Theory of relativity/Schwarzschild metric

In Schwarzschild coordinates, the Schwarzschild solution is

It is an exact vacuum solution to General relativity/Einstein equations, and according to the Birkoff theorem, all spherically symmetric exact vacuum solutions are equivalent to this solution, related through mere frame transformation.

Working out the solution


We are looking for a static spherically symmetric vacuum solution so it is reasonable to write down as a trial solution


But note that a mere r coordinate transformation can be done to drop the   absorbing it into the radial coordinate. So we will start with a trial solution of


The Einstein tensor   for this line element has a time time element of


For this to be an exact vacuum solution we must have


The   that is the solution to that differential equation is


where   comes in as an integration constant. So thus far our trial solution is


For this line element the metric tensor yields an r-r element of


For this to be an exact vacuum solution we must have


The   that is the solution to that differential equation is


where   comes in an an integration constant. But   can be absorbed by the time in a time coordinate transformation, a mere scaling of time, so doing so thus far our trial solution is


Now it turns out that for this line element all of the Einstein tensor elements are zero, so we have an exact vacuum solution. Finding the Riemann tensor though several elements are not zero and so what we have has spacetime curvature and is thus not a mere frame transformation of the metric of special relativity.

For the moment lets define a constant   according to


The exact vacuum Schwarzschild solution is then written


An exact calculation of the radial case of Geodesic motion of a test mass yields


where   is a constant of the motion called the energy parameter, the conserved energy per   of the test mass, and   is the proper time for the test mass. Differentiating this equation with respect to proper time and simplifying results in


So we see that this mass   which was essentially an integration constant for this spacetime geometry is what we think of as the active gravitational mass.

Geodesic Motion


The exact equations of geodesic motion for a test mass in Schwarzschild coordinates with the test mass's proper time being   are


where   is a constant of the motion called the energy parameter, the energy per   for the test mass.


where   is the conserved angular momentum per mass   for the test mass.


and finally


For radial motion this reduces to


The final equation of motion prior to the radial motion case, looks much like a Newtonian gravitation conservation of energy equation with the exception of the   term multiplying the angular part and the time derivatives being with respect to the test mass time. This factor multiplying the angular part perturbs the motion of nearly elliptical orbits so that they process. Orienting the coordinates so that the motion of an orbit is equatorial and defining   the equations of motion yield


which in a weak field the solution can be approximated by


where   is the eccentricity So perihelion occurs at


which for the weak field can be approximated by


and given an orbital period of T this implies that after a time t the orbit will have processed by an amount given by


where   is the semi-major axis. This effect was first observed in nature for Mercury which processes 575" per 100 earth years, 534" of which are accounted for by the gravitational effects from other planets.

Geodesics For Light


Writing the geodesic motion in terms of Schwarzschild time t instead of time for the test mass   and taking the limit as   and   go to   yields the motion for a massless test particle such as a photon. Orienting the coordinates so that the orbital plane of the photon is equatorial and defining   results in




where   is the distance of closest approach for a deflected photon. For small deflection of light the first of these two yields a deflection angle of


And for circular orbit of a photon yields


which is a location referred to as the photon sphere. Integrating the second with appropriate weak field approximation and writting the result in terms of a lab's time instead of remote observer Schwarzschild time for a photon following geodesics between earth and another planet at superior conjunction yields the round trip Shapiro delay equation of


where   and   are the orbital distance from sun of the planets and the  s are curve fit parameters allowing for perturbances such as the gravitational time dilation from the earth's mass itself.

Untransversable Wormhole Structure


Consider a spacelike hypersurface described by the Schwarzschild solution as a constant Schwarzschild time slice given cutting through for example at   by


In order to get a conceptual image of the way this spacelike hypersurface is curved lets write it as an imbedding in a higher dimensional hyperspace with an extradimensional spatial axis of w.


This gives us the differential equation for the hypersurface as


The integration of which gives us the equation for the hypersurface as


Drawing this swept around for all   the surface looks like

  • The Einstein-Rosen Bridge
  • We now see that the surface is a connection between two different external regions. This connection was originally termed the Einstein-Rosen Bridge, but it has become more popular to term connections between external regions wormholes. A transformation to Kruskal-Szekeres coordinates we shall see manifests the different regions, but shows how the connection is not transversable. The Kruskal-Szekeres coordinate transformation is outside the event horizon


    and inside the horizon


    Mapping the Schwarzschild coordinates onto the Kruskal-Szekeres coordinates from these one gets the following picture

  • Schwarzschild coordinate extension showing nontransversible wormhole structure
  • Here we see the two external regions and a wormhole cross section example is drawn on, but one can see now that there is no way to transverse from one side to the other. In Kruskal-Szekeres coordinates the coordinate speed of radial moving light is everywhere c, the solution expressed as


    and light like paths described by ds=0, so on the diagram one can not travel a path more than 45 degrees from the vertical. Starting from inside one external region, one can't get to the other external region without dipping past 45 degrees and thus traveling faster than light with respect to nearby observers.



    See also