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Metric Tensor, Connection, Curvature and Geodesic
In this chapter we give a brief introduction to the elements of differential geometry as used in
general relativity. The main motivation that led Einstein to develop general relativity was the
equivalence principle. In all forces except gravity the acceleration of a particle depends on its
(inertial) mass, the lighter the particle is the more it will accelerate in the same field. For example a
particle in an electric field has acceleration a = mq E. Here q is the strength with which the particle
couples to the electric field also called the electric charge but m is the inertia of the particle. Thus
lighter particles will accelerate more. But for gravity the relation is a = qgrav
g, here g is the strength
m
of the gravitational field (acceleration due to gravity) and qgrav is the strength with which the particle
couples to the gravitational field. According to the equivalence principle, this is proportional to the
inertial mass m. This means qgrav /m is a universal constant usually taken to be unity. Thus all
particles accelerate with the same acceleration g. According to legend, this was demonstrated by
Galileo in his famous leaning tower of Pisa experiments. This led Einstein to suspect that gravity
was very different from all the other forces. Since all particle behave the same in gravity, Einstein
said that gravity simply changes the geometry of space and time and all particle then move freely
in this curved space time. It is as if the stage is modified and all actors have to act on the same
stage. With electromagnetic force the path travelled by a particle depends on its mass. Whereas
for gravity it does not. So Einstein said that if gravity is the only force present then all particles
will follow the same straight path (the shortest route) but in a curved space time. Thus we have
to understand how matter curves space and time. For this we have to understand how to describe
geometry of four-dimensional space times.
I.
METRIC TENSOR
We know that in special relativity in cartesian coordinates, the distance that is invariant under
both ordinary rotations in space and Lorentz transformations (rotation that mixes space and time
variables) is
ds2 = c2 dt2 − dx2 − dy 2 − dz 2 = ηµν dxµ dxν
This simple form is only valid in Cartesian coordinates. In other orthogonal coordinates systems it
will look like,
ds2 = c2 dt2 − h1 dr2 − h2 dθ2 − h3 dφ2 = gµν dxµ dxν
where h1 = 1, h2 = r2 and h3 = r2 sin2 θ and if x = (ct, r, θ, φ) then g00 = 1, g11 = h1 , g22 =
h2 , g33 = h3 . One can also work in non-orthogonal coordinate systems. For example, we could
choose two linearly independent vectors ê1 and ê2 that are not mutually orthogonal to describe the
position of a particle in two space dimensions. ~r = x1 ê1 + x2 ê2 . Now if we calculate the square
of the distance between two neighboring points ds2 = d~r · d~r, we get d~r = dx1 ê1 + dx2 ê2 , hence
ds2 = d~r · d~r = (dx1 )2 + (dx2 )2 + 2(ê1 · ê2 ) dx1 dx2 . If the two unit vectors were orthogonal then
there would be no cross term like dx1 dx2 in the distance expression. For a general non-orthogonal
coordinate system then all kinds of cross terms would also be present. Thus the most general case is
that of non-orthogonal and curvilinear coordinates which means the metric (expression for distance
squared) has the general form,
ds2 = gµν (x)dxµ dxν
(1)
2
Now we can verify many properties of the metric. First we have to define the covariant coordinate
xµ . In special relativity it is dxµ = ηµν dxν , where ηµν = diag(1, −1, −1, −1), since this is sufficient
to ensure that dxµ dxµ = c2 dt2 − dx2 − dy 2 − dz 2 is a Lorentz invariant. In general relativity, we look
for four-distance such as ds2 that are invariant under general coordinate transformations including
Lorentz transformation. Thus we should write,
ds2 = gµν (x)dxµ dxν = dxµ dxµ
This means dxµ = gµν (x)dxν . Any object V µ (x) that transforms like dxµ under coordinate transformations is a contra-variant vector. Any object Vµ (x) that transforms like dxµ under coordinate
transformations is a co-variant vector.
Homework : (a) Find ds2 on a torus of small radius a and big radius b. (b) Find ds2 on an
2
2
ellipsoid of revolution (the surface obtained by rotating the ellipse xa2 + yb2 = 1 about the x-axis).
A.
Equation of a Geodesic from the Metric
In an earlier chapter we showed that the lagrangian of a free particle in relativistic physics is,
L = −mc
v
u
u
2t
2
~r˙
1− 2
c
we can calculate the action as,
S=
Now we write cdt =
√
Z tf
ti
v
u
Z tf u
t
2
L dt = −mc
ti
c2 dt2 and take it inside the square root.
S = −mc
But
2
~r˙
1 − 2 dt
c
Z tf q
ti
c2 dt2 − (d~r)2
q
c2 dt2 − (d~r)2 = ds. Thus,
S = −mc
Z f
i
ds
Thus the action of a free particle is proportional to the four dimensional distance between the initial
and final points (events). The principle of least action is same as finding the extremum of the distance
between two points in four dimensions (one must note that c2 dt2 − (d~r)2 may be positive, negative
or zero, hence ds, the distance between events is not always real). Hence the shortest possible route
between two points in space-time is also the path that minimizes the action and hence is the path
of the particle.
Now I want to repeat this exercise in curved space-time where the distance between points is given
by Eq.(1). The idea now is that I choose a clever way of parametrizing the path. I choose s itself
as the parameter. This is similar to describing the circle x2 + y 2 = a2 using x = a cos(s/a) and
y = a sin(s/a) where s is the distance along the circle s = aθ. Hence I will describe the path as
xµ (s). Thus,
S = −mc
Z f
i
ds = −mc
Z f
ds2
i
ds
= −mc
Z f
gµν (x)dxµ dxν
i
ds
= −mc
Z f
i
gµν (x)
dxµ dxν
ds
ds ds
3
We call
dxν
ds
= ẋν (s). Then,
S = −mc
Z f
i
µ
ν
gµν (x)ẋ (s)ẋ (s) ds =
Z f
i
L(x, ẋ) ds
where the lagrangian is given by L(x, ẋ) = −mc gµν (x)ẋµ (s)ẋν (s). Thus the path that minimizes
the action is that special path xµ (s) = cµ (s), that obeys the lagrange equation.
d ∂L(x, ẋ)
∂L(x, ẋ)
=
ρ
ds ∂ ẋ
∂xρ
(2)
After calculating the required derivatives of the lagrangian we get,
−2mc
d
∂gµν (c(s)) µ
(gρν (c(s))ċν (s)) = −mc
ċ (s)ċν (s)
ds
∂cρ
(3)
But,
dgρν (x)
= ẋµ (s)gρν,µ (x)
ds
Here the notation means f,ν (x) ≡
∂f (x)
.
∂xν
Or,
2 ċν (s)ċσ (s)gρν,σ (c(s)) + 2 gρν (c(s))c̈ν (s) = gσν,ρ (c(s))ċσ (s)ċν (s)
(4)
Now we show that, g µρ gρν = δνµ . To see this we first note that for any vector V µ and Vµ , V µ = g µν Vν .
Choose Vν = gνρ aρ and V µ = g µρ aρ hence g µρ aρ = g µν gνρ aρ . This means g µρ = g µν gνρ . It is easy to
see that g µρ = δ µρ since for any vector V µ = g µρ V ρ = δ µρ V ρ . Hence δ µν = g µρ gρν . We multiply both
sides of Eq.(4) by g µρ and divide by two to get,
c̈µ (s) + Γµνσ (c(s))ċν (s)ċσ (s) = 0
(5)
1
1
Γµνσ (x) = g µρ (x)gρν,σ (x) − g µρ (x)gσν,ρ (x) = g µρ (x)(gρν,σ (x) + gνρ,σ (x) − gσν,ρ (x))
2
2
(6)
where,
Homework : Find the objects Γ for the metric of points on a sphere (a is the radius of the sphere).
ds2 = a2 dθ2 + a2 sin2 (θ)dφ2
From this find the equation of the geodesic on the surface of the sphere.
II.
CONNECTION
Independent of the metric concept, one can introduce a concept called connection. Suppose I have
a vector field : V µ (x), that is a function at each point in space-time whose value is a vector (µ-th
0
component is chosen for illustration). Then one can ask if the two vector V (x) and V (x ) are parallel
0
at different points x and x ? In general the answer is no as the vector field can twist and turn as
it moves from point to point. But we can ask ourselves if there is some way we can move a vector
0
0
from x to x such that the vector Ṽ (x ) is now parallel to V (x). To do this we need a concept known
as connection. Connection is a quantity that tells us how to find a vector at some other point that
is parallel to a given vector at the starting point. To do this consider two neighboring points x and
x + dx (remember that x means (ct, x, y, z) or (ct, r, θ, φ) or anything with four entries). We want
4
Ṽ µ (x + dx) to be parallel to V µ (x). Let us now focus on the difference Ṽ µ (x + dx) − V µ (x), as
dx → 0, Ṽ → V , this is because as the points come closer and closer, the two vectors should coincide
: Ṽ → V . Thus for closely separated points the difference between these two vectors should be
proportional to the separation dx.
Ṽ µ (x + dx) − V µ (x) = dxν Hνµ (V (x))
(7)
where Hνµ (V (x)) is some coefficient. Now consider two such vectors V and W . Then we have,
W̃ µ (x + dx) − W µ (x) = dxν Hνµ (W (x))
(8)
Since V, W are vectors, they have to respect the principle of superposition. This means the same
relation should also hold for Z = V + W . Thus,
Z̃ µ (x + dx) − Z µ (x) = dxν Hνµ (Z(x))
(9)
Adding Eq.(7) and Eq.(8) and subtracting from Eq.(9) we get,
Hνµ (U (x) + V (x)) = Hνµ (U (x)) + Hνµ (V (x))
(10)
Hνµ (V (x)) = Γµνρ (x)V ρ (x)
(11)
This is possible only if,
Here Γµνρ (x) are known as the coefficients of a connection. They are more commonly known as
Christoffel symbols. If someone tells me what these are for some space-time only then I know how
to transport a vector parallel to itself from point to point. Hence,
Ṽ µ (x + dx) − V µ (x) = dxν Γµνρ (x)V ρ (x)
A.
(12)
Equation of the geodesic from the Connection
In three dimensional space a curve is described by ~r(t). The vector tangent to this curve is the
velocity vector ~r˙ (t). Similarly, in four dimensions if the curve is cµ (s), the vector tangent to this
d µ
c (s). Now if I transport a vector that is tangent to a curve along the curve
curve is ċµ (s) = ds
parallel to itself to another point on the curve, it will remain tangent to the curve provided the curve
is the straightest possible curve namely the geodesic. For this I choose the vector field,
V µ (x) =
Z f
i
0
0
0
ds ċµ (s ) δ 4 (x − c(s ))
(13)
where δ 4 (x − c(s)) = δ(x0 − c0 (s))δ(x1 − c1 (s))δ(x2 − c2 (s))δ(x3 − c3 (s)). This vector field has
the property that it represents the tangent to the geodesic c when x lies on the curve and is zero
otherwise. The vector Ṽ µ (x + dx) represents the vector that is parallel to V µ (x) but lives at point
x + dx. But we have argued that this vector is also tangent to the curve since V µ was a tangent
vector to begin with. Hence, Ṽ µ (x + dx) = V µ (x + dx). Substituting this into Eq.(12) we get,
V µ (x + dx) − V µ (x) = dxν
∂V µ (x)
= dxν Γµνρ (x)V ρ (x)
∂xν
(14)
We know from the definition that the vector V (x) is zero unless x lies on the geodesic c. For the same
reason x + dx also lies on the geodesic. Hence there exists a parameter s such that x = c(s). This
means s = c−1 (x). As x changes from one value to the next along the geodesic c, this parameter s
5
also changes. Now we divide both sides of Eq.(14) with ds. Then we substitute Eq.(13) into Eq.(14)
to get,
Z f
i
Z f
dxν ∂ 4
dxν µ
0
0
0
0
ds ċ (s )
δ (x − c(s )) =
ds ċρ (s ) δ 4 (x − c(s ))
Γνρ (x)
ν
ds ∂x
ds
i
0
0
µ
ν
(15)
ν
d
∂
∂
d
Now consider the identity ds
f (x(s)) = dx
f (x), or, dx
= ds
. Also the delta function forces
ds ∂xν
ds ∂xν
0
0
−1
x = c(s ) but we have argued that x = c(s) for a value of s = c (x). This means s = s . Hence
ν
d
= dsd 0 . On the right hand side of Eq.(15) we may substitute dx
= ċν (s) and x = c(s). Since x lies
ds
ds
on the geodesic c. After substituting all this into Eq.(15) we get,
Z f
i
Z f
d 4
0
0
0
0
ν
µ
ds ċ (s ) 0 δ (x − c(s )) = ċ (s) Γνρ (c(s))
ds ċρ (s ) δ 4 (x − c(s ))
ds
i
0
0
µ
(16)
Now we may integrate the left hand side by parts, to get,
Z f
i
0
0
ds ċµ (s )
Z f
h
is=sf
d 4
d
0
0
0
0
0
µ 0
4
δ
(x
−
c(s
))
=
ċ
(s
)
δ
(x
−
c(s
))
−
ds δ 4 (x − c(s )) 0 ċµ (s )
0
s=si
ds
ds
i
(17)
Here c(sf ) = xf is the end point of the curve (geodesic) and c(si )xi is the starting point. We assume
that x is on this curve but somewhere in between xi and xf so that x 6= c(si ), c(sf ). Thus the
boundary term drops out and we get,
Z f
i
2 µ
Z f
d 4
d2 cµ (s )
0
0
0
4
δ
(x
−
c(s
))
=
−
ds
δ
(x
−
c(s
))
ds0
ds0 2
i
0
0
0
ds ċµ (s )
0
(18)
c (s )
. Substituting Eq.(18) into Eq.(16) we obtain keeping in mind that s = s
since dsd 0 ċµ (s ) = d ds
02
and equating the coefficients of the delta functions on both sides we get,
0
−
d2 cµ (s)
= Γµνρ (c(s)) ċν (s)ċρ (s)
ds2
0
(19)
This equation Eq.(19) is the same as Eq.(5). Thus the Γ found in Eq.(5) written in terms of the
metric gµν is nothing but the geometric concept of a connection used to define parallel transport of
vectors.