time), we find that the two corresponding times ti and 12 differ by an amount 12- r1 ( This circumstance is called "failure of simultaneity at a distance, "and to make the idea a little clearer let us consider the following experiment Suppose that a man moving in a space ship( system S) has placed a clock at each end of the ship and is interested in making sure that the two clocks are in ynchronism. How can the clocks be synchronized? There One way, involving very little calculation, would be first to locate exactly the midpoint between the clocks. Then from this station we send out a light signal which will go both ways at the same speed and will arrive at both clocks, clearly It the same time. This simultaneous arrival of the signals can be used to syn chronize the clocks Let us the the man in S synchronizes his clocks by this particular method. Let us see whether an observer in system S would agree that the two clocks are synchronous. The man in S has a right to believe they are because he does not know that he is moving. But the man in S reasons that since the ship is moving forward, the clock in the front end was running away from the light signal, hence the light had to go more than halfway in order to catch up; the rear clock, however, was advancing to meet the light signal, so this distance was shorter. Therefore the signal reached the rear clock first, although the man in S thought that the signals arrived simultaneously. We thus see that when a man in a space ship thinks the times at two locations are simultaneous, equal values of t in his coordinate system must correspond to different values of t in the other coordinate system Let us interesting to note that the transformation between the xs and Is is analogous in of the rotation of coordinates. We then had x cos 8+ yin 8, (158) in which the new x' mixes the old x and y, and the new y' also mixes the old and y; similarly, in the Lorentz transformation we find a new x' which is a mixture of x and l, and a new t' which is a mixture of t and x. So the lorentz transforma- tion is analogous to a rotation, only it is a"rotation"in space and time, which appears to be a strange concept. a check of the analogy to rotation can be made c2r2=x2+y2+22-c242 (159) In this equation the first three terms on each side represent, in three-dimensional geometry, the square of the distance between a point and the origin(surface of a sphere) which remains unchanged (invariant) regardless of rotation of the co- ordinate axes. Similarly, Eq (15.9)shows that there is a certain combination which includes time that is invariant to a lorentz transformation. thus, the analogy to a rotation is complete, and is of such a kind that vectors, i. e, quantities time, are also useful in connection with relativity Thus we contemplate an extension of the idea of vectors, which we hay ar considered to have only sp include a time component. That is, we expect that there will be vectors with four components, three of which are like the components of an ordinary vector, and with these will be associate a fourth component, which is the analog of the time part