## What amount of energy is required to maintain constant acceleration?

Given is a body $\mathcal{B}$ that has a constant acceleration. Let the acceleration be $a$. Let us assume that the body starts at rest. If so then the travelled distance is given by $s(t) = \frac{1}{2} a t^2$ and the velocity is given by $v(t) = at$. So there is a critical time $t_c$ for what the velocity becomes the speed of light. This critical time is given by $\displaystyle t_c = \frac{c}{a}$. However, to maintain the constant acceleration $a$, energy is required. By definition, the amount of energy is given by

(1) $\displaystyle W = \int_{s_0}^{s_t} d\vec{s} \vec{F}$

The force is given by $\displaystyle \vec{F} = \frac{d\vec{p}}{dt}$, valid both for Classical Mechanics and Special Relativity. The difference is however, that the momentum is different for both Classical Mechanics and Special Relativity. For Classical Mechanics, the momentum is given by $\vec{p} = m \vec{v}$ and for Special Relativity, the momentum is given by $\displaystyle \vec{p} = \frac{m_0 \vec{v}}{\sqrt{1 - v^2/c^2}}$. Let us assume that the force is acting in the direction of the velocity, then we obtain

(2.1) $\vec{F}_\textrm{C} = m \vec{a}$ and

(2.2) $\displaystyle \vec{F}_\textrm{R} = \frac{m_0 \vec{a}}{\sqrt{1 - v^2/c^2}^3}$

Then it is clear that

(3.1) $\displaystyle W_\textrm{C} = \int_{s_0}^{s_t} ds m a = \left[ \frac{1}{2} m v^2 \right]_{v_0}^{v_t}$ and

(3.2) $\displaystyle W_\textrm{R} = \int_{s_0}^{s_t} ds \frac{m_0 a}{\sqrt{1 - v^2/c^2}^3} = \left[ \frac{m_0 c^2}{\sqrt{1 - v^2/c^2}} \right]_{v_0}^{v_t}$.

Using that $v_0=0$ and $v_t = a t$ we obtain

(4.1) $\displaystyle W_\textrm{C} = \frac{1}{2} m a^2 t^2$ and

(4.2) $\displaystyle W_\textrm{R} = \frac{m_0 c^2}{\sqrt{1 - \left( \displaystyle \frac{at}{c} \right)^2}} -1$

Then we obtain a special time for Special Relativity – where the amount of energy goes to infinity. It is clear that this time is given by $\displaystyle t_{W_\infty} = \frac{c}{a}$. Then we see that $t_c = t_{W_\infty}$, meaning that the critical time is reached when an infinite amount of energy is required to maintain the constant acceleration of the body $\mathcal{B}$. Therefore in Special Relativity, a body $\mathcal{B}$ can have a constant acceleration $a$ during some time-interval that is given by $[0,c/a]$; assuming that the body $\mathcal{B}$ starts at rest. Since an infinite amount of energy is already required to maintain this interval, exceeding the time $c/a$ is not possible. In Special Relativity a body can have a constant acceleration, but not for ever.