This is quite a long page. It is designed to be read in sequence but you can jump to whichever section you are interested in by clicking on the menu.

• Introduction.
• What do the Letters Stand For?
• E = Energy.
• m = Mass.
• c = the Speed of Light.
• What Does the Equation Mean?
• Einstein’s Explanation of his Equation.

Without doubt E = mc² is the world’s most famous equation. This page explains in simple terms what it means and some of its consequences. The equation is derived directly from Einstein’s Special Theory of Relativity and other pages in this series deal with the mathematical and logical derivation. Here though, we will examine the equation as it stands and keep the mathematics to a minimum.

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Each of the letters of E = mc² stands for a particular physical quantity. Writing them out in full we get:

In other words:

• E = energy (measured in joules, J)
• m = mass (measured in kilograms, Kg)
• c = the speed of light (measured in metres per second, ms^-1)

Note that the case of each letter is important and it would be incorrect to show the equation as, for example, e = MC². This is because physicists use the case of letters as well as the letters themselves to denote particular physical entities, quantities and constants in equations.

In order for the equation to be correct we need to "square" the term c (the speed of light), i.e. we multiply the speed of light by itself: hence c² is the same as c times c. This allows us to be write the equation in another, slightly unusual, but equally correct way:

As a matter of interest, the equals sign was only invented during the 16th century, by the Welsh mathematician Robert Recorde. Apparently he was fed up having to write out "is equal to" in his work. He could of chosen any number of symbols but chose two parallel lines because, as he himself put it "noe 2 thynges can be moare equalle".

We will now examine each term (letter) in the equation in turn before addressing the question of what the equation means.

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The word "energy" is actually quite new. Its modern use dates from around the middle of the nineteen-century, when it was beginning to be realised that the power that drove many different processes could be explained by the concept of energy being transferred from one system and form to another. For example, the trains of the day were powered by coal. The coal was burned under a water-filled boiler to produce steam, which in turn pushed pistons attached to the wheels of the train, the wheels turned and the train was set in motion. In this example we start with locked up ("latent") chemical energy in the coal. The chemical energy is turned into heat energy (sometimes called "thermal" energy) by burning the coal and boiling the water. Finally, the thermal energy is turned into the energy of movement (kinetic energy) by forcing the steam into pistons to drive the wheels.

There are many other forms of energy, such as electrical, gravitational, nuclear, and strain energy. However, as different as all these types of energy seem they can all be measured in the same way and thought of as the same "thing". The unit that we use to measure energy, from whatever energy source, is the joule (J). Two ways in which we use this unit in everyday terms are:

• The total amount of energy in a system.

An example is a lump of coal, which when burned will release a certain number of joules of energy. Another example is food. A calorie (more formally called the small or gram calorie) contains almost 4.2 joules of energy, so if we eat a piece of chocolate that has 100 calories we can expect to get around 420 joules of energy from it.

Note: The labelling on food products in Europe shows kJ meaning kilo (1 thousand) joules, and kcal meaning 1000 calories, while in the U.S.A. the labelling on food products shows either "calories" or "Calories", with both taken to mean kilocalories. To add further confusion, "Calories" can also to be taken to mean kilocalories in Europe.

• Energy used up over time.

Most electrical devices have their power consumption rated in watts. A watt is a rate of energy consumption of one joule per second. So, if you have a light bulb in your room that is rated at 100W it is using energy at a rate of 100 joules every second. To go back to the example in the previous bullet point, a piece of chocolate with 100 calories isn’t very big but could easily ruin a planned daily diet. However, if we turned the chemical energy in our piece of chocolate into electrical energy (a process that can be done) it would only have enough energy to keep our light bulb shining for 4.2 seconds. Personally, I’d rather eat the chocolate.

So, to summarise, energy can come in many forms, and it can be transferred from one system to another. The basic unit of measurement for energy is the joule.

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Mass is strictly defined as a measure of a body’s inertia, i.e. its resistance to acceleration. Another and simpler way of defining mass is to say it is the total amount of matter in an object. This latter definition isn’t strictly true, but is good enough for our purposes here. Mass is measured in kilograms (kg).

Note that mass isn’t the same as weight, although it is often thought to be. Weight is actually a measure of the gravitational force (pull) felt by a body and is measured in newtons (N)  (note that scientific units that are named after people are almost always in lower case when spelled out fully, hence newtons and not Newtons). For example, an astronaut walking on the surface of the Moon has the same mass as he or she does on the Earth but only weighs one sixth what they would do back home. The reason for this is that while the mass of the astronaut hasn’t changed, the pull of the Moon’s gravity is only one sixth of what the Earth’s gravitational pull is.

As with energy, the idea that mass is common to all objects is relatively new and dates back to around the nineteenth century. Before that time, different solids, liquids and gases were all thought to be only loosely connected in conceptual terms. As with energy we now consider that mass is neither created nor destroyed, but is merely changed from one form to another, e.g. we can turn water from a solid (ice) into a gas (steam), but its total mass doesn’t change.

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We use the letter c to represent the speed of light. The ‘c’ comes from the Latin word "celeritas", meaning swift, and it’s a very apt definition - there is nothing faster than light. In a vacuum, such as space, it travels at close to 186,300 miles per second (300,000 km per second). That’s about seven times around the Earth every second!

The speed of light was first accurately estimated by the Danish astronomer, Ole Roemer (sometimes written as Rømer) during the 1670s. Up until that time everyone assumed that the speed of light was infinite, i.e. that light arrived at its destination instantly. This isn’t such an unreasonable assumption given that when we look around us light does indeed appear to reach us instantly.

During the seventeenth century it was discovered that there was a problem in calculating the orbital time of Io, the innermost moon of Jupiter. It sometimes took "too long" to make an orbit of the planet and at other times was "too quick". It was thought that the problem must be due to a wobble in the orbit of Io, but Roemer took a different, and very radical, view of the matter. He argued that light, instead of being everywhere instantly, had a finite speed and that this would explain the problem of Io. The Earth was known to travel around the Sun and this meant that sometimes the Earth was closer to Jupiter and sometimes further away. Roemer realised that when the Earth was on the opposite side of the Sun from Jupiter the light from Io would take longer to reach us than when the two bodies were on the same side.

This means that the light has to travel further and therefore takes longer, providing, of course, that light has a speed in the first place. During a meeting of the new Academy of Science in Paris in 1676 Roemer demonstrated that the amassed observational data of the astronomer Cassini indicated that Io would next appear at 5.25pm on November the 9th of that year. He himself predicted that it wouldn’t appear until 10 minutes and 45 second later, using his theory that light had a finite speed. The day came and virtually every major observatory in Europe was ready to test the prediction. At 5.25pm, the time predicted by Cassini, Io wasn’t visible. Even at 5.35pm Io wasn’t visible. But at exactly 5.35pm and 45 second it appeared, just as Roemer said it would. From this, it was also possible to estimate the speed of light and Roemer put it at about 670,000,000mph – very close to what we know it to be today.

You may think that that was the end of the matter and that Roemer was celebrated as a scientific genius, showered with honours and given a secure future. Sadly, that’s far from what happened. He was only 21 when he made his discovery, while Cassini was a well-respected if egotistical elder scientist, who used his powerful friends to back him up to rubbish Roemer’s ideas. Scientists, it seems, are human after all and this wasn’t the first, or sadly, the last time that an ego got in the way of a new discovery. Roemer eventually gave up science completely and later became the director of the port of Copenhagen and then head of the state council of the realm. It wasn’t until 50 years later that further experiments convinced the scientific community that Roemer had been right all along.

That the speed of light is so fast is difficult to believe. However, there is a simple experiment that can be done to get some idea of how fast it really is.

You will need two mobile (cell) phones and a friend in order to carry out this experiment. Sit on the other side of a large room to your friend and phone him or her. Ask your friend to speak loudly enough into the phone so that you can hear him or her across the room, as well as through the phone. On certain occasions you will hear your friend’s voice sooner through the phone than you do through the room.

When you speak into a mobile phone your voice is converted into electrical signals and beamed at (or very close to) the speed of light to the receiving phone via satellites or other similar equipment. The total distance of the signal path can be many thousands of miles, but it still arrives faster than the sound of your friend’s voice in the room. This is because the speed of sound, and so the speed at which your friend’s words travel across the room, is around 750mph while the electromagnetic (radio) signal from the phone is travelling very close to the speed of light, i.e.186,300 miles per second.

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The equation tells us that energy and mass are the same thing, and how much energy is contained in a given mass, or vice versa. That energy and mass are really the same thing is quite an extraordinary claim and seems to go against two laws that had been established by scientists before Einstein came along:

The Law of the Conservation of Mass.
As we have seen, mass can be thought of as the quantity of matter in an object. The law of the conservation of mass states that mass is always conserved. That is, whatever we do with matter in a closed system we will always have the same amount of substance at the end. For example, if we burn a log, the wood gets lighter as the fuel in it is used up. However, if we gather together the ashes, all the tiny smoke particles and the water vapour produced and then weigh everything we find that the mass is exactly equal to the mass of the log that was burned. Mass is just mass, or so it seems, and while it can be chemically altered, such as burned, the total amount in any system remains the same.

The Law of the Conservation of Energy.
But what about the energy released in burning the log? The energy released in the burning process is "chemical energy", i.e. the breaking and reforming of chemical bonds between particles. Burning the wood released the energy locked up in it. No energy was created in the process and none was destroyed, it was just changed from one sort of energy (chemical bonds) to other forms of energy (heat and light). In other words the total amount of energy, just like the total amount of mass, remained the same.

After many experiments, notably by the scientist for whom the unit of energy is named, James Prescott Joule (1818-1889), it was established that the total amount of energy in a closed system always remains the same. This is known as the law of the conservation of energy.

What Einstein showed via his now famous equation was that mass and energy are in fact the same thing. Converting one into the other doesn’t therefore violate either of the two conservation laws. Both quantities are conserved, although the state of the mass/energy may have changed. Each atom of a substance can be thought of as a little ball of tightly packed energy that can be released under certain circumstances. Likewise, we can take energy (such as particles of light, called photons) and turn it into matter. This was first achieved in the 1930s.

The picture below shows the first successful experiment in which energy was converted into matter:

The picture shows the tracks of two particles that have been "created" after a photon annihilated, i.e. "fell apart" in a cloud chamber. The high-energy photon is not in the visible range and has entered the chamber from the bottom of the picture. 

The subject of turning matter into energy is dealt with in other pages in this series.

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Poor Einstein must have spent a great deal of his life explaining his most famous equation (he had many others but most people weren’t as interested in those!). Thankfully, though, someone recorded him explaining it and you can hear it in his own words by downloading the file here (right click on the icon and choose Save As..;. or Save Target As... to download the file):

Einstein Speaking about the Equation E = mc² (1191Kbytes, .wav file) From the soundtrack of the film, Atomic Physics Copyright © J. Arthur Rank Organisation, Ltd., 1948.

The recording is old, and that, together with Einstein’s accent, sometimes makes it difficult to hear the words properly. This is a transcript of the recording:

"It followed from the Special Theory of Relativity that mass and energy are both but different manifestations of the same thing - a somewhat unfamiliar conception for the average mind. Furthermore, the equation E is equal to mc², in which energy is put equal to mass, multiplied with the [by the] square of the velocity of light, showed that very small amounts of mass may be converted into a very large amount of energy and vice versa. The mass and energy were in fact equivalent, according to the formula mentioned before [E = mc²]. This was demonstrated by Cockcroft and Walton in 1932, experimentally."

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E=mc^2 The Basics

© Jim Doyle
Date created: 18 Oct 00
Last updated: 25 Feb 06