Notes on Relativity and Cosmology - Physics Department, UCSB
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Notes on Relativity and Cosmology - Physics Department, UCSB

Notes on Relativity and Cosmology - Physics Department, UCSB Notes on Relativity and Cosmology - Physics Department, UCSB

web.physics.ucsb.edu
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248 CHAPTER 9. BLACK HOLES r = R s r = 0 r = R s This part of the Eternal Black Hole is not relevant to a black hole that forms from the collapse of matter. Outside Edge of the matter Here, the Schwarzschild solution correctly describes the spacetime r = 0 r = R s r = R s We see that the ‘second exterior’ and the ‘past interior’ are in the part of the diagram with no direct relevance to relevance to black holes that form from collapsing matter. A careful study of the Einstein equations shows that, inside the matter, the spacetime looks pretty normal. A complete spacetime diagram including both then region inside the matter and the region outside would look like this: r = 0 Schwarzschild here Horizon Not Schwarzschild here r = 0 Center of the matter r < R s r = R s along Horizon from here on out Outside Edge of the matter

9.3. BEYOND THE HORIZON 249 9.3.5 Visualizing black hole spacetimes We have now had a fairly thorough discussion about Schwarzschild black holes including the outside, the horizon, the inside, and the “extra regions” (second exterior and past interior). One of the things that we emphasized was that the spacetime at the horizon of a black hole is locally flat, just like everywhere else in the spacetime. Also, the curvature at the horizon depends on the mass of the black hole. The result is that, if the black hole is large enough, the spacetime at the horizon is less curved than it is here on the surface of the earth, and a person could happily fall through the horizon without any discomfort (yet). It is useful to provide another perspective on the various issues that we have discussed. The idea is to draw a few pictures that I hope will be illustrative. The point is that the black hole horizon is an effect caused by the curvature of spacetime, and the way that our brains are most used to thinking about curved spaces is to visualize them inside of a larger flat space. For example, we typically draw a curved (two-dimensional) sphere) as sitting inside a flat three-dimensional space. Now, the r, t plane of the black hole that we have been discussing and drawing on our spacetime diagrams forms a curved two-dimensional spacetime. It turns out that this two-dimensional spacetime can also be drawn as a curved surface inside of a flat three-dimensional spacetime. These are the pictures that we will draw and explore in this section. To get an idea of how this works, let me first do something very simple: I will draw a flat two-dimensional spacetime inside of a flat three-dimensional spacetime. As usual, time runs up the diagram, and we use units such that light rays move along lines at 45 o angles to the vertical. Note that any worldline of a light ray in the 3-D spacetime that happens to lie entirely in the 2-D spacetime will also be the worldline of a light ray in the 2-D spacetime, since it is clearly a curve of zero proper time. A pair of such crossed light rays are shown below where the light cone of the 3-D spacetime intersects the 2-D spacetime.

248 CHAPTER 9. BLACK HOLES<br />

r = R s<br />

r = 0<br />

r = R s<br />

This part of the<br />

Eternal Black Hole is<br />

not relevant to a black<br />

hole that forms from<br />

the collapse of matter.<br />

Outside Edge of<br />

the matter<br />

Here, the Schwarzschild<br />

soluti<strong>on</strong> correctly describes<br />

the spacetime<br />

r = 0<br />

r = R s<br />

r = R s<br />

We see that the ‘sec<strong>on</strong>d exterior’ <strong>and</strong> the ‘past interior’ are in the part of the<br />

diagram with no direct relevance to relevance to black holes that form from<br />

collapsing matter. A careful study of the Einstein equati<strong>on</strong>s shows that, inside<br />

the matter, the spacetime looks pretty normal. A complete spacetime diagram<br />

including both then regi<strong>on</strong> inside the matter <strong>and</strong> the regi<strong>on</strong> outside would look<br />

like this:<br />

r = 0<br />

Schwarzschild here<br />

Horiz<strong>on</strong><br />

Not Schwarzschild<br />

here<br />

r = 0<br />

Center<br />

of the<br />

matter<br />

r < R s<br />

r = R<br />

s<br />

al<strong>on</strong>g Horiz<strong>on</strong> from here <strong>on</strong> out<br />

Outside Edge of the matter

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