General Relativity Physics Honours 2009 Prof. Geraint F. Lewis Rm 560, A29 Lecture Notes 3 Lecture Notes 3 Schwarzschild Geometry When faced with the field equations, Einstein felt that it analytic solutions may be impossible. In 1916, Karl Schwarzschild derived the spherically symmetric vacuum solution, which describes the spacetime outside of any spherical, stationary mass distribution; Ch. 9 This is in Geometrized units, where G=c=1. Note, that the geometrized mass has units of length and so the curved terms in the invariant above are dimensionless. Lecture Notes 3 Schwarzschild Geometry An examination of the Schwarzschild metric reveals; Time Independence: The metric has the same form for all values of t. Hence we have a Killing vector; Spherical Symmetry: This implies further Killing vectors, including one due to the independence of Weak Field Limit: When M/r is small, the Schwarzschild metric becomes the weak field metric we saw earlier. Lecture Notes 3 Schwarzschild Geometry Something clearly goes wrong at r=0 (the central singularity) and r=2M (Schwarzschild radius or singularity). More on this later. Remember: the (t,r,,) in this expression are coordinates and r is not the distance from any centre! If we choose a t=constant & r=constant we see the resulting 2-dimensional surface is a sphere (in 3-dimensions). So we can simply relate the area to r, but not the volume! Lecture Notes 3 Particle Orbits Massive particles follow time-like geodesics, but understanding their motion is aided by identifying conserved quantities. Given our two Killing vectors we obtain At large r the first conserved quantity is the energy per unit mass in flat space; While the second is the angular momentum per unit mass. Lecture Notes 3 Particle Orbits The conservation of l implies particles orbit in a plane. Consider an time-like geodesic passing through the point =0, with d/d=0. The conservation of l ensures d/d=0 along the geodesic and so the particle remains in the plane =0. But the spherical symmetry implies this is true for all orbits. Hence we will consider equatorial orbits with =/2 and u =0. Defining the 4-velocity of the particle to be Lecture Notes 3 Particle Orbits Substituting in our conserved quantities we get where This result differs from the Newtonian picture (found in any classical mechanics text) with the addition of the r -3 term in the potential! At large r the orbits become more Newtonian. Remember, while orbits are closed in r -1 potentials, they are not in general potentials. Lecture Notes 3 Particle Orbits Considering the relativistic and Newtonian potentials, we see that while they agree at large radii, they are markedly different at small radii. The Newtonian has a single minima, while the relativistic has a minimum and maximum; Lecture Notes 3 Particle Orbits For l/M>2M; Also for large r, then And so d=b and b is the impact parameter of the orbit! Lecture Notes 3 Light Ray Orbits The effective potential has a peak, and so unstable circular orbit, at Considering light rays starting from infinity, those with b -1 less than this scatter back to infinity, while those with more than this exceed the potential barrier and fall into the centre. Lecture Notes 3 Escaping to Infinity Consider a source at r