer, theologian and author (1642–1726).

<sup>2</sup> James Clerk Maxwell: Scottish mathematician and physicist (1831–1879).

<sup>3</sup> Charles-Augustin de Coulomb: French engineer and physicist (1736–1806).

<sup>4</sup> André-Marie Ampère: French physicist and mathematician (1775–1836).

<sup>5</sup> Michael Faraday: English physicist (1791–1867).

*Perspective Chapter: Slowing Down the "Internal Clocks" of Atoms – A Novel Way… DOI: http://dx.doi.org/10.5772/intechopen.102931*

enormous success of both Newton's and Maxwell's frameworks in describing mechanical objects and light, respectively, they were inconsistent with each other. Maxwell's equations predicted a constant speed of light and Newton's equations suggest that objects moving at different speeds should measure the speed of light to be different. On a more rigorous level, Newton's equations are invariant under Galilean<sup>6</sup> transformations while Maxwell's equations are invariant under Lorentz<sup>7</sup> transformations. So, each set of equations suggested a different symmetry present in nature. Moreover, in 1887, on the experimental side, Michelson<sup>8</sup> and Morley<sup>9</sup> tried to detect a difference in the speed of light for observers moving at different speeds but their results were decidedly negative. There were some attempts to resolve this consistency but it was Einstein,<sup>10</sup> who, in 1905, successfully presented the correct solution through a radical theory that would change our understanding of nature forever.

To resolve the inconsistency Einstein suggested that:

1.The laws of physics are invariant in all inertial frames of reference.

2.The speed of light in vacuum is constant in all frames of reference.

Simply put, he suggested that Newton's equations were fundamentally wrong and he replaced them with the equations of SR that were invariant under Lorentz transformations. The implications were dramatic: different observers experience different rates of time, lengths can shrink or elongate and many other peculiar effects take place as objects approach the speed of light. Newton's equations were only a very good approximation as long as objects moved slowly compared to the speed of light.

#### **2.2 Mathematical framework**

There are many ways to approach SR from a mathematical point of view. In this chapter we will present the mathematical framework of SR in a simple manner.

#### *2.2.1 Galilean transformations*

Limiting our consideration to one spatial dimension for simplicity, the most general way one can transform between two coordinate systems *O* and *O*<sup>0</sup> , where *O*<sup>0</sup> is moving with speed *v* in the positive *x*-direction compared to *O*, is the following:

$$
\begin{bmatrix} x' \\ t' \end{bmatrix} = \begin{bmatrix} a & b \\ e & f \end{bmatrix} \begin{bmatrix} x \\ t \end{bmatrix} \tag{1}
$$

Assuming that space is homogeneous and noticing that the principle of relativity requires that *O* moves at speed -*v* compared to *O*<sup>0</sup> , we are restricted to linear transformations. A deeper mathematical analysis of this is outside the scope of this chapter.

<sup>6</sup> Galileo Galilei: Italian astronomer, physicist, engineer, philosopher, and mathematician (1564–1642).

<sup>7</sup> Hendrik Antoon Lorentz: Dutch physicist (1853–1928).

<sup>8</sup> Albert Abraham Michelson: German-born American physicist (1852–1931).

<sup>9</sup> Edward Williams Morley: American physicist (1838–1923).

<sup>10</sup> Albert Einstein: German physicist (1862–1943).

Taking all this into account the forward and backward transformations are calculated to be:

$$
\begin{bmatrix} \mathbf{x'}\\ t' \end{bmatrix} = \begin{bmatrix} a & -av\\ \mathbf{1} - a^2 & a \end{bmatrix} \begin{bmatrix} \mathbf{x} \\ t \end{bmatrix} \tag{2}
$$

$$
\begin{bmatrix} \mathbf{x} \\ \mathbf{t} \end{bmatrix} = \begin{bmatrix} a & av \\ \frac{a^2 - 1}{av} & a \end{bmatrix} \begin{bmatrix} \mathbf{x'} \\ \mathbf{t'} \end{bmatrix} \tag{3}
$$

The Galilean transformations coincide with our everyday intuition (*a* = 1). Velocities are additive, acceleration is invariant and time is the same for all observers.

$$\mathbf{x}' = \mathbf{x} - vt \tag{4}$$

$$t'=t\tag{5}$$

#### *2.2.2 Lorentz transformations*

Going back to the general form of the transformations, we have:

$$\mathbf{x}' = a\mathbf{x} - bt\tag{6}$$

$$
\infty = a\mathfrak{x}' + bt'\tag{7}
$$

By setting *x*<sup>0</sup> ¼ 0, we can calculate the relative velocity of *O*<sup>0</sup> with respect to *O*

$$V'=b/\mathfrak{a}\equiv\mathfrak{v}\tag{8}$$

and similarly, by setting *x* ¼ 0, we calculate

$$V = -b/\mathfrak{a} = -v \tag{9}$$

Assuming the speed of light is constant in all inertial frames of reference, we consider a light signal when the origins coincide (*t* <sup>0</sup> ¼ *t*, *x*<sup>0</sup> ¼ *x*). The propagation of the light signal in both frames is:

$$\mathfrak{x} = \mathfrak{ct} \tag{10}$$

$$\mathfrak{x}' = \mathfrak{ct}'\tag{11}$$

Substituting Eqs. (10) and (11) into Eqs. (6) and (7) yields:

$$ct'=(ac-b)t\tag{12}$$

$$ct = (ac + b)t'\tag{13}$$

Substituting Eq. (13) into Eq. (12) and using Eq. (8) yields:

$$a = \frac{1}{\sqrt{1 - v^2/c^2}} \equiv \gamma \tag{14}$$

$$b = av = \eta v \tag{15}$$

*Perspective Chapter: Slowing Down the "Internal Clocks" of Atoms – A Novel Way… DOI: http://dx.doi.org/10.5772/intechopen.102931*

Substituting Eqs. (14) and (15) into Eqs. (6) and (7) yields:

$$\mathbf{x}' = \boldsymbol{\gamma}(\mathbf{x} - \boldsymbol{\nu}t) \tag{16}$$

$$\mathbf{x} = \mathbf{y}(\mathbf{x}' + \mathbf{v}t') \tag{17}$$

By substituting Eq. (16) into Eq. (17) the transformation of time is obtained:

$$t' = \chi \left( t - \frac{v\varkappa}{c^2} \right) \tag{18}$$

$$t = \chi \left( t' + \frac{\nu \chi'}{c^2} \right) \tag{19}$$

#### *2.2.3 Time dilation*

Consider an observer in the frame of reference O at the origin, so *x* ¼ 0 with a clock that has a period Δ*t* ¼ *t*<sup>2</sup> � *t*1. For the observer in *O*<sup>0</sup> that period is much longer, namely

$$
\Delta t' = \chi \Delta t.\tag{20}
$$

Hence, the clock is ticking slower for the moving observer.

#### *2.2.4 Length contraction*

Similar to the peculiar effect of time dilation, a moving observer experiences a contraction in length along the direction of movement. Consider an observer in the frame of reference O that measures a moving ruler to be of length Δ*x*. This measurement happens instantly at one point in time such that *t*<sup>2</sup> ¼ *t*1. This is an important detail as simultaneous times in one frame are not simultaneous in the other one. For the observer in *O*<sup>0</sup> that length transforms to

$$
\Delta \mathbf{x}' = \mathbf{y} \Delta \mathbf{x}.\tag{21}
$$

Now, by symmetry, i.e., a ruler at rest in *O*, that is measured to be of length *Lo* in *O*, is measured to be of length

$$L'\_o = \frac{1}{\mathcal{Y}} \Delta L\_o \tag{22}$$

in *O*<sup>0</sup> . Alternatively, we could have considered an object at rest in *O* at the beginning but this would have added an extra step in the derivation as we would have considered the transformations both in space and time. Either way we arrive at the same result; moving objects experience length contraction along the direction of motion.

With that knowledge in mind, we are now ready to discuss the proposed experiment.
