No Hair Theorem

Although most people have some notion of what a black hole is, you can’t decide on how to characterize one without a precise definition.  Such a precise definition is known in general relativity, but the details are too complex for this essay.  Roughly, however, a black hole is a region of space from which nothing, including light, can escape to infinity.  This definition is closely linked to the idea of an event horizon, which is the surface of the region of space mentioned above.

The no hair theorem states that mass, angular momentum, and electric charge completely characterize a black hole.  That is, no matter how two black holes are created, if the mass, angular momentum, and electric charge are the same, there are no measurable differences between them.  This theorem hasn’t actually been proven in full generality, but is known to be true for black holes which are surrounded by vacuum.

The proof of the vacuum solution part of the theorem has essentially two parts.  The work of numerous researchers has demonstrated that in general relativity, the only axisymmetric vacuum solutions to Einstein’s field equations are the Reissner-Kerr-Newman solutions.  These solutions have only three parameters – mass, angular momentum, and charge. 

The second part of the proof was introduced in 1972 by Richard Price.  Roughly speaking, Price’s theorem states that any non-axisymmetric protrusion in the spacetime geometry external to a black hole will radiate away as gravitational radiation, sometimes simplified to state that ‘whatever can be radiated will be radiated’.   In its original form Price’s theorem is rather limited in scope, but 15 years of work by the Cambridge relativity group proved the theorem in full generality. 

The no-hair theorem follows easily.  If non-axisymmetric collapse events occur, Price’s theorem states that eventually all structure by the axisymmetric part will radiate away, or collapse into the black hole.  The axisymmetric black hole which is finally attained after all nonsymmetric structure is radiated away is completely characterized by three parameters; mass, angular momentum, and electric charge. 

What we have presented to this point is founded in classical general relativity.  However, it is reasonable to ask if the same black hole characterization holds when quantum effects are included. 

It is difficult to precisely address this question, because a satisfactory quantum theory of gravity has not yet been invented.  However, considerable work has been done in a semiclassical approximation.  Roughly, the assumption is that the black hole is described by classical geometry, but matter and radiation fields are treated quantum mechanically.  That is, the black hole is located in a quantum vacuum.  Bekenstein and Hawking established a link between the laws of black hole mechanics and the laws of thermodynamics.  In this description, a black hole is characterized by its area (analogous to entropy), its surface gravity (analogous to temperature), and its mass (analogous to energy).  The resulting first law is essentially a statement of the no-hair theorem. 

These thermodynamic analogies have been confirmed in what are essentially statistical mechanical calculations carried out in fully quantum theories of gravity by Ashtekar and Krasnov.  Even though those theories are not fully satisfactory, this result does lend credibility to the universal applicability of the no-hair theorem.  If so, we can be sure that a black hole can be characterized by no more than three physical parameters; mass, angular momentum, and electric charge.