Does General Relativity actually satisfy the General Principle of Relativity?

Question

The “General Principle of Relativity” being “All systems of reference are equivalent with respect to the formulation of the fundamental laws of physics”. To my knowledge, this is related historically to Einstein’s perception of Mach’s principle, a somewhat ill-defined conjecture that seems to postulate the relativity of all forms of motion.

I understand that this is related to the idea of “general covariance” although given the different names i assume they are not synonymous. Along with that, i have also heard that even Newtonian Mechanics can be represented in a Generally Covariant form mathematically, yet Newtonian Mechanics needs an external cause in a non-inertial reference frame to explain inertial forces, which seems to violate the general principle of relativity as stated above as this distinguishes a non-inertial frame for an inertial one.

Am i missing something here? Does general relativity really follow this idea that the laws of physics are the same in every frame of reference, or does it merely have general covariance in a a way similar to newtonian mechanics?

Answer 1

General covariance simply gives a more mathematical expression to the general principle of relativity.

• General covariance: Local laws of physics are the same in all coordinate systems.

General covariance is applied by expressing laws using tensors (which is why it is called covariance). This does not apply to Newtonian Absolute Space and Absolute Time, because the principle has always been expressed in term of four dimensional tensors. On the other hand, it is perfectly true that, once Newtonian absolute space and time have been replaced by a local inertial reference frame, Newtonian mechanics holds in general relativity.

Underlying both the general principle and general covariance is

• Hume’s principle of uniformity in nature: the fundamental behaviour of matter is always and everywhere the same.

As observed by Hume, this is needed for the very existence of science. If fundamental behaviours could change, observation and empirical evidence would become meaningless. It follow that all observers can set up reference frames in the same way, and thus

• The general principle of relativity: Local laws of physics are the same irrespective of the reference matter which a particular observer uses to quantify them.

This goes into general covariance, because coordinate systems are set up by observers from reference matter.

None of this has any direct bearing on Mach's principle. Mach's principle was just something which Einstein thought about a great deal on his way to producing a precise mathematical statement from which general relativity can be derived.