PHYSICAL REVIEW VOLUME 136, NUMBER 2B 26 OCTOBER 1964

Relativistic Equations for Adiabatic, Spherically Symmetric Gravitational Collapse*

CHARLEs W. MIsNER
Department of Physics and Astronomy, University of 3faryland, College Park, Maryland
AND

DAVID H. SHARPt
Palmer Physical I.aboratory, Princeton University, Princeton, Sezo Jersey
(Received 15 June 1964)

The Einstein equations for a spherically symmetrical distribution of matter are studied. The matter is
described by the stress-energy tensor of an ideal Quid (heat Qow and radiation are therefore excluded). In this
case, the Einstein equations give a generalization of the Oppenheimer-VolkoR equations of hydrostatic
equilibrium so as to include an acceleration term and a contribution to the eRective mass of a shell of matter
arising from its kinetic energy. A second equation also appears in this time-dependent case; it gives the rate of
change of an appropriate "total energy" m(r, t) of each Quid sphere in terms of the work done on this sphere
by the Quid surrounding it. These equations would be an appropriate starting point for a study of relativistic
gravitational collapse in which an adiabatic equation of state more realistic than the p =0 form of Oppen-
heimer and Snyder could be used.

I. INTRODUCTION AND SUMMARY where nI' is the four-velocity field of the Quid, e is the
internal energy of the Quid per unit proper rest volume,
HE original discussion of an idealized problem of
and p is the pressure. Because this tensor is diagonal in
gravitational collapse due to Oppenheimer and
the local rest frame of the Quid, it cannot describe the
Snyder' assumes a spherically symmetric distribution
of matter, adiabatic flow (no viscosity, heat conduction, energy Qow associated with heat conduction or radia-
tion. Using Eq. (1.1) in the statement u„T&"., „=0 of
or radiation), the equation of state p=O, and simple
local energy conservation shows that the entropy of
initial conditions. In this note we maintain the assump-
each particle in the Quid is constant, m&s, „=0.We sum-
tions of spherical symmetry and adiabatic Qow, and
marize here the equations in the isentropic case, where
consider the introduction of pressure gradient forces
one further assumes s, „=0, so that the specific entropy
into the equations. Our purpose is to cast the equations
is constant throughout the volume of the Quid.
into as simple and physically transparent a form as we
The metric is chosen to have the diagonal form
can, preliminary to their numerical solution.
Much of the recent interest2 in gravitational collapse ds'= +R'dQ'
e'&dfs~ e"dr'— (1.2)
centers about the possibility (in a stage of collapse where where
the gravitational binding energy GM'/R becomes com- dQs=- d8'+sin'ed&ad'. (1.3)
parable to the rest energy Mcs) of a large energy outPut
of a star, a discussion of which falls outside the scope Here p, X, and R are each functions of r and t to be
of the equations derived here. Nevertheless, a study of determined by the Einstein held equations. We shall
these equations may provide a useful first step in a worl» in a system of coordinates moving at each point
more realistic analysis of the gravitational collapse of with the material located at that point (comoving or
—
stars which would presumably include the effects of Lagrangian coordinates). The components of the four-
rotation, departures from spherical symmetry, and velocity are thus
radiation— as well as some insight into the issues of
principle involved in gravitational collapse. '
In the remaining paragraphs of this section we will
summarize our results. These are derived in the succeed-
Then the hydrodynamic equations „=0 give
TI"",. the
ing sections.
result
Associated with an ideal Quid is a stress energy given
by the tensor
T "= (p+e)u u"+pg" (1.1) where h=u+pv= (e+p)/n is the specific enthalpy or
* Supported in part by NASA Grant No. NsG 436. heat function for a unit amount of fluid (the amount
f' NSF Postdoctoral Fellow, 1963—64. containing a mole of baryons). LThe specific internal
' J. R. Oppenheimer and H. Snyder, Phys. Rev. 56, 455 (1939).
'W. A. Fowler, Rev. Mod. Phys. 36, 549 (1964); F. Hoyle,
energy I and the specific volume v are related to the
matter density or baryon number density n(r, f) by
W. A. Fowler, G. R. Burbidge and E. M. Burbidge, Astrophys. =uaensd v=1/n. We choose units of n(r, t) so that
J. 139, 909 (1964); H. Y. Chin Ann. Phys. 26, 364 (1964); F. C.
Michel, Astrophys. J. 138, 109I (1963); S. A. Colgate and R. H.
White, Bull. Am. Phys. Soc. 8, 306 {1963).
e— + n and h —+ 1 as p ~ ]
0. In order to compute h, it is
sufficient to specify the adiabatic equation of state
J. A. Wheeler, in Gravitation and Relativity, edited by H. V.
Chiu and W. F. Hoffmann (W. A. Benjamin Company, Inc. ,
1964), Chap. 10. 6=6 N (1.6)
8571
 C. K. M ISNER AN D D. H. SHARP
for then the pressure equation p(n) can be deduced via m(r, 0), and U(r, 0). Equation (1.13) then defines e(r, 0)
the thermodynamic relation which gives the values of p, n, and ts through an equa-
tion of state and thus allows the time derivatives of E,
m, and U to be obtained from Eqs. (1.12). One thus
(1 7) obtains a solution for all times without invoking Eq.
(1.14); but this equation merely defines dA/dr initially,
and from h= (e+ p)/n one finds that and it is possible to show that the time derivative of the
left member of Eq. (1.14) vanishes as a consequence
of Eqs. (1.12), (1.13), (1.5), and (1.7). Thus, Eq. (1.14)
is a first integral of this system of equations.
The above system of equations is to be solved subject
The remaining Geld equations for this problem take to the boundary condition that
a simple form if one defines a quantity V which gives
the relative velocity Vd8 of adjacent Quid particles on p =0 at r = r, = constant, (1.15)
the same sphere of constant r, where r, deGnes the outer boundary of the distribution
—e of matter. It is then evident from. Eq. (1.12-m) that
V=DgE= &R.
m(r. , t) =M (1.16)
Here D& is the comoving proper-time derivative
is a constant and, in fact, the interior metric (1.2)
B can be joined smoothly at the surface r, to an exterior
D& u" e—&l
=-— —
trB)
.
l
(1.10) Schwarzschild metric whose mass 3f is given by Eq.
Bx" ~ Bt), (1.16).
It is also necessary to require that at r=o the func-
One also uses in place of X(r, t) a function m(r, t) defined tions R, m, and V all vanish.
by
2m(r,t)--' BR)'
e"&''&=g = 1+U' — . (1.11) II. THERMODYNAMIC PRELIMINARIES
Brj
l

R Local properties of a Quid such as pressure, tempera-

ture, specific entropy, internal energy density, etc. ,
The full set of Geld equations are then the three Grst-
which are scalars in nonrelativistic physics can all be
order dynamical equations
defined in special and general relativity so that they are
D,E.= V, (1.12-R.) again scalars. For, to be scalars, they need merely

D,m = —4srR'p U,

1+U'
e+P
2mR ' tBp —
(BR(
)-g
(1.12-m)
have a well-defined value at any event, independent of
every arbitrary choice of a coordinate system. One
achieves this by defining these quantities to have (in any
coordinate system) the values measured by an observer
who is at rest relative to the chosen small piece of Quid
at the time in question.
(m+4srR'p)

two equations free from time derivatives, namely,

(1.12-U)

Eq.
The basic law of thermodynamics4
dl = Tds pd8—
applies to a fixed amount of matter which, for con-
(2. 1)

(1.5) and the equation venience, we take to be a unit amount. The fact that
the amount of matter does not change can be expressed
(Bm) by introducing the particle number density n= (1/n)
4srR'e, (1.13)
EBRi, and requiring it to satisfy a continuity equation:

and the equation of continuity (nn&), „=0. The con- (nu&)., „=0. (2.2)
tinuity equation can be written in a form This law of co+servation of matter in hydro'dynamics
4srR'n BR dA can be derived from the microscopic law of conservation
i (1.14) of baryons.
(1+U' —2mR ')'t' Br dr j g=s
4The discussion in Secs. II and III is based on that of L.
Landau and E. Litshits, Fled 3Iechanscs (Addison-Wesley
appropriate to our comoving coordinates, where the Publishing Company, Inc. , Reading, Massachusetts, 1959), Chap.
amount of matter dA in any spherical shell deGned by a XV, especially in its emphasis on the continuity equation (2.2).
Gxed coordinate range dr is independent of time. For more complete discussions of the subject see: (i) A, Lich-
nerowicz, Theories Relativistes de Le &avitation et de L'E/ectro-
Solutions to the above system of equations can be mugnetisme (Masson et Cie, Paris, 1955); (ii) J. L. Synge, Rela-
obtained by specifying arbitrary initial values for R (r, 0), tivistic Hydrodynamics, Proc. London Math. Soc. 43, 376 (1937).
 SPHERI CALLY S YM METRI C GRAVITATIONAL COLLAPSE
When Eq. (2.1) is rewritten in terms of the energy and the surrounding empty space in order that the
density e=u/v and particle number density e, it reads interior metric (1.2) can be joined smoothly to the
exterior Schwarzschild metric
de = NTds+ (e+ p) s drz (2.3)
and gives Eq. (1.7) in the case ds=0. Thus, e=e(s, e) is ds'= —(1 2M—R ') dt'+ +R'dQ'. (4.3)
a convenient fundamental thermodynamic relationship 1 —2MB. '
for describing a Quid; it immediately gives the pressure
equation p(s, rt) via Eq. (1.7). By differentiating the These conditions will serve to relate the exterior coordi-
definition of specific enthalpy, h=u+pv=(e+p)/ri, nates R and t to the interior t coordinate and interior
and employing Eq. (2.1) or (2.3), one obtains a relation metric component ggg=R'(r, t). Assume that in the
exterior E,t coordinates the interface is described by
dh dp Tds an equation
(2.4)
(4 4)
h (e+p) h
The metric on the interface is obtained by inserting
which will be useful later.
this in Eq. (4.3), or alternatively by setting r =r, = const
in Eq. (1.2). By equating these two expressions,
III. HYDRODYNAMICS REVIEW
2M
The equations of motion of a Quid described by the (ds'). „,s= —(1— ds'+ +R, 'dB'
stress-energy tensor4 Tl"" of Eq. (1.1) and an equation R, i —2MR, '
of state e= e(s, e) are T&", „=0. One of . these four equa-
tions„namely, I„'lI"".„=0, reduces as a consequence of = —(e'4') dt'+R'(r t)dn' (4 5)
Eqs. (2.2) and (2.3) to the heat transfer equation for
an ideal Quid, which is the condition of adiabatic Qow we find an equation for the interface in the exterior
coordinates. It reads
NI s q=0. (3.1)
R=R, (t) =R(r„t), (4.6)
The remaining equations can be reduced to the form of
relativistic Euler equations: provided that we insist that the interior and exterior
time coordinates agree on the surface. This boundary
condition on t then leads to one on e&, namely,
u"= —(g "+u u") p,
e
u (3.2)
(e&) „,, = —
L1 =2MR, 'jL1+ U,2 —2MR 'j—'t', (4.7)
In the special case of isemtropic gou(, where one where we have defined
assumes the specific entropy s to be constant throughout
the Quid s, „=0, the Euler equation can be rewritten as = BR)
U. = (e e).B.— (e e (4g)
u' u" = —(g""+u"u") (lnh), , (3.3) at),
by use of Eq. (2.4). The function U, (t) is the rate of change of R, with
It is evident that in the isentropic case we may respect to the proper time of a comoving observer. The
consider e, p, and h as functions of the particle number conditions derived from the continuity of the deriva-
density n alone, i.e. , e= e(rs), p= p(u), h=h(e). tives of the metric can best be considered later.

IV. COORDINATES AND METRIC V. EULER EQUATION

The metric (1.2) must satisfy certain conditions at In the comoving coordinates delned by Eq. (1.4),
the origin r=0 to assure regularity there. The first is one obtains from Eq. (3.2) only one nontrivial Euler
equation, which reads:
R(0, t) =0. (4.1)
cjoy'/~r = —L1/( +P) j~p/~r. (5.1)
Next, in order for the usual I,orentz-Minkowski
geometry to be valid in an infinitesimal neighborhood In the isentropic case we may use Eq. (2.4) to integrate
of the origin, we must require that the circumference Eq. (5.1). With the boundary condition (4.7), one finds
2+E. of an infinitesimal sphere about the origin be just
2m times its proper radius e~"dr, or
1—/2M/R, (t)]
(5 2)
e" = (8R/Br)' at —0.
r=
Other conditions must hold at the interface between when h is normalized so that h= 1 at the surface r= r, .
the region occupied by matter (defined by a certain However, the coordinate conditions (1.4) and the
constant coordinate value r = r,, for the inf:erior solution) diagonal form of the metric (1.2) are preserved by
 C. K. MISNER AN D D. H. SHARP
transformations of the interior time coordinate t of the of Oppenheimer and Volkoff ~ in the static case,
form t — + f(t), so it is possible to change e4' by a factor where e "= 1 —2' ', indicates something of the form
which is an arbitrary function of time. Consequently, the solution might take. But the boundary condition
the solution (1.5) is also acceptable. IJse of Eq. (1.5) (4.2) suggests some modifications of this. The form
synchronizes the interior time coordinate with the e ~=X' 2 —2m' ' satis6es these boundary conditions if
proper time of a comoving observer at the interface (m/R) ~0 as r —+0, but it is inappropriate for di-
r=-r„and prevents the interior time coordinate from mensional reasons: The Lagrangian coordinate r is
inheriting the singularities of the exterior time coordi- arbitrary at time /=0, and therefore can be assigned
nate when the surface falls through the Schwarzschild a dimension independent of all other quantities in the
"singularity" (R, (t) —2M) ~0. problem, while m/R is dimensionless. This leads us to
When different layers in the body are allowed to have try the form e "=R' '(1+f) which simplifies Eq. (6.5)
different adiabatic equations of state e(is), it is not so that it reads
possible to integrate Eq. (5.1) in terms of the specific
enthalpy, but an integrated form such as 87reR'=8 (RU' Rf)—
/BR (6.6)

g=+
g
—1

e+p
—
Bp
BR
dR and thus yields the solution U' —
f= 2mR ' as given in
Eqs. (1.11) and (1.13). To interpret Eq. (1.13) it is
best to rewrite the integral
can of course be written. The boundary condition in-
corporated into Eq. (5.3) makes e&=1 at the surface
R=R, as in Eq. (1.5). The analogous generalization of m(r, t) = 4irR'edR (6.7)
Eq. (5.2) is evident.
VL INITIAL VALUE EQUATIONS in terms of the element of proper volume

The Einstein equations corresponding to the metric V3V =4~x~~»2'

be found in Landau and Lifshitz. ' Since one
(6.8)
(1.2) can to obtain
knows' that the I'0' and T equations will contain no
second time derivatives, one may hope to 6nd some- (6.9)
thing simple in them for a starting point. In the present
case, the T„' equation is the simplest. It reads
This last form reminds us that when considered as an
e 6.=2U'/R', (6 1)
energy, m includes contributions from the kinetic energy
where we use dots and primes to indicate the partial and the gravitational potential energy.
derivatives with respect to f and r, respectively, and It is now possible to rewrite the constraint (6.4) in
define U by Eqs. (1.6) and (1.7). We may use the an interesting form by substituting for X from Eq.
differential operators (1.11). The computation involves interchanging the
operators Di and 8/Br to write
ci/aR= (1/R') (8/Br), (6.2)
and BU Bp
— = +U
D, =e e(8/Bt)„ (6.3) D, lnR'— (6.10)
8E I9R
to rewrite the initial value equation (6.1) in the form
Dies=2(BU/BR) .
Using this identity and Eq. (5.1) gives

This equation may then be used to eliminate A from all ( 2' 'I'
= —U
1 Bp
the other Einstein equations. When it is used in the To' D, ln~ 1+U' — . (6.11)
equation, one 6nds e+p aR
87reRs= 1+ U +R(BUs/ciR) This equation is a useful erst integral in the cases con-
—L2RR"+R"].-~ —RR'(e-&)'. (6.5) sidered by Oppenheimer and Snyder' and by Bondi
where p=0. For then, since e&= 1 by Eq. (1.5), it reads
Since this equation is of erst order, and even linear, -,'8' — (nz/R) =E=const and will give Newtonian free
in e " we try to solve it for this function. The work fall for R(i) when we later discover that m(t) is constant
with this special p=0 equation of state.
'L. Landau and E. Lifshitz, The Classical Theory of Fields
(Addison-Wesley Publishing Company, Inc. , Reading, Massa-
chusetts, 1951), Sec. 11-7, Problem 5. 7
J. R. Oppenheimer and G. M. Volkoff, Phys. Rev. 55, 374
Y. Bruhat, in Gravitation: An Introduction to Current Re- (1939).
1962l, Eq. (4-1.8l.
¹w
seorch, edited by L. Witten (John Wiley 8r Sons, Inc. , York, 8 H. Bondi, 3Eonthly Notices
107, 410 (1947).
of the Royal Astronomical Society
 SPHERICALLY SYMMETRIC GRAVITATIONAL COLLAPSE
VII. EQUATION OF MOTION Because we have rather thoroughly reshu8. ed the
It is known' that the Einstein equations Einstein equations of the metric (1.2) in obtaining a
system of equations for independent field variables E., m,
R'r = 8~(T'i 'i T".)
kg— (7.1) and U, it may be of interest to prove directly that for
each r, Eq. (8.3) gives a constant of motion for the
for i, j=1, 2, 3 contain
as leading terms just BK;;/Bt,
system of equations (5.1), (1.12), and (1.13) supple-
where K;; is the second
fundamental form of the
mented by an adiabatic equation of state e(m) for
t=const surface. Equivalently, the only second time
each r. The computation begins by forming the loga-
derivative that appears in Eq. (7.1) is B'g;;/BP Th.us,
rithmic derivative of Eq. (8.3)
the R„„equation will contain just X and will be an
identity since we have eliminated X from our scheme D]A' D,n 2D, E D,E' 2m
by solving Eq. (6.5). The Rg& and R«eq uati ons will be + +—-,'B, lr (l.+ U'— (8 4)
equivalent (by symmetry) and each will contain just A' e R E.' R)
B. They read In the second term of Eq. (8.4) we can write by (1.12-R)
6
— that D&R= U; the third term has been rewritten in
4irR (e —p) =e &B(RRe &)/Bt+~e '&RRX Eq. (6.10); the derivatives in the last term can all be
c'
evaluated from Eqs. (1.12) and give Eq. (6.11) which,
+1 e"'2B(—
RR'e "~')/Br e ERR—
'rt'
by (5.1), reads
= Reg = (sin'8) —'R„„. (7.2)
2m)'~' BP
In this equation we introduce the operator of Eq.
D& D l ~1+U — =U (8.5)
(1.10) and, for some intermediate computations, the Rj ~

BR
operator
%e thus obtain the reduced form
D —
e x/2 ] +U2 (7 3)
DA'
Bt' R gR Dm
—
+Z2 (R'U) . (8.6)
Then X is eliminated using Eq. (6.4), is replaced by 8 aZ
U via Eq. (1.9), and p' with p' via Eq. (5.1).This result
is Eq. (1.12-U) which includes the well-known Oppen- Because of the adiabatic condition D~s=O, changes in
heimer-Volkov~ equation of hydrostatic equilibrium in density are related according to Eq. (2.3) by
the limiting case U=0.
Using the main equation (1.12-U) we can carry out De D]c
(8.7)
some of the differentiations in Eq. (6.11) to reduce it rl e+p
to the form (1.12-m). Et is this form which shows that
m=0 in the case of a p=0 equation of state. In the reduced system of equations for E, m, and U,
the density e is given by Eq. (1.13) which we differ-
VIII. EQUATIGN GF CGNTINUITY entiate to obtain D&e.
The continuity equation (2.2) implies quite generally
81Ã
that the integral = 8rrRUe+4rrR2Dge.
Di
A = iiu'( —g)'t'd'x (8.1)
(9E.

To evaluate the left-hand side of this equation we need

taken over a /=const surface is independent of t. Its the commutator
value A is analogous to the mass number of a nucleus
B- Bg(
and represents the total amount of matter, or total
number of baryons, in the system. In comoving coordi- D„= BR BR&
i
Di —U
B

BR
BU B
BR BR
(8.9)
nates satisfying Eq. (1.4) the corresponding integral
over any fixed domain of the spatial coordinates in which BP/BR can be eliminated using Eq. (5.1). We
x'(i=1, 2, 3) is time-independent, since Eq. (2.2) then find then with the use of Eqs. (1.12-m) and (1.13) that
reads
B (riu'g —g)/Bt = 0.
For our problem we can insert expressions for
(8.2)
I' and Di
Bm

BR
— 8+RpU 4mR'(e+— p)— BU
R
(8.10)
Q —g here to obtain the statement that
4~m~R'/tt1+ U' 2mR rjrt2=A'(r)—- which allows us to rewrite Eq. (8.8) in the form

1 8
is time-i~deperiderit.
D, e = —(e+ p) —' (R'U) . (8.11)
9 Reference 6, Eq. (4-1.9). E. BR
 C. K. M ISNEk AK 0 D. H. SHAPEUP

Combining this with Eqs. (8.7) and (8.6) then gives hypersurface r=r, one has
DfA'=0 (8.12) Kg. g. = —R, 'L1+ U,' —(2gg4/R. )]t", (9.3)
as we wished to show. while the exterior metric gives, for the hypersurface
R=R, (1),
IX. BOUNDARY CONDITIONS
K ~ = —R 'L1+ U ' —(2M/R, )]" (9.4)
The condition, previously discussed, that the metric
or first fundamental form of the boundary surface Matching these components of 4 therefore gives M = m,
should be the same whether obtained from the interior which is Eq. (1.16). Since M is a constant this equation
or exter ior metric, guarantees that for some coordinate can be differentiated with respect to t with r = r, to
system the metric components g„„will be continuous give m, =0 which implies, through Eq. (1.12-m), that
across the surface. In order to guarantee that coordi- p, U, = 0. The correct boundary condition is more
nates can be introduced for which the first derivatives specifically
of the metric, g„„, , are continuous, it is sufhcient that the —p(r„t) =0,
p, = (9 5)
second fundamental form be the same whether the
as can be seen by comparing the interior and exterior
boundary surface is considered imbedded in the in-
components K«and using the field equation (1.12-U).
terior or the exterior space-time. ' For any hypersurface
The interior computation gives
s with unit normal vector eI", the second fundamental
form 4 is defined as" 2M~'I' 1 rip
Ks c
= — 1+U'— (9.6)
4= ( ) e+pBR
I

gg„;,dx "dg—
g"), , (9.1) R
while the exterior gives
where the subscript s means that one of the coordinate
differentials is to be eliminated using the equation of
the surface. For example, one sets (dR A, dt), =0— in the IC +
R1
2M~-'~' M
+(1+U' — =I R'
+D, U . (9.7)—
exterior coordinates of our problem. For comparison
purposes, we write The difference of these two, using Eq. (1.12-U) and
m M is
@=Kg g. (egdt)s+Kg g DR, d8)s+ (R, sin8dq)'] (9 )
K, ,+ K, , = —L— U,— s
(2M/R, —
1+ 'l'4grP, R, . (9.8)
)]
and compute from the interior solution that for the
ACKNOWLEDGMENTS
' D. L. Beckedorff, thesis, Princeton University, Mathematics
Department, 1961 (unpublished); and C. W. Misner and D. L. We thank Dr. Stirling Colgate, Dr. Richard Lind-
Beckedorff (unpublished).
"E. Cartan, Lecons szsr la Geometric des EsPaces de Riemunn quist, and Professor J. A. Wheeler for discussions con-
(Gauthier-Villars, Paris, 1951), Sec. 207. cerning this problem.