Lorentz Transformations 27produced in cosmic ray collisions in the upper atmosphere, they appear to have a
longer lifetime by the factor훾.
Another example is furnished by particles of mass푚and charge푄circulating with
velocity푣in a synchrotron of radius푟. In order to balance the centrifugal force the par-
ticles have to be subject to an inward-bending magnetic field density퐵. The classical
condition for this is
푟=m푣∕QB.The velocity in the circular synchrotron as measured at rest in the laboratory frame
is inversely proportional to푡, say the time of one revolution. But in the particle rest
frame the time of one revolution is shortened to푡∕훾. When the particle attains relativis-
tic velocities (by traversing accelerating potentials at regular positions in the ring), the
magnetic field density퐵felt by the particle has to be adjusted to match the velocity
in the particle frame, thus
푟=푚푣훾∕QB.This equation has often been misunderstood to imply that the mass푚increases by
the factor훾, whereas only time measurements are affected by훾.
Relativity and Gold. Another example of relativistic effects on the orbits of circulat-
ing massive particles is furnished by electrons in Bohr orbits around a heavy nucleus.
The effective Bohr radius of an electron is inversely proportional to its mass. Near the
nucleus the electrons attain relativistic speeds, the time dilation will cause an appar-
ent increase in the electron mass, more so for inner electrons with larger average
speeds. For a 1s shell at the nonrelativistic limit, this average speed is proportional to
Z atomic units. For instance,푣∕푐for the 1s electron in Hg is 80∕ 137 = 0 .58, implying
a relativistic radial shrinkage of 23%. Because the higher s shells have to be orthog-
onal against the lower ones, they will suffer a similar contraction. Due to interacting
relativistic and shell–structure effects, their contraction can be even larger; for gold,
the 6s shell has larger percentage relativistic effects than the 1s shell. The nonrela-
tivistic 5d and 6s orbital energies of gold are similar to the 4d and 5s orbital energies
of silver, but the relativistic energies happen to be very different. This is the cause of
the chemical difference between silver and gold and also the cause for the distinctive
color of gold [2].
Light Cone. The Lorentz transformations [Equations (2.1), (2.2)] can immediately
be generalized to three spatial dimensions, where the square of the Pythagorean dis-
tance element
d푙^2 ≡dl^2 =d푥^2 +d푦^2 +d푧^2 (2.6)is invariant under rotations and translations in three-space. This is replaced by the
four-dimensional space-time ofHermann Minkowski(1864–1909), defined by the
temporal distancectand the spatial coordinates푥,푦,푧. An invariant under Lorentz