Question

Start with a railroad car on a frictionless track and use the example of a pulse of light with energy ?E traveling from one end of the car to the other to derive ?E = ?mc 2 . You must define all terms carefully and explain each step.

Answer 1

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 many experiments all over the world every week of the year. However, 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 that special relativity is correct? Probably the most spectacular "proof" is nuclear weapons. These pages are not about the morality of such weapons (but that is not to say 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.

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 (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 look at this as a graph we can see that at speeds close to the speed of light the mass increases greatly, 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.

n 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 (the exact arguments and mathematics required to derive this are quite advanced, but an example can be found 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:

We know that E - mc² is approximately equal to the Newtonian kinetic energy when v is small, so we can use E - mc²as the definition of relativistic kinetic energy:

We have now removed the Newtonian part of the equation. Note that we haven