By E. Poisson
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Additional resources for A Relativist's Toolkit - The Math of Black Hole Mechanics
2– (ω11 e1 + ω21 e2 + ω31 e3 ) · e1 = 0 (2–82) (ω12 e1 + ω22 e2 + ω32 e3 ) · e2 = 0 (2–83) (ω13 e1 + ω23 e2 + ω33 e3 ) · e3 = 0 (2–84) From Eqs.
F=⎢ . ⎣ ⎦ fm (t) ⎡ ⎤ α1 ⎢ . ⎥ ⎥ α=⎢ ⎣ .. ⎦ αp (1–155) (1–156) Then the functions (x1 (t), . . , xn (t)) form a system of diﬀerential equations if they can be written as ˙, . . , x(n) , t; α, f) = 0 (1–157) G(x, x where x(i) is the ith derivative of the function x with respect to the parameter t and ⎡ ⎤ g1 ⎢ . ⎥ ⎥ G=⎢ (1–158) ⎣ .. ⎦ gn is a vector function of x and any derivatives of x with respect to t, the scalar t itself, and the vector f. It is noted that we can write Eq. (1–157) in terms of the scalars g1 , .
In particular, suppose we choose a ﬁxed point O as the origin of a ﬁxed coordinate system and deﬁne a basis vector Ex to be in the direction of positive motion as shown in Fig. 2–6. In terms of the coordinate system deﬁned by x O Figure 2–6 P Ex Rectilinear motion of a particle P . O and Ex , the position of the particle can be written as r = xEx (2–43) where x = x(t) is the displacement of the particle from the point O. Given r from Eq. (2–43), the velocity and acceleration of the particle are given, respectively, as F v(t) F F = F a(t) = dr ˙ = x(t)E x = v(t)Ex dt (2–44) d2 r F ¨ ˙(t) = x(t)E r(t) = F v ≡ ¨ x = a(t)Ex dt 2 (2–45) where we note once again that, because x(t) is a scalar function of time, its rate of change is independent of the reference frame.