Reissner-Nordstr m Metric and Charged Black Holes

1. Reissner-Nordstr�m Metric andCharged Black Holes Jordan Gallagher and Jenn LeNestour

2. Reissner-Nordstr�m Metric Generalized Schwarzschild metric for a black hole that has An electric charge No angular momentum

3. Reissner-Nordstr�mMetric - Derivation Derivation process is similar to that of the Schwarzschild metric We assume spherical symmetry By Birkhoff�s Theorem, the metric will have the familiar form of:

4. Reissner-Nordstr�mMetric - Derivation However, we cannot set the Stress-Energy tensor to zero as in the Schwarzschild derivation There is electric charge present, and hence a non-zero Maxwell Tensor By equating the Ricci Tensor with the non-zero Maxwell Tensor, and after working through a lot of math, one arrives at the Reissner-Nordstr�m Metric

5. What is a Black Hole? Object with a very large gravitational field To escape you must be traveling faster than the speed of light Since no light escapes, black holes will always appear dark

6. Static Black Hole Structure Photon Sphere at 1.5 RS Singularity in the centre Event Horizon at RS

7. Static Black Hole Structure Photon Sphere Lowest possible orbit around a black hole Speed of c required to maintain orbit Light will orbit temporarily Orbits usually disturbed by other photons or particles and the photons will fall to the event horizon

8. Static Black Hole Structure Singularity Black hole�s mass is compressed into a region with zero volume Density, gravitational pull, and curvature of space-time are then infinite Can be thought of as a �place in time� since space and time change roles at the event horizon

9. Static Black Hole Structure Event Horizon Physically �Point of no return�Nothing that passes this sphere can return Required escape velocity greater than c Time and space effectively switch roles

10. Static Black Hole Structure Event Horizon Mathematically Coordinate singularity of the metricCan choose alternate coordinates (Eddington-Finkelstein) to show it isn�t a physical singularity Point after which the coefficients of dr and dt change signs

11. Event Horizons The �Horizon Function� H(r) is given by, H(r) = (1 - 2M/r + Q2/r2) and is quadratic with 2 distinct roots These two roots are given by; r+ = M + (M2 + Q2)1/2 r- = M + (M2 + Q2)1/2 These two roots correspond to two different event horizons, one at r+ and the other at r-

12. Event Horizons Schwarzschild black hole r+ = 2M r- = 0

13. Event Horizons The outer horizon at r+ is much like the event horizon of 2M for a Schwarzschild black hole Space and time change roles upon crossing the outer horizon as �normal� At the inner horizon, also known as the Cauchy horizon, something remarkable happens, the space and time co-ordinates change roles again. Inside the Cauchy horizon, time and space behave as �normal�, and the singularity is space-like

14. Spacetime Diagram for an R-N Black Hole Yellow = Radially inward light rays Orange = Radially outward light rays Purple (-) = Constant Time Purple (|) = Constant Radius Red = Event Horizons

15. Entering a Black Hole Static Once past the event horizon, you would be stretched due to tidal forces Due to time-space changing roles, the singularity is a point in time, and cannot be avoided As you approached the singularity the gravitational force will increase and you will be torn apart You become part of the black hole

16. Entering a Black Hole Penrose Diagram Space-time diagram Shows movement in a black hole

17. Entering a Black Hole Reissner-Nordstrom You would still be broken apart by the tidal forces Within the Outer Horizon the singularity would appear as a place in time Within the Inner Horizon time and space change roles again and so the singularity appears as a place in space, which is avoidable

18. Entering a Black Hole Reissner-Nordstrom Possible to pass �through� the singularity Find yourself in another black hole Would cross both event horizons and end up with the same velocity leaving as you did entering mirror-image of your worldline

19. Entering a Black Hole Penrose Diagram Space-time diagram Shows movement in a black hole Possible to transfer into other Universes

20. Do Black Holes ExistWithout Angular Momentum? Black Hole Formation Gravitational collapse of a star Collisions between neutron stars Pressure from the Big Bang (small Primordial Black Holes) Black holes can then absorb mass from interstellar gas and dust, as well as other stars and planets to become larger

21. Do Black Holes ExistWithout Angular Momentum? All of these methods will result in a black hole with some angular momentum Black holes with no angular momentum are only theoretical

22. Existence of Charged Black Holes Electric charge, over large scales, is pretty effective at �neutralizing� itself That is, macroscopic objects, especially astrophysical ones, tend to be electrically neutral The size of the Cauchy horizon depends directly on the magnitude of the charge of the black hole For a 3 solar mass black hole, a charge of~ 1019 C is required to create a Cauchy horizon of radius 0.001M Such a charge would require ~1038 electrons, and the resultant electric field would rip apart atoms with the greatest of ease

23. Do Reissner-Nordstr�m Black Holes Exist? No, they are just theoretical!