How Can Black Holes Be Described In Terms of Spacelike and Timelike Curves?
Vocabulary of Spacelike and Timelike Curves
Stephen Hawking and Roger Penrose developed a syntax and visual means of describing spacelike and timelike curves, both components of Einstein’s relativity. It is a little dense but I think it does a great job of showing what exactly is happening when we take relativity to the extreme, like say a black hole (Hawking 5).
They start by defining p as a present moment in spacetime. If we move around a space we are said to follow a spacelike curve but if we move forward and backward in time then we are on a timelike curve. We all move on both in our day-to-day lives. But there are ways to talk about movement in each direction alone. I+(p) as all the possible events that can occur in the future based on what p was. We get to these new points in spacetime by following a “future-directed timelike curve,” so this doesn’t discuss past events at all. Therefore, if I chose a new point in I+(p) and treated it as my new p, then it would have its own I+(p) emanating from it. And I-(p) would be all the past events that could have resulted in point p (Ibid).
And like I+(p), there is I+(S) and an I-(S), which is the spacelike equivalent. That is, it is the set of all future locations I can arrive at from set S and we define the boundary of “the future of set S” as i+(S). Now, how does this boundary operate? It isn’t timelike because if I picked a point q outside of I+(S), then to transition to the future would be a timelike maneuver. But i+(S) isn’t spacelike either, for it were looking at set S and I chose a point q within I+(S), then by moving to i+(S) I would pass it and go…before the future, in space? Doesn’t make sense. Therefore, i+(S) is defined as a null set because if I was on that boundary I wouldn’t be in set S. If true, then “a past-directed null geodesic segment (NGS) through q lying in the boundary” will exist. That is, I can travel along the border some distance. More than one NGS certainly can exist on i+(S) and any point I chose on it would be the “future endpoint” of the NGS. A similar scenario arises when talking about i-(S) (6-7).
Now, to make i+(S), we need some NGSs to construct it so that q will be that endpoint and also that i+(S) will indeed be that desired boundary for I+(S). Simple, as I am sure many of you are thinking! To make a NGS, one makes a change to Minkowski Space (which is our three dimensions mixed in with time to create 4-D space where reference frames shouldn’t impact how physics works) (7-8).
Okay, new vocab term. We define an open set U as globally hyperbolic if we have a rhombus region that is defined by a future point q and a past point p, with our set U being I+(p) ᴖ I-(q), or the set of points which fall into the future of p and the past of q. We also need to make sure that our region has strong causality, or that no closed or nearly closed timelike curves inside U. If we did have those, then we could get back to a point in time we already had been to. Causality that isn’t strong could be a thing, so watch out! (Hawking 8, Bernal)
Another term we will want to become familiar with in our discussion of extreme relativity is a Cauchy surface, denoted as Σ(t) by Hawking and Penrose, which is a type of spacelike or null surface which will cross the path of every timelike curve only once. It is similar the idea of being somewhere at an instantaneous moment of time, and only there at that time. Therefore, it can be used to determine the past and/or future of a point in set U. And that is how the global hyperbolicity condition implies that Σ(t) can have a family of surfaces for a given point t, and that has some definite quantum theory implications going on (Hawking 9).
If I have a globally hyperbolic space, then there exists a geodesic (a generalization of a straight line in different dimensions) of max length for points p and q that is joined as a timelike or null curve, which makes sense because to go from p to q one would have to move inside of U (timelike) or along the boundaries of set U (null). Now, consider a third point r which lies on a geodesic called γ which can be altered by using “an infinitely neighboring geodesic” in conjunction with it. That is, we would use r as something “conjugate to p along γ” so that our journey from p to q would be altered as we took a side route through r. By bringing conjugates into play, we are approaching the original geodesic but not matching it (10).
But do we have to stop at just one point r? Can we find more such deviations? As it turns out, in a globally hyperbolic spacetime we can show that this scenario plays out for any geodesic formed by two points. But then a contradiction results, for that would mean the geodesics we had formed initially are not “geodesically complete” because I would be unable to describe every geodesic that could form in my region. But we do get conjugate points in reality, and they are formed by gravity. It bends geodesics towards it, not away. Mathematically, we can represent the behavior with the Raychaudhuri-Newman-Penrose (RNP) Equation in its amplified form:
dρ/dv = ρ2 + σijσij + (1/n)*Rablalb
Where v is the defined parameter (simply a different way of relating variables together) along a congruence of geodesics with tangent vector la which is hypersurface orthogonal (that is, our vectors will emanate at a right angle to the surface which is one dimension lower than that which the geodesic is moving through), ρ is the “average rate of the convergence of the geodesics,” σ is the shear (a type of math operation), and Rablalb is the “direct gravitational effect of the matter on the convergence of the geodesics.” When n=2, we have null geodesics and for n=3 we have timelike geodesics. So, in an attempt to summarize the equation, it stats that the change in our convergence of geodesics with respect to the defined parameter (or our choosing) is found by taking the average rate of the convergence and adding both the shear terms with respect to i and j as well as the gravitational contributing the matter along the geodesics supplies (11-12).
Now, let’s mention the weak energy condition:
Tabvavb≥0 for any timelike vector va
Where Tab is a tensor that helps us describe how dense the energy is at any moment and how much is passing through a given area, va is a timelike vector and vb is a spacelike vector. That is, for any va, the matter density will always be bigger than zero. If the weak energy condition is true and we have “null geodesics from a point p begin to converge again” at ρo (the initial rate of convergence of the geodesics), then the RNP equation shows how the geodesics converge at q as ρ approaches infinity so long as are in parameter distance ρo-1 and the “null geodesic” along our boundary “can be extended that far.” And if ρ = ρo at v = vo then ρ≥1/(ρo-1 +vo –v) and a conjugate point exists before v = vo +ρ-1, otherwise we have a denominator of 0 and thus a limit approaching infinity just as the prior sentence predicted(12-13).
What all of that implies is that we can now have “infinitesimally small neighboring null geodesics” that intersect at q along γ. Point q is therefore conjugate to p. But what about points beyond q? On γ, many possibly timelike curves are possible from p, so γ cannot be in the boundary I+(p) anywhere past q because we would have infinitely many boundaries close together. Something in the future endpoint of γ will become the I+(p) we are looking for, then (13). This all leads up to the generators of black holes.
Black Holes by Hawking and Penrose
After our discussion on some of the basics of spacelike and timelike curves, it is time to apply them to singularities. They first arose in solutions to Einstein’s field equations in 1939, when Oppenheimer and Snyder found one could form from a collapsing dust cloud of sufficient mass . The singularity had an event horizon but it (along with the solution) only worked for spherical symmetry. Therefore, its practical implications were limited but it did hint at a special feature of singularities: a trapped surface, where the path light rays can travel decreases in area due to the gravity conditions present. The best the light rays can hope to do is move orthogonal to the trapped surface, otherwise they fall into the black hole. See the Penrose Diagram for a visual. Now, one may wonder if finding something has a trapped surface would be sufficient evidence for our object to be a singularity. Hawking decided to investigate this and looked at the situation from a time-reversed viewpoint, like playing a movie backwards. As it turns out, a reverse-trapped surface is huge, like on a universal scale (maybe like a Big Bang?) and people have often associated the Big bang with a singularity, so the possible connection is intriguing (27-8, 38).
So these singularities form from a spherically based condensation, but they don’t have any dependence on θ (angles measured in the xy plane) nor on φ (angles measured in the z plane) but instead on the r-t plane. Imagine 2 dimensional planes “in which null lines in the r-t plane are at ±45o to the vertical.” A perfect example of this is flat Minkowski space, or 4-D reality. We notate I+ as the future null infinity for a geodesic and I- as the past null infinity for a geodesic, where I+ has a positive infinity for r and t while I- has a positive infinity for r and a negative infinity for t. At each corner where they meet (notated as Io) we have a two-sphere of radius r and when r=0 we are at a symmetrical point where I+ is I+ and I- is I- . Why? Because those surfaces would extend forever (Hawking 41, Prohazka).
So we now have some basic ideas down, hopefully. Let us now talk about black holes as developed by Hawking and Penrose. The weak energy condition states that the matter density for any timelike vector must always be bigger than zero, but black holes would seem to violate that. They take matter in and at seem to have infinite density, so geodesics that are timelike would seem to converge at the singularity that is making the black hole. What if black holes merged together, something we know to be a real thing? Then the null geodesics we have used to define out boundaries I+(p) which have no endpoints would suddenly meet and…have endings! Our story would end and matter density would fall below zero. To ensure that the weak energy condition is upheld, we rely on an analogous form of the second law of thermodynamics labeled the second law of black holes (rather original, no?), or that δA≥0 (the change in the area of the event horizon is always bigger than zero). This is rather similar to the idea of the entropy of a system always increasing aka the second law of thermodynamics and as a researcher on black holes will point out, thermodynamics has led to many fascinating implications for black holes (Hawking 23).
So I have mentioned a second law of black holes, but is there a first? You bet, and it too has a parallel with its thermodynamic brethren. The first law states that δE = (c/8π)δA + ΩδJ + ΦδQ where E is the energy (and therefore the matter), c is the speed of light in a vacuum, A is the area of the event horizon, J is the angular momentum, Φ is the electrostatic potential, and Q is the charge of the black hole. This is similar to the first law of thermodynamics (δE = TδS + PδV) which relates energy to temperature, entropy, and work. Our first law relates mass to area, angular momentum, and charge, yet parallels do exist between the two versions. Both have changes in several quantities but as we mentioned earlier a connection exists between entropy and area of the event horizon, as we see here too. And that temperature? That will come back in a big way when the discussion of Hawking radiation entered the scene, but I am getting ahead of myself here (24).
Thermodynamics does have a zeroth law and so the parallel is extended to black holes also. In thermodynamics, the law states that the temperature is constant if we exist in a thermoequilibrium system. For black holes, the zeroth law states that “κ (the surface gravity) is the same everywhere on the horizon of a time-independent black hole.” No matter the approach, the gravity around the object should be the same (Ibid).
Cosmic Censorship Hypothesis
Something that is often left aside in much black hole discussion is the need of an event horizon. If a singularity doesn’t have one then it is said to be naked and is therefore not a black hole. This stems from the cosmic censorship hypothesis which implies the existence of an event horizon, aka “the boundary of the past of future null infinity.” Translated, it is the boundary where once you cross over, your past is no longer defined as everything up to this point but instead once you cross the event horizon and forever fall into the singularity. This boundary is made up of null geodesics and this composes a “null surface where it is smooth” (aka differentiable to a desired amount, which is important for the no-hair theorem). And for places where the surface isn’t smooth, a “future-endless null geodesic” will start from a point on it and keep going into the singularity. Another feature about event horizons is that the cross-sectional area never gets smaller as time goes on (29).
I briefly mentioned the cosmic censorship hypothesis in the previous section. Can we talk about it in a more specialized vernacular? We sure can, as developed by Seifert, Geroch, Kronheimer, and Penrose. In spacetime, ideal points are defined as places where singularities and infinites in spacetime can occur. These ideal points are a past set containing itself, and thus cannot be split into different past sets with one another. Why? We could get sets with the ideal points replicating and that leads to closed timelike curves, a big no-no. It is because of this inability to be broken down that they are referred to as indecomposable past-set, or an IP (30).
Two main types of ideal points exist: a proper ideal point (PIP) or a terminal ideal point (TIP). A PIP is the past of a spacelike point while a TIP is not the past of a point in spacetime. Instead, TIPs determine future ideal points. If we have an infinity TIP where our ideal point is at infinity, then we have a timelike curve that has “infinite proper length,” because that’s how far out the ideal point is. If we have a singular TIP, then it results in a singularity, where “every timelike curve generating it has a finite proper length” because it terminates at the event horizon. And for those wondering if ideal points have future counterparts, indeed they do: indecomposable future-sets! So we also have IFs, PIFs, infinite TIFs, and singular TIFs. But for any of this to work, we must assume no closed timelike curves exist aka no two points can have the exact same future AND the exact same past (30-1).
Alright, now onto naked singularities. If we have a naked TIP we are referring to a TIP in a PIP and if we have a naked TIF we are referring to a TIF in a PIF. Basically, the “past” and “future” parts are now intermingling without that event horizon. The strong cosmic censorship hypothesis says that naked TIPs or naked TIFs don't happen in general spacetime (a PIP). This means that any TIP cannot suddenly appear from nowhere into the spacetime we see (vertex of a PIP aka the present). If this was violated, then we could see something fall directly into the singularity where physics breaks down. You see why that would be a bad thing? Conservation laws and much of physics would be thrown into chaos, so we are hoping that the strong version is right. There is a weak cosmic censorship hypothesis out there too, which states that any infinite TIP cannot suddenly appear from nowhere into the spacetime we see (PIP). The strong version implies we can find equations governing our spacetime where no naked, singular TIPs exist. And in 1979, Penrose was able to show that not including the naked TIPs was the same as a globally hyperbolic region! (31)
That implies that spacetime can be some Cauchy Surface, which is great because that means we can create a spacelike region where every timelike curve is passed over only once. Sounds like reality, no? The strong version also has time symmetry behind it, so it works for IPs and IFs. But something called a thunderbolt could also exist. This is where a singularity has null infinities coming out of the singularity because of a change in surface geometry and therefore destroys spacetime, meaning global hyperbolicity comes back because of quantum mechanics. If the strong version is true, then thunderbolts are an impossibility (Hawking 32).
So…is cosmic censorship even true? If quantum gravity is real or if black holes blow up, then no. The biggest factor in the probability of the cosmic censorship hypothesis being real is that Ω or the cosmological constant (Hawking 32-3).
Now, for some more details on the other hypotheses I mentioned earlier. The strong cosmic censorship hypothesis is essentially stating that generic singularities are never timelike. This means we only examine spacelike or null singularities, and they will be either past TIFs or future TIPs so long as the hypothesis is true. But if naked singularities exist and cosmic censorship is false, then they could merge and be both of those types, for it would be a TIP and a TIF at the same time (33).
Thus, the cosmic censorship hypothesis makes it clear we cannot see the actual singularity or the trapped surface around it. Instead, we have only three properties we can measure from a black hole: its mass, its spin, and its charge. One would think that would be the end of this story, but then we explore quantum mechanics more and find out we could not be further from a reasonable conclusion. Black holes have some other interesting quirks we have missed in this discussion thus far (39).
Like for example, information. Classically, nothing is wrong about having matter fall into a singularity and never return to us. But quantumly it is a huge deal, because if true then information would be lost and that violates several pillars of quantum mechanics. Not every photon gets pulled into a black hole that surrounds it, but enough do make the plunge so that information is lost to us. But is it a big deal if it is just trapped? Queue the Hawking radiation, which implies that black holes will eventually evaporate and therefore that trapped info will actually be lost! (40-1)
Bernal, Antonio N. and Miguel Sanchez. “Globally hyperbolic spacetimes can be defined as ‘causal’ instead of “strongly causal’.” arXiv:gr-qc/0611139v1.
Hawking, Stephen and Roger Penrose. The Nature of Space and Time. New Jersey: Princeton Press, 1996. Print. 5-13, 23-33, 38-41.
Ishibashi, Akirhio and Akio Hosoya. “Naked Singularity and Thunderbolt.” arXiv:gr-qc/0207054v2.
Prozahka et al. “Linking Past and Future Null Infinity in Three Dimensions.” arXiv:1701.06573v2.
© 2018 Leonard Kelley