Black holes are sometimes viewed as some of the simplest objects in the universe. The no-hair theorem states that black holes can only have three independent properties: mass, charge and angular momentum . These properties, in principle, could be measured outside the black holes. (A black hole of a particular charge will repel or attract particles around it; mass can be measured through its gravitational field strength; and angular momentum by the amount of distortions in spacetime.)
The different properties of black holes and their respective types are derived as solutions to the Einstein field equations. Karl Schwarzschild was the first person to solve the Einstein field equation and it revealed the existence of black holes. He derived the solution while on the battlefield during the 1st World War!
The field equation is the ultimate mathematical expression of Einstein’s general theory of relativity. This is what it looks like:
Astronomy 130 version:
“Spacetime, with its curvature, tells masses how to move; masses tell spacetime how to curve.”
The solution to the field equation is a function that tells us how much distance or time is involved for unit displacements in the spacetime coordinates. It may be easiest to visualize this in terms of how close concentric circles of spacetime are near a black hole, as shown in the figure below. Notice that the distances between the circles are different for each radius because spacetime is warped.
To connect these circles with segments of these “too long” lengths, one can consider them to be offset from one another along some imaginary dimension that is perpendicular to x and y but is not z . (If it were z , the circles wouldn’t appear to lie in a plane!). We call such additional dimensions hyperspace (any space with more than 3 dimensions).
Schwarzschild’s solution to the Einstein field equations explains the curvature of spacetime outside a massive star. For a given, fixed star mass M , he examined how space and time are curved, if the star is made smaller and smaller in size.
If the star is made smaller than a certain critical size, the gravitational redshift of light (time dilation, remember) predicted by Schwarzschild's solution was infinite! This means that there is a singularity in the center of a black hole.
Once the Einstein field equation was solved, many other things could be inferred. For example, there are four different ‘types’ of black holes based on their mass, charge and spin (angular momentum).
- The simplest black hole is called the Schwarzschild black hole , which only has mass, no charge, no angular momentum (non-rotating) and it is spherically symmetric.
- Solutions to the Einstein field equations also exist for non-rotating black holes with charge and mass – Reissner-Nordstrom black hole.
- Non-charged, rotating black hole – Kerr metric or Kerr black hole .
- Stationary black hole with charge and angular momentum – Kerr-Newmann metric.
For non-rotating black holes, the singularity , where all the matter is concentrated, is surrounded by an imaginary sphere, the event horizon . The size of the event horizon is determined by the Schwarzschild radius.
For rotating black holes, the singularity is an infinitely thin ring! It has an inner and outer horizon. The light inside the horizon cannot escape.
A rotating BH drags spacetime around it like a whirlpool. A region called the ergosphere surrounds the horizon and has a bulging shape. At ergosphere you are forced to rotate in the same sense as the black hole although you can still escape.
The speed of this “frame dragging” is greater than the local speed of light.
The boundary of the ergosphere is called the static limit . It is called this because at this point an object that is counter-rotating around the BH at the speed of light, would appear stationary with respect to the rest of the universe. The space at the static limit is dragged at exactly the speed of light. Outside of it the space is dragged at speeds less than the speed of light.
Since it is possible for objects to escape the ergosphere, an object can potentially gain energy from BH’s rotation. This is called the Penrose process (proposed by Roger Penrose 1969). Up to 29% of BH’s total energy can be removed by this process. Eventually, the BH no longer spins and the ergosphere disappears.
- When a BH swallows matter, its mass and spin increase.
- The area of the black hole grows as mass2 for a Schwarzschild BH.
- When crashing two BHs together, the resulting BH has a bigger area than the two original BH’s added together.
- The area of a Kerr hole depends on both its spin and mass. It is possible to extract energy from a BH! For example with jets!
The Penrose process is well explained in this video: https://www.youtube.com/watch?v=ulCdoCfw-bY
Notice how the fact that objects are forced to co-rotate with the ergosphere is exactly what can be used to extract energy from the black hole. (Pay close attention around minute 3 of the video).