General Relativity Physics Honours 2007 A/Prof. Geraint F. Lewis Rm 557, A29 [email protected] Lecture Notes 8

General RelativityPhysics Honours 2007

A/Prof. Geraint F. LewisRm 557, [email protected]

Lecture Notes 8

Lecture Notes 8 http://www.physics.usyd.edu.au/~gfl/

Lecture

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 raditiation.


Lecture

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.


Lecture

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?


Lecture

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)


Lecture

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;


Lecture

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.


Lecture

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


Lecture

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.


Lecture

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?


Lecture

Slowly Rotating SpacetimeSo far, our massive objects have been static, but what if the body is rotating? We won’t have time to consider the derivation of the effect in general, but you can read it in Section 23.3. For a slowly rotating spherical mass

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;


Lecture

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;


Lecture

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.


Lecture

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.


Lecture

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


Lecture

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 2007 A/Prof. Geraint F. Lewis Rm 557, A29 [email protected] Lecture Notes 8