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4.4 The Penrose Process

Back

Suppose that a particle approaches a Kerr black hole along a geodesic. If p

is its 4-momentum we can identify the constant of the motion

E = 􀀀p _ k (4.48)

as its energy (since E = p0 at 1). Now suppose that the particle decays

into two others, one of which falls into the hole while the other escapes to

......................................................................................

...................... .... .... ... ....

.....................................

......... . . . .. . . . .

horizon

E1 = 􀀀p1 _ k

E2 = 􀀀p2 _ k

By conservation of energy

E2 = E 􀀀 E1 (4.49)

Normally E1 > 0 so E2 < E, but in this case

E1 = 􀀀p1 _ k (4.50)

which is not necessarily positive in the ergoregion since k may be spacelike

there. Thus, if the decay takes place in the ergoregion we may have E2 > E,

so energy has been extracted from the black hole.

4.4.1 Limits to Energy Extraction

For particles passing through the horizon at r = r+ we have

􀀀 p _ _ _ 0 (4.51)

Since _ is future-directed null on the horizon and p is future-directed timelike

or null. Since _ = k + Hm,

E 􀀀 HL _ 0 (4.52)

where L = p_m is the component of the particle's angular momentum in the

direction de_ned by m (only this component is a constant of the motion).

Thus

L _

(4.53)

If E is negative, as it is for particle 1 in the Penrose process then L is also

negative, so the hole's angular momentum is reduced. We end up with a

hole of mass M + _M and angular momentum J + _J where _M = E and

_J = L so

_J _

M2 + pM4 􀀀 J2

_M (4.54)

from formula for H. This is equivalent to (Exercise)

M2 +

M4 􀀀 J2

_ 0 (4.55)

(This quantity must increase in the Penrose process).

Lemma A = 8_

M2 + pM4 􀀀 J2

is the `area of the event horizon', of a

Kerr black hole (i.e. area of intersection of H+ with partial Cauchy surface,

e.g. area of v = constant, r = r+ in Kerr coordinates (See Question III.5).

Corollary Energy extraction by Penrose process is limited by the requirement

that _A _ 0. This is a special case of the second law of black hole

mechanics.

4.4.2 Super-radiance

The Penrose process has a close analogue in the scattering of radiation by a

Kerr black hole. For simplicity, consider a massless scalar _eld _. Its stress

tensor is

T__ = @__@__ 􀀀

g__(@_)2 (4.56)

Since D_T_

_ = 0 we have

D_ (T_

_k_) = T__D_k_ = 0 (4.57)

so we can consider

j_ = 􀀀T_

_k_ = 􀀀@__k _ @_ +

k_(@_)2 (4.58)

as the future directed (􀀀k _ J > 0) energy ux 4-vector of _. Now consider

the following region, S, of spacetime with a null hypersurface N _ H+ as

one boundary.

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..... ...... ...... ..... ...... ...... ...... ..... ...... ...... ..

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.....................................

.........................................................

.....................................

􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀

􀀀􀀀􀀀

􀀀􀀀

􀀀􀀀􀀀

􀀀􀀀

􀀀 􀀀􀀀 􀀀􀀀 􀀀􀀀 􀀀 􀀀􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 i0

􀀀n

Note: 􀀀_ is `outward

directed' normal to N, as

determined by continuity

H+

􀀀_

Assume that @_ = 0 at i0. Since D_j_ = 0 we have

0 =

d4xp􀀀gD_j_ =

dS_ j_ (4.59)

dS_ j_ 􀀀

dS_ j_ (4.60)

= E2 􀀀 E1 􀀀

dS_ j_ (4.61)

where Ei is the energy of the scalar _eld on the spacelike hypersurface _i.

The energy going through the horizon is therefore

_E = E1 􀀀 E2 = 􀀀

dS_ j_ (4.62)

= 􀀀

dA dv __j_; (v is Kerr coordinate) (4.63)

The energy ux lost/unit time (power) is therefore

P = 􀀀

dA __j_ =

dA (_ _ @_)(k _ D_) (4.64)

(since _ _ k = 0 on horizon by previous Lemma)

_ + H

(4.65)

For a wave-mode of angular-frequency !

_ = _0 cos (!v 􀀀 __) ; _ 2 Z (angular quantum no.) (4.66)

The time average power lost across the horizon is

P =

_20

A!(! 􀀀 _) (4.67)

where A is the area of the horizon.

P is positive for most values of !, but for ! in the range

0 < ! < _H (4.68)

it is negative, i.e. a wave-mode with !; _ satisfying the inequality is ampli_ed

by the black hole.

Remarks

i) Process is positive only for _ 6= 0 because the ampli_ed _eld must also

take away angular momentum from the hole.

ii) Process is similar to stimulated emission in atomic physics, which suggests

the possibility of a spontaneous emission e_ect. This can be

shown to occur in the quantum theory so any black with an ergoregion

cannot be stable quantum mechanically.

iii) We have neglected the back-reaction of _ on the metric. When corrected

for back-reaction the metric can be stationary only if @_=@_ =

0, but then j_ = 0 and the black hole energy doesn't change, i.e.

strictly speaking super-radiance is incompatible with stationarity.

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