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Gravitational Wave Astronomy

Gravitational Wave Astronomy. Dr. Giles Hammond Institute for Gravitational Research SUPA, University of Glasgow. Universität Jena, August 2010. Mathematical Tools of GR. The spacetime of gravity is described by a Riemann metric

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Gravitational Wave Astronomy

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  1. Gravitational Wave Astronomy Dr. Giles Hammond Institute for Gravitational Research SUPA, University of Glasgow Universität Jena, August 2010

  2. Mathematical Tools of GR • The spacetime of gravity is described by a Riemann metric • A continuous space which can be differentiated everywhere and is locally flat • In GR we need to define how vectors and their basis sets transform in this space.

  3. y x Mathematical Tools of GR • A Riemann space can be differentiated everywhere • Point X has coordinates • Can define the gradient of a function  at X as the space can be differentiated • Can think of as a set of basis vectors which span the space • In general the basis at X and Y will be different => given a coordinate system we can construct a natural basis for a vector field

  4. Vectors and One-Forms • It is necessary to introduce briefly vectors and one-forms • Consider a 2-D rotation • Transformation rule for a vector (denoted by superscript):

  5. Vectors and One-Forms • Now consider a scalar field (r) : How does  transform under a rotation? • For example: • Transformation rule for a one-form (denoted by subscript):

  6. Vectors and One-Forms • We can show that • In 2-D Euclidean space the same results are obtained • This is not the case in general spaces (e.g change of basis from Cartesian to Polar coordinates) • Any vector space can be written in terms of a dual-basis (covariant/contravariant). The metric tensor maps between them. vector one-form

  7. Parallel Transport • Differentiation of a vector field involves subtracting vector components at different points (i.e. P and Q) • Need a procedure which parallel translates components of A to point Q • This is called Covariant Differentiation • where • are the Christoffel symbols which are used to connect the basis vectors at Q to those at P (i.e. stitch together spacetime)

  8. Christoffel Symbols • The Christoffel symbols can be written in terms of the metric • and thus we can define how vector fields transport over the surface of a general curved space • A geodesic in spacetime is a curve along which the tangent vector is parallel translated. Thus it also includes the Christoffel symbols • Material particles under the influence of gravity follow geodesics

  9. Curvature II • The curvature of gravitational spacetime is described by the Riemann tensor => components can be obtained by parallel translation of a vector Flat space ( = 0) Curved space ( 0) • The Riemann tensor provides the mathematical tool to describe curvature • It is fairly simple to write programs which calculate the components of the Riemann tensor for general spaces

  10. Simple Example • Consider polar coordinates dr and d. The line element is • The only non-zero Christoffel symbols are dr r ds d

  11. Simple Example • All the components of the Riemann tensor are zero => Flat space. The 1st term for example is • And the geodesic equations are • The equations of motion can also be derived from the Lagrangian of the interval

  12. More Complicated Examples • There are add-on packages to Mathematica which turn the program into a powerful symbolic manipulator of tensors • http://www.inp.demokritos.gr/~sbonano/RGTC/ • Eg. 1 The polar coordinate space already examined => Flat space with all components of the Riemann tensor=0 • Eg 2. The surface of a 2-D sphere defined by polar coordinates  and  => Curved space (constant curvature) with some non-zero components of the Riemann tensor

  13. Ricci Tensor/Scalar • The Riemann tensor in 4-D has 256 components (20 independent components) • We can produce a 2-index tensor (16 components) by contraction. This gives the Ricci tensor; • And the Ricci scalar or curvature scalar is • The Ricci scalar of a sphere, radius a, is

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