Natural Science Forum / Physics / Particle Physics / July 2005

Indeed, but this is also true of almost everything else in physics:
nobody really knows what energy is, or what a particle is. However, we
can still take measurements of quantities that we can observe, even if
we don't know what they really are. We can then *define* quantities in
terms of their (mathematical) relationships to each other and to
observables. You don't need to know what inertia "really is" in order
to perform calculations involving it. What's important is that those
calculations will result in predictions of things that can be observed.

The mathematical model may be no more than a metaphor for what reality
really is, but it can still mirror its workings. If its results
parallel what happens in the real world then we call it a good theory.
If it deviates then we go back to the drawing board. Einstein's genius
was to accept that light *really does* travel at the same speed for all
observers (instead of trying to find a mechanism to explain it away),
and then to examine the implications thereof without prejudice. One of
those implications was the result e=mc^2.

The annoying but true answer is because it's a good theory. Think of it
this way: if you're going to construct a physical theory that is to be
accurate, it must account for everything that has been observed up till
now. It must not contradict past observations. You must tie all your
variables together in such a way that they mimic the *currently* known
behaviour of the universe. By 1905, Newton's laws were failing in that
respect because of this speed of light business. Special relativity
took account of it (ignoring gravity for the moment).

Now, it seems unlikely that there will be very many mathematically
inequivalent ways of doing this. So if it works correctly for what's
gone before then the chances are fair that it will correctly predict
things that haven't been thought of yet, simply because there can only
be so many different ways of putting it together. e *must* be equal to
mc^2 if the rest of the theory is to work properly.

 They can't. Since accelerators don't precipitate
 anything, except in physics papers.

 And then only if they
 have television screens.

 We already have outsmarted him. Since his equations
 are second order tensors, which immediatly
 implies that DNA doesn't float in
 Twin paradoxes.

The general relationship
              P^2 - 2MH + z H^2 = 0
              M = m + zH
              (H = kinetic energy, m = rest mass,
               M = total mass, P = momentum)
comes out of the study of the properties of the
Galilei/Poincare'/Euclidean symmetry group, and could have been
inferred any time from the late 19th century onward (particularly since
symmetry groups were the running theme around that time in geometry).

Applying Hamilton's equation (dH/dP = V) gives you the velocity:
               2P - 2MV - 2zH^2 + 2zH^2 = 0
or P = MV, V = P/M.

For Galilei, z = 0, and you get
                  P^2 = 2mH, M = m
which yields the kinetic energy H = P^2/2m and, thus,
                  P = mv, H = 1/2 mv^2.

For the Euclidean group z < 0, or the Poincare' group, you could define
the total energy E = M/z, and write down
                 E = H + m/z
                 E^2 = P^2/z + (m/z)^2;
and you get (in all 3 cases, including Galilei)
                 M = m/sqrt(1 - zv^2)
                 P = mv/sqrt(1 - zv^2)
                 H = mv^2/(1 + sqrt(1-zv^2))
and for Euclid and Poincare'
                 E = M/z = m/(z sqrt(1-zv^2))

For Euclid (z < -1/K^2), this would yield
                   E = H - mK^2
                   E^2 + (PK)^2 = (mK^2)^2,
the rest energy being -mK^2, in terms of the rest mass m, the total
energy decreasing with increasing momentum.

For Poincare' (z = 1/c^2 > 0), this would yield
                   E = H + mc^2
                   E^2 - (Pc)^2 = (mc^2)^2,
with the rest energy being mc^2, the total energy going up with
increasing momentum.