If we let f 2 = 0, h denote the height up to f 1 , and g denote the gravitational accelera- tion near the ground and then substitute f 1 = gh and f 2 = 0 into equation {, we obtain the following: 1 - 2 1 - 2 gh - 0 gh = = 2 1 c 2 c 2 c The clock at the higher altitude measures a time Dt 1 that is ahead of the clock below, which reads Dt 2 . We know this to be true, because in the equation above, gh/c 2 and Dt 1 are both greater than zero, and therefore 1 - 2 > 0 ; hence, 1 > 2 . The True Nature of Gravity in General Relativity Space-time surrounding the existence of matter is warped, as explained in the manga. It is apparent that this warping of space-time has the same effect as gravity in attracting sur- rounding matter. Einstein unified these effects in a set of equations called the Einstein field equations. The Einstein field equations showed that time and space (space-time), which were previously thought to exist as a framework for measuring the motion of matter, were fundamentally connected with matter itself. Phenomena Discovered from General Relativity This section introduces the following phenomena, which were discovered from general relativity: · · · Gravitational lensing Anomalous perihelion precession of Mercury Black holes Bending of Light (Gravitational Lensing) Near a Large Mass (Such as the Sun) Gravitational lensing is the phenomenon that when light passes in the vicinity of the Sun, the path of that light bends. Space is bent in the vicinity of the Sun because of the large mass of the Sun, as shown in Figure 4-4. Since light advances along that curvature, the light from a distant star bends, and the direction of the star is observed with a slight shift. This effect was verified during a total solar eclipse. It is noted as the first proof discovered of general relativity. Also, when light is coming from a distant galaxy, as shown in Figure 4-5, if a massive object (such as a galaxy) lies at an intermediate point, it will bend the light from the distant galaxy as though there were a condensing lens at that intermediate point. This bend may make the distant galaxy seem distorted. Many instances of this effect have been observed. This is another instance of gravitational lensing. 162 Chapter 4