Module I

∗
Lecture-1: Newton’s Gravitation: Part-I

1 Introduction
The viewpoint we will adopt in this course is that when one tries to make Newton’s scalar theory of
gravitation compatible with special relativity, one is automatically led to Einstein’s general relativity.
Right at the outset, it must be pointed out that the label General Relativity is somewhat misleading
because it is not a framework that generalizes or extends special relativity to noninertial frames.
However, rather, it is a theory that generalizes Newton’s nonrelativistic theory of gravitation to a
relativistic theory of gravitation. Perhaps a more suitable name for this theory would be Einstein’s
relativistic theory of gravitation, but due to historical inertia, we will persist with the misnomer
“General Relativity” or “GR” for short.
With this mindset, we will begin this course with a thorough review of Newton’s nonrelativistic
theory of gravitation, highlighting the concepts we will need later when we construct GR field equations
for gravity. This will be followed by a review of special relativity, emphasizing how the laws of classical
nonrelativistic physics (mechanics of particles and fluids, Electrodynamics, Thermodynamics, etc.) are
made relativistic (compatible with special relativity). Finally, we will try to construct the relativistic
generalization of Newton’s scalar (potential) theory of gravitation, which will lead us to a relativistic
two-index symmetric tensor potential (namely the Fierz-Pauli field), and adding interactions will lead
to general coordinate invariance and the realization that the gravitational field distorts the Minkowski
metric. Once we get there, we will switch to the more elegant and economical differential geometric
formulation of the theory (Riemannian Geometry), write down the Einstein field equations, and explore
some simple, exact solutions, such as the Schwarzschild solution.

Comment
Historically, Einstein based general relativity on two postulates, namely the principle of general co-
ordinate invariance (general covariance) and the principle of equivalence. However, developments in
quantum field theory in the 1960s established that these principles/postulates follow from consider-
ations of locality, Lorentz invariance, and unitarity if one identifies gravitational interactions to be
mediated by a massless spin-2 particle. In other words, general relativity is the unique (quantum)
theory of interacting massless spin-2 particles respecting unitarity and Lorentz invariance 1 .

Convention
We denote all the vectors by boldfaced symbols. We mostly work in 3 (spatial) dimensions. Indices
i, j = 1, 2, 3 label three components/coordinates and repeated indices are summed over.
∗
Please report typos and errors to sroy@phy.iith.ac.in / abhattacharyya@iitgn.ac.in.
1
Refer to Ch. 10, Sec. 8 of Weinberg’s text Gravitation and Cosmology.

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2 Review of Newton’s theory of gravitation
The founding postulate of Newton’s theory of gravitation is the well-known “universal law of gravita-
tion”, namely the force between two point objects of gravitational charges m1 and m2 is
m1 m2
F = −GN (x1 − x2 ). (1)
|x1 − x2 |3
Here x1 and x2 are the position vectors of the two point objects. GN is the Newton’s constant. Some
important points to notice are:

• The force is instantly transmitted. If the first object moves to a new position x1 , then the second
object instantly feels a change in the force of attraction. If a signal is transmitted, it must be
infinitely fast.
• This force/interaction is dubbed “action at a distance”, meaning it requires no medium and
violates the speed bound in special relativity.
• Newton imagined that the gravitational charge mG depends on the “quantity of matter” it
possesses. We distinguish it from the inertial mass mi (measured using Newton’s second law:
F = mI a). Typically,
F
mI = , and mG = f (mI ).
a

3 Gravitational Field and Potential

Now let’s consider a situation in which we place a “test object” of a small gravitational charge, m at
a position x in a region of space surrounded by N other objects located respectively at positions,x′ i ,
i = 1, ..., N with their respective gravitational charges being mi . Then the net force on the test mass
m due to the other objects, mi is of course given by the vector sum,
N
X mmi
F(x) = − GN (x − x′i ).
|x − x′i |3
i=1

Now the force experienced per unit point charge for a vanishingly small test mass placed at x is
N
F(x) X mi
g(x) = lim =− GN (x − x′i ). (2)
m→0 m |x − x′i |3
i=1

We refer to this vector quantity, g(x) as the gravitational field at the point, x .
It is easy to see that the gravitational force, F (x) is conservative, i.e. ∇ × F(x) = 0 . It is also
obvious that then the same would hold for the gravitational field,
∇ × g(x) = 0.
This implies immediately that the gravitational field, g(x) can be expressed as a gradient of a scalar
Φ(x, Thus, we can define a scalar potential Φ(x):
g(x) = −∇Φ(x),
This scalar field, Φ(x) is referred to as the scalar potential. From the expression of the gravitational
field in (2), it follows that the potential is given by
N
X mi
Φ(x) = − GN . (3)
|x − x′i |
i=1

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4 Continuum Case: Gauss Law and Poisson Equation
If instead of N discrete point masses we have a single massive lump which is characterized by gravita-
tional charge density, ρ(x′ ) at point x′ in the lump, then we can generalize the discrete case expressions
(2) and (3) to the continuum case (lump) as follows,
ˆ
ρ(x′ )d3 x′
g(x) = − GN (x − x′ ), (4)
|x − x′ |3
ˆ
ρ(x′ )d3 x′
Φ(x) = − GN . (5)
|x − x′ |

All we have done is to replace the point mass, mi at location x′i by the elementary gravitational
charge, δm(x′i ) = ρ(x)d3 x′ , and instead of summing over index i we have integrated over position of
the mass elements, x′i . From the continuous mass distribution case expressions (4, 5) it follows that
the gravitational field obeys the Gauss law,

∇ · g(x) = −4πGN ρ(x),

and that the scalar potential satisfies the Poisson equation:

∇2 Φ(x) = 4πGN ρ(x). (6)

x−x′ ′
Here we have used the vector identity, ∇ · |x−x′ |3
= −∇2 |x−x
1 3
′ | = 4πδ (x − x ) .

5 Poisson and Laplace equations: Solution in a bounded region

A lot of effort was devoted in the 18th and early 19th centuries to developing methods to solve the
Poisson or Laplace equations, both in the context of Gravitation and Electrostatics. Given a (partial)
differential equation, one has to consider the issues of:
• Existence: Does there exist a solution?

• Uniqueness: Under what boundary conditions is the solution unique?

• Stability: If one starts with finite or regular initial/boundary data, does the solution diverge
or develop singularities as one evolves the equation in finite time or finite distance?
We will briefly discuss these here. Since Φ obeys Poisson’s equation, just as the electrostatic
potential does, the theorems of Uniqueness, Mean-Value, and Earnshaw also hold for the gravitational
potential Φ.

• Uniqueness Theorem: The solution to Poisson’s equation in a region is unique if the potential
is specified at the boundary of the region (Dirichlet condition) or if the normal derivative is
specified at the boundary (Neumann condition).

• Mean Value Theorem: If Φ(x) satisfies Laplace’s equation (i.e., for regions without/outside
a mass distribution), then the value of the potential at any point P (which can be taken to be
the origin without loss of generality), is given by the mean value of the potential over a sphere
(of any radius) centered at P :
‹
1
Φ(xP ) = dS Φ(x′ ),
4πR2
where dS is the area element on the surface of a sphere of radius R centered at P .

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 • Earnshaw’s Theorem: This is a fundamental result concerning potentials obeying Laplace’s
equation (i.e., in regions outside sources). It states that a system consisting of such bodies cannot
be in stable static equilibrium. Earnshaw’s theorem implies that neither a planetary system nor
an atom can be composed of static bodies.

Proofs of these three well-known theorems are left for you as exercises. Also, try to construct a
general solution of the Poisson equation in a bounded region.

6 Gravitational Potential Energy and Field Energy

Newton’s gravitation is not a field theory but an action-at-a-distance theory, which means the gravi-
tational energy is only present in/possessed by the masses, and no energy is stored in the empty space
outside the sources. However, in the relativistic theory of gravitation, one will eventually have to
introduce an energy-momentum carrying entity, “gravitational field,” in the region surrounding (and
inside) massive bodies. By a heuristic argument, one can arrive at the energy (density) contained in
a static gravitational field from Newton’s theory despite the fact that Newton’s theory is not a field
theory. Here, we “derive” this energy stored in the region of space around massive bodies. To begin
with, we recall that the gravitational potential energy of a pair of (point) gravitational masses mi and
mj located at xi and xj respectively is,
mi mj
Uij = −GN .
|xi − xj |

This is a negative definite quantity. Now, when we have a collection of such point masses then, we
need to sum over all such pairs,
X 1X 1 X mi mj
U= Uij = Uij = − GN .
2 2 |xi − xj |
i>j i,j i,j

where we are summing over all masses labeled by indices i and j, but we maintain the condition, i > j
so that we count a pair only once. If we remove this restriction and sum over all i and j then we are
counting each pair twice, so we need to put a half,
1X 1 X mi mj
U= Uij = − GN .
2 2 |xi − xj |
i,j i,j

Instead of a point mass distribution, if we have a continuous distribution of mass, this turns into
ˆ
1 ρ(x)ρ(x′ )
U = − GN d3 x d3 x′ .
2 |x − x′ |

Note that since every quantity inside the integral sign is positive, the potential energy is negative
definite (because this is a bound system held together by gravitational attraction, and it will take
positive work done to rip it apart). Now recall that the gravitational potential at x is given by,
ˆ
ρ(x′ )
Φ(x) = −GN d3 x′ ,
|x − x′ |

so the energy can be rewritten as, ˆ

1
U= d3 x ρ(x)Φ(x). (7)
2

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This is a negative definite quantity as well since the starting point as negative. To extract the energy
stored in the region outside the massive objects or sources, which is positive definite, let us split up
this energy expression in the following manner 2
ˆ  ˆ 
3 1 3
U = d x ρ(x)Φ(x) + − d x ρ(x)Φ(x) .
2
| {z } | {z }
Umatter Ufield

Next, we use the Poisson equation in the second term to obtain,

ˆ ˆ
1 3 1
d3 x ∇2 Φ Φ


Ufield = − d x ρ(x)Φ(x) = −
2 8πG
ˆ
1
d3 x ∇ · (Φ∇Φ) − (∇Φ)2
 
=−
8πG
‹ ˆ
1 1
=− dS · (Φ∇Φ) + d3 x(∇Φ)2 .
8πG ∞ 8πG
where we have used the Gauss divergence theorem to convert the first term into a surface integral
at (spatial) infinity. Now assuming the mass distribution is localized (i.e. does not extend all the way
1
to infinity), one has Φ ∼ |x| , and so the integrand of the surface integral term falls off as Φ∇Φ ∼ |x|1 3 ,
while the elemental surface area grows as dS ∼ |x|2 . As a result, the surface integral at infinity
vanishes, and we get, ˆ
1
Ufield = d3 x(∇Φ)2 ,
8πG
Since this is a volume integral, we can write down an energy density of the form,
1 1 2
ufield = (∇Φ)2 = g .
8πG 8πG
This result holds even at regions where there is no matter, i.e., ρ = 0, but a nonvanishing gravitational
field, g. So that is why it is natural to think of this as the density of energy contained in the region
external to the massive bodies i.e., the gravitational fields in the relativistic theory (this explains the
subscript, “field”).

7 Multipole Expansion of the Newtonian Potential

Newton proved that for a perfectly spherical massive body of mass M, the gravitational potential
(and hence the gravitational field) at any point outside the sphere is identical to the expression of
potential (and field) for a point mass sitting at the center of the sphere. However, for non-spherical
sources/objects the gravitational potential or field does look like that of point mass, and instead, there
are corrections due to finite size/extension of the massive source- all of those corrections are captured
by the Multipole expansion of the potential. The gravitational potential at the point x due to the
extended source is given by, ˆ
ρ(x′ )
Φ(x) = −GN d3 x′ . (8)
|x − x′ |
The origin of the coordinate system is fixed to some point within the source, let’s say the center
of mass, i.e., x′CM = 0 Now if the point x is far from the source, the vector x is very large when
2
Right now, there is no legit justification for this way of splitting up the energy into a contribution Ufield purely
located in the (static) gravitational field region, outside the masses and a contribution, Umatter which is localized in the
mass distribution and this is indeed totally ad hoc. However, when we get to Fierz-Pauli theory, we will see that indeed,
in the non-relativistic approximation, Ufield will turn out to be the energy contained in the gravitational field outside
matter-energy sources.

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compared to a point within the source, say x′ . So we Taylor expand |x−x 1 ′
′ | around x = 0,, i.e., the
center of mass of the extended source:
   
1 1 ′i ∂ 1 1 ′i ′j ∂ ∂ 1
= +x + x x
|x − x′ | |x| ∂x′i |x − x′ | x′ =0 2! ∂x′i ∂x′j |x − x′ | x′ =0
 
1 ∂ ∂ ∂ 1
+ x′i x′j x′k + ...
3! ∂x′i ∂x′j ∂x′k |x − x′ | x′ =0
xi δ ij xi xj
 
1 1
= + x′i 3 + x′i x′j − 3 + 3 5
|x| |x| 2 |x| |x|
δ x + δ x + δ xj
ij k jk i ki xi xj xk
 
1 ′i ′j ′k
+ x x x −3 + 15 + ....
3! |x|5 |x|7

Substituting into Eq. (8), we obtain the multipole expansion:

ˆ i δ ij xi xj
  
3 ′ ′ 1 ′i x 1 ′i ′j
Φ(x) = −GN d x ρ(x ) +x + x x − 3 +3 5
|x| |x|3 2 |x| |x|
δ ij xk + δ jk xi + δ ki xj xi xj xk
  
1
+ x′i x′j x′k −3 + 15 + . . .
3! |x|5 |x|7
´ 3 ′ ′ i ˆ ˆ
d x ρ(x ) x 3 ′ ′ ′i G N xi xj
d3 x′ ρ(x′ ) −δ ij x′2 + 3x′i x′j


= −GN − GN 3 d x ρ(x )x − 5
|x| |x| 2 |x|
 ij k jk i ki j i j k
ˆ
GN δ x +δ x +δ x xx x
− −3 5
+ 15 d3 x′ ρ(x′ )x′i x′j x′k + . . .
3! |x| |x|7
M x · D GN ij xi xj GN ijk xi xj xk
= −GN − GN − Q − Q + ..., (9)
|x| |x|3 2 |x|5 3! |x|7

where
ˆ
M= d3 x′ ρ(x′ ),
ˆ
i
D = d3 x′ ρ(x′ ) x′i ,
ˆ
ij
d3 x′ ρ(x′ ) 3x′i x′j − x′2 δ ij ,


Q =
ˆ   
Qijk = d3 x′ ρ(x′ ) 15x′i x′j x′k − x′2 x′i δ jk + x′j δ ki + x′k δ ij .

The first term is called the monopole contribution and corresponds to the potential of a point
mass M located at the origin (center of mass). The second term is the dipole term, proportional
to the dipole moment Di , which vanishes when calculated about the center of mass by definition,
Di = 0. The third term is the quadrupole term involving the quadrupole moment Qij , a symmetric
rank-2 tensor, and is the first nonvanishing correction to the monopole term. The fourth term is the
octupole term, with a symmetric rank-3 tensor Qijk , and so on.

Q: Show that the quadrupole moment tensor vanishes for a spherical mass distribution.