What Happens When a Black Hole Evaporates Below its Schwarzschild Radius?

[Chris Hillman]

Static Spherically Symmetric Perfect Fluids (ssspf), Anyone?

We are still talking about gtr, aren't we?

Maximum gravitational curvature is acquired at the sun's surface. Any deeper points have less curvature, just like any other solid spheroidal object.

If you mean path curvature of the world lines of the fluid particles, while the details depend upon what ssspf solution you adopt for the interior (in favorable cases this comes down to choice of equation of state), in general that is not quite correct: the acceleration vector of the fluid particles always points radially outward as you would expect; its magnitude is of course the "surface gravity" at the surface and typically increases to a maximum inside the surface and then decreases to zero at the center--- as must happen by symmetry. (There are some solutions in which the maximal path curvature does occur at the surface, but they are the exception.)

If you mean spacetime curvature or curvature of the spatial hyperslices (the ones orthogonal to the world lines of the fluid particles), then in general the maximal curvatures occur at the center. In particular, if you consider the three-dimensional Riemann tensor of the spatial hyperslices orthogonal to the static timelike congruence (which corresponds to the world lines of bits of fluid inside and to the world lines of static observers who use their rocket engines to "hover" outside), then with components taken wrt the natural frame field, typically 0 < r_{2323} = r_{2424} < r_{3434} with both increasing to a maximum at the center, where they agree.

The simplest model of an isolated object is probably the Schwarzschild stellar model constructed by K. Schwarzschild a few months after Einstein published his field equations. KS "matched" his vacuum solution across the world sheet of a static sphere of a certain radius to his "constant density" ssspf solution. Then if we consider the static congruence, the hyperslices orthogonal to this congruence consist of part of the Flamm paraboloid matched to a spherical cap, which you can visualize embedded in E^3 with the tangents agreeing at the sphere where the matching is carried out, which physically corresponds to the zero pressure surface. In this model, the density is indeed constant, and the pressure increases from zero at the surface to a maximum at the center of the star. Here is a Wikipedia article which was unfortunately left incomplete when I left Wikipedia, but which does include a plot of this embedding. See also [post=146274]this post[/post]. While it is not obvious from the plot, the curvature is in fact uniform and constant in the interior, but this is exceptional. For most ssspf solutions, the curvature in the interior is nonuniform, although at the center it does approximate the Schwarzschild fluid (by the remark at the end of the preceding paragraph).

I once started to write a review paper of ssspf solutions (a task for which I cheerfully confess myself utterly lacking in qualification!), so I am familiar with the behavior of pressure and density wrt "radius" (typically the Schwarzschild radial coordinate is used) for two dozen or so distinct ssspf solutions. On the order of a hundred such solutions have been published (the general ssspf is in some sense "known"), but Kayll Lake has shown in his own review that most of these were wrong or physically unacceptable for any values of their parameters. Lake's review predates important advances by Matt Visser and his coworkers. I also found simple formulas for the tidal tensor and hyperslice curvature of the relativistic polytrope, which has often been considered intractable. Unfortunately motivating my derivation requires knowledge of Lie's methods of symmetry analysis.

Among the "good" solutions, one whose virtues stand out is also one of the very earliest such solutions discovered, the Tolman IV ssspf solution. This turns out to admit an equation of state and provides a fairly good empirical match to observations of neutron stars, the most compact astrophysical objects (in the sense of "made of stuff") which we currently know about, but not to ordinary stars.

I could say a lot more about this subject, so much so that it would definitely belong in another thread (in the relativity forum).

 

[aman malik]

can anybodu pls. tell me what really is schwarzchild radius??

 

[Chris Hillman]

Hi, aman,

In curved spacetimes, there are many coordinate systems we may and do use to represent a given exact solution (model of some specific physical scenario involving gravitation and perhaps other physics).

The Schwarzschild radial coordinate, usually written r, is a particular coordinate which can be defined in spherically symmetric spacetimes and whose geometric meaning is comparable to the usual radial coordinate as used in the polar spherical coordinates familiar from vector calculus in euclidean three-space.

The Schwarzschild vacuum is a Lorentzian four-manifold which is an exact vacuum solution of the Einstein field equation (EFE) and which provides a simple model of a black hole. In the most commonly used coordinate chart, its line element is
<br /> ds^2 = -(1-2m/r) \, dt^2 + \frac{dr^2}{1-2m/r} <br /> + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),<br /><br /> -\infty &lt; t &lt; \infty, \; 2 m &lt; r &lt; \infty, \; 0 &lt; \theta &lt; \pi, \; -\pi &lt; \phi &lt; \pi<br />
where r is the Schwarzschild radial coordinate. As you can see, the case m=0 reduces to the usual polar spherical chart for Minkowski vacuum (flat spacetime; no matter or fields present)
<br /> ds^2 = -dt^2 + dr^2<br /> + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),<br /><br /> -\infty &lt; t &lt; \infty, \; 0 &lt; r &lt; \infty, \; 0 &lt; \theta &lt; \pi, \; -\pi &lt; \phi &lt; \pi<br />

It turns out that in gtr, any uncharged nonrotating black hole can be modeled by the Schwarzschild vacuum solution. The Schwarzschild radius of an uncharged nonrotating black hole is the value of the Schwarzschild radial coordinate which locates a sphere known as the event horizon, which forms the inner boundary of the chart above, namely r=2 m. Here, m is a parameter describing the model which turns out to correspond to mass (for example, the mass you would use in Keplers laws, far from the hole).

There is a similar model of a charged nonrotating black hole known as the Reissner-Nordstrom electrovacuum, which can also be written using a Schwarzschild radial coordinate, and which also has an event horizon. The Schwarzschild radius of such a hole is again the value of the Schwarzschild radial coordinate which locates the event horizon. It is approximately
r \approx 2 m - \frac{q^2}{2 m}
where m is the mass and q is the charge of the hole (I assumed that m \gg q, a condition which holds for astrophysical black holes). This approximation is derived from the exact value, the positive root of r^2 - 2 m r + q^2 = 0.

A bit more about coordinates: in the theory of manifolds, a coordinate is simply an increasing function u defined on some local neighborhood U of our manifold. That is, u has nonzero gradient; in symbols, we require that du \neq 0 on U. If we have another coordinate defined on a neighborhood in a two-manifold M, and if du, \; dv, \; du \wedge dv are all nonzero on U, then u,v define a coordinate chart with domain U on M. Here, the condition du \wedge dv \neq 0 ensures that the gradients of u,v are never aligned inside U, which means that the level surfaces (here one dimensional) of the coordinates, u = u_0, \, v=v_0, are nowhere tangent inside U. Similarly in higher dimensions. These conditions ensure that each point in U has a unique "address" as a tuple of values of the coordinate functions. (Indeed, a street address in Manhattan is pretty much an integer valued pair of numbers which are the values of two coordinates. One can imagine locating a tree on the sidewalk by using a tuple of real numbers.)

If there are symmetries present, it is possible to define in a "coordinate free manner" particular functions which can serve as "preferred coordinates" which respect the symmetries and which have convenient interpretations. In the case of the Schwarzschild radial coordinate on a static spherically symmetric spacetime, such a spacetime contains a family of nested static spheres labeled by different values of r, and each of these has surface area A = 4 \pi \, r^2. That is, the defining characteristic of the Schwarzschild radial coordinate is that has the same relation to surface area of these spheres as in euclidean three-space, but it does not have the usual relationship to distance measured along radii from :"the origin". Indeed, "the origin" might not exist at all in a curved but spherically symmetric space, and in fact does not in the Schwarzschild vacuum or Reissner-Nordstrom electrovacuum.

So now you know!