Black Holes : Where God Divided by Zero

1. Black Holes : Where God Divided by Zero Suprit Singh Black Holes : Where God Divided by Zero Suprit Singh

2. What’s a hole anyway? • Well, it’s a ‘place’ in the spacetime where the curvature blows up… called Singularity • Can’t Specify ‘where’, as it doesn’t exist on the manifold • So we are concerned with r>0

3. Structure of Black Holes Work of Israel, Carter, Robinson and Hawking All but M, Q and J information is lost for bodies falling into the Black Holes “Black Holes have No Hair” - Wheeler Although Mathematically Complex, structurally quite Simple.

4. Black Hole Types Only 3 Special Unique solns: Schwarzschild Reissner – Nordstrom Kerr

5. Okay…How do you get ‘em???? • Erm..brutally… • Einstein's Field =ns relate Geometry at a point to the Energy-momentum density at that point… ( Zey r Local) • So you crank up the RHS and solve it up… • And yeah..’Smart’ guys use Symmetries to make it a lot easier… Is it Static? Holy Symmetry : Spherical Any Axial Symmetry Expect Minkowski structure in the far-off limit Smart Guy’s Box

6. Schwarzschild Black Hole The Coordinates Metric Singularities Falling In Formation Conformal Diagram • Flat Minkowski space-time gets modified such that Spherical Symmetry is preserved. • Note that, now, r is not radial coordinate. We define it as Circumference/2pi. Hence sometimes also called ‘Reduced Circumference’ and is the shell-observer distance between adjacent circles and is more than dr for r > 2M • Coordinate t is the far-off observer’s time. The shell time is therefore related as giving local shell structure as

7. Schwarzschild Black Hole The Coordinates Metric Singularities Falling In Formation Conformal Diagram Schwarzschild Black Hole • The Coordinates Metric Singularities Falling In Formation Penrose Diagram • First Singularity is regarding the Spherical Coordinate system and is Removable • Second Apparent Singularity concerns, r = 2M which also removable but has interesting implications. • Lastly, we have singular structure at r = 0 which is an honest tear in the spacetime for Curvature Scalar Blows up ‘there’

8. Schwarzschild Black Hole The Coordinates Metric Singularities Falling In Formation Conformal Diagram Schwarzschild Black Hole • The Coordinates Metric Singularities Falling In Formation Penrose Diagram • Exploring the Causal Structure of the Schwarzschild Metric • Consider radial null curves ( well , radial Photons , OK  ) from which we note that • which is the slope of Light Cones. Note that for far away r, it is +/- 1 however it becomes infinite for r = 2M, i.e., the cones shrivel up as they get near 2M. For the far observer, it never gets there… • However, the light gets in • alright in its own frame, • everything’s smooth • Lets get inside the horizon. First nothing’s stationary there. • For static particle we’ll have an impossibility...

9. Schwarzschild Black Hole The Coordinates Metric Singularities Falling In Formation Conformal Diagram • Then How do particles move inside??? We can get the answer if we know how light cones behave in there.. And (t, r, theta, phi) are inadequate for r<2M. Hence we introduce new time coordinate not singular at 2M and express the geometry in Eddington- Finkelstein coordinates • The surface r = 2G M, while being locally perfectly regular, globally functions as a point of no return.

10. Schwarzschild Black Hole The Coordinates Metric Singularities Falling In Formation Conformal Diagram

11. Schwarzschild Black Hole The Coordinates Metric Singularities Falling InFormation Conformal Diagram • Review of the Kruskal-Szekeres Map : • Region I and II correspond to Outside • and inside of the S - Black hole. • Region IV and III correspond to Outside • and inside of the time - symmetric • S - White hole. • A black hole is a “region of spacetime from which no signal can escape to infinity” (Roger Penrose) • This is unsatisfactory because ‘infinity’ is not part of the spacetime. However the ‘definition’ concerns the causal structure of spacetime which is unchanged by conformal compactification

12. Schwarzschild Black Hole The Coordinates Metric Singularities Falling InFormation Conformal Diagram • Choose Λs.t. • i.e., all points at ∞ in the original metric are at finite affine parameter in the new metric • Elements of the diagram : • a. spacetime null lines are oriented • at 45 to the vertical • b. ‘infinity’ is represented as finite • boundary to the picture • Fig (b) is the Diagram for Minkowski • spacetime. • Fig (c) is the Diagram for Spherically symmetric collapse to Black Hole, the horizon lies at 45. Any material particle’s world line cannot tilt at more that 45 to the vertical, so it cannot escape from the interior region behind the horizon once it has crossed into it. Moreover, once inside that region, it is forced into the singularity .

13. Schwarzschild Black Hole The Coordinates Metric Singularities Falling InFormation Conformal Diagram The Program for Minkowski :

14. Reissner- Nordström Black Hole The Coordinates Metric Singularities M2 < P2 - Q2 M2 > P2 - Q2 M2 = P2 - Q2 • Generalized Schwarzschild metric for a black hole that has • An electric charge • No angular momentum Δ(r) = (1 - 2M/r + P2/r2 +Q2/r2 )

15. Reissner- Nordström Black Hole The Coordinates Metric SingularitiesM2 < P2 - Q2 M2 > P2 - Q2 M2 =P2 - Q2 • The Horizon Function, Δ(r) is given by, • Δ(r) = (1 - 2M/r + P2/r2 +Q2/r2 ) and is quadratic with 2 distinct roots • These two roots are given by; • r+ = M + (M2 - P2 - Q2)1/2 • r- = M - (M2 - P2 - Q2)1/2 • The two roots correspond to two different event horizons, one at r+ and the other at r-

16. Reissner- Nordström Black Hole The Coordinates Metric Singularities M2 < P2 - Q2 M2 > P2 - Q2 M2 = P2 - Q2 • Case One: M2 < P2 - Q2 • In this case the coefficient Δ is always positive (never zero) • The metric is completely regular all the way down to r = 0. • The coordinate t is always timelike and r is always spacelike. The r = 0 singularity is a timelike line. • Since there is no event horizon, there is no obstruction to an observer • traveling to the singularity and returning to report on what was observed. • The singularity is repulsive-timelike, geodesics never intersect r = 0; approach and reverse course (Null geodesics can reach the singularity) • As r tends to infinity the solution approaches flat spacetime and the causal structure seems normal everywhere. The conformal diagram is just like that of Minkowski , except that now r = 0 is a singularity

17. Reissner- Nordström Black Hole The Coordinates Metric Singularities M2 < P2 - Q2 M2 > P2 - Q2 M2 = P2 - Q2 However ,

18. Reissner- Nordström Black Hole The Coordinates Metric Singularities M2 < P2 - Q2M2 > P2 - Q2 M2 = P2 - Q2 • Case Two: M2 > P2 - Q2 • Space and time change roles upon crossing the outer horizon as “normal” • Fall to inner Horizon is inevitable. • All Phenomenon same as P = Q = 0 • At the inner horizon, the space and time co-ordinates change roles again. • The singularity being timelike now can be avoided (Not necessarily in your future) • Choose r = 0 or back to increasing r through r(-), • r will again be timelike, in reverse and will increase to spit past r(+) • Then you can choose to go back into the black hole – this time a different one and • repeat the voyages as many times you want.

19. Reissner- Nordström Black Hole The Coordinates Metric Singularities M2 < P2 - Q2M2 > P2 - Q2 M2 = P2 - Q2

20. Reissner- Nordström Black Hole The Coordinates Metric Singularities M2 < P2 - Q2 M2 < P2 - Q2M2 = P2 - Q2 • Case Three: M2 = P2 - Q2 • The Extreme Solution, Unstable as adding just a little • matter will bring it to Case Two • Single event horizon r = M but r is never timelike, it • becomes null at M, but is spacelike on the other side • Singularity r=0 is timelike as in other cases and again • avoidable • However, it stays on left though you can move to extra • copies of Asymptotical flativerse

21. Kerr Black Hole The Coordinates Event Horizon/s Ring Singularity Conformal Diagram • Non-zero angular momentum • Axial symmetry • Uses Boyer-Lindquist coordinates • Take limit (a,0), we are left with Schwarzschild • Take limit (M,0), Its flativerse but not polar but Ellipsoidal ( Expected ??)

22. Kerr Black Hole The Coordinates Event Horizon/s Ring Singularity Conformal Diagram • As in the Reissner-Nordstrom solution, there are three possibilities: • M > a : Physical case • M = a : Unstable • M < a : Naked Singularity • The Event Horizons are : • A surface of infinite gravitational red shift • can be determined by The region between the Outer Horizon and Outer static limit is termed Ergosphere (Energy??)

23. Kerr Black Hole The Coordinates Event Horizon/s Ring Singularity Conformal Diagram • The True Singularity doesn’t occur at r=0 in this case • But at • r = 0 is not a point in space but a disc and singularity is a ring at the edge of this disc • Around the ring are CTCs and if you go round, • you can meet your past • Go through the ring and Exit to another Asymptote • flativerse but not an identical copy • The new spacetime is described by Kerr metric • with r < 0, as a result there are no horizons

24. Kerr Black Hole The Coordinates Event Horizon/s Ring Singularity Conformal Diagram Penrose Process : Living off a Rotating Black Hole

25. References • Exploring Black Holes : an introduction to general relativity ,Taylor and Wheeler • Black Holes and Time Warps, Kip S Thorne • Spacetime and Geometry , Sean Carrol • Black Holes , Paul Townsend http://arxiv.org/abs/gr-qc/9707012v1 • The Nature of Space and Time, Hawking and Penrose arxiv.org/abs/hep-th/9409195 • The Road to Reality, Roger Penrose • Gravity, Black Holes, and the Very Early Universe , T L Chow • Introducing Einstein’s Relativity, Ray D’Inverno • Hope you Enjoyed the ride…