67151981 Penrose Diagrams

8/3/2019 67151981 Penrose Diagrams

1/9


2/9

ds 2 = du 2 1 2M

r 2dudr + r 2 d22 (3)

At r = 2 M all metric coefficents are nite and we have a method of considering, smoothly, the motionof objects above, on and below the event horizon. Setting ( ,,u ) = constant we have a null space-timeinterval, ds 2 = 0, for all r . Thus such a choice represents a lights path in a radial direction. Unlike inspherical polar coordinates, we are able to draw a space-time diagram for this choice of coordinates torepresent the behaviour of light cones.

Figure 1: Light-cone shapes around r = 2 M in (u,r,, ) coordinates

These coordinates are not physically viable for a gravitational system within relativity because theyexhibit anti-gravity, as can be seen in Figure (1). The light cones tip over in such a manner that anytime-like curve within the region 0 < r < 2M is inevitably forced out of the event horizon. Instead,these allow use to consider past horizons with a space-time completely described by a Schwarzchildmetric.

Since the Schwarzchild metric is stationary (invariant under t t ) there also exists advancedEddington-Finklestein coordinates, v giving the more familiar description,

v = t + r + 2 M ln |r 2M | (4)

ds 2 = 1 2M

rdv 2 + 2 dvdr + r 2 d22 (5)

These give a similar, but slightly different space-time diagram around the event horizon.

Figure 2: Light-cone shapes around r = 2 m in (v,r,, ) coordinates

With r = 0 on the left hand side of Figure (2) and the vertical line representing the event horizon, it

2


3/9

can be seen that the light cones are again tipped over, but in this system so that any time-like curveis constrained to remain in the region r < 2M once it has entered it.

While these diagrams are helpful in our consideration of the local space-time around an event horizon,there is still the question of global structure as well as the fact Figures (1) and (2) both show that thelight cones still close up somewhere, within the event horizon. To counter this, some kind of conformalrepresentation of the light cones is needed, so that the light cones remain open (preferably at 45 degreeseverywhere) throughout considerations. These two requirements lead through to Penrose Diagrams.

Unphysical Space-time

If we are willing to put aside our requirement for specic details of the space-time but rather justwanting an overview of the global structure which remains conformally valid, descriptions are possiblethrough the use of unphysical metrics, ds 2 , related to our physical metric by

ds 2 = 2 (x )ds 2 (6)

(x ) is our conformal factor, not to be confused with the S 2 line element d2 from Equations (1), (3)and (5). Causal structure is preserved since ds 2 = 0 ds 2 = 0 in all but a few cases (simple to dealwith) and angles are preserved since

cos2 =(gab V a W b )2

(gab V a V b )(gcd W c W d )

(gab V a W b )2

(gab V a V b )( gcd W c W d )(7)

Since the orders of the metric is equal in the numerator and denominator, the factors of have can-celled.

The purpose of this factor is to absorb the innities. One metric might be singular while the othermight not, depending on . We now have more freedom in our choice of coordinates for a system sothat our variables r and t (or whatever our choice of coordinates may be) no longer have an innitedomain but a nite one. With only a nite domain, a pictorial representation of the entirity of thespace-time may be possible.

Minkowski space-time and the Einstein Static Universe

ds 2 = dt 2 + dr 2 + r 2 d22 (8)

Since we are in spherical coordinates, we have the domains < t < and 0 r < , as well asthe usual (already nite) S 2 metric for a sphere of radius r . Choosing null coordinates u = t r andv = t + r gives us

ds 2 = dudv +(u v)2

4d22 (9)

The domain is now < u v < . An educated guess at the choice of new variables would be afunction (or functions) which goes singular at a nite value of its argument. In this case u = tan p

and v = tan q give us

ds 2 = sec2 p sec2 q dpdq +14

sec2 sec2 p sin2 (q p)d22

= sec 2 p sec2 q dpdq +14

sin2 ( p q)d22 (10)

The domain for the variables is now 2 < p q > 2M the Schwarzchild and Minkowski diagrams should effectivelybe the same. Another clue to its layout can be gleaned from Figures 1 and 2 along with the factPenrose diagrams do not change the angle light rays move at (though Figures 1 and 2 certainly do).Since light rays behave normally out in the asymptotically at region of the space-time, they mustdo the same near, on and in the event horizon. However, Figures 1 and 2 tell us that on the r = 2 M surface/line (depending on dimensional supression) the light rays move along the line of constant r .Therefore, in the Schwarzchild Penrose diagram, the line r = 2 M , which represents the event horizon,is null! Figure 3 shows us that in Minkowski space-time, two possible null lines represent space-timenull innities. Since this within the area that the Schwarzchild and Minkowski diagrams coincide,those two null lines could not possibly be the r = 2 m line in the Schwarzcild diagram. This leaves twoother possibilities.

By considering the same smooth variation of the geodesics as was done for the Minkowski case, itcan be seen that part of the Schwarzchild diagram is (at least supercially) the same as the EinsteinStatic Universe, in Figure 5, except that the two left hand diagonals no longer represent null innities,but the surface r = 2 M .

But what of 0 < r < 2M ? We now can revert back to using the original Schwarzchild metric inEquation (1), since its value in the range 0 < r < 2M , but the signature has changed from + ++to + ++. The radial direction is now time-like and the time direction is now space-like. This can

6


7/9

be seen to be expected if wed continued varying our time-like geodesics through the event horizon,though wed not know where they end up. This changing of signature is the reason you cannot avoidthe singularity. To move in a time-like fashion, as all objects with positive mass must, you must havemotion along the radial line and youre only choice of direction is towards the singularity.

But there is a limit to how far you can vary our time-like geodesics, what of the r = 0 case? Just ast = 0 gives us the horizontal geodesic line for Minkowski space-time, r = 0 gives the same situationinside the event horizon due to the signature change. Therefore r = 0 is a horizontal line, which cannotbe crossed because it is a physical singularity. This gives us the partly completed Penrose diagramshown below.

Figure 7: Partial Penrose diagram for a Schwarzchild black hole

To tie up the loose geodesics can be done by realising that a photon which is related exactly on r = 2 M surface will remain there for eternity. This means that we can extend the r = 2 M line till it hits thesingularity, where the meeting place will be a future time-like innity, just as the other one. The time-like geodesics within the event horizon must all start or nish there as well. This has now completelydetermined the shape of the black hole region. By noting the t t symmetry, this completion canbe extended to the white hole region. Through the use of Kruskal coordinates (omitted) the left handside of the diagram is completed with a second, asymptotically at space-time, giving the nal diagramas Figure 8.

Figure 8: Complete Penrose diagram for a Schwarzchild black hole

Red lines for time-like, blue for space-like. An addition point of interest would be the central pointwhere all four regions meet. Taking a surface of constant t (since too fast a time-like motion will resultin you ending up in the black hole) gives the metric of the right hand region as

ds 2 = 1 2M

r

1

dr 2 + r 2 d22 (14)

7


8/9

Setting one of the polar coordinates, , to a constant 2 , leaves a metric form which admits the followingchange of variables

r = 2 M cosh2 (15) = 2 M + M sinh2 (16)

The value of depends on the region you are in, with = 0 if r = 2 M , > 0 in the right hand regionand < 0 in the left hand region. The metric now becomes

ds 2 = d 2 + 4 M 2 cosh2 d 2 (17)

This is a throat, a cylinder with a varying radius, minimised at = 0, signifying the wormholebetween the two asymptotically at space-times.

Figure 9: Wormhole between universes

Reissner-Nordstrom black holes

The Schwarzchild metric is the simplest black hole solution to the Einstein Field Equations (its alsounique by Birkhoffs Theorem). The next simplest is a charged but non-spinning black hole, foundby Reissner and Nordstrom and with the metric

ds 2 = 1 2M

r+

Q 2

r 2dt 2 + 1

2M r

+Q 2

r 2

1

dr 2 + r 2 d22 (18)

Q is the charge of the black hole and it can be seen that the metric asymptotes to Schwarzchild andthen Minkowski so similarities between those metrics Penrose diagrams and a R-N diagram is expected.

One difference between the R-N and Schwarzchild solutions is that the R-N solution has 3 distinctcases, depending on the value of Q . Since the event horizon in polar coordinates is when grr , itis dened to computed to be at

r = M M 2 Q 2 (19)This goes to r = 2 M if Q = 0, but its assumed not to be. If M > |Q | , there are two solutions, sotwo event horizons. If M = |Q | , the black hole is extremal and if M < |Q | were left with a nakedsingularity, there is no event horizon to shroud it. This is excluded by the cosmic censorship hypothesisand is thought to be physically unviable due to the self limiting of charge in a black hole.

For simplicity of dicussion and drawing, the rst case only is considered and the Penrose diagram

8


9/9

67151981 Penrose Diagrams

Documents

Transcript of 67151981 Penrose Diagrams