General Relativity Physics Honours 2011 Prof. Geraint F. Lewis Rm 560, A29 [email protected] Lecture Notes 2

General RelativityPhysics Honours 2011

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

Lecture Notes 2

http://www.physics.usyd.edu.au/~gfl/LectureLecture Notes 2

Curved SpacetimeThere are many coordinate systems we could use to describe flat spacetime;

Chapter 7

While these look different, the underlying geometry is the same. (How do you tell?) We can simply map between one coordinate system and another.

The goal of a good coordinate system is to uniquely label each point. Most coordinate systems fail to do this; what is the coordinate of the origin if using polar coordinates?


Coordinate SingularitiesConsider a flat plane described by polar coordinates. The line element is given by;

We can make a simple coordinate transformation;

The line element now blows up at r’=0, but the geometry of the surface is unchanged, it is still flat!

While you may think we made a silly coordinate transformation, when it comes to curved spacetimes, choosing the right coordinate system without such singularities is not straight-forward.


Mixing it upIn relativity, time and space can be mixed together. We can take our spherical polar flat spacetime and make a coordinate transformation;

We can make another coordinate change;

We now have the entire universe on the page.


The MetricThe metric is central to studying relativity. In general;

The metric is symmetric and position dependent. The metric for flat spacetime in spherical polar coordinates is;

The metric has 10 independent components, although there are 4 functions used in transforming coordinates, so really there are 6 independent functions in the metric.

7.3 - indices


Local Inertial FramesThe equivalence principle states that the local properties of curved spacetime should be indistinguishable from flat spacetime. Basically, this means that at a specific point in a general metric g(x) we should be able to introduce a new coordinate such that;

So we have a locally flat piece of spacetime in which the rules of special relativity hold. This defines a local inertial frame.


Back to LightCones

If we consider a world line of the form X(T)=A cosh(T), the path is timelike and hence always locally traveling less than the speed of light. But this spacetime is actually flat and we can make a coordinate transformation back to

Consider a spacetime with an interval of the form

It is straight-forward to calculate the paths of light rays (as ds2=0). Note that these appear to be distorted.


Curving Spacetime

The Alcubierre spacewarp has a metric of the form;

where Vs=dxs/dt and f is a function which is unity at the ship and falls to zero at rs. Globally, we are traveling faster than light, but locally we never exceed c.

7.6 - Volumes


CurvatureSo, the interval describes intrinsically curved surfaces. We can visualize the curved surface with an embedding diagram. Consider a wormhole with the interval

We can take a constant time slice, and a constant angle (the metric is spherically symmetrical), choosing =/2.

This is an axisymmetric 2-dimensional surface. We can embed this in 3-dimensional space. Let’s use cylindrical coordinates (, , z) and choose =.


Curvature

The result is a surface given by

The result is a wormhole which joins two infinite asymptotically flat universes!


Vectors (again)Now we have curved spacetime, we need to look at what this means for vectors. It is important to remember that vectors are local quantities, and obey usual vector rules at that point.

When considering a vector at a point, we need to consider its components in two different, but important, coordinate systems; the coordinate basis and the orthonormal basis


Vectors (again)In the coordinate basis

These are the vectors you transport around the manifold.

In the orthonormal basis

These are the vectors as measured by an observer.

Remember


Vectors (again)We can connect the vector components in the orthonormal and coordinate frames by projecting each basis onto each other (i.e. we express the unit vectors of one frame as vectors in the other). Given this;

For a diagonal metric, we can simply construct the orthonormal frame from the coordinate frame with

Etc…


Vectors (again)Consider polar components on a plane. In the coordinate frame the unit vectors depend upon position, where as in the orthonormal frame they have unit length everywhere.We can define the basis vectors in each frame;

7.9 - Surfaces


More MathsSuppose we have a function on a manifold f(x), and a curve x(), we can define the derivative along the curve as

Ch 20.1-20.4

The vector t is the tangent vector to the curve and has the components

And the directional derivative to be


Transforming VectorsHow do we transform the components of a vector from one coordinate system to another?

And so;

And;


Dual VectorsA dual vector (or covector) is a linear map from a vector to a real number;

Where are the components of the covector. As with vectors, we can express a covector in terms of dual basis vectors;

The basis {e} is dual to the basis {e} if;


Moving indicesAny “thing” can be written in terms of its vector or dual basis and hence we have an interpretation of the dual mapping;

And defining the inverse metric through


TensorsTensors generalize the linear mapping of vectors to reals. The metric tensor maps two vectors to a number (it’s a rank 2);

This can be easily generalized to any rank;

And we can move the indices around with the metric tensor

Respect you indices!


To work effectively with tensors, you need to know a couple more operations;

Simple construction:

Vector results:

Contraction:

Tensors


Tensor ConversionTo convert between a coordinate basis and an orthonormal basis, we can generalize what we learnt for vectors;

Similarly, we can convert between two different coordinate bases by again generalizing;


Variational Principle

Ch. 3.5

The laws of Newtonian physics can be expressed in terms of the variational principle;

A particle moves between a point in space at one time and another point in space at a later time so

as to extremize the action in between

The action is the path integral over the Lagrangian. In 1-dimension, we can write this as;

With


Variational PrincipleIt is possible to show (textbook) that paths that extremize the action are solutions to the Lagrange equation;

Plugging in our 1-d Newtonian Lagrangian we find;

This is just Newton’s second law!


Variational PrincipleImportantly, the variational principle can be extended to n-dimensions (or coordinates) and we can write;

This results in a series of coupled differential equations which can be solved to describe the motion of a particle.

However, you should be asking yourself “how does this apply to relativity as we have spatial coordinates and time in there as separate things and we know that they are not really separate”…


Variational PrincipleWith a slight modification, we can write the variational principle in a form suitable for special relativity;

Ch. 5.4

The worldline of a free particle between two timelike separated points extremizes the

proper time between them.

The proper time between two points is simply the interval

We can parameterize any path between A and B as a series of coordinates x = x() and then the proper time is;


Variational PrincipleWe are faced with the same problem as the Newtonian variational principle and so the extremal paths are the solution to the Lagrange equation;

Where the Lagrangian is


Variational PrincipleLet’s try the x1 component;

As the Lagrangian L=d/d and multiplying by L;

This, of course holds for all coordinates =0,1,2,3 and so it tells us that in Special Relativity, free massive particles follow straight-line paths through spacetime and these extremize the proper time (in fact they are maxima!). These paths are called geodesics.


Variational PrincipleAnother slight modification & we can use the variational principle to determine the paths through curved spacetime;

Chapter 8

The worldline of a free test particle between two timelike separated points extremizes the

proper time between them.

Again the proper time between two points is the interval

And we can take the Lagrangian to be;


A simpler LagrangianDe’Inverno (pg. 100) shows how you can simplify the Lagrangian picture;

And the Lagrange equation is

For a massive particle, K is the 4-velocity and we have the additional constraint that;


An exampleThe invariant for a wormhole is

And hence the Lagrangian is

With the result;

Try this with the K formulation of the Lagrangian.


Geodesic EquationIt should be apparent that we can write these equations as;

The are known as Christoffel Symbols (and are related to

the covariant derivative). These are symmetric in their lower two indices, so

Remembering the definition of 4-velocity, then we can write the geodesic equation as


Christoffel SymbolsFor the wormhole example, the only non-zero Christoffel symbols are;

Remember that these are symmetric in their lower indices. Christoffel symbols are not tensors and there is no implicit summation over repeated indices.

We are now armed with everything we need to calculate paths through arbitrary spacetime geometries, but things can be made a little simpler.


Christoffel SymbolsThe Christoffel symbols can be expressed in terms of the underlying metric (see Hartle’s wepage if you are interested in the details) and

Note that this is slightly different to the version given in the text book as it makes use of the inverse metric. This makes calculating the Christoffel symbols more straightforward when handling non-diagonal metrics.

While this looks messy (and don’t forget the implicit summations) there are symbolic mathematics packages that greatly simplify these calculations (GRTensor and one on Hartle’s website).


Through a wormholeGiven the wormhole metric seen earlier, how much time does an observer experience traveling from R to –R, assuming the initial radial velocity is ur=U. First determine the 4-velocity;

You can calculate the geodesic equations and you’ll find the symmetry means there are no angular influences. The result is that;

And so the result is that r() = U .


How does a vector move over a manifold? We need to use the concept of parallel transport, but a lot can be gained from understanding conserved quantities. One of the most powerful is the concept of a Killing vector.

Simply put, a Killing vector results from a symmetry of a metric, and this symmetry implies a conserved quantity along a geodesic path.

Suppose a metric has no time dependence, so moving from one time coordinate x0, to another x0+constant, results in the metric being unchanged, then we have a Killing vector;

Killing Vectors


Killing VectorsSo what? Take a step back to the Lagrangian. With no time dependence (=0) we see that;

And so we have;

Therefore, this quantity is conserved along a geodesic path. Hence, we can calculate how the components of a vector change across a manifold using such conservation laws.


Gravitational RedshiftHere we will consider the Schwarzschild metric

Chapter 9.2

If two observers are at different radial locations and exchange photons, what would each observer measure the energy to be?

We know each observer has a normalized 4-velocity so


Gravitational RedshiftAs the observers are fixed spatially;

The Schwarzschild metric is time-independent and so we have a Killing vector of the form

And so


Gravitational RedshiftTherefore, the energy of a photon as measured by an observer can be written as

But the final quantity in this expression is conserved and so we can simply calculate the ratio of frequencies from the above.

It is important to note that symmetries are often not obvious when looking at the metric (consider cartesian verses spherical polar coordinates), but one can used Killing’s equation to identify hidden symmetries.


Null GeodesicsSo far, we have only considered timelike geodesics. But the geodesic formulism can be simple transferred to null paths for photons;

With the constraint on the 4-velocity of

Where is an affine parameter. So, now we have the machinery to calculate the path of massless and massive particles.

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General Relativity Physics Honours 2011 Prof. Geraint F. Lewis Rm 560, A29 [email protected] Lecture Notes 2