E=mc2 proof

Introduction: The "Proof" of Special Relativity:

When Einstein first proposed his Special Theory of Relativity in 1905 few people understood it and even fewer people believed it. It wasn't until 1919 that the Special Theory was "proved by inference" from an experiment carried out on his General Theory of Relativity. Physicists now routinely use relativity in experiments all over the world every day of the year. However, many of these experiments are highly specialised and usually require a great deal of knowledge and training in order to understand them. So what evidence is there for the general public? Probably the most spectacular "proof" is nuclear weapons. These pages are not about the morality of such weapons (but that's not to say that the question of their existence or use is not an important one). However, whether one "likes" nuclear weapons or not no one would deny that they exist.

Nuclear weapons (such as A- and H-bombs) are built on one principle; that mass can be turned into energy, and the equation that exactly predicts that conversion is E = mc². So what has that to do with Special Relativity? The answer is that E = mc² is derived directly from Special Relativity. If relativity is wrong, then nuclear weapons simply wouldn't work. Any theory or point of view that opposes Special Relativity must explain where E = mc²comes from if not relativity. Other models of relativity that contain E = mc² exist but here we are concerned with the "standard" model as proposed by Einstein.

This page explains, with minimal mathematics, how E = mc² is derived from Special Relativity. In doing so it follows the same theoretical arguments that Einstein used.

The Two Postulates:

The whole of special relativity is based on just two "rules", or as they are called in physics, postulates:

Postulate I: The principle of relativity: The laws of physics are the same in same in all inertial frames. Postulate II: The principle of the constancy of the speed of light: The speed of light (in a vacuum) has the same constant value c in all inertial frames.

Jumping from these postulates to E = mc² requires a little work. In order to understand the following arguments it helps to be familiar with Special Relativity, and in particular how moving at very high speeds dramatically changes the properties of mass and time. If you aren't familiar with these ideas you can read about the basics here.

An Apparent Increase in Mass due to Speed

One of the consequences of Special Relativity is that mass appears to increase with speed. The faster an object goes, the "heavier" it seems to get. This isn't noticeable in everyday life because the speeds we travel at are far too small for the changes to be apparent. In fact, an object needs to be moving at an appreciable percentage of the speed of light (186,300 mph, or 300,000 kilometres per second) before any mass increase starts to become noticeable in everyday terms. The equation that tells us by how much mass appears to increase due to speed is:

  Where:

• m = relativistic mass, i.e. mass at the speed it is travelling.
• m₀ = "rest mass", i.e. mass of object when stationary.
• v = speed of object.
• c = speed of light.

If we plot the equation as a graph we can see that at speeds close to the speed of light the mass greatly increases, tending towards (but never getting to) infinity:

Note that the graph shows that the mass can never be smaller than unity (1). This may seem a trivial point. After all, we can't just make the mass vanish into nothing. However, while seemingly unimportant, we will return to this point later and see that it is in fact essential to an understanding of how the equation E = mc² is derived.

Also note that the mass increase isn't felt by the object itself, just as the time dilation of Special Relativity isn't felt by the object. It's only apparent to an external observer, hence it is "relative" and depends on the frame of reference used. To an external observer it appears that the faster the object moves the more energy is needed to move it. From this, an external, stationary observer will infer that because mass is a resistance to acceleration and the body is resisting being accelerated, the mass of the object has increased.

Kinetic Energy

Next, we need to look at the energy involved in very high speed movement. We have seen that as an object gets faster its mass increases, and the more mass an object has the more energy is required to move it. The standard equation for the energy of movement (kinetic energy) is:

That is, kinetic energy is equal to half the mass multiplied by the velocity squared. This is often called Newtonian kinetic energy. Note that the velocity term is squared. This means, for example, that it takes far more than twice the energy to travel at twice any particular speed. We can see this by working through the equation for two values of v where: v = 50ms^-1 and v = 100ms^-1 respectively, both with the same mass of 10kg:

This equation is fine at "low" speeds, i.e. the speeds we encounter in everyday life. However, we know that mass appears to increase as the speed increases and so the Newtonian equation for kinetic energy must start to become inaccurate at speeds comparable to the speed of light. How do we compensate for the observed mass increase?

Relativistic Kinetic Energy and Mass Increase

In order to compensate for the apparent mass increase due to very high speeds we have to build it into our equations. We know that the mass increase can be accounted for by using the equation:

From this equation we know that mass (m) and the speed of light (c) are related in some way. What happens if we set the speed (v) to be very low? Einstein realised that if this is done we can account for the mass increase by using the term mc² . Using this term we now have an equation that takes into account both the kinetic energy and the mass increase due to motion, at least for low speeds:

This equation seems to solve the problem. We can now predict the energy of a moving body and take into account the mass increase. What's more, we can rearrange the equation to show that:

This result is fine for low speeds, but what about speeds closer to the speed of light? We know that mass increases at high speeds, but according to the Newtonian part of the equation that isn't the case. Therefore, we need to replace the Newtonian part of the formula in order to make the equation correct at all speeds. How can we do this? We know that E - mc² is approximately equal to the Newtonian kinetic energy when v is small, so we can use E - mc² as the definition of relativistic kinetic energy:

We have now removed the Newtonian part of the equation. Note that we haven’t given a formula for relativistic kinetic energy. The reason for this will become apparent in a moment. Rearranging the result shows that:

It can now be seen that relativistic energy consists of two parts. The first part is kinetic and depends on the speed of the moving body. The second part is due to the mass increase and does not depend on the speed of the body. However, both parts must be a form of energy, but what form? We can simplify the equation by setting the speed (i.e. the relativistic kinetic energy) of the moving body to be zero, thereby removing it from the equation: