Schwarzschild metric

"Es ist immer angenehm, über strenge lösungen einfacher Form zu verfügen." (It is always pleasant to avail of exact solutions in simple form) Karl Schwarzschild, 1916, Sitzber.Deut. Akad. Wiss, p.189

In the theory of gravity, general relativity, the Schwarzschild metric is a static spherically symmetric solution of Einstein's field equations of the vacuum (empty space). It defines the gravitational field outside a point mass or outside a spherical non-rotating mass. It was found by Karl Schwarzschild in 1916, only a few month after the publication of the Einstein field equations. It was the first exact solution of these equations. Schwarzschild's solution showed how the predictions of general relativity for the gravitational field of the Sun and stars (for example) would deviate from the predictions obtained with Newtonian gravity. Using his solution, three classical tests of general relativity have been predicted, which for about half a century were the only experimental verification of general relativity. They are the gravitational redshift, the gravitational deflection of light and the perihelium shift of the planet Mercury.

The Schwarzschild solution defines a characteristic radius associated with every mass, called the Schwarzschild radius or gravitational radius. The solution is only valid for radii larger than this radius, which is proportional to the mass. For the Sun it is about 3 km, for the Earth 3 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 sqeezed from a radius of 700,000 km to 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, like the radial coordinate become timelike and the time coordinate becomes spacelike, which do not allow an orbserver or particle to remain at a constant radius. Causality requires a particle to fall inwards. For a long time the inner-Schwarzschild solution was considered as not physical. One now thinks that such objects within the Schwarzschild radius exist and calls them black holes.

In 1960 it was found by Kruskal and Szekeres, the Kruskal-Szekeres coordinates, that the singularity at the Schwarzschild radius was not a physical singularity. For example the components of the Riemann tensor, which describe the tidal forces that a falling observer experiences, are not singular if expressed in an invariant form. So the Schwarzschild radius is only a coordinate singularity (compare the North Pole of Earth, a point which in certain coordinate systems is also singular and may stretch to a line).

Schwarzschild has had little time to think about his solution. He died shortly afterwards as a result of disease contracted in the German army in World War I.

Table of contents

1 Solution of the Einstein equations 2 Symmetry considerations 2.1 spherical symmetry 2.2 static solution 2.3 Schwarzschild or geometrical coordinates 3 Schwarzschild metric 4 Embedding Schwarzschild space in Euclidean space

Solution of the Einstein equations

Since the Schwarzschild solution is the simplest solution of the Einstein field equations (see general relativity), it is worthwhile to review a few considerations in some detail.

Solving the Einstein equations, means that one solves for the metric tensor for a given mass distribution. With the metric tensor the path of light and particles can be calculated. The Schwarzschild metric, for example, leads to predictions such as the bending of light near the Sun and a change in the path of the planet Mercury as compared to the result of Kepler's laws of planetary motion.

The metric is in a certain way the counterpart of the gravitational potential in Newtonian physics which is determined by the mass-ditribution. In Newtonian physics, however, gravity is determined by a scalar potential (one real value in every point in spacetime). The acceleration of a particle is given by the first order derivative of the potential; it is a force. And the tidal force (a difference in forces that a weightless astronaut in orbit really feels and can measure), is given by the second derivatives of the potential.

In general relativity, the gravity for a certain mass distribution is given by a 4 by 4 tensor, the metric tensor, which has in principle 16 values in every point in spacetime. The acceleration of a particle involves a kind of first order derivative (see covariant derivative). The equivalence of tidal forces in general relativity is the curvature tensor which can be expressed in a second order differential equation of the metric components.

The 16 values of the metric tensor give a measure of the "length" (distance) between two nearby points in spacetime, say a point with time coordinate and three spatial coordinates , denoted as

and a point  a little further away at 

The distance is expressed as and in components of the metric tensor (a and b run from 0 to 3) as

where the are the 16 functions of the spacetime coordinates, which Schwarzschild determined for the case of a point mass.

Symmetry considerations drastically reduce the number of components. He showed that all diagonal terms are zero, ; for so one is left with 4 unknowns and a metric of the form

The boundary condition to the equations is that the solution for large radii should approach to the Newtonian situation. This is how the mass and the gravitational constant enters into the Schwarzschild metric.

Symmetry considerations

In general relativity, the choice of coordinate system is arbitrary and the laws of physics should not depend on this choice. However, many results look simpler, if described in a particular type of coordinates. Calculated observables do not contain this choice (they are invariant).

Schwarzschild exploited the symmetries of a static spherical mass first by choosing spherical coordinates, so in stead of 4 coordinates

we write , a time t plus three spatial coordinates: the radius r, an azimuthal angle and a lattitude measured from a chosen equator.

spherical symmetry

A spherical symmetric solution implies that the resulting metric should not depend on cross terms like , because otherwise the lengths of curves in the positive -direction would be different from curves in the negative -direction, violating spherical symmetry. So all terms

and

should be zero. That is:

In the same manner all terms containing only one factor should be zero. The angle dependence of the metric tensor could thus be written in the familiar spherical coordinate form

which as a whole could still be multiplied by a function of and , but the metric tensor components should be independent of

and .

static solution

Since we are seeking static solutions, alle components should not depend on t. Under these conditions the most general line element which we write in terms of coordinates is

where and can be arbitrary functions of (we write the function before the term as and we may do so since

is an arbitrary function).

We want to keep the spherical symmetry, but we may still choose freely a new and coordinate as a function of

and  respectively.

Without loss of generality we can get rid of the cross term by choosing a new time-coordinate .

The most general static spherical symmetric metric becomes

for three arbitrary functions

Schwarzschild or geometrical coordinates

Schwarzschild made an additional choice, now called Schwarzschild coordinates or geometrical coordinates. He assumed a choice of the radius r such that the proper circumference of a circle is equal to the Euclidean value . (In general relativity, gravitation is described as a change in the geometry of space and a simple statement: circumference of a circle is , does not hold in general). With this choice, r is not anymore the proper radius of a circle.

So in terms of the arbitrary function above, Schwarzschild used the freedom in radial coordinates, by fixing the radial coordinate so that such that the angular part can be written in the suggestive and familiar form

.

Note that this form does not imply is the "radius" (proper radius) of a circle or of a sphere that has as a radial cooordinate. It is merely a handy choice of an arbitrary radial coordinate, which has the nice property that for it approaches the value which is equal to the circumference of a circle divided by With this choice the metric becomes

Note that other, entirely reasonable, choices could have been made. One could have chosen for spatial coordinates such that the spatial part of the metric is written by an arbitrary function times . Such a choice would also be spherically symmetric. This choice is called isotropic coordinates, since the three spatial coordinates are treated the same way. Note that a radial coordinate from is a different one than the Schwarzschild coordinate defined above. Physical results are invariants and should not depend on the coordinate system used.

Schwarzschild metric

The two remaining unknowns, the function and and can be solved using Einstein's field equations' in empty space

where is the Ricci tensor which depend the metric tensor . The two components that appear of relevance are

and . For these two coupled differential equations one may solve for the two remaining functions
and  in the Schwarzschild static 

and spherically-symmetric line element given above, subjected to two boundary conditions: for the solution should describe Newtonian gravity and for the metric should approach the Lorentz metric of special relativity.

The Schwarzschild solution is given explicitly as

where is the gravitational radius (or Schwarzschild radius) and is the mass of the source of the field.

For a spherical symmetric star, the solution is valid outside the radius

of the star, provided that  is replaced  by
for  and by the constant 
for . The function  is related to the mass distribution within the star and can be thought off as the mass within the radius .

Indeed, for the terms with

vanishes and one is left with the 

Lorentz metric of special relativity in spherical coordinates

Embedding Schwarzschild space in Euclidean space

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

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 constant and and map this in 3 dimensions with the Euclidean metric

We will get a curved surface by identifying (using and rewriting

with the Schwarzschild metric for the plane (constant)

This is the case for

and especially for , the radial coordinate of the radius of the star for which we write

The geometry of the plane inside the star (using the simplifying assumption for a constant density 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-Eclidean geometry..

See also Kruskal-Szekeres coordinates, Eddington coordinates