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Minimal Surface Approach to Static Gravitational Fields in General Relativity.

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~~
Annalen der Physik. 7. Folge, Band 47, Heft 2/3, 1990, S. 177-182
J. A. Berth, Leipzig
Minimal Surface Approach t o Static Gravitational Fields
in General Relativity
By G. NEUGERAUER
Sektion Physik ider Friedrich-Schiller-Universit&t,
Jena, DDlZ
Dedicated to Armin Uhlmann on the Occasion of h i s Sixtieth Birthday
A b s t r a c t . The gravitational fields of rigidly rotating perfect fluids are minimal surfaces in
Riemannian Potential space with the metric
dS2 = -2dadFV
-
2x,We2"-2updW2
D
4u
+ %CVdU2 - e-dda2.
2w
Here the minimal surface approach is applied t o static gravitational fields.
Minimalflachenbeschreibungstatischer Gravitationsfelder
in der allgemcinen Relativitatstheorie
I n h a l t s u b e r s i c h t , Die Gravitationsfelder einer starr rotierenden perfekten Flussigkcit sind
Minimdflichen in einem Riemannschen Potentinlranm mit der Metrik
I/
dS2,= - 2 d a d W -
2 % o W e 2 J - 2 " p d V 2 +.'WtEU2--dda2.
2 II/
I n der vorlicgenden Arbeit wird die iCIinimalflachenbcschveibLing auf statische Gravitationsfelder angcwandt.
1. Introduction
To solve the rotating body problem in (:enera1 Relativity more effort must be made
to analyse Einstein's equations for the interior of the body.
We are hopeful about our chances of improving the situation by using a strictly
geometrical approach. It can be shown [I, 21 that the Einstein equations for rigidly and
uniformly rotating perfect fluids in terms of the (slightly modified) Lewis-Papapetrou
metric
+
+
(j
s2 -- e 21'(e2'W,GW,DSC')g;III
dx" dx"
W 2 d y 3 )- e2"(dt
(I d q ~ ) ~
,
(1.1)
where the gravitational potentials U , ~ 1 ,W , 01 and gills ( A , B = I, 2) do not depend on
the time t (stationarity)and the azimuth y (axisymmetry),are equivalent to the differential equations of the two-dimensional minimal surfaces of the four-dimensional Riemannian Potential space S,4 with the line element
e4 1J
dS2= -2dndW
--
2 x , W e 2 ' ~ 2 " p ( V ) d W 2 +2 W d U 2 - --ddaZ,
2w
(1.2)
where the gravitational potentials U , I ( , W , a ariso as coordinates ("second geometrisation of the gravitational field"). g A B is the metric of the minimal surface aridxo is the
Ann. Physik Leipzig 47 (1990) 213
178
gravitational constant. p as a function of the co-rotating Newtonian potential V can be
calculated from the Euler equations
if the energy density
E
is given as a function of p (“equation of state”)
& = &(p).
(1.4)
(Throughout this paper we shall assume that the source consists of a perfect fluid).
For physical reasons, the pressure a t the surface of the fluid has to vanish
p ( V )= 0 H P = V,,
V , being a zero of p , whereas the pressure inside the source must be positive
p ( V )> 0 H P < V , .
(1.5)
(1.6)
There is no pressure outside the source
p ( V ) = o @ v > V,.
(1.7)
As an example, we consider the incompressible fluid with a constant energy density &
&(eV0-‘
- 1) inside the source ( V < V , ) ,
outside the source ( V > V,) .
The example demonstrates that p ( V ) iu (1.2) can be considered as a given function of V .
This paper is meant to test the minimal surface formalism by applying it t o static
sources (a = 0 in ( l . l ) , V = U ) . Though there is no general proof that all static isolated sources are spherically symmetric we will presume this symmetry. I n this case, it
can be shown [I] that the minimal surface describing the interior and exterior gravitational field must have the form
+
1 WZf(U).
(1.9)
(In proof of (1.9) choose d = U , x2 = W and note that (1.1) must reduce to the line
element of a 2-sphere for constant values of t and U ) . Thus we have to discuss the function f = f ( U ) solely.
Obviously, the metric (1.1) is known if f ( U ) is known. (The coordinates in (1.1) are
U (= xl), W (= z2),y , t. a = 0, and o( = ol(zl, 9 )is given by (1.9). Inserting (1.9) into
(1.2) one obtains the metric of the minimal surface).
e-“
=
2. Minimal Surface Equation
Minimal surfaces of the Potential space Si are defined by the variational principle
6 J’I/;d2z’O,
(2.1)
.d
where d is a 2-space in S; with the line element
as; = g A B dxA dxB
(2-2)
and
9 == det g A B
is the determinant of the metric of that 2-space.
From (1.9) and (1.2) ( a = 0, V = U ) one obtains
(2.3)
-
1/q = 2 ~ e 2 J?/,
“
(2.4)
G . NEUGEBAUER,
Static Gravitational Fields
179
where
1
y = e - y f - ’ton)- - WZf‘2,
16
n
d
:= p e - - 2 U , f ’:= -f ( U ) .
(2.6)
dU
To characterise the surface d we have to translate space-time data (“physical information of the body”) into the minimal surface language. We will presume that the body has
an equatorial plane (which implies an additional discrete symmetry of the gravitational
field). Then we may consider a space-time sector q ~ t, = constant bounded by symmetry
axis and equatorial plane. Obviously, the sector metric is conformally equivalent t o the
metric of the minimal surface. Calculating the lint: element a t the minimal surface
houndary from the boundary conditions of the metric coefficients of the sector ( W = 0
a t the symmetry axis, reflection symmetry a t the cxpatorial plane, vanishing values
of OL and U a t infinity) one obtains dSi = 0 a t the boundary of d.Hence the surface
a2 is characterised by d S i 2 0
d:dSi>O.
I n U - W coordinates, d can be described by
(2.7)
d : 0 5 W 5 A ( U ) , U , 2 U 5 U,,
(2.8)
where A solely depends on U , and Uc and U , are the values of U in the center and a t
infinity, resp. W = A ( U ) holds a t the equatorial plane, where d g = 0. Combining both
conditions one obtains
ylw=a
=
(I
+ A2f) (f - xon)
1
-
7A*f’2 =
16
0.
(2.9)
We are now able to calculate the variational integra,l
By means of the substitution
13 = fC, C :== A2,
(2.11)
the Euler-Lagrange equation
3
+ 8xon - 4f) = 4
It
2f‘n‘
2
together with (2.9) can be transformed into the first order system
(f
-
’ton)(f”
C ’ ( U ) = 2C
J:R
1+B
- xonC
f’2 -
(2.12)
9
B ’ ( U ) = (4’tonC - 2B) - y B :+’to:C
(2.13)
’
with n ( U ) as in (2.6).
Because of the first inequality in (2.8) we may replace W / A by a new variable 6
W
= Asin
B,O 5 8 I
n,
(2.14)
Ann. Physik Leipzig 47 (1990) 213
180
whence
+
1 B sin2 6.
Then the space-time line element takes the form
ePzn =
(2.15)
(2.16)
3. Solutions to the Rhimal Surface Equations
E.
As is well known, the classical Schwareschild solution has a constant energy density
Thus (1.8) applies ( V = U ! )and the equations (2.13) have the solution
I
_-
12
xne(3w, -
xonA2
2W"P
X%'(3wo
+
-
36
w2(3w0
-
w,, - w ) (w
-
w,)
inside the source
(w
5 WrJ)
2w,l2 w-W2 inside the source (w2 wo)
)
48(w0 - w , ) ~( 2 ~ 0 w,)~
-c-1
xOE(3w0- 2 4 4
outside the source ( w > wo),
(3.2)
where
w := e(J,wo:= eUa,w, := eue,w,
:= e"m,
(3.3)
U,. being the gravitational potential in the center. The functions C and B are regular
within the interval U,, 5 U < U,. For U ---f U,: C -+ co arid B -+ 0, whereas B ( U J =
0 = C ( U , ) .Thus the solution (2.16), (3.1)-(3.3) is indeed the Schwarzschild solution
formulated in the minimal surface language.
The first order system (2.13) seems to be well-suited for numerical solutions in view
of relativistic star models with realistic equations of state F = ~ ( p )However,
.
an initial
value problem with C ( U , )= 0 = R ( U , )would fail, since the right hand sides of the eqnatioris (2.1 3) are undefined. Nevertheless, one could calculate C'(U,,)and B'(U,.)by hand
and start the numerical procedure a t U , + dU (dU > 0), that means with initial values
U(U,. d U ) = B'(U,) dU and C(l7,. d U ) = C'(U,)d U .
Near the center any star model with a reasonable eqoation of state woiild behave
like the Schwarzschild solution
+
+
c=
(10 -
whcrc the functions
(2
w,)
Cc(2U))
13 = xozc
+
(20
and b were regular for w
- w,)2 b ( w ) ,
= zcc
(3.4)
From (3.4) and (2.13) one finds
(3.5)
Then a numerical algorithm could consist in the following steps :
(i) Prescribe the equation of state E = s ( p ) to get n = n ( U ) (n= p ~ - - ~ from
" ) (1.3)
U!).
(ii) Calculate the initial values a t U,
dU from (3.5) and set U , = 0 (or choose a
different value).
(V
=
+
181
G. NEUQEBAUER,
Static Gravitational Fields
(iii) Integrate (2.13) within the interval 0 5 U 5 Uo for suitable values of the stirface gravitational potential Uo t o get the interior solution B ( U ) , C ( U ) (the exterior
solution U > U , is always Schwarzschild).
(iv) Rescale U to have U , = 0. This can be done by means of the relations B(w,)
= 0 ( H w, = 37u0 - 2wC) and B(wo)= sinhz ( U , - U,), where B(wo) was found by
integration. Thus one obtains
+ + iu,)+
+
U
u, In ( i B o I
U't,
where D" is the rescaled gravitational potential ([I&
(3.6)
-= 0).
4. Gravitational Thermodynamics
The hydrostatic pressure p contains the complete information of the source. I n a
thermodynamic context one would expect that p is something like a thermodynamical
potential. It can be shown [ 3 ] that this idea leads to a thermodynamical description of
the gravitational system as a whole (i.e. t o a thermodynamics comparable with Black
Hole thermodynamics). As a thermodynamical potential, p depends on temperature T
and chemical potential ,u
+
dp =sdT
e dp,
(4.1)
where sand e denote the densities of entropy and mass, resp. Inserting Tolmaris equilibrium conditions
Y'eU
To,fie" = Po
where To and po are space-time constants, one obtains
(4.2)
9
+
+
dp = e e - u d,uO se-" d T o - ( E
p ) dlli.
(4.3)
Here one has to compare different states {po,To,U } and fpo dpO,To d T o , U d U } ) .
Using this formula one can show [ 3 ] that the minimal surface area of a given solution
+
+
+
(4.4)
defines a Gibbs relation
d2' = - Y d T o - . A d P o ,
(4.5)
where 9
'and A! are entropy and baryonic mass, ~ e s p(which
.
may be calculated from
s and e, resp., in the usual way). Because of (4.5), 3 is a thermodynamical potential
describing the gravitational system as a whoIe. It turns out that
+
+
M := 9 TOY p0.A
is just the (Schwarzschild) mass of the system. Then
1
(dM-PodA').
TO
A particular case is Black Hole thermodynamics,
S = k1112 ( k a constant).
dS=-
(4.6)
(4.7)
(4.8)
5?(!7'o,po)or S ( M , A!)can be used to discuss bifurcation phenomena (e.g. critical masses
for star models) in the framework of thermodynamics. The thermodynainical potential 9
of the Schvvarzschild solution is given by
(4.9)
182
Ann. Physik Leipzig 47 (1990) 2/3
where
(4.10)
,6 arid y being arbitrary functions of To/po.
References
[l]NEUGEBAUER,
G.; HERLT,E.: Class. Quantum Grav. 1(1984) 695.
[2] NEUGEBAUER,
G. : Proceedings of the Balatonszeplak Relativity Workshop (Ed. Z. PERJES)
(1985) 103.
[3] NEUCEBAUER,
G.: in: Relativity Today (ed. Z. PERJES),
Singapore 1988.
Bei der Redaktion eingegangen am 28. Juli 1989.
Anschr. d. Verf.: Dr. G. NEUUEBAUER
Sektion Physik
Jena
der Friedrich-Schiller-Universitat
Max-Wien-Platz 1
Jena
DDR-6900
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