How extreme Black Holes defy the ‘no-hair theorem’

The no-hair theorem of black holes describes how these spacetime events have few outwardly measurable characteristics — but extreme black holes could defy this rule.

One of the most famous quotes about black holes is also, on its surface, one of the most bizarre: “black holes have no hair.” When physicist John Wheeler spoke these words he was using ‘hair’ as an analogy for outwardly discernable and measurable information.

But much like pop-culture’s most famous bald man, Homer Simpson, black holes have three strands of informational ‘hair’ pulled across their dome. These characteristics are mass, angular momentum or ‘spin,’ and electric charge. Three qualities which can exist both in isolation and in conjunction with each other, and which can be measured by outside observers.

Homer Simpson is perfect analogy for information and black holes. Despite being bald he has three strands of hair, just as black holes “have no hair” but have three characteristics. (Matt Groening and Fox Broadcasting)

Black holes that share the same values for these characteristics are indistinguishable from one other, as there are no further qualities to use in order to tell them apart.

Carrying the ‘baldness’ analogy further — perhaps further than is sensible — much like a bald man, a black hole may occasionally slip on a toupee to hide its lack of hair. In this case, that ‘toupee’ is a massless scalar field that can be donned by a black hole with the maximum possible electric charge — referred to as an ‘extreme black hole.’ This scalar field ‘toupee’ can be observed from great distances.

This finding of a massless scalar field would initially seem to defy proof provided in the 1970s by Jacob Bekenstein for the nonexistence of scalar field ‘hair’ which relied on a particular set of assumptions.

“Since Bekenstein’s proof, several papers found examples for scalar hair, and all these examples violate one or another of the assumptions made by Bekenstein,” explains Lior Burko of Theiss Research, an Associate Professor at Georgia Gwinnett College. “But in all cases, the hair was made of the scalar field itself.”

But, what Burko discovered — building upon the work of Angelopoulos, Aretakis, and Gajic — was that this additional ‘hair’ is different than expected.

“A massless scalar hair does not violate any of the assumptions underlying Bekenstein’s proof,” the researcher says. “It is hair in a different sense than the kinds of hair that were found before. It is not the scalar field itself, but a certain integral on a derivative of the scalar field that is to be calculated on the surface of the black hole, on its event horizon.”

This toupee or new hair can be seen by observers at a great distance by utilising another quality of the black hole at this distance, and therefore, at an infinitely later time.

“These would be observers who are very distant from the black hole, and who make the measurements in the infinite future,” Burko says. “We wanted to see what happens at late but finite times, to see the time dependence of the measurement and how it approaches its asymptotic value.”

Burko adds that another special thing about this new hair is that initial models suggest that it applies only for exactly extreme black holes. This led the team to wonder what happens when the black hole is nearly — but not exactly — extreme (ie — possessing close to the maximum charge allowable).

Non-extreme black holes can slip on a toupee — but only temporarily

Together with his colleagues Gaurav Khanna of the University of Massachusetts Dartmouth and his former student Subir Sabharwal. Burko shows that measurements from a great distance are approaching the ‘hair value’ — with the difference between them decaying with inverse time.

Then, going beyond the model put forward by Angelopoulos, Aretakis, and Gajic, the team considered not just black holes with near to maximum charge, but also black holes with maximum, and close to maximum, angular momentum. In other words, black holes spinning at precisely — or near to — the maximum rate.

“In addition to a maximal value of charge, there is also a limit for how fast a black hole can spin,” Burko explains. “Black holes that spin at the maximal allowed rate are also called extreme black holes. We describe both maximally charged and maximally spinning black holes by the name extreme black holes, as there are many similarities between the two.”

Gargantua, a fictional black hole created for the movie Interstellar is a great testing ground for new models exploring the no-hair theorem

The new hair was originally found for a very useful model for black holes — those that are spherically symmetric and electrically charged — but, in reality, black holes are neither. “Instead, we wanted to find out if this hair can be found also for spinning black holes,” says Burko. “In the movie Interstellar the monster black hole is nearly extreme.

“We wanted to see if Gargantua has hair.”

Gargantua is a fictional black hole encountered by NASA probes at the other side of a wormhole in Christopher Nolan’s 2014 movie Interstellar — a film acclaimed for its scientific accuracy relative to other Hollywood sci-fi offerings. It was suitable for the team to use as a model as it is both very massive and, rapidly spinning as well as being very precisely described by physicist Kip Thorne, a science advisor to the film and author of The Science of Interstellar.

The team used very intensive numerical simulations to generate their results involving dozens of the highest-end Nvidia graphics-processing-units (GPUs) with over 5,000 cores each, in parallel. “Each of these GPUs can perform as many as 7 trillion calculations per second,” adds Khanna. “Even with such computational capacity, the simulations took many weeks to complete.”

Thus, the team was able to demonstrate that for the nearly extreme spinning black holes the ‘toupee donning’ or adoption of new hair is a transient behaviour.

At intermediate times nearly extreme black holes behave as extreme black holes would, but at late times they behave like regular, non-extreme black holes.

“Nearly extreme black holes can pretend that they are extreme for only so long. But eventually, their non-extremality becomes manifest,” says Burko. This means that early extreme black holes that don their toupee, must eventually remove it and become bald again.