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General Relativity Physics Honours 2007

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Title: General Relativity Physics Honours 2007


1
General Relativity Physics Honours 2007
  • A/Prof. Geraint F. Lewis
  • Rm 557, A29
  • gfl_at_physics.usyd.edu.au
  • Lecture Notes 5

2
Light Ray Orbits
  • The equivalence principle tells us that light
    rays should be influenced as they pass through a
    gravitational field. We can use the same geodesic
    formulism to study this.
  • Firstly, we still have two conserved quantities
    due to the symmetries of the Schwarzschild metric.

Note that the derivatives are with respect to the
affine parameter, not the proper time. Secondly
we have the normalization of the 4-velocity
Ch. 9.4
3
Light Ray Orbits
  • So

again assuming ??/2. Using the conserved
quantities
And so we can write
4
Light Ray Orbits
  • This has the same form as the massive particle
    orbits if we assume an effective potential of the
    form

And treating b-2 as an energy term. What is the
physical meaning of b? Consider orbits that start
at rgtgt2M
Also for large r, then
And so db and b is the impact parameter of the
orbit!
5
Light Ray Orbits
The effective potential has a peak, and so
unstable circular orbit, at
Considering light rays starting from infinity,
those with b-1 less than this scatter back to
infinity, while those with more than this exceed
the potential barrier and fall into the centre.
6
Escaping to Infinity
  • Consider a source at rlt3M emitting light in all
    directions. Some light will escape to infinity,
    while some will fall into the black hole. What is
    the critical angle at which light barely escapes?
  • We need to consider the light ray as seen in the
    orthonormal basis. Again, working in a plane
    where ??/2, then

Example 9.2
7
Escaping to Infinity
  • As the metric is diagonal, we can simply define
    the orthogonal basis vectors as

Hence, the photon 4-velocity in the orthonormal
frame is
(you should convince yourself that the 4-velocity
of the photon in the orthonormal frame is null!)
8
Escaping to Infinity
  • Each angle corresponds to a different value of
    b-1 and an examination of the potential shows
    that rays which escape to infinity have
    energies greater than the potential barrier.
    Hence the critical angle occurs at b2 27M2 and

9
Deflection of Light
  • How much is light deflected by a massive,
    spherical object?
  • From our conserved and geodesic equations we have

And so
10
Deflection of Light
  • From infinity, the photon travels to a radius r1,
    before heading out again. This radius occurs at

And the angle swept out is
The smaller the impact parameter, the larger the
deflection angle, to the point where the photon
enters a circular orbit or falls into the centre.
11
Deflection of Light
  • In solving for the deflection angle, introduce a
    new variable

Note, if M0 then the resulting integral is ?, no
deflection. Considering a light ray grazing the
surface of the Sun
We can write the deflection angle as
12
Deflection of Light
  • We can expand this expression out in the lowest
    order terms of 2M/b and get

Remember, w1 is the root of the denominator. The
result is that the deflection is given by
13
Shapiro Time Delay
  • The Shapiro time delay is apparent when photons
    are send on a return path near a massive object.
    In the Solar System, this involves bouncing
    radar off a reflector (space ship or planet)
    located on the other side of the Sun, and seeing
    how long the signal takes to return. The result
    is different to what you would expect in flat
    (special relativisitic) spacetime.

14
Shapiro Time Delay
  • As with the deflection of light, we can write

And the total time taken for the trip is
where
and
15
Shapiro Time Delay
  • As with the deflection of light, we can find the
    weak field limit of this integral which would
    apply in the Solar System

The first term in this expression is simply the
expected Newtonian time delay, and the other
terms are a relativistic correction (but what is
wrong with the above?). For photons grazing the
Solar surface we get
16
Solar System Tests
  • Chapter 10 discusses Solar System tests of
    general relativity, including measurements of the
    Parameterized-Post-Newtonian (PPN) parameters
    these add higher terms to the metric and extend
    relativity. For Einsteins theory of relativity,
    these parameters must be exactly unity.
  • While interesting, the contents of this chapter
    will not be examinable. However, you should read
    through the material.
  • We will summarize the solar system tests.

17
Perihelion Shift of Mercury
  • Mercury is the closest planet to the Sun, with a
    semi-major axis of 58106 km and eccentricity of
    0.21. The orbit of Mercury has been known to
    precess for quite a while. The vast majority of
    the precession is due to Newtonian effects.
    However, a residual precession of 42.980.04
    /century could not be explained.

The prediction from Einsteins analysis of the
orbit in the weak-field limit predicts
18
Gravitational Lensing
  • Einsteins first prediction was that light would
    be deflected as it passed by massive objects. He
    calculated that a light ray grazing the Sun would
    be deflected by

Made in 1916, this prediction could not be tested
until the end of WWI.
Eddington organized two expeditions to observed
an eclipse in 1919. With the Moon blocking out
the Sun, the positions of stars could be
measured, agreeing (roughly) with Einsteins
prediction. Now measured to an accuracy of 1.
19
Gravitational Redshift
  • The final test proposed by Einstein in 1916 was
    the gravitational redshift. This was finally
    measured by the Pound-Rebka experiment in 1959 by
    firing gamma rays up and down a 22m tower at
    Harvard.
  • Measuring a frequency change of 1 part in 1015,
    their measurement agreed with t Einstein
    prediction with an uncertainty of 10.
  • Five years later, the accuracy was improved to a
    1 agreement and now measurements can accurately
    agree to less than a percent accuracy.

20
Shapiro Time Delay
  • The Shapiro delay has also been measured using
    space probes, including Mariner in 1970 and
    Viking in 1976.
  • The expected delay is of order 100s of
    microseconds over a total journey time of hrs,
    but atomic clocks are accurate to 1 part in 1012.
  • A recent measurement using the Cassini space
    probe found the agreement to be

(Bertotti et al 2003)
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