Most physicists admit that they don’t know what mass is but they all say that they know how to measure it. That’s an oversimplification. The measurement of mass is a non-trivial subject. Millions of man-hours and millions of dollars have been dedicated to the measurement of particle masses? Why? Because our theories cannot calculate the particle masses from first principles. At this point we have no choice but to use the measured values of particle masses as inputs into our theories.

**Uncertainties and error bars**

Most measurements in particle physics are statistical measurements. In other words, we measure the same quantity over and over again. Each measurement yields a slightly different number. We then take the average of these numbers and present it as the measured value of that particular quantity. Measured values always come with error bars. It is very important to understand the error bars.

Why does each measurement yield a slightly different number? There are experimental errors associated with instrumentation. There are also intrinsic uncertainties arising from the quantum nature of particles. Intrinsic (quantum) uncertainties cannot be eliminated. Instrumentation can be improved but there are limits. There are also statistical errors that can be improved by collecting more data (repeating the experiment).

The way forward is to repeat the experiment as many times as possible. That’s what particle physicists do. They design better colliders and detectors to increase the frequency of measurements. More data yields smaller statistical errors.

How do we measure the mass of electrically charged particles? How do we measure the mass of particles that carry no electric charge? How do measure the mass of unstable particles that live for a fraction of a second? How do we measure the mass of quarks inside the protons and neutrons? These are very serious questions. Experimental physicists have been working on perfecting the measurement techniques for years.

**Measurement of mass**

If we know the total energy and momentum of a particle we can then extract the mass of that particle.

The total energy of elementary particles are measured by calorimeters which are specialized instrumentation inside the large particle detectors like CMS and ATLAS at the Large Hadron Collider.

The momentum of electrically charged particles are determined from the curvature of their paths in uniform magnetic fields.

For example, we can shoot a proton into a field of magnetic field that we control. We know the field strength of this magnetic field precisely. We then measure the curvature of the proton path in this magnetic field.

The bending radius of a particle moving in a uniform magnetic field is

where is the momentum of the particle, is the speed of light in vacuum and is the electrical charge of the particle.

The formula above is saying that the bending radius is directly proportional to the particle’s momentum and inversely proportional to the magnitude of the charge and the strength of the magnetic field

the momentum will be

Once we know the total energy and momentum we can determine the mass using this relationship

where is the “invariant mass.”

**Relativistic mass**

After Einstein demonstrated the mass-energy equivalence popularly known as

the term in the formula above was referred to as the “relativistic mass” but the actual formula is

where is the momentum of the particle and is the speed of light in vacuum.

Particle physicists do not use the term “relativistic mass” because it is just another name for the total energy except for a proportionality constant.

**Invariant mass**

When particle physicists use the term “mass” they mean the “invariant mass” which is represented by in the formulas.

The formula

clarifies that the “relativistic mass” is a combined effect of the momentum and the “invariant mass.” The “relativistic mass” can increase as we push the particle harder and harder towards higher energies but the “invariant mass” never changes.

In the past the term “invariant mass” was known as the “rest mass.”

**We don’t know what “invariant mass” is**

What is the ontological status of ? That’s the million dollar question. We really don’t know what “invariant mass” is. We have been extracting the “invariant mass” of elementary particles from experiments. For example, in units of energy, electron “invariant mass” is 0.510998910 Mev. The muon “invariant mass” is 105.6583715 Mev. We don’t have a theory that can explain these numbers. We don’t know why the muon “invariant mass” is 206.7683 times more than the electron’s “invariant mass.”

**Why ?**

The reason Einstein advocated the instead of was the following. According to Einstein’s General Relativity theory the gravitational force is an outcome of the total energy distorting the local space-time. In other words, the gravitational force is proportional to the . Gravitational force acts on the total energy not on the “invariant mass” per se. Photons have zero “invariant mass” but they still feel the gravitational force.

**Gravitation is so different**

Magnetic field acts on the electric charge of the particle. Photons are not influenced by magnetic fields because photons do not carry any electric charge. Gravitational force, on the other hand, does not care about the electric charge but acts on the total energy of the particle instead. All known forces except gravity act on various charges but gravity only cares about the total energy .

**Inertial mass**

The resistance to acceleration is known as the inertial mass. In other words, the inertial mass is the mass that appears in Newton’s second law

where is the acceleration.

**Equivalence of inertial and gravitational masses**

You may remember from science classes that when you drop different objects with different weights from a high place they reach the ground at the same time. This is a well established fact. Experiments with many different objects have been performed countless times. We are assuming of course that the air resistance is negligible. In the absence of air resistance, all objects – light or heavy – accelerate towards the ground at the same rate and reach the ground at the same time.

We also know that gravitational force is proportional to the mass of the test object (as expressed by the Newton’s law of gravitation).

where is the gravitational mass. The can be interpreted as the gravitational acceleration. The has the same units as acceleration.

This means that, when is the the inertial mass is the same as .

When Einstein developed his General Relativity theory as a replacement for Newton’s theory of gravity, he did not want to derive this principle from a more fundamental principle. Instead, he took the principle of the equivalence of inertial and gravitational masses as an axiom and developed the General Relativity on this foundation. Einstein, in effect, claimed that it is virtually impossible to differentiate the artificial gravity felt by an astronaut inside a circular space station as shown below (standing on the outside wall of the station- inside of course) and the gravity we feel on Earth (assuming the rotation speed of the space station is properly set).

Image credit: NASA (Wernher von Braun explains his concept for a wheel-shaped space station that would spin to create artificial gravity)

The “Equivalence Principle” is a deep subject. There is much more to the “Equivalence Principle” than I can discuss in this tutorial. I recommend that you take a look at

http://www.mathpages.com/home/kmath622/kmath622.htm

http://mathpages.com/home/kmath582/kmath582.htm

I also like this cool statement from Wikipedia describing the Equivalence Principle:

“The local effects of motion in a curved space (gravitation) are indistinguishable from those of an accelerated observer in flat space, without exception.”

**Is the same as ?**

In principle no. When the object is moving at relativistic speeds (close to the speed of light in vacuum) the answer is certainly no. The invariant mass is constant, it does not change. At the relativistic speeds Newton’s second law is not meaningful. Instead of “force” we have to think in terms of “momentum” and “energy.”

**Quantum uncertainty in the measurement of mass**

The total energy is measured by calorimeters as mentioned in the “Measurement of mass” section above. Total energy measurement is never a precise measurement. Apart from the instrumentation errors there is an intrinsic uncertainty. There is a minimum for the product of the uncertainties of the energy and time.

For short lived unstable particles uncertainty in the duration of measurement will be small, therefore the uncertainty in the total energy will be large. Since the uncertainty in the “invariant mass” will be large too.

**Measurement of quark masses**

Quarks are always bound to the inside of a proton or a neutron. Quarks can exist for a short time as pairs of quarks too. Free quarks have never been observed. We can never measure the curvature of their path in a uniform magnetic field because they are never free. So, quark invariant masses can only be inferred using some theory, and comparing with experimental results that depend on the quark mass. Different theories can give different values for the quark invariant mass.

**Measurement of masses of the electrically neutral particles**

Magnetic field has no effect on the electrically neutral particles. Therefore we cannot measure the radius of their curved path. They move straight. Their invariant masses can only be measured by careful reconstruction of masses from the appearance of these electrically neutral particles in interactions involving charged particles.

**Mass due to Higgs Field**

There is a universal field known as the Higgs field. We know that the Higgs field exists because experiments at the Large Hadron Collider clearly demonstrated the existence of such a field. We don’t know the properties of this universal field yet but there is strong evidence that all elementary particles get their invariant masses from this Higgs field.

There are many caveats. In his article titled “Why the Higgs and Gravity are Unrelated” and “Does the Higgs Field Give the Higgs Particle Its Mass or Not? ” Matt Strassler reminds us that

*“The Higgs field does not give an atomic nucleus all of its mass, and since the nucleus is the vast majority of the mass of an atom, that means it does not provide all of the mass of ordinary matter.”**“Black holes appear at the centers of galaxies, and they appear to be crucial to galaxy formation; but the Higgs field does not provide all of a black hole’s mass. In fact the Higgs field’s contribution to a black hole’s mass can even be zero, because black holes can in principle be formed from massless objects, such as photons.”**“There is no reason to think that dark matter, which appears to make up the majority of the masses of galaxies and indeed of all matter in the universe, is made from particles that get all of their mass from the Higgs field.”**“The Higgs field, though it provides the mass for all other known particles with masses, does not provide the Higgs particle with its mass.”*

**Mass due to binding energy (negative energy)**

Protons and neutrons are not elementary particles. They are composite particles consisting of quarks and gluons. The “invariant mass” of a proton or neutron is due to the binding energy that keeps the quarks and gluons together. What is that binding energy? We refer to this binding energy as “strong nuclear force.” As you might have guessed we don’t really know what “strong nuclear force” is ontologically but we clearly know that it is different from the universal Higgs field. So, it is important to mention that the composite particles such as protons and neutrons DO NOT get their invariant masses from the Higgs field.

**What is the point of this tutorial?**

Students are confused with the terms “invariant mass”, “rest mass”, relativistic mass”, “gravitational mass”, “inertial mass”, “mass due to binding energy” and “mass due to Higgs field.” I don’t blame them. My purpose was 3-fold:

- Wanted to clarify the terms.
- Wanted to show the importance of measurement errors and the role of intrinsic (quantum) uncertainty
- Wanted to point out the importance of the ontological problem regarding the “invariant mass,” not to mention the ontological status of energy.