1-10 - Fysikum - Stockholms universitet

GENERAL RELATIVITY
1 Introduction
Lecture notes
by
Kjell Rosquist
Fysikum, Stockholms universitet
VT 2011
Galileo demonstrated in his famous experiment in Pisa that particles move on identical spacetime
trajectories in the gravitational field independently of their mass. Newton went further and showed
that the gravitational acceleration of a body does not depend on its composition. This follows from
the formula m1a = Gm1m2r −2 which gives the acceleration (a) of a mass m1 in the gravitational field
of another mass m2 at a distance r (G is Newton’s gravitational constant). By dividing both sides of
the equation by m1 we see that the acceleration of m1 does not depend on the mass m1 itself. The
equality between inertial and gravitational mass is noted more or less as a curious fact in Newtonian
gravity. Einstein, however, regarded this observation as a special case of a more general postulate,
the principle of equivalence, which he took as the basis of general relativity (GR). The principle of
equivalence states that all physical laws if written in tensor form have the same form in any freely
falling reference system. Such a reference frame is by definition a system which is unaffected by all
forces except gravity. Mathematically this means that the theory should be invariant under arbitrary
transformations of the spacetime coordinates. The fundamental observation that the gravitational
interaction does not depend on the mass or composition of a body makes it natural to interpret
gravity as an aspect of the geometry of the spacetime rather than as an ordinary force. In Newtonian
or special relativistic mechanics free particles move along straight lines. Two free particles do not
influence each other (unless they collide). Therefore, in order to describe for example the motion of
the earth in the gravitational field of the sun in geometrical terms we clearly need a more general
concept of geometry than that of ordinary flat Euclidean space. It is here that Riemann’s theory of
1

Kjell Rosquist 8
Figure 4: Differentials in curvilinear coordinates. The vector with components v α at x α
has components v α + dv α at x α + dx α . To take the differential we first parallel transport
the vector from x α to x α + dx α . The transported vector has components v α + δv α .
A scalar is unchanged by parallel transport, δφ = 0, i.e. Dφ = dφ or φ;α = φ,α. We can use this
to derive the change in the components of a covector under parallel transport. Using δ(u α vα) =0we
obtain
Then since the u α are arbitrary it follows that
u α δvα = −vαδu α
= vαu γ Γ α γβdx β
= vγu α Γ γ αβdx β .
(31)
δuα = uγΓ γ αβdx β . (32)
It follows from this that the covariant derivative of a covector is given by
uα;β = uα,β − uγΓ γ αβ . (33)
Next we show that the connection coefficients are symmetric in their lower indices, Γ α βγ =Γ α γβ.
To that end let φ be a scalar and define a covector with components uα = φ,α. Thenuα,β = uβ,α from
which it follows
uα;β − uβ;α = −(Γ γ αβ − Γ γ βα)uγ . (34)
The left hand side of this expression is a tensor. Therefore the right hand side must also be a tensor.
But then the right hand side being zero in Cartesian coordinates must be zero in any coordinate
system. Thus the connection coefficients are symmetric as claimed.
Let us now take a look at the relation between the connection and the metric. The covariant
derivative of the metric is a tensor with components given by
gαβ;γ = gαβ,γ − gδβΓ δ αγ − gαδΓ δ βγ . (35)
In Cartesian coordinates the right hand side is zero. Thus since it is a tensor gαβ;γ = 0 in any
coordinate system. In other words the metric is covariantly constant. We shall express the connection
in terms of the metric by solving (35) as a system of linear equations in Γ α βγ. Definingfirst
we write (35) in three equivalent ways by permuting indices as
Γαβγ = gαδΓ δ βγ , (36)
gαβ,γ =Γαβγ +Γβαγ ,
gγα,β =Γγαβ +Γαγβ ,
−gαβ,γ = −Γβγα − Γγβα ,
(37)