1. An Overview of General
Relativity and An
Introduction to Cosmology
Pratik Tarafdar
M.Sc. 1st Year
Dept. of Physics
IIT Bombay

2. Introduction
GR is Einstein’s theory of gravitation that
builds on the geometric concept of space-
time introduced in SR.
Is there a more fundamental explanation of
gravity than Newton’s law ?
GR makes specific predictions of deviations
from Newtonian gravity.

3. Curved space-time
Gravitational fields alter the rules of
geometry in space-time producing “curved”
space
For example the geometry of a simple
triangle on the surface of sphere is different
than on a flat plane (Euclidean)
On small regions of a sphere, the geometry is
close to Euclidean

4. How does gravity curve space-time?
•With no gravity, a ball thrown upward continues upward
and the worldline is a straight line.
•With gravity, the ball’s worldline is curved.
No gravity gravity
t t
x x
•It follows this path because the spacetime surface on
which it must stay is curved.
•To fully represent the trajectory, need all 4 space-time
dimensions curving into a 5th dimension.
•Hard to visualize, but still possible to measure

5. Principle of Equivalence
A uniform gravitational field in some direction is
indistinguishable from a uniform acceleration in the
opposite direction, i.e. inertial mass = gravitational
mass.
At every space-time point in an arbitrary
gravitational field it is possible to find a “locally
inertial coordinate system”, such that within a
small region of the point in question, the laws of
nature take the same form as in unaccelerated
Cartesian coordinate systems in the absence of
gravitation.

6. Einstein was bothered by what he saw as a dichotomy in the
concept of "mass." On one hand, by Newton's second law
(F=ma), "mass" is treated as a measure of an object’s
resistance to changes in movement. This is called inertial mass.
On the other hand, by Newton's Law of Universal Gravitation,
an object's mass measures its response to gravitational
attraction. This is called gravitational mass. Einstein resolved
this dichotomy by putting gravity and acceleration on an equal
footing.
The principle of equivalence is really
a statement that inertial and
gravitational masses are the same
for any object.

7. The General Principle of Covariance
A physical equation holds in any arbitrary gravitational field if two
conditions are met –
1. The equation must reduce to the SR equation of motion in the
absence of gravitational field.
2. The equation must preserve its form under a general coordinate
transformation.

8. The Equation of Motion
The equation of motion of a particle moving freely under the influence of
purely gravitational forces is given by
and
In the absence of any gravitational field, Γνμσ = 0 and gμν = ημν . Thus, the
above equations reduce to the usual SR equations of motion.
Moreover, under a general coordinate transformation x  x’ , it can be
shown that the LHS of the first equation transforms like a tensor, i.e. the
equation of motion preserves its form.
Thus, according to the principle of general of covariance, the equations
of motion are true in any general gravitational field.

9. Algorithm to assess effects of gravitation on physical
systems
1. Write the appropriate SR equations of motion that hold in
the absence of gravitation.
2. Replace ημν with gμν.
3. Replace all derivatives appearing in the equation with
covariant derivatives.

10. Einstein’s Field Equations
The field equations relate the curvature of spacetime with
the energy and momentum within the spacetime (Matter tells
spacetime how to curve, and curved space tells matter how to move).
Where µ and ν vary from 0 to
Gμν = - 8πGTμν = Rμν – 1/2gμνR 3,
Ricci curvature tensor - Rµν
how space is curved location and motion of matter Metric coefficients - g
µν
Riemann-Christoffel Tensor Curvature Scalar – R
Gravitational Constant – G
Stress Energy Tensor - Tµν
Curvature Tensor
Ricci Tensor

11. Tests of General Relativity
Orbiting bodies
GR predicts slightly different paths than
Newtonian gravitation
Most obvious in elliptical orbits where
distance to central body is changing and
orbiting object is passing through regions of
different space-time curvature
The effect - orbit does not close and each
perihelion has moved slightly from the
previous position

12. Effect is greatest for Mercury -
closest to Sun and high
eccentricity of orbit
•Mercury’s perihelion position
advances by 5600 arcsec per
century.
•All but 43 arcsec can be
accounted for by Newtonian
effects and the perturbations of
other planets.
•Einstein was able to explain
the 43 arcsec exactly via GR
calculations.

13. Bending of Light
Einstein said that the warping of space-
time alters the path of light as it passes
near the source of a strong gravitational
field (i.e. photons follow geodesics).
When viewing light from a star, the
position of the star will appear different if
passing near a massive object (like the
Sun).
θ = 4GM/bc2
Where θ angle is in radians and b is
distance from light beam to object of
mass M
If b is radius of Sun (7x1010cm), θ is 8.5x10-6 rad or 1.74 arcseconds

14. Meaurements must be made
during a solar eclipse, when light
from Sun is blocked and stars
near the Sun’s edge can be
seen.
Sir Arthur Eddington headed the
attempt to verify Einstein’s
prediction during an eclipse in
1919 and did so with only a 10%
error.
Since then, the same experiment has been done with radio sources
(better positional accuracy) with much lower error and higher accuracy.
Similarly, the bent path of light also means a delay in the time for a
signal to pass the Sun. This effect has been measured by bouncing
radio waves off Mercury and Venus as they pass behind the Sun, and
observing signals from solar system space craft. GR effects have been
confirmed to an accuracy of 0.1% using these measurements.

15. Gravitational Lensing
Any large galaxy or galaxy cluster
can act as a gravitational lens; the
light emitted from objects behind the
lens will display angular distortion
and spherical aberration. Measuring
the degree of lensing can be used to
calculate the mass of the intervening
body (galaxy clusters usually).
Good way to detect dark matter..
Light waves passing through areas of
different mass density in the
gravitational lens are refracted to
different degrees. Produces double
galaxy images and Einstein Rings (if
observer, lens, and source are aligned
correctly).

16. Gravitational Radiation
Massive objects distort spacetime and a
moving mass will produce “ripples” in
spacetime which should be observable
(e.g. two orbiting or colliding neutron
stars).
Quite natural, because Einstein’s
equations just like Maxwell’s
equations give us radiative
solutions….!! So, just as
accelerated charged particles give
off EM radiation, GR predicts that
accelerated mass should emit
gravitational radiation.

17. Man’s Quest to detect Gravitational Radiation
Laser Interferometer Gravity
Observatory
LIGO - Will try to detect the
ripples in space-time using
laser interferometry to
measure the time it takes
light to travel between
suspended mirrors. The
space-time ripples cause
the distance measured by a
light beam to change as the
gravitational wave passes
by.

18. The Cosmological Principle
A large portion of the modern cosmological theory is based on the
Cosmological Principle. It is a hypothesis that all positions in the
universe are essentially equivalent.
The universe is HOMOGENOUS. It has symmetry in the distribution of
matter and energy.
The universe is ISOTROPIC. It has directional symmetry.

19. Newtonian Cosmology
Newton’s theory of Gravitation leads to a simple model of the expanding
homogenous and isotropic universe. It leads to the derivation of the
evolution equations of Newtonian Cosmology. The homogeneity and
isotropy also consequently establish the Hubble’s Law.
Assumptions in Newtonian Cosmology
1. Matter filling the universe is non-relativistic.
2. Scales are much smaller than the Hubble radius.
Limitations
1. Early hot universe was radiation dominant and hence relativistic
treatment is necessary.
2. General Relativistic considerations become vital at super-Hubble
scales.

20. Derivation of Hubble’s Law from Newtonian Principle
Choose a coordinate system with origin O such that matter is at
rest there. Let v be the velocity field of matter all around. Let us
assume another observer at O’ with radius vector r0’ moving with
velocity v(r0’) with respect to the observer O. If the velocity of matter
at a point p with respect to O and O’ are v(rp) and v’(r’p)
respectively.
r’p = rp – ro’
v’(r’p) = v(rp) – v(r0’)
According to the Cosmological Principle, since the universe is
homogenous therefore the velocity field should have the same
functional form at every point irrespective of the coordinate system.
v(r’p) = v(rp) – v(r0’)  Velocity field needs to a linear function of the
radius vector.
v(r,t) = T(t)r
where T(t) is a 3x3 matrix

21. T(t) can be diagonalised to H(t) and since the universe is isotropic, therefore
T(t) reduces to Tij = H(t)δij ,i.e. v(r,t) = H(t)r , which is the Hubble’s Law.
The Hubble’s Law
Solving for r(t) we know how the distance between two points
in space changes with time, given the expansion rate H(t),
known as the Hubble parameter.
This equation shows how distances in a homogenous and
isotropic universe scale with the scale factor a(t)

22. Cosmic Evolution Equation from Newtonian Approach
Total energy of the particle on the
surface of the sphere
Incorporating Hubble’s Law
Friedmann Equation…!!
Using the equation of continuity for non-relativistic fluid and then using
Hubble’s law, one can calculate the mass density
The subscript and superscript 0 represents
quantities at present epoch.

23. Wait a minute….!!!
In the energy expression, potential energy is assumed to be
zero at infinity. But, in a homogenous space with uniform
matter density, the total mass of the universe diverges as r 3.
If we assume the density to vanish for large r, then we are
in conflict with the concept of homogeneity.
Conservation of energy is difficult to understand in an
infinite, homogenous universe.

24. Newtonian Cosmology itself gives rise to an evolving
model of the universe.
GTR not required at all..! Still concept of static universe
was so deep rooted that when Friedmann found non-
static solutions to Einstein’s equations, Einstein himself
could not believe in it, and tried to reconcile his theory
with the perception of non evolving universe by
introducing the Cosmological constant, which he later
himself withdrew.

25. The Newtonian force of gravitation on a particle of unit mass on the surface of the
homogenous sphere of radius r is given by
Substituting for r(t) on both sides in terms of the scale factor a(t), we get
the acceleration equation.
Observations on Cosmic Microwave Background Radiation indicate that
K is very nearly equal to zero. Putting K=0 in Friedmann equation and
substituting for the expression for matter density, we find that (da/dt) 2 is
proportional to a-1. Integrating we get,
The above solution is known as Einstein-de-Sitter solution which can be
used to estimate the age of universe.
Without gravity, universe would expand at a constant rate H0. Using Hubble’s
Law in that case, age of the universe would be given by
which is the maximum limit for the age of the universe in the hot big bang model.

26. Attempt to reconcile Newtonian gravity with the picture of a static universe can be
made by adding a repulsive term in the force law in the form of a linear force,
because inverse square force and linear force are the only two central forces that
give rise to stable circular orbits.
The modified acceleration equation is given by
where Λ is the cosmological constant. The integrated form of the above equation
gives the modified Friedmann equation.
The modified force law was proposed by Neumann and Seeliger in 1895-96, much
before Einstein gave his theory of gravity. It can be noted from the modified force
law that the cosmological constant term is equivalent to a constant matter density
term.
Pressure Corrections – Cosmological constant belongs to the category of
relativistic systems. Relativistic fluids have non-zero pressure. Hence, pressure
corrections are required. Let us consider adiabatic expansion of a unit comoving
volume in the expanding universe. According to the first law of thermodynamics,
Pb(t) is the pressure of the background fluid .

27. Now, the energy density of the fluid can be expressed through the mass density
Substituting for the energy in the first law of thermodynamics, we get the continuity
equation
Thus, comparing with the previous continuity equation, we observe that it
corresponds to a pressure correction with an additional term Pb/c2. For non-
relativistic fluids energy density dominates over pressure. The early hot universe
needs to tbe treated relativistically. Newtonian approach becomes valid at later
stage when matter became dominant. Choose c =1, so that relativistic energy
density and pressure become the same.
We claim that the correct equation of acceleration for a background fluid with
energy density ρb and pressure Pb is
Check that multiplying both sides by da/dt and using the continuity equation to
express Pb, we get
Integrating the above equation we get back the modified Friedmann equation

28. Now, if we introduce the cosmological constant as a perfect fluid with constant
energy density then from the corrected continuity equation we find that it
contributes to a negative pressure. Then from the corrected acceleration equation
we observe that while positive pressure leads to deceleration, the negative
pressure term aids in the acceleration of the universe.
Dark Energy
The previous equations establish Cosmological Constant as a perfect barotropic
fluid with density ρΛ = Λ/8πG and pressure PΛ = -Λ/8πG
Including the cosmological constant in the background fluid, we see that expansion
(positive acceleration) is characterised by a large negative pressure, accounted for
by an exotic fluid dubbed “Dark Energy”.

29. Dark Energy
Very large negative pressure due to an exotic fluid, responsible for the
late time expansive acceleration of the universe. Dark energy can be
theoretically accounted for by the cosmological constant or by the
introduction of scalar fields.
The Cosmological constant can be a candidate for the dark energy.
However, its very small value leads to fine tuning and coincidence
problem.
Scalar field models are good alternatives that act like cosmological
constant at late times and also provide viable cosmic dynamics at the
early hot epoch. However, there are a large number of scalar field
models and we need considerable amount of data to narrow down to
certain specific ones.

30. Future Plans of Study
Pressure in cosmology is a relativistic effect which can be
understood properly only in the framework of Einstein’s General
Theory of Relativity.
Homogeneity and isotropy of universe is an example of generic
symmetry in space time, which admits analytical solutions of the
otherwise complicated non-linear field equations.
Cosmological constant does not need any adhoc assumption for its
introduction in relativistic cosmology, unlike Newtonian cosmology.
Deep theoretical issues related with cosmological constant. For
example, it can be associated with vacuum fluctuations in QFT. It is
no longer a free parameter in this scheme.
Motivation for moving on to Relativistic Cosmology.

31. Bibliography
S. Weinberg , ‘Gravitation and Cosmology : Principles and
Applications of the General Theory of Relativity’
R.M.Wald , ‘General Relativity’
Misner, Thorne, Wheeler, ‘Gravitation’
Hartle , ‘Gravity : An introduction to Einstein’s General
Relativity’
P.A.M. Dirac , ‘General Relativity’
‘A Primer on Problems and Prospects of Dark Energy’, M.Sami,
Centre of Theoretical Physics, Jamia Millia Islamia, New Delhi

32. Acknowledgement
Dr. Archan S. Majumdar, Dept. of Astrophysics and
Cosmology, SNBNCBS
Mr. Sunish Kr. Deb, Deputy Registrar (Academic)
The whole staff and all my friends and co-learners
at SNBNCBS
THANKS TO YOU ALL….!!