Questioning Relativity 5: General Covariance

· physics, theory of relativity

A basic postulate of Einstein’s general theory of relativity named general covariance, states that the mathematical representations of a law of physics in different coordinate systems must be connected by the mathematics of coordinate transformations.

With this interpretation general covariance is a postulate which is vacuous of physical content and does not say anything about reality, only about mathematical rules of coordinate transformations. This criticism of general relativity was formulated by Erich Kretschmann in 1917 shortly after Einstein in 1915 had presented his generalization of his special theory of relativity after 10 years of hard struggle. The story is recalled in

by John D. Norton, 2001.

A mathematician immediately understands that a statement about physics, for instance about the physical size of a certain physical object, can be be expressed in different units or coordinate systems and that a corresponding coordinate transformation does not say anything about the physical size as such. To say that 1 meter is 100 centimeters says nothing about e.g. your actual physical height.

Einstein as a non-mathematician apparently missed this point by attributing physical meaning to coordinate transformations without physics. In particular, it was Einstein who gave the transformed time coordinate of the Lorentz transformation a physical meaning as real time dilation with clocks in motion slowing down, something which was explicitly forbidden by Lorentz as the inventor of the Lorentz transformation.

Einstein’s relation to mathematics is discussed in Did Einstein Not Understand Mathematics?

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