No hair theorem and Klein-Gordon equation

Question

The no-hair theorem states that we can't detect scalar fields outside a black hole, meaning that the solution for the KG equation is trivial, but in fact, we can solve it (for instance for a Schwarzchild BH) and the solution is not trivial, what am I missing?

Answer 1

… what am I missing?

You are missing the meaning of (various) no-hair theorems. Those theorems are talking about equilibrium configurations corresponding to end points of gravitational collapse and thus about static or stationary solutions to Einstein equations with various kinds of matter fields.

As an example consider the following “no hair” result due to Bekenstein:

A black hole in its final (static or stationary) state cannot be endowed with any exterior scalar, vector, or spin-2 meson fields.

Note the phrase “final (static or stationary) state”. This result does not exclude all non-trivial solutions to KG equation on a black hole background but only static or stationary ones. Physically this means that if we specify non-trivial initial data for a scalar field near a black hole, then with enough time this field excitation would either disperse to the asymptotic region or be absorbed by the black hole.

Answer 2

There is no singularity at the event horizon, its a perfectly smooth boundary, a light cone, that in its inner future and past has time terminating hyperboloids tangent to the cone in future.

In the Kruskal coordinates radial light rays and light cones have 45° slope. The infalling massive particle path ends at the singularity with light velocity.

This is the case of the Schwarzschild equations for a spherical BH in the vacuum Einstein equations, with two hyperboloids serving assingularities deforming the geometry everywhere. Inside a spherical black hole other topologies are possible. In any case, the horizon is simply the future inner singularity absorbing any material or light path, that somehow has crossed the horizon cone in space time.

The Klein Gordon equation, therefore, does not need a spatial boundary condition at the horizon double cone, because locally its no special place in general coordinates.

Following the philosophy of wave equations, a start conditon for time forward frequency modes on the past singularity hav tie be provided, which would be sent with its content of particle mode structure entities like mass, spin, momentum, to all of space in future, albeit very dark redshifted.

Answer 3

What you are talking about seems to be a perturbation. You have a scalar field that exists in the Schwarzchild background.

No hair theorem is about back reaction. You impose that the scalar field backreacts to the metric so you have to solve Einstein's equation simultaneously with the Klein Gordon equation.

One can show then that the scalar field is trivial if not supported by a negative scalar potential.

By hair we mean that at the final state of gravitational collapse,where the black hole has formed, we have non trivial (with this word I mean the mass angular momentum and charges) information about the black hole.

Answer 4

Belongs to the question of external quantum fields in the Schwartzschikd geometry of a static black hole.

Since there is no boundary condition on the horizon like its is eg. outside a solid crowded with occupied states, its an open question if the vacuum state outside is a vaccum state or a temperature state. The temperatur would be governed by the mass, that gives the surface of a sphere with no outer knowledge what happens inside. Perfect chaos at the surface assumed.

The latter assumption brought Hawking to postulate the Hawking radiation for all physical existent modes, thereby evaporating the central mass.

Its of course all speculative mathematical physics.