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### Derivation of E=mc²

16 Sep 2016
Introduction: The “Proof” of Special Relativity
When Einstein first proposed his Special Theory of Relativity in 1905 few people understood it and even fewer 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 specialized 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 = mc2 comes 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.

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 (186,000 miles per second, or 300,000 kilometres per second) before any apparent 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 examine a table of representative values for the speed of light (below) we see that mass hardly increases at all until we reach about 75% of c, but then starts to climb very rapidly. The second set of columns show that beyond 99.9% of c the mass increase is very rapid indeed for even just a very tiny increase in speed, tending towards (but never quite getting to) infinity.
Note that the mass can never be smaller than unity (i.e.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 appears to increase, 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. So, 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 realized that if this is done we can account for the mass increase by using the term mc2 (the exact arguments and mathematics required to derive this are quite advanced, but an example is provided here). 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:

We now have the famous equation in the form it’s most often seen in, but what does it mean?
We have seen that a moving body apparently increases in mass and has energy by virtue of its speed (the kinetic energy). Looking at the problem another way we can say that as the speed of a body gets lower there will be less and less kinetic energy until at rest the body will have no kinetic energy at all. So far so good, but what about the mass due to the speed of the body? Again, as the body slows down the mass will become progressively smaller but it can’t reach zero. As noted near the start of the page, the lowest the mass can be is unity (1) and we can’t just make the body disappear into nothing. The lowest possible mass the body can have is its “rest mass”, i.e. the mass the body has when it is at rest. But the equation we have derived (E = mc2) isn’t for mass, it’s for energy. The energy must somehow be locked up in the mass of the body.
Einstein therefore concluded that mass and energy are really different manifestations of the same thing, i.e. that any mass is really tightly packed energy. At the time he saw no mechanism for releasing the energy from the mass, and was in fact skeptical of the idea that it could ever be achieved. This didn’t really matter to Einstein, however. As a theoretical physicist he was happy that his equations were consistent and he had a model to predict what happens to a body moving at very high speeds.

The Full Form of the Equation
So far we have referred to the energy of very high speeds as “relativistic kinetic energy”. This is fine for allowing us to work quickly through the equations and to keep them simple, but there has to be a more formal way of expressing what we mean. As with the term mc2, the detailed derivation of the full equation for E = mc2 is quite complex. However, to those familiar with the basic mathematics of Special Relativity the way in which we take into account the kinetic energy of E = mc2 will come as no surprise:
For any non-stationary body the total energy is given as:

This equation takes into account the total energy (E), the mass of the body (m), and the speed of the body (v). As such it accounts for both the relativistic mass increase and the relativistic kinetic energy.

Conclusion
The subject of this page is quite difficult to understand, even though much of the more difficult mathematical considerations have been left to one side. The conceptual jump from the two postulates of Special Relativity to the equivalence of mass and energy is certainly not obvious, and it’s extraordinary that Einstein proposed it long before there were any experimental results to indicate the true nature of the relationship between mass and energy.
The results of Einstein’s work in this area are far more widespread than is usually thought and affects everyone on the planet. As with all science we can use the results for “good” and “bad”. The reason the words “good” and “bad” are in quotes is because it all depends on your point of view. For example, you may think that nuclear power stations (which use E = mc2 directly) are either a good or a bad thing. Likewise, depending on your point of view, nuclear weapons are either a good or a bad thing; they either ended one war and prevented another, or they are immoral and bound to fall into the wrong hands sooner or later. On the other hand, in recent years there have been great advances in using E = mc2 in the medical field, particularly to treat cancer. Again though, this can be seen as either a good thing (i.e. curing a disease) or a bad thing (i.e. overburdening an already over-populated planet).
Although these issues are undoubtedly important it’s not for pages such as this to hold an opinion either way, but to merely explain some of the science behind them. It’s far too late to “un-invent” E = mc2 and the best we can do is to use it in an informed way for the things that we believe are worthwhile.