Mathematics of Physics and Engineering
106 Theory of Relativity relativity, the frames can move with variable relative velocities, for exam- ple, rotate, and there are ...
Einstein's Field Equations 107 ^•[fl]-^[fl]fl« = ^T«[fl], i,j = l,..A, (2.4.22) where Rij, R, and T^- are non-linear partial dif ...
108 Theory of Relativity (2.4.25). The operator R in (2.4.22) is denned as follows: RB=aij(*)Rijg- (2-4.28) EXERCISE 2.4.6. (a ...
Einstein's Field Equations 109 eral relativity is that the physical laws are independent of a particular frame, inertial or not, ...
110 Theory of Relativity The tensor Rij[g] — {l/2)R[g)gij, which is the left-hand side of (2.4.22), is called Einstein's tensor ...
Einstein's Field Equations 111 It turns out that a geodesic is always parameterized by its arc length s, although not necessaril ...
112 Theory of Relativity (c) Integrate by parts and use y%{a) = yl(b) = 0 to conclude that dF(x(s),x'(s)) d dF(x(s),x'(s))\ i( W ...
Einstein's Field Equations 113 From now on, we will consider (2.4.34), which is still a system of 10 equations, and call this sy ...
114 Theory of Relativity in (2.4-35) approaches the flat Minkowski metric in spherical coordinates (dr)^2 +r^2 (sin^2 ip(d9)^2 + ...
Einstein's Field Equations 115 (cte)^2 = (dr)^2 + r^2 sin^2 <p (d6)^2 + r^2 {dip)^2 - c^2 (dt)^2. Conclude that A = -c^2 , wh ...
(^116) Theory of Relativity equal to the corresponding R 0. As the preceding exercise shows, a black hole can be produced either ...
Einstein's Field Equations 117 Step 4- Put u(9(s)) = l/r(s), using r{s) = -(u'(9)/u^2 (9))6(s) = -cm'(9), to get {u'(9))^2 + u^2 ...
118 Theory of Relativity solution. On an intuitive level, this shift can be explained without using the Schwarzschild solution o ...
Einstein's Field Equations 119 gravitational field of the Earth is much weaker in space. With communica- tion signals travelling ...
120 Theory of Relativity to conclude that At 0 A£i R(h) RihY Hint: r(ti) - r(t 0 ) = r(h + Ati) - r(t 0 + At 0 ). (2.4.47) By ob ...
Chapter 3 Vector Analysis and Classical Electromagnetic Theory 3.1 Functions of Several Variables 3.1.1 Functions, Sets, and the ...
122 Functions of Several Variables at A and radius b. A point in a set is called interior if there exists a neighborhood of the ...
Functions, Sets, Gradient 123 known as a Jordan curve) divides the plane into two domains, and the domain enclosed by the curve ...
124 Functions of Several Variables The function / is called dif f erentiable at the point A if / is de- fined in some neighborho ...
Functions, Sets, Gradient 125 EXERCISE 3.I.3.^5 Show that if V/ = 0 in a domain G, then f is constant in G. Hint: For point P g ...
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