We wrapped up the Introduction to Black Holes online class last week, finishing with some of the more bizarre results that have come out of trying to understand black hole behavior. In a course this short (four one-hour meetings), we’re only able to scratch the metaphorical surface of black hole science, but I wanted my students to see both the astronomy and the theoretical physics sides. The physics of accretion and jets is the stuff we have more-or-less direct access to, thanks to powerful telescopes, but the theory side is where we can get close to the workings of gravity — including possibly a regime where general relativity breaks down and must be supplemented with a new set of ideas.

One interesting result arising from the study of gravitational collapse is the “no-hair” theorem (named by John Archibald Wheeler, obviously not someone untroubled by receding hairlines). Hair isn’t literal: it refers to anything that might poke out through the black hole’s event horizon, the barrier past which nothing can return to the outside Universe. Lumpy stars result in smooth black holes; strong magnetic fields get radiated away during gravitational collapse; chemical composition is lost behind the event horizon. The no-hair theorem states that the only properties a black hole presents to the cosmos are its mass, spin, and (possibly) electric charge.[**1**]

The theorem is fairly simple, but its implications are astounding. Two black holes with the same mass and rotation are identical, even if the stars that birthed them were different in chemistry, size, and magnetic properties. In that sense, they might seem more like elementary particles such as electrons, which are indistinguishable from each other. The analogy doesn’t go too far: black holes have a continuous range of masses and spins that can change over the life of the object, while each electron has precisely the same mass and spin. However, compared with stars, pulsars, white dwarfs, and other astronomical bodies, black holes are remarkably simple. (For questions about what happens inside black holes, see my most recent article at *Nautilus*.)

The no-hair theorem and gravitational collapse are both natural consequences of general relativity, but nearly nobody believes our current theory is the last word. If nothing else, the infinite density predicted at the heart of black holes and at the Big Bang itself have made many suspect that general relativity contains the sign of its own limitations. Others, motivated by inflation or dark energy, have introduced new gravitational theories to explain those phenomena. Modifications to gravity can lead to testable consequences, and perhaps the no-hair theorem fails as well.

### A theoretical hair apparent

That happens in a class of modified gravity theories known (unfortunately) as “scalar-tensor” models. Obviously nobody was thinking of non-physicists when they proposed the name, so as a popularizer I’m already at a major disadvantage. “Scalar-tensor” refers to the mathematical objects used in the theories. Standard general relativity is a “tensor” theory, which expresses how gravity shapes the paths of particles and also how neighboring trajectories deviate from each other. (I’m not entirely sure, but I believe the name itself comes from the mathematical description of the internal mechanics of solids, including tension.)

Tensors turn out to be a particularly elegant way to describe gravity, but the details of their working are something I’ll leave for another day; you can see a bit of how they work in the appendix of an earlier post. Scalar fields, on the other hand, are fairly simple things: they’re like the density of air, which is just a number that fluctuates from place to place. The Higgs field, which gives masses to many particles as they interact with it, is a scalar field: it doesn’t depend on how fast the particle is moving or what direction, unlike forces like gravity or electromagnetism.

Scalar-tensor theories of gravity start with general relativity and add one or more extra scalar fields. Typically, those fields don’t connect directly to matter — they don’t constitute a new force, in other words — but affect the strength of gravity behind the scenes. Inflation is often described as a scalar field, driving the rapid expansion of spacetime during the Universe’s first instants, but scalar-tensor inflation theories couple this field to gravity in a way that alters gravity’s behavior. Other scalar-tensor models are motivated by string theory, or to explain dark energy.

A recent paper in *Physical Review Letters*[**2**] worked out the details of the no-hair theorem in a very general version of scalar-tensor gravitational models. The authors didn’t pick one theory, but considered a whole class of them together, and they found that the extra scalar field changed the behavior of black holes. In a way, that’s utterly unsurprising: a different theory predicts new results! However, the specific changes due to the scalar field lead to a breakdown of the no-hair theorem for realistic black holes.

A solitary isolated black hole in scalar-tensor theories is no different than what general relativity predicts. However, under some circumstances the presence of matter nearby — perhaps in the form of matter flowing from the accretion disk swirling around a black hole, which is how we detect them in nature — an instability in the scalar field, making it behave like a massive particle. (Weirdly, in some cases that mass is described by an imaginary number, which could be problematic. No consistent theory involving such imaginary masses exists to my knowledge.) That makes the black hole “hairy”: the scalar field sticks out of the event horizon, providing an extra bit of physical detail beyond mass and spin.

The paper doesn’t discuss specific observational consequences, but we do have some information about them nevertheless. Scalar-tensor theories have been around for decades, and thanks to a multitude of observations, we can constrain their predictions pretty well.[**3**] As the authors of the current paper point out, this doesn’t rule out the possibility of scalar-tensor models, but systems like binary pulsars put the kibosh on many strong scalar field effects.

Any black hole “hair” would have to abide by these constraints, and if we don’t see the predicted effects, it could have consequences for modified gravity models for dark energy or inflation. In other words, black holes with their strong gravity could prove to be a lab for testing theories of cosmology.

### Notes

- It’s likely that most black holes do have some electric charge simply by eating more of one type of particle than another, but it takes a lot of excess charge to make a measurable difference. I’ll ignore charged black holes for the rest of this post.
- Vitor Cardoso, Isabella P. Carucci, Paolo Pani, and Thomas P. Sotiriou, “Black Holes with Surrounding Matter in Scalar-Tensor Theories”.
*Phys. Rev. Lett.***111**, 111101 (2013). DOI: 10.1103/PhysRevLett.111.111101, ArXiV: 1308.6587 - Those versed in gravitational physics should see Clifford M. Will’s excellent book
*Theory and experiment in gravitational physics*(Cambridge University Press, 1993). I’m not sure if there’s a book explaining the same concepts on a popular level.