The Consequence of General Relativity
Newton's theory of gravity has been proven to be quite adequate to
describe such gravitational phenomena as orbital motions of the planets
in the solar system and trajectories of objects in Earth's gravitational
field. What the general relativity predicts ofr these usual phenomena
differently from Newton's theory have to be quantitatively small.
In his paper of 1916 Einstein presented the quantitive predictions
for the following phenomena; the deflection of starlight passing
near the Sun, the precession of Mercury's orbit around the Sun and
the redshift of radiation from distant stars. All these predictions
have been observationally confirmed within the experimental errors.
One can consider two situations where the general relativistic predictions
can be significantly different from those of Newton's theory; one is
the situation with an unusually strong gravitational field and the other
is when one deals with very large scales in time and space so that,
although the gravitational field is not very strong, small general
realtivistic effects are accumulated to give the final results
significantly different from those of Newton's theory. The former
is the case of the phenomena around a compact stellar objects and the
latter is the case of cosmology. In the rest of this section we will
consider the case of compact stellar objects and then describe the
standard cosmology briefly.
The geometry in the vacuum around a spherically symmetric source is
given by the Schwarzschild metric which is the solution of the
Einstein field equation with
,
.
When on observer stationed sufficiently far away from the
object (
) receives this radiation, the observed frequency of the radiation
is given by
Then the value of the redshift factor is
When
This value for the Sun for example is calculated to be very
small, O(10-6).
It is about 10-4 for a typical white
dwarf and a few tenths for a typical neutron star. We can see that,
when dealing with as compact an object as a neutron star,
general relativistic effects become quite large and Newtonian
treatment of gravity becomes inadequate. As r0
approaches 2GM, the value of z grows to infinity. For an object so
compact that its surface retreats inside r=2GM, no light emitted
from it can reach an outside receiver. This object is called a
black hole and the surface at r=2GM, the Schwarzschild radius,
is called the event horizon.
Let us now review cosmology. After publishing his general theory of
relativity in 1916, Einstein in 1917 presented a new cosmological
model based on the concept of space in the Riemann geometry. The
cosmological space in this model is a 3 dimensional space corresponding
to the surface of a 4 dimensional sphere in a 4 dimensional Euclidean
space. The radius of the sphere determines the size of the Universe
which is finite and without a boundary anywhere. Matter distribution
in the space is assumed to be uniform. Einstein also assumed the Universe
to be static with fixed size. When this model is put into the Einstein
field equation, it turns out that the energy density or pressure has
to take a negative value. Einstein considered this unacceptable and
, in order to remedy the problem, he modified his field equation by
inserting a new term, called "cosmological term", into it.
However, with the help of observations by Hubble in the later part of 1920,
it is eventually etablished that the Universe is not static but actually
expanding. Acknowledging this new development, Einstein in 1931 erased the
cosmological term and restored his equation back to the original form.
Many studies in cosmology followed eventually leading to the establishment
of the "standard model cosmology". In the standard model cosmology,
a very high degree of symmetry is assumed. The so-called "cosmological
principle" assumes that the Universe, on a large scale to include several
clusters of galaxies in one unit, is uniform and isotropic. Under this
symmetry, the spacetime of the Universe can be described by the
Robertson-Walker metric;
where R(t) is the cosmological scale factor and k is some constant.
At an instant of a given t, the spatial geometry of the Universe is
determined by the sign of k; it is a close (finite) space corresponding
to the 3 dimensional surface of a sphere in a 4 dimensioanl Euclidean
space if k>0, an open (infinite) space with a negative curvature if
k<0 and a flat space if k=0.
In the spacetime described by the Robertson-Walker metric, it can be
checked that the orbit with constant space coordinates
is a free-fall orbit, and objects in the cosmological space is
interpreted to follow such orbits on the average. Then the coordinate t
is the time that the clocks attatched to such objects would indicate.
Even though the spatial coordinates of objects remain fixed, the
distances between objects increase with time as the value of the
cosmic scale factor R(t) increases with time. The proper distance
between an object at r=0 and another at r=r1 is
Then the speed at which the two objects move away from each other is
calculated as follows;
The speed is proportional to the distance D and the proportionality
constant is
which is the Hubble constant. The Hubble constant is spatially constant
but changes with time. Its present value is usually expressed as
where h, known to have the value between about 0.5 and 1.0, represents
the uncertainty in our knowledge about its exact value.
Puting the Robertson-Walker metric into the Einstein field equation and
assuming the perfect-fluid form for the energy-momentum tensor.
with
,
one obtains the following equations
The second of the above equations is the energy conservation equation.
For non-relativistic matter with
,
the solution of the equation is 
(the subscript m represents matter), while for radiation with
,
it is
(the subscript r represents radiation). Since
one finds
,
that is, the temperature of the Universe is inversely proportional to the
cosmic scale factor.
From the first of the above equations, one can see that the future
of the Universe depends critically on whether the present value of the
energy density
is greater or smaller than the critical value
If 
,
k is positive and R will stop growing in some finite future and
start decreasing , that is, the Universe will stop expanding
in some finite future and contract again. If
,
k is negative and R will keep growing forever. The
case is the boundary between the above two cases. Which of the
three is the actual case for our Universe is not definitely known yet.
However, in all three cases, if one goes back to the past, R keeps
decreasing until it reaches zero in some finite past. Universe started
in some finite past in an infinite energy density and temperature,
somehow exploded to begin expanding. The expansion still continues
although the rate of expansion has been decreasing. Another name for
the standard model cosmology is the "Big-Bang cosmology". We conclude
this section with a remark about the big bang. It is a point of
singularity at which the laws of physics cannot be applied. We
reach that point only if we assume that the general theory of
relativity and other usual physical laws are valid under such
extreme conditions as those near big bang. It is generally believed
that, under those extreme conditions, the current physical laws
including the general theory of relativity have to be replaced by
some more fundamental laws that are not yet known.