5.4 Energy Conditions

Back

T__ satis_es the dominant energy condition if for all future-directed timelike vector _elds

v, the vector _eld

j(v) _ 􀀀v_T _

_ @_ (5.48)

is future-directed non-spacelike, or zero.

All physically reasonable matter satis_es this condition, e.g. for massless

scalar _eld _ (with T__ = @__@__ 􀀀 1

2g__(@_)2):

j_(v) = 􀀀v _ @_@__ +

1

2

v_(@_)2 (5.49)

j2(v) =

1

4

v2 􀀀

(@_)2_2

| {z }

_0

_ 0 if v2 < 0 (5.50)

so j(v) is timelike or null if v is timelike. Since v is assumed future-directed,

j(v) will be too if 􀀀v _ j > 0. Allowing for j = 0 means that we have to

prove that 􀀀v _ j _ 0. Now

􀀀 v _ j = (v _ @_)2 􀀀

1

2

v2(@_)2 (5.51)

=

1

2

(v _ @_)2 +

1

2

􀀀

􀀀v2__

@_ 􀀀

v(v _ @_)

v2

_2

(5.52)

But (v _ @_)2 _ 0 and 􀀀v2 > 0 for timelike v, so we have to prove that

_

@_ 􀀀

v(v _ @_)

v2

_2

_ 0 (5.53)

i.e. that

_

@_ 􀀀 v(v_@_)

v2

_

is spacelike or zero. This follows from

v _

_

@_ 􀀀

v(v _ @_)

v2

_

= 0 (5.54)

99

since v _V < 0 for any non-zero timelike or null vector for timelike v (choose

coordinates s.t. v = (1;~0)). So if v _V = 0 then V cannot be timelike or null.

Since 􀀀v _ j = v_v_T__ , the dominant energy condition implies that

v_v_T__ _ 0 for all timelike v. By continuity it also implies the

Weak energy condition

v_v_T__ _ 0 8 non-spacelike v (5.55)

There is also the

Strong energy condition

v_v_

_

T__ 􀀀

1

2

g__T

_

_ 0 8 non-spacelike v (5.56)

Note, Dominant 6, Strong.

The strong energy condition is needed to prove the singularity theorems,

but the dominant energy condition is the physically important one. (An

inationary universe violates the strong energy condition). For example it

is needed for the

Positive Energy Theorem (Shoen & Yau, Witten)

The ADM energy of an asymptotically-at spacetime satisfying G__ =

8_GT__ is positive semi-de_nite, and vanishes only for Minkowski spacetime

with T__ = 0, provided that

i) 9 an initially non-singular Cauchy surface (otherwiseM < 0 Schwarzschild

would be a counter-example).

ii) T__ satis_es the dominant energy condition (clearly, some condition

on T__ is necessary).

iii) Some other technical assumptions which we ignore here.

100

Навигация