# Phenomenologies in Hypersphere Soliton and String Photon Models
## 4 SPM Predictions  
In this section, we will predict the physical quantities such as the photon intrinsic frequency and photon size in the SPM [13]. To do this, we will exploit the Nambu-Goto string action [17; 18] and its extended rotating bosonic string model in \(D=3+1\) dimensions of spacetime [29]. Note that in the \(D=26\) dimension open string theory, which will be briefly discussed below, it is well known that there exists the vector boson with 24 independent polarizations [10; 11], corresponding to the photon in the stringy photon model defined in the \(D=3+1\) dimensions of spacetime [13] considered in this paper. Before we construct the SPM, we pedagogically summarize a mathematical formalism for the Nambu-Goto open string, which is related with a photon. In order to define the action on a curved manifold, we introduce \((M,g_{ab})\), which is a \(D\) dimensional spacetime manifold \(M\) associated with the metric \(g_{ab}\). Given \(g_{ab}\), we can have a unique covariant derivative \(\
abla_{a}\) satisfying [30]  
\[\
abla_{a}\bar{g}_{bc} = 0,\] \[\
abla_{a}\omega^{b} = \partial_{a}\omega^{b}+\Gamma^{b}_{\,ac}  
Moreover, we proceed to construct the world sheet currents  
\[p_{0} = \frac{\kappa}{f}x_{i}^{\prime}x_{i}^{\prime},\quad\ p_{i}=-\frac{ \kappa}{f}[(\dot{x}_{j}x_{j}^{\prime})x_{i}^{\prime}-(x_{j}^{\prime}x_{j}^{ \prime})\dot{x}_{i}],\] \[\pi_{0} = -\frac{\kappa}{f}\dot{x}_{i}x_{i}^{\prime},\ \ \ \pi_{i}=-\frac{\kappa}{f}[(\dot{x}_{j}x_{j}^{\prime})\dot{x}_{i}+(1-\dot{x}_{j} \dot{x}_{j})x_{i}^{\prime}]. \tag{62}\]  
Now, exploiting (57), we obtain the string equation of motion  
\[\frac{\partial p_{\mu}}{\partial\tau}+\frac{\partial\pi_{\mu}}{\partial\sigma }=0, \tag{63}\]  
and the string boundary condition  
\[\pi_{\mu}(\tau;\sigma=0)=\pi_{\mu}(\tau;\sigma=\pi)=0. \tag{64}\]  
Inserting \(p_{\mu}\) and \(\pi_{\mu}\) in (62) into the string equation of motion in (63), we find  
\[\frac{\partial}{\partial\tau}\left(\frac{x_{i}^{\prime}x_{i}^{ \prime}}{f}\right)-\frac{\partial}{\partial\sigma}\bigg{(}\frac{\dot{x}_{i}x_ {i}^{\prime}}{f}\bigg{)} = 0,\] \[\frac{\partial}{\partial\tau}\left[\frac{(\dot{x}_{j}x_{j}^{ \prime})x_{i}^{\prime}-(x_{j}^{\prime}x_{j}^{\prime})\dot{x}_{i}}{f}\right]+ \frac{\partial}{\partial\sigma}\bigg{[}\frac{(\dot{x}_{i}x_{j}^{\prime})\dot{ x}_{i}+(1-\dot{x}_{j}\dot{x}_{j})x_{i}^{\prime}}{f}\bigg{]} = 0. \tag{65}\]  
Exploiting the boundary conditions in (64), we also obtain at \(\sigma=0\) and \(\sigma=\pi\)  
\[\dot{x}_{i}x_{i}^{\prime}=0,\ \ \ (1-\dot{x}_{i}\dot{x}_{j})x_{i}^{\prime}=0. \tag{66}\]  
Next, in order to describe an open string, which is rotating in the \((x_{1},x_{2})\) plane and residing on the string center of mass, we take an ansatz [29]  
\[x_{i}^{rot}=(r(\sigma)\cos\omega\tau,r(\sigma)\sin\omega\tau,0). \tag{67}\]  
Here, we propose that \(r(\sigma)\) and \(\omega\) represent respectively the diameter and angular velocity of the photon with solid spherical shape, which is delineated by the open string. Note that \(r(\sigma=\pi/2)\) denotes the center of the diameter of the string. More specifically, \(r(\sigma=\pi/2)\) is located in the center of the solid sphere, which describes the photon. The first boundary condition in (66) is trivially satisfied and the second one yields  
\[r^{\prime}(\sigma=0,\pi)=0. \tag{68}\]  
We then obtain \(r(\sigma)\), which fulfills the above condition in (68)  
\[\
abla_{a}\omega^{b} = \partial_{a}\omega^{b}+\Gamma^{b}_{\,ac}  
Moreover, we proceed to construct the world sheet currents  
\[p_{0} = \frac{\kappa}{f}x_{i}^{\prime}x_{i}^{\prime},\quad\ p_{i}=-\frac{ \kappa}{f}[(\dot{x}_{j}x_{j}^{\prime})x_{i}^{\prime}-(x_{j}^{\prime}x_{j}^{ \prime})\dot{x}_{i}],\]0  
Note that the photon has a finite size, which is filled with mass.