Schwarzschild Spacetime

Abstract

The Schwarzschild solution describing spacetime outside a spherical mass distribution is deduced. In this deduction we give a detailed prescription of how one calculates the components of Einstein’s curvature tensor using differential forms as decomposed in an orthonormal basis . The predictions for the classical tests of Einstein’s theory —gravitational frequency shift and time dilation, deflection of light passing the Sun and the perihelion shift of Mercury —are deduced. Finally the Reissner–Nordström solution describing spacetime outside a charged particle is deduced.

Exercises

• 8.1. Non-relativistic Kepler motion

In the first part of this exercise we will consider the gravitational potential at a distance r from the Sun, $V(r) = - {\text{GM}}/r$, where M is the mass of the Sun.

• (a) Write down the classical Lagrangian in spherical coordinates $(r,\theta ,\phi )$ for a planet with mass m moving in this field. The Sun is assumed to be stationary.

  What is the physical interpretation of the canonical momenta $p_{\phi } = \ell$?

  How is it possible, by just looking at the Lagrangian, to state that $p_{\phi }$ is a constant of motion?

  Find the Euler equation for ϑ and show that it can be written into the form

  $$ \frac{\text{d}}{{{\text{d}}t}}\left( {mr^{4} \dot{\theta }^{2} + \frac{{\ell^{2} }}{{m\sin^{2} \theta }}} \right) = 0. $$

  (8.168)

  Based on the above equation, show that the planet moves in a plane by choosing a direction of the z-axis so that at a given time, $t = 0$, we have that $\theta = \pi /2$ and $\dot{\theta } = 0$.

• (b) Write down the Euler equation for r and use this equation to find $u = 1/r$ as a function of ϕ. Show that the orbits that describe bound states are elliptic. Find the period T₀ for a circular orbit in terms of the radius R of the circle.

• (c) If the Sun is not entirely spherical, but rather a little deformed (i.e. more flat near the poles), the gravitational field in the plane where the Sun has its greatest extension will be modified into

  $$ V(r) = - \frac{\text{GM}}{r} - \frac{S}{{r^{3} }}, $$

  (8.169)

  where S is a small constant. We now assume that the motion of the planet takes place in the plane where the expression of is V (r) valid. Show that a circular motion is still possible. What is the period T now, expressed by the radius R?

We now assume that the motion deviates slightly from a pure circular orbit, that is $u = \frac{1}{R} + u_{1}$, where $u_{1} \ll \frac{1}{R}$. Show that u₁ varies periodically around the orbit,

$$ u_{1} = k\sin (f\phi ). $$

(8.170)

• (d) Find f and show that the path rotates in space. What is the size of the angle $\Delta \phi$ that the planetary orbit rotates per round trip?

The constant S can be written as $S = \frac{1}{2}J_{2} {\text{GM}}R_{\text{Sun}}^{2}$ where J₂ is a parameter describing the quadrupole moment and $R_{\text{Sun}}$ is the radius of the Sun. Observational data indicate that $J_{2} { \lesssim }3\cdot10^{ - 5}$. Calculate how large the rotation $\Delta \phi$ of the orbit of Mercury this can cause. Is this sufficient to explain the discrepancy between the observed perihelion motion of Mercury and that predicted by Newtonian theory?

• 8.2. The Schwarzschild solution in isotropic coordinates

• (a) We introduce a new radial coordinate ρ so that the Schwarzschild metric (with units so that $c = 1$),

$$ {\text{d}}s^{2} = - \left( {1 - \frac{{R_{S} }}{r}} \right){\text{d}}t^{2} + \left( {1 - \frac{{R_{S} }}{r}} \right)^{ - 1} {\text{d}}r^{2} + r^{2} {\text{d}}\Omega ^{2} , $$

(8.171)

gets the following form

$$ {\text{d}}s^{2} = - \left( {1 - \frac{{R_{R} }}{r(\rho )}} \right){\text{d}}t^{2} + f^{2} (\rho )({\text{d}}\rho^{2} + \rho^{2} {\text{d}}\Omega ^{2} ), $$

(8.172)

where ${\text{d}}\Omega ^{2} = {\text{d}}\theta^{2} + \sin^{2} \theta {\text{d}}\phi^{2}$. Find the functions $r(\rho )$ and $f(\rho )$, and write down the explicit expression of the line element with ρ as the radial coordinate.

• (b) What is the value of ρ at the Schwarzschild horizon $r = R_{S}$ and at the origin, $r = 0$? The Schwarzschild coordinates t and r interchange their roles as $r < R_{S}$. What is the behaviour of ρ inside the horizon?

• 8.3. Proper radial distance in the external Schwarzschild space

Calculate the proper radial distance from a coordinate position r to the horizon R_S in the external Schwarzschild space .

• 8.4. The Schwarzschild–de Sitter metric

The Einstein equations for empty space with a cosmological constant $\varLambda$ are

$$ R_{\mu \nu } - \frac{1}{2}Rg_{\mu \nu } +\Lambda g_{\mu \nu } = 0. $$

(8.173)

• (a) Use curvature coordinates and solve the Einstein field equations with a cosmological constant for a static spacetime with a spherically symmetric 3-space outside a spherical body with mass M.

• (b) The solution of Einsteins field equations with a cosmological constant in globally empty space, i.e. with $M = 0$, is called the de Sitter spacetime . Introduce a de Sitter radius $R_{\Lambda } = \sqrt {3/\Lambda }$. Give a physical interpretation of this radius and calculate how large it is in a universe where the value of the cosmological constant corresponds to a density of LIVE equal to the average density of the masse and vacuum energy of the universe, $\Lambda = 10^{ - 52}$ m⁻².

• 8.5. The perihelion precession of Mercury and the cosmological constant

• (a) Show that the orbit equation for free particles moving outside a spherically symmetric body with mass M has the form

  $$ \frac{{{\text{d}}^{2} u}}{{{\text{d}}\phi^{2} }} + u = \frac{M}{{L^{2} }} + 3Mu^{2} - \frac{\Lambda }{{3L^{2} u^{3} }}, $$

  (8.174)

  where $u = 1/r$, and L is the angular momentum per unit mass for the particle.

• (b) Assume that the orbit can be described as a perturbation of a circle and calculate the precession angle per round trip.

• (c) Estimate the contribution to the precession of Mercury’s perihelion from the cosmological constant if we assume that the value of the cosmological constant is $\Lambda \approx 10^{ - 52}$ m⁻².

• 8.6. Relativistic time effects and GPS

Calculate the magnitude of the kinematical and gravitational time effects upon the GPS satellite clocks. Are standard clocks on the GPS satellites going slower or faster than a standard clock at rest on the surface of the Earth.

To compute the position of an object by means of the GPS system with a precision of 1 m, the GPS satellite clocks must measure time with a precision of one part in 10¹³.

Are the relativistic effects so small that they can be neglected?

• 8.7. The photon sphere

The photon sphere is defined as a spherical shell made up of light moving horizontally in the Schwarzschild spacetime. Calculate the radius of the photon sphere.