LIGO – Laser Interferometer Gravity Observatory – Washington State and Louisiana – separated by space to allow to check false signals Mirrors separated by 2.5 miles LISA – Laser Interferometer Space Antenna – mirror separated by 3 million miles! About 30 million miles from Earth
Overview of GTR and Introduction to Cosmology
An Overview of General Relativity and AnIntroduction to Cosmology Pratik Tarafdar M.Sc. 1st Year Dept. of Physics IIT Bombay
IntroductionGR is Einstein’s theory of gravitation thatbuilds on the geometric concept of space-time introduced in SR.Is there a more fundamental explanation ofgravity than Newton’s law ?GR makes specific predictions of deviationsfrom Newtonian gravity.
Curved space-timeGravitational fields alter the rules ofgeometry in space-time producing “curved”spaceFor example the geometry of a simpletriangle on the surface of sphere is differentthan on a flat plane (Euclidean)On small regions of a sphere, the geometry isclose to Euclidean
How does gravity curve space-time?•With no gravity, a ball thrown upward continues upwardand 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 onwhich it must stay is curved.•To fully represent the trajectory, need all 4 space-timedimensions curving into a 5th dimension.•Hard to visualize, but still possible to measure
Principle of EquivalenceA uniform gravitational field in some direction isindistinguishable from a uniform acceleration in theopposite direction, i.e. inertial mass = gravitationalmass.At every space-time point in an arbitrarygravitational field it is possible to find a “locallyinertial coordinate system”, such that within asmall region of the point in question, the laws ofnature take the same form as in unacceleratedCartesian coordinate systems in the absence ofgravitation.
Einstein was bothered by what he saw as a dichotomy in theconcept of "mass." On one hand, by Newtons second law(F=ma), "mass" is treated as a measure of an object’sresistance to changes in movement. This is called inertial mass.On the other hand, by Newtons Law of Universal Gravitation,an objects mass measures its response to gravitationalattraction. This is called gravitational mass. Einstein resolvedthis dichotomy by putting gravity and acceleration on an equalfooting. The principle of equivalence is really a statement that inertial and gravitational masses are the same for any object.
The General Principle of CovarianceA physical equation holds in any arbitrary gravitational field if twoconditions 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 of a particle moving freely under the influence ofpurely 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 systems1. 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.
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 - Rµνhow 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
Tests of General RelativityOrbiting bodiesGR predicts slightly different paths thanNewtonian gravitationMost obvious in elliptical orbits wheredistance to central body is changing andorbiting object is passing through regions ofdifferent space-time curvatureThe effect - orbit does not close and eachperihelion has moved slightly from theprevious position
Effect is greatest for Mercury -closest to Sun and higheccentricity of orbit•Mercury’s perihelion positionadvances by 5600 arcsec percentury.•All but 43 arcsec can beaccounted for by Newtonianeffects and the perturbations ofother planets.•Einstein was able to explainthe 43 arcsec exactly via GRcalculations.
Bending of LightEinstein said that the warping of space-time alters the path of light as it passesnear the source of a strong gravitationalfield (i.e. photons follow geodesics).When viewing light from a star, theposition of the star will appear different ifpassing near a massive object (like theSun). θ = 4GM/bc2Where θ angle is in radians and b isdistance from light beam to object ofmass MIf b is radius of Sun (7x1010cm), θ is 8.5x10-6 rad or 1.74 arcseconds
Meaurements must be madeduring a solar eclipse, when lightfrom Sun is blocked and starsnear the Sun’s edge can beseen.Sir Arthur Eddington headed theattempt to verify Einstein’sprediction during an eclipse in1919 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 asignal to pass the Sun. This effect has been measured by bouncingradio waves off Mercury and Venus as they pass behind the Sun, andobserving signals from solar system space craft. GR effects have beenconfirmed to an accuracy of 0.1% using these measurements.
Gravitational LensingAny large galaxy or galaxy clustercan act as a gravitational lens; thelight emitted from objects behind thelens will display angular distortionand spherical aberration. Measuringthe degree of lensing can be used tocalculate the mass of the interveningbody (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 RadiationMassive objects distort spacetime and amoving mass will produce “ripples” inspacetime which should be observable(e.g. two orbiting or colliding neutronstars). 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.
Man’s Quest to detect Gravitational Radiation Laser Interferometer Gravity ObservatoryLIGO - Will try to detect theripples in space-time usinglaser interferometry tomeasure the time it takeslight to travel betweensuspended mirrors. Thespace-time ripples causethe distance measured by alight beam to change as thegravitational wave passesby.
The Cosmological PrincipleA large portion of the modern cosmological theory is based on theCosmological Principle. It is a hypothesis that all positions in theuniverse are essentially equivalent.The universe is HOMOGENOUS. It has symmetry in the distribution ofmatter and energy.The universe is ISOTROPIC. It has directional symmetry.
Newtonian CosmologyNewton’s theory of Gravitation leads to a simple model of the expandinghomogenous and isotropic universe. It leads to the derivation of theevolution equations of Newtonian Cosmology. The homogeneity andisotropy also consequently establish the Hubble’s Law.Assumptions in Newtonian Cosmology1. Matter filling the universe is non-relativistic.2. Scales are much smaller than the Hubble radius.Limitations1. Early hot universe was radiation dominant and hence relativistictreatment is necessary.2. General Relativistic considerations become vital at super-Hubblescales.
Derivation of Hubble’s Law from Newtonian PrincipleChoose a coordinate system with origin O such that matter is atrest there. Let v be the velocity field of matter all around. Let usassume another observer at O’ with radius vector r0’ moving withvelocity v(r0’) with respect to the observer O. If the velocity of matterat 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 ishomogenous therefore the velocity field should have the samefunctional form at every point irrespective of the coordinate system.v(r’p) = v(rp) – v(r0’) Velocity field needs to a linear function of theradius vector. v(r,t) = T(t)rwhere T(t) is a 3x3 matrix
T(t) can be diagonalised to H(t) and since the universe is isotropic, thereforeT(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)
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 usingHubble’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 bezero at infinity. But, in a homogenous space with uniformmatter density, the total mass of the universe diverges as r 3.If we assume the density to vanish for large r, then we arein conflict with the concept of homogeneity.Conservation of energy is difficult to understand in aninfinite, homogenous universe.
Newtonian Cosmology itself gives rise to an evolvingmodel of the universe.GTR not required at all..! Still concept of static universewas so deep rooted that when Friedmann found non-static solutions to Einstein’s equations, Einstein himselfcould not believe in it, and tried to reconcile his theorywith the perception of non evolving universe byintroducing the Cosmological constant, which he laterhimself withdrew.
The Newtonian force of gravitation on a particle of unit mass on the surface of thehomogenous sphere of radius r is given bySubstituting for r(t) on both sides in terms of the scale factor a(t), we getthe acceleration equation.Observations on Cosmic Microwave Background Radiation indicate thatK is very nearly equal to zero. Putting K=0 in Friedmann equation andsubstituting for the expression for matter density, we find that (da/dt) 2 isproportional to a-1. Integrating we get,The above solution is known as Einstein-de-Sitter solution which can beused 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 bemade 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 thatgive rise to stable circular orbits.The modified acceleration equation is given bywhere Λ is the cosmological constant. The integrated form of the above equationgives the modified Friedmann equation.The modified force law was proposed by Neumann and Seeliger in 1895-96, muchbefore Einstein gave his theory of gravity. It can be noted from the modified forcelaw that the cosmological constant term is equivalent to a constant matter densityterm. 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 densitySubstituting for the energy in the first law of thermodynamics, we get the continuityequationThus, comparing with the previous continuity equation, we observe that itcorresponds to a pressure correction with an additional term Pb/c2. For non-relativistic fluids energy density dominates over pressure. The early hot universeneeds to tbe treated relativistically. Newtonian approach becomes valid at laterstage when matter became dominant. Choose c =1, so that relativistic energydensity and pressure become the same.We claim that the correct equation of acceleration for a background fluid withenergy density ρb and pressure Pb isCheck that multiplying both sides by da/dt and using the continuity equation toexpress Pb, we getIntegrating the above equation we get back the modified Friedmann equation
Now, if we introduce the cosmological constant as a perfect fluid with constantenergy density then from the corrected continuity equation we find that itcontributes to a negative pressure. Then from the corrected acceleration equationwe observe that while positive pressure leads to deceleration, the negativepressure term aids in the acceleration of the universe. Dark EnergyThe previous equations establish Cosmological Constant as a perfect barotropicfluid 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 forby an exotic fluid dubbed “Dark Energy”.
Dark EnergyVery large negative pressure due to an exotic fluid, responsible for thelate time expansive acceleration of the universe. Dark energy can betheoretically accounted for by the cosmological constant or by theintroduction of scalar fields.The Cosmological constant can be a candidate for the dark energy.However, its very small value leads to fine tuning and coincidenceproblem.Scalar field models are good alternatives that act like cosmologicalconstant at late times and also provide viable cosmic dynamics at theearly hot epoch. However, there are a large number of scalar fieldmodels and we need considerable amount of data to narrow down tocertain specific ones.
Future Plans of StudyPressure in cosmology is a relativistic effect which can beunderstood properly only in the framework of Einstein’s GeneralTheory of Relativity.Homogeneity and isotropy of universe is an example of genericsymmetry in space time, which admits analytical solutions of theotherwise complicated non-linear field equations.Cosmological constant does not need any adhoc assumption for itsintroduction in relativistic cosmology, unlike Newtonian cosmology.Deep theoretical issues related with cosmological constant. Forexample, it can be associated with vacuum fluctuations in QFT. It isno longer a free parameter in this scheme.Motivation for moving on to Relativistic Cosmology.
BibliographyS. Weinberg , ‘Gravitation and Cosmology : Principles andApplications of the General Theory of Relativity’R.M.Wald , ‘General Relativity’Misner, Thorne, Wheeler, ‘Gravitation’Hartle , ‘Gravity : An introduction to Einstein’s GeneralRelativity’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
AcknowledgementDr. Archan S. Majumdar, Dept. of Astrophysics andCosmology, SNBNCBSMr. Sunish Kr. Deb, Deputy Registrar (Academic)The whole staff and all my friends and co-learnersat SNBNCBS THANKS TO YOU ALL….!!