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.