What about space warping? Imagine sitting in the laboratory at the outer shell and measuring the distance to a nearby lower shell using a ruler. If the lower shell is close by, the measured distance on the ruler between the two nearby shells is given by the second term in Schwarzschild’s equation:
Here dR would be the distance between the shells if space were flat. Since the factor (1 – RS/R) is now in the denominator, the distance between the nearby shells as measured by an observer on one of the shells is larger than it would have been in flat space. This means that space is being stretched and time is being slowed down as we get closer to a star.
To get a feel for the size of these effects, we can put the numbers in for the case of the Sun. The Sun’s Schwarzschild radius is approximately 3 kilometres and its radius is approximately 700,000 kilometres. This gives a distorting factor of 1.000002 at the surface of the Sun. This means that, for two Sun-sized shells whose radii differ by 1 kilometre in flat space, the measured distance between them would be 2 millimetres longer than 1 kilometre. Likewise, an observer far away from the Sun would see a watch at the Sun’s surface run slow by 2 microseconds every second, which is around a minute per year.
The Schwarzschild black hole: just remove the star
Schwarzschild’s solution was originally used to study the region outside of a star or planet (the region inside the star is filled with matter and his solution is not valid there). The remarkable thing is that the same solution can also be used to describe a black hole. All we need to do is to ignore the star. Schwarzschild’s solution then describes an infinite, eternal Universe in which the spacetime becomes more and more distorted as we head inwards towards the singularity at R = 0: a perfect eternal black hole.
Figure 4.5. Schwarzschild ‘space diagram’ with the star removed. There is no matter anywhere.
We’ve sketched Schwarzschild’s space without a star in Figure 4.5. The two imaginary shells we considered before are still there, but the star has disappeared, leaving only empty Schwarzschild spacetime. We’ve also drawn a shell at the Schwarzschild radius, which previously lay inside the star. Looking back at our equations, something very strange happens on the shell at the Schwarzschild radius: the (1 – RS/R) factors are equal to zero. Even more dramatically, as we continue further inwards, these factors become negative. What does this mean? From the perspective of someone in freefall across the shell at the Schwarzschild radius the Equivalence Principle informs us that nothing untoward happens. And yet, from a distant perspective, the shell is a place where clocks stop and space has an infinite stretch.
To understand what is happening, it helps to draw some pictures. Before plunging in with Penrose diagrams, we can learn something from a spacetime diagram. As with the diagrams of flat spacetime we’ve already met, there are many ways to construct these diagrams (corresponding to different choices of grid). We will use Schwarzschild’s coordinate grid since we have just seen that something interesting happens at the Schwarzschild radius. Figure 4.6 shows the light cones at each point in Schwarzschild spacetime. This should be contrasted with the corresponding diagram in flat spacetime. If spacetime is flat, the light cones are all aligned and point vertically upwards, but that is not the case in Schwarzschild spacetime. Far from the Schwarzschild radius, the light cones do look like those in flat spacetime, but as we approach the Schwarzschild radius the cones get narrower and narrower. At the Schwarzschild radius the light cones are infinitely narrow, which means an outgoing beam of light can only travel in the time direction and can never climb away from the hole.* Now we can appreciate that the Schwarzschild radius is also the event horizon: an outgoing beam of light emitted at the Schwarzschild radius stands still.
Inside the horizon, the light cones have flipped round. This is because the (1 – RS/R) factor has become negative, which means the factor in front of dt2 gets a minus sign and dR2 gets a plus sign. It is as if space and time have switched roles, but in fact what has switched roles is our interpretation of the Schwarzschild t and R coordinates. Because light cones open out around the Schwarzschild R direction, this is the direction of ‘time’ for anything inside the horizon, and the Schwarzschild t direction is now ‘space’. Since Schwarzschild coordinates correspond to measurements made using clocks and rulers for someone far from the black hole, this means that what is time for someone inside the black hole is space for someone far away, and vice versa. As we’ve been at pains to emphasise, the coordinates we use don’t have to correspond to anyone’s idea of space and time: to quote Einstein again ‘they do not have to have an immediate metrical meaning’. The Schwarzschild R and t coordinates do happen to have a nice interpretation far away from a black hole, but inside the horizon their roles flip. The startling consequence is that an object inside the horizon moves inexorably towards the centre of attraction at R = 0, just as surely as you move inexorably towards tomorrow.
Figure 4.6. Schwarzschild spacetime. The t and R coordinates are those used by Schwarzschild. Notice how the light cones tip over at values of R smaller than the Schwarzschild radius.
We haven’t said much about the centre of attraction yet. Inside a black hole, this is the singularity, the ‘place’ where Einstein’s theory and Schwarzschild’s solution break down. The quotation marks are appropriate because the singularity isn’t really a place in space. It is a moment in time: the end of time that lies in the future for all who dare to cross the horizon. Figure 4.6 illustrates very nicely that the singularity lies in everything’s future inside the horizon, because all the light cones point towards it. It’s also evident from Figure 4.6 that the singularity is not a point in space, which is what we are tempted to think when we look at Figure 4.5. The time and space role reversal means that it is an infinite surface at a moment in time. Let’s explore this remarkable claim in more detail by plunging in and drawing the Penrose diagram for a Schwarzschild black hole.
The Penrose diagram for the eternal Schwarzschild black hole
The Penrose diagram for the eternal Schwarzschild black hole is shown in Figure 4.7. It is built from two portions: the diamond shape to the right corresponds to the universe outside the black hole. The triangle at the top corresponds to the interior of the black hole and the dividing line between the two is the event horizon. It is a 45-degree line because the horizon is lightlike, which means that light can ‘get stuck’ on it. The singularity is the horizontal line at the top edge of the triangle. It is horizontal because it corresponds to the inexorable future of anything that falls inside the horizon. To see all of this, recall that light cones are always oriented vertically upwards on a Penrose diagram, and worldlines always head into future light cones.