# Laser-Accelerated Plasma-Propulsion System
## 2 Relativistic Propellant  
Although only the exhaust flow may move at relativistic velocity, it is convenient to introduce the problem using a fully covariant formalism. A relativistic reaction engine (basically a thruster based on a particle accelerator) expels, in an infinitesimal time interval, an infinitesimal exhaust mass at relativistic velocity with respect to the rocket rest frame. As presented in Figure 1, the problem can be schematized as a 1D system in which the four-velocity of the rocket is given by:  
\[u^{u}=\gamma(u)(c,u), \tag{1}\]  
where \(c\) is the speed of light in vacuum while \(\gamma(u)=(1-(u/c)^{2})^{-1/2}\) is the relativistic factor associated with the scalar rocket velocity \(u\). We will use the standard Einstein notation in which the 1D covariant four-velocity is given by \(u_{\mu}=\eta_{\
u\mu}u^{\
u}=\gamma(u)(c,-u)\), where \(\eta_{\mu\
u}\) is the Minkowski tensor with trace (\(+,-\)). The product is given by \(u^{u}u_{\mu}=\gamma^{2}(u)(c^{2}-u^{2})=c^{2}\) (summation over indices). From the differential of the four-velocity \(du^{\mu}=\gamma^{3}(u)(u/c,1)du\), we can easily construct the scalar relation:  
\[du^{u}du_{\mu}=-\gamma^{4}(u)(du)^{2}=-(cd\theta)^{2}, \tag{2}\]  
where the rapidity \(\theta\equiv\tanh^{-1}(u/c)\) has been introduced to simplify the notation. Sometimes, the quantities \(\sigma=c\theta\), which is referred as the proper speed of the rocket, is used instead of the rapidity. If an infinitesimal mass \(dm\) is expelled from the rocket, the instantaneous conservation of the four-momentum implies:  
\[d(Mu^{\mu})+dmv^{\mu}=0, \tag{3}\]  
where \(M\) is the instantaneous rocket mass while \(v^{\mu}\) is the propellant four-velocity. Differently from \(dm\), \(dM\) does not correspond to the propellant mass but, more correctly, represents the mass \"lost\" by the rocket. In fact, a fraction of \(dM\) is also converted into kinetic energy according to relativistic principles. The limit is represented by the so-called photonic rocket [1] (a photon-based engine), in which the propellant mass is zero. In this case, the total photons four-momentum is given by \(dmv^{\mu}\equiv\sum hk^{\mu}\), where \(k^{\mu}\) is the wave four-vector. An other independent equation can be retrieved directly from Equation (3) once contracted with \(u_{\mu}\). Considering that \(du^{\mu}u_{\mu}=0\), \(u^{\mu}u_{\mu}=c^{2}\) and \(v^{\mu}u_{\mu}=c^{2}\gamma(v_{r})\), we obtain:  
\[dmc^{2}+dmc^{2}\gamma(v_{r})=0, \tag{4}\]  
where \(d(Mu^{\mu})u_{\mu}=dmu^{\mu}u_{\mu}+Mdu^{\mu}u_{\mu}\) has been used. As expected, the relevant speed is represented by the propellant relative velocity \(v_{r}\), which is measured from the rocket rest frame. In more detail, the term \(v^{\mu}u_{\mu}=\gamma(u)\gamma(v)(c^{2}-uv)\), which is invariant under Lorentz transformations, has been rearranged to \(c^{2}\gamma(v_{r})\), expressed in the rocket frame. Thus, Equation (4) can be used to eliminate \(dm\) from Equation (3):  
\[Mdu^{\mu}+dM(u^{\mu}-v^{\mu}/\gamma(v_{r}))=0. \tag{5}\]  
From Equation (5), once the terms are separated and squared, we finally obtain the differential invariant rocket equation:  
\[M^{2}(-cd\theta)^{2}=-(dM)^{2}v_{r}^{2}, \tag{6}\]  
where the relation (2) and \(u^{\mu}u_{\mu}=v^{\mu}v_{\mu}=c^{2}\) have been used. It is interesting to observe that Equation (6), which is valid also for photonic rockets, does not contain any explicit propellant relativistic factor. Considering that the rapidity \(d\theta\) is an additive function, Equation (6) can be directly integrated once chosen \(Md\theta=-dMv_{r}/c\) as a physical solution, and gives:  
\[\theta=\frac{v_{r}}{c}\ln\left[\frac{M_{i}}{M}\right], \tag{7}\]  
where \(M_{i}\) and \(M\) are the initial rocket mass and the instantaneous rocket mass respectively, calculated in the rocket rest frame. Using the proper speed \(\sigma=c\theta\) instead of rapidity, we find that Equation (7) is formally identical to the classical non-relativistic [PERSON]'s rocket equation [28].