What is mass?

The more you learn about physics, the more confusing it gets.

What is mass? Should be easy enough, right? Turns out that it's not that simple.

Classical Physics

There are two kinds of mass: gravitational mass and inertial mass.

Gravitational mass is the mass in Newton's Law of Universal Gravitation.

There. basically this formula describes the (attractive) force between two
bodies of mass m_{1} and m_{2}, at a distance r (between the
centers of the masses). There's a funky constant (G) in there
as well. The m1 and m2 are *gravitational* masses.

Why do you care? Well, this law of
gravitation thing keeps your feet on the ground. You see, one of the masses
often represents the mass of the earth. And the earth has quite a bit of mass,
roughly 6.0 x 10^{24} kg.
That's a 6 followed by 24 zeros. The other mass often represents your own mass (* not*
weight). The result is that the earth attracts you (and you attract the earth).

OK. Now for the other type of mass: *inertial*
mass. This is the mass in another one of Newton's laws (Newton was a rather
bright guy):

More precisely, that should be written as **F**
= d**p**/dt = d(m**v**)/dt = **v**dm/dt + md**v**/dt. Since dm/dt is
0 is most practical scenarios, the equation reduces to **F** = m**a** with
**a** being d**v**/dt of course.

The formula describes the relationship between a force (F) applied to an object with a certain mass (m), and the acceleration (a) that is a result of this.

Inertia can be described informally as
resistance against acceleration (sometimes described as resistance against
motion, but that's just plain wrong). The more mass an object has the harder you
have to push it to give it a certain acceleration. It's easier to throw a golf
ball than it is to throw a basketball. Note that this is **not** because the
basketball has a greater *weight*, it is because the basketball has a
greater *mass*. All this applies in the International Space Station (where
stuff is *weightless*) as well as on earth. Although (at least on earth)
mass and weight are intimately connected, they are fundamentally two *very*
different things.

This whole thing about ** weight** is
a can of worms which I shall leave closed for now...

If you paid attention, you probably noticed that this force thingy (F) appeared in both of the previous formulas. Let's manipulate them a little, and define the following:

m_{1} = M_{E} = the mass of the
earth

m = m_{2} = the mass of a person being attracted by the earth's gravity.

So, then we get:

Now, if we equate the two expressions for F to find a person's acceleration (while jumping down, for example) due to gravity, we get:

or

So, what is so interesting about this? Well, the acceleration of an object due to the earth's gravity is independent of the objects mass! Every object is accelerated the same, heavy or light.

Everyday experience seems to contradict this. A
feather doesn't fall nearly as fast as a brick. That's because the atmosphere
(air) messes things up here. On the moon (which has no atmosphere worth
mentioning) a feather and brick **do** fall at the same speed.

Galileo (another bright man) was one of the first, if not the very first, to realize this. In his days people were convinced that heavier objects fall faster than light objects. Galileo's argument against this went something like this: say I have a metal ball that weighs 1 kilogram. A ball that weighs 2 kilograms is supposed to fall twice as fast. Now, what if I take two balls of 1 kilogram and connect them with a piece of string? Will the balls fall twice as fast just because they are now connected with a piece of string?

If you paid very close attention you may have
noticed that we did something that is somewhat questionable in order to get to
our formula for a. We deleted m from both sides of the equation. But on
the left side, m represented *inertial* mass and on the right side it
represented *gravitational* mass! What we did is only allowed if the two
are identical (and not zero, but that's splitting hairs).

**In other words: is inertial mass the same as
gravitational mass?**

As far as science knows, the answer to that is: yes.

Just for fun let's calculate the acceleration a due to the earth's gravity, usually indicated with "g". This is a simple matter of substituting:

G = 6.67 x 10^{-11} m^{3} kg^{-1}
s^{-2 }, M_{E} = 5.97 x 10^{24} kg, r = 6.37x 10^{6}
m, and we find for g ( = a): 9.8134 ms^{-2}

The official value (apparently) is : 9.8067 ms^{-2}.
Close enough.

I'm an engineer. One of the things you need to learn as an engineer is to decide when things are "good enough".

Relativity

Leave it to Al Einstein to complicate things.

Let's look at the most famous equation in the history of physics.

This equation relates the *energy* (E) of
an object to it's *mass* (m), where c is the speed of light in a vacuum.

That c indicates the speed of light in a *
vacuum* is rather important, since light waves propagate at different
speeds in different media. The speed of light in vacuum is the universal
speed limit. Nothing can go faster. As far as we know.

Einstein's famous equation is part of the
theory of relativity. Note that there are two theories of relativity, the *
special* theory of relativity and the *general* theory of relativity.
While mere mortals at least have *some* chance of comprehending special
relativity, general relativity requires a degree in higher mathematics before
you can even start to try to figure it out.

Anyhow, I always interpreted E = mc^{2}
as a formula to calculate the relationship between mass and energy when mass is
converted to energy and energy is converted to mass. I always considered mass as
a "frozen" form of energy, so to speak. I'll get back to that later on.

Nuclear Physics

Not that I know much about nuclear physics, but anyway.

An atom's nucleus consists of *nucleons*.
A nucleon can either be a *proton*, which has a positive electric charge,
or a *neutron*, which has no electric charge. Nucleons are made up of
(three) quarks. There's also gluons that keep the quarks together. Gluons have
no mass as far as scientists know, so we are ignoring them for now.

A proton is made up of two up quarks and one down quark. A neutron is made up of two down quarks and one up quark.

Quarks actually do have mass. For example, an
up quark weighs about 1.5 - 4.0 MeV/c^{2}

Whoa. Let's explain this a little. Particle
physicists prefer to express masses of particles in electron Volts (eV). An
electron Volt is nothing other than the charge of an electron multiplied by
one Volt. This gives a result that has the physical dimension of *energy*
(Joules). Energy and mass are related by E = mc^{2}, so
it the correct way is to say that mass has a physical dimension of eV/c^{2},
but I guess that's just too cumbersome, since physicists typically express
masses in eV. The electron Volt is a bit of an impractical unit, so
particle masses are typically expressed as MeV (millions of electron Volts)
or GeV (billions of electron Volts).

Then, a down quark weighs about 3.5 - 6.0 MeV/c^{2}

Note that these masses are *not* very well
known at all. Fortunately, that doesn't matter for the point I'm going to
make.

If we add up the quark masses for a proton, we get a range of 6.5 - 14.0 MeV.

If we look up the (rather well known) mass of a
proton, we find that it is 938.272 MeV/c^{2}

Errr, that doesn't add up at all, now does it? A proton is between 67 times and 144 times heavier than the sum of its parts.

**Where does the difference come from?**

Binding energy. The ("strong") nuclear force
that binds the quarks together into a proton is rather strong. Guess that's why
it's called the *strong* force. Strong forces result in high (binding)
energies.

So, the vast majority of the mass in a proton
(or neutron for that matter) is actually *energy*.

So, my earlier assumption that mass and energy
can be converted into each other is at least half inaccurate. "Energy can be
converted into mass" No! Energy **is** mass! Both gravitational and inertial
mass as a matter of fact, since the two are identical.

So, does the opposite: "mass **is** energy"
hold true? Short answer: don't know.

All the above was inspired by the fact that I
was reading a book (called *The Lightness of Being)* about physics,
matter and the like. The writer is a man named Frank Wilczek, and he must know
what he is talking about since they gave him the Nobel Prize for physics in
2004.

I have a problem with books about physics.
They are either too much aimed at the general public, and as a consequence
hardly contain any equations, since conventional wisdom states that every
equation halves the sale of a popular physics book. Or, they are textbooks
aimed at university physics students, and they make your head spin after two
pages. You have to spend six months brushing up on your math just to be able
to read them. There seems to be no middle ground. One notable exception is
*Relativity* by Einstein himself. If I read that one four more times
maybe I'll understand a little of it.

One thing is for certain: mass **can** be *
converted* into energy, the process obeying E = mc^{2}. Enter the
bizarre world of antimatter. This stuff is not just a brain child of the writers
of Star Trek, antimatter (or at least antiparticles) is (are) very real.

Take for example the antiparticle of the
electron, the *positron*. This is a particle that is identical to an
electron, but with an opposite electrical charge: the positron has a positive
electrical charge. If you bring an electron and a positron together close
enough, they *annihilate*: poof, gone, both of them. All that's left is
pure energy, and yes, the relationship between the combined masses and the
produced energy does indeed obey E = mc^{2}

So, could the produced energy be *binding*
energy, just like in the case of the proton discussed earlier? The answer is:
unlikely. The established opinion in particle physics is that an electron is *
not* built up of sub-particles like the nucleons are. Also, the quarks
discussed earlier do have *some* mass "of their own", the mass of a nucleon
is not totally made up of energy. There is no reason to suspect that quarks can
be split.

Wilczek's book addresses the same question in a way. The answer to "what is mass" is basically: "well, in the case of nucleons it's mostly energy, and as far as the quark and electron masses are concerned, we don't know".

Relativistic Mass

I'm gong to go crazy now and put one of my own conclusions here. It may be totally wrong.

If you read about relativity, one thing you'll find out that, as mentioned earlier, there is an absolute speed limit in the universe and that is the speed of light in vacuum (c). This fact is one of the main postulates of the theory (theories) of relativity. Apparently it was necessary to assume this, since Maxwell's laws of electromagnetism would not make any sense at all without it. Don't ask me why.

One of the results of relativity is that the
mass of an object increases as its velocity increases. This is called the *
relativistic* mass. The relationship between *relativistic* mass (m_{r})
and *rest* mass (m_{0}) is as follows.

Where v is the velocity of the object, and c is the light speed in vacuum.

Smarter guys than me have figured this out, so why question it?

Anyway, that relationship neatly enforces the universal speed limit: as an object's velocity approaches the speed of light, it's relativistic mass becomes larger, hence it is harder to accelerate the object even more. For v = c, the relativistic mass is infinite, so that cannot happen.

Great. Now consider a particle in a particle accelerator. These things typically travel at more than 99% of the speed of light. If we put more energy in them (with magnetic fields) in order to accelerate them even more, where does the energy go?

Well, in the classical, non-relativistic case,
the *kinetic* energy of an object (or particle) is given by E_{k}
= 0.5 x mv^{2}

So, if we apply a force to an object, we
increase its velocity, thus increasing its kinetic energy. Back to the particle
that is already at 99% of the speed of light. We cannot increase its speed much
more, since the particle is already very close to the universal speed limit. So,
where does the energy go that we add to it? Well, the only way is to increase
its **mass**! As per the equation above.

Then again, I cannot help but wonder if I am confusing cause and effect here. Like I said, it may be totally wrtong. Must think more about it...

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