Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

72 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
Note that represents the disk. On the disk, Eq. (12) reduces to
i.e. we may identify the constant from (11) as
The solution depends on two parameters, e.g. and The surface
mass–density calculated according to (9) is given by
and the mass of the disk turns out to be
A comparison with Eqs. (5), (6) shows that this is exactly the Maclaurin
disk.
It is interesting to insert numerical values characteristic for the Milky
Way system into the last equation. Of course, is not constant in the
Galaxy. For a value of corresponding to the revolution time of the
sun about the galactic center (approximately 240 million years) and
50,000 light years, the quite reasonable mass value of
is obtained ( : mass of the sun).
3. GENERAL–RELATIVISTIC
CONTINUATION
In Einstein’s theory of gravitation, the Poisson equation
and the Euler equation [p: pressure, V: “corotating”
potential, cf. (11)] have to be replaced by Einstein’s field equations
with the Ricci tensor the Ricci scalar R, the
metric tensor the energy–momentum tensor and
(c: velocity of light). The advantage of the disk limit remains the same:
The global (interior + exterior + matching) problem can be formulated
as a boundary–value problem for the vacuum equations
For axially symmetric and stationary configurations, the
Einstein vacuum equations are equivalent to the well–known
Ernst equation