### Lei Mao

Machine Learning, Artificial Intelligence, Computer Science.

# 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.

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.

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.

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.

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.