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The light from clock 1 (below) is “red-shifted”. Conversely, light from the upper clock traveling down to the lower clock is measured to have a higher frequency than the local clock 1; it is “blue-shifted”. The principle of equivalence states that the above situation, in a closed system, cannot be distinguished by physical measurements, from that in which the rocket is at rest in a uniform gravitational field. The field must produce an acceleration of magnitude |a|, on all masses placed in it. y′ G G is a non-accelerating frame with a uniform 2 gravitational field present.
It is written for those with a flair for Mathematics. A1. Some useful mathematics: transformations and matrices Let a point P[x, y] in a Cartesian frame be rotated about the origin through an angle of 90°; let the new position be labeled P′[x′, y′] +y P′[x′, y′] P[x, y] -x +x -y We see that the new coordinates are related to the old coordinates as follows: x′ (new) = −y (old) and y′ (new) = +x (old) where we have written the x’s and y’s in different columns for reasons that will become clear, later.
The algebraic rule for carrying out the “matrix multiplication” is obtained directly by noting that the single symbolic equation is the equivalent of the two original equations. We must therefore have 58 x′ = a times x + b times y and y′ = c times x + d times y. We multiply rows of the matrix by columns of the coordinates, in the correct order. 2 x 2 matrix operators will be seen to play a crucial role in Einstein’s Special Theory of Relativity. A2. Galilean-Newtonian relativity revisited The idea of matrix operators provides us with a useful way of looking at the equations of classical relativity, discussed previously.