Time Travel Research Center © 2005 Cetin BAL - GSM:+90 05366063183 - Turkey / Denizli General Theory of Relativity - Curved Space-Time(Note: Some of the material on this handout involves trigonometry,
which you are not required to know for this course. Just try to understand
what you can from this handout.) We have seen that space can be non-Euclidean in General Relativity. But
exactly what does it mean to say that space or, for that matter, space-time
is curved? A few examples can help to illustrate things. At first, we will
consider only two-dimensional surfaces. Imagine that we have a flat sheet.
If we draw a right triangle with sides denoted by
where
where
g11 = g22 = 1 and
g12 = g21 = 0. The g's
are called the metric of the surface. They tell you the distance
between two points that are close together. The reason for expressing
Now consider the surface of a cylinder. We can draw a right triangle, one
of whose sides is a circular arc of the cylinder and the other side is
parallel to the axis of the cylinder. The hypotenuse is the shortest
distance that connects the sides. Is this a flat space? YES! How do you know?
It is possible to cut the cylinder parallel to its axis, open it up and fold
it out flat. Although the surface of a cylinder is curved, it is
geometrically equivalent to a flat surface. Again we can write
The situation changes dramatically when we look at a spherical surface. There is no way to cut the surface and lay it flat without lumps. This is why it is impossible to represent the spherical Earth on a flat projection without distortion. A spherical surface has an absolute curvature. We can look at triangles on the surface of a sphere. ``Straight Lines'' on a sphere are defined as the shortest distance between two points on the sphere. Such lines are arcs of great circles. A great circle is a circle on a sphere whose diameter is equal to a diameter of the sphere. [For example, lines of longitude (running from the North Pole to the South Pole) are great circles, but the only line of latitude that is a great circle is the Equator.] As and example, a triangle can be formed by two lines of longitude that run from the Equator to the North Pole and the part of the Equator that connects them. The sum of the angles of this triangle is greater than 180 degrees - the geometry is non-Euclidean. We can also draw a small right triangle on the surface of the sphere in
which one side is an arc of latitude (of length
This expression gives us the distance traveled for a given (small)
displacement on the surface of the sphere specified by
Here we note two important points. The first is that for the sphere, no
matter what coordinates we choose, it is impossible to express the
metric in the form of Eq. (2) with
g11 = g22 = 1 and
g12 = g21 = 0 everywhere on
the sphere. The other point is that if we are only interested in a
small region of the sphere, it is possible to find coordinates in
which the metric is in the form of Eq. (2)
with
g11 = g22 = 1 and
g12 = g21 = 0 in that local
region. As an example of this, we could choose new coordinates
x The examples above illustrate the criterion necessary for a two-dimensional
space to obey the laws of Euclidean geometry. If a system of coordinates
can be found in which the line element (hypotenuse of a small right
triangle) can be written as
with
g11 = g22 = g33 = 1
and
g12 = g21 = g13 =
g31 = g32 = g23 = 0,
the geometry is Euclidean. If not the geometry is not Euclidean. It is
convenient to introduce a notation in which
where the symbol
The extension to four dimensional space-time is straightforward. We have
already seen that the space-time interval in special relativity (where space-time
is flat and Euclidean) is equal to
-
with g11 = g22 = g33 = 1, g44 = - 1, and all the other gij's equal to zero. It then follows that four dimensional space-time is Euclidean if a coordinate system can be found in which -
with g11 = g22 = g33 = 1, g44 = - 1, and all the other gij's equal to zero. If no such coordinate system can be found, space-time is said to be curved. Space-time in a rotating coordinate system is not curved since one can transform into an inertial frame where the g's will have the Euclidean values. However, whenever mass is present, it is impossible to find a coordinate system in which g11 = g22 = g33 = 1, g44 = - 1, and all the other gij's equal to zero - the presence of mass creates curved space-time. In fact, it is the mass distribution that determines the g's, which, in turn, determine the geometrical properties of space-time! One final note. It is possible for space to be non-Euclidean, but for space-time to be Euclidean. This is true in accelerating reference frames in the absence of mass, such as in the example with the rotating platform. In different reference frames, the curvature of space can be different, but the curvature of space-time is absolute. (Adapted from the notes of P.R. Berman) Introduction to General Relativity Hiçbir yazı/ resim izinsiz olarak kullanılamaz!! Telif hakları uyarınca bu bir suçtur..! Tüm hakları Çetin BAL' a aittir. Kaynak gösterilmek şartıyla siteden alıntı yapılabilir. The Time Machine Project © 2005 Cetin BAL - GSM:+90 05366063183 -Turkiye / Denizli
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