QuantumPhysics.dvi

From this it is clear thatβcmust be interpreted as the relative velocity between the twoframes.
Indeed, the pointx′= 0, which is fixed in frameR′, travels with velocityv=x/t=βcfrom the
point of view of frameR, which is one way we may define the relative velocity between the frames.

The relativistic transformation properties of momentum, energy, mass, and of the electro-
magnetic fields may be derived analogously. It is much more efficient, however, to obtain such
relations using the Lorentz vector and tensor notation, which we shall provide first.

21.2 Lorentz vector and tensor notation

Just as we use vector notation in 3-dimensional space to collect the three coordinates (x,y,z) into
a vectorx, so we use also 4-vector notation to collect the four coordinates of an event (ct,x,y,z) =
(ct,x) into a 4-vectorx, denotedwithout bold face or arrow embellishment. Actually, one mostly
uses a slight variant of the 4-vector notation, with an indexadded,

xμ≡(x^0 ,x^1 ,x^2 ,x^3 ) = (ct,x,y,z) μ= 0, 1 , 2 , 3 (21.8)

The time direction being special through its special signature in the Minkowski distance, one
reserves the index “0” to denote it. The Minkowski distance may be easily cast in this notation,

s^2 =

∑

μ,ν=0, 1 , 2 , 3

ημν(xμ 1 −xμ 2 )(xν 1 −xν 2 ) (21.9)

where the Minkowski metricημνis defined as follows,

ημν≡







−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1







μν

(21.10)

Einstein introduced therepeated index summation convention, which instructs one to sum over one
repeated upper and one repeated lower index. With the Einstein index convention, we thus have,

s^2 =ημν(xμ 1 −xμ 2 )(xν 1 −xν 2 ) (21.11)

The precise distinction between upper and lower indices andtheir separate roles will be spelled out
later.

A relativistic transformation may be expressed in 4-index notation as well. A general affine
transformation between the coordinatesxμandx′μmay be parametrized as follows,

xμ→x′μ= Λμνxν+aμ (21.12)

Here, Λ is a 4×4 matrix with real entries, and the repeatedν-index is to be summed over. The
4-vectoraμis real, and parametrizes translations in time and space. Invariance of the Minkowski
distances^2 is tantamount to

ημνxμxν = ημνx′μx′ν

= ημνΛμρxρΛνσxσ

= ηρσΛρμxμΛσνxν (21.13)