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Hermann Minkowski invented a “space-time” diagram that shows

Einstein’s formulas in a very visual way.

Minkowski was one of Einstein’s mathematics professors at the ETH.

A little “extra” about Relativity…

Minkowski graphed events by where in space they happen and at what time.

Two events with a particular spatial separation and happening a certain time apart would then be separated by a particular “space-time interval,” shown as the hypotenuse of this triangle.

Minkowski graphed events by where in space they happen and at what time.

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Minkowski graphed events by where in space they happen and at what time.

The same events as measured in a second (moving) frame of reference is graphed with the axes tilted relative to the first one.

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Minkowski graphed events by where in space they happen and at what time.

The same events as measured in a second (moving) frame of reference is graphed with the axes tilted relative to the first one.

Notice that the spatial separations are different in the two frames of reference, and so are the time separations – just as Einstein had explained.

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But now, look at the space-time intervals for both frames of reference…

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But now, look at the space-time intervals for both frames of reference…

The space-time intervals for both frames of reference – the hypotenuses for both triangles – are identical!

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The “space-time interval” between any two events is an invariant quantity!

Despite the apparent strangeness of “relative” measurements in different frames of reference, the combination of space and time measurementsgives an over-all separation between the events that does not change from frame to frame.

Einstein at first thought of Minkowski’s “space-time” as a cute little exercise.

Later, he realized that this geometrical way of thinking would be crucial to the generalization of his relativity theory – the curvature due to gravity is not to space by itself, but to space-time.

…and now, the finish of this series of lectures…