The theory of **special relativity**, (developed by
*Albert Einstein*) embodies several concepts which are very important to
particle physicists. Amongst other things, the theory tells us that mass is a
form of energy. You may also have heard that the mass of an object increases
with its velocity. These effects are only significant at very high velocities
(approaching the speed of light).

**Frames of reference**

The velocity of an object can only be measured *relative* to another
object. No single point in the Universe is truly stationary. This means that
when we measure the velocity (or mass!) of an object, we should say which *
frame of reference* the measurement was taken in. To measure in an object's
frame of reference means to measure velocities relative to that object, which is
taken to be stationary. This frame of reference is called the object's *rest
frame.*

**The invariant mass**

At the extreme velocities reached by
particles in an accelerator, the effects of relativity become important and
complicate measurements and comparisons of the masses and momenta of particles.
Thankfully, a property of an object, or a system of
objects, called the *invariant mass *is the same whatever frame of
reference we measure it in.

The invariant mass is defined as the energy of an object in its rest frame
(in appropriate units). This quantity is used when calculating and comparing masses and momenta in
particle physics to remove the complications of relativity. It is particularly
useful since:

*The invariant mass of a particle's decay products equals the
rest mass of the decaying particle.*

For any system of
particles:

- W is the invariant mass of the system

- ∑ E is the sum of
*mass energies*, E =
.m.c², of
particles in the system

- ∑
__p__ is the sum of linear momenta, __p__
= .m.v, of
particles in the system

- = (1-(v²/c²))
^{-½
}
- c is the speed of light

- v is the magnitude of the velocity
__v__

Note:

The in the
expressions for mass energy and momentum are *relativistic *corrections and
represent the effect of special relativity. Under normal, non-relativistic,
conditions is
very close to 1 and is usually omitted.

__v__ must be measured in the same *frame of reference* for each
particle.

The tasks on this site calculate the invariant mass for you.