In general relativity a black hole is a relatively simple object.  It can be described by three basic quantities: its mass, its rotation (angular momentum), and its charge. No matter what type of material collapses into a black hole, in the end it’s reduced to mass, rotation and net charge. This property is known as the no-hair theorem, because unlike other astronomical objects like stars and planets, black holes should have no features (hair). But is the theorem too simplistic?

One of the criticisms of the no-hair theorem is that it’s only been formally proven in the case of isolated black holes. The problem is that black holes tend to do things like build up disks of matter around them, or orbit with other black holes and the like. Does the no-hair theorem apply then, or is it just for lonely black holes? A new paper in Physical Review Letters argues that the no-hair theorem does apply even for black holes surrounded by matter, at least for a broad class of physically reasonable cases.

The Earth with it’s gravitational variations shown greatly magnified. Credit: ESA

The paper looks at an aspect of gravity known as multipoles. A perfectly spherical mass would have a gravitational field that is the same in all directions. But an object like the Earth isn’t perfectly spherical, so the Earth’s gravitational field is slightly distorted. But the Earth is approximately spherical, so one can approximate Earth’s gravitational field as a series of perturbations from perfectly spherical. The spherical part is sometimes called the monopole, and the deviation along its axis the dipole, then the quadrupole, octopole, etc. With each successive multipole your model of the gravitational field better approximates the actual field. This is often done in relativity because after a few terms the deviations are so small that you can basically ignore them.  It’s like saying the value of pi is 3.14159. For most applications that’s close enough.

According to the no-hair theorem, the gravitational field of a stationary black hole should be a monopole. Other multipole terms would be due to deviations from a spherical shape, and thus be “hair.” In this new work the author showed that the gravitational field of stationary black hole is just a monopole, even if there is matter surrounding it.

This is rather surprising. Suppose there were a black hole with a dense accretion disk surrounding it (which is actually rather common). The mass of the accretion disk would exert a gravitational force on the black hole, and the event horizon of the black hole should distort accordingly. This would mean the black hole isn’t perfectly spherical, and so should have multipole gravity terms. But it turns out that in general relativity the distortion of the event horizon actually cancels out the multipole gravity terms due to the non-spherical shape of the black hole. So even when a stationary black hole isn’t spherical, its gravity only has a monopole term.

This is a nice bit of theoretical work, but it could also have observational consequences. Monopole gravity doesn’t produce gravitational waves, for example, so any gravitational waves produced by such a black hole would be due to the surrounding matter, not the black hole itself. There are also alternatives to general relativity where the no-hair theorem doesn’t apply, so there might be a way to use this to distinguish general relativity from other models.

For now, though, we know that for black holes it really is that simple.

Paper: Norman Gürlebeck. No-Hair Theorem for Black Holes in Astrophysical Environments. Phys. Rev. Lett. 114, 151102 (2015)

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According to general relativity, a black hole has three measurable properties: mass, rotation (angular momentum), and charge. That’s it. If you know those three things, you know all there is to know about the black hole. If the black hole is interacting with other objects, then the interactions can be much more complicated, but an isolated black hole is just mass, rotation and charge.

In general relativity this is known as the no-hair theorem. The basic idea of the no-hair theorem is that the material properties of any object (referred to as “hair” because a physicist named John Wheeler once coined the phrase “a black hole has no hair”) become unmeasurable (hence unknowable) as the object collapses into a black hole.

On the surface this seems fairly reasonable. If a neutron star collapses into a black hole, for example, all the neutrons and their interactions become trapped inside the black hole’s event horizon when the black hole forms. The same would be true for an object that was lopsided (say with a mountain range on one side). As it collapses into a black hole, any irregularities would be squashed flat as it approaches the black hole limit.

But there are also difficulties with the no-hair theorem. For one, even though it’s referred to as a theorem, it has never been proved in general relativity. So it really should be called the no-hair hypothesis. There have been lots of demonstrations that the theorem is reasonable, and computer simulations tend to agree that black holes stabilize to a structure defined by mass, rotation and charge. But none of these reach the level of proof.

Then there is the problem that if a black hole really is just defined by mass, charge and rotation, then it has no temperature, and that means that its entropy is zero. This violates the principles of thermodynamics. Of course when we try to include quantum theory into our black hole description we know that black holes do have a temperature. In Hawking’s theory, the temperature of a black hole depends upon its mass, so even a Hawking black hole would be definable by mass, rotation and charge. It’s possible that the no-hair theorem is valid even for a quantum black hole.

But there is a more subtle mystery that hides within the no-hair theorem, because it would seem that a black hole is much simpler than other massive objects such as planets, stars and the like. If you think about an object like the Sun, it has a certain chemical composition, and it’s giving off light with different wavelengths having varying intensities. There are sunspots, solar flares, convection flows that create granules, and the list goes on. The Sun is a deeply complex object that we have yet to fully understand. And yet, if our Sun were compressed into a black hole, all that complexity would be reduced to mass, rotation and charge. So what happens when a complex object like a star collapses into a black hole? Where does all that complexity go?

In physics we refer to that complexity as the physical information of a system. According to quantum theory, physical information is never lost, but according to general relativity and the no-hair theorem, physical information that enters a black hole is lost forever. This contradiction is known as the black hole information paradox, or sometimes the firewall paradox. Now you might think that the easy answer is just to presume the no-hair theorem is wrong.

But it’s not that simple, and if we started exploring that paradox, things would get a bit hairy.

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