General Relativity Physics Honours 2005 Dr Geraint F. Lewis Rm 557, A29

Slides:



Advertisements
Similar presentations
General Relativity Physics Honours 2009 Prof. Geraint F. Lewis Rm 560, A29 Lecture Notes 4.
Advertisements

Black Holes. Underlying principles of General Relativity The Equivalence Principle No difference between a steady acceleration and a gravitational field.
Newton’s Law of Universal Gravitation
Physics 55 Friday, December 2, Quiz 7 (last quiz) 2.Relativistic mass, momentum, energy 3.Introduction to General Relativity 4.Einstein’s equivalence.
General Relativity Physics Honours 2006 A/Prof. Geraint F. Lewis Rm 557, A29 Lecture Notes 9.
15.1Tenets of General Relativity 15.2Tests of General Relativity 15.3Gravitational Waves 15.4Black Holes General Relativity CHAPTER 15 General Relativity.
More on time, look back and otherwise t in R(t) starts from beginning of big bang Everywhere in universe starts “aging” simultaneously Observational.
Extragalactic Astronomy & Cosmology First-Half Review [4246] Physics 316.
Developing a Theory of Gravity Does the Sun go around the Earth or Earth around Sun? Why does this happen? Plato } Artistotle } Philosophy Ptolemy }& Models.
General Relativity Physics Honours 2008 A/Prof. Geraint F. Lewis Rm 560, A29 Lecture Notes 3.
Optical Scalar Approach to Weak Gravitational Lensing by Thick Lenses Louis Bianchini Mentor: Dr. Thomas Kling Department of Physics, Bridgewater State.
How do we transform between accelerated frames? Consider Newton’s first and second laws: m i is the measure of the inertia of an object – its resistance.
1. White Dwarf If initial star mass < 8 M Sun or so. (and remember: Maximum WD mass is 1.4 M Sun, radius is about that of the Earth) 2. Neutron Star If.
General Relativity Physics Honours 2006 A/Prof. Geraint F. Lewis Rm 557, A29 Lecture Notes 6.
General Relativity Physics Honours 2007 A/Prof. Geraint F. Lewis Rm 557, A29 Lecture Notes 8.
Semester Physics 1901 (Advanced) A/Prof Geraint F. Lewis Rm 560, A29
General Relativity Physics Honours 2005 Dr Geraint F. Lewis Rm 557, A29
General Relativity Physics Honours 2006 A/Prof. Geraint F. Lewis Rm 557, A29 Lecture Notes 5.
General Relativity Physics Honours 2007 A/Prof. Geraint F. Lewis Rm 557, A29 Lecture Notes 4.
R F For a central force the position and the force are anti- parallel, so r  F=0. So, angular momentum, L, is constant N is torque Newton II, angular.
Physics 133: Extragalactic Astronomy ad Cosmology Lecture 4; January
The Properties of Stars Masses. Using Newton’s Law of Gravity to Determine the Mass of a Celestial Body Newton’s law of gravity, combined with his laws.
Lecture 16 GR – Curved Spaces ASTR 340 Fall 2006 Dennis Papadopoulos.
General Relativity Physics Honours 2007 A/Prof. Geraint F. Lewis Rm 557, A29 Lecture Notes 7.
Chapter 12 Gravitation. Theories of Gravity Newton’s Einstein’s.
1 The Origin of Gravity ≡ General Relativity without Einstein ≡ München 2009 by Albrecht Giese, Hamburg The Origin of Gravity 1.
Special Relativity Speed of light is constant Time dilation Simultaneity Length Contraction Spacetime diagrams.
MANY-BODY PROBLEM in KALUZA-KLEIN MODELS with TOROIDAL COMPACTIFICATION Astronomical Observatory and Department of Theoretical Physics, Odessa National.
General Relativity Physics Honours 2010
Newton and Kepler. Newton’s Law of Gravitation The Law of Gravity Isaac Newton deduced that two particles of masses m 1 and m 2, separated by a distance.
Stationary Elevator with gravity: Ball is accelerated down.
Gravity & orbits. Isaac Newton ( ) developed a mathematical model of Gravity which predicted the elliptical orbits proposed by Kepler Semi-major.
Typical interaction between the press and a scientist?!
Acceleration - rate of change of velocity (speed or direction), occurs any time an unbalanced force is applied.
Central Force Motion Chapter 8
Chapter 26 Relativity. General Physics Relativity II Sections 5–7.
Black Holes and General Relativity
Apsidal Angles & Precession A Brief Discussion! If a particle undergoes bounded, non-circular motion in a central force field  There will always be radial.
General Relativity Physics Honours 2011 Prof. Geraint F. Lewis Rm 560, A29 Lecture Notes 2.
Two-Body Systems.
GRAVITATIONAL LENSING
General Relativity (1915) A theory of gravity, much more general than Newton’s theory. Newtonian gravity is a “special case”; applies when gravity is very.
Bending Time Physics 201 Lecture 11. In relativity, perception is not reality Gravity affects the way we perceive distant events For example, although.
A Short Talk on… Gravitational Lensing Presented by: Anthony L, James J, and Vince V.
Gravitation. Gravitational Force and Field Newton proposed that a force of attraction exists between any two masses. This force law applies to point masses.
General Relativity Physics Honours 2005 Dr Geraint F. Lewis Rm 557, A29
General Relativity and the Expanding Universe Allan Johnston 4/4/06.
Gravitational Lensing
NOTES FOR THE DAY: General Theory of Relativity (a gravity theory): Ingredients: The Principle of Equivalence: acceleration ~ gravity Riemann's Curved.
Fundamental Principles of General Relativity  general principle: laws of physics must be the same for all observers (accelerated or not)  general covariance:
Lecture 27: Black Holes. Stellar Corpses: white dwarfs white dwarfs  collapsed cores of low-mass stars  supported by electron degeneracy  white dwarf.
We use Poinsot’s construction to see how the angular velocity vector ω moves. This gives us no information on how the angular momentum vector L moves.
Lecture 6: Schwarzschild’s Solution. Karl Schwarzschild Read about Einstein’s work on general relativity while serving in the German army on the Russian.
General Relativity Physics Honours 2008 A/Prof. Geraint F. Lewis Rm 560, A29 Lecture Notes 10.
Principle of Equivalence: Einstein 1907 Box stationary in gravity field Box falling freely Box accelerates in empty space Box moves through space at constant.
Influence of dark energy on gravitational lensing Kabita Sarkar 1, Arunava Bhadra 2 1 Salesian College, Siliguri Campus, India High Energy Cosmic.
10/5/2004New Windows on the Universe Jan Kuijpers Part 1: Gravitation & relativityPart 1: Gravitation & relativity J.A. Peacock, Cosmological Physics,
Module 6Aberration and Doppler Shift of Light1 Module 6 Aberration and Doppler Shift of Light The term aberration used here means deviation. If a light.
H8: Evidence for general relativity
General Relativity Physics Honours 2008 A/Prof. Geraint F. Lewis Rm 560, A29 Lecture Notes 9.
Astronomy 1143 – Spring 2014 Lecture 19: General Relativity.
The Meaning of Einstein’s Equation*
General Relativity and Cosmology The End of Absolute Space Cosmological Principle Black Holes CBMR and Big Bang.
General Relativity Physics Honours 2009 Prof. Geraint F. Lewis Rm 560, A29 Lecture Notes 3.
Celestial Mechanics I Introduction Kepler’s Laws.
Celestial Mechanics IV Central orbits Force from shape, shape from force General relativity correction.
Einstein’s Tests of General Relativity through the Eyes of Newton
Curvature in 2D… Imagine being an ant… living in 2D
General Relativity Physics Honours 2006
A theory of gravity, much more general than Newton’s theory.
Presentation transcript:

General Relativity Physics Honours 2005 Dr Geraint F. Lewis Rm 557, A29

ecture Where are we? With the geodesic equation or the Euler-Lagrange approach, you are now armed with the mathematical tools necessary to calculate the vast majority of tests of General Relativity. Perihelion shift of Mercury Deflection of light Redshift of light in a gravitational field The Shapiro time-delay

ecture Schwarzschild Metric The Schwarzschild metric is famous for describing the space-time of a black hole, but it also describes the space-time outside any spherical mass distribution (i.e. the Sun). In spherical polar coords; were G=c=1. Hence, we can use this metric to test general relativity within the Solar System.

ecture Classical Keplerian Motion (15) Planetary motion was a great success of Newtonian Mechanics. For a test particle orbiting in the field of a massive, spherically symmetric body of mass , Newton’s second law is; Angular momentum is conserved due to the spherical symmetry (Nother’s theorem) and so the particle orbits in a plane. Taking the polar angle to be  =  /2, then where h is the specific angular momentum and R is the distance from the origin.

ecture Classical Kelperian Motion We can therefore derive the radial equation of motion Introducing a new variable u=R -1 then (exercise) This is Binet’s equation.

ecture Classical Keplerian Motion The solution to Binet’s equation is Often the solution is written as This is the polar equation of a conic with eccentricity e (<1 for an ellipse), orientation  o & semi-latus rectum l. The closest approach to the Sun (  =  o ) is the perihelion.

ecture Planetary Motion in GR If we treat a planet orbiting the Sun as a particle, we expect it to follow a time-like geodesic. Hence the “Lagrangian” is Remember, dot here is differentiation is with respect to the proper time . We can then apply the Euler-Lagrange;

ecture Planetary Motion in GR This results in three additional equations of motion These 4 equations allow us to calculate x a (  ). Let us assume the motion is equatorial (  =  /2) with d  /d  =0. We get

ecture Planetary Motion in GR Similarly Substituting into the “Lagrangian” we find With the angular momentum equation, and setting u=1/r;

ecture Planetary Motion in GR Differentiating with respect to u This is the relativistic Binet equation. This can be solved in terms of elliptical functions, but 3mu 2 is » for Mercury. Setting  =3m 2 /h 2 gives (with differentiation wrt to  )

ecture Planetary Motion in GR Treating the relativistic correction as a perturbation; Then the zeroth order term is just the non-relativistic orbit and

ecture Planetary Motion in GR Adopting the ansatz; we find (follow argument in textbook) The dominant term is  sin  (why?) so neglecting other terms

ecture Planetary Motion in GR Thus the orbit is periodic in  with a period of T » 2  (1-  ), but perihelion is not reached at the same value of  each orbit. Hence, the perihelion advances with This precession of the perihelion. For Mercury, this predicts 43” per century (note that Mercury suffers » 5600” per century from Newtonian gravity).

ecture Planetary Motion in GR While perturbation methods are applicable in the Solar System, it is simple to examine planetary motion in stronger GR environments; Solve Christoffel symbols Integrate equations of motion

ecture Light in GR In a similar fashion we can calculate the Binet equation for light In the non-gravitational limit (m ! 0) the solution is Where D is the distance of closest approach. The above is an equation of a straight line as  =  o !  o +  (i.e. light in special relativity travels in straight lines).

ecture Light in GR Starting with  o =0 and seeking an approximate solution with it can be seen (exercise) Considering the asymptotic limits (u !§ 0) then (Fig 15.6)

ecture Light in GR Eddington’s 1919 eclipse observations “confirmed” Einstein’s relativistic prediction of  = 1.78 arcseconds. Later observations have provided more accurate evidence of light deflection due to the influence of GR.

ecture Light in GR Cosmologically, a large number of “gravitational lens” systems exist. In these optical illusions, multiple images of the same background source produced by the gravitational field of an intervening galaxy.

ecture Light in GR

ecture Light in GR All of the previous examples use weak-field (and hence small angle) approximations. In strong gravitational fields, the paths of light can be complex and analytic solutions difficult to find. Armed with the metric, it is possible to integrate the geodesic equation and hence calculate the light paths in strong gravity. The image of a flat accretion disk about a rotating black hole.

ecture Light in GR One of the greatest successes of GR (IMHO) was the Paczynski curve, the prediction of the shape of light curve of a more distant star when compact mass (another star passes in front). The dots in these pictures are the data, whereas the solid curves are the theoretical model.

ecture Gravitational Redshift A simple thought experiment (Ch 15.5) shows us that a gravitational redshift is required for energy conservation. We can propose a simple argument that the energy of a photon at a radius r is and the conservation of energy as a photon travels from r 1 ! r 2

ecture Gravitational Redshift Making the approximation on the RHS we get Note: a more detailed (but not detailed enough) argument is given in the text. In 1960 Pound and Rebka exploited the Mossbauer effect to measure the gravitational redshift over 22.5m, obtaining a result which was § times the expected fractional frequency shift of £

ecture Shapiro Time Delay (Ch 15.6) If we calculate the time taken for light or radio waves to reach us from a satellite on the far side of the Sun we find they are delayed with respect to the case with the Sun not present. Again assuming  =  /2, the equation for a null geodesic is in the approximation that the geodesic is straight (exercise). Here D is the distance of closest approach to the Sun.

ecture Shapiro Time Delay Expanding in powers of m/r we find Integrating from a planet’s radius D p to the Earth radius D E yields a difference in time travel compared to flat space of

ecture Shapiro Time Delay The Shapiro time delay has been tested by bouncing radar off planets and over communications with space probes. For Venus, the GR contribution to the time delay is » 200  s and has been verified to better than 5%. Hence to an external observer, time appears to slow down in a gravitational field, although an observer close to the Sun would see the photon pass at c!

ecture Shapiro Time Delay The Shapiro Time Delay is a significant contributor in gravitational lens systems. These are the light curves for two images (black & white dots resp.) of the gravitationally lensed quasar Q0957 (at two wavelength g & r). To align the light curves of each image, one has been temporally shifted by ~420days. This is a combination of geometric and Shapiro time delay.