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General Relativity
Physics Honours 2005
Dr Geraint F. Lewis
Rm 557, A29
[email protected]
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
http://www.physics.usyd.edu.au/~gfl/L
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.
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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.
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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.
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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.
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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;
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Planetary Motion in GR
This results in three additional equations of motion
These 4 equations allow us to calculate xa(). Let us assume the
motion is equatorial (=/2) with d/d=0. We get
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Planetary Motion in GR
Similarly
Substituting into the “Lagrangian” we find
With the angular momentum equation, and setting u=1/r;
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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 3mu2 is »10-7 for Mercury.
Setting =3m2/h2 gives (with differentiation wrt to )
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Planetary Motion in GR
Treating the relativistic correction as a perturbation;
Then the zeroth order term is just the non-relativistic orbit
and
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Planetary Motion in GR
Adopting the ansatz;
we find (follow argument in textbook)
The dominant term is  sin  (why?) so neglecting other terms
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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).
http://www.physics.usyd.edu.au/~gfl/L
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
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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).
http://www.physics.usyd.edu.au/~gfl/L
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)
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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.
http://www.physics.usyd.edu.au/~gfl/L
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.
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Light in GR
http://www.physics.usyd.edu.au/~gfl/L
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.
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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.
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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 r1! r2
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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 0.9990§0.0076 times the expected fractional
frequency shift of 4.905£10-15.
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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.
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Shapiro Time Delay
Expanding in powers of m/r we find
Integrating from a planet’s radius Dp to the Earth radius DE yields
a difference in time travel compared to flat space of
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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!
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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.
http://www.physics.usyd.edu.au/~gfl/L