## 3 Materials and Methods
### Recovered Frequency Standard Defined from a Special Relativity Perspective  
When ground stations and satellites are in non-inertial motion relative to an inertial frame of reference, clock synchronization requires a definition distinct from that used for inertial or stationary states within the same frame. This synchronization framework should be interpreted in the context of the Minkowski spacetime. Since bidirectional signaling between satellites and ground stations occurs at the speed of light, different coordinate times taken during the uplink and downlink processes result in picosecond-order asymmetry in the bidirectional satellite link. The concept of \"satellite Einstein synchronization\" has previously been introduced in our work and is briefly explained herein [25; 27]. Figure 4 shows the schematic of the bidirectional information exchange between a satellite and a ground station. The ground station antenna's position vector, projected onto the equatorial plane, interacts dynamically with the Earth's rotation and the satellite's synchronized orbit. In the Minkowski spacetime, this creates equidistant spirals, denoted as G (ground station) and S (satellite). The signal transmission from the ground station is represented by the coordinates \(t_{1}\), \(G_{1}\), and \(S_{1}\). Here, \(t_{1}\) denotes the coordinate time of the event, \(G_{1}\) denotes the ground  
Figure 4: Schematic diagram of round-trip frequency transfer from earth station to satellite. station's projection on the two-dimensional spatial plane, and \(S_{1}\) denotes the position of the GEO satellite within this plane. For the event where the signal arrives at the satellite, we use \(t_{2}\), \(G_{2}\), and \(S_{2}\). The retransmission of the signal by the satellite is denoted by \(t_{3}\), \(G_{3}\), and \(S_{3}\). The time taken for the signal to pass through the satellite's transponder is negligible compared to the transmission time within the satellite link; hence, the reception and retransmission of the signal by the satellite is considered to occur simultaneously. Finally, the event of the signal returning to Earth and being received by the ground station is denoted by \(t_{4}\), \(G_{4}\), and \(S_{4}\). Assuming that on the GEO satellite a time reference (\(\hat{T}^{(s)}\)) exists, which is perfectly synchronized with the local time (\(T_{2}^{(g)}\)) observed at the ground station whenever the satellite receives a signal, multiple ground stations can independently define their local time references on the satellite. This enables instant determination of the time and frequency comparisons of these local times directly on the satellite. Clearly, the introduced time reference (\(\hat{T}^{(s)}\)) has a unique significance. This concept is named satellite-based Einstein synchronization. The formula for \(\hat{T}^{(s)}\) is derived using approximate geometric calculations based on the principles of special relativity, expressed as follows:  
\[\left\{ \begin{array}{l}\hat{T}^{(s)}=\frac{1}{2}\Big{(}T_{1}^{(g)}+T_{4}^{( g)}\Big{)}-\Sigma\cdot\gamma\\ \Sigma=-\frac{\omega_{0}\overrightarrow{r}_{e}\times\overrightarrow{r}_{s} }{c^{2}-\omega_{0}^{2}\overrightarrow{r}_{e}\cdot\overrightarrow{r}_{s}} \\ \gamma=\frac{\sqrt{c^{2}-\omega_{0}^{2}\overrightarrow{r}_{e}\cdot \overrightarrow{r}_{e}}}{c}\end{array} \right. \tag{4}\]  
where \(T_{1}^{(g)}\) and \(T_{4}^{(g)}\) denote the local times recorded at the ground station for the signal emission and reception events during the bidirectional exchange, \(c\) denotes the speed of light, \(\omega_{0}\) denotes the angular velocity of Earth's rotation, \(\overrightarrow{r}_{e}\) denotes the vector from Earth's center to the ground station projected onto the equatorial plane, and \(\overrightarrow{r}_{s}\) denotes the vector from Earth's center to the satellite projected onto the equatorial plane.