Relativistic Mechanics
In this chapter we will formulate the basic concepts of kine(ma)tics and the basic dynamical laws, taking care to satisfy the Einsteinian version of the principle of relativity. The formulation thus should be compatible with the postulate that inertial fr
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Relativistic Mechanics
In this chapter we will formulate the basic concepts of kine(ma)tics and the basic dynamical laws, taking care to satisfy the Einsteinian version of the principle of relativity. The formulation thus should be compatible with the postulate that inertial frames connected by Poincare transformations be on equal footing. Mathematically this means that the laws are to be Lorentz covariant, i.e., we should be able to formulate them in such a way that they take the same mathematical form in all inertial frames. This postulate is certainly fulfilled if we are able to write these laws as equalities between four-vectors. Thus, technically, we shall illustrate in this chapter the use of four-vectors and their scalar products. In most applications (but not always!) this technique offers great advantages over the Lorentz transformation method used in chap. 2.
4.1
Kinematics
Consider a point mass whose motion relative to an inertial frame I is given by x = x(t). Its velocity is dx v= dt' (4.1.1) and we assume that Ivl < 1. From eq. (2.9.2) we know its rather complicated behavior under Lorentz transformations (4.1.2) stemming from the fact that the denominator in eq. (4.1.1) has also to be transformed. We cannot expect that this velocity concept will allow the formulation of manifestly Lorentz covariant laws. However, if we parametrize the world line of the point mass by its (Lorentz invariant) proper time s as Xi = Xi(S), a suitable substitute for v comes to mind immediately, namely the four-velocity u with components
. dx i u'·=. ds·
(4.1.3)
Here the coordinates enter symmetrically as they do in eq. (4.1.2), and it is obvious that the u i form the components of a four-vector, since the dx i were the prototype of four-vector components. We therefore can write abstractly u = dx/ds. Because of eq. (2.6.2) we have i U
=
( dt dx )
ds' ds
T
dt T T = ds (1, v) = 'Y (1, v) .
(4.1.4)
This shows that u does not contain more information than v; in the (so-called 'nonrelativistic', N.R.) limiting case where Ivl «: 1 relative to the frame considered, we R. U. Sexl et al., Relativity, Groups, Particles © Springer-Verlag Wien 2001
64
4 Relativistic Mechanics
have "I ~ 1 and therefore u i ~ (1, v) T. U is just a new packing of the ordinary velocity concept with a better Lorentz transformation behavior of its components. In terms of Minkowski geometry, u is nothing but the unit tangent vector to the world line at the point considered, since we have for its four-square (4.1.5)
It is timelike and future-directed (dxO > 0, ds > 0). The fact that there is no absolute speed smaller than 1 here appears in the mathematical fact that the only independent Lorentz invariant quantities associated with a timelike vector u are its four-square and sign( uO)-and those are the same for all four-velocities. Our definition suggests associating with our point mass a four-momentum (4.1.6)
p:=mu,
where m is the (inertial) mass as measured in the usual ways in low velocity situations. N.R. we have pi ~ (m,mv)T, so that the
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