30 Special Relativity
to aDoppler redshiftfor a receding source, and to a correspondingblueshiftfor an
approaching source.
Actually, the light cones in Figure 2.1 need to be modified for an expanding uni-
verse. A scale factor푎(푡)that increases with time implies that light will travel a dis-
tance greater thanctduring time푡. Consequently, the straight lines defining the cone
will be curved outwards.
2.2 Metrics of Curved Space-time
In Newton’s time the laws of physics were considered to operate in aflat Euclidean
space, in which spatial distance could be measured on an infinite and immovable
three-dimensional grid, and time was a parameter marked out on a linear scale run-
ning from infinite past to infinite future. But Newton could not answer the question
of how to identify which inertial frame was at rest relative to this absolute space. In
his days the solar frame could have been chosen, but today we know that the Solar
System orbits the Galactic center, the Galaxy is in motion relative to the local galaxy
group, which in turn is in motion relative to the Hydra–Centaurus cluster, and the
whole Universe is expanding.
The geometry of curved spaces was studied in the nineteenth century by Gauss,
Riemann and others. Riemann realized that Euclidean geometry was just a partic-
ular choice suited to flat space, but not necessarily correct in the space we inhabit.
And Mach realized that one had to abandon the concept of absolute space alto-
gether. Einstein learned abouttensorsfrom his friend Marcel Grossman, and used
these key quantities to go from flat Euclidean three-dimensional space to curved
Minkowskian four-dimensional space in which physical quantities are described by
invariants. Tensors are quantities which provide generally valid relations between dif-
ferent four-vectors.
Euclidean Space. Let us consider how to describe distance in three-space. The path
followed by a free body obeying Newton’s first law of motion can suitably be described
by expressing its spatial coordinates as functions of time:푥(푡),푦(푡),푧(푡).Timeisthen
treated as an absolute parameter and not as a coordinate. This path represents the
shortest distance between any two points along it, and it is called ageodesicof the
space. As is well known, in Euclidean space the geodesics are straight lines. Note that
the definition of a geodesic does not involve any particular coordinate system.
If one replaces the components푥,푦,푧of the distance vectorlby푥^1 ,푥^2 ,푥^3 ,this
permits a more compact notation of the Pythagorean squared distance푙^2 in themetric
[Equation (2.6)]
푙^2 =(푥^1 )^2 +(푥^2 )^2 +(푥^3 )^2 =
∑^3
푖,푗= 1푔ij푥푖푥푗≡푔ij푥푖푥푗. (2.10)The quantities푔ijare the nine components of themetric tensorg, which contains all
the information about the intrinsic geometry of this three-space. In the last step we