47
General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 [email protected] Lecture Notes 1

General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 [email protected] Lecture Notes 1

Embed Size (px)

Citation preview

Page 1: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

General RelativityPhysics Honours 2009

Prof Geraint F. LewisRm 560, [email protected]

Lecture Notes 1

Page 2: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Why are we here?

Page 3: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Why are we here?G

PS

Extr

em

e S

tars

Bla

ck Hole

sTh

e U

niv

erse

Chapter 1

Page 4: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Prior KnowledgeThere is no escaping the fact that General Relativity is a mathematically challenging physical theory. However, this course is structured differently to typical courses on relativity; the goal will be to develop a physical understanding of underlying theory and learn how to apply the mathematical framework in various physical situations.

To tackle this course effectively, prior knowledge includes

Differential Equations Special Relativity Lagrangian Mechanics (desirable) Maxwell’s Equations (desirable) Tensors (desirable)

Page 5: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Textbook

Gravity: An Introduction to Einstein’s GR Course BookJ. B. Hartle

You should obtain a copy of the textbook as it contains required reading and some of the assignment questions. Note that we will not step through the book linearly! Other useful texts are;

Introducing Einstein’s Relativity by R. D’Inverno

General Relativity by Hobson, Efstathiou & Lazenby

A First Course in GR by B. F. Schutz

Spacetime and Geometry by S. M. Carroll

Page 6: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Additional Resources

There are many additional general relativity resources that I encourage you to explore. These include;

http://www.physics.usyd.edu.au/~gfl/Lecture/ http://spacetimeandgeometry.net/ “A no-nonsense introduction to general relativity” by Sean

Carroll (http://pancake.uchicago.edu/~carroll/notes) “Living Reviews in Relativity” (http://www.livingreviews.org) “The meaning of Einstein’s equations” by John Baez and

Emory Bunn (http://arxiv.org/gr-qc/0103044) Preprint archive (http://www.arxiv.org) Type “general relativity” etc into google & wikipedia

Page 7: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Assessment for this course will consist of;

Three assignments 30%Final exam 70%

Assignments are to be handed into the Student Support Office on the due date. Late assignments will be penalized 20% for each day they are late. Assignments more than one week late will not be accepted without a formal special consideration.

You can bring one hand written A4 page into the exam. Postgraduate students must achieve >70% on all assignments and will sit the exam in an open book environment.

Assessment

Page 8: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

For the assignments, students are encouraged to work on problems in groups, although the submitted assignment must reflect a student’s work.

Analytical solutions may not exist for some of the differential equations you encounter. In this case, you will be required to obtain a numerical solution. This can be obtained using canned routines, such as ode23/ode45 in Matlab or NSolve in Mathematica. Alternatively, you can use canned routines in C/Fortran/Whatever code, or write your own integrator. You must state clearly which method you employ.

Assessment

Page 9: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

All questions relating to this course (including those on the assignments) will be addressed in the WebCT discussion forum. Students are encouraged to post and answer questions on the forum. Please follow the rules.

WebCT Discussion Forum

Page 10: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Geometry and PhysicsWe will hear a lot about geometry, especially the geometry of curved surfaces. This can be quite different to Euclidean geometry (flat surface) and requires a way to characterize the curvature (the metric).

Chapter 2

We will rely on differential geometry, whereby geometry can be understood in terms of a line element.

Cartesian coordinates

Polar coordinates

Page 11: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Geometry and PhysicsIf we consider the surface of a sphere, the line element is more complicated (remember the surface of a sphere is 2-dimensional). If we use spherical polar coordinates;

In relativity we will be using the concept of spacetime, in which we consider 4-dimensional surfaces;

[This is the line element for the Schwarzschild black hole]

Page 12: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Coordinates and InvarianceWhen discussing distances, motions etc through any general curved surface (a manifold), we rely on coordinates. Such coordinates are not fundamental to the surface and different coordinate systems can be used over the surface.

For example, we can cover a flat plane with a Cartesian or Polar coordinate system.

The physical predictions we make should be invariant and not depend on the choice of coordinates. We should be able to transform from one coordinate system to another.

What we will see is that we can write all physical laws in a tensor form that provides a method to transform between the differing coordinate systems.

Page 13: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Newtonian RelativityNewtonian mechanics assumes flat space and absolute time.

To describe the motion of particles we can use a flat-space coordinate system (e.g. Cartesian), and events have a unique set of coordinates (t,x,y,z).

Newtonian mechanics gives us a special set of observers, those in inertial reference frames, who will see Newton’s laws hold. These are stationary, or move with constant velocity, with respect to one another.

Chapter 3

All inertial observers are equivalent as far as dynamical experiments are concerned.

Page 14: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Galilean TransformationsObservers can transform between the coordinates of an event in different inertial frames with Galilean Transformations. If the frames S and S’ have collinear x-axes, and S’ moves along the x-axis with constant velocity v, then;

Review 3.3-3.5

How do you use the rules of Newtonian mechanics to make predictions in a particular physical situation?

Page 15: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Relativity in a Nutshell

Einstein Field Equation

Geodesic Equation

Page 16: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Special RelativityThe foundation of Special Relativity comes from Einstein’s understanding of Maxwell’s equations. He realised that while they predicted the velocity of light, they do not say with respect to what this velocity should be measured.

In 1905, Einstein proposed the postulate of the constancy of the speed of light.

The velocity of light in free space is the same for all inertial observers.

Chapter 4

Note that while the famous null result of the Michelson-Morley experiment can be explained in terms of this postulate, the experiment was not Einstein’s motivation!

Page 17: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Special RelativityIf the speed of light c is the same in all inertial frames, then something else has to give. Einstein proposed that the Newtonian ideas of space and absolute time had to be abandoned. As well as carrying their own spatial coordinates, all inertial observers also carry their own clock. Observers now disagree on where an event happens, and also when it happens.

http://en.wikibooks.org/

If we consider two nearby events for a particular observer, we can define an invariant interval;

[Review section 4.3]

Page 18: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Light ConesThe negative sign in ds2 has some interesting effects. Intervals in flat space are always positive, but this is not the case with spacetime.

This breaks spacetime into distinct regions. These are very important to the notion of causality.

Page 19: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

World Lines

Massless particles travel along null world lines.

Massive particles travel along timelike world lines. Note that their path always lies within their past and future light cones.

Page 20: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

The Interval Again!Consider a person moving along a particular world line.

To them, they are at rest and so dx=0.

So, the only thing that changes along a path is the time they experience (i.e. what is seen on their watch).

This is known as the proper time d.

Page 21: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

The Interval Again!Remember that the interval is invariant, so everyone agrees on its value.

This allows us to connect how spacetime is experienced between observers.

Rewriting;

Therefore there is time dilation between the observers.

Page 22: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

The Twin ParadoxAn awful lot has been written about the twin paradox, much of it complete rubbish.

The important point to note is that the straight-line path on a spacetime diagram represents the longest 4-distance (i.e. through spacetime).

All other paths are shorter.

ww

w.p

hys.

vt.

edu

Wikipedia!

Page 23: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Lorentz BoostsTo transform between inertial frames, we use Lorentz Boosts.

where

Page 24: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Simultaneity & ContractionConsider the green dots as two separate events. In the primed frame, these occur on a line of constant ct’ and are simultaneous. In the unprimed frame, these two events occur on two different lines of ct, and so are not simultaneous.Furthermore, if we take the events to denote the ends of a rod, clearly the spatial length as measured in the two frames will be different and we get length contraction.

Page 25: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Addition of VelocitiesThe Lorentz boosts also give relativistic velocity addition.

and similarly

/

Page 26: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

4-VectorsWhen dealing with 4-dimensional spacetime, it is natural to consider ‘things’ in terms of 4-vectors. In an inertial frame, we can define basis 4-vectors of unit length, pointing along (t,x,y,z). Then a 4-vector a can we written as;

Chapter 5

We will be dealing with the components of the vector a=(at,ax,ay,az) when considering relativity. Note that the components are often written as a=(a0,a1,a2,a3). We can write a 4-vector as;

The last term here is the Einstein Summation Convention, his greatest contribution to maths!

Page 27: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Lorentz Boosts The components of a 4-vector can be transformed between inertial frames using Lorentz Boosts.

Here we have set c=1.

It should be clear that we can write Lorentz Boosts as matrix multiplications.

Page 28: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Scalar ProductsThe concept of a scalar product is very important for 4-vectors

Here we have an implicit summation over and .

Here, is the metric of flat spacetime (in Cartesian coordinates).

Page 29: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Scalar ProductsAs with 3-vectors, two 4-vectors are orthogonal if

The length (or magnitude) of a 4-vector is

Note, the length of a vector can be positive (spacelike), zero (null) or negative (timelike).

Importantly, the invariant interval can be written as

Page 30: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

OrthogonalityConsider two unit 4-vectors in the primed frame, one along t’ and the other along x’.

This is orthogonal as a.b= 0

a & b in the unprimed frame;

But;

Orthogonality is preserved!

Page 31: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

KinematicsAny particle is constantly moving through spacetime (even if it is stationary in space).We can describe location in spacetime in terms of parameterized coordinates;

For massive objects, the natural choice of parameter is the proper time .

We can now define the velocity of a particle through spacetime;

Page 32: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

KinematicsThe components of the 4-velocity are;

So, 4-velocity is

Note;

All massive particle have the same speed through spacetime.

Page 33: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Newton’s lawsNewton’s first law holds in relativistic mechanics;

We also have Newton’s second law;

Here, m is the rest mass of the particle. But we are now dealing with the 4-force and 4-acceleration.

You should convince yourself that f and u are orthogonal

Page 34: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Other 4-VectorsAnother important 4-vector which we will see is the 4-momentum;

And we can write Newton’s second law as;

The components of the 4-momentum are

Review 5.4

The magnitude of p is

Page 35: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Light RaysLight rays do not have a proper time (as ds=0) and so we cannot use it to parameterize path through spacetime.We can use an affine parameter which we use to describe the motion, but it has no physical meaning.As a light rays has a straight-line path in a spacetime diagram, we can parameterize the path as;

Where u = dx/d. This is a null vector and u.u=0.

Again, the 4-momentum is important and

The 4-momentum and wave 4-vector k are null!

Page 36: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Doppler ShiftConsider a source emitting photons of frequency in all directions (in its rest frame). The wave 4-vector is;

Suppose to an observer, this source is moving along the x’-axis with velocity V. What frequency will this observer see a photon detected at angle ’? In the primed frame

We can connect the two using a Lorentz Boost

But kx’ = ’ cos ’ so

Page 37: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Observers

The orthonormal basis is an important concept in relativity. This consists of one time and three spatial unit vectors, with the time vector tangent to the world line (i.e. it points along the 4-velocity u). The components of the orthonormal basis must be orthogonal to each other. What an observer measures is relative to the orthonormal basis.

Page 38: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

ObserversSuppose a particle with 4-momentum p passes through a lab with a defined orthonormal basis. We can express the 4-momentum in terms of the orthonormal basis;

The first component in this basis is the energy of the particle as seen by the observer. We can extract the components of the 4-momentum via

So the energy of the particle can be written as

Work through the examples in Chapter 5.

Page 39: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Steps to General RelativityWhile special relativity overturned established ideas in physics, Einstein realised that it was incompatible with gravity

The problem is positions according to who, and at what time?

Einstein’s revelation began with the realization of equivalence of gravitational and inertial mass. Simply put, all masses fall at the same rate. What happens if I drop a laboratory?

Chapter 6

Clearly everything falls together. What Einstein realized is that effectively gravity has vanished.

Page 40: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Equivalence Principle

Einstein’s “happiest thought” came from the realization he could take the equivalence principle further.

Simply put, Einstein reasoned that;

There is no experiment that can distinguish between uniform acceleration and a uniform

gravitational field.

Page 41: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

ImplicationsThere were several implications of Einstein’s view of the equivalence principle, including the fact that light should be deflected in a gravitational field.

http://physics.syr.edu/courses/modules/LIGHTCONE/equivalence.html

Page 42: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Time & AccelerationSuppose the rocket is accelerating at g. The location of the two astronauts is

Suppose a pulse is fired from A to B at t=0. Then

Now a second pulse is emitted from A at A and is received at t1+B

Page 43: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Time & AccelerationIf we assume is small, and so we can expand these expressions in only the linear terms, we find

But the equivalence principle tells us that this should also occur in a uniform gravitational field, and so we would expect

Where is the Newtonian gravitational potential. This immediately suggests that the rocket and gravitational field should see photons red/blue-shifted.

GPS 6.4

Page 44: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Spacetime is CurvedWe could suppose that spacetime is flat and the gravitational potential somehow influence the running of the clock. This is a little like measuring distances on a flat map, but having to change the length of the ruler.

While some aspects of relativity do adopt this view, it is “simpler, more economical and ultimately more powerful” to keep our measuring devices (rulers and clocks) fixed and assume the geometry in which they sit is curved.

Newtonian gravity can be expressed in the Static Weak-Field Metric in which time and space are curved

Page 45: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Time DilationConsider two observers, xA and xB, in a gravitational field. A fires two photons to B; these do not travel as straight lines in this picture, but they will have the same shape. Hence, if photons from A are separated with a coordinate time t, then B will receive them with the same separation.

But how much time will each measure on their watches (their proper time). Neither move spatially, so x=y=z=0 and so the interval is;

Page 46: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Time DilationTaking the square root then;

As t is common for both observers, we can equate their proper times and find that;

Again, we recover the time dilation, but find it is due to the geometry of spacetime.

Page 47: General Relativity Physics Honours 2009 Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au Lecture Notes 1

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 1

Spacetime Diagrams

Understanding the paths of light rays over a spacetime diagram is an important aspect of relativity.

In general, they do not look like those seen in special relativity, as they can be squeezed & rotated.

Here we have a spacetime diagram for a white hole in Eddington-Finkelstein Coordinates.