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6.1 Geodesic Congruences

Back

De_nition A congruence is a family of curves such that precisely one

curve of the family passes through each point. It is a geodesic congruence

if the curves are geodesics.

The equations of a geodesic congruence may be written as x_ = x_ (y_; _)

where the parameters y_; _ = 0; 1; 2 label the geodesic and _ is an a_ne

parameter on the geodesic, i.e.

t =

@x_

@_ (6.1)

is the tangent to the geodesics such that t _ Dt_ = 0. Since the parameter

_ is a_ne, t2 _ 􀀀1 for timelike geodesics (while t2 _ 0 for null geodesics).

The vectors

__ =

dy_ =

@x_

@y_ @_ (6.2)

may be considered as a basis of `displacement' vectors across the congruence:

101

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

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

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

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

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

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

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

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neighboring geodesics

constant _

Note that t and __ commute (since we could choose coordinates x_ s.t.

t = @=@_ and __ = @=@y_), so

0 = t_@___

_ 􀀀 __

_@_ t_ (6.3)

= t_ (@___

_ + 􀀀_

____

_) 􀀀 __

_ (@_ t_ + 􀀀_

__ t_) (6.4)

= t_D___

_ 􀀀 __

_D_ t_ (by symmetry of connection) (6.5)

t_D___

_ = B_

___

_ (6.6)

where

_ = D_t_ (6.7)

measures the failure of the displacement vectors __ to be paralelly-transported

along the geodesics, i.e. it measures geodesic deviation.

A geodesic nearby some _ducial geodesic may now be speci_ed by a

displacement vector _, but this speci_cation is not unique because _0 = _+at

(a = constant) is a displacement vector to the same geodesic.

102

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

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

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

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

_ + at

neighbouring

geodesic

_ducial

gedoesic

For timelike geodesics we can remove this ambiguity by requiring _ to be

orthogonal to t, i.e.

_ _ t = 0 (6.8)

Strictly, speaking we can only make such a choice at a given value of _, by

choosing the origin of _ across the congruence. However

(_ _ t) = (t _ D__) t_ (since t _ Dt_ = 0) (6.9)

= B_

___t_ = (__D_ t_) t_ (6.10)

_ _ @t2 = 0 ; (6.11)

since t2 _ 􀀀1 for timelike congruences, so if _ _ t is chosen to vanish at one

value of _ it will do so for all _.

For null congruences the condition _ _ t = 0 is not su_cient to eliminate

the ambiguity in the choice of _ because

_0 _ t = (_ + at) _ t = _ _ t + at _ t (6.12)

= _ _ t (6.13)

when t2 = 0, which means that _0 _ t = 0 whenever _ _ t = 0. The problem

is that the 3-dim space of vectors orthogonal to t now includes t itself, so

the displacement vectors _ orthogonal to t specify only a two-parameter

family of geodesics. Displacement vectors to the other null geodesics in the

congruence have a component in the direction of a vector n that is not

103

orthogonal to t. The choice of n is otherwise arbitrary (it is analogous to

the choice of gauge in electrodynamics), but it is convenient to choose it

such that

n2 = 0; n _ t = 􀀀1 (6.14)

e.g. if t is tangent to an outgoing radial null geodesic, then n is tangent to

an ingoing one.

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

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

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

outgoing radial

null geodesic

t =

(1;􀀀1)

n =

(1; 1)

Consistency of the choice of n requires that n2 and n _ t be independent of

_, which is satis_ed if

t _Dn_ = 0 (6.15)

i.e. we choose n to be parallely-transported along the geodesics.

Having made a choice of the vector n, we may now uniquely specify a twoparameter

subset of geodesics of a null geodesic congruence by displacement

vectors _ orthogonal to t by requiring them to also satisfy

_ _ n = 0 (6.16)

The vectors _ now span a two-dimensional subspace, T?, of the tangent

space, that is orthogonal to both t and n, i.e. P_ = _, where

_ = __

_ + n_t_ + t_n_ (6.17)

projects onto T?.

104

Proposition P_ = _ ) t _ D__ = ^B_

___, where

_ = P_

_B_

_P_

_ (6.18)

i.e. if _ 2 T? initially, it remains in this subspace.

Proof

t _ D__ = t _ D(P_

___) (if P_ = _) (6.19)

= P_

_t _ D__ (since t _ Dn = t _ Dt = 0) (6.20)

= P_

_B_

___ (by de_nition) (6.21)

= P_

_B_

_P_

___ (since P_ = _) (6.22)

= ^B _

___ 􀀀 : (6.23)

is e_ectively a 2_2 matrix. We now decompose it into its algebraically

irreducible parts

_ =

_P_

_ + ^__

_ + ^!_

_ (6.24)

where

_ = ^B _

_ (trace) expansion

^___ = ^B(__) 􀀀 1

2P__ ^B_

_ (symmetric, traceless) shear

^!__ = ^B[__] (anti-symmetric) twist

Notation:

(__) =

__ + ^B__

[__] =

__ 􀀀 ^B__

Lemma t[_

__] = t[_B__]

Proof Using t _ Dt = 0 and t2 = 0, we have

_ = B_

_ + t_

n_B_

_ + n_B_

_n_t_

􀀀

_n__

t_ (6.25)

Hence result. ([ ] indicates total anti-symmetrization on enclosed indices).

Proposition The tangents t are normal to a family of null hypersurfaces i_ ^! = 0.

105

Proof If ^! = 0, then

0 = t[_ ^!__] _ t[_

__] (6.26)

= t[_B__] (by Lemma) (6.27)

= t[_D_t_] (6.28)

so t is normal to a family of hypersurfaces by Frobenius' theorem. (In this

case we can take t = l).

Conversely, if t is normal to a family of null hypersurfaces, then Frobenius'

theorem implies t[_D_t_] = 0. Then, reversing the previous steps we

_nd that,

0 = t[_ ^!__] =

(t_ ^!__ + t_ ^!__ + t_ ^!__) (6.29)

Contract with n. Since n _ t = 􀀀1 and n^! = ^!n = 0 (because ^! contains

the projection operator P), we deduce that ^! = 0.

If ^! = 0 we have a family of null hypersurfaces. The family is parameterized

by the displacement along n

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

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

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

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

n t

family of

null hypersurfaces

6.1.1 Expansion and Shear

Two linearly independent vectors _(1) and _(2) orthogonal to n and t determine

an area element of T?. The shear ^_ determines the change of shape of

this area element as _ increases. The magnitude of the area element de_ned

by _(1) and _(2) is

a = "____ t____(1)

_ _(2)

_ (6.30)

Since t _ Dt = 0 and t _ Dn = 0, we have

= t _ @a = t _ Da = "____t_n_

t _ D_(1)

_ _(2)

_ + _(1)

_ t _D_(2)

(6.31)

106

= "____ t_n_

_ _(1)

_ _(2)

_ + _(1)

_ _(2)

(6.32)

= 2"____t_n_ ^B _

_ _(1)

[_ _(2)

_] (6.33)

= _a (see Question IV.2) (6.34)

i.e. _ measures the rate of increase of the magnitude of the area element. If

_ > 0 neighboring geodesics are diverging, if _ < 0 they are converging.

Raychaudhuri's equation for null geodesic congruences

= t _ D

􀀀

_P_

(6.35)

= P_

_t _ DB_

_ (since t _ Dt = 0 and t _ Dn = 0) (6.36)

= P_

_t_D_D_t_ (6.37)

= P_

_t_D_D_t_ + P_

_t_ [D_;D_] t_ (6.38)

= P_

4D_ (t _ Dt_) | {z }

􀀀(D_t_) (D_t_)

5 + P_

_t_R _

__ _t_ (6.39)

= 􀀀P_

_B_

_ 􀀀 t_R__t_ (using symmetries of R) (6.40)

= 􀀀P_

_B_

_P_

_B_

_ + P_

_B_

_t_n_B_

_ + P_

_B_

_n_t_B_

_ 􀀀 t_t_R__

= 􀀀^B _

_ 􀀀 t_t_R__ (using t _ Dt _ 0 and t2 _ 0) (6.41)

= 􀀀

_2 􀀀 ^___ ^___ + ^!__ ^!__ 􀀀 R__t_t_ (6.42)

This is Raychaudhuri's equation for null geodesic congruences.

Some consequences of Raychaudhuri's equation for null hypersurfaces

Proposition The expansion _ of the null geodesic generator of a null hypersurface, N,

obeys the di_erential inequality

d_ _ 􀀀

_2 (6.43)

provided the spacetime metric solves Einstein's equations G__ = 8_GT__ and T__ satis_es

the weak energy condition.

107

Proof ^_2 _ 0 because the metric in the orthogonal subspace T? (to l and

n) is positive de_nite. ^!2 _ 0 also, but this comes in with wrong sign,

however ^! = 0 for a hypersurface. Thus Raychaudhuri's equation implies

d_ _ 􀀀

_2 􀀀 R__ l_l_ (6.44)

_ 􀀀

_2 􀀀 8_gT__l_l_ (by Einstein's eq.) (6.45)

_ 􀀀

_2 by weak energy condition (6.46)

Corollary If _ = _0 < 0 at some point p on a null generator of a null hypersurface,

then _ ! 􀀀1 along within an a_ne length 2= j_0j.

Proof Let _ be the a_ne parameter, with _ = 0 at p. Now

d_ _ 􀀀

_2 ,

􀀀

_􀀀1_

2 ) _􀀀1 _

_ + constant (6.47)

where, since _ = _0 at _ = 0, the constant cannot exceed _􀀀1

0 . Thus

_􀀀1 _

_ + _􀀀1

0 ) _ _

1 + 1

2__0

(6.48)

If _0 < 0 the right-hand-side ! 􀀀1 when _ = 2= j_0j, so _ ! 􀀀1 within

that a_ne length.

Interpretation When _ < 0 neighboring geodesics are converging. The

attractive nature of gravitation (weak energy condition) then implies that

they must continue to converge to a focus or a caustic.

Proposition If N is a Killing horizon then ^B__ = 0 and

= 0 (6.49)

Proof Let _ be the Killing vector s.t. _ = fl (l _ Dl = 0) on N for some

non-zero function f. Then

__ = ^B(__) (since ^! = 0 for family of hypersurface) (6.50)

108

= P _

_ B(__)P_

_ _ P _

_ D(_l_)P_

_ (6.51)

= P _

􀀀

@(_f􀀀1_

__)P_

_ (since D(___) = 0) (6.52)

= 0 (since P_ = _P = 0) (6.53)

In particular _ = 0, everywhere on N, so d_=d_ = 0.

Corollary For Killing horizon N of _

R______jN

= 0 (6.54)

Proof Using d_=d_ = 0 and ^B__ = 0 in Raychaudhuri's equation.

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