Taylor M.E. Partial Differential Equations III: Nonlinear Equations
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658 18. Einstein’s Equations
In particular, @
r
. C / D 0. Thus, as in the argument following (2.56), we can
fix the t-coordinate so that C D 0, and hence the metric is again in the static
form (2.57):
(5.7) ds
2
De
.r/
dt
2
C e
.r/
dr
2
C r
2
d!
2
:
Now the right side of (5.6) is a function of r alone, so E D E.r/;thatis,the
electromagnetic field F has the form
(5.8) F D E.r/ dt ^ dr:
Then the equation d
F
F D 0 is equivalent to @
r
p
gF
01
D 0,whereg D
e
e
r
2
r
2
Dr
4
,sowehave
(5.9) E.r/ D
q
r
2
;
for some constant q. If we substitute this formula for E in (5.6), we obtain the
ODE
(5.10) r
0
.r/ D
1
q
2
r
2
e
1:
If we set .r/ D e
.r/
, this becomes
(5.11)
r
d
dr
C 1
.r/ D 1
q
2
r
2
;
a nonhomogeneous Euler equation with general solution .r/ D 1 K=r C
q
2
=r
2
,so
(5.12) e
.r/
D 1
K
r
C
Q
2
r
2
;
where we have set Q
2
D q
2
. Hence we obtain the metric
(5.13) ds
2
D
1
K
r
C
Q
2
r
2
dt
2
C
1
K
r
C
Q
2
r
2
1
dr
2
C r
2
d!
2
:
This is known as the Reissner–Nordstr
¨
om solution. It becomes the Schwarzschild
solution when Q D 0.
It remains to check that G
22
D 8T
22
, which by (5.5) is equal to E.r/
2
r
2
D
q
2
=r
2
. In this case, (2.54) yields
G
22
D
1
2
r
2
e
00
.r/ C
0
.r/ C
2
r
0
.r/
;
6. Relativistic fluids 659
so we need
(5.14)
00
.r/ C
0
.r/
2
C
2
r
0
.r/ D
2q
2
r
4
e
.r/
:
This can be obtained from (5.10) in a fashion similar to the deduction of (2.60)
from (2.58), so we can conclude that (5.8), (5.9), (5.13) is our desired spherically
symmetric solution to the Maxwell–Einstein equations (5.1).
Exercises
1. Using the method of 3 and 4, study the timelike geodesics (i.e., possible paths of
an uncharged particle in free fall) for a spacetime with the Reissner–Nordstr¨om metric
(5.13).
Exercises 2–4 deal with the Kerr–Newman metric, given in .t;r;;'/-coordinates by
(5.15) ds
2
D
A
2
dta sin
2
'd
2
C
2
A
dr
2
C
2
d'
2
C
sin
2
'
2
.r
2
Ca
2
/da dt
2
;
where
A D r
2
Kr C a
2
C Q
2
;
2
D r
2
C a
2
cos
2
':
Here, K>0;a,andQ are constants. Note that the case a D 0 gives the Reissner–
Nordstr¨om metric (5.13) while the case Q D 0 gives the Kerr metric (3.56).
2. Show that (5.15), together with
(5.16)
F D
Q
4
.r
2
a
2
cos
2
'/dr ^ .dt a sin
2
'd/
C
2Q
4
ar.cos '/.sin '/ d' ^
.r
2
C a
2
/d adt
;
provides a solution to (5.1).
3. Show that (5.15) is stationary, but not static, if a ¤ 0.
4. Study the timelike geodesics on a spacetime with the metric (5.15).
6. Relativistic fluids
In general relativity, the motion of an ideal fluid is governed by Einstein’s equation
(6.1) G
jk
D 8T
jk
;
where T
jk
has the form
(6.2) T
jk
D . C p/u
j
u
k
C pg
jk
:
660 18. Einstein’s Equations
This is the stress-energy tensor of a fluid, with 4-velocity u, satisfying hu; uiD1.
The pressure is p, and the density is , both of these quantities being measured
by an observer traveling with velocity u.
The condition that div G D 0 leads to fluid equations. In fact, a computation
gives
(6.3) div T D . C p/r
u
u C L
u
. C p/ u C . C p/.div u/u C grad p:
Note that, since hu; uiD1; u ?r
u
u. Thus we can separate (6.3) into compo-
nents orthogonal to and parallel to u and conclude that div T D 0 if and only if
(6.4)
. C p/r
u
u D….u/ grad p;
div.u/ Dp div u;
where ….u/ denotes projection orthogonal to u with respect to the Lorentz metric.
The case p D 0 is that of a dust;then(6.4) reduces to
(6.5) r
u
u D 0; div.u/ D 0:
For an isentropic fluid, the pressure p is a function of , so there is an equation of
state
(6.6) p D p./:
One can compare (6.4) with the nonrelativistic fluid equations, (5.12)–(5.13),
of Chap.16, which are, (with X
00
./ Drp=),
(6.7)
@v
@t
Cr
v
v
Drp;
@
@t
C div.v/ D 0:
This is an approximation to (6.4)wheng
jk
jk
(the Minkowski metric), u
.1; v/; jvj << 1,andjpj << .
As in the study of nonrelativistic fluids in Chaps. 16 and 17, it is of interest to
consider the vorticity of a fluid flow. First, if Qu is the 1-form corresponding to u
via the Lorentz metric, we consider the 2-form
(6.8)
Q
D d Qu:
We can express this in terms of the linear map
(6.9) „
u
W T
p
M ! T
p
M; „
u
X Dr
X
u:
6. Relativistic fluids 661
In fact,
(6.10)
Q
.X;Y/ Dhr
X
u;Yihr
Y
u;XiDh.„
u
„
u
/X; Y i:
In particular,
Q
D 0 if and only if „
u
D „
u
. Note that since hu; uiD1,wehave
0 D Xhu; uiD2hr
X
u; ui,soh„
u
X; uiD0,thatis,
(6.11) „
u
W T
p
M ! †
p
;†
p
D .u
p
/
?
:
We also define
(6.12) A
u
W †
p
! †
p
;A
u
D „
u
ˇ
ˇ
†
p
:
Part of the significance of A
u
is in determining whether the subbundle † of TM
is integrable, as shown by the following:
Lemma 6.1. The bundle † is integrable if and only if A
u
D A
u
.
Proof. If X and Y are sections of †,then
(6.13) Qu
ŒX; Y
Dd Qu.X; Y / Dh„
u
X; Y ihX; „
u
Y i;
the last identity by (6.10). By (6.11)–(6.12), we obtain
(6.14) Qu
ŒX; Y
D
˝
.A
u
A
u
/X; Y
˛
;
whenever X and Y are sections of †. The lemma follows, by Frobenius’s theorem.
It is useful to remark that the following formula holds:
(6.15) div u DTr A
u
;
whenever hu; uiD1. To see this, pick fe
j
W 1 j 3g togivealocal
orthonormal frame field for †.Wehave
(6.16) div u D
3
X
j D1
hr
e
j
u;e
j
ihr
u
u; uiD
3
X
j D1
hr
e
j
u;e
j
i;
the last identity holding since 2hr
u
u; uiDuhu; uiD0.Thisgives(6.15).
The vorticity is the vector field W , defined as follows. If ! is the volume form
on M (a 4-form), W is uniquely specified by the identity
(6.17)
W
! DQu ^d Qu:
662 18. Einstein’s Equations
Note that if we wedge both sides of (6.17) with Qu and use the anticommutator
identity ^
Qu
W
C
W
^
Qu
DhW; QuiI , we obtain
(6.18) hW; QuiD0; i.e., W
p
2 †
p
:
We can restate Lemma 6.1 in terms of the behavior of W :
Lemma 6.2. The bundle † is integrable if and only if W D 0.
Proof. By (6.13), we see that † is integrable if and only if
(6.19) d Qu.X; Y / D 0; 8 X; Y 2 †
p
;
for all p 2 M . If we pick the basis ff
0
DQu;f
1
;f
2
;f
3
g of T
p
M to be the dual
basis to fu;e
1
;e
2
;e
3
g, and write d Qu.p/ as a linear combination of f
j
^ f
k
,we
see that (6.19) holds if and only if d Qu.p/ DQu ^ ˛,forsome˛ 2 T
p
; in turn this
holds if and only if Qu ^d Qu D 0, which holds if and only if W D 0.
We can derive a “vorticity equation,” via calculations parallel to those used in
(5.21)–(5.26) of Chap.16. First, (6.3)–(6.4)imply
(6.20) . C p/r
u
Qu C.L
u
p/Qu C dp D 0:
Next, we have L
u
Qu r
u
Qu D .1=2/d hu; uiD0,so
(6.21) L
u
Qu C B Qu C dq D 0;
where, assuming p D p./,
(6.22) dq D
dp
C p
;BD
L
u
p
C p
D L
u
q:
Applying the exterior derivative to (6.21) yields
(6.23) L
u
Q
Dd.B Qu/:
We next produce an equation for L
u
.Qu ^
Q
/. If we start with
L
u
.Qu ^
Q
/ DQu ^ L
u
Q
C .L
u
Qu/ ^
Q
;
and use (6.21)–(6.23), we obtain
(6.24) L
u
.Qu ^
Q
/ Dd.q
Q
/:
6. Relativistic fluids 663
We can also produce an equation involving L
u
W . Note the following
characterization of W , equivalent to (6.17):
(6.25) .Qu ^
Q
/ ^ ˛ DhW;˛i!;
for every 1-form ˛. Beginning with
L
u
.Qu ^
Q
/ ^ ˛ D L
u
.Qu ^
Q
^ ˛/ Qu ^
Q
^L
u
˛;
and making a computation parallel to that of (5.25)–(5.26) of Chap.16, one
obtains
(6.26) L
u
W C .div u/W D#.q
Q
/;
where the vector field #.q
Q
/ is uniquely defined by
(6.27) h#.q
Q
/;˛i! D d.q
Q
/ ^ ˛;
for all 1-forms ˛. The equation (6.26) can be compared with (5.26) of Chap. 16;
the primary difference is that the right side of (5.26) is zero.
Note that d.q
Q
/ D d.Qu ^dq/ and #.q
Q
/ D #.Qu ^ dq/, so another way to
write (6.24)isas
(6.28) L
u
.Qu ^
Q
/ D d
Qu ^ dp
C p
;
and another way to write (6.26)isas
(6.29) L
u
W C .div u/W D #
Qu ^ dp
C p
:
We can produce a vorticity equation of A. Lichnerowicz as follows. Note that
(6.30) .L
u
C B/Qu D e
q
L
u
.e
q
Qu/;
so (6.21) yields
(6.31) L
u
.e
q
Qu/ C d.e
q
/ D 0;
and applying the exterior derivative gives
(6.32) L
u
D 0; D d Qw; Qw D e
q
Qu:
Note that D e
q
.
Q
C dq ^Qu/, and hence
(6.33) Qu ^ D e
q
Qu ^
Q
:
664 18. Einstein’s Equations
We can rewrite the form (6.31) of the first part of the Euler equations (6.4),
using L
u
.e
q
Qu/ D
u
d.e
q
Qu/ d.e
q
/. Thus (6.31) is equivalent to
(6.34)
u
D 0:
The second part of the Euler equations (6.4) can be rewritten as
(6.35) div u D
L
u
C p
;
which is also equivalent to
(6.36) div w D L
w
‰./;
with
(6.37) w D e
q
u;‰./D log e
2q
. C p/:
This in turn is equivalent to
(6.38) d
F
Qw D L
w
‰:
Note that if we multiply (6.34)bye
q
and then apply the exterior derivative, we
obtain the following variant of (6.32):
(6.39) L
w
D 0:
One relation between Qu ^
Q
and is given in (6.33). If the Euler equation, in
the form (6.34), holds, we can deduce another relation, via the anticommutator
relation ^
Qu
u
C
u
^
Qu
DI . Applying this to and using (6.33), we obtain
(6.40)
u
D 0 H) De
q
u
.Qu ^
Q
/:
Putting (6.33)and(6.40) together, we get
(6.41) D 0 ”Qu ^
Q
D 0;
whenever satisfies (6.34). This enables us to prove the following:
Proposition 6.3. Assume u solves the relativistic Euler equation. Let S be a
spacelike hypersurface, and assume the vorticity W vanishes on O S.Then
W vanishes on the union U of the integral curves of u through points in O.
Proof. That vanishes on U follows from (6.41)and(6.39); applying (6.41)
again, we have Qu ^
Q
D 0 on U, hence W D 0 on U .
6. Relativistic fluids 665
Note incidentally, that, when
u
D 0,
(6.42) D e
q
W
u
!:
Next we will derive a second-order PDE for Qw.Todothis,weuse(6.32)and
(6.38) to compute Qw,where Ddd
F
d
F
d .Wehave
(6.43) Qw Dd.L
w
‰/ d
F
:
It is convenient to write
(6.44) ‰ D ˆ.e
2q
/; e
2q
Dhw; wi;
so
(6.45) L
w
‰ D2ˆ
0
.hw; wi/ hr
w
w; wi:
Since
˝
X; dhw; wi
˛
D Xhw; wiD2hr
X
w; wi,wehave
(6.46) d hw; wiD2
w
rQw:
Similarly,
(6.47) d hr
w
w; wiD
w
r.r
w
Qw/ C
.r
w
w/
rQw;
so
(6.48) d.L
w
‰/ D2ˆ
0
w
r.r
w
Qw/ C A.w; rw/;
where
(6.49) A.w; rw/ D2ˆ
0
.r
w
w/
rQw C 4ˆ
00
hr
w
w; wi
w
rQw:
Hence we have the coupled system
(6.50) Qw 2ˆ
0
w
r.r
w
Qw/ C A.w; rw/ Dd
F
; L
w
D 0:
A computation using (6.44), (6.37), and (6.22)gives
(6.51) ˆ
0
D
0
.p/ 1
2e
2q
:
In component notation, Q D
w
r.r
w
Qw/ is given by
(6.52) Q
`
D .w
j
w
k
w
j Ik
/
I`
D w
j
w
k
w
j IkI`
C R
`
;
666 18. Einstein’s Equations
where R
`
involves only first-order derivatives. Now
(6.53)
w
D 0 H) w
j
w
j I`
D w
j
w
`Ij
:
Using this, one sees that
(6.54) Q
`
D w
j
w
k
w
`Ij Ik
C
e
R
`
;
where
e
R
`
involves only first-order derivatives. Thus, we replace the system
(6.50)by
(6.55) Qw 2ˆ
0
r
2
w;w
Qw C
e
A.w; rw/ Dd
F
; L
w
D 0:
The left side of the first equation in (6.55) is a second-order, quasi-linear
operator acting on Qw; its principal symbol is scalar, and provided
0
.p/ 1 (i.e.,
provided p
0
./ 1), it is hyperbolic, and every hypersurface that is spacelike
for the Lorentz metric g
jk
is also spacelike for this operator. Of course, we have
d
F
on the right, and a second equation involving w and .SinceL
w
in-
volves first-order derivatives of w as well as of , the question of well-posedness
of the initial-value problem for (6.55) requires further investigation. Following
[CBr3], we clarify this by applying r
w
to both sides of the first equation.
Since the operator r
w
has scalar principal symbol,
(6.56) r
w
d
F
D d
F
r
w
C B
0
.w; rw; r/:
Meanwhile,
(6.57) .r
w
/.X; Y / D .L
w
/.X; Y / .r
X
w; Y / .X; r
Y
w/;
so
(6.58) L
w
D 0 H) r
w
d
F
D B.D
2
w; r/:
Thus we replace the system (6.55)by
(6.59) r
w
. 2ˆ
0
r
2
w;w
/ Qw
e
B.D
2
w; r/ D 0; L
w
D 0;
which is analytically more tractable. Note that the first equation here contains no
higher derivatives on than the first equation of (6.55).
Now the fluid velocity is also coupled to the gravitational field, via (6.1)–(6.2),
so all of these equations have to be treated simultaneously. We will discuss this
further in 8. In preparation for that, let us mention that the first equation in (6.59),
when written in local coordinates, involves the metric tensor and derivatives of the
metric tensor, up to third order.
6. Relativistic fluids 667
We next construct some examples of static, spherically symmetric solutions
to (6.1)–(6.2), which provide models for stable stars. We look for a solution
involving a metric of the form
(6.60) ds
2
De
.r/
dt
2
C e
.r/
dr
2
C r
2
d!
2
;
and use x
0
D t; x
1
D r,asin 2. For the fluid to be static, we need
(6.61) u
0
D e
=2
; u
1
D u
2
D u
3
D 0;
so
(6.62) T
jk
D . C p/e
ı
j0
ı
k0
C pg
jk
:
Using (2.51)–(2.55), we see that (6.1) is equivalent to the following set of equa-
tions, recording G
jj
D 8T
jj
for j D 0; 1,and2, respectively:
(6.63)
1 r
r
e
D8r
2
e
;
1 C r
r
e
D 8pr
2
e
;
rr
C
1
2
2
r
C
1
r
.
r
r
/
1
2
r
r
D 16p e
:
If we assume p and are related by an equation of state, p D p./,asin(6.6),
then (6.63) is a system of three equations, in three unknowns: .r/;.r/,and
p.r/. The system can be simplified a bit.
If we apply e
.d=dr/e
to the middle equation in (6.63) and subtract r times
the last equation, we get
(6.64)
0
.
0
C
0
/ D16re
p
0
:
Meanwhile, taking the difference of the first two equations in (6.63)gives
(6.65)
0
.
0
C
0
/ D 8re
.p C /
0
:
Comparing these two equations, we have
(6.66) p
0
.r/ D
1
2
.p C /
0
.r/:
It is instructive to rederive this last equation, as a consequence of the vanishing
of T
jk
Ik
.Infact,ifu is given by (6.61), then div u D 0,soby(6.3) the vanishing
of div T is equivalent to
(6.67) . C p/r
u
u C L
u
. C p/ u C grad p D 0: