# Massive Issues

We all know that many objects (atoms, cats, us) have mass.  What you probably don’t know is that there are multiple different types of mass, and this has real physical (and astrophysical) consequences.

The most familiar type of mass is probably the “quantity” version.  That is, an object like a car has a certain amount of “stuff” (metal, plastic, glass), and that quantity of matter can be measured by mass.  Of course, some things are small and dense, while others are large and light.   So with quantity mass it’s not just the size that matters, but the density.

In our daily lives, we usually determine mass by an object’s weight.  Heavier things have more mass.  The weight of an object depends on how it interacts with a gravitational field, such as the Earth’s gravitational field.  This interaction is due to a type of mass known as passive gravitational mass.  The more passive gravitational mass, the heavier an object will be in the Earth’s gravitational field.

Of course, a passive gravitational mass can interact with the Earth because the Earth has a gravitational field.  This field is produced by the active gravitational mass of Earth.  The active and passive gravitational masses are what allow gravitational interaction (at least in classical physics).

Another type of mass determines how easy or difficult it is to move an object.  This is known as inertial mass.  The inertial mass is the m in Newton’s second law (F = ma), and is why a baseball is easier to throw than a bowling ball.

So all massive objects have a certain quantity of matter, a passive gravitational mass that interacts with gravitational fields, an active gravitational mass that produces a gravitational field, and an inertial mass that determines how the object will move when forces act on it.  In Newtonian physics, all these different types of masses are the same.  When we use the term mass, we generally mean the Newtonian version, which is why we don’t distinguish between them.

But this simple, Newtonian view of mass can lead to some confusion when it comes to things like special relativity.  Special relativity is derived from the fact that the speed of light is a constant.  This means that if I’m travelling at 90% of the speed of light relative to you, and shine a flashlight ahead of me, I will see the light move from me at the speed of light, not 10% of the speed of light.  This strange behavior leads to things like time dilation.  From your vantage point my time appears to move more slowly.  But it also means that my mass appears to get larger.

This is sometimes referred to as relativistic mass, and is a very real effect.  For example, in the Large Hadron Collider protons are accelerated to *nearly* the speed of light.  They are moving so fast that their relativistic mass is much larger than their regular Newtonian mass.  This means we have to push them harder to keep them moving in a circular path. So as the protons are sped up, the magnetic fields used to keep them moving in a circle have to be strengthened.  The closer the protons get to the speed of light, the bigger their relativistic mass, and the harder they are to move.  This is also, by the way, why they can’t be accelerated to the speed of light.

In physics we tend to avoid the term “relativistic mass”, because it isn’t really the same as the other masses.  Relativistic mass is an object’s apparent inertial mass from your vantage point.  So if you see a proton zip past you at a large fraction of the speed of light, the proton would appear to have a large inertial mass.  But if I’m zipping along with the proton, I would say its mass is a normal proton mass.  Relativistic mass is dependent on who’s doing the observing, while the other masses are an inherent property of the object.

In Newtonian physics, the equivalence of the inertial and gravitational masses is why everything falls at the same rate.  Even though a baseball and a cannonball have different masses, they fall at the same speed (barring air resistance).  Bigger masses are harder to move, but they also feel a stronger gravitational force.  This has been known since the time of Galileo, but it was used by Einstein as the foundation of general relativity.

In order to formulate general relativity in terms of general covariance, Einstein later strengthened this argument to yield what is known as the strong equivalence principle:  The ratio between the inertial mass of a particle and its gravitational mass is a universal constant.

Einstein saw a parallel between the relative nature of motion in his theory of special relativity and the relative nature of gravity, and so he worked to generalize relativity to include both gravity and motion. This theory of general relativity is what he published in 1915. It was a radical proposal. In his theory Einstein argued that gravity was not a force in the way Newton had thought. Instead, gravity was an effect of a curvature of space and time.

In general relativity, the active gravitational mass of an object curves space around it.  But this leads to another type of mass, what you might call “curvature mass”.  As an example, consider the mass of a black hole.  A black hole is so dense that any matter it once had is now trapped forever behind its event horizon.  We can’t observe that matter, but we can determine the mass of a black hole by measuring the curvature of space around it.  So a black hole has mass even though it isn’t made of “stuff”.

In our daily lives we can treat mass as a single property of an object.  But in astrophysics that simple view can lead to massive problems.

Cartoon of (likely mythical) Galileo dropping objects from the tower of Pisa. Credit: San Diego State University