Page 161 - Intro to Tensor Calculus

156

where γ is given by equation (1.5.151). These equations are also known as the Lorentz transformation.
v
Note that for v<< c,then ≈ 0, γ ≈ 1 , then the equations (1.5.153) closely approximate the equations
c 2
(1.5.144). The equations (1.5.153) also satisfy the equations (1.5.146) and (1.5.147) identically as can be
readily verified by substitution. Further, by using chain rule differentiation we obtain from the relations
(1.5.148) that
dx
dx dt + v
= . (1.5.154)
dt dx
dt v
1+
c c
The equation (1.5.154) is the Einstein relative velocity addition rule which replaces the previous Newtonian
rule given by equation (1.5.145). We can rewrite equation (1.5.154) in the notation of equation (1.5.145) as
V P/Q + V Q/R
V P/R = . (1.5.155)
1+ V P/Q V Q/R
c c
Observe that when V P/Q << c and V Q/R << c then equation (1.5.155) approximates closely the equation
(1.5.145). Also as V P/Q and V Q/R approach the speed of light we have

V P/Q + V Q/R
lim = c (1.5.156)
V P/Q →C 1+ V P/Q V Q/R
V Q/R →C c c
which agrees with Einstein’s hypothesis that the speed of light is an invariant.
Let us return now to the viewpoint of what gravitation is. Einstein thought of space and time as being
related and viewed the motion of the planets as being that of geodesic paths in a space-time continuum.
Recall the equations of geodesics are given by

2 i
j
d x i dx dx k
+ =0, (1.5.157)
ds 2 jk ds ds
where s is arc length. These equations are to be associated with a 4-dimensional space-time metric g ij
i
where the indices i, j take on the values 1, 2, 3, 4and the x are generalized coordinates. Einstein asked
the question, ”Can one introduce a space-time metric g ij such that the equations (1.5.157) can somehow
2
d ~ ρ GM
reproduce the law of gravitational attraction dt 2 + ρ 3 ~ρ = 0?” Then the motion of the planets can be
viewed as optimized motion in a space-time continuum where the metrices of the space simulate the law of
gravitational attraction. Einstein thought that this motion should be related to the curvature of the space
which can be obtained from the Riemann-Christoffel tensor R i jkl . The metricwedesire g ij ,i, j =1, 2, 3, 4
ij
i
has 16 components. The conjugate metric tensor g ij is defined such that g g jk = δ and an element of
k
2
i
j
arc length squared is given by ds = g ij dx dx . Einstein thought that the metrices should come from the
Riemann-Christoffel curvature tensor which, for n = 4 has 256 components, but only 20 of these are linearly
independent. This seems like a large number of equations from which to obtain the law of gravitational
attraction and so Einstein considered the contracted tensor

∂ n ∂ n m n m n
t
G ij = R ijt = − + − . (1.5.158)
∂x j in ∂x n ij in mj ij mn
Spherical coordinates (ρ, θ, φ) suggests a metric similar to
2
2
2
2
2
2
2
2
ds = −(dρ) − ρ (dθ) − ρ sin θ(dφ) + c (dt) 2