Chemistry - Schwarzschild metric

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3 Embedding Schwarzschild space in Euclidean space

Introduction

In Einstein's theory of general relativity, the Schwarzschild metric is the most general static, spherically symmetric solution of the vacuum field equations. It defines the gravitational field outside a point mass or outside a spherical, non-rotating mass. The Schwarzschild metric is named in honour of its discoverer Karl Schwarzschild (in 1916). Schwarzschild discovered the solution only a few months after the publication of Einstein's theory of general relativity. It was the first exact solution of these equations. The Schwarzschild metric is derived in the article deriving the Schwarzschild solution.

Schwarzschild's solution showed how the predictions of general relativity would deviate from the predictions obtained from Newtonian gravity. Using his solution for the gravitational field of the Earth and the Sun, the outcome of three classical tests of general relativity has been predicted. For about half a century they were the only experimental verification of general relativity. The classical tests are the gravitational redshift, the gravitational deflection of light and the perihelion shift of the planet Mercury. In fact, the perihelion shift of Mercury was one of the major problems that astronomers were trying to understand; when Einstein used Schwarzschild's solution to calculate the observed shift, he found that it was exactly (within experimental errors) the observed shift. For Einstein, this was the first major triumph of general relativity.

The Schwarzschild metric

The Schwarzschild solution is:

$ds^{2} = \left(1-\frac{r_{s}}{r}\right)^{-1}dr^2+ r^2 (d\theta^2+\sin^2\theta d\phi^2)-\left(1-\frac{r_{s}}{r} \right)c^2 dt^2$

where

$r_{s}=\frac{2Gm}{c^2}$

defines the Schwarzschild radius or Schwarzschild horizon, with $G$ the gravitational constant, $c$ the speed of light and $m$ the mass of the gravitating object. For a spherically symmetric object, the solution is valid outside the radius $R$ of the object, provided that $m$ is replaced by $m (r)$ for $r < R$ and by the constant $m (R)$ for $r \ge R$ . The function $m (r)$ is related to the mass distribution within the star and can be thought of as the mass within the radius $r$ . Indeed, as $r \rightarrow\infty$ the terms with $r s / r$ vanish and one is left with the Minkowski metric:

$ds^{2} = dr^2 + r^2 \left(d\theta^2+\sin^2 \theta d\phi^2 \right) - c^2 dt^2$

The Schwarzschild radius is clearly proportional to the mass of the gravitating object. The solution is only valid for radii larger than the Schwarzschild radius. For the Sun, it is approximately 3 km and for the Earth, approximately 1 cm. Schwarzschild realized that his solution was singular (it becomes infinite for radii approaching the Schwarzschild radius). He tended to ignore this and simply let the radial coordinate not start at 0 but at the Schwarzschild radius, because a normal star would never be so compact as to fall entirely within its Schwarzschild radius (the Sun would have to be squeezed from a radius of approximately 700,000 km to approximately 3 km).

The region with radii smaller than the Schwarzschild radius is also a valid solution of the Einstein equations. It has some odd properties. The Schwarzschild radial coordinate r becomes timelike and the time coordinate t becomes spacelike. A curve at r= constant is not anymore a possible world line of a particle or observer. Causality requires a particle to fall inwards. For a long time the inner-Schwarzschild solution was considered non-physical. One now thinks that such objects within the Schwarzschild radius exist and calls them black holes.

The Schwarzschild solution appears to have a singularity at $r = r s$ . Consideration of invariants gives quantities that are independent of the observer. One such important invariant is Kretschmann invariant $R a b c d R a b c d$ . It turns out that:

$R_{abcd}R^{abcd}= \frac{12 r_s ^2}{r^6}$

Hence, at $r = r s$ , there is no physical singularity. However, there is a genuine physical singularity at $r = 0$ (a coordinate singularity is like the North Pole of the Earth, a point which in certain coordinate systems is also singular and may stretch to a line; yet, no one at the North Pole experiences a local (physical) singularity).

Schwarzschild had little time to think about his solution. He died shortly after his paper was published, as a result of a disease he contracted while serving in the German army during World War I.

Embedding Schwarzschild space in Euclidean space

The Schwarzschild metric can be visualized in a so called embedding diagram.

In general relativity mass changes the geometry of space. Space with mass is "curved", whereas empty space is flat (Euclidean). In some cases we can visualize the deviation from Euclidean geometry by mapping a 'curved' subspace of the 4-dimensional spacetime onto a Euclidean space with one dimension more.

Suppose we choose the equatorial plane of a star, at a constant Schwarzschild time $t =$ constant and $θ = π / 2$ and map this in 3 dimensions $z, r,φ$ with the Euclidean metric

d s 2 = d z 2 + d r 2 + r 2 d φ 2

We will get a curved surface $z = z (r)$ by identifying (using $dz = \frac{dz}{dr}dr$ and rewriting

$ds^2 = \left\{ 1 + \left( \frac {dz(r)} {dr}\right)^2 \right\} dr^2 + r^2d\phi^2$

with the Schwarzschild metric for the plane ( $θ = π / 2, t =$ constant)

$ds^2 = \left(1-\frac{2m}{r} \right)^{-1} dr^2 + r^2d\phi^2$

This is the case for

$z(r) = \int_0^r \frac{dr}{\left( \frac{r}{2m(r)}-1\right)^{1/2}} \;\; (0 < r < \infty)$

and especially for $r > R$ , the radial coordinate of the radius of the star for which we write $m (R) = M$

$z(r) = \left( 8M (r- 2M) \right)^{1/2} + \mbox{ a constant} \;\; (R < r < \infty)$

The geometry of the plane inside the star (using the simplifying assumption $m(r) = \frac{4\pi}{3}\rho r^3$ for a constant density $ρ$ inside the star) is drawn in the figure.

Embedding the 2-dimensional (non-Euclidean) equatorial plane of the Schwarzschild geometry around a star into an assumed 3-dimensional Euclidean space. Note that the 3-dimensional space has nothing to do with the physical world: the space outside the plane has no physical meaning and merely serves to overcome the mental difficulty to imagine non-Euclidean geometry..

References

Schwarzschild, K. (1916). Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften 1, 189-196.
Ronald Adler, Maurice Bazin, Menahem Schiffer, Introductin to General Relativity (Second Edition), (1975) McGraw-Hill New York, ISBN 0-07-000423-4 See chapter 6.
Lev Davidovich Landau and Evgeny Mikhailovich Lifshitz, The Classical Theory of Fields, Fourth Revised English Edition, Course of Theoretical Physics, Volume 2, (1951) Pergamon Press, Oxford; ISBN 0-08-025072-6. See chapter 12.
Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0.
Steven Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory or Relativity, (1972) John Wiley & Sons, New York ISBN 0-471-92567-5. See chapter 8.

Schwarzschild metric

Introduction

The Schwarzschild metric

Embedding Schwarzschild space in Euclidean space

References

See also