More on Schwarzschild Black Holes

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Abstract. We take a closer look at the maximally extended version of Schwarzschild spacetime, and investigate the geometry and topology beyond the event horizon, leading us to the notion of “white holes”. We discuss the concept of wormhole, its implications, and choices of coordinate system which allow us to quantify them. We then consider the laws of thermodynamics in the presence of event horizons, and find that Schwarzschild black holes have entropy and temperature, and undergo an evaporation process through the emission of thermal radiation. We introduce the holographic principle, and briefly discuss the implications it has for our understanding of the interior region enclosed by the horizon surface.

In my last article, I have introduced to you one of the simplest and most fundamental solutions to the Einstein equations, Schwarzschild spacetime, and we have been discussing the basics of Schwarzschild black holes. Today we will be taking a closer look at those, and tie up a few loose ends.

Recall first the metric of this spacetime in terms of what is called Schwarzschild coordinates :

(1)   \begin{equation*} \displaystyle{ds^2=\left ( 1-\frac{2M}{r} \right )dt^2-\left ( 1-\frac{2M}{r} \right )^{-1}dr^2-r^2d\Omega ^2} \end{equation*}

A Schwarzschild black hole results from the collapse of matter ( such as a star ), and can be conceptualised like so [1] :

Fig 1. Gravitational Collapse

The effect of gravity during and after the collapse is that light cones attached to events in the vicinity of the black hole become tilted towards the hole, such that at the event horizon, no future-oriented geodesics will lead back out.

However, there is a problem here – what is below the horizon ? We have learned in my previous article that, for an outside observer, the in-fall time of a test particle to the horizon is infinite :

(2)   \begin{equation*} \displaystyle{\Delta t=\int_{r_0}^{r_s}\frac{\mathrm{d} t}{\mathrm{d} r}dr\rightarrow \infty } \end{equation*}

This means that there is no 4-tuple of Schwarzschild coordinates {t,r,\theta ,\phi} that can physically refer to any point at or below the horizon, since t would have to be infinitely large. The Schwarzschild coordinate system is hence incomplete, and cannot describe the region of space at or below the horizon.

Schwarzschild Coordinates. This coordinate system is valid only outside the event horizon.

In order to investigate the structure of spacetime at and below the horizon, we therefore have to resort to a different set of coordinates, one that is not singular at the horizon, and extends throughout the entire spacetime. The principle of general covariance guarantees that we are free to choose any coordinate basis we like, without affecting the physics; in other words, we are free to choose different labels for events in the same spacetime, without affecting the relationships between those events. There are many ( in fact infinitely many ) possibilities of how to do this, but the coordinate system most suitable for our purposes here are called Kruskal-Szekeres coordinates. The Schwarzschild metric, written in those coordinates, looks like this :

(3)   \begin{equation*} \displaystyle{ds^2=\frac{32G^3M^3}{r}e^{-r/2GM}\left ( -dT^2+dX^2 \right )+r^2d\Omega ^2} \end{equation*}

At first glance this looks very different from (1), however, the two metrics are actually related by a coordinate transformation, and describe the same spacetime with the same geometry. The transformation itself is not just any arbitrary transformation, but chosen so that, when plotted on a diagram ( see below ), a ray of light radially falling into a black hole will be a perfectly straight line.

Kruskal-Szekeres coordinates are nonetheless a somewhat artifical way of labelling events in our spacetime, in that the new T and X coordinates do not correspond to any physical observer; however, they do have several advantages that allow us to more closely examine the geometry we are dealing with. For one thing, the coordinate system now spans the entire spacetime, from curvature singularity all the way to infinity; the event horizon is located at

(4)   \begin{equation*} \displaystyle{T=\pm X} \end{equation*}

and the central singularity is given by

(5)   \begin{equation*} \displaystyle{T^2-X^2=1} \end{equation*}

It is now explicitly obvious that the event horizon is just another, ordinary region of space – there is no singularity here. The fact that Schwarzschild coordinates became singular at the horizon was just an artefact of the coordinate system itself, but not a physical curvature singularity; we have eliminated it simply be relabelling events in our spacetime. The same is not true for the central curvature singularity – it is still there, even in our new coordinate system.

Kruskal-Szekeres Coordinates. This coordinate system spans the entire Schwarzschild spacetime, including the horizon and the inner region below the horizon. The horizon itself is – as expected – an ordinary region of spacetime without any singularities.

But wait – what is going on in (4) and (5) ? There seem to be two horizons, and two singularities ( at r=0, one for T and one for -T ) ! This spacetime seems to be a lot more complex than we initially suspected. To understand what is going on, let us draw a diagram of how the coordinates behave [2] :

Fig 2. Kruskal-Szekeres Diagram

As mentioned previously, a ray of light falling radially in will be represented as a perfectly straight line in these coordinates ( blue line ), which is the defining characteristic of this way of labelling events. However, other characteristics of the above diagram are far more interesting for us :

  1. There are two singularities
  2. There are two event horizons
  3. There are four different regions
  4. The ray of light ( blue line ) is of finite length

The fact that the geometric length of the world line of a ray of light ( or that of any infalling test particle ) is finite, is something we already know, and doesn’t come as a surprise; however, in our new coordinate system this is now immediately obvious, even without any complicated calculations.

What is far more surprising is that we find two horizons and two singularities here; it is as if we have “glued together” two black holes. Well, not exactly – if we look at the above diagram, we see that the horizons and the singularities are mirror images, with signs reversed; that means we are dealing not with two black holes, but with our “normal” black hole plus a time-reversed mirror image of it. This time-reversed mirror image is generally called a white hole, and it behaves in the exact opposite manner than a black hole does – whereas nothing can escape once a black hole horizon has been crossed, nothing can return once it has left a while hole’s horizon. It is as if time is running backwards, or gravity has become repulsive. We can now see the meaning of the four regions in the diagram :

Region 1 : Black hole exterior

Region 2 : Black hole interior

Region 3 : White hole exterior

Region 4 : While hole interior

Since most people find the diagram in fig 2 initially confusing, it may be helpful to plot the same thing in a slightly different way [3] :

Fig 3. Schwarzschild Spacetime

It is important to understand that the black hole and the white hole are not two different solutions to the Einstein equations, but both togther form part of the same Schwarzschild spacetime. When we talk about the “Schwarzschild solution”, we hence mean the entire construct, both the black hole and the white hole part. The graphic in fig 3 depicts only one singularity, but technically speaking there are actually two, distinguished by the sign on the T coordinate ( but at the same “location”, in so far as such a concept is even meaningful ).

Schwarzschild Spacetime. If we choose a set of coordinates that spans the entire spacetime of this solution, we find that there are in fact two singularities and two event horizons; one belongs to a black hole, and the other to its time-reversed counterpart, a white hole.

If we generate a proper embedding diagram of this situation, it will look something like this [4] :

Fig 4. Einstein-Rosen Bridge

What we have here is two regions of space-time connected by something that looks a bit like a tunnel – this topological construct is called an Einstein-Rosen Bridge, or a wormhole. It is a natural feature of many types of black hole solutions to the Einstein field equations, even seemingly simple ones like the Schwarzschild solution.

So exactly what is being connected here ? Where does such a wormhole “lead” ? Interestingly, there are two possible answers to this, and the theory itself is unable to distinguish between the two :

  1. The wormhole connects two distant regions of the same universe
  2. The wormhole connects two otherwise completely separate regions of spacetime, i.e. two different universes

The reason why both of these are valid possibilities is that the Einstein field equations by themselves constrain only the local geometry of spacetime, but not its global topology – this means that one multiply connected spacetime ( possibility 1 ) is just as valid as two singly connected spacetimes ( possibility 2 ).

The question that immediately comes to mind here is whether or not it is possible to actually send a rocket into a black hole, somehow avoid the central singularity, traverse the wormhole, only for it to be spat out – fully intact – by a white hole in some distant region, or even in another universe.

Unfortunately, the answer is no. To understand why, we need to recall that the Schwarzschild solution is obtained from a number of very restrictive assumptions – namely that we are dealing with an isolated, static, stationary, non-rotating and electrically neutral object in otherwise completely empty space. It turns out that, while the exterior region outside the horizon is pretty stable even under perturbations such as adding a small test particle, the interior region including the wormhole is not. In fact, even adding something the mass of which is very much smaller than the mass of the black hole – such as a rocket – will perturb the geometry of spacetime below the horizon in such a way that the wormhole will “pinch off” in a time much shorter than is required to successfully traverse it; this can be conceptualised like so [5] :

Fig 5. Schwarzschild Wormhole

Observe carefully the two yellow arrows – neither is able to enter, traverse, and escape the wormhole. This means that, not only is Schwarzschild spacetime more topologically complex than it at first appears, but it is also dynamic and unstable if test particles are added to it. This type of wormhole is therefore an example of a non-traversable wormhole, i.e. a wormhole that one cannot travel through. General Relativity does permit solutions which describe wormholes that are traversable, such as for example the Ellis Bridge; however, such solutions suffer from other problems, most notably the need for a substance called “exotic matter” to stabilise and hold open the wormhole throat. Furthermore, wormholes can connect regions which are distant in spatial terms, but they could also be used to connect a region to itself at a different time – they could therefore be used to construct a time machine ( at least in principle ). This is highly problematic from the point of view of possible causality violations, which is why most physicists believe that traversable wormholes are unlikely to exist in the real universe.

Schwarzschild Spacetime. The “maximally extended” description of Schwarzschild spacetime ( i.e. the version that spans the entire spacetime ) has the topological structure of a bridge connecting two distant or separate regions of spacetime. The bridge is highly unstable under perturbations, and cannot be traversed. The “throats” of the wormhole are hidden behind event horizons, one of which constitutes a black hole, and the other a white hole. White holes themselves are hypothetical constructs, and have not been observed.

I should, at this point, remind you again that the Schwarzschild solution to the field equations is highly idealised, and based on largely unphysical assumptions. Most notably, black holes that exhibit the topology and geometry described above would have had to exist eternally, in an otherwise completely empty universe; this is not a condition we find in the real world, so Schwarzschild black holes of the exact form described here do not exist. However, they are useful approximations to many real-world physical scenarios, and are hence of academic interest when studying the physics involved, due to their relative simplicity.

Let us now perform a very simple gedankenexperiment. Remember first that the event horizon of a black hole forms a natural boundary, past which there is no longer a causal connection between events – you can enter a black hole, but once you have entered, you cannot leave or exchange any kind of information with the outside world. Let us now take a small object composed of interacting particles – an object such as a rock, or a box filled with gas, or anything else of that nature. According to the laws of thermodynamics, such a system has a certain amount of degrees of microscopic freedom ( given by the statistical mechanics or quantum mechanics describing the system ). Let us now consider the composite system “black hole + object”. Because of the degrees of freedom inherent in that composite system, we can ascribe it a finite entropy, and the second law of thermodynamics tells us that in a closed system ( we assume the black hole + object are isolated in space and time ), entropy can never decrease :

(6)   \begin{equation*} \displaystyle{\frac{\mathrm{d} S}{\mathrm{d} t}\geq 0} \end{equation*}

So far so good. But what happens if we allow the object to fall below the horizon ? There no longer is a causal connection between the object and any external observer, so for the rest of the universe the microscopic degrees of freedom that made up the original object have become causally disconnected, and effectively no longer exist. That means that, from the point of view of the rest of the universe ( i.e. any external observer ), the entropy of the “black hole + object” system ( let’s denote it by “S” )  must have decreased :

(7)   \begin{equation*} \displaystyle{\frac{\mathrm{d} S}{\mathrm{d} t} < 0} \end{equation*}

in clear violation of the second law of thermodynamics ! Now, the laws of thermodynamics follow in a very fundamental way from statistical mechanics and other considerations, and are extremely well verified both theoretically and empirically; a violation of those laws is hence not acceptable. So how can we save the day ? Somehow we must find a way to preserve the entropy S, even though the object has fallen through the horizon and is no longer causally connected to us. The most obvious resolution to this is to postulate that the black hole itself carries entropy – when the object falls through the horizon, its entropy becomes part of the black hole’s overall entropy, thereby avoiding any violations of the second law. This has two fundamentally important consequences :

  1. If the black hole – more specifically, its event horizon – carries entropy, then by the laws and principles of thermodynamics we are forced to conclude that it must have a non-zero, finite temperature as well
  2. Since the interior of the black hole is not causally connected to the external universe, the only place the entropy can meaningfully “reside” is on the horizon surface. That physically means that information about degrees of freedom within the 3-dimensional volume enclosed by the horizon is encoded on the 2-dimensional horizon surface itself.

Point (1) implies that it is physically reasonable and meaningful to define the temperature of a black hole; detailed ( and complicated ) calculations from statistical mechanics and quantum field theory yield the surprisingly simple result for the horizon entropy

(8)   \begin{equation*} \displaystyle{S_{BH}=\frac{kA}{4\sqrt{G\hbar/c^3}}} \end{equation*}

wherein k is the Boltzmann constant, and A is the surface area of the event horizon. This is called the Bekenstein-Hawking entropy. The associated temperature is

(9)   \begin{equation*} \displaystyle{T_{H}=\frac{\hbar c^3}{8\pi GMk}} \end{equation*}

which is called the Hawking temperature of the black hole. If the black hole has a thermodynamic temperature, then it must emit thermal radiation, just like any other warm body; in fact, it turns out that the radiation emitted ( which is called Hawking radiation ), has the spectral composition of a perfect black body.

This has another consequence – from the point of view of an outside observer, a black hole emitting thermal radiation must loose energy ( in accordance with the laws of thermodynamics ), and the only measure of energy which a Schwarzschild black hole possesses is its mass ( there are no other free parameters in the metric ). That means that a radiating Schwarzschild black hole will continuously loose mass and shrink – it undergoes evaporation. The total time it would take for a black hole of mass M to completely evaporate is

(10)   \begin{equation*} \displaystyle{t = \frac{5120\pi G^2 M^3}{\hbar c^4}} \end{equation*}

This is a very long time – to put it in perspective, the evaporation time for a black hole of one solar mass is of the order 10^67 years, which is many orders of magnitude above the age of the universe.

This of course leaves the question just what remains if a black hole completely evaporates – this is a good and interesting question, to which modern physics has no answer ( yet ), but it is beyond the scope of this article.

Black Hole Thermodynamics. In order to preserve the laws of thermodynamics, black hole event horizons must carry entropy, and hence have a finite temperature. Schwarzschild black holes therefore emit black body radiation ( Hawking radiation ), and undergo a process of continuous evaporation.

The second consequence we have mentioned above is of a more abstract nature – because the 3-dimensional region enclosed by the horizon is not causally connected to the rest of the universe, all relevant information about its microscopic degrees of freedom must be encoded on its 2-dimensional boundary, the event horizon surface. This is remarkable, because firstly this is a highly non-local phenomenon, and secondly because this hints at a fundamental principle called the holographic principle :

Holographic Principle. The description of an n-dimensional region can be encoded on a lower-dimensional boundary to that region. For the case of Schwarzschild black holes, the degrees of freedom contained in the 3-volume enclosed by the horizon are encoded on the 2-dimensional horizon itself.

It is called the “holographic principle” because this is how holograms work – they encode information about 3-dimensional objects on a 2-dimensional surface. It turns out that this principle finds application in many areas of modern physics, such as quantum field theory, quantum gravity, string theory etc etc.

If we consider the implications of this, we find even more remarkable conclusions – the entropy of the black hole horizon surface is finite. The physical meaning of this entropy is that it is a measure of the microscopic states contained in the volume of space enclosed by the horizon; because the entropy is finite, there are only a finite number of possible microstates associated with our 3-volume. But because the 3-volume is just empty space ( remember that the central singularity is not a point on our manifold ), we are forced to conclude that this space cannot be smooth and continuous, or else the entropy on the horizon surface would be infinitely large. Hence, by elementary considerations, we find definitive hints that the classical spacetime of General Relativity itself must break down below the horizon, and be replaced by something that has only a finite number of degrees of freedom. One possibility would be that spacetime ceases to be smooth and continuous, and becomes discrete. This is a very far-ranging conclusion based on a very straightforward thought process – see above -, which is really quite remarkable; we are seeing hints here of how a ( as per yet unknown ) theory of quantum gravity must agree with already known classical laws.

And this concludes our investigation of Schwarzschild black holes. We have found that the seemingly simple Schwarzschild solution actually possesses a fairly complex and non-trivial geometry and topology, and that by combining it with the laws of thermodynamics, we can deduce some remarkable and quite fundamental principles of nature.

However, we are still only at the beginning of our journey, because Schwarzschild black holes are very restrictive in the sense that the only parameter we can play around with is its total mass. In the following articles we will investigate other solutions to the field equations, which permit more attributes, such as angular momentum, electric charge, and infalling/outgoing matter and radiation.

As always – stay tuned !







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