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Diagramatic Representations of Black Hole Spacetimes 1 Kruskal Diagramatic Representations of Black Hole Spacetimes PHYS 471 Our studies of the Schwarzschild spacetime have shown us that there are two types of singularities: ones which are actual artifacts of the geometry (r = 0), and ones that are artifacts of the coordinate system in which they occur (r = 2GM). The former cannot be removed by a coordinate transformation, while the latter can. The physics that occurs at the event horizon, however, is still consistently described in the new coordinate system (i.e. clocks appearing to stop as they approach the horizon, infalling observers seeing nothing special, but being unable to communicate outward once through, etc...). In the following pages, we'll discuss different ways of representing black hole spacetimes in different coordinates mathematically, as well as through spacetime diagrams. 1 Kruskal-Szekeres Coordinates In class, and on your assignment, we discussed a \simple" transformation that re- moves the horizon singularity from the metric. These are the Kruskal-Szekeres transformations, which are represented by new \time" (v) and \space" (u) coordi- nates 1 1 r 2 t r 2 t v = 1 er=(4GM) sinh ; u = 1 er=(4GM) cosh 2GM − 4GM 2GM − 4GM in the external region r > 2GM, and 1 1 r 2 t r 2 t v = 1 er=(4GM) cosh ; u = 1 er=(4GM) sinh − 2GM 4GM − 2GM 4GM for r < 2GM. We can show this transforms the Schwarzschild metric into the form 32G3M 3 ds2 = e−r=(2GM)(dv2 du2) + angular stuff r − which obviously is non-singular and continuous everywhere except at r = 0 (the true geometric singularity). Rearranging these coordinates, we can further define 2 2 r r=2GM u + v v u = 1 e ; t = 2GM ln − − 2GM u v − These expressions tell us very important things about the meaning of v and u, and how we can use them to represent the actual physics of the Schwarzschild black hole spacetime. A number of characteristics emerge, including: Photons travel along 45◦ lines. Since ds2 = 0 for a photon, we can consider • a radially-directed photon to obey the equation dv ds2 = 0 = dv2 du2 = 1 − −! du ± That is, lines of slope 1 represent photon paths. ± Massive particles follow worldlines with slope greater than 1. In this • case, ds2 > 0 for lightlike worldlines (massive particles). By the same reasoning, we could say " # dv 2 dv ds2 dv2 du2 = du2 1 > 0 > 1 ∼ − du − −! du du But this doesn't necessarily preclude dv < 1, so alternatively we could calcu- late − r dv 1 r t r = 1 cosh e 4GM dt 4GM 2GM − 4GM r du 1 r t r = 1 sinh e 4GM dt 4GM 2GM − 4GM and then dv dv cosh t = dt = 4GM > 0 du du t dt sinh 4GM for all values of t. Timelike worldlines therefore have slope greater than 1, and will always be contained inside a light cone. The event horizon is defined by the line v = u. The event horizon is the • point where r = 2GM, so r v2 u2 = 1 er=2GM at r = 2GM v2 u2 = 0 − − 2GM −! −! − and so v = u represents the event horizon. ± At the event horizon, the time coordinate is t , indicating that • according to outside observers, it takes an infinite! 1 amount of time to reach the horizon. Again, at r = 2GM, or u = v, the equation for coordinate time tells us u + v t = 2GM ln at r = 2GM t u v −! −! ! 1 − Lines of constant r are defined by the hyperbolae v2 u2 = a2, where • a2 is a function of the radius r. For any constant value− R± > 2GM, we have R v2 u2 = 1 eR=2GM < 0 − − 2GM which represents a hyperbola oriented in the vertical plane. Similarly, inside the event horizon, we have R < 2GM and R v2 u2 = 1 eR=2GM > 0 − − 2GM which is a horizonally-oriented hyperbola. The singularity is the hyperbola v2 u2 = +1. From the equations above, • its easy to see that r = 0 defines the given− hyperbola. As there is no r < 0, the region above v2 u2 = 1 is undefined. − Figure 1: Kruskal-Szekeres coordinate diagram of a Schwarzschild black hole, showing regions of spacetime inside and outside horizon. Figure 2: The worldline of an infalling observer as they pass through the event horizon. The observer's lightcone is dislayed at various stages, including outside the horizon, and at the horizon. Note that in the latter case, the lightcone coincides with the horizon, indicating that no signal can be sent back to the outside universe, and also that the only thing in the observer's causal future is the singularity. 2 Penrose Diagrams The advantage of the Kruskal-Szekeres coordinates is that they remove the singular behavior at the event horizon, while preserving the essence of our spacetime diagrams from special relativity: light cones and worldlines. We can go one better that this, however, by defining a transformation that brings the entire spacetime structure into a compact diagram. That is, we can define a set of transformations that take the ranges < t < + and 0 < r < and map them to finite ranges. We will call these transformations−∞ 1conformal, because1 they preserve angles between lines. The resulting diagrams are called conformal diagrams, and are also known as Penrose diagrams, named after their inventor: Sir Roger Penrose. Transformations that take infiinite ranges and make them finite are no mystery to you: in fact, you learned about them well before University. The function tan(θ), π π for example, takes the range 2 < θ < 2 and maps it to < tan(θ) < . So, the inverse function arctan(x)− must take an infinite range range−∞ to a finite one.1 This process is called compactification. Another function that shares this property is tanh(x), which takes < x < to the finite range 1 < tanh(x) < 1. Either of these can be used to construct−∞ Penrose1 diagrams. − 2.1 Minkowski Spacetime Let's first describe the Penrose diagram for Minkowski (flat) spacetime. Recall the metric is ds2 = dt2 dr2 r2dθ2 r2 sin2 θdφ2 − − − We first note that this can be written ds2 = (dt dr)(dt + dr) r2dθ2 r2 sin2 θdφ2 − − − and so the essence of the lightcone { and causality { is contained in those terms. So, we choose Y − = tanh(t r) and Y + = tanh(t+r) as our conformal transformations. Let's examine the behavior− of these functions at all points in spacetime, while referencing Figure 3: Figure 3: Penrose diagram for Minkowski spacetime. When t + r , we have tanh(t + r) 1. There are two extremes to this: • the point (t!= 1 ; r = 0) and (t = 0;! r = ). We call these ι+ (\future timelike infinity")1 and ι0 (\spacelike infinity"),1 respectively. Future timelike infinity represents every point in space far in the future that can be reached by timelike worldlines. Spacelike infinity represents every point in space infinitely far away that can't be reached by anything (even photons). The two points ι+ and ι0 are connected by the line defined by the transformation, • along which tanh(t + x) = 1. This line represents points in spacetime that only light can reach. So, we call this line future lightlike infinity, or +. I If we consider tanh(t x) = 1, we get a \mirror" image of the previously • defined region. Again,− when t−= , the point represents spacetime in the distant past. We call this past lightlike−∞ infinity, or ι−. One can imagine this is where \everything" (timelike) came from, just as the distant future everything goes to ι+. One can show that, in a similar fashion to the KS transformations, these con- • formal transformations turn lines of constant r = R into curves that begin at ι− and end at ι+. As they are timelike geodesics, their slopes must be greater than 1. Lastly, the line of points where tanh(t x) = 1 is called past lightlike • infinity, −, and represents the hypothetical− place− in spacetime where light seems to haveI \come from". Note that in Minkowski spacetime, there are no singularities, so there's nothing particularly special about the coordinate r = 0, but... 2.2 Schwarzschild Spacetime ... in this case, the singular behavior at r = 0 does appear on the Penrose diagram. The generic shape of the Schwarzschild Penrose diagram is the same as for Minkowski (lines of constant R are curves, light travels at 45◦ lines, etc...), but of course there are the event horizon and singularity to account for. If one performs similar conformal transformations on the KS coordinates (v; u), we get a diagram that looks like that of Figure 4. It contains pretty much all the properties of the original KS diagram, with the added advantage of the Penrose points ι+, ι0,ι−, +, and −. One difference between the KS diagram and the Penrose diagram is thatI the singularityI at r = 0 is now represented by a constant horizontal line, and not a hyperbola, as a result of the transformation. Although r = 0 was a vertical line for flat spacetime, it is now a horizontal one by virtue of the fact that time and space have swapped roles inside the horizon. 2.3 Maximally-Extended Penrose Diagrams Technically, the Penrose diagrams above represent the spacetime for only positive spatial positions r 0. Although physically the region r < 0 is meaningless, there is nothing mathematically≥ wrong with considering it. In fact, the equations still work for ι+ Future timelike infinity (t = ) ∞ r = 0 (Singularity) + Future lightlike infinity Region II I (Horizon) GM =2 r ι0 Spacelike infinity (r = ) Constant r<2GM Region I ∞ − Past lightlike infinity I Constant r>2GM ι− Past timelike infinity Figure 4: Penrose diagram for Schwarzschild spacetime, showing all spacetime outside the black hole (Region I, 2GM < r < ) and inside (Region II, 0 < r < 2GM).
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