The Minkowski Metric
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There's one more thing I want to do with the Lorentz Transformation before we get into relativistic dynamics. This will not, in my opinion, significantly improve your understanding of Relativity. In fact, I believe that this feature of Relativity may have actually worked as more of a distraction than a help in the development of the complete theory of Relativity.
Multiply the fourth equation of the Lorentz Transformation by the speed of light (which converts duration units into distance units), square it, then subtract the square from the sum of the squares of the other three equations. After you apply the appropriate algebraic simplification procedure you should obtain
Certainly the phrase "sum of the squares" seems familiar to you, because it echoes the key phrase in the conventional statement of the Pythagorean theorem. You can see the algebraic expression of that key phrase incorporated, in a somewhat unfamiliar way, in Equation 1. We have just obtained from the procedure above a rather strange version of what Pythagoras deduced over twenty-five centuries ago, a version known as Minkowski's theorem. We call the equation that corresponds to the theorem a metric equation because it reveals the "metric" or "measure" of the space in which our observers make their measurements.
Consider the case of two right triangles that share a common hypotenuse. The sum of the squares of the sides of one triangle must equal the sum of the squares of the sides of the other triangle. If we call the length of one side of the first triangle A, the length of the other side B, and the length of the hypotenuse C, then we have the classical Pythagorean equation,
And if we call the length of one side of the second triangle a, the length of the other side b, then, because that triangle has the same hypotenuse C as does the first triangle, we also have
Because the sums in those two equations come out equal to the same thing, then they must, as Euclid noted, equal each other; that is,
which is just the metric equation equivalent to two Pythagorean equations for the same hypotenuse. In a sense, the hypotenuse offers us a straight line that probes the structure of space.
Just as we derived Equation 1 from the Lorentz Transformation, so too we can derive Equation 4 from the analogous Pythagorean Transformation. We have the equations of the Pythagorean Transformation (actually, the inverse of that transformation) as
and we can use them to illustrate anew the workings of a transformation of coordinates.
As an example of such a transformation, let's consider how we might calculate the distances along the sides of two paths going from the intersection of Barrington Avenue and Olympic Boulevard to the intersection of Westwood Boulevard and Santa Monica Boulevard in West Los Angeles. You might measure along Olympic Boulevard to Westwood Boulevard and thence along Westwood Boulevard to Santa Monica Boulevard and I, on the other hand, might measure due east from Barrington Avenue and Olympic Boulevard to a point and thence due north to Westwood Boulevard and Santa Monica Boulevard. Our measurements will differ from each other, but we must both calculate the same straight line distance between the two intersections. The transformation equations shown above will convert your measurements into mine and we can test the validity of those equations by squaring them and adding them together: we can only take them as valid transformations if that procedure yields the Pythagorean theorem as it was stated above.
To obtain the figures to fill out the current example I took a ruler and a protractor to a map of Los Angeles and took the appropriate sines and cosines from the mathematical tables in the 1961-62 Handbook of Chemistry and Physics. You can attribute discrepancies in the numbers to the fact that the scale on my map is 2800 feet to the inch and that I was using rather crude tools: if I had used a proper drafting table and instruments with an aerial photograph at a scale of 100 feet to the inch, the numbers would have come out more accurate. Nonetheless, I obtained this:
1) The straight-line distance between the intersections is C = 6350 feet in the direction 61 degrees north of due east. The square of that distance, C2 = 40,322,500;
2) The distances that you would measure along Olympic Boulevard and thence along Westwood Boulevard, which runs perpendicular to Olympic are a = 5780 feet and b = 2630 feet. Olympic Boulevard runs at an angle of 36.5 degrees north of due east. The sum of the squares is equal to 40,325,300 and the square root of that is 6350.22 feet.
3) The distances that I would measure due east and thence due north are, respectively, A = 3080 feet and B = 5555 feet. The sum of the squares is 40,344,425 and the square root of that is 6351.726 feet.
4) The angle between your a-line and my
A-line is 36.5 degrees. The cosine of that angle is 0.80386 and the sine is
0.59482, so the transformation equations yield
Those figures come close enough to my direct measurement to illustrate the transformation.
As I said before, the hypotenuse of a right triangle offers us a probe of the structure of space. You have seen how the transformation equations contain weird coefficients, the Lorentz factor in the Lorentz Transformation and the sines and cosines in the Pythagorean Transformation. But those coefficients go away in the metric equations: the Minkowski equation and the Pythagorean equation contain only the coordinates representing the lengths of the lines in the right triangles or in the spatio-temporal analogues of right triangles in Relativity. The Pythagorean metric equation has the simple form of a sum of squares without any weird coefficients because we lay out our triangles on a surface that is more or less flat (albeit slightly tilted). In a similar sense we can say that space and time as represented in Minkowski's theorem are also flat because the metric equation also comprises a simple sum and difference of squares without any weird coefficients involved.
If we had drawn our triangles on a surface with noticeable curvature, if we had, for example, drawn our lines between intersections in Los Angeles and New York City, then our metric equation would have included coefficients to modify the squares of the coordinates. In that case the distortion of the metric equations would come from the fact that we have constrained ourselves to measure distances on Earth's surface. But Earth exists in a space that is effectively Euclidean, so any major distortions in the metric equations come from such artificial constraints or from the use of inherently curved coordinate systems, such as spherical coordinates (radius from some chosen center along with latitude and longitude). However, the metric equations of General Relativity, in contrast, modify the squares with coefficients that represent a true warping of space and time away from flatness. We will come to those equations later in this treatise.
In deducing Einstein's postulates of Relativity (as two Cosmic Theorems) I deduced a theorem to the effect that space has a finite extent that has the character of an infinite distance. I want ultimately to deduce the metric equation that encodes that theorem and work out the transformation equations that go with it. That whole-space metric will compel us to see Minkowski's metric as an approximation that is only valid over small expanses of space and short elapses of time, just as our Pythagorean metric is only valid over small areas of Earth's surface (such as West Los Angeles) and is only an approximation to the metric that must be applied over areas of Earth's surface for which Earth's curvature becomes noticeable.
The fact that he could make an analogy between his metric equation and the algebraic representation of the Pythagorean theorem led Hermann Minkowski to declare in 1908 that he made an analogy between Special Relativity and Euclidean geometry. Just as we work out Euclid's geometry, which includes Pythagoras' theorem, on flat planes, so we must work out Special Relativity on a flat continuum analogous to a plane. Minkowski called that continuum "spacetime" and got so carried away in his enthusiasm that he predicted that "space and time by themselves are doomed to fade away...." That prediction has not come true to Reality, but that fact should not astonish us: Relativity offers nothing that obliges us to give up our notion that space and time are separate things, so our experience of space and time as different things overrides any mystical impulse to see them unified.
We have another, more picturesque, if somewhat unenlightening, way in which we can represent a metric equation. Draw a straight line and then construct on that line all of the right triangles that have that line as their hypotenuse. The line that connects the free vertices of those triangles makes a circle (Euclid proves a theorem to that effect). To make the equivalent construction of a diagram that represents the Minkowski metric you must select one direction on a piece of graph paper to be the x-direction and the other to be the time-direction (we usually suppress the y- and z-directions in discussions of Relativity), pick a point to represent one event, and then plot all of the points that represent the distance and duration from that first point to a second event as observers occupying all the inertial frames moving in the x-direction would measure them. Those points will trace out an hyperbola, which we can see as the line connecting the free vertices of a set of obtuse triangles whose common base represents the time interval between the two events in the inertial frame in which the events occur in the same place. We know that the circle and the hyperbola, being conic sections, are related to each other, but they are also quite different from each other, one being closed on itself and the other being open to infinity. Likewise, Euclidean geometry and Special Relativity differ significantly from each other and an overemphasis on their similarities is not a good thing.
Why do I make such a claim? In 1907, the year before Minkowski presented his geometric interpretation of Special Relativity, Einstein conceived the basic idea behind his theory of General Relativity. He did so in response to reading about an incident that could have inspired a long list of imaginary experiments of the kind he had used to work out the features of Special Relativity. Those imaginary experiments would have led him to create a version of General Relativity that uses the same concepts and mathematics (essentially first-year calculus) that he used in Special Relativity, a version much like the one that I have been working out for the past sixteen years (as of 2003 Nov 27). Instead, Minkowski's geometric vision inspired him to develop a version based on the use of non-Euclidean geometry and tensor calculus, thereby making it the most difficult theory in physics.
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