An Overview of General Relativity and An Introduction to

Cosmology

An Overview of General Relativity and An Introduction to

Cosmology

Pratik TarafdarM.Sc. 1st Year

Dept. of PhysicsIIT Bombay

Pratik TarafdarM.Sc. 1st Year

Dept. of PhysicsIIT Bombay

IntroductionIntroduction

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.

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.

Curved space-timeCurved 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

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

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.

•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

t t

x x

No gravity gravity

Principle of EquivalencePrinciple 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.

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.

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.

The General Principle of Covariance

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.

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.

The Equation of MotionThe Equation of Motion

The equation of motion of a particle moving freely under the influence of

purely gravitational forces is given by

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.

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.

Where and vary from 0 to 3,Ricci curvature tensor - R Metric coefficients - g Curvature Scalar – RGravitational Constant – GStress Energy Tensor - Tµν

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).

Gμν = - 8πGTμν = Rμν – 1/2gμνR

how space is curved location and motion of matter

Einstein’s Field Equations

Ricci Tensor

Riemann-Christoffel Tensor

Curvature Tensor

Tests of General RelativityTests of General Relativity

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

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

Orbiting bodies

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.

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).

= 4GM/bc2

Where angle is in radians and b is distance from light beam to object of mass M

If b is radius of Sun (7x1010cm), is 8.5x10-6 rad or 1.74 arcseconds

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.

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.

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).

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.

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.

Man’s Quest to detect Gravitational Radiation

Laser Interferometer GravityObservatory

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.

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.

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

Derivation of Hubble’s Law from Newtonian Principle

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)

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.

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.

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 r3. 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.

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.

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.

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 .

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

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/8πGIncluding 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”.

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.

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.

BibliographyBibliography

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

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

AcknowledgementAcknowledgement

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

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….!!