By Prof. Theodore Frankel, Physics

A excellent selection for complicated scholars of arithmetic, this quantity also will entice mathematicians drawn to physics. It stresses the worldwide facets of cosmology and is appropriate for autonomous examine in addition to for classes in differential geometry, relativity, and cosmology. necessities contain a historical past in Riemannian geometry and uncomplicated physics or a familiarity with relativity thought. history chapters, with derivations, disguise designated relativity, continuum mechanics, and electromagnetism.

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**Additional resources for Gravitational Curvature: An Introduction to Einstein's Theory**

**Example text**

6 c along the X-axis relative to frame S, the two events occur at the same place. 25 ç 20 è ø÷ c2 In S¢, the events are (0, 0) and (0, 16 s). In S¢, the time interval between the two events is 16s. This result could have been derived from time dilation or from invariance of ds2. Example 3 In an inertial frame S event A occurs at x = 0 at t = 0 and event B occurs at x = 20 c and t = 12 s. Find a frame S¢ in which the two events are simultaneous. Solution In frame S, ds2 = (x2 – x1)2 – c2(t2 – t1)2 = (20 c)2 = c2(12)2 = 256 c2.

2) becomes \ or x¢ = ct¢ ... (3) or x¢ = ct¢ + d¢ say ... (4) vx ¢ ö æ G(x¢ + vt¢) = cG ç t ¢ + ÷ + d è c2 ø or x¢ = ct¢ + d v æ G 1- ö è cø Eqns. (3) and (4) show that in S¢ the pulses travel with speed c (as they should) and the distance between them is æ v2 ö d ç1 ÷ è c2 ø d d¢ = = v æ1 - v ö G æ1 - ö è cø è cø 1/2 æ c + vö =dç è c - v ø÷ 1/2 Example 10 A rigid rod of length L makes an angle q with the X-axis of the system in which it is at rest in the X – Y plane. Show that for an observer moving with respect to the rod with speed v along the positive X-direction; the apparent length L¢ and the angle q ¢ are given by 1/2 é æ cos q ö 2 ù L¢ = L ê ç + sin 2 q ú ; tanq ¢ = G tanq ÷ ëê è G ø ûú 30 A Primer of Special Relativity Solution Suppose the rod is at rest in frame S as shown in Fig.

1(a). Let us see what it means to observer in S and S¢ when we say that clocks in S and S¢ are synchronized to read t = t¢ = 0 at the instant origins O and O¢ coincide. Of course from the point of view of observer S , all the clocks in his frame are synchronized. However, v ö æ he finds that clocks in frame S¢ show a time given by t¢ = G ç t x ÷ which reduces to (when t = 0) è c2 ø v x. c2 Hence the clocks read different times depending on their location. Clocks to the left of origin (x is negative) are ahead (t¢ > 0) and those to the right are behind (t¢ < 0).

### Gravitational Curvature: An Introduction to Einstein's Theory by Prof. Theodore Frankel, Physics

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