Relativistic Particle Dynamics in Gravitational Fields
In Chap. 1 we learned that Newtonian gravitation can be considered as a geometry of space–time. In addition we learned the fundamentals of the corresponding mathematical apparatus, differential geometry. In this chapter we want to apply those methods to r
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Relativistic Particle Dynamics in Gravitational Fields
2.1 Relativistic Gravity In Chap. 1 we learned that Newtonian gravitation can be considered as a geometry of space–time. In addition we learned the fundamentals of the corresponding mathematical apparatus, differential geometry. In this chapter we want to apply those methods to relativistic particle dynamics. In contrast to the non-relativistic case, this will lead to a new theory, the so-called general relativity, which will modify the theory of special relativity as well as the theory of Newtonian gravitation. This means that general relativity predicts new physical phenomena.
2.2 Geometry of Minkowski Space–Time In this section we want to formulate well-known properties of the relativistic space– time in terms of differential geometry. Special relativity is assumed to be known (for an introduction, see [1]). We choose the units such that c = 1. Special relativity is based on inertial frames, exactly as is the case in Newton’s theory. These privileged reference frames are again determined by free motion. Physically they can in principle be realized by the following devices: 1. a freely floating radar equipment, 2. an ideal clock, 3. non-rotating orthonormal axes of coordinates (measuring rods, gyroscopes, etc.). The coordinates {x¯μ }, which correspond to such an inertial frame and describe an event in this inertial frame, can be measured by means of this equipment. A transformation between the coordinates, which correspond to two different inertial frames, determines an element of the Poincar´e group μ
xμ = Λν xν + aμ ,
H´aj´ıcˇ ek, P.: Relativistic Particle Dynamics in Gravitational Fields. Lect. Notes Phys. 750, 39–92 (2008) c Springer-Verlag Berlin Heidelberg 2008 DOI 10.1007/978-3-540-78659-7 2
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2 Relativistic Particle Dynamics μ
where Λν is a Lorentz matrix, i.e., μ
ημν Λρ Λσν = ηρσ . Vice versa, every element of this group gives rise to an inertial frame if we apply this transformation to an arbitrary inertial frame. The time intervals (in particular the simultaneity) and the distances of events in Minkowski space–time are relative: they depend on the choice of the inertial frame. The only absolute quantity (i.e., the only quantity that does not depend on the chosen reference frame) that is related to time intervals and distances is the socalled interval. The interval I between two events with coordinates {x¯μ } and {y¯μ } with respect to an inertial frame is 2 2 2 2 I = x¯0 − y¯0 − x¯1 − y¯1 − x¯2 − y¯2 − x¯3 − y¯3 . The geometric meaning of the interval is the following. If the interval between two events is positive (I > 0), then there exists an inertial frame where the differences of the spatial coordinates vanish x¯1 − y¯1 = x¯2 − y¯2 = x¯3 − y¯3 = 0 and I = T 2 , where T equals the time difference between these two events happening at the same position. If I < 0, there exists an inertial frame where the two events occur at the same time, x¯0 − y¯0 = 0, such that I = −d 2 , where d is the distance of these two simultaneou
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