General Relativity Physics Honours 2009 Prof. Geraint F. Lewis Rm 560, A29 [email protected] Lecture Notes 4

General RelativityPhysics Honours 2009

Prof. Geraint F. LewisRm 560, [email protected]

Lecture Notes 4

http://www.physics.usyd.edu.au/~gfl/Lecture

Lecture Notes 4

Gravitational LensingThere are many astrophysical applications using what we have already learnt. Here we will examine two; gravitational lensing and relativistic accretion disks.

Chapter 11

Given that massive objects can deflect the path of a beam of light, this means that objects can behave as a gravitational lens. Such lenses resemble glass lenses in many ways, giving multiple images which can be magnified.


Lecture Notes 4

Accretion DisksAccretion disks are important in astrophysics as they efficiently transform gravitational potential energy into radiation.Accretion disks are seen around stars, but the most extreme disks are seen at the centre of quasars.

These orbit black holes with masses of ~106-9 M, and radiate up to 1014 L, outshining all of the stars in the host galaxy.

If we assume the black hole is not rotating, we can describe its spacetime with the Schwarzschild metric and make predictions for what we expect to see.


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Accretion Disks

Suppose we have radiating material orbiting a black hole. The photons leave the disk and are received by a distant observer. How much redshifting of the photons does the observer see?


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Accretion DisksHow fast is the material orbiting? Assuming circular orbits

So we write the 4-velocity of the emitting matter to be

And using the normalization of the 4-velocty


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Accretion DisksWe can now us the Killing vector conserved quantities;

Remember, the conserved quantities and the impact parameter are related via


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Accretion DisksRemember we are in a plane where =/2. What redshift do we see? Let’s consider the matter moving transversely to the line-of-sight. For these, b=0 and so

So, there is no Doppler component. What about the matter moving directly towards or away from the observer? Now we explicitly need to consider the value of b


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Accretion DisksRemembering that photons are null

And so

The resultant frequency ratio is


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Accretion DisksFor small values of M/r then

where


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Accretion DisksAccretion disks are hot an emit in highly ionized species.Remembering that the radius for the last stable orbit occurs at r=6M, the maximum redshift for an edge-on disk is

For a face-on disk, this is ~0.7. Above is the X-ray spectrum of the 6.4keV Fe line in a Seyfert galaxy. The solid line is a fit to the data using a relativistic accretion disk seen at 30o. The extent of the redshift reveals this must be a black hole.


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Black HolesWe have seen previously that something weird happens when we fall towards the origin of the Schwarzschild metric; while the proper motion for the fall is finite, the coordinate time tends to r=2M as t->∞.

Chapter 12

Clearly, there is something weird about r=2M (the Schwarzschild radius) for massive particles. But what about light rays? Remembering that light paths are null so we can calculate structure of radial light paths.


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Black Holes

Clearly the light cones are distorted, and within r=2M all massive particles are destined to hit the origin (the central singularity). In fact, once within this radius, a massive particle will not escape and is trapped (and doomed)!But how do we cross from inside to outside?

We can calculate the gradients of light rays from the metric

Again, light curves tell us about the future of massive particles.


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Eddington-FinkelsteinWhile Eddington figured out the solution in the early 1900s, but it was Finkelstein in the late 1950s who rediscovered the answer. Basically, we will just make a change of coordinates;

And the Schwarzschild metric can be written as

The geometry is the same, but the metric now contains off-axis components. But notice now that the metric does nothing weird at r=2M (but still blows up at r=0).


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Eddington-FinkelsteinLight rays are still null, so we can find the light cones

Clearly, v=const represents a null geodesic (and these are ingoing rays). There is another that can be integrated;

http://casa.colorado.edu/~ajsh/

So, the change in coordinates removes the coordinate singularity and now light (and massive particles) happily fall in.


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Eddington-FinkelsteinTo create this figure, we define a new time coordinate of the form;

This straightens in-going light rays (making it look like flat space time).

However, outgoing light rays are still distorted.


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White HolesAnother coordinate transformation can straighten outgoing light rays.The result is a white hole and massive particle at r<2M are destined to be ejected and cannot return.Note that while this is still the Schwarzschild solution, this behaviour is not seen in the original solution.


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Collapse to a Black HoleConsider the collapse of a pressureless (dust) star. The surface of the will collapse along time-like geodesics. We know that the proper time taken for a point to collapse to r=0 is

Surfaces B12.2

While the Schwarzschild metric blows up at the horizon, the E-F coordinates remain finite and you can cross the horizon.

Once across the horizon, the time to the origin is

Which is 10-5s for the Sun.


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Collapse to a Black HoleRemember that in E-F coordinates, outgoing light rays move along geodesics where

Let’s consider an emitter at (vE,rE) and receiver at (vR,rR). If the receiver is at large distances, then the log term can be neglected. When the emitter is close to r=2M the log term dominates. For the distant observer then

Where t is the Schwarzschild time coordinate (which will equal the proper time of the observer as the spacetime far from the hole is flat).


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Collapse to a Black HoleKeeping the dominant terms, the result is

As rE2M, tR∞; Photons fired off at regular intervals of proper time are received later and later by the observer.

These photons are also redshifted. If the photons are emitted at regular intervals of (proper time) then


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Collapse to a Black HoleRemembering that the frequency of the emitted and received photons are inversely related to the ratio of the emitted and received time between photons, then

So the distant observer sees the surface of the star collapsing, but as it approaches r=2M it appears to slow and the received photons become more and more redshifted.

The distant observer never sees the surface cross the Schwarzschild radius (or horizon) although we see from the E-F coordinates, the mass happily falls through.


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Kruskal-Szekeres We can take the game of changing coordinates even further with Kruskal-Szekeres coordinates. Starting again with the Schwarzschild metric, and keeping the angular coordinates unchanged, the new coordinates are given by

Where c & s are cosh & sinh with the first combination used for r>2M and the second for r<2M. We also find that


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Kruskal-SzekeresThe resultant metric is of the form


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Kruskal-Szekeres

Lines of constant t are straight, while those at constant r are curves. Light cones are at 45o, as in flat spacetime. We have our universe, plus a future singularity (black hole) and past singularity (white hole). There also appears to be another universe over to the left.


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Kruskal-Szekeres We can examine the radial infall of matter in these coordinates. The distant observer moves along a line of constant radius, while the matter falls in emitting photons. Again, the distant observer sees the photons arriving at larger intervals and never sees the matter cross the horizon. Note that once inside the horizon, the matter must hit the central singularity.


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PenrosePenrose mapped the Kruskal coordinates further, such that now we get two entire universes on a single page. This is an example of maximal compactification.


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Astrophysical Black HolesUnfortunately, we don’t have time to cover astrophysical black holes in the lectures, but you should read through Chapter 12 of the textbook.

Chapter 13

You should ensure you read in detail Section 13.3 which explains how black holes can evaporate via Hawking radiation.


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Rotation

Chapter 14

So far, we have considered the motion of point like particles through spacetime. These travel along timelike (massive) or null

(massless) geodesics;

This equation actually parallel transports the tangent vector to the geodesic (the 4-velocity) along the path. Clearly, in flat Minkowski spacetime, the Christoffel symbols are zero.

Other vectors can undergo parallel transport. The spin of a gyroscope can be represented as a spacelike spin 4-vector.


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RotationIn our rest frame,

And the 4-spin and 4-velocity are orthogonal. However, this must hold in all frames. The 4-spin is parallel transported along the geodesic using

As we will see later, this is related to the covariant derivative, and the Christoffel symbols take into account the change in the coordinates over the manifold. In flat space, the spin of the gyroscope remains fixed. What about curved spacetime?


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RotationLet’s consider a gyroscope orbiting in the static Schwarzschild metric. We will see that General Relativity predicts that, relative to the distant stars, the orientation of the gyroscope will change with time. This is geodetic precession.If the gyroscope is on a circular orbit then

And so the 4-velocity is given by

(Remember we are orbiting in a plane where =/2)


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RotationWe can use the orthogonality to find

And solving we find

We can use the Christoffel symbols for the Schwarzschild metric and the gyroscope equation to find;


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RotationCombining these, we can derive the equation

This is just the usual wave equation with a frequency

Clearly, there is something weird here. The components of the spin vector change with time, but the frequency is not the same as the orbital frequency. So, the spin angle and the orbital frequency are out of phase.


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RotationThe solution to these equations are quite straight-forward and

Where s.s=s2*. Let’s consider t=0 and align the spin along the

radial direction and we need to look at the orthonormal frame of the observer; as the metric is diagonal, we can define the radial orthonormal vector is


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RotationThe period of the orbit is P=2/ and therefore the component of spin in the radial direction after one orbit is

So, after each orbit, the direction of the spin vector has been rotated by

This is the shift for a stationary observer at this point in the orbit. We could also set up a comoving observer at this point (orbiting with the gyro) but the radial component of the spin vector would be the same.


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RotationFor small values of M/R the angular change is

For a gyroscope orbiting at the surface of the Earth, this corresponds to

While small, this will be measured by Gravity Probe B.

What would you expect if you were in an orbiting laboratory with windows that opened once per year?


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Slowly Rotating Spacetime

Where J is the angular momentum of the mass. Hence the form of the metric is deformed, but what is the effect of this distortion? Firstly, lets consider this metric in cartesian coordinates;


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Slowly Rotating SpacetimeLet’s drop a gyro down the rotational axis. We can approximate the Schwarzschild metric as being flat (as contributing terms would be c-5). For our motion down the z-axis (rotation axis) then

The non-zero Christoffel symbols for our flat + slow rotation metric are simply

When evaluated on the z-axis and retaining the highest order terms. From this, we can use the gyroscope equation to show;


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Slowly Rotating Spacetime

Again, we have a pair of equations that show that the direction of the spin vector change with time with a period of

This is Lense-Thirring precession. At the surface of the Earth this has a value of

While small, this will be observable by GP-B.


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Rotating Black HolesUnfortunately, we will not get time in this course to look at general black holes. Here we will just review the basic properties. If a black hole is rotating, its spacetime geometry is

described by the Kerr metric.

where

These are Boyer-Lindquist coordinates. This is asymptotically flat and the Schwarzschild metric when a=0.


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Rotating Black HolesWe can analyze particle motion in this metric using the same approach as we have used previously. The horizon is flattened

and the singularity is distorted into a ring.

http://www.engr.mun.ca/~ggeorge/astron/shad/index.html


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Charged Black HolesThe only thing left that can influence the form of black holes is charge (black holes have no hair). The geometry of a charged black hole is described by the Reissner-Nordstrom metric.

Where is the charge of the black hole.

Again, when the charge is zero, we recover the Schwarzschild solution. Again, the metric results in weird structure not seen in the simple solution. If 2>m2, then we have a central singularity but no shielding horizon, it is naked.

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General Relativity Physics Honours 2009 Prof. Geraint F. Lewis Rm 560, A29 [email protected] Lecture Notes 4