The Schwarzschild Metric

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1 The Schwarzschild Metric The Schwarzschild metric describes the distortion of spacetime in a vacuum around a spherically symmetric massive body with both zero angular momentum and electric charge. It is the first non-trivial exact solution to the Einstein field equations to ever be published. It is of practical importance in astrophysics and cosmology as an approximation to the gravitational behavior between slowly moving planets such as the Sun and Earth.

2 Historical Background The Schwarzschild metric was found by Karl Schwarzschild in 1915 just over a month after Einstein s publication of the theory of general relativity. Schwarzschild s 1916 paper which contained his results caused controversy in the physics community due to its strange properties, especially its singularities at the points r=0 and r=r S, the Schwarzschild radius. In Schwarzschild s original derivation of his solution, the point rs is located at the origin of the coordinate system. David Hilbert performed an analysis of the Schwarzschild metric in 1917 and identified the two singularities, but the behavior of the solution at r S remained unclear. Several authors in the period between 1920 and 1940 were able to show that the singularity at the Schwarzschild radius was a coordinate singularity and can be removed by a suitable choice of coordinates. However, the singularity at r = 0 cannot be removed by any coordinate transformation and describes the simplest type of black hole (i.e. one that is not rotating and is electrically neutral).

3 Schwarzschild Coordinates Schwarzschild s original results used what are now known as Schwarzschild coordinates. These coordinates are a type of coordinate known as circumferential radial coordinates, meaning that if we integrate the spacetime line element with all differentials besides dr being zero we recover the ordinary circumference but if we measure the difference between two different values of r we do not get a simple distance between the two. The geometry is stretched in the radial direction depending on the value of the Schwarzschild radius. See handout for details. Schwarzschild s solution can be expressed in many different coordinate systems, each of which are capable of highlighting different interesting properties of the solution.

4 Derivation of the Metric Conditions and Assumptions The textbook lists the assumptions that Schwarzschild made in his derivation on page 239. The unknown functions U,V, and W must only depend on the radial coordinate to satisfy spherically symmetric initial and boundary conditions on the behavior of a massive object in Schwarzschild spacetime. Think of the solution to a standard heat equation with no sources where the temperature is prescribed on the boundary of a spherically symmetric object as an analogy for why these functions must only depend on the radial coordinate.

5 Birkhoff s Theorem Birkhoff s theorem (in the context of general relativity, there is a more general version of it) states that the Schwarzschild solution is the only spherically symmetric solution to the Einstein field equations in a vacuum. The physical intuition behind this idea is that a spherically symmetric gravitational field should be produced by a massive object at the origin of a spherically symmetric coordinate system. If there were mass-energy located anywhere besides the origin, then the solution would no longer be spherically symmetric which is a key condition on the desired solution.

6 Comments on zero Ricci tensor and scalar Roughly speaking, the Ricci curvature tensor represents how much the volume of a geodesic ball in a curved space differs from a Euclidean ball of the same dimension in flat space. If the scalar curvature is zero, then the volume of a geodesic ball in the curved space is the same as the volume of the ball in Euclidean space. The scalar curvature is the Lagrangian density which forms the concept of an action in general relativity. Lagrangian density is a Lagrangian per differential volume element, and integration of this density gives the action of that volume element. A zero Lagrangian density means that any volume element in the relevant spacetime will have zero action, meaning that the underlying gravitational field is static in time (because the energy is non-zero in the spactime we are considering).