General Relativity
Physics Honours 2010
Prof. Geraint F. Lewis
Rm 560, A29
gfl@physics.usyd.edu.au
Lecture Notes 3
Schwarzschild Geometry
When faced with the field equations, Einstein felt that it
analytic solutions may be impossible. In 1916, Karl
Schwarzschild derived the spherically symmetric vacuum
solution, which describes the spacetime outside of any
spherical, stationary mass distribution;
This is in Geometrized units, where G=c=1.
Note, that the geometrized mass has units of length and so
the curved terms in the invariant above are dimensionless.
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Ch. 9
Schwarzschild Geometry
An examination of the Schwarzschild metric reveals;
Time Independence: The metric has the same form for all
values of t. Hence we have a Killing vector;
Spherical Symmetry: This implies further Killing vectors,
including one due to the independence of ;
Weak Field Limit: When M/r is small, the Schwarzschild
metric becomes the weak field metric we saw earlier.
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Schwarzschild Geometry
Something clearly goes wrong at r=0 (the central singularity)
and r=2M (Schwarzschild radius or singularity). More on this
later.
Remember: the (t,r,,) in this expression are coordinates
and r is not the distance from any centre!
If we choose a t=constant & r=constant we see the resulting
2-dimensional surface is a sphere (in 3-dimensions). So we
can simply relate the area to r, but not the volume!
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Particle Orbits
Massive particles follow time-like geodesics, but understanding their
motion is aided by identifying conserved quantities. Given our two
Killing vectors we obtain
At large r the first conserved quantity is the energy per unit
mass in flat space;
While the second is the angular momentum per unit mass.
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Particle Orbits
The conservation of l implies particles orbit in a plane.
Consider an time-like geodesic passing through the point =0,
with d/d=0. The conservation of l ensures d/d=0 along the
geodesic and so the particle remains in the plane =0. But the
spherical symmetry implies this is true for all orbits. Hence we
will consider equatorial orbits with =/2 and u=0.
Defining the 4-velocity of the particle to be
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Particle Orbits
Substituting in our conserved quantities we get
where
This result differs from the Newtonian picture (found in any
classical mechanics text) with the addition of the r-3 term in
the potential! At large r the orbits become more Newtonian.
Remember, while orbits are closed in r-1 potentials, they are
not in general potentials.
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Particle Orbits
Considering the relativistic and Newtonian potentials, we see
that while they agree at large radii, they are markedly
different at small radii. The Newtonian has a single minima,
while the relativistic has a minimum and maximum;
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Particle Orbits
For l/M<121/2, the extrema in the effective potential vanish,
and so the particle, even if it has an angular component to its
4-velocity, will fall to the origin. This is contrast to the
Newtonian potential as the present of any angular velocity
ensures that it will miss the origin.
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Example Orbits
These example orbits are for l/M=4.3, with differing values of
total energy. The first presents two circular orbits, with the
inner one being unstable. In the next one we see that a noncircular orbit can precess, while the latter two show motion
that is not seen in Newtonian r-1 potentials.
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Radial Plunge Orbits
Radial plunge orbits have no angular momentum (l=0) and
follow a strictly radial path. If we assume that particle is at
rest at r=, then
The equation of motion becomes
Using the time-like Killing vector, the 4-velocity is
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Radial Plunge Orbits
It’s quite straightforward to solve for r();
Which can be integrated to give
Where
So, according to its own clock, a particle falls from a
coordinate r to the origin of the coordinate system in a finite
amount of proper time (note this result is the same as the
Newtonian!).
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Radial Plunge Orbits
What about r(t), where t is the coordinate time.
Integrating gives;
In terms of coordinate time, the position of the particle
asymptotes to r=2M as t, and so the particle never crosses
the Schwarzschild radius!
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Stable Circular Orbits
These occur at the minimum of the effective potential. As l/M
becomes smaller, then the location of the minimum moves
inwards. When l/M=121/2, this orbit is the Innermost Stable
Circular Orbit and occurs at;
The angular velocity in terms of coordinate time is
For a circular orbit, dr/d=0 and E = Veff and
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Stable Circular Orbits
With this, and noting the location of the stable circular orbits
are minima of the effective potential, then
and
Remembering the 4-velocity is
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Precession
The orbit of a test mass around a massive object do not
precess in Newtonian physics. As we have seen, this is not
true in relativity.
We can take two equations;
And assuming =/2, then
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Precession
In Newtonian physics, the angle swept out between aphelion
and perihelion (the closest & furthest distance) is .
We can calculate the corresponding angle in relativity by
integrating over this formula between the turning points on the
relativistic orbit. The peri- and aphelion are those points in the
orbit where dr/d=0 and so it is where
And so
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Precession: Newtonian Limit
We can examine this in the Newtonian limit to see what we
would expect for Solar System planets. Putting the speed of
light back in we get;
The first term can be related to the Newtonian energy through
Again, relativity introduces an additional term into the
expression (the final term in the integral).
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Precession: Newtonian Limit
Without this final term, the angle swept out per orbit is always
 = 2. Including it results in a shift of
to first order in 1/c2 (See assignment).
Remember that l is related to the conservation of angular
momentum and so we can rewrite the above expression in
terms of the semimajor axis a and eccentricity e so
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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
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Ch. 9.4
Light Ray Orbits
So;
again assuming =/2. Using the conserved quantities
And so we can write;
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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 r>>2M;
Also for large r, then
And so d=b and b is the impact parameter of the orbit!
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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.
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Escaping to Infinity
Consider a source at
r<3M 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;
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Example 9.2
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!)
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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
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Deflection of Light
How much is light deflected by a massive, spherical object?
From our conserved and geodesic equations we have;
And so
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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.
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Deflection of Light
In solving for the deflection angle, introduce a new variable;
Note, if M=0 then the resulting integral is , no deflection.
Considering a light ray grazing the surface of the Sun;
We can write the deflection angle as
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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
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Shapiro Time Delay
The Shapiro time delay is apparent when photons are sent 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.
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Shapiro Time Delay
As with the deflection of light, we can write
And the total time taken for the trip is
where
and
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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
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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 Einstein’s 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.
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Perihelion Shift of Mercury
Mercury is the closest planet to the Sun,
with a semi-major axis of 58x106 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
42.98±0.04 “/century could not be
explained.
The prediction from Einstein’s analysis of the orbit in the weakfield limit predicts;
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Gravitational Lensing
Einstein’s 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 Einstein’s
prediction. Now measured to an accuracy of ~1%.
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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.
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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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