Speed of light and infinite energy

Question

While browsing some physics websites, I saw that to make an object reach the speed of light, it requires infinite energy and talked about its relation with Einstein's famous equation $E=mc²$. However, they didn't show how they reached the conclusion that it requires infinite energy to reach the speed of light. I would like to know how it was proved.

Answer 1

It has been observed that the momentum of an object is

$$p=\frac{m_0v}{\sqrt{1-\frac{v^2}{c^2}}}$$

So as $v$ approaches $c$, the bottom term approaches zero and therefore the momentum approaches infinity.

To increase the momentum of an object you give it additional kinetic energy. To increase the momentum infinitely takes infinite energy.

Answer 2

However, they didn't show how they reached the conclusion that it requires infinite energy to reach the speed of light.

In the context of special relativity (SR), an object with invariant mass $m$ (the mass of the object as measured in the inertial reference frame (IRF) in which it is at rest) has energy

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

where $v$ is the speed of the object in the IRF in which the energy $E$ is measured (that is, the energy of an object is reference frame dependent).

This follows straightforwardly from the Lorentz transformations and has been experimentally confirmed.

Clearly, one cannot set $v = c$ in the above equation because the expression would be undefined (the denominator is zero for $v = c$).

However, we can note that as $v \rightarrow c$, the energy $E \rightarrow \infty$ thus the expression infinite energy would be required to accelerate to the speed of light.

On my view, this is a clumsy way of saying that massive objects cannot have speed $c$ in any IRF since, according to the Lorentz transformation, an object with speed $c$ in an IRF has speed $c$ in all IRFs, i.e., there is no IRF in which the object is at rest (this is crucial).

But this contradicts the premise that the object is massive since, by stipulating that the object has invariant mass $m$, we imply that there is an IRF in which the object is at rest.