General Relativity Physics Honours 2008 A/Prof. Geraint F. Lewis Rm 557, 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;