**Appendix A. Relativistic frequency shifts**

We will derive the relativistic frequency shift formula in two steps. First we will consider a classical example to derive the classical Doppler shift then we will consider the relativistic effects involved in the second step.

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

#### **A.1 Classical Doppler shift**

Consider a resting source emitting a photon with frequency *f* <sup>0</sup> and period *T*<sup>0</sup> collinear with an observer moving with speed *u* in the same direction as the emitted photon. We denote the frequency observed as *f <sup>S</sup>*. The observed wavelength *λ<sup>S</sup>* is given by

$$
\lambda\_{\mathbb{S}} = (\mathfrak{c} - \mathfrak{u}) T\_0. \tag{44}
$$

Hence,

$$\frac{c}{f\_S} = \frac{c-u}{f\_0},\tag{45}$$

and so,

$$f\_{\mathfrak{d}} = (\mathfrak{1} - \mathfrak{beta}) f\_{\mathfrak{S}},\tag{46}$$

where *<sup>β</sup>* <sup>¼</sup> *<sup>u</sup> c* .

#### **A.2 Relativistic Doppler shift**

