“Imagine sitting on the surface of the Earth, where your location is given by three dimensions: X, Y, and Z are your coordinates (see figure 5). As you travel anywhere along the surface of the Earth, the radius of the Earth, R, remains the same, where R2 = X2 + Y2 + Z2. This is a three-dimensional version of the Pythagorean theorem. *
*[To see this, let us take Z = 0. Then the sphere reduces down to a circle in the X and Y plane, just as before. We saw that as you move around this circle, we have X2 + Y2 = R2. Now, let us gradually increase Z. The circle gets smaller as we rise in the Z direction. (The circle corresponds to the lines of equal latitude on a globe.) R remains the same, but the equation for the small circle becomes X2 + Y2 + Z2 = R2, for a fixed value of Z. Now, if we let Z vary, we see that any point on the sphere has coordinates given by X, Y, and Z, such that the three-dimensional Pythagorean theorem holds. So in summary, the points on a sphere can all be described by the Pythagorean theorem in three dimensions, such that R remains the same, but X, Y, and Z all vary as you move around the sphere. Einstein’s great insight was to generalize this to four dimensions, with the fourth dimension being time. ]
Now, if we take Einstein’s equations and then rotate space into time and time into space, the equations remain the same. This means that the three dimensions of space are now joined with the dimension of time, T, which becomes the fourth dimension. Einstein showed that the quantity X2 + Y2 + Z2 − T2 (with time expressed in certain units) remains the same, which is a modified version of the Pythagorean theorem in four dimensions. (Notice that the time coordinate has an additional minus sign. This means that although relativity is invariant under rotations in four dimensions, the time dimension is treated slightly differently from the other three spatial dimensions.) So Einstein’s equations are symmetric in four dimensions.”
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