General Relativity Physics Honours 2007. A/Prof. Geraint F. Lewis Rm 557, A29 [email protected] Lecture Notes 1. Why are we here?. Why are we here?. GPS. Black Holes. The Universe. Extreme Stars. Chapter 1. Prior Knowledge.
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There 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
Gravity: An Introduction to Einstein’s GR Ã Course Book
J. B. Hartle
You should obtain a copy of the textbook as it contains required reading and the assignment questions. Note that we will not step through the book linearly! Other useful texts are;
Introducing Einstein’s Relativity byR. D’Inverno
General Relativity by Hobson, Efstathiou & Lazenby
A First Course in GR by B. F. Schutz
Spacetime and Geometry by S. M. Carroll
There are many additional general relativity resources that I encourage you to explore. These include;
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.
We 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).
We will rely on differential geometry, whereby geometry can be understood in terms of a line element.
If 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]
When 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 of 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.
Newtonian 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.
All inertial observers are equivalent as far as dynamical experiments are concerned.
Observers 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;
How do you use the rules of Newtonian mechanics to make predictions in a particular physical situation?
Einstein Field Equation
The 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.
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!
If 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.
If we consider two nearby events for a particular observer, we can define an invariant interval;
[Review section 4.3]
The 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.
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.
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.
Remember that the interval is invariant, so everyone agrees on its value.
This allows us to connect how spacetime is experienced between observers.
Therefore there is time dilation between the observers.
An 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.
To transform between inertial frames, we use Lorentz Boosts.
Consider 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.
The Lorentz boosts also give relativistic velocity addition.
When 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;
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!
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.
The 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.
As 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
Consider 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;
Orthogonality is preserved!
Any 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;
The components of the 4-velocity are;
So, 4-velocity is
All massive particle have the same speed through spacetime.
Newton’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
Another important 4-vector which we will see is the 4-momentum;
The components of the 4-momentum are
The magnitude of p is
And we can write Newton’s second law as;
Light 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!
Consider 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
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.
Suppose 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.
While 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?
Clearly everything falls together. What Einstein realized is that effectively gravity has vanished.
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.
There were several implications of Einstein’s view of the equivalence principle, including the fact that light should be deflected in a gravitational field.
Suppose 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
If 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.
We 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
Consider 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;
Taking 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.
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.