Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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so -function sources will in general be present at a

Lipschitz continuous matching of metrics), and

guarantee that the matched metric solves the Einstein

equations in the weak sense. The Lipschitz matching

of the metrics, together with the conservation

constraint, leads to a system of ordinary differential

equations (ODEs) that determine the shock position,

together with the TOV density and pressure at the

shock. Since the TOV metric depends only on

r,the

equations thus determine the TOV spacetime beyond

the shock wave. To obtain a physically meaningful

outgoing shock wave, we impose the constriant

p 





to ensure that the equation of state on the TOV side

of the shock is physically reasonable, and as the

entropy condition we impose the condition that the

shock be compressive. For an outgoing shock wave,

this is the condition >



, p >

p, that the pressure

and density be larger on the side of the shock that

receives the mass flux – the FRW side when the

shock wave is propagating away from the FRW

center. This condition breaks the time-reversal sym-

metry of the equations, and is sufficient to rule out

rarefaction shocks in classical gas dynamics (Smoller

1983, Smoller and Temple 2003). The ODEs,

together with the equation-of-state bound and the

conservation and entropy constraints, determine a

unique solution of the ODEs for every 0 < 1and



 0, and this provides the two-parameter family of

solutions discussed here (Smoller and Temple 1995,

2003). The Lipschitz matching of the metrics implies

that the total mass M is continuous across the

interface,andsowhenr



> 0, the total mass of the

entire solution, inside and outside the shock wave, is

finite at each time t > 0, and both the FRW and

TOV spacetimes emerge at the big bang. The total

mass M on the FRW side of the shock has the

meaning of total mass inside the radius

r at fixed

time, but on the TOV side of the shock, M does not

evolve according to equations that give it the

interpretation as a total mass because the metric is

inside the black hole. Nevertheless, after the space-

time emerges from the black hole, the total mass

takes on its usual meaning outside the black

hole, and time asymptotically the big bang ends

with an expansion of finite total mass in the usual

sense. Thus, when r



> 0, our shock wave refine-

ment of the FRW metric leads to a big bang of

finite total mass.

A final comment is in order regarding our overall

philosophy. The family of exact shock wave solutions

described here are rough models in the sense that

the equation of state on the FRW side satisfies the

condition  = const., and the equation of state on the

TOV side is determined by the equations, and

therefore cannot be imposed. Nevertheless, the

bounds on the equations of state imply that the

equations of state are qualitatively reasonable, and

we expect that this family of solutions will capture

the gross dynamics of solutions when more general

equations of state are imposed. For more general

equations of state, other waves, such as rarefaction

waves and entropy waves, would need to be present

to meet the conservation constraint, and thereby

mediate the transition across the shock wave. Such

transitional waves would be very difficult to model in

an exact solution. But, the fact that we can find

global solutions that meet our physical bounds, and

that are qualitatively the same for all values of  2

(0,1] and all initial shock positions, strongly suggests

that such a shock wave would be the dominant wave

in a large class of problems.

In the next section, the FRW solution is derived

for the case  = const., and the Hubble length is

discussed as a critical length scale. Subsequently,

the general theorems in Smoller and Temple (1994)

for matching gravitational metrics across shock

waves are employed. This is followed by a discus-

sion of the constr uction of the family of solutions in

the case r



= 0. Finally, the case r



> 0 is discussed.

(Details can be found in Smoller and Temple (1995,

2003, 2004).)

The FRW Metric

According to Einstein’s theory of general relativity,

all properties of the gravitational field are deter-

mined by a Lorentzian spacetime metric tensor g,

whose line element in a given coordinate system

x = (x

, ..., x

) is given by

¼ g

½4

(We use the Einstein summation convention,

whereby repeated up–down indices are assumed

summed from 0 to 3.) The components g

of the

gravitational metric g satisfy the Einstein equations

¼ T

; T

¼ðc

þ pÞw

þ pg

½5

where we assume that the stress-energy tensor T

corresponds to that of a perfect fluid. Here G is the

Einstein curvature tensor,

 ¼

8G

½6

is the coupling constant, G is Newton’s gravitational

constant, c is the speed of light, c

is the energy

density, p is the pr essure, and w = (w

, ..., w

) are

the components of the 4-velocity of the fluid (cf.

Weinberg 1972), and again we use the convention

that c = 1 and G= 1 when convenient.

562 Shock Wave Refinement of the Friedman–Robertson–Walker Metric

Putting the metric ansatz [1] into the Einstein

equations [5] gives the equations for the FRW metric

(Weinberg 1972),



 

½7

and

 ¼3ðp þ ÞH ½8

The unknown quantities R, ,andp are assumed to

be functions of the FRW coordinate time t alone, and

the ‘‘dot’’ denotes differentiation with respect to t.

To verify that the Hubble lengt h

crit

= 1=H is the

limit for FRW–TOV shock matching outside a black

hole, write the FRW metric [1] in standard

Schwarzschild coordinates x = (

t), where the

metric takes the form

¼Bð



tÞd



þ Að



tÞ

1



d

½9

and the mass function M(

t) is defi ned through the

relation

A ¼ 1 



½10

It is well known that a general spherically symmetric

metric can be transformed to the form [9] by

coordinate transformation (see Weinberg (1972) and

Groah and Temple (2004)). Substituting

r = Rr into

[1] and diagonalizing the resulting metric, we obtain

(see Smoller and Temple (2004) for details)

¼

1  kr

 H







1  kr

 H







d

½11

where is an integrating factor that solves the

equation



1  kr

 H



1  kr





1  kr



¼ 0 ½12

and the time coordinate

t =

t(t,

r) is defined by the

exact differential



t ¼

1  kr

 H



1  kr



dt þ



1  kr





r ½13

Now using [10] in [7], it follows that

Mðt ;



rÞ¼





ðtÞ s

ds ¼







½14

Since in the FRW metric,

r = Rr measures arclength

along radial geodesics at fixed time, we see from

[14] that M(t,

r) has the physical interpretation as

the total mass inside radius

r at time t in the FRW

metric. Restricting to the case of critical expansion

k = 0, we see from [7], [14],and[13] that

r = H

1

equivalent to 2M=

r = 1, and so at fixed time t, the

following equivalences are valid:



r ¼ H

1

iff



¼ 1 iff A ¼ 0 ½15

We conclude that

r = H

1

is the critical length scale

for the FRW metric at fixed time t in the sense that

A = 1  2M=

r changes sign at

r = H

1

, and so the

universe lies inside a black hole beyond

r = H

1

,as

claimed above. Now, we proved in Smoller and

Temple (1998) that the standard TOV metric out-

side the black hole cannot be continued into A = 0

except in the very special case  = 0. (It takes an

infinite pressure to hold up a static configuration at

the event horizon of a black hole.) Thus, shock

matching beyond one Hubble length requires a

metric of a different character, and for this purpose,

we introduce the TOV metric inside the black hole –

a metric of TOV form, with A < 0, whose fluid is

comoving with the timelike radia l coordinate

r (Smoller and Temple 2004).

The Hubble length

crit

= c=H is also the critical

distance at which the outward expansion of the FRW

metric exactly cancels the inward advance of a radial

light ray impinging on an observer positioned at the

origin of a k = 0 FRW metric. Indeed, by [1], a light

ray traveling radially inward toward the center of an

FRW coordinate system satisfies the condition

¼ R

½16

so that



Rr þ R

r ¼ H



r  c ¼ H



r 



> 0 ½17

if and only if



r >

Thus, the arclength distance from the origin to an

inward moving light ray at fixed time t in a k = 0

FRW metric will actually increase as long as the light

ray lies beyond the Hubble length. An inward moving

light ray will, however, eventually cross the Hubble

length and reach the origin in finite proper time, due

to the increase in the Hubble length with time.

We now calculate the infinite redshift limit in terms

of the Hubble length. It is well known that light emitted

at (t

, r

) at wavelength 

in an FRW spacetime will be

observed at (t

, r

) at wavelength 



Shock Wave Refinement of the Friedman–Robertson–Walker Metric 563

Moreover, the redshift factor z is defined by

z ¼



 1

Thus, infinite redshifting occurs in the limit R

! 0,

where R = 0, t = 0 is the big bang. Consider now a

light ray emitted at the instant of the big bang, and

observed at the FRW origin at present time t = t

Let r

denote the FRW coordinate at time t ! 0of

the furthest objects that can be observed at the FRW

origin before time t = t

.Thenr

marks the position

of objects at time t = 0 whose radiation would be

observed as infinitly redshifted (assuming no scatter-

ing). Note then that a shock wave emanating from

r = 0 at the instant of the big bang, will be observed at

the FRW origin before present time t = t

only if its

position r at the instant of the big bang satisfies the

condition r < r

.Toestimater

, note first that from

[16] it follows that an incoming radial light ray in an

FRW metric follows a lightlike trajectory r = r(t)if

r  r

¼

d

RðÞ

and thus

d

RðÞ

½18

Using this, the following theorem can be proved

(Smoller and Temple 2004).

Theorem 1 If the pressure p satisfies the bounds

0  p 

 ½19

then, for any equation of state, the age of the

universe t

and the infinite red shift limit r

are

bounded in terms of the Hubble length by

 t



½20

 r



½21

(We have assumed in Theorem 1 that R = 0 when

t = 0 and R = 1 when t = t

, H = H

The next theorem gives closed-form solutions of

the FRW equations [7], [8] in the case when

 = const. As a special case, we recover the bounds

in [20] and [21] from the cases  = 0 and 1/3.

Theorem 2 Assume k = 0 and the equation of state

p ¼  ½22

where  is taken to be constant,

0    1

then (assuming an expanding universe

R > 0), the

solution of system [7], [8] satisfying R = 0 at t = 0

and R = 1 at t = t

is given by

 ¼

3ð1 þ Þ

½23

R ¼



2=½3ð1þÞ

½24

½25

Moreover, the age of the universe t

and the infinite

red shift limit r

are given exactly in terms of the

Hubble length by

3ð1 þ Þ

½26

1 þ 3

½27

From [27] we conclude that a shock wave will be

observed at the FRW origin before present time

t = t

only if its position r at the instant of the big

bang satisfies the condition

r <

1 þ 3

Note that r

ranges from one-half to two Hubble

lengths as  ranges from 1 to 0, taking the

intermediate value of one Hubble length at  = 1=3

(cf. [21]).

Note that using [23] and [24] in [14], it follows

that

M ¼





ðtÞ s



9ð1 þ Þ

2=ð1þÞ

2=ð1þÞ

½28

M < 0if>0. It follows that if p = ,

 = const. > 0, then the total mass inside radius

r = const. decreases in time.

The General Theory of Shock Matching

The matching of the FRW and TOV metrics in the next

two sections is based on the following theorems that

were derived in Smoller and Temple (1994) (Theorems

3 and 4 apply to non-lightlike shock surfaces. The

lightlike case was discussed by Scott (2002).)

Theorem 3 Let  denote a smooth, three-dimen-

sional shock surface in spacetime with spacelike

564 Shock Wave Refinement of the Friedman–Robertson–Walker Metric

normal vector n relative to the spacetime metric g;

let K denote the second fundamental form on ; and

let G denote the Einstein curvature tensor. Assume

that the components g

of the gravitational metric g

are smooth on either side of  (continuous up to the

boundary on either side separately), and Lipschitz

continuous across  in some fixed coordinate

system. Then the following statements are

equivalent:

(i)[K] = 0 at each point of .

(ii) The curvature tensors R

jkl

and G

, viewed as

second-order operators on the metric compo-

nents g

, produce no -function sources on .

(iii) For each poin t P 2 , there exists a C

1,1

coordinate transformation defined in a neigh-

borhood of P, such that, in the new coordinates

(which can be taken to be the Gaussian normal

coordinates for the surface), the metric compo-

nents are C

1,1

functions of these coordinates.

(iv) For each P 2 , there exists a coordinate frame

that is locally Lorentzian at P, and can be

reached within the class of C

1,1

coordinate

transformations.

Moreover, if any one of these equivalencies hold,

then the Rankine–Hugoniot jump conditions,

[G]



= 0(which express the weak form of con-

servation of energy and momentum across  when

G = T), hold at each point on .

Here [f] denotes the jump in the quantity f across

 (this being determined by the metric separately on

each side of  because g

is only Lipsc hitz

continuous across ), and by C

1,1

we mean that

the first derivatives are Lipsch itz continuous.

In the case of spherical symmetry, the following

stronger result holds. In this case, the jump condi-

tions [G

= 0, which express the weak form of

conservation across a shock surface, are implied by a

single condition [G

= 0, so long as the shock is

non-null, and the areas of the spheres of symmetry

match smoothly at the shock and change mono-

tonically as the shock evolves. Note that, in genera l,

assuming that the angular variables are identified

across the shock, we expect conservation to entail

two conditions, one for the time and one for the

radial components. The fact that the smooth

matching of the spheres of symmetry reduces

conservation to one condition can be interpreted as

an instance of the general principle that directions of

smoothness in the metric imply directions of

conservation of the sources.

Theorem 4 Assume that g and

g are two spheri-

cally symmetric metrics that match Lipschitz con-

tinuously across a three-dimensional shock interface

 to form the matched metric g [

g. That is, assume

that g and

g are Lorentzian metrics given by

¼aðt; rÞdt

þ bðt; rÞdr

þ cðt; rÞd

½29

and



¼



að



rÞd



bð



rÞd



cð



rÞd

½30

and that there exists a smooth coordinate transforma-

tion  :(t, r) ! (

r), defined in a neighborhood of a

shock surface  given by r = r(t), such that the metrics

agree on .(We implicitly assume that  and ’ are

continuous across the surface.) Assume that

cðt; rÞ¼



cððt; rÞÞ ½31

in an open neighborhood of the shock surface , so

that, in particular, the areas of the 2-spheres of

symmetry in the barred and unbarred metrics agree

on the shock surface. Assume also that the shock

surface r = r(t) in unbarred coordinates is mapped to

the surface

r =

t) by (

t)) =(t, r

(t)). Assume,

finally, that the normal n to  is non-null, and that

nðcÞ 6¼ 0 ½32

where n(c) denotes the derivative of the function c in

the direction of the vector n. Then the following are

equivalent to the statement that the components of

the metric g [

g in any Gaussian normal coordinate

system are C

1,1

functions of these coordinates across

the surface :

½G

n

¼ 0 ½33

½G

n

¼ 0 ½34

½K¼0 ½35

Here again,[f ] =

f  f denotes the jump in the

quantity f across , and K is the second fundamental

form on the shock surface.

We assume in Theorem 4 that the areas of the

2-spheres of symmetry change monotonically in the

direction normal to the surface. For example, if

c = r

, then @c=@t = 0, so the assumption n(c) 6¼ 0is

valid except when n = @=@t, in which case the rays

of the shock surface would be spacelike. Thus, the

shock speed would be faster than the speed of light

if our assumption n(c) 6¼ 0 failed in the case c = r

FRW–TOV Shock Matching Outside the

Black Hole – The Case r



= 0

To construct the family of shock wave solutions for

parameter values 0 < 1 and r



= 0, we match

the exact solution [23]–[25] of the FRW met ric [1]

to the TOV metric [2] outside the black hole,

Shock Wave Refinement of the Friedman–Robertson–Walker Metric 565

assuming A > 0. In this case, we can bypass the

problem of deriving and solving the ODEs for the

shock surface and constraints discussed above, by

actually deriving the exact solution of the Einstein

equations of TOV form that meets these equations.

This exact solution represents the general-relativistic

version of a static, singular isothermal sphere –

singular because it has an inverse square density

profile, and isothermal because the relationship

between the density and pressure is

p =



,  = const.

Assuming the stress tensor for a perfect fluid, and

assuming that the density and pressure depend only

r, the Einstein equations for the TOV metric [2]

outside the black hole (i.e., when A = 1 2M=

r > 0)

are equivalent to the Oppenheimer–Volkoff system



¼ 4



 ½36



p ¼GM



 1 þ







 1 þ

4





1 

2GM





1

½37

Integrating [36], we obtain the usual interpretation

of M as the total mass inside radius

Mð



rÞ¼



4



ðÞd ½38

The metric component B B(

r) is determined from





and M through the equation



rÞ

¼2



rÞ



p þ





½39

Assuming



p ¼ 



;



ð



rÞ¼





½40

for some constants  and , and substituting into

[3], we obtain

Mð



rÞ¼4



r ½41

Putting [40] and [41] into [37] and simplifying yields

the identity

 ¼

2G



1 þ 6 þ 



½42

From [38] we obtain

A ¼ 1  8G<1 ½43

Applying [39] leads to

B ¼ B











2=ð1þÞ

¼ B





4=ð1þÞ

½44

By resc aling the time coordinate, we can take B

= 1

= 1, in which case [44] reduces to

B ¼



4=ð1þÞ

½45

We conclude that when [42] holds, [40]–[43] and

[44] provide an exact solution of the Einstein field

equations of TOV type, for each 0    1. (In this

case, an exact solution of TOV type was first found

by Tolman (1939), and rediscovered in the case

 = 1=3 by Misner and Zapolsky (cf. Weinberg

(1972 p. 320)).) By [43], these solutions are defined

outside the black hole, since 2M=

r < 1. When

 = 1=3, [42] yields  = 3=56G (cf. Weinberg

(1972, equation (11.4.13))).

To match the FRW exact solution [23]–[25] with

equation of state p =  to the TOV exact solution

[40]–[45] with equation of state

p =



 across a

shock interface, we first set

r = Rr to match the

spheres of symmetry, and then match the timelike

and spacelike components of the corresponding

metrics in standard Schwarzschild coordinates. The

matching of the d

coefficient A

1

yields the

conservation of mass condition that implicitly gives

the shock surface

r =

r(t),

Mð



rÞ¼

4

ðtÞ



½46

Using this together with [41] gives the following two

relations that hold at the shock surface:



r ¼

ﬃﬃﬃﬃﬃﬃﬃﬃ

3

ðtÞ

 ¼

4



rðtÞ

3



rðtÞ

¼ 3



 ½47

Matching the coefficient B of d

on the shock

surface determines the integrating factor in a

neighborhood of the shock surface by assigning

initial conditions for [44]. Finally, the conservation

constraint [T

= 0 leads to the single condition

0 ¼ð1 AÞð þ



pÞðp þ



Þ

þ 1 





 þ



pÞð þpÞ

þðp 



pÞð 



Þ

½48

which upon using p =  and

p =



 is satisfied

assuming the condition

 ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

9

þ 54 þ 49



 

 HðÞ½49

Alternatively, we can solve for  in [49] and write

this relation as

 ¼

ð þ 7Þ

3ð1  Þ

½50

566 Shock Wave Refinement of the Friedman–Robertson–Walker Metric

This guarantees that conservation holds across the

shock surface, and so it follows from Theorem 4 that

all of the equivalencies in Theorem 3 hold across the

shock surface. Note that H(0) = 0, and to leading

order  = (3/7) þ O(

)as ! 0. Within the

physical region 0  ,   1, H

() > 0, <,and

H(1=3) =

ﬃﬃﬃﬃﬃﬃ

 4  0.1231, H(1) =

ﬃﬃﬃﬃﬃﬃﬃﬃ

112

=2  5 

0.2915.

Using the exact formulas for the FRW metric in

[23]–[25], and setting R

= 1at = 

, t = t

,we

obtain the following exact formulas for the shock

position:



rðtÞ¼t ½51

rðtÞ¼



rðtÞ RðtÞ

1

¼ t

ð1þ3Þ=ð3þ3Þ

½52

where

 ¼ 3ð1 þ Þ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



1 þ 6 þ 

 ¼ 

ð1þ3Þ=ð3þ3Þ

3





1=ð3þ3Þ

½53

It follows from [41] that A > 0, and from [52] that



= lim

t!0

r(t) = 0. The entropy condition that the

shock wave be compressive follows from the fact

that  = H() <. Thus, we conclude that for each

0 < 1, r



= 0, the solutions constructed in

[40]–[53] define a one-parameter family of shock

wave solutions that evolve everywhere outside

the black hole, which implies that the distance

from the shock wave to the FRW center is less than

one Hubble length for all t > 0.

Using [51] and [52 ], one can determine the shock

speed, and check when the Lax characteristic

condition (Smoller 1983) holds at the shock. The

result is the following theorem. (Note that even

when the shock speed is larger than c, only the

wave, and not the sound speeds or any other

physical motion, exceeds the speed of light. See Scott

(2002) for the case when the shock speed is equal to the

speed of light.) The reader is referred to Smoller and

Temple (1995) for details.

Theorem 5 There exist values 0 <

<

< 1,

(

 0.458, 

ﬃﬃﬃ

=3  0.745), such that, for

0 < 1, the Lax characteristic condition holds at

the shock if and only if 0 <<

; and the shock

speed is less than the speed of light if and only if

0 <<

The explicit solution in the case r



= 0 can be

interpreted as a general-relativistic versio n of a

shock wave explosion into a static, singular,

isothermal sphere, known in the Newtonian case as

a simple model for star formation (Smoller and

Temple 2000). As the scenario goes, a star begins as

a diffuse cloud of gas. The cloud slowly contracts

under its own gravitational force by radiating energy

out through the gas cloud as gravitational potential

energy is converted into kinetic energy. This

contraction continues until the gas cloud reaches

the point where the mean free path for transmission

of light is small enough that light is scattered,

instead of being transmitted, through the cloud. The

scattering of light within the gas cloud has the effect

of equalizing the temperature within the cloud, and

at this point the gas begins to drift toward the most

compact configuration of the density that balances

the pressure when the equation of state is isother-

mal. This configuration is a static, singular, iso-

thermal sphere, the general-relativistic version of

which is the exact TOV solution beyond the shock

wave when r



= 0. This solution in the Newtonian

case is also inver se square in the density and

pressure, and so the density tends to infinity at the

center of the sphere. Eventually, the high densities at

the center ingnite thermonuclear reactions. The

result is a shock wave explosion emanating from

the center of the sphere, and this signifies the birth

of the star. The exact solutions when r



= 0

represent a general-relativistic version of such a

shock wave explosion.

Shock Wave Solutions Inside the Black

Hole – The Case r



> 0

When the shock wave is beyond one Hubble length

from the FRW center, we obtain a family of shock

wave solutions for each 0 < 1 and r



> 0by

shock matching the FRW metric [1] to a TOV

metric of form [2] under the assumpti on that

Að



rÞ¼1 

2Mð



rÞ



 1  Nð



rÞ < 0 ½54

In this case,

r is the timelike variable. Assuming that

the stress tensor T is taken to be that of a perfect

fluid comoving with the TOV metric, the Einstein

equations G = T, inside the black hole, take the

form (see Smoller and Temple (2004) for details)



p þ





N  1

½55

¼



þ 





½56

¼

N  1



þ 







½57

Shock Wave Refinement of the Friedman–Robertson–Walker Metric 567

The system [55]–[57] defines the simplest class of

gravitational metrics that contain matter, evolve

inside the black hole, and such that the mass function

r) < 1 at each fixed time

r. System [55]–[57] for

A < 0 differs substantially from the TOV equations

for A > 0 because, for example, the energy density

is equated with the timelike component G

when

A < 0, but with G

when A > 0. In particular, this

implies that, inside the black hole, the mass function

r) does not have the interpretation as a total mass

inside the radius

r as it does outside the black hole.

Equations [56], [57] do not have the same

character as [54], [55] and the relation

p =



 with

 = const. is inconsistent with [56], [57] together with

the conservation constraint and the FRW assumption

p =  for shock matching. Thus, instead of looking

for an exact solution of [56], [57] ahead of time, as in

the case r



= 0, we assume the FRW solution [23]–

[25], and derive the ODEs that describe the TOV

metrics that match this FRW metric Lipschitz-

continuously across a shock surface, and then impose

the conservation, entropy, and equation of state

constraints at the end. Matching a given k = 0 FRW

metric to a TOV metric inside the black hole across a

shock interface leads to the system of ODEs, (see

Smoller and Temple (2004) for details),

¼

ð1 þ uÞ

2ð1 þ 3uÞN





ð3u  1Þð  uÞN þ 6uð1 þ uÞ

ð  uÞN þð1 þ uÞ



½58



¼

1 þ 3u



½59

with conservation constraint

v ¼

ð1 þ uÞþð  uÞN

ð1 þ uÞþð  uÞN

½60

where

u ¼





; v ¼





;¼



½61

Here  and p denote the (known) FRW density and

pressure, and all variables are evaluated at the

shock. Solutions of [58]–[60] determine the

(unknown) TOV metrics that match the given

FRW metric Lipschitz-continuously across a shock

interface, such that conservation of energy and

momentum hold across the shock, and such that

there are no -function sources at the shock (Israel

1966, Smoller and Temple 1997). Note that the

dependence of [58]–[60] on the FRW metric is only

through the variable , and so the advantage of

taking  = const. is that the whole solution is

determined by the inhomogeneous scalar equation

[58] when  = const. We take as the entropy

constraint the condition that

0 <



p < p; 0 <



< ½62

and to insure a physically reasonable solution, we

impose the equation of state constriant on the TOV

side of the shock (this is equivalent to the dominant

energy condition (Blau and Guth 1987))

0 <



p <



 ½63

Condition [62] implies that outgoing shock waves

are compressive. Inequalities [62] and [63] are both

implied by the single condition (Smoller and Temple

2004),

1  u

1 þ u



  u

 þ u



½64

Since  is constant, eqn [58] uncouples from [59],

and thus solutions of system [58]–[60] are deter-

mined by the scalar nonautonomous equation [58].

Making the change of vari able S = 1=N, which

transforms the ‘‘big bang’’ N !1 over to a rest

point at S ! 0, we obtain

ð1 þuÞ

2ð1 þ3uÞS





ð3u1ÞðuÞþ6uð1þuÞS

ð  uÞþð1þuÞS



½65

Note that the conditions N > 1and0<

p < p

restrict the domain of [65] to the region 0 < u <

<1, 0 < S < 1. The next theorem gives the exis-

tence of solutions for 0 < 1, r



> 0, inside the

black hole (Smoller and Temple 2003).

Theorem 6 For every ,0<<1, there exists a

unique solution u



(S) of [65], such that [64] holds

on the solution for all S,0 < S < 1, and on this

solution,0< u



(S) <

u, lim

S!0



(S) =

u, where



u ¼ Min 1=3;

½66

and

lim

S!1



p ¼ 0 ¼ lim

S!1



 ½67

For each of these solutions u



(S), the shock position

is determined by the solution of [59], which in turn

is determined uniquely by an initial condition which

can be taken to be the FRW radial position of the

shock wave at the instant of the big bang,



¼ lim

S!0

rðSÞ > 0 ½68

Concerning the shock speed, we have

568 Shock Wave Refinement of the Friedman–Robertson–Walker Metric

Theorem 7 Let 0 <<1. Then the shock wave is

everywhere subluminous, that is, the shock speed



(S)  s(u



(S)) < 1 for all 0 < S  1, if and only if

  1=3.

Concerning the shock speed near the big bang

S = 0, the following is true:

Theorem 8 The shock sp eed at the big bang S = 0

is given by

lim

S!0



ðSÞ¼0;<1=3 ½69

lim

S!0



ðSÞ¼1;>1=3 ½70

lim

S!0



ðSÞ¼1;¼ 1=3 ½71

Theorem 8 shows that the equation of state

p = /3 plays a special role in the analysis when r



> 0,

and only for this equation of state does the shock

wave emerge at the big bang at a finite nonzero

speed, the speed of light. Moreover, [66] implies that

in this case, the correct relation



 = is also

achieved in the limit S ! 0. The result [67] implies

that (neglecting the pressure p at this time onward),

the solution continues to a k = 0 Oppenheimer–

Snyder solution outside the black hole for S > 1.

It follows that the shock wave will first become

visible at the FRW center

r = 0 at the moment

t = t

,(R(t

) = 1), when the Hubble length

1

= H

1

) satisfies

1 þ 3



½72

where r



is the FRW po sition of the shock at the

instant of the big bang. At this time, the number of

Hubble lengths

ﬃﬃﬃﬃﬃ

from the FRW ce nter to the

shock wave at time t = t

can be estimated by

1 

1 þ 3



ﬃﬃﬃﬃﬃ



1 þ 3

ﬃﬃﬃﬃ

3

ð1þ3Þ=ð1þÞðÞ

Thus, in particular, the shock wave will still lie

beyond the Hubble length 1=H

at the FRW time t

when it first becomes visible. Furthermore, the time

crit

> t

at which the shock wave will emerge from

the white hole given that t

is the first instant at

which the shock becomes visible at the FRW center,

can be estimated by

1 þ 3

=4



crit



1 þ 3

ﬃﬃﬃﬃ

3

=ð1þÞ

½73

for 0 < 1=3, and by the better estimate

ﬃﬃ



crit

 e

3=2

½74

in the case  = 1=3. Inequalities [73], [74] imply, for

example, that at the Oppenheimer–Sn yder limit  = 0,

ﬃﬃﬃﬃﬃﬃﬃ

¼ 2;

crit

¼ 2

and in the limit  = 1=3,

1:8 

crit

 4:5; 1 <

ﬃﬃﬃﬃﬃﬃﬃ

 4:5

We can conclude that at the moment t

when the

shock wave first becomes visible at the FRW center,

the shock wave must lie within 4.5 Hubble lengths of

the FRW center. Throughout the expansion up until

this time, the expanding universe must lie entirely

within a white hole – the universe will eventually

emerge from this white hole, but not until some later

time t

crit

,wheret

crit

does not exceed 4.5t

Conclusion

We believe that the existence of a wave at the

leading edge of the expansion of the galaxies is the

most likely possibility. The alternatives are that

either the universe of expanding galaxies goes on out

to infinity, or else the universe is not simply

connected. Although the first possibility has been

believed for most of the history of cosmology based

on the Friedmann univer se, we find this implausible

and arbitrary in light of the sho ck wave refinements

of the FRW metri c discussed here. The second

possibility, that the universe is not simply connected,

has received considerable attention recently (Klarreich

2003). However, since we have not seen, and

cannot create, any non-simply-connected 3-spaces

on any other length scale, and since there is no

observational evidence to support this, we view this

as less likely than the existence of a wave at the leading

edge of the expansion of the galaxies, left over from the

big bang. Recent analysis of the microwave back-

ground radiation data shows a cutoff in the angular

frequencies consistent with a length scale of around

one Hubble length (Andy Abrecht, private commu-

nication). This certainly makes one wonder whether

this cutoff is evidence of a wave at this length scale,

especially given the consistency of this possibility

with the case r



> 0 of the family of exact solutions

discussed here.

Acknowledgments

The work of JS was supported in part by NSF

Applied Mathematics Grant Number DMS-010-

3998, and that of BT by NSF Applied Ma thematics

Grant Number DMS-010-2493.

Shock Wave Refinement of the Friedman–Robertson–Walker Metric 569

See also: Black Hole Mechanics; Cosmology:

Mathematical Aspects; Newtonian Limit of General

Relativity; Symmetric Hyperbolic Systems and Shock

Waves.

Further Reading

Blau SK and Guth AH (1987) Inflationary cosmology. In: Hawking

SW and Israel W (eds.) Three Hundred Years of Gravitation,

pp. 524–603. Cambridge: Cambridge University Press.

Groah J, Smoller J, and Temple B (2003) Solving the Einstein

equations by Lipschitz continuous metrics: shockwaves in

general relativity. In: Friedlander S and Serre D (eds.)

Handbook of Mathematical Fluid Dynamics,vol.2,

pp. 501–597. Amsterdam: North Holland.

Groah J and Temple B (2004) Shock-Wave Solutions of the Einstein

Equations: Existence and Consistency by a Locally Inertial Glimm

Scheme. Memoirs of the AMS, 84 pages, vol. 172, no. B13.

Israel W (1966) Singular hypersurfaces and thin shells in general

relativity. IL Nuovo Cimento XLIV B(1): 1–14.

Klarreich E (2003) The shape of space. Science News 164–165.

Oppenheimer JR and Snyder JR (1939) On continued gravita-

tional contraction. Physical Review 56: 455–459.

Scott M (2002) General Relativistic Shock Waves Propagating at

the Speed of Light. Ph.D. thesis, UC-Davis.

Smoller J (1983) Shock Waves and Reaction Diffusion Equations.

New York, Berlin: Springer.

Smoller J and Temple B (1994) Shock-wave solutions of the

Einstein equations: the Oppenheimer–Snyder model of grav-

itational collapse extended to the case of non-zero pressure.

Archives for Rational and Mechanical Analysis 128: 249–297.

Smoller J and Temple B (1995) Astrophysical shock-wave solutions of

the Einstein equations. Physical Review D 51(6): 2733–2743.

Smoller J and Temple B (1997) Solutions of the Oppenheimer–

Volkoff equations inside 9/8’ths of the Schwarzschild radius.

Communications in Mathematical Physics 184: 597–617.

Smoller J and Temple B (1998) On the Oppenheimer–Volkov

equations in general relativity. Archives for Rational and

Mechanical Analysis 142: 177–191.

Smoller J and Temple B (2000) Cosmology with a shock wave.

Communications in Mathematical Physics 210: 275–308.

Smoller J and Temple B (2003) Shock wave cosmology inside a

black hole. Proceedings of the National Academy of Sciences

of the United States of America 100(20): 11216–11218.

Smoller J and Temple B (2004) Cosmology, black holes, and

shock waves beyond the Hubble length. Methods and

Applications of Analysis 11(1): 77–132.

Tolman R (1939) Static solutions of Einstein’s field equations for

spheres of fluid. Physical Review 55: 364–374.

Weinberg S (1972) Gravitation and Cosmology: Principles and

Applications of the General Theory of Relativity. New York:

Wiley.

Shock Waves see Symmetric Hyperbolic Systems and Shock Waves

Short-Range Spin Glasses: The Metastate Approach

C M Newman, New York University, New York, NY,

USA

D L Stein, University of Arizona, Tucson, AZ, USA

Introduction

Thenatureofthelow-temperaturespinglassphasein

short-range models remains one of the central problems

in the statistical mechanics of disordered systems (Binder

and Young 1986, Chowdhury 1986, Me´zard et al. 1987,

Stein 1989, Fischer and Hertz 1991, Dotsenko 2001,

Newman and Stein 2003). While many of the basic

questions remain unanswered, analytical and rigorous

work over the past decade have greatly streamlined the

number of possible scenarios for pure state structure and

organization at low temperatures, and have clarified the

thermodynamic behavior of these systems.

The unifying concept behind this work is that of

the ‘‘metastate.’’ It arose independently in two

different constructions (Aizenman and Wehr 1990,

Newman and Stein 1996b), which were later shown

to be equivalent (Newman an d Stein 1998a). The

metastate is a probability measure on the space of

all thermodynamic states. Its usefulness arises in

situations where multiple ‘‘competing’’ pure states

may be present. In such situations it may be

difficult to construct individual states in a measur-

able and canonical way; the metastate avoids this

difficulty by focusing instead on the statistical

properties of the states.

An important aspect of the metastate approach is

that it relates, by its very construction (Newm an and

Stein 1996b), the observed behavior of a system in

large but finite volumes with its thermodynamic

properties. It therefore serves as a (possibly indis-

pensable) tool for analyzing and understanding both

the infinite-volume and finite-volume properties of a

system, particularly in cases where a straightforward

interpolation between the two may be incorrect, or

their relation otherwise difficult to analyze.

We will focus on the Edwards–Anderson (EA)

Ising spin glass model (Edwards and Anderson

570 Short-Range Spin Glasses: The Metastate Approach

1975), although most of our discussion is relevant to

a much larger class of realistic models. The EA

model is described by the Hamiltonian

¼

hx;yi



½1

where J denotes a particular realization of all of the

couplings J

and the brackets indicate that the sum is

over nearest-neighbor pairs only, with x, y 2 Z

.We

will take Ising spins 

= 1; although this will affect

the details of our discussion, it is unimportant for our

main conclusions. The couplings J

are quenched,

independent, identically distributed random variables

whose common distribution  is symmetric about zero.

States and Metastates

We are interested in both finite-volume and infinite-

volume Gibbs states. For the cube of length scale L,



= {L, L þ1, ..., L}

,wedefineH

J, L

to be

the restriction of the EA Hamiltonian to 

with a

specified boundary condition such as free, fixed, or

periodic. Then the finite-volume Gibbs distribution



(L)

= 

(L)

J, 

on 

(at inverse temperature  = 1=T)is



ðLÞ

J;

ðÞ¼Z

1

exp H

J;L

ðÞ



½2

where the partition function Z

() is such that

the sum of 

(L)

J, 

over all  yields 1. (In this and all

succeeding definitions, the dependence on spatial

dimension d will be suppressed.)

Thermodynamic states are described by infinite-

volume Gibbs measures. At fixed inverse temperature

 and coupling realization J, a thermodynamic state



J, 

is the limit, as L !1, of some sequence of such

finite-volume measures (each with a specified bound-

ary condition, which may remain the same or may

change with L). A thermodynamic state 

J, 

can also

be characterized intrinsically through the Dobrushin–

Lanford–Ruelle (DLR) equations (see, e.g., Georgii

1988): for any 

, the conditional distribution of 

J, 

(conditioned on the sigma-field generated by

{

: x 2 Z

n

}is

(L), 

J, 

,where is given by the

conditioned values of 

for x on the boundary of 

Consider now the set G= G(J, ) of all thermo-

dynamic states at a fixed (J, ). The set of extremal,

or pure, Gibbs states is defined by

ex G¼Gn a

þð1  aÞ

a 2ð0; 1Þ; 

;

2G; 

6¼ 

½3

and the number of pure states N(J, )at(J, )isthe

cardinality jex Gjof ex G. It is not hard to show that, in

any d and for a.e. J, the following two statements are

true: (1) N = 1 at sufficiently low >0; (2) at any

fixed , N is constant a.s. with respect to the J’s.(The

last assertion follows from the measurability and

translation invariance of N, and the translation

ergodicity of the disorder distribution of J.)

A pure state 



(where  is a pure-state index) can

also be intrinsically characterized by a ‘‘clustering

property’’; for two-point correlation functions, this

reads

h







h





h





! 0 ½ 4

as jx  yj!1. A simple observation (Newman and

Stein 1992), with important consequences for spin

glasses, is that if many pure states exist, a sequence

of 

(L)

J, 

’s, with boundary conditions and L’s chosen

independently of J, will generally not have a

(single) limit. We call this phenomenon ‘‘chaotic

size dependence’’ (CSD).

We will be interested in the properties of ex G at

low temperatures. If the spin-flip symmetry present

in the EA Hamiltonian equation [1] is spontaneously

broken above some dimension d

and below some

temperature T

(d ), there will be at least a pair of

pure states such that their even-spin correlations

are ident ical and their odd-spin correlations have the

opposite sign. Assuming that such broken spin-flip

symmetry indeed exists for d > d

and T < T

(d), the

question of whether there exists more than one

such pair (of spin-flip related extremal infinite-

volume Gibbs distributions) is a central unresolved

issue for the EA and related models. If many such

pairs should exist, we can ask about the structure of

their relations with one another, and how this

structure would manifest itself in large but finite

volumes. To do this, we use an approach, introduced

by Newman and Stein (1996b), to study inhomoge-

neous and other systems with many competing pure

states. This approach, based on an analogy with

chaotic dynamical systems, requires the construction

of a new thermodynamic quantity which is called the

‘‘metastate’’ – a probability measure 

on the

thermodynamic states. The metastate allows an

understanding of CSD by analyzing the way in

which 

(L)

J, 

‘‘samples’’ from its various possible limits

as L !1.

The analogy with chaotic dynamical systems can

be understood as follows. In dynamical systems, the

chaotic motion along a deterministic orbit is

analyzed in terms of some appropriately selected

probability measure, invariant under the dynamics.

Time along the orbit is replaced, in our context, by

L and the phase space of the dynamical system is

replaced by the space of Gibbs states.

Newman and Stein (1996b) considered a ‘‘micro-

canonical ensemble’’ (as always, at fixed , which

will hereafter be suppressed for ease of notation) 

in which each of the finite-volume Gibbs states

Short-Range Spin Glasses: The Metastate Approach 571