Thereís no Such Thing as Gravity:
The World Just Sucks!

A slightly different approach to general relativity
for high school physics students

By Mark Sowers

Table of Contents

Chapter 3:
E=mc2

My goal in this paper is to use as little math as possible, and yet I just used an equation as the title of this chapter. That's because as equations go, E=mc2 is a pretty good one. Itís one of those rare equations that does explain some important concepts, if you know how to read it. So letís try to read it.

E

E stand for Energy. Most people understand what energy is. It is what we use to heat our homes, to power our cars, even to run our computers. Energy takes many different forms: electrical, heat, light, motion, etc. And it can be stored in just as many ways. The most common way to store energy is chemically, either as fuel, like wood or oil, or in batteries. When chemical bonds are formed or broken, energy is released and that is how we get energy out of these stored forms.

Take an average 6-volt lantern battery. The kind used in those big flashlights. One of these batteries is able to store and release about 0.1 kilowatt-hours of energy, the amount of energy it takes to light a 100-watt bulb for about 1 hour. After that hour, the battery is dead and most people throw it away. Ahem, I mean dispose of it properly so it doesnít end up in a landfill, right?

But thereís actually a lot (and I mean A LOT!) of energy still in that battery. Weíll get to that in a moment.

m

m stands for mass. A simple way of thinking of mass is weight. Our lantern battery weighs about 1 kilogram (about 2.5 pounds). The reason we say mass instead of weight is that weight depends on where you are; mass does not. On the surface of the Earth that battery weighs 2.5 pounds, but on the moon it weighs less than half a pound. Is the battery any different? No. It still has just as much Ďstuffí in it. It still has just as much mass. It just weighs different because gravity is different.

c

c stands for the speed of light, which as we saw earlier is about 186,000 miles per second. For this equation weíll use meters per second, so the speed of light is 299,792,458 m/s.

The speed of light is important because while the battery itself may be just sitting there, all the atoms, all the electrons, protons, neutrons that make up those atoms, and all the quarks that make up those protons and neutrons are nothing but little waves of energy. And those little waves of energy are each moving (in their own little space) at the speed of light

2 (squared)

Speed-squared is actually a simplification. Unfortunately because of this simplification the real meaning behind the physics is lost.

The idea of an object having energy on its own isnít really useful. Instead energy comes from an object interacting with another object. The idea of the energy of a billiard ball is meaningless until it collides with another billiard ball. The idea of the energy of a photon is meaningless until it interacts with your retina. It is the interaction with another object that allows energy to take form. And it is the speed of that second object that determines how much energy the first object has.

Imagine a car traveling down a highway at 60mph. An observer standing on the side of the road would say that car has a lot of energy. Now suppose that car rear-ends another car that was only going 58mph. Does that second car feel all the energy that the stationary observer said it had? No. According to an observer in the second car, the first car had only a little bit of energy, so it didnít do much damage.

Now imagine the second car headed in the opposite direction, towards the first car at 60mph. This time the observer in the second car thinks the first has a lot more energy, and when they hit, the first car will do a lot more damage.

The point is that different observers will measure different amounts of energy. Energy is relative to the observer. This is where the term Ďrelativityí comes from. The speed of the observer is just as important in determining the energy of an object as the speed of the object itself. More specifically, the energy of an object is determined by multiplying the speed of that object by the speed of the observer.

Letís try to put this into an equation. If you are in a car headed down the road at 60mph and another car is headed towards you at 60mph, the amount of energy you will feel is found by multiplying the mass of the oncoming car (m1), by its speed (v1), and by your speed (v2). Doing this gives the equation E=m1v1v2.

Notice the simplicity of this equation. The three elements that make up energy, each equally important, multiplied together.

Despite its simplicity, if you have been doing physics for a while you are probably a little uncomfortable with the above equation. This equation looks similar to, but not exactly like other equations you may be used to. To truly understand energy, it is important to see exactly what is going on, and the equation for energy written this way really does explain what is going on. It shows very clearly that the speed of the observer (v2) is just as important as the speed of the object (v1).

The reason you donít see this version of the equation very often (or at all), is that this equation comes with certain qualifications. Specifically, the measurements of the two speeds must be made from a specific frame of reference.

As youíve probably already figured out, the equation above does not work in all frames of reference. Letís look at the case where car 1 is headed east at 90mph (relative to the ground) and car 2 is headed west at 30mph (relative to the ground). If the observer is in car 2, he would measure the speed of the oncoming car (relative to himself) as 120mph, and his own speed (relative to himself) as 0mph. Of course anything multiplied by 0 is 0, so this answer wouldnít be helpful. The same would happen if the measurements were taken from car 1ís perspective. Even from a stationary position on the ground, the answer would be incorrect. So which frame of reference is the correct one?

Isaac Newtonís third law of motion states that for every action there is an equal and opposite reaction. When two objects collide, they will exert an equal and opposite force on each other. When two identical cars collide, they will sustain equal amounts of damage, regardless of which frame of reference their speeds were measured from.

So the correct frame of reference is the one in which the two speeds are equal and opposite. One car going east at 60mph hitting another car going west at 60mph (equal and opposite) is exactly the same as one car going east at 90mph hitting another car going west at 30mph, which is exactly the same as one car going east at 120mph hitting another car at rest.

To mathematically get to this specific frame of reference, you need to find the difference between the speeds of the two cars. Then you assign half of that difference to each car. This results in an equation for energy that works in all frames of reference.

E=m1

Of course for most physics problems v2, the speed of the observer, is assumed to be 0, in which case the equation becomes...

E=m1

...which simplifies to...

Remember this is the equation for the amount of energy you will feel. And remember that the other car sustained equal damage; it feels just as much. So the total energy from the collision is twice what you feel, or...

Squaring the speed of the object is just the result of a mathematical simplification. It works, but only when you assume that the speed of the observer is 0. What happens when it is not?

We found the speed of each of the little waves of energy within our lantern battery (v1) to be c. Which means we must also find that the speed of each of the little waves of energy within ourselves (v2) to be c. Because c is a lot different than 0, we canít use the simplified equation; we must go back to the original...

E=m1v1v2

...which becomes...

E=m1c1c2

...which under most circumstances can be very closely approximated as...

E=mc2

(note: We do not double the energy because if we squeeze every last drop of energy out of the lantern battery, the battery itself is annihilated. It no longer exists, so it canít feel any energy. Therefore the total energy is just the energy that you feel.)

E=mc2

So now we are ready to put some numbers into our equation to find the true energy of our lantern battery, which weighs 1 kilogram. Energy = 1kg(m) times 299,792,458m/s(c, the speed of each atom-part in the battery) times 299,792,458m/s(c, the speed of each atom-part in the observer). When we do the math we get 89,875,517,900,000,000 joules, which is a whole lot of energy. This inherent energy of the battery is also known as its rest energy.

89,875,517,900,000,000 joules equates to 24,970,000,000 kilowatt-hours of electricity, or enough to power all of New York City for 6 months. Thatís a lot of energy from one little battery.

Of course we have no way of getting all that energy out of that battery. But just knowing that the energy was there was enough to spur the race for the atomic bomb in the 1930s and 1940s. Even the most efficient bombs today extract only a small fraction of the energy contained within the bomb materials.

In a bomb, matter itself is converted into energy. A more general statement would be that matter itself is simply energy contained within a small area. Thatís the beauty of E=mc2. Just as electricity and magnetism are simply two different forms of the same thing, E=mc2 shows that matter and energy are simply two different forms of the same thing. Energy equals matter.

Because there is so much energy within ordinary objects, any miniscule change in the speed of light (c) will have a big effect on the energy measured. But can the speed of light really change?