Matter Cannot Travel Faster Than Light

Introduction

Einstein has once confidently pointed out that under the assumptions of special relativity, matter, anything that has weight, cannot travel faster than constant light speed $c$.

In this blog post, I would like to use two mathematical approaches to show that why this is the case.

Relativistic Momentum Approach

In my previous blog post, I have derived the expression of relativistic momentum.

$$
\begin{align}
p &= m_r v \\
&= \gamma m_0 v \\
\end{align}
$$

where $\gamma$ is the Lorentz factor, $m_r$ is the relativistic mass at the velocity of $v$, and $m_0$ is the rest mass which does not change with velocity.

$$
\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}
$$

Newton’s second law states that the rate of change of momentum of a body is directly proportional to the force applied, and this change in momentum takes place in the direction of the applied force.

$$
\begin{align}
F &= \frac{dp}{dt} \\
&= \frac{d(m_r v)}{dt} \\
&= \frac{d(\gamma m_0 v)}{dt} \\
&= m_0 \frac{d(\gamma v)}{dt} \\
&= m_0 \frac{d\bigg(\frac{v}{\sqrt{1 - \frac{v^2}{c^2}}}\bigg)}{dt} \\
&= m_0 \frac{d\bigg(\frac{1}{\sqrt{\frac{1}{v^2} - \frac{1}{c^2}}}\bigg)}{dt} \\
&= m_0 \frac{d ( v^{-2} - c^{-2} )^{-\frac{1}{2}}}{dt} \\
&= m_0 \frac{-\frac{1}{2} ( v^{-2} - c^{-2} )^{-\frac{3}{2}} (-2) v^{-3} dv}{dt} \\
&= m_0 \frac{ ( v^{-2} - c^{-2} )^{-\frac{3}{2}} (v^{2})^{-\frac{3}{2}} dv}{dt} \\
&= m_0 \frac{ \big( 1 - \frac{v^2}{c^2} \big)^{-\frac{3}{2}} dv}{dt} \\
&= m_0 \big( 1 - \frac{v^2}{c^2} \big)^{-\frac{3}{2}} \frac{dv}{dt} \\
\end{align}
$$

We could easily see that if we apply a constant force to matter, as the velocity $v$ of matter approaches $c$, $\big( 1 - \frac{v^2}{c^2} \big)^{-\frac{3}{2}} \rightarrow \infty$, the apparent acceleration $\frac{dv}{dt} \rightarrow 0$, meaning that the apparent acceleration becomes smaller with the constant force applied on the matter. When $v = c$, to make the matter travel faster than light, you would need to have a tiny apparent acceleration $\frac{dv}{dt}$. However, in this time, since $\big( 1 - \frac{v^2}{c^2} \big)^{-\frac{3}{2}} = \infty$, we would need to apply infinity large force $F$ to the matter, which is not plausible.

Mass-Energy Equivalence Approach

In my previous blog post, I have also derived the mass-energy equivalence.

$$
\begin{align}
E &= m_r c^2 \\
&= \gamma m_0 c^2 \\
\end{align}
$$

where $\gamma$ is the Lorentz factor, $m_r$ is the relativistic mass at the velocity of $v$, and $m_0$ is the rest mass.

We could easily see that as the velocity $v$ of matter approaches $c$, the Lorentz factor $\gamma \rightarrow \infty$, and the total energy of the matter $E \rightarrow \infty$. Based on the energy conservation, we would need to supply infinity energy to the matter to archive velocity $c$, which is not plausible.

References

• Relativistic Momentum and Relativistic Mass
• Mass-Energy Equivalence Derivation