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Lei Mao

Machine Learning, Artificial Intelligence, Computer Science.

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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