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

Lecture Notes 2 Curved Spacetime There 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?

Lecture Notes 2 Coordinate Singularities Consider 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.

Lecture Notes 2 Mixing it up In 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.

Lecture Notes 2 The Metric The 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 indices

Lecture Notes 2 Local Inertial Frames The 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.

Lecture Notes 2 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 ds 2 =0). Note that these appear to be distorted.

Lecture Notes 2 Curving Spacetime The Alcubierre spacewarp has a metric of the form; where V s =dx s /dt and f is a function which is unity at the ship and falls to zero at r s. Globally, we are traveling faster than light, but locally we never exceed c Volumes

Lecture Notes 2 Curvature So, 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 =.

Lecture Notes 2 Curvature The result is a surface given by The result is a wormhole which joins two infinite asymptotically flat universes!

Lecture Notes 2 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

Lecture Notes 2 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

Lecture Notes 2 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…

Lecture Notes 2 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; Surfaces

Lecture Notes 2 More Maths Suppose we have a function on a manifold f(x  ), and a curve x  (), we can define the derivative along the curve as Ch The vector t is the tangent vector to the curve and has the components And the directional derivative to be

Lecture Notes 2 Transforming Vectors How do we transform the components of a vector from one coordinate system to another? And so; And;

Lecture Notes 2 Dual Vectors A 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;

Lecture Notes 2 Moving indices Any “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

Lecture Notes 2 Tensors Tensors 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!

Lecture Notes 2 To work effectively with tensors, you need to know a couple more operations; Simple construction: Vector results: Contraction: Tensors

Lecture Notes 2 Tensor Conversion To 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;

Lecture Notes 2 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 Notes 2 Variational Principle It 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 Notes 2 Variational Principle Importantly, 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 Notes 2 Variational Principle With 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 Notes 2 Variational Principle We 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 Notes 2 Variational Principle Let’s try the x 1 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 Notes 2 Variational Principle Another 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 Notes 2 A simpler Lagrangian De’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 Notes 2 An example The invariant for a wormhole is And hence the Lagrangian is With the result; Try this with the K formulation of the Lagrangian.

Lecture Notes 2 Geodesic Equation It 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 Notes 2 Christoffel Symbols For 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 Notes 2 Christoffel Symbols The 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 Notes 2 Through a wormhole Given the wormhole metric seen earlier, how much time does an observer experience traveling from R to –R, assuming the initial radial velocity is u r =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 Notes 2 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 x 0, to another x 0 +constant, results in the metric being unchanged, then we have a Killing vector; Killing Vectors

Lecture Notes 2 Killing Vectors So 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 Notes 2 Gravitational Redshift Here 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

Lecture Notes 2 Gravitational Redshift As the observers are fixed spatially; The Schwarzschild metric is time-independent and so we have a Killing vector of the form And so

Lecture Notes 2 Gravitational Redshift Therefore, 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 Notes 2 Null Geodesics So 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.