## Archive for **October 2012**

## The measure of massiveness

This year’s Nobel Prize has not been awarded for the discovery of the Higgs. It rather celebrates achievements that will undoubtedly have practical applications. In order to compensate for this unfairness I wanted to dedicate a couple of hours of my time recollecting and reminiscing of some fundamental things that are related to the notion of **mass**. As I’m not formally trained in physics, this is based on what I learned in school and read in popular literature.

### Mass in Intuitive mechanics

- Mass measures the amount of stuff
- Mass is conserved
- Mass is additive
- Mass is a measure of inertia

### Mass in Newtonian mechanics

- All of the above
- Mass is invariant in all inertial frames of reference
- Mass is the source of gravity
- F=ma

Note that F=ma is nothing more than:

- Definition of force F = dp/dt
- Definition of acceleration a = dv/dt
- Definition of momentum p = mv
- Conservation of momentum (with no F there is no change of p)

Note that it is easy to guess that gravity is no different from inertia. Acceleration does not depend on the mass? Come on, it’s not a real force! It’s the same kind of force that gives the same acceleration to skinny and fat people on a bus that suddenly breaks.

Also note that it’s easy to predict that massless thing are affected by gravity: once you consider F=ma and F~mM, you see that a doesn’t depend on m. Which Laplace did. Laplace was brilliant, as I mentioned here and here, but this simplistic prediction did not match the observation quantitatively (by the factor of 2).

### Relativity

- We need to redefine momentum and energy so that they are still conserved
- Mass is still invariant.
- The rest energy is not zero. The mass is the rest energy.
- The definitions of force and acceleration are still the same, but F is not ma.
- Mass is not a measure of inertia. In fact, there is no invariant measure of inertia. Bodies resist forces depending on how they are moving.
- Mass does not define the gravitational force. In the same manner as inertia, gravity depends on how the bodies are moving – it’s defined by the energy and momentum.
- Mass is not additive. If we have a system of moving or interacting bodies, its mass is not the sum of the masses of the constituents.
- Massless things cannot be at rest, in any frame of reference.
- Mass is not the amount of stuff. It is the amount of motion and interaction – just like energy. But motion is relative… Once again, the mass is the rest energy.

### The Higgs

- When it seems that there are no interacting parts, but the massiveness is still there – it is due to interaction with the Higgs field.