General Theory of Relativity - Curved Space-Time - Cetin BAL - GSM:+90 05366063183 -Turkiye / 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 $\Delta$ x and $\Delta$ y (where $\Delta$ x, and $\Delta$ y denote displacements along the x and y axes, respectively), and hypotenuse by $\Delta$ l, then by the Pythagorean theorem ( $\Delta$ l )² = ( $\Delta$ x)² + ( $\Delta$ y)². We can also write this as

$\displaystyle \Delta$ l² = $\displaystyle \Delta$ x² + $\displaystyle \Delta$ y², (1)

where $\Delta$ x² is a shorthand notation for ( $\Delta$ x)², $\Delta$ y² for ( $\Delta$ y)², etc. The fact that we can write $\Delta$ l² in the form of Eq. (1) is a statement that the space is Euclidean or ``flat''. Thus, a flat sheet is flat and Euclidean - nothing very profound here. Although it may seem a bit stupid, we can rewrite $\Delta$ l² as

$\displaystyle \Delta$ l² = g₁₁ $\displaystyle \Delta$ x² + g₂₂ $\displaystyle \Delta$ y² + g₁₂ $\displaystyle \Delta$ x $\displaystyle \Delta$ y + g₂₁ $\displaystyle \Delta$ y $\displaystyle \Delta$ x, (2)

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 $\Delta$ l² in the form of Eq. (2) is that it turns out that for any coordinates, it is always possible to express the distance $\Delta$ l in the form of Eq. (2). In general, however, the g's ( g₁₁, g₂₂, etc) will be more complicated than those given above (see the example below of the spherical surface).

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 $\Delta$ l² = $\Delta$ x² + $\Delta$ y², where $\Delta$ x = r $\Delta$ $\phi$ , r is the radius of the cylinder, $\Delta$ $\phi$ is an angle around the cylinder, and y is the direction of the axis of the cylinder. This can also be written as $\Delta$ l² = g₁₁ $\Delta$ x² + g₂₂ $\Delta$ y² + g₁₂ $\Delta$ x $\Delta$ y + g₂₁ $\Delta$ y $\Delta$ x, where g₁₁ = g₂₂ = 1 and g₁₂ = g₂₁ = 0.

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 $\Delta$ x) and the other side an arc of longitude (of length $\Delta$ y). If we choose coordinates of longitude (an angle $\phi$ , measured from some arbitrary longitude line), and latitude (an angle $\theta$ , which we measure from the Equator), than we can express the length of the arc of longitude as $\Delta$ y = r $\Delta$ $\theta$ , and the length of the arc of latitude as $\Delta$ x = r cos( $\theta$ ) $\Delta$ $\phi$ , where r is the radius of the sphere. The cos( $\theta$ ) in the previous equation accounts for the fact that as you get closer to the North Pole, the distance you travel ( $\Delta$ x) for a given change in longitude ( $\Delta$ $\phi$ ) decreases. [ cos( $\theta$ ) = 1 at the Equator ( $\theta$ = 0), and decreases as one moves toward the poles ( $\theta$ = ±90^o).] The length of the hypotenuse of the triangle $\Delta$ l is given by $\Delta$ l² = $\Delta$ x² + $\Delta$ y². Expressing this in terms of $\theta$ and $\phi$ , we find that

$\displaystyle \Delta$ l² = r²cos²( $\displaystyle \theta$ ) $\displaystyle \Delta$ $\displaystyle \phi^{2}_{}$ + r² $\displaystyle \Delta$ $\displaystyle \theta^{2}_{}$ . (3)

This expression gives us the distance traveled for a given (small) displacement on the surface of the sphere specified by $\Delta$ $\theta$ and $\Delta$ $\phi$ . We can express this in terms of the metric as $\Delta$ l² = g₁₁ $\Delta$ $\phi^{2}_{}$ + g₂₂ $\Delta$ $\theta^{2}_{}$ + g₁₂ $\Delta$ $\phi$ $\Delta$ $\theta$ + g₂₁ $\Delta$ $\theta$ $\Delta$ $\phi$ , where g₁₁ = r²cos²( $\theta$ ), g₂₂ = r² and g₁₂ = g₂₁ = 0. Note that g₁₁ is now a function of theta and is no longer a constant equal to unity.

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 = r $\phi$ , and y = r $\theta$ . Then Eq. (3) could be written $\Delta$ l² = cos²( $\theta$ ) $\Delta$ x² + $\Delta$ y². At the Equator, cos( $\theta$ ) = 1, so on the Equator only the metric can be expressed in the form of Eq. (2) with g₁₁ = g₂₂ = 1 and g₁₂ = g₂₁ = 0.

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 $\Delta$ l² = g₁₁ $\Delta$ x² + g₂₂ $\Delta$ y² + g₁₂ $\Delta$ x $\Delta$ y + g₂₁ $\Delta$ y $\Delta$ x, with g₁₁ = g₂₂ = 1 and g₁₂ = g₂₁ = 0 everywhere in that space, then the geometry is Euclidean. If not the geometry is not Euclidean. This is easily generalized to three-dimensional space. If a system of coordinates can be found in which the line element (hypotenuse of a small right triangle) can be written as

$\displaystyle \Delta$ l² = g₁₁ $\displaystyle \Delta$ x² + g₂₂ $\displaystyle \Delta$ y² + g₃₃ $\displaystyle \Delta$ z² + g₁₂ $\displaystyle \Delta$ x $\displaystyle \Delta$ y + g₂₁ $\displaystyle \Delta$ y $\displaystyle \Delta$ x + g₂₃ $\displaystyle \Delta$ y $\displaystyle \Delta$ z + g₃₂ $\displaystyle \Delta$ z $\displaystyle \Delta$ y + g₃₁ $\displaystyle \Delta$ z $\displaystyle \Delta$ x + g₁₃ $\displaystyle \Delta$ x $\displaystyle \Delta$ z, (4)

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 $\Delta$ x₁ $\equiv$ $\Delta$ x, $\Delta$ x₂ $\equiv$ $\Delta$ y, and $\Delta$ x₃ $\equiv$ $\Delta$ z (the symbol `` $\equiv$ '' means ``defined as''). Then the line element can be written as

$\displaystyle \Delta$ l² = $\displaystyle \sum_{i,j = 1}^{3}$ g_ij $\displaystyle \Delta$ x_i $\displaystyle \Delta$ x_j, (5)

where the symbol $\sum$ means to sum the thing to its right over all combinations of i and j equaling 1 through 3. Equation (5) is simply a short hand notation for Eq. (4).

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 $\Delta$ s² = - $\Delta$ l² = (c $\Delta$ t)² - $\Delta$ x² - $\Delta$ y² - $\Delta$ z². If we set (c $\Delta$ t) = $\Delta$ x₄, then this can be rewritten as

- $\displaystyle \Delta$ s² = $\displaystyle \Delta$ l² = $\displaystyle \sum_{i,j = 1}^{4}$ g_ij $\displaystyle \Delta$ x_i $\displaystyle \Delta$ x_j. (6)

with g₁₁ = g₂₂ = g₃₃ = 1, g₄₄ = - 1, and all the other g_ij's equal to zero. It then follows that four dimensional space-time is Euclidean if a coordinate system can be found in which

- $\displaystyle \Delta$ s² = $\displaystyle \Delta$ l² = $\displaystyle \sum_{i,j = 1}^{4}$ g_ij $\displaystyle \Delta$ x_i $\displaystyle \Delta$ x_j

with g₁₁ = g₂₂ = g₃₃ = 1, g₄₄ = - 1, and all the other g_ij'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 g_ij'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.

Ana Sayfa / index / Roket bilimi / E-Mail /CetinBAL/ Astronomy / UFO Technology