How can we know if a theory is a physical theory about some real physics or a mathematical theory which does not say anything about physical reality?
Easy enough: Look at the fundamental postulates of the theory and check if they connect to real physics or not. If there is no physics there, then there is no physics anywhere in the theory.
A postulate can take one of the following two forms:
1. A definition is empty of content and only prescribes how to use certain words. A definition is true by its construction, as long as it is not contradictory. Example: There are 100 centimeters on a meter.
2. An axiom makes a statement about the components of the theory, which is not empty of content. Example from Euclidean geometry: Through two distinct points there is a unique straight line. If point and straight line are given a physical meaning, e.g. as single dots and collection of dots drawn by a ruler on a blackboard, then this axiom makes a statement about physical reality: Given two distinct dots it is possible to draw a unique line through the points by the ruler.
This is the distinction made in logic between analytic statements (true by definition or tautologies) and synthetic statements about reality which may be true or false depending on reality.
Let us subject Newtonian mechanics and Einstein’s theories of relativity to this test:
The basic postulate is Newton’s Law
where is the mass of a body moving with velocity and acceleration subject to a gravitational force . This is an axiom with physical content which connects two different physical aspects: motion and gravitational force.
The axiom states that acceleration is proportional to gravitational force. This is a statement about physics which may be true or not true. It is not a definition which is true by its construction.
Newtonian mechanics is a physical theory which makes definite predictions about physical reality. Galileo prepared the theory by experiments.
The basic postulate of special relativity is
- The speed of light is the same for all observers.
Today this is a definition since the length unit is defined as a fraction of lightsecond according to the 1983 SI standard. The speed of light is today defined to be equal to 1 lightsecond per second, the same for all observers.
Basic postulates of general relativity are
- Principle of Relativity: Physical laws are the same in all reference systems.
- Principle of Equivalence: Inertial mass is equal to heavy mass.
The Principle of Relativity is a definition or truism only stating what is self-evident, namely that the expression of a physical law in different coordinate systems must connect according mathematical laws of coordinate transformation (general covariance).
The Principle of Equivalence is a definition since it defines inertial mass to be equal to gravitational mass.
The physics of general relativity enters through the right-hand side of Einstein’s equations which acts as a stress-energy source to the differential equation for the space-time geometry tensor.
Result of Test
- Newtonian mechanics is a physical theory.
- Special relativity is a non-physical theory based on a definition.
- General relativity is based on two definitions plus an axiom connecting stress-energy to space-time curvature similar to Newton’s Law.