General Relativity Physics Honours 2006

General RelativityPhysics Honours 2006

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

Lecture Notes 3

mailto:[email protected]

Lecture Notes 3 http://www.physics.usyd.edu.au/~gfl/

Lecture

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


Lecture

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!


Lecture

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”…


Lecture

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;


Lecture

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


Lecture

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.


Lecture

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;


Lecture

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;


Lecture

An exampleThe invariant for a wormhole is

And hence the Lagrangian is

With the result;

Try this with the K formulation of the Lagrangian.


Lecture

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


Lecture

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.


Lecture

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).


Lecture

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 .


Lecture

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


Lecture

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.


Lecture

Gravitational RedshiftThe spacetime outside of a spherically symmetric mass distribution is given by 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


Lecture

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


Lecture

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.


Lecture

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.

General Relativity Physics Honours 2006

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