Spacetime, Curvature of

The concept of curvature can be approached through examination of two-dimensional surfaces. The floor of a room is flat, the surface of a ball or of the earth is curved. What does this mean more exactly? Let us imagine two wanderers going from place P to place Q along two different paths and carrying pointers which, in the beginning, were oriented in the same direction. Let the wanderers regulate the directions of their pointers in such a way that they do not change in the plane tangent to the surface of the earth (it is possible to envision this with the help of large-scale maps). It is easy to ascertain with the aid of a globe that, in the case of repeated meeting of the wanderers whose paths taken together enclose a large area, the directions of the pointers will be different (see Figure 1).

The path of a wanderer continuing in the initial direction of the pointer will be the most upright line on the earth's surface, the geodesic. Two initially parallel geodesies (principal circles on the spherical surface) will gradually approach each other and will finally intersect. But the direction of the parallel transported pointers would be independent on the path (see Figure 2), and the initially parallel geodesies would remain parallel in the case of flat, noncurved space.

Great mathematicians of the 18 th and 19th centuries (Gauss, Riemann, Levi-Civita, and others) elaborated the concept of curvature and extended it to the case of spaces with arbitrary dimension. In the 20th century new, more abstract formulations succeeded, but in this entry the original, more intuitive definitions are retained.

Figure 1

Let us have in the space two “infinitesimally near” points P, Q with the coordinates x', x' + dx'. Let the vector (depicted by the arrow) have the component V', given in the point P. By its parallel transport into the point Q, the vector V' arises with the components

The coordinate dependent quantities F'-k are called the components of affine connection. Their prescription makes it possible to transport vectors in a parallel manner along an arbitrary path and to find the parametrically expressed geodesies (most direct lines) x' (σ) by solving the equation

The geometrically natural request of closeness of infinitesimal parallelograms constructed by parallel transport leads to the symmetry of the components of the connection or, in the other words, to the zero value of the torsion tensor field

In the case of the Riemannian spaces with the infinitesimal distance ds defined as

(gik are the components of the metrical field, or the metrics), it is natural to demand the inalterability of the lengths of parallel transported vectors and the angles between them. Thus in the case of zero [Page 1181]torsion (3), the relation can be derived

Figure 2

determining the components of the connection by help of the metrics (here, g∗ is the inverse matrix to the matrix gah). Then we speak about the Levi-Civita connection and the components of the connection (5) are called the Christoffel symbols. It is possible to introduce coordinate systems where these quantities are equal to zero in a prescribed point, with the effect that the equations of geodesies locally merge into the equations of right line. Such systems are “locally geodesic.” Thus the Christoffel symbols do not determine the curvature of the space, only the curvature of the coordinate system.

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