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 $E=mcÂ˛$. 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.

# [Physics] Speed of light and infinite energy

energymass-energyspecial-relativityspeed-of-light

#### Related Solutions

It is well established by now that the speed of light barrier does not apply to quantum particles, and this property makes the construction of relativistic quantum field theories and other relativistic quantum systems tightly constrained. The argument that you can't transmit signals faster than light is fine, but particles are not necessarily signals, because if you make a particle over here, and measure a particle over there, you might not be measuring the particle you created, but another identical particle you created from the vacuum.

So relativistic field theory, with its faster-than-light particles, requires that there are no unique particles, that all particles have identical copies. Further, the faster than light motion can be back-in-time in different frames, and back-in-time motion means that every particle must have a back-in-time partner, called its antiparticle.

The quantum field restores locality. So even though quantum particles can propagate faster than light, information can't propagate faster than light. The quantum fields are the quantities which tell you what information you can gain locally by experiments. The particles in a Hamiltonian formulation are nonlocally related to the quantum fields (but the two are related more simply in a particle path integral formulation).

The proof that quantum particles cannot be restricted to less than light speed is simple: the restriction that the energy is positive means that the frequency is positive, while the restriction to forward propagation inside the light cone means that the propagator vanishes outside the future lightcone, so in particular, into the past. and there is no function which vanishes in a half-plane whose Fourier transform also vanishes in a big unbounded region like this. This is covered in this answer: the causality and the anti-particles

For a basic treatment of the Michelson-Morley experiment please see 1. It's not important to know the technical details of the experiment to answer your questions though. The only relevant thing is the result, let me put it in basic terms since you seem to struggle with the "physics slang":

*While the total velocity of a ball thrown from a truck is the sum of the velocity of the ball relative to the truck and the velocity of the truck relative to the observer, the velocity of a light beam emitted from the truck is not. Much more the velocity of the light beam seems completely independent of the velocity of the truck.*

Michelson and Morely didn't have a truck, they had the earth orbiting the sun.

Please make it clear to yourself that this experimental fact can be explained by **stating** that the speed of light is constant. If I say to you the speed of light is constant in every frame of reference, then the above result isn't surprising at all to you.

But you want more. You want me to prove to you that the speed of light is universally constant. I cannot. There will never be an experiment that shows that this **axiom** is universally true. How should one ever construct such an experiment, how should one, for example, test the theory in the Andromeda galaxy? It's impossible, but it doesn't matter: Why not just stick with the axiom, as long as we can explain everything we see around us with it?

As you already said there's an interesting connection between the invariance of the speed of light and Maxwell's equations. One can indeed prove that the speed of light has to be constant, otherwise, Maxwell's theory can't be true for all inertial frames. But this is no proof that can convince you either, since accepting Maxwells equations is no different to accepting the invariance of the speed of light. Furthermore, the basis of Einstein's theory is not the invariance of the speed of light, but the invariance of the speed of action. Which cannot be concluded from Maxwell's theory, even though it's a reasonable guess.

**Physical theories are not provable. But as long as they comply with reality, we accept them as truths.**

Addendum: I recommend this short lecture for layman by R. Feynman on the topic. Feynman and I present a very similar line of reasoning.

## Best Answer

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.